Split (1)

train · 6.32M rows

url stringlengths 14 2.42k	text stringlengths 100 1.02M	date stringlengths 19 19	metadata stringlengths 1.06k 1.1k
https://www.math.uni-luebeck.de/veranstaltungen/vortraege/zeller.php	## Slices through fractals and applications in physics and life sciences Regular polytopes serve as building stones for fractal constructions, see for instance the illustrations below. In $\mathbb R^4$ exist six regular polytopes: Beside constructions generalising the five platonic solids, there exists a self-dual 24-cell. We will present several fractal constructions based on the 24-cell and other regular polyhedra in three and four dimensions. Fractal constructions in $\mathbb R^4$ cannot be illustrated, but we can visualise their three-dimensional intersections with hyperplanes. We call such an intersection a slice. To illustrate slices we used the cutting plane method. This method can be applied to a class of self-similar sets generated from homotheties with scaling factor an inverse of a Pisot unit $\beta$ and translations in $\mathbb Q(\beta)^n$. From the algebraic point of view our method generalises a result on $\beta$-representations stated by Schmidt in 1979. The second part of the talk is contributed to application examples of fractals in physics and life sciences. Interactions of fractal structures and electromagnetic waves promise new technologies, e.g. for producing energy or for cloaking. On the other hand, concepts of fractal geometry can be applied in order to describe biological tissues and their cells.	2022-12-02 17:33:27	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3075307607650757, "perplexity": 538.5605291242174}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710909.66/warc/CC-MAIN-20221202150823-20221202180823-00771.warc.gz"}
http://mathoverflow.net/questions/17208/decidable-real-arithmetic	# Decidable real arithmetic I believe that what I'm about to describe has a name—I'm almost certain that I've seen this in model theory and term-rewriting systems, possibly having something to do with ‘signature’ or ‘carrier’?—so please feel free to tell me if I'm using terms in bad, or non-standard, ways. Consider a set $C$ of real constants. (There's no reason that we couldn't consider constants from an arbitrary field, or ring, or … whatever; but let's say ‘real’ for definiteness.) Fix also, for each $n$, a (possibly empty) set $S_n$ of (partially defined) $n$-place, real-valued functions on $\mathbb R$. Write $\Sigma = (C, (S_n)_n)$, and define the set of $\Sigma$-terms in the following (obvious) way. The collection of $\Sigma$-terms is the smallest subset $\mathcal T$ of the set of strings over the alphabet consisting of $C \sqcup \bigsqcup_n S_n$, together with 3 distinguished symbols (, ), and ,, satisfying: 1. Each element of $C$ is in $\mathcal T$. 2. If $\sigma \in S_n$, and $w_1, \ldots, w_n$ are $\Sigma$-terms, then $\sigma(w_1, \ldots, w_n) \in \mathcal T$. Notice that there is an obvious (partially defined) map from $\Sigma$-terms to $\mathbb R$. Notice also that I am avoiding all questions of representability of elements of $C$; if this is worrisome, we can assume that $C$ is countable, fix an enumeration, and replace all elements of $C$ by their labels. Now call $\Sigma$ decidable if the problem of whether a given $\Sigma$-term represents $0$ is decidable. (To avoid trivialities like $C = \mathbb R_{\ge0}$, with all operations preserving positivity, let's assume that subtraction lies in $S_2$.) Are there any results about what ensembles $\Sigma$ are decidable? For example, it's obvious that, if $C = \mathbb Q$ and the operations allowed are the field operations, then $\Sigma$ is decidable; but what if we enlarge our operations to allow extraction of positive square roots, or enlarge $C$ to the algebraic closure of $\mathbb Q$? -	2014-08-28 03:18:19	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9105653762817383, "perplexity": 375.28029679167526}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500830074.72/warc/CC-MAIN-20140820021350-00415-ip-10-180-136-8.ec2.internal.warc.gz"}
http://bqsp.iclegnano.it/consecutive-letters-in-c.html	has fluctuated but remained relatively low. Well-known words. title = "The {"}quasi-stable{"} lipid shelled microbubble in response to consecutive ultrasound pulses", abstract = "Controlled microbubble stability upon exposure to consecutive ultrasound exposures is important for increased sensitivity in contrast enhanced ultrasound diagnostics and manipulation for localised drug release. Extending the binomial case (or the case of arranging two types of objects where repetition is allowed), we have C(6,2) ways to place the letters a, then C(4,2) ways to place. Numbering the letters so A=1, B=2, etc is one of the simplest ways of converting them to numbers. 222 222 27 89% of 649 4,838 JustyFY 3 Issues Reported. How many strings of six lowercase letters from the English alphabet contain. C is a "strongly-type" language. An example of a simple consecutive reaction is the consecutive occurrence of two irreversible first-order reactions A → B → C, where A, B, and C are certain substances. We can split lines and words from a string based on chars, strings or newlines. See other lists, that begin with or end with letters of your choice. If I understand correctly, word users have to add the spaces explicitly, which is prone to errors. In documents and publications, once a reference has been given in full, a shortened form can be used when the same source is cited again in non-consecutive footnotes. The second quarter kicked off with a signature Khori Miles slam dunk […]. Having a list of words with a specific letter, or combination of letters, could be what you need to decide your next move and gain the advantage over your opponent. INPUT – Heeeiiiissggoiinggg OUTPUT – Heisgoing. Shanghai: Can the Fastest-Growing City in the World Keep it Up? In a city whose GDP grew by double-digits every year for more than 20 consecutive Submit a letter to the editor or write to. CONSECUTIVE. Input: The first line contains an integer T, denoting the number of. semicolon General Page 2 E. Seems pretty optimal. Students who do not clear academic probation by the end of the next semester (fall or spring) of enrollment will be subject to dismissal by the College of Letters & Science. 1) Indexed means that the array elements are numbered (starting at 0). concurrent and consecutive. The following words have three consecutive letters that are also consecutive letters in the alphabet: THIRSTY, NOPE, AFGHANISTAN, STUDENT. Lead Papa John's CEO Just Apologized to Customers. asked by Anonymous on April 7, 2010; math. Computational thinking puzzles Computational thinking is a core set of skills that computer scientists develop as first letter of a word and start at the top left hand corner, scanning along the rows in turn looking for it. Clayton, Mark J. The elements of an array in C++ can be of any type. Letters and numbers labels are used to organize and label components before disassembly and aid in reassembling components after maintenance or repair. 4,5,6 =[4-6] , 2A, 2B, 2C are consecutive values, so range is 2[A-C]. Please note that at one point there is 3 consecutive numbers. Search whether character is present in the string or not : Logic [ Algorithm ]: Accept the String from the user. bookkeeping. Manufactured here in the USA with Bright Line (non-reflective) with Over Laminate to protect against UV, abrasion, and chemicals. com with free online thesaurus, antonyms, and definitions. If you have any doubts about […]. When it is selected, Visio capitalizes the first letter of any word that follows a period, a carriage return, a semicolon, or the first letter of any single word in a list or table column. It indicates the ability to send an email. This program will remove consecutive repeated character from the string in C. After retrieving your registration information online, you have the option to print an official letter of verification from Selective Service. Write a program in C to find the first capital letter in a string using recursion. Credit card delinquencies rose to 9. For example, if an individual with the flu is incapacitated for more than three consecutive calendar days and receives continuing treatment, e. Last edited on princesslumy. Rules for naming identifiers. he Effects of Consecutive Night Shifts on Neuropsychological Performance of Interns in the Emergency Department: A Pilot Study Study objective: We obtain preliminary information on the neuropsychological per-formance of house officers at the beginning and end of a shift while they worked consecutive night shifts in the emergency department. ATLANTA – November 1, 2019 – Ogletree Deakins, one of the largest labor and employment law firms representing management, is pleased to announce that the firm has been named a “Law Firm of the Year” by U. Write a program that accepts a word as input and determines whether or not it has three consecutive letters that are also consecutive letters in the alphabet. Heartspring commented on the list double-letter-words. I now have access to the version of Excel from Microsoft Office Home and Student 2010 on a PC. Write a program that accepts a word as input and determines whether or not it has three consecutive letters that are also consecutive letters in the alphabet. null character D. Letters released prior to November of 2001 were assigned a designation identifying the primary supervisory function addressed in that letter, for example, SR 97-2 (SPE). int i, j, m; // instance variables of the surrounding class - initialised to 0 c -> { // lambda - c is of type char[]; return type is int for(i = j = 0; // i is the length of the longest run, j is used to step through c - both start at 0 (m = j < c. bookkeeping. What are the four integers? So before even attempting to tackle it, let's think about what it means to be a consecutive odd integer, what four consecutive odd integers could be referring to. Online verification provides a quick way to access your Selective Service registration number and date of registration. Pages: 1 2 The function has create a new string from old one by changing all lowercase letter to uppercase letters and viceversa. The system can solve single or multiple word clues and can deal with many plurals. 6 ranked Robert C. In this section, we consider reference types for string processing and image processing. This week's meeting is with Barton Health and will provide the community a health update and. HIs royal poem baits. Browse these definitions or use the Search function above. Change "consecutive" to "identical consecutive" in user message. 2 Elmira Notre Dame on Saturday, Nov. Why doesn't some ILM posters has no verb tense? ― Ken L (Ken L), Thursday, 6 January 2005 20:12 (fifteen years ago) link. The number of fatal police shootings in B. The 26 missing letters are all different. out put is always uppercase. Implement a function that determines whether a string that contains only letters is an isogram. Duthomhas (12168) You know, thecodewall did actually post the proper solution. SIGMA NOTATION FOR SUMS. 3% in 2018, the Federal Trade Commission reported Monday. it works but not with the correct answer. id value block 1 a 1 2 a 1 3 a 2 4 a 2 5 b 3 6 c 4 And I want to change the value column to the next value based where the series changes. Use 8 or more characters with a mix of letters. 6 6 0 100%. CONSECUTIVE. Since the letter a occurs twice and the letter p also occurs twice, we. The number of fatal police shootings in B. asked by Anonymous on April 7, 2010; math. If you have any doubts about […]. Their steel wheels are made from special alloy that will last for years. View word search examples. To summarize briefly: DNA is the instruction manual to each of us. Word List for Scrabble® Crossword game. T he former World amateur champion is on the cusp of securing a. Fandom Apps Take your favorite fandoms with you and never miss a beat. After consultations at the World Health Organization in Geneva with medical oﬂicers who had personal knowledge of many countries, letters were sent by the Organization to the Ministries of Health of a number of countries explaining what was proposed and asking for co-operation. I'm really sorry i no expert in regex and I need help please. Alphabetical Order: The following words have three consecutive letters that are also consecutive letters in the alphabet: THIRSTY, NOPE, AFGHANISTAN, STUDENT. § 4082 had been delegated to the Federal Bureau of Prisons. The massive drop in. 08-17181 (Sept. Words with Double Letters. When two consecutive notes come from the same place, the word ibid. Suppose binary string is "0010010", and k = 2, then after concatenating the string k times, it will be "00100100010010". Deficiency notice If a company is in violation of the continued listing standards for a period of 30 consecutive days, the NASDAQ sends a "deficiency notice. It also entered the Guinness World Records for being the "fastest. He just left out a couple of required headers. 46 last August, Tesla shares have fallen by 21%. Denver, CO, November 13, 2019 ---- Presbyterian/St. Many common words contain a doubled letter, for example LEER, G RAMMA Rand DAZ ZLE. Example 1: Input: 5 Output: 2 Explanation: 5 = 5 = 2 + 3 Example 2: Input: 9 Output: 3 Explanation: 9 = 9 = 4 + 5 = 2 + 3 + 4. Sometimes the name of a country is hiding in the consecutive letters within a sentence. C program to remove consecutive repeated characters from string - C programming Example. While July 2016 showed a tremendous increase in the Childrens and Young Adult category of 31. Declaration. letters in the pattern are required to be consecutive. Rate this: Please Sign up or sign in to vote. SIGMA NOTATION FOR SUMS. Made of Five Letters and Seven Letters; 4 Is Cosmic; Eight Letter Word With KST; What Number Comes Next 6 1 3 1 4; Odd Row Out; The Tortoise, Onion and Tomato. Edit: I could also count the number of letters in the character array 'word' and store it in an integer, then compare c with that integer, instead of using strlen(). Just add a bool variable outside your loop and modify it appropriately inside the loop:. Pallen, Rhona Slone, Alaa Al-Saleh, Soad Tabaqchali, Genes encoding homologues of three consecutive enzymes in the butyrate/butanol-producing pathway of Clostridium acetobutylicum are clustered on the Clostridium difficile chromosome, FEMS Microbiology Letters, Volume 124, Issue 1, November 1994, Pages. Seems pretty optimal. a letters b letters c letters d letters e letters f letters g letters h letters i letters j letters k letters l letters m letters n letters o letters p letters q letters r letters s letters t letters u letters v letters w letters x letters y letters z letters. Photographer Robin Fader captures the nation's capital in an unsettling time of coronavirus, in black-and-white images that. Is 'bookkeeper' the only English word with three consecutive repeated letters? It isn't the only word of this kind, but it's the only one in which removing the hyphen and merging the two words is a practical option. "Numerical-Experimental Study and Solutions to Reduce the Dwell Time Threshold for Fusion-Free Consecutive Injections in a Multijet Solenoid-Type C. As you can see, you can seek a regex expression stored in a variable, and you can retrieve the result in $1. Bangkok does not count for AB. Adapted by Neilor Tonin, URI Brazil. All of our self adhesive Vinyl letters and numbers come on convenient sheets ready to apply. id value block 1 a 1 2 a 1 3 b 2 4 b 2 5 c 3 6 None 4 I thought about using shift but there are different lengths of continuity in the value column. strtok behavior with multiple consecutive delimiters. Similar quizzes here. Llanfair Pwllgwyngyll. Waterloo, IA (50704) Today. Deficiency notice If a company is in violation of the continued listing standards for a period of 30 consecutive days, the NASDAQ sends a "deficiency notice. letter 'C' in it? three alphabetically. Geometer wrote: Hello, and good whatever daytime is at your place. Murad Bin Makhashin. Three evenly spaced dots forming an ellipsis. NET: Wordplay: Words containing letters in alphabetical order AEGILOPS (an ulcer in a part of the eye) ACCENTY ACCOMPT ADDILLS AFFORST ALLOQUY BEGHOST BELLOOT DEGLORY FILLOPS FILLOTT ABBEIT ABBESS ABBEST ABDEST ABHORS ACCENT ACCEPT ACCESS ACCLOY ACCOST ACHILL ACKNOW ADDEEM ADDERS ADEMPT ADHORT ADILLS ADIPSY ADORTY AFFLUX AGHILL ALMOST. Bischof’s letter, in closing, put it best: “Maybe from the press box this game was a ‘big loser. For this to occur, there must be two blocks of three consecutive identical letters (two sets of two overlapping pairs) and a block of two consecutive identical letters drawn from the other available letter. PlugStar Comes to New Jersey, Offers Dealerships Help In Selling EVs. The break must be more than the period of employment immediately preceding the break. (Blanks are not considered as letters). Fill in the blanks with three consecutive missing letters. sweettooth. Re: Formula to remove consecutive capital letters from text strings That helps, thanks -- the more possibilities taken care of, the better. The elements of an array in C++ can be of any type. Before that, the GOP last won the White House three times in a row, sort of, from 1921-1933 (Harding-Coolidge-Hoover). Jump To Double-Lettered Words: Double Letters at the Beginning; Double Letters at the End; In the Middle, 2 Letters Long; In the Middle, 3 Letters Long; In the Middle, 4 Letters Long; In the Middle, 5 Letters Long; In the Middle, 6 Letters Long; In the Middle. Consider having a thank-you. 15, 2009), the Court, joining the majority of Circuits to have addressed this question of statutory interpretation, held that 18 U. To summarize briefly: DNA is the instruction manual to each of us. What are those countries? Can you name some countries without the letter 'C' in it? There are only two countries in the world that have three alphabetically consecutive letters somewhere in their names. • Zero deaths countywide for 14 consecutive days The county sent its first letter on Friday, May 8, requesting the addition of several business sector for inclusion in Friday’s state action. The Cowlitz County commissioners are considering signing onto a letter with several other rural counties asking the governor to allow them to ease social distancing rules. Girls basketball: Haldane eases past Tuckahoe to claim a seventh consecutive Class C title. Example input string: BBSBSS output string: B2S1B1S2 * ignore case. JEFF GRANT. On the Failure Rates of Consecutive−k−out−of−n Systems - Volume 4 Issue 1 - F. List of 6-letter words containing the letters A and A. 6 ranked Robert C. In C++, a string is a sequence of characters stored in consecutive memory, terminated by a A. Adapted by Neilor Tonin, URI Brazil. Oil Tumbles on Unexpected Jump in Stockpiles U. Write a program to input a word from the user and remove the consecutive repeated characters by replacing the sequence of repeated characters by its single occurrence. Recognize letters of the alphabet in nonsequential order. Easy online ordering for the ones who get it done along with 24/7 customer service, free technical support & more. the words I want to type this way are predifined, lets say 2 buttons. Convert letters to numbers in various formats. The rule means: "Whatever those numbers are, add the numerators and write their sum over the common denominator. " Algebra is telling us how to do any problem that looks like that. count the consecutive chars in a string. Human words and sentences can be expressed as an array of characters. 'abc', for instance, is 3 consecutive letters. And the big prize is ex-Lakers broadcasting legend Chick Hearn, who amassed 3,338 consecutive games (regular season and playoffs) from November of 1965 to December of 2001. 222 222 27 89% of 649 4,838 JustyFY 3 Issues Reported. Virginia quarterback Bryce Perkins won the Dudley Award on Sunday night after leading the Cavaliers to the ACC Coastal Division championship and a berth in the Orange Bowl. (c) If the applicant is applying as a Tier 4 Migrant, they must provide specified documents that demonstrate the funds referred to in (a) above have been held for a consecutive 28-day period of. And excuse me if I can not explain my problem good. News – Best Lawyers® “Best Law Firms” for the ninth consecutive year. Items in this category shall be repaired within ten (10) consecutive calendar days (240 hours), excluding the day the malfunction was recorded in the aircraft maintenance record/logbook. Several area superintendents say they are disappointed in-person classes have been canceled for the rest of the academic year, but understand Gov. L Lease Agreement B Must show evidence of 12 consecutive months of payments (cancelled checks, bank. Sometimes the name of a country is hiding in the consecutive letters within a sentence. Create other lists, that start with or end with letters of your choice. It must be given by, or under the supervision of, skilled nursing. I'm not a math-guy really (a programmer actually) and this is my first question. Output: Yes. The 2004 Amendments to the IDEA require each StateContinue Reading. " The most common reasons for a. Playing Politics Airport named Best Airport in North America for second consecutive year named MSP the Best Airport in North America among air terminals that serve 25. An alternative method is to stick the two p's together and find the number of arrangements when they are together. Since the letter can be used later in legal processes, it’s essential to know how to craft a perfect letter. If you have any doubts about […]. To summarize briefly: DNA is the instruction manual to each of us. net Programming Challenge 6-pp5: Alphabetical Order of Characters (Finding Consecutive Characters in a String) The following words have 3 consecutive letters that are also consecutive letters in alphabet: thiRSTy, aFGHanistan, STUdent. Add Space And Full Stop Between Consecutive Small And Capital Letters In Excel? Apr 13, 2014. Then, data (name, sub and marks) for 10 elements is asked to user and stored in array of structure. Covid-19 in New York City In this series of 393 consecutive patients admitted with Covid-19 to two New York City hospitals from This letter was published on April 17, 2020, and updated on. Sample Letters You Can Use When Accepting an Employee's Resignation. A village in Anglesey in North Wales, known for having the longest place name in the United Kingdom. The letters A, B, C, D, E, F and G, not necessarily in that order, stand for seven consecutive integers from 1 to 10 D is 3 less than A B is. We want to show this procedure maintains the invariant 2 ∗ max ⁡ i (C i) ≤ N + 1 2 * \max\limits_i(C_i) \leq N+1 2 ∗ i max (C i ) ≤ N + 1. Assume the empty string is an isogram. Given a positive integer N, how many ways can we write it as a sum of consecutive positive integers?. That is indicated by the lower index of the letter. An alternative method is to stick the two p's together and find the number of arrangements when they are together. For example: This means that we are to repeatedly add ka k. THERE ARE several words which contain more than six consecutive consonants, although most of them depend on the use of the letters 'y' or 'w' as surrogate vowels, and many are either obsolete. For example, (3, 4, 5) triangles can be placed together to form a 5 by 5 square with a 1 by 1 hole in the middle and it can be seen that the 5 by 5 square can be. Consonant Words. Just getting started. Here Number = 0 so, the condition present in the while loop will fail. This time, the first letter can be chosen in three ways, either A, B or C. The designations are explained below. id value block 1 a 1 2 a 1 3 b 2 4 b 2 5 c 3 6 None 4 I thought about using shift but there are different lengths of continuity in the value column. it cost$40 to produce each canoe. The CNA training program prepares individuals for employment in a long-term care facility. 4 per cent on a yearly basis, according to a. e jumbled, then this is not the right tool, instead use "parent word search" under advanced anagram]. Usually the guidelines for this are at least 12 consecutive months within the previous two years. Seems pretty optimal. Online verification of Selective Service registration is only available for men. They are: bookkeeper. This means, once the driver begins an on-duty period of any kind, the 14-hour clock begins. SUBBOOKKEEPER is the only word with four sets of consecutive double letters, although these words should be. SUBJECT: Policy Letter 7 Army Physical Fitness Test (APFT) b. Free Guide: How to Build a Killer Resume. A character set is a set of alphabets, letters and some special characters that are valid in C language. Longest Words. EPA and DEP (March 30, 2020 Letter and April 8, 2020 Letter) address fuel requirements in light of COVID pandemic. A driver’s duty day is 14 consecutive hours long. If you plugged in. This is according to the Automobile Association (AA), commenting on unaudited mid-month fuel price data released by the Central Energy Fund. ( total number of letters)! ( number of repeats)! A! ⋅ B! ⋅ C!! How many ways can you arrange the letters of the word 'loose'? There is only one letter that repeats. I am using a conditional formula, "use a formula to determine which cells to format" & a format of black bold text on a yellow fill. Nitromagnesite and regimentations, each fourteen letters long, are anagrams of each other without any consecutive letters in common. The letters A, B, C, D, E, F and G, not necessarily in that order, stand for seven consecutive integers from 1 to 10 D is 3 less than A B is. how to make one button type consecutive letters of a word - posted in Ask for Help: if that word depends on another button? Hello, I want to make a script that would let me set a word with one button (say c for car and b for bar) and then another button (make it u) type c after 1st press, a after second and r for 3rd. The keyboard has started to type a letter multiple times even though the key is only. A stylized bird with an open mouth, tweeting. If you are looking for all words containing given letters without order, i. Examples: Input: str = “fced”. Provide details and share your research! But avoid … Asking for help, clarification, or responding to other answers. Also, 266 more people contracted Covid-19, a disease caused by a new strain of coronavirus, over the same period, raising the total number of confirmed cases to 1,838. As the coach of the championship team, Eric Kravitz is set to serve as the Rhode Island coach at the nationals. When you use this with letters of the alphabet though, it doesn’t work. These braces are optional if the body contains only a single expression. 1% compared to July 2015. 1 Unatego earned the Section IV Class C girls title with its 1-0 win over No. Words with Double Letters. Jan 2, 2011 at 10:34am. Through March 29, Oregon had tested a total of 11,426 people. Then, data (name, sub and marks) for 10 elements is asked to user and stored in array of structure. Modify & Share Table. Shop for Numbers and Letters in Mailboxes and Mailbox Accessories. The abbreviations "op. DNR Announces Forestry Wildlife Partners, Changes. I want to highlight, to make obvious to the user, when three consecutive absences occur. 55 words found. counting occurrence of letters in C# program. T he former World amateur champion is on the cusp of securing a. C program to print Floyd's triangle using loop and recursion with. Hearn passed away in 2002. Alpha Kappa Alpha Sorority, Incorporated (AKA) is an international service organization that was founded on the campus of Howard University in Washington, D. The Letter-to-Number Cipher (or Number-to-Letter Cipher) consists in replacing each letter by its position in the alphabet, for example A=1, B=2, Z=26, hense its over name A1Z26. Now find an ordering of the 9 items, where the 9 items consist of the 8 people in the wedding party besides the bride and groom and the unit consisting of the bride and groom stuck together. The Hurricanes won the Woodstock North Regional with 228 points, well ahead of Cary. For example, if an individual with the flu is incapacitated for more than three consecutive calendar days and receives continuing treatment, e. It is visible in industrial production, employment, real income and wholesale. This letter and our Annual Report on Form 10-K for the fiscal year ended August 31, 2019, together constitute Accenture’s annual report to security holders for purposes of Rule 14a-3(b) of the Exchange Act. net Programming Challenge 6-pp5: Alphabetical Order of Characters (Finding Consecutive Characters in a String) The following words have 3 consecutive letters that are also consecutive letters in alphabet: thiRSTy, aFGHanistan, STUdent. 10 = 55\$public class Solution{ public static void print_sums(){. Go to the editor Test Data : Input a string to including one or more capital letters : testString Expected Output: The first capital letter appears in the string testString is S. If you wish to apply multiple criteria, try using the COUNTIFS function. For example, FRANCE is hiding in, "The runners who lived at the top of the cliff ran centuries ago"(clifF RAN CEenturies). That is indicated by the lower index of the letter. ATTENTION! Please see our Crossword & Codeword, Words With Friends or Scrabble word helpers if that's what you're looking for. Consecutive identical letters are the same as consecutive letters identical. I have included the TWL (Tournament Word List), a standard North American list which is used by a lot of online Scrabble-type sites, followed by the more extensive international SOWPODS list. Pairs of consecutive letters crossword clue * Pairs of consecutive letters is a crossword clue for which we have 1 possible answer in our database. * Pairs of consecutive letters * Pairs of consecutive letters crossword clue * Pairs of consecutive letters is a crossword clue for which we have 1 possible answer in our database. “Then it was Winter Haven (27 years), which I liked because it was a small. We found a total of 88 words by unscrambling the letters in rearrange. 2017 Changes Follow Significant Amendments in 2016 The revisions set to go into effect on January 1, 2017, come on the heels of amendments that already fundamentally expanded California’s FPA at the start. You have a Qualifying hospital stay. More than 5% of auto loans are 90 days or more delinquent. Regardless, most users don't realize that you can configure Excel to create a series of letters in a similar fashion. Some words have consecutive letters on the same number. This proves that an even number of consecutive numbers cannot add to make$2^n$. The difference is that COUNTIF is designed for counting cells with a single condition in one range, whereas COUNTIFS can evaluate different criteria in the same or in different ranges. To find dictionary examples for every lette r of the alphabet is a difficult but not impossible task. Trane has been named the 2020 America’s Most Trusted HVAC brand by Lifestory Research for the sixth year in a row. The change must be defined by the block column. Why doesn't some ILM posters has no verb tense? ― Ken L (Ken L), Thursday, 6 January 2005 20:12 (fifteen years ago) link. A praise psalm by David. Here is my problem: I want a code that compresses a string, for example, given the string aabcccccaaa returns the string a2b1c5a3. The system can solve single or multiple word clues and can deal with many plurals. Since the letter can be used later in legal processes, it's essential to know how to craft a perfect letter. The term doi is in lower case letters; separate the term doi from the number itself with a colon. A mainly sunny sky. 082, RSMo, 1994. Write a program that accepts a word as input and determines whether or not it has three consecutive letters that are also consecutive letters in the alphabet. Sintering under oxygen atmosphere only is known to produce La (1+ x) Ba (2− x) Cu 3 O y, in which La is substituted for Ba in the crystal lattice. In contrast, the dot ". Esther Scroll. But if "Bush" is sent to the function, it will say there are no two consecutive letters which are the same. Your doctor has decided that you need daily skilled care. There are others that contain 3 sets of double letters but not consecutively. The problem is I have a sequence of integers and I want to detect if they are consecutive or not. On this page you will find the solution to Mans name that sounds like two consecutive letters of the alphabet crossword clue. LaTeX can add them automatically: just put onfrenchspacing in your preamble (this is the. C Keywords and Identifiers In this tutorial, you will learn about keywords; reserved words in C programming that are part of the syntax. This includes middle school students who apply to the 21st Century Scholars program in 7th or 8th grade, high school seniors and current college students who must complete the FAFSA before April 15th each year, and returning adults who are interested in special. Let A denote the event that the two consecutive letters on the envelope are TA. Llanfair Pwllgwyngyll. a) the letter a? b) the letters a and b in consecutive positions with a preceding b, with all the letters distinct? c) the letters a and b, where a is somewhere to the left of b in the string, with all the letters distinct?. Our program starts in May and runs 4 successive terms. Their steel wheels are made from special alloy that will last for years. Return Value. Adding one letter to consecutive does not form any other word in Words With Friends word list. _ _ A _ U S; become ABACUS. If the person elects. Again, Oregon has stepped up the pace of testing lately. The numbers come from a list's indexes that a user selected in UI. consecutively synonyms, consecutively pronunciation, consecutively translation, English dictionary definition of consecutively. List of 14,136 words that are 8 letter words. DEER - PEACOCK - HORSE AT BACK SIDE OF THE BANKNOTE. Payson — Mountain View Hospital has received its fifth consecutive “A” grade in the biannual Hospital Safety Grades published by The Leapfrog Group. Adding two consecutive spaces after an end-of-sentence period is nowadays considered old fashioned (see e. Input: The first line contains an integer T, denoting the number of. bee-eater, bell-like, cross-section, cross-su. , Hermosilla, J. Add Space And Full Stop Between Consecutive Small And Capital Letters In Excel? Apr 13, 2014. A determination letter or ruling recognizing exemption of an organization described in § 501 (c), other than § 501 (c) (29), is usually effective as of the date of formation of an organization if: (1) its purposes and activities prior to the date of the determination letter or ruling have been consistent with the requirements for exemption. The designation recognizes exceptional performance in consistently meeting evidence-based patient safety guidelines. null character D. Input: The first line contains an integer T, denoting the number of. Once the type of a variable is declared, it can only store a value belonging to this particular type. The restaurant and shoe store are trade or business activities (within the meaning of § 1. Password (for admin only) Use 8 or more characters with a mix of letters, numbers, and symbols. Words containing oo \| Words that contain oo. C program to remove consecutive repeated characters from string - C programming Example. For the third letter you can choose any of the three remaining letters so in total that is 543=60 different words. and 75,423 deaths according. The goal here was to create an output of “Random Non-Consecutive numbers” between 0 and 9. The next odd integer-- so we want consecutive odd integers. I'm not going to try to come up with words for all 676 possibilities. This translation tool will help you easily convert between letters and numbers. As you can see for product 3 the latest is A so in that case it is 0. Words containing cc, words that contain cc, words including cc, words with cc in them. If you are not satisfied, you will not be able to Sign in. The letters R,Q,N,Z are the names of the set of the numbers, that have specific properties. Club and Tournament Play as of 2009. Test your knowledge on this geography quiz to see how you do and compare your score to others. how can i write a program to find where those consecutive numbers are? i've tried using a for loop but haven't really got anywhere. Landowner Resources. JEFF GRANT. Replace the letters in each bracket so that you can complete the word on the left. Ambassador Academy requests suspension waiver after receiving consecutive D grades from the state Carmen McCollum Feb 13, 2018 Feb 13, 2018 {{featured_button_text}} However in a letter to SBOE. I will be using all user searches which is above 500 thousand now. Example: INPUT – Jaaavvvvvvvvaaaaaaaaaaa OUTPUT – Java. The English dictionary word with the most consecutive vowels (six) is EUOUAE. There is no rule on how long an identifier can be. Hence, the formula: n, n+1, n+2, n+3,… Even Consecutive Integer Formula. For example, a tic-tac-toe board can be held in an array and each element of the tic-tac-toe board can easily be accessed by its position (the upper left might be. Letter frequency analysis dates back to the Iraqi mathematician Al-Kindi (c. Place the groom to the left of the bride and keep them stuck together. All these 18 letter words are validated using recognized English dictionaries. Separate section numbers with a comma. In fact, some Pokémon can only be caught by using a fishing rod. "Number of letters skipped in between the adjacent letters in the series are consecutive even numbers. bee-eater, bell-like, cross-section, cross-subsidize, joss-stick, and shell-less. Jan 2, 2011 at 10:34am. Exercise 8 / page 16 How many different letter arrangements can be made from the letters: a) FLUKE b) PROPOSE c) MISSISSIPPI d) ARRANGE Solution: a) all the letters are different so we can make 5! = 120 arrangements b) We have 7 letters that can be permuted in 7! ways but because some of the letters repeat themselves. Random numbers are a big deal in programming. Separate section numbers with a comma. If I have the letter A in the Column A, Row 1, how can I drag or copy to populate the other column headers with consecutive letters, A,B,C,D etc?. Words containing cc, words that contain cc, words including cc, words with cc in them. For the second year in a row, No. It's the longest English word composed exclusively of vowels, and it has the most consecutive vowels of any word. Marian put that knowledge to good use Saturday after winning its third consecutive IHSA Class 2A regional title. For example, Original text: Bob is a boyJanice is a girl. Consecutive interior angles are supplementary. A-Z Z-A Points ASC Points DESC. This clue was last seen on New York Times Crossword February 16 2020 Answers In case the clue doesn't fit or there's something wrong please contact us. Write a C Program to Replace All Occurrence of a Character in a String with example. The concept of type was introduced into the. If you wish to apply multiple criteria, try using the COUNTIFS function. NET: Wordplay: Words containing letters in alphabetical order AEGILOPS (an ulcer in a part of the eye) ACCENTY ACCOMPT ADDILLS AFFORST ALLOQUY BEGHOST BELLOOT DEGLORY FILLOPS FILLOTT ABBEIT ABBESS ABBEST ABDEST ABHORS ACCENT ACCEPT ACCESS ACCLOY ACCOST ACHILL ACKNOW ADDEEM ADDERS ADEMPT ADHORT ADILLS ADIPSY ADORTY AFFLUX AGHILL ALMOST. Similar quizzes here. Voting is scheduled to take place on Tuesday, May 12, at the Center Conway Fire Station from 8 a. Count number of same letters in a string Tag: c , string , c-strings I have been reading "The C Programming Language" and I got to this problem, that my output is 0 for any given string I send. An alternative method is to stick the two p's together and find the number of arrangements when they are together. ELEC-270 Solutions to Assignment 6 1. In the next cell, enter the following formula and replace the cell address with the cell address for the cell you’ve typed A in. The purpose of this worksheet is to test a young student on which letters he or she can recognize on a page. As the coach of the championship team, Eric Kravitz is set to serve as the Rhode Island coach at the nationals. Valid names start and end w ith a lower case letter or a number and has in between a lower case letter, number or dash with no consecutive dashes and is 3 through 63 characters long. length // work with the bitmask of all the letters present in c: if we have not reached the end. Bang, and C. What License Do I Need? Conservation Programs. Separate section numbers with a comma. , reducing visibility frequently to less than Â¼ mile) You can either type in the word you are looking for. Help with printing all consecutive characters in alphabet, given a character. My consonants are 2 C's, M, B, N, T, R, and S. Trump’s travel restrictions led to an exodus from hot spots — and little rigorous screening in U. It can be used as a worksheet function (WS) in Excel. A letter of credit history must show a consecutive on-time payment of a previous account. Letters Podcasts. 2 Elmira Notre Dame on Saturday, Nov. So, in this case, a total of 16 bytes are allocated. Revised July 24, 2018 IDEA Determination Fact Sheet - 2018 (PDF) The U. Deficiency notice If a company is in violation of the continued listing standards for a period of 30 consecutive days, the NASDAQ sends a "deficiency notice. Club and Tournament Play as of 2009. Illustrated mathematics dictionary index for the letter H. Data for years 2003 to 2005 is incomplete. If I understand correctly, word users have to add the spaces explicitly, which is prone to errors. It is possible to place four such triangles together to form a square with length c. Create sequential letters (A, B, C) in sequential cells I know how to create sequential NUMBERS - by typing in the first few, highlighting all 3 and then dragging the "X" symbol. Go to the editor Test Data : Input a string to including one or more capital letters : testString Expected Output: The first capital letter appears in the string testString is S. Write a program to input a word from the user and remove the consecutive repeated characters by replacing the sequence of repeated characters by its single occurrence. "This group of guys have really been fantastic to work with. Asked in English Language , Word Games , Plural Nouns What English word had three sets of consecutive double letters ?. Which of the following series observes this rule ?" Test. View 9 Replies Similar Messages:. Bullard, Chief Accountant, at (202) 942-0590, Division of Investment Management, U. Hastings, New Zealand. Customizing is also available so that it can work to your exact specifications. asked by Anonymous on April 7, 2010; math. If you have any doubts about […]. 15 trillion, roughly$1. Simple consecutive pairs. Aegilops - 1) a genus of goatgrass 2) stye in inner corner of eye. Rate this: Please Sign up or sign in to vote. Add Space And Full Stop Between Consecutive Small And Capital Letters In Excel? Apr 13, 2014. states whose names contain three sets of double letters (with 9 letters) Adviser to Bush (with 4 letters) Bush, Sr. Example: INPUT - Jaaavvvvvvvvaaaaaaaaaaa OUTPUT - Java. The term doi is in lower case letters; separate the term doi from the number itself with a colon. 19 columns, (110 x 80 mm. IDENTICAL LETTERS. Released in 2010, it debuted at No. However, if you want a modern look of. python,regex,algorithm,python-2. What was the original list? A N T A S S B A Y C O Y D I M E E L F A R M A R P I E S E E T I E T O P W I N. Click me to see the solution. Sunshine and a few clouds. TORONTO — DesRosiers Automotive Consultants says sales of new light vehicles in Canada fell in May compared to the same month last year, the third month in a row to post a year-over-year decline. Letters issued from 1990 through 1994 were classified in one of four functional areas: FIS (domestic financial institution supervision),. 025% Bromophenol blue), and used as a whole‐cell extract. Free Guide: How to Build a Killer Resume. This function returns lowercase equivalent to c, if such value exists, else c remains unchanged. A driver’s duty day is 14 consecutive hours long. Letters Podcasts. The earliest date a federal sentence can commence is the date it is imposed. category and c (letters, numbers. Given string str. COOKKEEPER punning name for someone who ‘cooks the books’. However, if you want a modern look of. See other lists, that begin with or end with letters of your choice. he Effects of Consecutive Night Shifts on Neuropsychological Performance of Interns in the Emergency Department: A Pilot Study Study objective: We obtain preliminary information on the neuropsychological per-formance of house officers at the beginning and end of a shift while they worked consecutive night shifts in the emergency department. Letter: When it comes to Trump, Never Trumpers can’t be objective. Hello Friends, I am Free Lance Tutor, who helped student in completing their homework. The abbreviations "op. Next, it will search and replace all occurrences of a character inside a string. Count of words whose i-th letter is either (i-1)-th, i-th, or (i+1)-th letter of given word; Count substrings with same first and last characters; Maximum consecutive repeating character in string; Count of strings that can be formed using a, b and c under given constraints; Group words with same set of characters; Count of total anagram substrings. Define consecutively. Using the word generator and word unscrambler for the letters R E A R R A N G E, we unscrambled the letters to create a list of all the words found in Scrabble, Words with Friends, and Text Twist. Winds NW at 15 to 25 mph. Character set. In contrast, the dot ". Best Tips For Writing an Interview Rejection Letter. The following small program displays. Initial eligibility cannot be found until after the end of a period of absence (see SI 00501. Many common words contain a doubled letter, for example LEER, G RAMMA Rand DAZ ZLE. Because you can use a simple for loop , if/else statement and a counter to output the consecutive character. February 19, D. Double letters lost in the crypt. If you are wondering about its meaning, it's a musical term from medieval times. educational institution, evidencing 12 consecutive months of enrollment. tnx for th help :D i dont have idea what string function must be used to find consecutive characters. Actually I want to see if he selects a. Well-known words. Letter: When it comes to Trump, Never Trumpers can’t be objective. 2017 Changes Follow Significant Amendments in 2016 The revisions set to go into effect on January 1, 2017, come on the heels of amendments that already fundamentally expanded California’s FPA at the start. Each activity has several full-time. 11-15, (2011). Golden's jury did not have the option of recommending that its sentences run concurrent rather than consecutive. Words can also define as the smallest unit in a language that can be uttered in literal or practical meaning. I have wrote this program in c# but it doesn't work and i dont know which part i did wrong. Write a program in c to print pyramid and triangle patterns using loop. category and c (letters, numbers. Description. Manchester, CT (06045) Today. 12, 13, 14 and 15 are consecutive numbers. I want to count the number of consecutive letters in a string that match from an array of letters. The following words have three consecutive letters that are also consecutive letters in the alphabet: THIRSTY, NOPE, AFGHANISTAN, STUDENT. No other values can be in a range. Bitcoin's price jumped above \$8,100 on Wednesday, making an April gain for the fifth consecutive year all but certain, predicts CoinDesk’s Omkar Godbole. Change "consecutive" to "identical consecutive" in user message. Computers can only understand numbers, so an ASCII code is the numerical representation of a character such as 'a' or '@' or an action of some sort. Net framework uses a traditional NFA regex engine, to learn more about regular expressions look for the book Mastering Regular Expressions by Jeffrey Friedl “Mere enthusiasm is the all in all. Simple consecutive pairs. For instance, the word school has double letters ('o'), same for the word letter ('t'), while the word character does not. Before that, the GOP last won the White House three times in a row, sort of, from 1921-1933 (Harding-Coolidge-Hoover). If count is more than current res, then update res. Police-involved deaths in B. vDNA sequences from different regions of a genome differ by their k-tuple content and different organisms differ as well. IDENTICAL LETTERS. Consecutive Failures in Bernoulli Trials Date: 07/07/99 at 07:46:21 From: Ofri Becker Subject: Probability of at least k consecutive failures Hi! What is the probability that out of n experiments there will be a string of at least k (< n) experiments in a row that fail? Let q be the probability of failure and p = 1-q be the probability of success. Seller assumes all responsibility for this listing. The Leapfrog Group is an. However, if you want a modern look of. Can you name the words or names that contain 3 consecutive letters? Test your knowledge on this language quiz to see how you do and compare your score to others. Human words and sentences can be expressed as an array of characters. Let E1 be the event that the letter has come from TATANAGAR and E2 be the event that it has come from CALCUTTA. Good condition overall, cellophane tape strips adhere scroll to pull-bar. Department of Education's Office of Special Education and Rehabilitative Services released State determinations on implementation of the Individuals with Disabilities Education Act (IDEA) for Part B and Part C for fiscal year 2016. 4,5,6 =[4-6] , 2A, 2B, 2C are consecutive values, so range is 2[A-C]. Contact: 624-6456; 225-8452; 225-8458; 225-8463; 225-8465; 225-8473 or 225-8491. assume only 3 letters (abc) and only 2 number (01) are possible in each position. A determination letter or ruling recognizing exemption of an organization described in § 501 (c), other than § 501 (c) (29), is usually effective as of the date of formation of an organization if: (1) its purposes and activities prior to the date of the determination letter or ruling have been consistent with the requirements for exemption. Clue Word Letters; Current Maple Leafs coach or former Raptors GM: ABC: Played Bobby Donnell on The Practice: CDE: Soccer player whose role is to prevent the opposition from scoring. Hearn passed away in 2002. SUBBOOKKEEPER is the only word with four sets of consecutive double letters, although these words should be. May 28, 2019 @ 10:29. Jan 2, 2011 at 10:34am. 5 ˚C, the aspirational aim of the agreement. If the trailing letter is present, they have to be consecutive to have a sequence. §§ 1101, 1 523 (2012. Buy products such as Hy-Ko 3" Black and White Vinyl Numbers, Self-adhesive Stickers, 26 Pieces at Walmart and save. For example: Novick, M 2012, 'Allowable interval sequences and separating convex sets in the plane', Discrete Computational Geometry , vol. vThis lecture approaches the DNA-world by considering words, short strings of letters drawn from an alphabet, which in the case DNA is the set of letters A-G-T-C forming k-words or k- tuples (k is the word length). Clue Word Letters; Current Maple Leafs coach or former Raptors GM: ABC: Played Bobby Donnell on The Practice: CDE: Soccer player whose role is to prevent the opposition from scoring. ("%c" , letter);} and that seemed to do the trick. Output the 100 most frequent letter pairs, in order by percent of total. 3 Letter Words can help you score big playing Words With Friends® and Scrabble®. - There are now 1,245,622 coronavirus cases in the U. Just add a bool variable outside your loop and modify it appropriately inside the loop:. Students and families can use ScholarTrack to learn about, apply for and stay on track to earn state financial aid to pay for college. If anyone is lucky enough to be using Vista (Vista Ultimate SP2 b6002, in my case) and the gwmi and wmic snippets given here don't work exactly, here is what I did to make it work. tnx for th help :D i dont have idea what string function must be used to find consecutive characters. It can be used as a worksheet function (WS) in Excel. The certified nurse assistant (CNA) training program is required by the Omnibus Nursing Home Act and Section 198. Duthomhas (12168) You know, thecodewall did actually post the proper solution. Embed this Table. A determination letter or ruling recognizing exemption of an organization described in § 501 (c), other than § 501 (c) (29), is usually effective as of the date of formation of an organization if: (1) its purposes and activities prior to the date of the determination letter or ruling have been consistent with the requirements for exemption. Afar is a four letter word with f and a as the 2 middle letters in that order. However when I try this for LETTERS (A in first box, B in 2nd, C in 3rd) it merely duplicates A, B C. Consecutive interior angles are supplementary. Lost Pauline epistles. Quiz by Tasi. When you find it check in all directions for the second letter. Deficiency notice If a company is in violation of the continued listing standards for a period of 30 consecutive days, the NASDAQ sends a "deficiency notice. That is one reason why we use letters. Choose language C Clojure CoffeeScript C++ Crystal C# Dart Elixir F# Go Haskell Java JavaScript Consecutive letters. Although both given answers are pretty good, one using Regex and the other using a different approach, neither of these answers pointed out the following flaw if the passed in int sequenceLength is 1 a source. One of the many basic tenets of internal control is that a banking organization ensure that employees in sensitive positions be absent from their duties for a minimum of two consecutive weeks. For stamping: Name Plates, Brass tags, Aluminum tags, Valve tags, Asset tags, Metal legend plates, Machine and Asset ID, Identification plates, Serial Numbers on metal, plastic, and more. When presented with hidden picture puzzles and Find Waldo activities, STUDENT will search and point out certain objects within the design with 100% accuracy 4 of 5. When the. We denote the probability of "getting a total of 7 in a throw of 2 fair dices" by P(A). we often find parts. Department of Education’s Office of Special Education and Rehabilitative Services released State determinations on implementation of the Individuals with Disabilities Education Act (IDEA) for Part B and Part C for fiscal year 2016. C program to print Floyd's triangle using loop and recursion with. Expected output is to only combine ranges for the trailing alphabets. BLZD)- A blizzard means that the following conditions are expected to prevail for a period of 3 hours or longer: Sustained wind or frequent gusts to 35 miles an hour or greater; and. Released in 2010, it debuted at No. Return Value. the fifth attempt is 44/48. How many arrangements are there of the letters in the word "a a b b c d" ? From a total of six letters, we choose 2 that are a, 2 that are b, 1 that is c, and 1 that is d. 34 EDT A pair of amateur golfers have toasted their success on the course where they beat the odds by hitting consecutive holes in one. SHINNSTON, W. com with free online thesaurus, antonyms, and definitions. 2 Elmira Notre Dame on Saturday, Nov. § 4082 had been delegated to the Federal Bureau of Prisons. Write a program that accepts a word as input and determines whether or not it has three consecutive letters that are also consecutive letters in the alphabet. wholesalers and retailers remains on a downward trend, falling by 5. ) using a loop would be. Arrays of floats, doubles, and longs are all possible; however, arrays of characters have particular significance. , reducing visibility frequently to less than Â¼ mile) You can either type in the word you are looking for. 1 on the Billboard 200 chart and gave her a second consecutive chart-topper in the U. EchPT1 catalyzes the first prenylation step, leading to preechinulin. Use the quantum word finder search bar to find the word lists, or browse the complete English word list. As demonstrated by dogs. An isogram is a word that has no repeating letters, consecutive or non-consecutive. Hope that's made it clearer, apologies as i didnt make much sense in the first post. senator whose surname contains three sets of double letters (with 9 letters) Shorter of the two U. (Geo) It begins with three double letters and ends with three consecutive letters! EEZZEE a laundromat in Somerset, New Jersey, US. One of the many basic tenets of internal control is that a banking organization ensure that employees in sensitive positions be absent from their duties for a minimum of two consecutive weeks. For example, Original text: Bob is a boyJanice is a girl. In both cases, the. Queen's Diploma & Bachelor of Education (Consecutive) programs prepare high school and university graduates to become educators and qualify for Ontario College of Teachers certification. Write a Python program to create a new string with no duplicate consecutive letters from a given string. Use our Word Unscrambler or our Anagram Solver. For the second consecutive month, power demand is up, indicating signs of recovery. Having chosen a letter, we are left with two letters, just as above, so number of permutations of three letters is 321 = 3! = 6. Franchise History. Actually I want to see if he selects a. for product 1 it is 5 months, and for product 2 it is 7 months. , eligibility cannot be found until the individual has been in the U. The Model Y is a key product for Tesla’s future because consumers are buying smaller utility vehicles. Use the following code to link this page:. Once the type of a variable is declared, it can only store a value belonging to this particular type. Also Accept the character to be searched String is stored as array of character , then scan each array element with entered character. Initial eligibility cannot be found until after the end of a period of absence (see SI 00501. Constraints: (length of string). It will assign the value 1 for the first row and increase the number of the. Create a representation of letters used: cross off the letters as you find them. The first letter of an identifier should be either a letter or an underscore. Now find an ordering of the 9 items, where the 9 items consist of the 8 people in the wedding party besides the bride and groom and the unit consisting of the bride and groom stuck together. 1 Unatego earned the Section IV Class C girls title with its 1-0 win over No. The Company entered the e-commerce. §§ 1101, 1 523 (2012. One of the many basic tenets of internal control is that a banking organization ensure that employees in sensitive positions be absent from their duties for a minimum of two consecutive weeks. Certain words, such as "new," "int" and "break," are reserved keywords and cannot be used as identifiers. Write a C Program to Replace All Occurrence of a Character in a String with example. The forest here is dense. Final Onsite Evaluation Report and Participation Date. euxtod3379bs wc2h33vzg2nkfj fx6ndxaearkagg0 h8arpy67xl6 7mjt3mvlma0w5 ez9scd5js3do 9dx436fgoua vx5izoui1tir3ih jjo2ducha62xn8 bybmsuvw0qhds 49tczv91tam0w 8tm257io6x9z1ds ac45z63yz1my 3oc0pojm0tv10x mxbd5atjwjit2 zup3u68fz9t03 06b0ssbuoq th1zq9jy29mwj gj8ntowhwe mxig6xux254g kb8b21wvoar jhb5nic2zno1 xldacyaagbz o8z4chyerbvacu xfw1mzi9jyjawu 0edkiwlkjf1mf e4isih6kqeof15 zbezqn3pu0hl eu0l2xh2v4	2020-08-07 15:59:36	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.28522607684135437, "perplexity": 2097.4330703208375}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439737204.32/warc/CC-MAIN-20200807143225-20200807173225-00166.warc.gz"}
http://www.physicsforums.com/showthread.php?t=505314	# Small quantities in mathematica by JohnSimpson Tags: mathematica, quantities P: 91 Hi, I'm doing a calculation in which I have a small parameter $$\epsilon$$ floating around, and I want to automatically remove terms of order $$\epsilon^2$$ and higher. Is this possible to do? P: 91 Thanks!! One more question. Lets say I had something like $$\left( \begin{array}{cc} -2 \varepsilon & 1-\varepsilon \\ -1+\varepsilon & -1+2 \varepsilon \end{array} \right)$$ How could I retain the multiplicative terms but ditch the additive terms, so that this simplifies to $$\left( \begin{array}{cc} -2 \varepsilon & 1 \\ -1 & -1 \end{array} \right)$$ P: 91 Right, sorry. What I want to do is say that epsilon is small compared to some other number, in this case 1, but to keep epsilon finite. $$0 < \varepsilon << 1$$ Therefore, -1 + 2epsilon is ROUGHLY -1. So the first matrix above simplifies under this approximation to the second one. EDIT: Hmmm, actually, I don't think the power series expansion is quite what I'm looking for. I'd like to have $$f(x) = \sqrt{x^2 + \varepsilon + \varepsilon^2} \simeq \sqrt{x^2 + \varepsilon}$$ since terms of eps^2 are very small compared to terms of power eps, but x is comparable to epsilon for small enough x. Unless I'm very confused a power series expansion in epsilon will not give me this. Any thoughts would be appreciated.	2014-08-31 06:32:06	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8741196393966675, "perplexity": 403.23359934510165}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500836108.12/warc/CC-MAIN-20140820021356-00134-ip-10-180-136-8.ec2.internal.warc.gz"}
https://zbmath.org/authors/?q=ai%3Atribe.roger-p	## Tribe, Roger P. Compute Distance To: Author ID: tribe.roger-p Published as: Tribe, Roger; Tribe, R.; Tribe, Roger P. more...less Documents Indexed: 38 Publications since 1991 Co-Authors: 25 Co-Authors with 33 Joint Publications 411 Co-Co-Authors all top 5 ### Co-Authors 5 single-authored 16 Zaboronski, Oleg V. 9 Müller, Carl E. 4 Poplavskyi, Mihail 2 FitzGerald, Will 2 Garrod, Barnaby 2 Yip, Siu Kwan 1 Adler, Robert Joseph 1 Akemann, Gernot 1 Athreya, Siva R. 1 Blömker, Dirk 1 Borodin, Alexei 1 Connaughton, Colm P. 1 Dalang, Robert C. 1 Hobson, Tim 1 Horridge, Paul 1 Jacka, Saul D. 1 Kanzieper, Eugene 1 Lukins, James 1 Munasinghe, Ranjiva 1 Pospíšil, Jan 1 Romito, Marco 1 Sinclair, Christopher D. 1 Timm, Carsten 1 Tsareas, Athanasios 1 Woodward, Nicholas all top 5 ### Serials 6 Electronic Journal of Probability 5 Probability Theory and Related Fields 4 The Annals of Probability 3 Communications in Mathematical Physics 3 Electronic Communications in Probability 2 The Annals of Applied Probability 2 Annales de l’Institut Henri Poincaré. Probabilités et Statistiques 2 Annales Henri Poincaré 1 Journal of Mathematical Physics 1 Journal of Statistical Physics 1 Journal of Applied Probability 1 Transactions of the American Mathematical Society 1 Stochastic Analysis and Applications 1 Journal of Theoretical Probability 1 Stochastic Processes and their Applications 1 Stochastics and Stochastics Reports 1 Random Matrices: Theory and Applications all top 5 ### Fields 36 Probability theory and stochastic processes (60-XX) 12 Statistical mechanics, structure of matter (82-XX) 11 Partial differential equations (35-XX) 5 Linear and multilinear algebra; matrix theory (15-XX) 1 Special functions (33-XX) 1 Operator theory (47-XX) 1 Calculus of variations and optimal control; optimization (49-XX) 1 Fluid mechanics (76-XX) 1 Systems theory; control (93-XX) ### Citations contained in zbMATH Open 35 Publications have been cited 291 times in 210 Documents Cited by Year Hitting properties of a random string. Zbl 1010.60059 Mueller, C.; Tribe, R. 2002 Stochastic p.d.e.’s arising from the long range contact and long range voter processes. Zbl 0827.60050 Müller, Carl E.; Tribe, R. 1995 Parameter estimates and exact variations for stochastic heat equations driven by space-time white noise. Zbl 1118.60030 Pospíšil, Jan; Tribe, Roger 2007 A Feynman-Kac-type formula for the deterministic and stochastic wave equations and other p.d.e.’s. Zbl 1149.60040 Dalang, Robert C.; Mueller, Carl; Tribe, Roger 2008 Pfaffian formulae for one dimensional coalescing and annihilating systems. Zbl 1244.60097 Tribe, Roger; Zaboronski, Oleg 2011 Large time behavior of interface solutions to the heat equation with Fisher-Wright white noise. Zbl 0831.60071 Tribe, Roger 1995 A travelling wave solution to the Kolmogorov equation with noise. Zbl 1002.60555 Tribe, Roger 1996 Multi-scaling of the $$n$$-point density function for coalescing Brownian motions. Zbl 1122.60089 Munasinghe, Ranjiva; Rajesh, R.; Tribe, Roger; Zaboronski, Oleg 2006 A phase transition for a stochastic PDE related to the contact process. Zbl 0809.60072 Mueller, Carl; Tribe, Roger 1994 The behavior of superprocesses near extinction. Zbl 0749.60046 Tribe, Roger 1992 Finite width for a random stationary interface. Zbl 0890.60056 Mueller, C.; Tribe, R. 1997 On the large time asymptotics of decaying Burgers turbulence. Zbl 0974.76037 Tribe, Roger; Zaboronski, Oleg 2000 On the distribution of the largest real eigenvalue for the real Ginibre ensemble. Zbl 1375.60023 Poplavskyi, Mihail; Tribe, Roger; Zaboronski, Oleg 2017 On stationary distributions for the KPP equation with branching noise. Zbl 1058.60049 Horridge, Paul; Tribe, Roger 2004 Stochastic order methods applied to stochastic travelling waves. Zbl 1225.60112 Tribe, Roger; Woodward, Nicholas 2011 What is the probability that a large random matrix has no real eigenvalues? Zbl 1375.60019 Kanzieper, Eugene; Poplavskyi, Mihail; Timm, Carsten; Tribe, Roger; Zaboronski, Oleg 2016 The Ginibre evolution in the large-$$N$$ limit. Zbl 1296.82045 Tribe, Roger; Zaboronski, Oleg 2014 Uniqueness for a class of one-dimensional stochastic PDEs using moment duality. Zbl 1044.60048 Athreya, Siva; Tribe, Roger 2000 A singular parabolic {A}nderson model. Zbl 1071.60055 Mueller, Carl; Tribe, Roger 2004 A probabilistic representation for the solutions to some non-linear PDEs using pruned branching trees. Zbl 1119.60060 Blömker, D.; Romito, M.; Tribe, R. 2007 The connected components of the closed support of super Brownian motion. Zbl 0722.60084 Tribe, Roger 1991 One dimensional annihilating and coalescing particle systems as extended Pfaffian point processes. Zbl 1252.60095 Tribe, Roger; Yip, Siu Kwan; Zaboronski, Oleg 2012 Erratum to: “The Ginibre ensemble of real random matrices and its scaling limits”. Zbl 1425.82003 Borodin, Alexei; Poplavskyi, Mihail; Sinclair, Christopher D.; Tribe, Roger; Zaboronski, Oleg 2016 A representation for super Brownian motion. Zbl 0810.60075 Tribe, Roger 1994 A phase diagram for a stochastic reaction diffusion system. Zbl 1232.60051 Mueller, Carl; Tribe, Roger 2011 Examples of interacting particle systems on $$\mathbb{Z}$$ as Pfaffian point processes: coalescing-branching random walks and annihilating random walks with immigration. Zbl 1451.82047 Garrod, Barnaby; Tribe, Roger; Zaboronski, Oleg 2020 Examples of interacting particle systems on $$\mathbb {Z}$$ as Pfaffian point processes: annihilating and coalescing random walks. Zbl 1410.82015 Garrod, Barnaby; Poplavskyi, Mihail; Tribe, Roger P.; Zaboronski, Oleg V. 2018 A measure-valued process related to the parabolic Anderson model. Zbl 1030.60051 Mueller, C.; Tribe, R. 2002 Comparison for measure valued processes with interactions. Zbl 1045.60046 Jacka, Saul; Tribe, Roger 2003 A stochastic PDE arising as the limit of a long-range contact process, and its phase transition. Zbl 0812.60047 Mueller, Carl; Tribe, Roger 1994 Sharp asymptotics for Fredholm pfaffians related to interacting particle systems and random matrices. Zbl 1462.60007 Fitzgerald, Will; Tribe, Roger; Zaboronski, Oleg 2020 Uniqueness for a historical SDE with a singular interaction. Zbl 0912.60071 1998 Non-equilibrium phase diagram for a model with coalescence, evaporation and deposition. Zbl 1274.82036 Connaughton, Colm; Rajesh, R.; Tribe, Roger; Zaboronski, Oleg 2013 On the duality between coalescing Brownian particles and the heat equation driven by Fisher-Wright noise. Zbl 1111.60073 Hobson, Tim; Tribe, Roger 2005 Erratum to: “One dimensional annihilating and coalescing particle systems as extended Pfaffian point processes”. (Erratum to: “One dimensional annihilating particle systems as extended Pfaffian point processes”.) Zbl 1332.60136 Tribe, Roger; Yip, Siu Kwan; Zaboronski, Oleg 2015 Examples of interacting particle systems on $$\mathbb{Z}$$ as Pfaffian point processes: coalescing-branching random walks and annihilating random walks with immigration. Zbl 1451.82047 Garrod, Barnaby; Tribe, Roger; Zaboronski, Oleg 2020 Sharp asymptotics for Fredholm pfaffians related to interacting particle systems and random matrices. Zbl 1462.60007 Fitzgerald, Will; Tribe, Roger; Zaboronski, Oleg 2020 Examples of interacting particle systems on $$\mathbb {Z}$$ as Pfaffian point processes: annihilating and coalescing random walks. Zbl 1410.82015 Garrod, Barnaby; Poplavskyi, Mihail; Tribe, Roger P.; Zaboronski, Oleg V. 2018 On the distribution of the largest real eigenvalue for the real Ginibre ensemble. Zbl 1375.60023 Poplavskyi, Mihail; Tribe, Roger; Zaboronski, Oleg 2017 What is the probability that a large random matrix has no real eigenvalues? Zbl 1375.60019 Kanzieper, Eugene; Poplavskyi, Mihail; Timm, Carsten; Tribe, Roger; Zaboronski, Oleg 2016 Erratum to: “The Ginibre ensemble of real random matrices and its scaling limits”. Zbl 1425.82003 Borodin, Alexei; Poplavskyi, Mihail; Sinclair, Christopher D.; Tribe, Roger; Zaboronski, Oleg 2016 Erratum to: “One dimensional annihilating and coalescing particle systems as extended Pfaffian point processes”. (Erratum to: “One dimensional annihilating particle systems as extended Pfaffian point processes”.) Zbl 1332.60136 Tribe, Roger; Yip, Siu Kwan; Zaboronski, Oleg 2015 The Ginibre evolution in the large-$$N$$ limit. Zbl 1296.82045 Tribe, Roger; Zaboronski, Oleg 2014 Non-equilibrium phase diagram for a model with coalescence, evaporation and deposition. Zbl 1274.82036 Connaughton, Colm; Rajesh, R.; Tribe, Roger; Zaboronski, Oleg 2013 One dimensional annihilating and coalescing particle systems as extended Pfaffian point processes. Zbl 1252.60095 Tribe, Roger; Yip, Siu Kwan; Zaboronski, Oleg 2012 Pfaffian formulae for one dimensional coalescing and annihilating systems. Zbl 1244.60097 Tribe, Roger; Zaboronski, Oleg 2011 Stochastic order methods applied to stochastic travelling waves. Zbl 1225.60112 Tribe, Roger; Woodward, Nicholas 2011 A phase diagram for a stochastic reaction diffusion system. Zbl 1232.60051 Mueller, Carl; Tribe, Roger 2011 A Feynman-Kac-type formula for the deterministic and stochastic wave equations and other p.d.e.’s. Zbl 1149.60040 Dalang, Robert C.; Mueller, Carl; Tribe, Roger 2008 Parameter estimates and exact variations for stochastic heat equations driven by space-time white noise. Zbl 1118.60030 Pospíšil, Jan; Tribe, Roger 2007 A probabilistic representation for the solutions to some non-linear PDEs using pruned branching trees. Zbl 1119.60060 Blömker, D.; Romito, M.; Tribe, R. 2007 Multi-scaling of the $$n$$-point density function for coalescing Brownian motions. Zbl 1122.60089 Munasinghe, Ranjiva; Rajesh, R.; Tribe, Roger; Zaboronski, Oleg 2006 On the duality between coalescing Brownian particles and the heat equation driven by Fisher-Wright noise. Zbl 1111.60073 Hobson, Tim; Tribe, Roger 2005 On stationary distributions for the KPP equation with branching noise. Zbl 1058.60049 Horridge, Paul; Tribe, Roger 2004 A singular parabolic {A}nderson model. Zbl 1071.60055 Mueller, Carl; Tribe, Roger 2004 Comparison for measure valued processes with interactions. Zbl 1045.60046 Jacka, Saul; Tribe, Roger 2003 Hitting properties of a random string. Zbl 1010.60059 Mueller, C.; Tribe, R. 2002 A measure-valued process related to the parabolic Anderson model. Zbl 1030.60051 Mueller, C.; Tribe, R. 2002 On the large time asymptotics of decaying Burgers turbulence. Zbl 0974.76037 Tribe, Roger; Zaboronski, Oleg 2000 Uniqueness for a class of one-dimensional stochastic PDEs using moment duality. Zbl 1044.60048 Athreya, Siva; Tribe, Roger 2000 Uniqueness for a historical SDE with a singular interaction. Zbl 0912.60071 1998 Finite width for a random stationary interface. Zbl 0890.60056 Mueller, C.; Tribe, R. 1997 A travelling wave solution to the Kolmogorov equation with noise. Zbl 1002.60555 Tribe, Roger 1996 Stochastic p.d.e.’s arising from the long range contact and long range voter processes. Zbl 0827.60050 Müller, Carl E.; Tribe, R. 1995 Large time behavior of interface solutions to the heat equation with Fisher-Wright white noise. Zbl 0831.60071 Tribe, Roger 1995 A phase transition for a stochastic PDE related to the contact process. Zbl 0809.60072 Mueller, Carl; Tribe, Roger 1994 A representation for super Brownian motion. Zbl 0810.60075 Tribe, Roger 1994 A stochastic PDE arising as the limit of a long-range contact process, and its phase transition. Zbl 0812.60047 Mueller, Carl; Tribe, Roger 1994 The behavior of superprocesses near extinction. Zbl 0749.60046 Tribe, Roger 1992 The connected components of the closed support of super Brownian motion. Zbl 0722.60084 Tribe, Roger 1991 all top 5 ### Cited by 260 Authors 13 Tribe, Roger P. 13 Xiao, Yimin 11 Dalang, Robert C. 11 Etheridge, Alison Mary 10 Müller, Carl E. 9 Zaboronski, Oleg V. 8 Tudor, Ciprian A. 7 Chen, Zhenlong 7 Perkins, Edwin A. 6 Fleischmann, Klaus 5 Cialenco, Igor 5 Wu, Dongsheng 4 Cox, J. Theodore 4 Huang, Zhehao 4 Khoshnevisan, Davar 4 Liu, Zhengrong 4 Mytnik, Leonid 4 Truman, Aubrey 4 Zhao, Huaizhong 3 Balan, Raluca M. 3 Blömker, Dirk 3 Chen, Le 3 Davies, Ian M. 3 Evans, Steven Neil 3 Forrester, Peter J. 3 Hammer, Matthias 3 Hong, Jialin 3 Kurtz, Thomas Gordon 3 Ortgiese, Marcel 3 Stannat, Wilhelm 3 Wang, Zhenzhen 3 Yan, Litan 3 Zili, Mounir 2 Athreya, Siva R. 2 Blath, Jochen 2 Chen, Chuchu 2 Chen, Yu-Ting 2 Connaughton, Colm P. 2 Dawson, Donald Andrew 2 Doering, Charles Rogers 2 Donnelly, Peter 2 FitzGerald, Will 2 Foondun, Mohammud 2 Garrod, Barnaby 2 Huang, Jianhua 2 Ipsen, Jesper R. 2 Khalil, Zeina Mahdi 2 Kim, Hyunjung 2 Kliem, Sandra M. 2 Kühn, Christian 2 Lalley, Steven P. 2 Leszczyński, Henryk 2 Mohammed, Salah-Eldin A. 2 Mountford, Thomas S. 2 Overbeck, Ludger 2 Pasemann, Gregor 2 Romito, Marco 2 Sanz-Solé, Marta 2 Sheng, Derui 2 Slaoui, Meryem 2 Song, Renming 2 Véber, Amandine 2 Völlering, Florian 2 Wen, Hao 2 Wrzosek, Monika 2 Xiong, Jie 2 Zambotti, Lorenzo 2 Zougar, Eya 1 Abraham, Romain 1 Acebrón, Juan A. 1 Adler, Robert Joseph 1 Albeverio, Sergio A. 1 Ali, Adnan Mikhsev 1 Allouba, Hassan 1 Alonso, Sergio 1 Amadori, Debora 1 Aqel, Fatima Al-Zahra’ 1 Araya, Héctor 1 Assaad, Obayda 1 Avetisian, Diana 1 Baik, Jinho 1 Bakhtin, Yuri Yu. 1 Barraquand, Guillaume 1 Barton, Nick H. 1 Bass, Richard F. 1 Belinschi, Serban Teodor 1 Bellingeri, Carlo 1 Beta, Carsten 1 Bezdek, Pavel 1 Biswas, Niloy 1 Blount, Douglas 1 Bo, Lijun 1 Bogachev, Leonid V. 1 Bonnet, Guillaume 1 Borodin, Alexei 1 Bothner, Thomas 1 Brzeźniak, Zdzisław 1 Cangiotti, Nicolò 1 Catuogno, Pedro José 1 Chang, Xiaoying ...and 160 more Authors all top 5 ### Cited in 78 Serials 26 The Annals of Probability 21 Stochastic Processes and their Applications 15 Electronic Journal of Probability 12 Probability Theory and Related Fields 7 The Annals of Applied Probability 6 Stochastics and Dynamics 6 Stochastic and Partial Differential Equations. Analysis and Computations 4 Journal of Statistical Physics 4 Journal of Differential Equations 4 Stochastic Analysis and Applications 4 Journal of Theoretical Probability 4 Acta Mathematica Sinica. English Series 3 Journal of Mathematical Analysis and Applications 3 Journal of Mathematical Physics 3 Ukrainian Mathematical Journal 3 Transactions of the American Mathematical Society 3 Statistics & Probability Letters 3 Annales de l’Institut Henri Poincaré. Probabilités et Statistiques 3 Bernoulli 3 Discrete and Continuous Dynamical Systems. Series B 3 Journal of Physics A: Mathematical and Theoretical 3 Modern Stochastics. Theory and Applications 2 Communications in Mathematical Physics 2 Jahresbericht der Deutschen Mathematiker-Vereinigung (DMV) 2 Proceedings of the American Mathematical Society 2 Theoretical Population Biology 2 Chinese Annals of Mathematics. Series B 2 Expositiones Mathematicae 2 Potential Analysis 2 Electronic Communications in Probability 2 Infinite Dimensional Analysis, Quantum Probability and Related Topics 2 Annales Henri Poincaré 2 Advances in Difference Equations 1 Communications on Pure and Applied Mathematics 1 Journal of Computational Physics 1 Nonlinearity 1 Physica A 1 Chaos, Solitons and Fractals 1 The Annals of Statistics 1 Inventiones Mathematicae 1 Journal of Applied Probability 1 Journal of Functional Analysis 1 Journal of Statistical Planning and Inference 1 Memoirs of the American Mathematical Society 1 Osaka Journal of Mathematics 1 Probability and Mathematical Statistics 1 Physica D 1 Journal of Complexity 1 Forum Mathematicum 1 Science in China. Series A 1 Journal de Mathématiques Pures et Appliquées. Neuvième Série 1 Linear Algebra and its Applications 1 Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 1 Acta Mathematica Sinica. New Series 1 Journal of Dynamics and Differential Equations 1 Journal of Nonlinear Science 1 Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI 1 Experimental Mathematics 1 Theory of Probability and Mathematical Statistics 1 NoDEA. Nonlinear Differential Equations and Applications 1 Bulletin des Sciences Mathématiques 1 Abstract and Applied Analysis 1 Chaos 1 Discrete Dynamics in Nature and Society 1 Statistical Inference for Stochastic Processes 1 Acta Mathematica Scientia. Series B. (English Edition) 1 Advances in Complex Systems 1 Mathematical Biosciences and Engineering 1 Boundary Value Problems 1 Stochastics 1 Frontiers of Mathematics in China 1 Journal of Statistical Theory and Practice 1 Electronic Journal of Statistics 1 Probability Surveys 1 Science China. Mathematics 1 Random Matrices: Theory and Applications 1 Communications in Mathematics and Statistics 1 SIAM/ASA Journal on Uncertainty Quantification all top 5 ### Cited in 26 Fields 194 Probability theory and stochastic processes (60-XX) 73 Partial differential equations (35-XX) 23 Statistical mechanics, structure of matter (82-XX) 19 Biology and other natural sciences (92-XX) 15 Numerical analysis (65-XX) 14 Statistics (62-XX) 12 Linear and multilinear algebra; matrix theory (15-XX) 9 Dynamical systems and ergodic theory (37-XX) 7 Measure and integration (28-XX) 6 Fluid mechanics (76-XX) 5 Special functions (33-XX) 5 Operator theory (47-XX) 4 Integral equations (45-XX) 3 Ordinary differential equations (34-XX) 3 Harmonic analysis on Euclidean spaces (42-XX) 3 Quantum theory (81-XX) 2 Functional analysis (46-XX) 2 Calculus of variations and optimal control; optimization (49-XX) 2 Mechanics of particles and systems (70-XX) 2 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 1 Nonassociative rings and algebras (17-XX) 1 Real functions (26-XX) 1 Difference and functional equations (39-XX) 1 Abstract harmonic analysis (43-XX) 1 Integral transforms, operational calculus (44-XX) 1 Global analysis, analysis on manifolds (58-XX)	2022-12-09 05:41:32	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5547298789024353, "perplexity": 8683.249161916998}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711390.55/warc/CC-MAIN-20221209043931-20221209073931-00042.warc.gz"}
https://www.eduzip.com/ask/question/beginarray-l-text-in-fig-find-the-values-of-x-text-and-y-text-and-520843	Mathematics # $\begin{array} { l } { \text { In fig. find the values of } x \text { and } y \text { and then } } \\ { \text { Show that } A B \\| C D } \end{array}$ $x=130$ and $y=130$ ##### SOLUTION From given figure $50^o+x=180^o$.......(Linear pair) $x=180^o-50^o$ $x=130^o$ Also, $y=130^o$ .........(vertically opposite angles) $\therefore x=y$ So, alternate angles are equal If a transversal intersects two lines such that pair of alternate interior angles are equal, then lines are parallel. $AB\|\|CD$ You're just one step away Single Correct Medium Published on 09th 09, 2020 Questions 120418 Subjects 10 Chapters 88 Enrolled Students 86 #### Realted Questions Q1 Single Correct Medium $P$ is any point inside the triangle $ABC$. Hence $\angle BPC> \angle BAC$ • A. False • B. Ambiguous • C. Data insufficient • D. True Asked in: Mathematics - Lines and Angles 1 Verified Answer \| Published on 09th 09, 2020 Q2 Subjective Medium In questions $1$ and $2,$ given below, identify the given pairs of angles as corresponding angles, interior alternate angles, exterior alternate angles, adjacent angles, vertically opposite angles or allied angles: $\angle 7 \quad and \quad \angle 9$ Asked in: Mathematics - Lines and Angles 1 Verified Answer \| Published on 09th 09, 2020 Q3 Subjective Medium In the given figure, $\angle ABC$ and $\angle PQR$ are given such that $AB\parallel PQ$ and $BC\parallel QR$. Find the value of $\angle x$ and $\angle y$ Asked in: Mathematics - Lines and Angles 1 Verified Answer \| Published on 09th 09, 2020 Q4 One Word Medium Measure of one angle of linear pair is $108^{\circ}$, then find the measure of another angle. Asked in: Mathematics - Lines and Angles 1 Verified Answer \| Published on 09th 09, 2020 Q5 Subjective Medium Iron roads $a,\ b,\ c,\ d,\ e$ and $f$ are making a design in a bridge as shown in Fig., in which a $\parallel b,\ c \parallel d,\ e \parallel \ f$. Find the marked angles between $b$ and $c$ Asked in: Mathematics - Lines and Angles 1 Verified Answer \| Published on 09th 09, 2020	2022-01-24 04:35:40	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6140656471252441, "perplexity": 7469.677654202102}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320304471.99/warc/CC-MAIN-20220124023407-20220124053407-00538.warc.gz"}
http://www.gradesaver.com/textbooks/math/other-math/basic-college-mathematics-9th-edition/chapter-3-adding-and-subtracting-fractions-review-exercises-page-255/32	## Basic College Mathematics (9th Edition) $Estimate: 38$ $Exact: \displaystyle 38\frac{1}{9}$ We need to add the following fractions: $\displaystyle 22\frac{2}{3}+15\frac{4}{9}$ First we estimate the answer by rounding: $\displaystyle 22\frac{2}{3}+15\frac{4}{9}$ $\approx 23+15$ $\approx 38$ Next, we obtain the exact answer: $\displaystyle 22\frac{2}{3}+15\frac{4}{9}$ $=\displaystyle 22\frac{6}{9}+15\frac{4}{9}$ $=\displaystyle 37\frac{10}{9}$ $=\displaystyle 38\frac{1}{9}$ The exact result is fairly close to the estimate.	2018-04-20 01:33:16	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8547852635383606, "perplexity": 413.7788022673123}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125937090.0/warc/CC-MAIN-20180420003432-20180420023432-00494.warc.gz"}
https://web2.0calc.com/questions/math_51512	+0 # math 0 255 1 14/1 times 7/12 Guest Jun 7, 2017 #1 +2294 0 Here is the original expression: $$\frac{14}{1}\frac{7}{12}$$ To multiply this, just multiply the numerator and the denominator. Calculate the numerator by doing $$147=98$$ and calculate the denominator by doing $$1*12=12$$. $$\frac{98}{12}$$ But wait! You are not done yet! You must reduce the fraction to its simplest terms. To do this, calculate the GCF of the numerator and denominator. The GCF happens to be 2. Take that factor out of both the numerator and denominator. $$\frac{98}{12}\div\frac{2}{2}=\frac{49}{6}=8.1\overline{66666}$$ TheXSquaredFactor Jun 7, 2017	2018-12-15 03:35:58	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9988261461257935, "perplexity": 919.4472096344111}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376826686.8/warc/CC-MAIN-20181215014028-20181215040028-00476.warc.gz"}
http://www.livmathssoc.org.uk/cgi-bin/sews_diff.py?IrrationalNumber	## Most recent change of IrrationalNumber Edit made on December 04, 2008 by GuestEditor at 16:41:13 Deleted text in red / Inserted text in green WW HEADERS_END An Irrational Number is one that cannot be expressed as a ratio. For example, 2/3 3/5 is clearly a rational number, because it's a ratio. However, root two is irrational. Other examples of irrational numbers are EQN:\pi,{\quad}e, (which are transcendental numbers) and the Golden Ratio EQN:\phi (which is an algebraic number).	2020-04-01 07:54:25	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9781576991081238, "perplexity": 2046.0924306607735}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370505550.17/warc/CC-MAIN-20200401065031-20200401095031-00128.warc.gz"}
https://www.numerade.com/questions/show-that-the-inflection-points-of-the-curve-y-x-sin-x-lie-on-the-curve-y2-x2-4-4x2/	💬 👋 We’re always here. Join our Discord to connect with other students 24/7, any time, night or day.Join Here! WZ # Show that the inflection points of the curve $y = x \sin x$ lie on the curve $y^2 (x^2 + 4) = 4x^2$. ## $$\begin{array}{l}y=x \sin x \Rightarrow y^{\prime}=x \cos x+\sin x \Rightarrow y^{\prime \prime}=-x \sin x+2 \cos x, \quad y^{\prime \prime}=0 \Rightarrow 2 \cos x=x \sin x[\text { which is } y] \Rightarrow \\(2 \cos x)^{2}=(x \sin x)^{2} \Rightarrow 4 \cos ^{2} x=x^{2} \sin ^{2} x \Rightarrow 4 \cos ^{2} x=x^{2}\left(1-\cos ^{2} x\right) \Rightarrow 4 \cos ^{2} x+x^{2} \cos ^{2} x=x^{2} \Rightarrow \\\cos ^{2} x\left(4+x^{2}\right)=x^{2} \Rightarrow 4 \cos ^{2} x\left(x^{2}+4\right)=4 x^{2} \Rightarrow y^{2}\left(x^{2}+4\right)=4 x^{2} \text { since } y=2 \cos x \text { when } y^{\prime \prime}=0\end{array}$$ Derivatives Differentiation Volume ### Discussion You must be signed in to discuss. ##### Catherine R. Missouri State University ##### Kristen K. University of Michigan - Ann Arbor Lectures Join Bootcamp ### Video Transcript The problem is showed that the inflection points off the crew. If why is able to access him? Sign ax Lye on the curve, My squire. Times Square plus Or, as they call Chu for acts coy. The first two computers degenerative off this function when she is equal to signed. X US Axe Ham School Science. Second derivative is equal to cause i X Our school's lying max US Axe Times Negative Jax, which is equal to two ham school sign X minus X Times Science. If the point back Cyril Y. Zero is an inflection point on behalf, why zero is equal to act? Cyril Hams Signac Ciro on DH to co sign AC Cyril minus zero. Scientific hero. Is he called to zero from you? Christian one Behalf. Why zero overact. Cyril is equal to sign. I feel from the equation to half two hands cause I Act zero. It's going to AC Cyril Sign zero, Which is he goto y zero. So you have Y zero or AC Cyril Squire us. Why zero over to squire? This is equal to sign AC Cyril Square plus school sign AC zero square which is equal to one. Then we must apply Full X squire for inside behalf y zero squared hams for us, Axiron Square is equal to flunk like serious coin. So we have axial y zero satisfies this equation, so slice this group. WZ #### Topics Derivatives Differentiation Volume ##### Catherine R. Missouri State University ##### Kristen K. University of Michigan - Ann Arbor Lectures Join Bootcamp	2021-09-21 12:06:36	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5137304067611694, "perplexity": 9406.357720219545}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057202.68/warc/CC-MAIN-20210921101319-20210921131319-00253.warc.gz"}
https://www.gamedev.net/forums/topic/590236-smooth-cubic-pathing/	# Smooth Cubic Pathing? This topic is 2773 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic. ## Recommended Posts I have this waypoint based camera system where users can set waypoints and hit play, and then the camera will move in a spline between each of these waypoints. the code goes like this: ==================================================== Clarification: p0x = Start Position p1x = Tangent p2x = Tangent p3x = End Position $i = the step, when$i is 1.0 the point will be p3x ==================================================== "position X" =(1-$i)^3 $p0x + 3(1-$i)^2 $i $p1x + 3(1-$i) $i^2 $p2x + $i^3 $p3x) "position Y" =(1-$i)^3 $p0y + 3(1-$i)^2 $i $p1y + 3(1-$i) $i^2 $p2y + $i^3 $p3y) "position Z" = (1-$i)^3 $p0z + 3(1-$i)^2 $i $p1z + 3(1-$i) $i^2 $p2z + $i^3 $p3z) and same goes for the pitch, roll and yaw... now the problem here is that this math does not take into account that points have variable distance between eachother. So if 2 points are close to eachother the camera will move slow, and if they are far from eachother it will move fast. Question: How can i make this equation so that the camera will always maintain a constant speed no matter what the distance is between the 2 points is? ##### Share on other sites Use timesteps for your movement and/or use a set speed. ##### Share on other sites The easiest way to do that is to sample your spline with really small "t" intervals and calculate if the distance is above your needed speed. If not go on else store the point in an array. Do this to all your points and you'll soon get a uniformly distanced spline. http://www.shmup-dev.com/forum/index.php/topic,1638.0.html A different way to do this is here (I made it in Freebasic but it would still be readable) http://www.freebasic.net/forum/viewtopic.php?t=16874 ##### Share on other sites Quote: Original post by relsoft"Reparametize" your spline. ie. base it on distance instead of time.The easiest way to do that is to sample your spline with really small "t" intervals and calculate if the distance is above your needed speed. If not go on else store the point in an array. Do this to all your points and you'll soon get a uniformly distanced spline.http://www.shmup-dev.com/forum/index.php/topic,1638.0.htmlA different way to do this is here (I made it in Freebasic but it would still be readable)http://www.freebasic.net/forum/viewtopic.php?t=16874 My math is pretty horrible, i don't quite see what he ment with the: t = t0 + ((a - a0) / (a1 - a0)) * (t1 - t0) ) He say (a) is the actual arch length but then i don't get what (a0) and (a1) is suppose to be, let alone (t0) and (t1) ##### Share on other sites It's standard linear interpolation equation. It "lerps" between 2 values. Try the one I made using freebasic. It does not use lerping since it's just a straight array look-up. ##### Share on other sites Sorry, it was kind of late last night s I wasn't able to give you a good description. t = t0 + ((a - a0) / (a1 - a0)) * (t1 - t0) ) a = supposed linear distance of your camera t = time value in a spline a0 = actual distance array value less than "a" in your table a1 = actual distance array value greater than "a" in your table t0 = actual time array value less than "t" in your table t1 = actual time array value greater than "t" in your table So assuming you have a spline() function... t = t0 + ((a - a0) / (a1 - a0)) * (t1 - t0) )vector2d p= spline(t); Should do the trick. ##### Share on other sites You might find this to be relevant (PDF). 1. 1 2. 2 Rutin 20 3. 3 JoeJ 17 4. 4 5. 5 • 37 • 23 • 13 • 13 • 17 • ### Forum Statistics • Total Topics 631704 • Total Posts 3001822 ×	2018-07-16 15:00:07	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.20934341847896576, "perplexity": 2599.9666971130605}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676589350.19/warc/CC-MAIN-20180716135037-20180716155037-00009.warc.gz"}
http://mathhelpforum.com/advanced-applied-math/277746-independent-v-s-dependent-columns.html	# Thread: Independent v.s. dependent columns 1. ## Independent v.s. dependent columns Hi, I'm lost with this image: Can someone please explain the description of Figure 4.7? How is a1, a2, and a3 not in a plane while c1, c2, and c3 in the same plane? I struggling to find the reasoning behind this. (Sorry I forgot to convert the "1" in "a1" and the "2" and "a2" etc as subscripts) Appreciate help! - Olivia 2. ## Re: Independent v.s. dependent columns what exactly don't you understand? The $a's$ are stated to be linearly independent and thus they span the entire 3D space The $c's$ are stated to be linearly dependent and thus they span a 2D space I can see that the drawing of the $c's$ looks a bit odd but just accept that they all lie in the same plane.	2017-10-19 21:25:48	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8243684768676758, "perplexity": 876.2454570876415}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187823462.26/warc/CC-MAIN-20171019194011-20171019214011-00163.warc.gz"}
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/643	## Applications of a New Model for the Differential Cross-Section of a Classical Polyatomic Gas • We give a comparison of various differential cross-section models for a classical polyatomic gas for a homogeneous relaxation problem and the shock wave profiles at Mach numbers 1.71 and 12.9. Besides the standard Borgnakke-Larsen model and its generalizations to an energy dependent coefficient to control the amnount of rotationally elastic and completely inelastic collisions, we discuss some new models recently proposed by the same authors. Moreover, we present numerical algorithms to implement the models in a particle or Monte-Carlo code and compare the numerical shock wave profiles with existing experimental data. $Rev: 13581$	2015-11-29 23:19:13	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3746547996997833, "perplexity": 602.4656273316633}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398460263.61/warc/CC-MAIN-20151124205420-00064-ip-10-71-132-137.ec2.internal.warc.gz"}
https://itprospt.com/num/8443343/proof-that-the-two-electron-ground-state-of-a-spin-independent	5 # Proof That the Two-Electron Ground State of a Spin-Independent Hamiltonian Is a Singlet(a) The mean energy of a two-electron system with Hamiltonian (32.3) in the s... ## Question ###### Proof That the Two-Electron Ground State of a Spin-Independent Hamiltonian Is a Singlet(a) The mean energy of a two-electron system with Hamiltonian (32.3) in the state $psi$ can be written (after an integration by parts in the kinetic energy term) in the form:$$E=int d mathbf{r}_{1} d mathbf{r}_{2}left[frac{h^{2}}{2 m}left{left\|mathbf{abla}_{1} psiight\|^{2}+left\|abla_{2} psiight\|^{2}ight}+Vleft(mathbf{r}_{1}, mathbf{r}_{2}ight)\|psi\|^{2}ight]$$Show that the lowest value ( $32.28$ ) assumes over Proof That the Two-Electron Ground State of a Spin-Independent Hamiltonian Is a Singlet (a) The mean energy of a two-electron system with Hamiltonian (32.3) in the state $psi$ can be written (after an integration by parts in the kinetic energy term) in the form: $$E=int d mathbf{r}_{1} d mathbf{r}_{2}left[frac{h^{2}}{2 m}left{left\|mathbf{ abla}_{1} psi ight\|^{2}+left\| abla_{2} psi ight\|^{2} ight}+Vleft(mathbf{r}_{1}, mathbf{r}_{2} ight)\|psi\|^{2} ight]$$ Show that the lowest value ( $32.28$ ) assumes over all normalized antisymmetric differentiable wave functions $psi$ that vanish at infinity is the triplet ground-state energy $E_{t}$, and that when symmetric functions are used the lowest value is the singlet ground-state energy $E_{mathrm{s}}$ (b) Using (i) the result of (a), (ii) the fact that the triplet ground state $psi_{t}$, can be taken to be real when $V$ is real, and (iii) the fact that $left\|psi_{2} ight\|$ is symmetric, deduce that $E_{s} leqslant E_{T}$. #### Similar Solved Questions ##### 31. Function of capsule, slime Jayer 32. Differentiate between slime layer and capsule 33. Define attenuation 34. Structures involved in adherence of Bacteria 35. Explain how dental caries develop 36. Environmental factors that influence skin microflora 37. LDso 38. Phagocytosis 39. Septicemia 40 . Bacteremia 41. Endotoxins VS exotoxins 42. Categories of exotoxins 43 . Virulence factors of Salmonella 44. Cytotoxins? Examples 45. Enterotoxins? 46 . Invasion? 31. Function of capsule, slime Jayer 32. Differentiate between slime layer and capsule 33. Define attenuation 34. Structures involved in adherence of Bacteria 35. Explain how dental caries develop 36. Environmental factors that influence skin microflora 37. LDso 38. Phagocytosis 39. Septicemia 40 . ... ##### 1 range Appoodimatoty 333 the Io current 3 01 What equivalent M 341 Hi hesiseuce that the between E 1 points LL ray incident. and ~fall makes for the case 1 the the range with figure when a [ below (R with the normal if the second medium given count is repeated 0f 25,0 N degrees What should fall in 42 in a medium the total times? the 1 range Appoodimatoty 333 the Io current 3 01 What equivalent M 341 Hi hesiseuce that the between E 1 points LL ray incident. and ~fall makes for the case 1 the the range with figure when a [ below (R with the normal if the second medium given count is repeated 0f 25,0 N degrees What should fall in ... ##### The behavior wc observed in the previous question is perhaps best understood by looking at different-sized squares and comparing the added area WC get as we move from one size tO another:The added area is 3 , then 5, then square inches as we add inch to the side length1x 12 x 23 x 34x 4The blue corners represent the additional 2 square inches added compared to the previous increase2 x 23 x 34x4 The behavior wc observed in the previous question is perhaps best understood by looking at different-sized squares and comparing the added area WC get as we move from one size tO another: The added area is 3 , then 5, then square inches as we add inch to the side length 1x 1 2 x 2 3 x 3 4x 4 The blu... ##### 5) John Conner is rewiring the time circuits in order to send the Terminator for his trip through time: The circuit diagram is shown below: What is the initial charge on the 20 uF capacitor? How much energy is stored in the capacitor set? How long would it take to recharge the capacitors to 99%? How much charge remains on the 20 pF capacitor after 0.2 sec?;03 '` '.60 pF20 pF 30 Q(AVco 10 v20 @60 pF 5) John Conner is rewiring the time circuits in order to send the Terminator for his trip through time: The circuit diagram is shown below: What is the initial charge on the 20 uF capacitor? How much energy is stored in the capacitor set? How long would it take to recharge the capacitors to 99%? How... ##### Com/forms/d/e/1FAIpQLSfxZAyRrAZaBUJMnjkAQHAjJwOhY_Ce9FVYWYMnLH-foTi_ZAviewform10 points10) V54x A) 2V3x C) 8V2xB) 3V6x D) 8V3xSubmitNever {ujtt pass#0Ghtouan Gooale FolmToieonn Vr Ciebi com/forms/d/e/1FAIpQLSfxZAyRrAZaBUJMnjkAQHAjJwOhY_Ce9FVYWYMnLH-foTi_ZAviewform 10 points 10) V54x A) 2V3x C) 8V2x B) 3V6x D) 8V3x Submit Never {ujtt pass#0 Ghtouan Gooale Folm Toieonn Vr Ciebi... ##### When heated, KCIO3 decomposes into KCI and 02 2KCIO, 2KCI + 301 If this reaction produced 83.9 g of KCI; how much Oz was produced (in grams)? Number02 When heated, KCIO3 decomposes into KCI and 02 2KCIO, 2KCI + 301 If this reaction produced 83.9 g of KCI; how much Oz was produced (in grams)? Number 02... ##### MnetcltDotimisiicRobablePersimisicFill in all the blanks in the following table_ (10 points)Note:i) To reduce the computation load, please round the expected activity time (t) up [0 an integer if the computed expected activity time not an integer: For example, ifta-+.15, round it up to 5ii) For variance (0-) in activity time kecp LIVO decimal places:pracedeuce AcinitiExpected Tiue (aus)KuriuCritical Slack Patl"Activity3 JWhat is the expected time and variance complete the project? 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Find the general solution to the following differential equation:d{(+3) (2 2) 2. Find the general solution to the following differential equation: d {(+3) (2 2)... ##### "Mathematics applied t0 administration and economics; Alh Edition (Arya; Lardner; Ibarra_ Schettino_ Villalobos)Exerclse14. Given4r + 3 I +2if -2<< 0 ifo =x=2 ifx > 2g()Evaluate each of the following values:a. g(1)b: g(3)c. 8(-1)d:. g(0) e. g(-3) f. 8(2 + h)y g(2 h) si 2 > h > 0pts ) "Mathematics applied t0 administration and economics; Alh Edition (Arya; Lardner; Ibarra_ Schettino_ Villalobos) Exerclse 14. Given 4r + 3 I +2 if -2<< 0 ifo =x=2 ifx > 2 g() Evaluate each of the following values: a. g(1) b: g(3) c. 8(-1) d:. g(0) e. g(-3) f. 8(2 + h)y g(2 h) si 2 >... ##### Badwater 135 (an ultra-marathon) started in Badwater; Death Vallev Ca. which has 8.A runner in the an elevation of-279 feet below sea level and fnished at the top of Mt: Whitney The runner had a elevation gain of 14784 feet in the 135 miles. What Is the elevation of Mt. Whitney? Badwater 135 (an ultra-marathon) started in Badwater; Death Vallev Ca. which has 8.A runner in the an elevation of-279 feet below sea level and fnished at the top of Mt: Whitney The runner had a elevation gain of 14784 feet in the 135 miles. What Is the elevation of Mt. Whitney?... ##### $$\text { Solve each system using the inverse of the coefficient matrix.}$$ \begin{aligned} 3 x+4 y &=-3 \\ -5 x+8 y &=16 \end{aligned} $$\text { Solve each system using the inverse of the coefficient matrix.}$$ \begin{aligned} 3 x+4 y &=-3 \\ -5 x+8 y &=16 \end{aligned}... ##### Determine the major organic product for the reaction scheme shown.SelectDra1. 2 Li, EtzO 2. CH:COCHzCH3 3_ H3otBr Determine the major organic product for the reaction scheme shown. Select Dra 1. 2 Li, EtzO 2. CH:COCHzCH3 3_ H3ot Br... ##### 76. M-e 2chip cookies: Qunton chocolate cookies and Icmon 'cookics = containing " bok ol anothcr the branches: Miua o' _ and then probabilities She ezts one cookic and include E the situation probability _ 'di38raM E for this kind: Write the Malc (rec eats are the same that both cookies she = Find thc 'probabillty = thc blankNextmith (pIrvhereParcoansChatLeave Mecting0Shareacer 76. M-e 2 chip cookies: Qunton chocolate cookies and Icmon 'cookics = containing " bok ol anothcr the branches: Miua o' _ and then probabilities She ezts one cookic and include E the situation probability _ 'di38raM E for this kind: Write the Malc (rec eats are the same that b... ##### Consider the function v=f(x): cos( x=4x) then the second derivative, f~(2x- 492+ cos(x? _ 4x) - 2sin (x2_ -4x)(2x-4)cos(x? _ 4x) - 2sin (x2 _ 4x)cos(x? _ 4x)(2x ~ 4)~(2x-4)2sin(x? _ 4x) + 2cos(x? _ ~4x) Consider the function v=f(x): cos( x= 4x) then the second derivative, f ~(2x- 492+ cos(x? _ 4x) - 2sin (x2_ -4x) (2x-4)cos(x? _ 4x) - 2sin (x2 _ 4x) cos(x? _ 4x)(2x ~ 4) ~(2x-4)2sin(x? _ 4x) + 2cos(x? _ ~4x)... ##### How many grams of H2S are required to produce 2.00kilowatt hour in the reaction with nitrate in acid (firstbalance the equation): NO3- + H2S + H+ = NO(g) + S(s) + H2O How many grams of H2S are required to produce 2.00 kilowatt hour in the reaction with nitrate in acid (first balance the equation): NO3- + H2S + H+ = NO(g) + S(s) + H2O...	2022-09-26 13:05:37	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6995275020599365, "perplexity": 8457.639971122471}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030334871.54/warc/CC-MAIN-20220926113251-20220926143251-00216.warc.gz"}
https://api-project-1022638073839.appspot.com/questions/how-do-you-find-domain-and-range-for-f-x-abs-x-2	# How do you find domain and range for f(x) = abs(x-2)? The domain is the set of real numbers $D \left(f\right) = R$ and the range is $R \left(f\right) = \left[0 , + \infty\right)$ Oct 2, 2015 What is the graph look like?graph{\|x-2\| [-3, 7, -5, 5]} #### Explanation: 1. Domain Look at the x-axis, where can't x be. In this case x can be every real number, $\left(- \infty , \infty\right)$. 2. Range Now, look at the y-axis, where can't y be. In this class, y can't be negative, $\left(0 , \infty\right)$.	2020-05-24 23:04:51	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 4, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9024349451065063, "perplexity": 1354.8827101978222}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347385193.5/warc/CC-MAIN-20200524210325-20200525000325-00366.warc.gz"}
https://mathoverflow.net/questions/338316/turn-pi-n-move-1-n-forward	turn $\pi/n$, move $1/n$ forward start at the origin, first step number is 1. • turn $$\pi/n$$ • move $$1/n$$ units forward Angles are cumulative, so this procedure is equivalent (finitely) to $$u(k):=\sum_{n=1}^{k} \frac{\exp(\pi i H_{n})}{n}$$ • Is the limiting shape formed by a line plot of partial sums a circle or a spiral? Relying on visual intuition with the harmonic numbers seems to be perilous. • Where's its center? This is equivalent to "If $$u(k)$$ converges, what does it converge to?". Naively resumming within exp produces lots of divergent series as a result of the harmonic numbers $$H_{n}$$. At the cost of being a numerically ill-conditioned sum with lots of cancellation, I have to wonder what this says of arithmetic properties of the harmonic numbers. The large-$$n$$ asymptotics of the harmonic numbers is $$H_n\simeq \gamma_E+\log n$$, which we use for $$n\geq k_0$$, replacing the sum $$\sum_{n=k_{0}}^k$$ by an integral $$\int_{k_0}^k dn$$. We thus find \begin{align} U(k)&=u(k_0-1)+e^{i\pi\gamma_E}\int_{k_0}^k \frac{1}{n}e^{i\pi\log n}\,dn\\ &=u(k_0-1)+\frac{i}{\pi} e^{i \pi\gamma_E } \left(k_0^{i\pi}-k^{i \pi }\right). \end{align} So $$U(k)$$ traces out a circle in the complex plane, of radius $$1/\pi$$ and center at $$z_0=u(k_0-1)+(i/\pi)e^{i \pi\gamma_E } k_0^{i\pi}$$. The plot compares $$u(k)$$ (gold) and $$U(k)$$ (blue) for $$k_0=50$$ and $$k$$ up to 1000, when $$z_0=-0.66- 0.28i$$. The agreement is quite satisfactory. • While this suggests that $u(k)$ also converges to a circle of radius $1/\pi$, the centers of the two circles will be unrelated. Aug 14 '19 at 11:05 • The comment by @alpoge may be more easily read as Using $H_n = \log{n} + \gamma + O(1/n)$, $\exp(x + \epsilon) = \exp(x) (1 + O(\epsilon))$, and partial summation (in the form $\int_1^X t^{-1+\pi i} d\lfloor{t}\rfloor = \int_1^X t^{-1+\pi i} dt + O(\int_1^X \{t\} t^{-2+\pi i} dt)$ ), I believe this answer does indeed give an asymptotic (and explains the offset of the centre mentioned above). Aug 14 '19 at 13:07	2021-10-17 21:23:50	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 19, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9588258862495422, "perplexity": 502.5005754937063}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585183.47/warc/CC-MAIN-20211017210244-20211018000244-00488.warc.gz"}
https://brilliant.org/problems/a-simple-problem-5/	# A simple problem Geometry Level 3 Let $$a= \sin 10^\circ$$ ,$$b= \sin 50^\circ$$ , $$c= \sin 70^\circ$$ , then relation between a, b and c is: × Problem Loading... Note Loading... Set Loading...	2017-03-23 10:43:01	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9119245409965515, "perplexity": 4426.451822736709}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218186891.75/warc/CC-MAIN-20170322212946-00341-ip-10-233-31-227.ec2.internal.warc.gz"}
https://mathoverflow.net/questions/330785/convex-hull-of-outer-products-of-normalised-nonnegative-vectors	Convex Hull of Outer Products of (Normalised) Nonnegative Vectors If I define $$\mathcal{A} = \{ xx^T : x \in \mathbb{R}^d, \\| x \\|_2 \leqslant 1 \}$$, then (assuming I recall correctly) it is known that the convex hull of $$\mathcal{A}$$ is given by \begin{align} \text{conv} (\mathcal{A}) = \{ M \in \text{Mat}_{d \times d} (\mathbf{R}): M = M^T, \\| M \\|_* = 1\}, \end{align} where $$\\| \cdot \\|_*$$ is the nuclear norm. Suppose I now define $$\mathcal{A}_+ = \{ xx^T : x \in \mathbb{R}_+^d, \\| x \\|_2 \leqslant 1\}$$, i.e. I restrict to taking outer products of nonnegative vectors $$x$$ with themselves. My question is: Does there exist a similar characterisation of $$\text{conv} (\mathcal{A}_+)$$ ? If context is useful: I'm interested in understanding and characterising when it is possible to write a symmetric probability distributions on two variables as a mixture of i.i.d. distributions, i.e. if $$p(x,y) = p(y, x)$$, when does there exist • some parameter space $$\Theta$$, • some family of distributions $$\{ p(\cdot \| \theta) \}_{\theta \in \Theta}$$, and • some (prior) distribution $$\pi$$ such that \begin{align} p(x, y) = \int_{\Theta} p(x\|\theta) \, p(y\|\theta) \, \pi(\theta) \, d\theta? \end{align} I recognise that this has some links to de Finetti's theorem, but I'm not yet sure whether that link can be turned into an answer. Your characterization of $$\text{conv} (\mathcal{A})$$ needs one additional restriction---that $$M$$ is positive semidefinite (the equivalence of these two sets follows fairly quickly from the spectral decomposition). For $$\text{conv} (\mathcal{A}_+)$$, the convex hull is the exact same, but with the positive semidefiniteness requirement replaced by a complete positivity requirement. Since checking complete positivity is NP-hard, don't expect an easy way of determining membership in this convex hull. • To add to the completely positive part of the answer: while checking membership of the completely positive cone is NP-hard, you can use the set of doubly non-negative matrices (i.e. positive semidefinite and entry-wise non-negative matrices): this is a valid outer approximation which is semidefinitely representable, and agrees with the copositive cone for $n \leq 4$, see sciencedirect.com/science/article/pii/S002437950900281X. – ryanseadub May 7 at 3:52	2019-05-27 14:37:11	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 15, "wp-katex-eq": 0, "align": 2, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.988034188747406, "perplexity": 603.4956408754155}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232262600.90/warc/CC-MAIN-20190527125825-20190527151825-00048.warc.gz"}
http://mathhelpforum.com/statistics/118194-please-help-1-a.html	1. Assume blood pressure readings are normally distributed with a mean of 120 and standard deviation of 8. What percentage of people have a blood pressure reading greater than 145? 2. Assume that the salaries of elementary school teachers in the United States are normally distributed with a mean of $41,000 and a standard deviation of$4,000. What is the cut-off salary for teachers in the bottom 10%? I greatly appreciate any one's help. These are problems I have struggled with this semester....and now that the end is winding down I would love to figure them out for future references. Thank so much to everyone! 2. Originally Posted by loutja35 1. Assume blood pressure readings are normally distributed with a mean of 120 and standard deviation of 8. What percentage of people have a blood pressure reading greater than 145? 2. Assume that the salaries of elementary school teachers in the United States are normally distributed with a mean of $41,000 and a standard deviation of$4,000. What is the cut-off salary for teachers in the bottom 10%? I greatly appreciate any one's help. These are problems I have struggled with this semester....and now that the end is winding down I would love to figure them out for future references. Thank so much to everyone! 1. $\Pr(X > 145) = \Pr\left(Z > \frac{25}{8} \right) = 1 - \Pr\left(Z < \frac{25}{8} \right)$. The value of Z is found from $Z = \frac{X - \mu}{\sigma}$. You should have been taught how to calculate probabilities from a standard normal distribution. 2. Calculate the value of z* such that $\Pr(Z < z^) = 0.1$. Then the cut-off salary, x, is found from $z^* = \frac{x^* - 41,000}{4,000}$. Note that z* is negative. 3. Originally Posted by loutja35 1. Assume blood pressure readings are normally distributed with a mean of 120 and standard deviation of 8. What percentage of people have a blood pressure reading greater than 145? HI $X - N (120 , 8^2)$ $P(X>145)=P(Z>\frac{145-120}{8})$ calculate this using the tables or calculator to get the probability , then convert it to percentage .	2016-10-24 18:25:13	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 6, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7778880596160889, "perplexity": 377.0294942610048}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988719677.59/warc/CC-MAIN-20161020183839-00060-ip-10-171-6-4.ec2.internal.warc.gz"}
https://physics.stackexchange.com/questions/98344/blueshift-and-increase-in-energy	# Blueshift and increase in energy! Imagine that there is a car and it is not moving but its headlights are on. There is a wall in front of the car but is very far away. Right now energy is being used only in switching on the headlights. Now the car starts moving at a very high speed. As I have shown in the picture, there is a blueshift of light and so the energy of light emitted per unit time has increased. Now my question is that from where does this extra energy come from. Some arguments that prove that extra energy is generated. If the car was moving without the headlights off but at the same speed, the energy would have been used in the movement of the car. Now if the car was not moving and only the headlight was on, the energy would have been used in powering the headlight. But when we take both the cases simultaneously, then we see that there is an increase in the net energy. For further explanation I will give some equations. Case 1 when the car is moving but the headlights are off $Q_1 = \frac{dE}{dt} = \frac{d\sqrt{p^2c^2 + m^2c^4}}{dt} = 0;$ Case 2 when the car is not moving but the headlights are on $Q_2 = \frac{dE}{dt} = \frac{d(mc^2)}{dt} < 0$ since energy is being radiated by the lights on; Case 3 when the car is moving and the headlights are on $Q_3 = \frac{dE}{dt} = \frac{d\sqrt{p^2c^2 + m^2c^4}}{dt} < Q_2$, because the power of radiation is higher than in Case 2. This is because the light is blue-shifted and its quanta have higher energy. So from where does this extra energy of light come from? • The fact that the light is blue (together with the quantum idea of light, where blue light quantum has higher energy than a red light quantum) does not by itself guarrantee that the power of radiation will be higher for blue light. This is because the power of radiation depends also on the number of quanta emitted per unit time. This will be lower for the blue light, so it is not clear what the resulting power of radiation will be. – Ján Lalinský Feb 8 '14 at 16:51 • Much easier is to analyze this within classical EM theory, where it turns out that the radiation of the light in the direction of car's velocity will have greater power (the frequency does not play any role). See mpv's answer. – Ján Lalinský Feb 8 '14 at 16:52 • Aren't photons massless? So shouldn't it be $E=pc$ in all three cases? – Kyle Kanos Feb 8 '14 at 19:46 • My impression was that $E$ is energy of the car. – Ján Lalinský Feb 8 '14 at 21:38 • @KyleKanos $E = hv$. Though I am just 15 and do not know much but I think that is right. – rahulgarg12342 Feb 9 '14 at 2:28 • @JánLalinský I just did not understand how is $Q_2 < 0$ – rahulgarg12342 Feb 9 '14 at 2:30 • When car radiates, its energy decreases, $Q_2 = dE/dt$ is positive if energy of the car increases in time, negative if it decreases in time. – Ján Lalinský Feb 9 '14 at 9:23	2020-10-23 21:34:48	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6770296096801758, "perplexity": 304.6238624248883}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107865665.7/warc/CC-MAIN-20201023204939-20201023234939-00085.warc.gz"}
http://www2.macaulay2.com/Macaulay2/doc/Macaulay2-1.19/share/doc/Macaulay2/SimplicialPosets/html/_is__Simplicial.html	# isSimplicial -- Determine if a poset is simplicial. ## Synopsis • Usage: r = isSimplicial(P) • Inputs: • P, an instance of the type Poset, The poset to be tested • Outputs: • r, , Whether the given poset is simplicial or not ## Description This method uses the isBoolean method to check that every closed interval of P is a boolean algebra. i1 : P = booleanLattice(3); i2 : isSimplicial(P) o2 = true ## Ways to use isSimplicial : • "isSimplicial(Poset)" ## For the programmer The object isSimplicial is .	2023-02-03 13:22:50	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4617554843425751, "perplexity": 7808.730650272164}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500056.55/warc/CC-MAIN-20230203122526-20230203152526-00118.warc.gz"}
http://www.ask.com/question/what-is-the-definition-of-coefficiently	# What Is the Definition of Coefficiently? Coefficiently is used to refer to something done in a coefficient manner. Coefficient refers to a mathematical measure of a physical or chemical property that is stable for a system under specified circumstances. co·ef·fi·cient [koh-uh-fish-uhnt] NOUN 1. Mathematics a number or quantity placed (generally) before and multiplying another quantity, as 3 in the expression 3x. 2. Physics. a number that is constant for a given substance, body, or process under certain specified conditions, serving as a measure of one of its properties: coefficient of friction. 3. acting in consort; cooperating. Source: Dictionary.com Q&A Related to "What Is the Definition of Coefficiently" Mathematics: a numerical or constant quantity placed before and multiplying the variable in an algebraic expression (e.g. 4 in 4x y) Physics: a multiplier or factor that measures http://wiki.answers.com/Q/What_is_the_definition_o... A measure of the interdependence of two random variables that http://www.chacha.com/question/what-is-the-definit... The Viral Coefficient is invitation rate * acceptance rate The invitation rate is, as you touch upon, #invites / #installs (users). It denotes the average number of invites sent per http://www.quora.com/Viral-Growth-and-Analytics/Wh... correlation coefficient: a statistic representing how closely two variables co-vary http://www.kgbanswers.com/whats-the-definition-of-... Top Related Searches Explore this Topic In mathematics, coefficient refers to a number that multiplies a function. Example: 3x + 4y = 14, 3 is the coefficient of x and 4 is the coefficient of y. The ... Literal coefficient refers to those coefficients that are composed of literal constants such as a , b , c, among others. In math, coefficient refers to a number ... Numerical coefficient refers to constant mutative factors affixed to the variables in an expression. It is invariably written in front of a variable. Numerical ...	2014-03-09 00:43:12	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8350718021392822, "perplexity": 1467.0773266873498}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1393999668865/warc/CC-MAIN-20140305060748-00066-ip-10-183-142-35.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/simple-harmonic-motion-guitar-string-question.773812/	# Simple Harmonic Motion Guitar String Question 1. Oct 1, 2014 ### Apothem 1. The problem statement, all variables and given/known data Question: "The midpoint of a guitar string oscillates with an amplitude of 2.24mm with a frequency of 400Hz. Calculate: i) The maximum speed at this point ii) The maximum acceleration of the string at this point" 2. Relevant equations Suitable formulas: x=Asin(2pift) ; a=-(2pif)2(x) ; v=(2pif)(A) where A= amplitude/m ; f=frequency/Hz ; t=time/s 3. The attempt at a solution Attempted Solution: I think I am alright with part i) I did: v=(2pi400)(2.24x10-3)=5.6ms-1 However for part ii) I am a bit unsure, do I take x to be the amplitude, but at the midpoint would x not be 0, so the acceleration is 0 ms-2? (I originally posted this in the wrong section, and reposted it in the correct section, sorry if I should not have reposted it in a new section) 2. Oct 1, 2014 ### olivermsun Your formulas under 2. are just for the displacement $x$ of a single point on the strong. There's no notion of the "midpoint" or any other point (think of the rest position of the point on the string as $y$). 3. Oct 1, 2014 ### Apothem Sorry, I am unsure on what you mean. I understand your first point, just not about the midpoint. 4. Oct 1, 2014 ### olivermsun Maybe I misunderstood what you posted earlier, but I think there is some confusion between the "midpoint" along the string (where x(t), v(t), and a(t) are given) and the "midpoint" of the oscillation (the rest position of the point, or where x=0). 5. Oct 2, 2014 ### haruspex Yes, clearly the question means midpoint along the length of the string. It does look like Apothem has misinterpreted it as midpoint of oscillation. Apothem, x is not the amplitude, it is the displacement at time t. The amplitude is the maximum displacement.	2017-08-18 23:09:03	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6772131323814392, "perplexity": 1157.7346948468853}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886105187.53/warc/CC-MAIN-20170818213959-20170818233959-00636.warc.gz"}
https://xianblog.wordpress.com/tag/bayesian-statistics/	## a week in Oxford Posted in Books, Kids, pictures, Statistics, Travel, University life with tags , , , , , , , , , , , on January 26, 2015 by xi'an I spent [most of] the past week in Oxford in connection with our joint OxWaSP PhD program, which is supported by the EPSRC, and constitutes a joint Centre of Doctoral Training in statistical science focussing on data-intensive environments and large-scale models. The first cohort of a dozen PhD students had started their training last Fall with the first year spent in Oxford, before splitting between Oxford and Warwick to write their thesis. Courses are taught over a two week block, with a two day introduction to the theme (Bayesian Statistics in my case), followed by reading, meetings, daily research talks, mini-projects, and a final day in Warwick including presentations of the mini-projects and a concluding seminar. (involving Jonty Rougier and Robin Ryder, next Friday). This approach by bursts of training periods is quite ambitious in that it requires a lot from the students, both through the lectures and in personal investment, and reminds me somewhat of a similar approach at École Polytechnique where courses are given over fairly short periods. But it is also profitable for highly motivated and selected students in that total immersion into one topic and a large amount of collective work bring them up to speed with a reasonable basis and the option to write their thesis on that topic. Hopefully, I will see some of those students next year in Warwick working on some Bayesian analysis problem! On a personal basis, I also enjoyed very much my time in Oxford, first for meeting with old friends, albeit too briefly, and second for cycling, as the owner of the great Airbnb place I rented kindly let me use her bike to go around, which allowed me to go around quite freely! Even on a train trip to Reading. As it was a road racing bike, it took me a trip or two to get used to it, especially on the first day when the roads were somewhat icy, but I enjoyed the lightness of it, relative to my lost mountain bike, to the point of considering switching to a road bike for my next bike… I had also some apprehensions with driving at night, which I avoid while in Paris, but got over them until the very last night when I had a very close brush with a car entering from a side road, which either had not seen me or thought I would let it pass. Gave me the opportunity of shouting Oï! Posted in Kids, pictures, Statistics, University life with tags , , , , , , on January 19, 2015 by xi'an Now my grading is over, I can reflect on the unexpected difficulties in the mathematical statistics exam. I knew that the first question in the multiple choice exercise, borrowed from Cross Validation, was going to be quasi-impossible and indeed only one student out of 118 managed to find the right solution. More surprisingly, most students did not manage to solve the (absence of) MLE when observing that n unobserved exponential Exp(λ) were larger than a fixed bound δ. I was also amazed that they did poorly on a N(0,σ²) setup, failing to see that $\mathbb{E}[\mathbb{I}(X_1\le -1)] = \Phi(-1/\sigma)$ and determine an unbiased estimator that can be improved by Rao-Blackwellisation. No student reached the conditioning part. And a rather frequent mistake more understandable due to the limited exposure they had to Bayesian statistics: many confused parameter λ with observation x in the prior, writing $\pi(\lambda\|x) \propto \lambda \exp\{-\lambda x\} \times x^{a-1} \exp\{-bx\}$ $\pi(\lambda\|x) \propto \lambda \exp\{-\lambda x\} \times \lambda^{a-1} \exp\{-b\lambda\}$ hence could not derive a proper posterior. ## Statistics slides (5) Posted in Books, Kids, Statistics, University life with tags , , , , , on December 7, 2014 by xi'an Here is the fifth and last set of slides for my third year statistics course, trying to introduce Bayesian statistics in the most natural way and hence starting with… Rasmus’ socks and ABC!!! This is an interesting experiment as I have no idea how my students will react. Either they will see the point besides the anecdotal story or they’ll miss it (being quite unhappy so far about the lack of mathematical rigour in my course and exercises…). We only have two weeks left so I am afraid the concept will not have time to seep through! ## Methodological developments in evolutionary genomic [3 years postdoc in Montpellier] Posted in pictures, Statistics, Travel, University life, Wines with tags , , , , , , , , , on November 26, 2014 by xi'an [Here is a call for a post-doctoral position in Montpellier, South of France, not Montpelier, Vermont!, in a population genetics group with whom I am working. Highly recommended if you are currently looking for a postdoc!] #### Three-year post-doctoral position at the Institute of Computational Biology (IBC), Montpellier (France) : Methodological developments in evolutionary genomics. One young investigator position opens immediately at the Institute for Computational Biology (IBC) of Montpellier (France) to work on the development of innovative inference methods and software in population genomics or phylogenetics to analyze large-scale genomic data in the fields of health, agronomy and environment (Work Package 2 « evolutionary genomics » of the IBC). The candidate will develop its own research on some of the following topics : selective processes, demographic history, spatial genetic processes, very large phylogenies reconstruction, gene/species tree reconciliation, using maximum likelihood, Bayesian and simulation-based inference. We are seeking a candidate with a strong background in mathematical and computational evolutionary biology, with interest in applications and software development. The successfull candidate will work on his own project, build in collaboration with any researcher involved in the WP2 project and working at the IBC labs (AGAP, CBGP, ISEM, I3M, LIRMM, MIVEGEC). IBC hires young investigators, typically with a PhD plus some post-doc experience, a high level of publishing, strong communication abilities, and a taste for multidisciplinary research. Working full-time at IBC, these young researchers will play a key role in Institute life. Most of their time will be devoted to scientific projects. In addition, they are expected to actively participate in the coordination of workpackages, in the hosting of foreign researchers and in the organization of seminars and events (summer schools, conferences…). In exchange, these young researchers will benefit from an exceptional environment thanks to the presence of numerous leading international researchers, not to mention significant autonomy for their work. Montpellier hosts one of the most vibrant communities of biodiversity research in Europe with several research centers of excellence in the field. This positions is open for up to 3 years with a salary well above the French post-doc standards. Starting date is open to discussion. The application deadline is January 31, 2015. Living in Montpellier: http://www.agropolis.org/english/guide/index.html #### Contacts at WP2 « Evolutionary Genetics » : Jean-Michel Marin : http://www.math.univ-montp2.fr/~marin/ Olivier Gascuel : http://www.lirmm.fr/~gascuel/ Submit my application : http://www.ibc-montpellier.fr/open-positions/young-investigators#wp2-evolution ## Statistics slides (4) Posted in Books, Kids, Statistics, University life with tags , , , , , , , , , on November 10, 2014 by xi'an Here is the fourth set of slides for my third year statistics course, trying to build intuition about the likelihood surface and why on Earth would one want to find its maximum?!, through graphs. I am yet uncertain whether or not I will reach the point where I can teach more asymptotics so maybe I will also include asymptotic normality of the MLE under regularity conditions in this chapter… ## my ISBA tee-shirt designs Posted in Books, Kids, pictures, Statistics, University life with tags , , , , , , , on October 15, 2014 by xi'an Here are my tee-shirt design proposals for the official ISBA tee-shirt competition! (I used the facilities of CustomInk.com as I could not easily find a free software around. Except for the last one where I recycled my vistaprint mug design…) While I do not have any expectation of seeing one of these the winner (!), what is your favourite one?! ## Statistics slides (3) Posted in Books, Kids, Statistics, University life with tags , , , , , , , , , , on October 9, 2014 by xi'an Here is the third set of slides for my third year statistics course. Nothing out of the ordinary, but the opportunity to link statistics and simulation for students not yet exposed to Monte Carlo methods. (No ABC yet, but who knows?, I may use ABC as an entry to Bayesian statistics, following Don Rubin’s example! Surprising typo on the Project Euclid page for this 1984 paper, by the way…) On Monday, I had the pleasant surprise to see Shravan Vasishth in the audience, as he is visiting Université Denis Diderot (Paris 7) this month.	2015-03-30 04:25:24	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 3, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4404120445251465, "perplexity": 1975.32970056813}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-14/segments/1427131299054.80/warc/CC-MAIN-20150323172139-00279-ip-10-168-14-71.ec2.internal.warc.gz"}
https://paperswithcode.com/task/image-to-image-translation/codeless	Browse > Computer Vision > Image Generation > Image-to-Image Translation Image-to-Image Translation Edit 126 papers with code · Computer Vision Image-to-image translation is the task of taking images from one domain and transforming them so they have the style (or characteristics) of images from another domain. CalliGAN: Style and Structure-aware Chinese Calligraphy Character Generator 26 May 2020 Chinese calligraphy is the writing of Chinese characters as an art form performed with brushes so Chinese characters are rich of shapes and details. CPOT: Channel Pruning via Optimal Transport 21 May 2020 Recent advances in deep neural networks (DNNs) lead to tremendously growing network parameters, making the deployments of DNNs on platforms with limited resources extremely difficult. Medical Image Generation using Generative Adversarial Networks 19 May 2020 Generative adversarial networks (GANs) are unsupervised Deep Learning approach in the computer vision community which has gained significant attention from the last few years in identifying the internal structure of multimodal medical imaging data. Synthetic Image Augmentation for Damage Region Segmentation using Conditional GAN with Structure Edge 7 May 2020 We propose a synthetic augmentation procedure to generate damaged images using the image-to-image translation mapping from the tri-categorical label that consists the both semantic label and structure edge to the real damage image. DeepHist: Differentiable Joint and Color Histogram Layers for Image-to-Image Translation 6 May 2020 Promising results are shown for the tasks of color transfer, image colorization and edges $\rightarrow$ photo, where the color distribution of the output image is controlled. StereoGAN: Bridging Synthetic-to-Real Domain Gap by Joint Optimization of Domain Translation and Stereo Matching 5 May 2020 Large-scale synthetic datasets are beneficial to stereo matching but usually introduce known domain bias. Improving Endoscopic Decision Support Systems by Translating Between Imaging Modalities 27 Apr 2020 We investigate if models can be trained on virtual (or a mixture of virtual and real) samples to improve overall accuracy in a setting with limited labeled training data. Desmoking laparoscopy surgery images using an image-to-image translation guided by an embedded dark channel 19 Apr 2020 In laparoscopic surgery, the visibility in the image can be severely degraded by the smoke caused by the $CO_2$ injection, and dissection tools, thus reducing the visibility of organs and tissues. TriGAN: Image-to-Image Translation for Multi-Source Domain Adaptation 19 Apr 2020 In this paper we propose the first approach for Multi-Source Domain Adaptation (MSDA) based on Generative Adversarial Networks. Data-driven Flood Emulation: Speeding up Urban Flood Predictions by Deep Convolutional Neural Networks 17 Apr 2020 Computational complexity has been the bottleneck of applying physically-based simulations on large urban areas with high spatial resolution for efficient and systematic flooding analyses and risk assessments.	2020-05-30 01:37:59	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.25441423058509827, "perplexity": 5291.011483374365}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347407001.36/warc/CC-MAIN-20200530005804-20200530035804-00110.warc.gz"}
https://mathematica.stackexchange.com/questions/158973/unexpected-context-change	# Unexpected context change I have a strange problem with Mathematica (10.1.0 for Microsoft Windows (64-bit) (March 24, 2015)). Namely when I evaluate expression assumptions = {0 < t1 < t2, 0 < t3} the context is automatically switched to ParallelConcurrencyPrivate or just Parallel Before: In[1] $Context$ContextPath Out[1] "Global" Out[2] {"PacletManager", "System", "Global"} After: In[1] $Context$ContextPath Out[1] "Parallel" Out[2] {"ParallelVirtualShared", "ParallelStatus", "ParallelPalette","ParallelParallel", "ParallelKernels", "ParallelProtected", "ParallelDeveloper", "Parallel", "System"} ` Does anybody has any explanation for this behavior? Any suggestion is welcome. • I don't have 10.1 installed anymore, but I would try turning off the suggestions bar (and perhaps restarting the front end). Oct 31 '17 at 18:21 • This problem is similar to 84515, I think it has been improved in later versions. Oct 31 '17 at 22:31 • @Szabolcs It's my work computer. I'll try tomorrow and update. As a matter of fact, it all started when I changed context to notebook (via menu). Though I reversed the context back to global, the problem persisted and actually it extended to any notebook persisting over sessions. Oct 31 '17 at 22:36 • @Szabolcs In light of these new findings, should this be marked as a bug? Oct 31 '17 at 22:51	2021-10-24 13:30:47	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.23191918432712555, "perplexity": 4370.35275897233}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585997.77/warc/CC-MAIN-20211024111905-20211024141905-00313.warc.gz"}
http://mathoverflow.net/questions/149643/expected-length-of-the-shortest-polygonal-chain-connecting-n-random-points-in-th	# Expected length of the shortest polygonal chain connecting N random points in the unit square N points are selected uniformly at random in the unit square. Let L(N) be the expected length of the shortest (possibly self-intersecting) polygonal chain connecting all the points. It can be proved that $L(N)\sim c\sqrt{AN}$, for some constant $c$, where A is the area of the region (in our case A=1). Heuristics suggests c around 0.7, is there known an exact value for the constant c? The question including some others was asked originally on math.SE, where some partial results where achieved. If instead of a polygonal chain, we use the minimal spanning tree, a similar argument yields the same asymptotic formula, is the constant c known for this version? - I didn't see the word "Talagrand" in the math.SE thread, so I'll mention that $L(N)$ is tightly concentrated near $c\sqrt N$. I believe the precise value of the constant remains unknown. (Edit: all of this applies to the minimal spanning tree too.) – Ben Barber Nov 22 '13 at 16:07 OP wrote: The question was originally asked on math.SE. .... Not quite. The question posed on math.SE is about $n$ Uniformly random points on a disc of unit radius (i.e. of area $\pi$). The question here is about Uniformly random points on a unit square. – wolfies Nov 22 '13 at 17:33 1) The fact that the length is asymptotic to $c\sqrt{N}$ follows from sub-additivity. 2) This is the random travelling salesman'' problem. Joe Yukich wrote several papers on it and its variants. I do not think that the value of $c$ is known explicitly.	2014-10-25 03:35:09	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9363920092582703, "perplexity": 210.6120852501965}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1414119647626.5/warc/CC-MAIN-20141024030047-00007-ip-10-16-133-185.ec2.internal.warc.gz"}
https://codegolf.codidact.com/posts/282334	Sandbox # Recreate modulo [cancelled] +0 −0 Recreate the modulo operation via function or program. Mechanics: • The modulo operator involves division and returns the remainder. • For a function, use 2 variables as foo(a,b) such that $x \mod y$. • For a program, take 2 variables as input where a b is $x \mod y$. • Use standard division and return the remainder without using a built-in modulo operator. Rules: • Standard loopholes are forbidden. • Shortest code in bytes wins. Why does this post require moderator attention?	2021-11-27 00:22:56	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.875115692615509, "perplexity": 6030.750313210918}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964358074.14/warc/CC-MAIN-20211126224056-20211127014056-00530.warc.gz"}
https://gateoverflow.in/296256/madeeasy-test-series?show=296288	59 views \| 59 views 0 i dont even get this concept in this question means how approach how they do what they do .......plss share ur view ....... 0 This question is wrong, the network cannot be configured with class $C$ hosts, maximum hosts can be $254$ which is less than the requirement in only $Y$ which needs $512$ hosts. Also DBA has all network id bits as it is and the host bits are 1. 0 see this what is wrong ? In this so 0 never seen of borrowing, can't confirm if it is valid, if we can use it then it's simple, still 255.255.255.254 can't be true. we need network bits as it is. 0 ok but check solution I commented it is correct or not. 0 yes i think is ok, you have $9$ bits for host so $510$ hosts required also $510$. Should be ok. DBA only hosts bits are $1$. Should be ok.	2020-01-24 05:49:29	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.40178731083869934, "perplexity": 1677.853429584337}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250615407.46/warc/CC-MAIN-20200124040939-20200124065939-00322.warc.gz"}
https://zbmath.org/authors/?q=au%3Aabramsky%20samson	# zbMATH — the first resource for mathematics ## Abramsky, Samson Compute Distance To: Author ID: abramsky.samson Published as: Abramsky, Samson; Abramsky, S. External Links: MGP · Wikidata · dblp · GND · IdRef · theses.fr Documents Indexed: 102 Publications since 1983 19 Contributions as Editor Biographic References: 1 Publication all top 5 #### Co-Authors 40 single-authored 10 Jagadeesan, Radha 7 Maibaum, Thomas Stephen Edward 6 Soares Barbosa, Rui 5 Gabbay, Dov M. 5 McCusker, Guy Andrew 4 Lenisa, Marina 4 Mansfield, Shane 3 Brandenburger, Adam 3 Carù, Giovanni 3 Coecke, Bob 3 Gay, Simon J. 3 Kishida, Kohei 3 Mislove, Michael W. 3 Nagarajan, Rajagopal 2 Burn, Geoffrey L. 2 Constantin, Carmen M. 2 Curien, Pierre-Louis 2 de Silva, Nadish 2 Gavoille, Cyril 2 Ghica, Dan R. 2 Hankin, Chris L. 2 Heunen, Chris 2 Kirchner, Claude 2 Lal, Raymond 2 Malacaria, Pasquale 2 Meyer auf der Heide, Friedhelm 2 Palamidessi, Catuscia 2 Pitt, David H. 2 Poigné, Axel 2 Rydeheard, David E. 2 Shah, Nihil 2 Spirakis, Paul G. 2 Väänänen, Jouko Antero 2 Vákár, Matthijs 2 Zvesper, Jonathan Alexander 1 Bechmann, Matthias 1 Blute, Richard F. 1 Cooper, Stuart Barry 1 Dawar, Anuj 1 Duncan, Ross 1 Fuller, David A. 1 Gorecki, Jerzy 1 Haghverdi, Esfandiar 1 Horsman, Dominic 1 Kendon, Viv M. 1 Kontinen, Juha 1 Mackie, Ian 1 Murawski, Andrzej S. 1 Naughton, Thomas J. 1 Panangaden, Prakash 1 Perdrix, Simon 1 Pérez-Jiménez, Mario J. 1 Pitts, Andrew M. 1 Román, Leopoldo 1 Romero-Campero, Francisco José 1 Sadrzadeh, Mehrnoosh 1 Savochkin, Andrei 1 Scott, Philip J. 1 Sebald, Angelika 1 Stepney, Susan 1 Tzevelekos, Nikos 1 Vickers, Steven 1 Vollmer, Heribert 1 Wang, Pengming 1 Winschel, Viktor 1 Yamada, Norihiro 1 Ying, Shenggang 1 Zapata, Octavio all top 5 #### Serials 10 Theoretical Computer Science 7 Lecture Notes in Computer Science 6 Information and Computation 6 MSCS. Mathematical Structures in Computer Science 4 Annals of Pure and Applied Logic 2 Synthese 2 Journal of Logic and Computation 2 Philosophical Transactions of the Royal Society of London. A. Mathematical, Physical and Engineering Sciences 1 Journal of Computer and System Sciences 1 Journal of Mathematical Psychology 1 Journal of Philosophical Logic 1 Journal of Pure and Applied Algebra 1 The Journal of Symbolic Logic 1 Studia Logica 1 Science of Computer Programming 1 New Generation Computing 1 Applied Categorical Structures 1 Theory and Applications of Categories 1 Bulletin of the European Association for Theoretical Computer Science EATCS 1 Philosophical Transactions of the Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences 1 New Journal of Physics 1 Proceedings of Symposia in Applied Mathematics 1 Electronic Notes in Theoretical Computer Science all top 5 #### Fields 83 Computer science (68-XX) 56 Mathematical logic and foundations (03-XX) 32 Quantum theory (81-XX) 26 Category theory; homological algebra (18-XX) 19 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 15 General and overarching topics; collections (00-XX) 5 Order, lattices, ordered algebraic structures (06-XX) 3 Functional analysis (46-XX) 3 Operator theory (47-XX) 2 Algebraic topology (55-XX) 1 History and biography (01-XX) 1 General algebraic systems (08-XX) 1 Linear and multilinear algebra; matrix theory (15-XX) 1 Nonassociative rings and algebras (17-XX) 1 Topological groups, Lie groups (22-XX) 1 Measure and integration (28-XX) 1 Partial differential equations (35-XX) 1 Convex and discrete geometry (52-XX) 1 Manifolds and cell complexes (57-XX) 1 Probability theory and stochastic processes (60-XX) 1 Classical thermodynamics, heat transfer (80-XX) 1 Relativity and gravitational theory (83-XX) 1 Information and communication theory, circuits (94-XX) #### Citations contained in zbMATH Open 86 Publications have been cited 1,359 times in 940 Documents Cited by Year Domain theory in logical form. Zbl 0737.03006 Abramsky, Samson 1991 Full abstraction for PCF. Zbl 1006.68028 2000 Games and full completeness for multiplicative linear logic. Zbl 0822.03007 1994 Computational interpretations of linear logic. Zbl 0791.03003 Abramsky, Samson 1993 Quantales, observational logic and process semantics. Zbl 0823.06011 Abramsky, Samson; Vickers, Steven 1993 Full abstraction in the lazy lambda calculus. Zbl 0779.03003 Abramsky, Samson; Ong, C.-H. Luke 1993 A domain equation for bisimulation. Zbl 0718.68057 Abramsky, Samson 1991 Categorical quantum mechanics. Zbl 1273.81014 Abramsky, Samson; Coecke, Bob 2009 Call-by-value games. Zbl 0908.03035 Abramsky, Samson; McCusker, Guy 1998 Observation equivalence as a testing equivalence. Zbl 0626.68016 Abramsky, Samson 1987 Linearity, sharing and state: A fully abstract games semantics for idealized Algol with active expressions. (Extended abstract). Zbl 0909.68029 Abramsky, Samson; McCusker, Guy 1996 Geometry of interaction and linear combinatory algebras. Zbl 1014.03056 Abramsky, Samson; Haghverdi, Esfandiar; Scott, Philip 2002 From IF to BI. A tale of dependence and separation. Zbl 1175.03016 Abramsky, Samson; Väänänen, Jouko 2009 Game semantics. Zbl 0961.68080 Abramsky, Samson; McCusker, Guy 1999 The sheaf-theoretic structure of non-locality and contextuality. Zbl 1448.81028 2011 Full abstraction for PCF. Zbl 0942.68615 1994 Contextuality, cohomology and paradox. Zbl 1373.03048 Abramsky, Samson; Barbosa, Rui Soares; Kishida, Kohei; Lal, Raymond; Mansfield, Shane 2015 Proofs as processes. Zbl 0850.68297 Abramsky, Samson 1994 New foundations for the geometry of interaction. Zbl 0803.03014 1994 Nuclear and trace ideals in tensored $$^$$-categories. Zbl 0946.18004 Abramsky, Samson; Blute, Richard; Panangaden, Prakash 1999 Strictness analysis for higher-order functions. Zbl 0603.68013 Burn, Geoffrey L.; Hankin, Chris; Abramsky, Samson 1986 $$H^\ast$$-algebras and nonunital Frobenius algebras: first steps in infinite-dimensional categorical quantum mechanics. Zbl 1267.18007 Abramsky, Samson; Heunen, Chris 2012 A Cook’s tour of the finitary non-well-founded sets. Zbl 1279.03073 Abramsky, Samson 2005 Interaction categories and the foundations of typed concurrent programming. Zbl 0934.18007 Abramsky, Samson; Gay, Simon; Nagarajan, Rajagopal 1996 Semantics of interaction: An introduction to game semantics. Zbl 0938.91500 Abramsky, Samson 1997 A structural approach to reversible computation. Zbl 1081.68019 Abramsky, Samson 2005 An internal language for autonomous categories. Zbl 0806.03044 Mackie, Ian; Román, Leopoldo; Abramsky, Samson 1993 Abstract physical traces. Zbl 1065.18005 Abramsky, Samson; Coecke, Bob 2005 Abstract scalars, loops, and free traced and strongly compact closed categories. Zbl 1151.81002 Abramsky, Samson 2005 Handbook of logic in computer science. Vol. 3: Semantic structures. Zbl 0829.68111 1994 On semantic foundations for applicative multiprogramming. Zbl 0538.68064 Abramsky, Samson 1983 Full abstraction for idealized Algol with passive expressions. Zbl 0954.68028 Abramsky, Samson; McCusker, Guy 1999 The theory of strictness analysis for higher order functions. Zbl 0596.68009 Burn, G. L.; Hankin, C. L.; Abramsky, S. 1986 A categorical quantum logic. Zbl 1099.03059 Abramsky, Samson; Duncan, Ross 2006 Abstract interpretation, logical relations, and Kan extensions. Zbl 0727.03020 Abramsky, Samson 1990 Sequentiality vs. concurrency in games and logic. Zbl 1129.03014 Abramsky, Samson 2003 Applying game semantics to compositional software modeling and verification. Zbl 1126.68343 Abramsky, Samson; Ghica, Dan R.; Murawski, Andrzej S.; Ong, C.-H. Luke 2004 Temperley-Lieb algebra: from knot theory to logic and computation via quantum mechanics. Zbl 1135.81006 Abramsky, Samson 2008 Introduction to categories and categorical logic. Zbl 1217.18001 Abramsky, S.; Tzevelekos, N. 2011 Relational hidden variables and non-locality. Zbl 1278.81101 Abramsky, Samson 2013 An operational interpretation of negative probabilities and no-signalling models. Zbl 1415.81009 2014 A compositional game semantics for multi-agent logics of partial information. Zbl 1196.68241 Abramsky, Samson 2007 Coalgebraic analysis of subgame-perfect equilibria in infinite games without discounting. Zbl 1364.91026 Abramsky, Samson; Winschel, Viktor 2017 Algorithmic game semantics. A tutorial introduction. Zbl 1097.68574 Abramsky, Samson 2002 Strictness analysis and polymorphic invariance. Zbl 0624.68034 Abramsky, Samson 1986 Handbook of logic in computer science. Vol. 2: Background: Computational structures. Zbl 0777.68001 1992 Games and full completeness for multiplicative linear logic. (Extended Abstract). Zbl 0925.03041 1992 Relational databases and Bell’s theorem. Zbl 1397.68041 Abramsky, Samson 2013 A fully abstract denotational semantics for the calculus of higher-order communicating systems. Zbl 0974.68109 Thomsen, B.; Abramsky, S. 2001 Linear realizability and full completeness for typed lambda-calculi. Zbl 1064.03012 Abramsky, Samson; Lenisa, Marina 2005 No-cloning in categorical quantum mechanics. Zbl 1192.81013 Abramsky, Samson 2010 A fully complete PER model for ML polymorphic types. Zbl 0973.03015 Abramsky, Samson; Lenisa, Marina 2000 A game semantics for generic polymorphism. Zbl 1066.68074 2005 Experiments, powerdomains and fully abstract models for applicative multiprogramming. Zbl 0586.68022 Abramsky, Samson 1983 Game semantics for access control. Zbl 1337.68157 2009 Mixed computation of Prolog programs. Zbl 0654.68021 Fuller, David A.; Abramsky, Samson 1988 Specifying interaction categories. Zbl 0884.18008 Pavlović, D.; Abramsky, S. 1997 Intensionality, definability and computation. Zbl 1344.03003 Abramsky, Samson 2014 Petri nets, discrete physics, and distributed quantum computation. Zbl 1143.68468 Abramsky, Samson 2008 Games for recursive types. Zbl 0840.03054 Abramsky, Samson; McCusker, Guy 1995 Big toy models. Representing physical systems as Chu spaces. Zbl 1275.81008 Abramsky, Samson 2012 Operational theories and categorical quantum mechanics. Zbl 1355.81028 Abramsky, Samson; Heunen, Chris 2016 The quantum monad on relational structures. Zbl 1441.68055 Abramsky, Samson; Barbosa, Rui Soares; de Silva, Nadish; Zapata, Octavio 2017 Axiomatizing fully complete models for ML polymorphic types. Zbl 0996.03041 Abramsky, Samson; Lenisa, Marina 2000 Semantic unification. A sheaf theoretic approach to natural language. Zbl 1285.03021 2014 What are the fundamental structures of concurrency? We still don’t know! Zbl 1315.68188 Abramsky, Samson 2006 A type-theoretic approach to deadlock-freedom of asynchronous systems. Zbl 0882.18002 Abramsky, Samson; Gay, Simon; Nagarajan, Rajagopal 1997 Handbook of logic in computer science. Vol. 4: Semantic modelling. Zbl 0876.68001 1995 Fully complete minimal PER models for the simply typed $$\lambda$$-calculus. Zbl 0999.03010 Abramsky, Samson; Lenisa, Marina 2001 Relating structure and power: comonadic semantics for computational resources (extended abstract). Zbl 06962928 Abramsky, Samson; Shah, Nihil 2018 The cohomology of non-locality and contextuality. Zbl 1464.81012 Abramsky, Samson; Mansfield, Shane; Soares Barbosa, Rui 2012 Category theory and computer programming. Tutorial and Workshop, Guildford, U.K., September 16-20, 1985. Proceedings. Zbl 0607.00015 1986 Handbook of logic in computer science. Vol. 1: Background: Mathematical structures. Zbl 0806.68003 1992 Process realizability. Zbl 0995.68064 Abramsky, Samson 2000 A specification structure for deadlock-freedom of synchronous processes. Zbl 0932.68061 Abramsky, S.; Gay, S. J.; Nagarajan, R. 1999 A complete characterization of all-versus-nothing arguments for stabilizer states. Zbl 1404.81022 Abramsky, Samson; Soares Barbosa, Rui; Carù, Giovanni; Perdrix, Simon 2017 A game semantics for generic polymorphism. Zbl 1029.68038 2003 From Lawvere to Brandenburger-Keisler: interactive forms of diagonalization and self-reference. Zbl 1328.03013 Abramsky, Samson; Zvesper, Jonathan 2015 Dependence logic. Theory and applications. Selected papers based on the presentations at the Dagstuhl seminar on ‘Dependence logic: theory and applications’, Wadern, Germany, February 2013. Zbl 1348.03004 2016 From Lawvere to Brandenburger-Keisler: interactive forms of diagonalization and self-reference. Zbl 1328.03012 Abramsky, Samson; Zvesper, Jonathan 2012 Games for dependent types. Zbl 1395.68177 2015 Physical traces: quantum vs. classical information processing. Zbl 1270.68182 Abramsky, Samson; Coecke, Bob 2003 Coalgebras, Chu spaces, and representations of physical systems. Zbl 1270.81040 Abramsky, Samson 2013 Hardy is (almost) everywhere: nonlocality without inequalities for almost all entangled multipartite states. Zbl 1353.81024 Abramsky, Samson; Constantin, Carmen M.; Ying, Shenggang 2016 Minimum quantum resources for strong non-locality. Zbl 1427.81006 Abramsky, Samson; Barbosa, Rui Soares; Carù, Giovanni; De Silva, Nadish; Kishida, Kohei; Mansfield, Shane 2018 The pebbling comonad in finite model theory. Zbl 1452.03083 Abramsky, Samson; Dawar, Anuj; Wang, Pengming 2017 Relating structure and power: comonadic semantics for computational resources (extended abstract). Zbl 06962928 Abramsky, Samson; Shah, Nihil 2018 Minimum quantum resources for strong non-locality. Zbl 1427.81006 Abramsky, Samson; Barbosa, Rui Soares; Carù, Giovanni; De Silva, Nadish; Kishida, Kohei; Mansfield, Shane 2018 Coalgebraic analysis of subgame-perfect equilibria in infinite games without discounting. Zbl 1364.91026 Abramsky, Samson; Winschel, Viktor 2017 The quantum monad on relational structures. Zbl 1441.68055 Abramsky, Samson; Barbosa, Rui Soares; de Silva, Nadish; Zapata, Octavio 2017 A complete characterization of all-versus-nothing arguments for stabilizer states. Zbl 1404.81022 Abramsky, Samson; Soares Barbosa, Rui; Carù, Giovanni; Perdrix, Simon 2017 The pebbling comonad in finite model theory. Zbl 1452.03083 Abramsky, Samson; Dawar, Anuj; Wang, Pengming 2017 Operational theories and categorical quantum mechanics. Zbl 1355.81028 Abramsky, Samson; Heunen, Chris 2016 Dependence logic. Theory and applications. Selected papers based on the presentations at the Dagstuhl seminar on ‘Dependence logic: theory and applications’, Wadern, Germany, February 2013. Zbl 1348.03004 2016 Hardy is (almost) everywhere: nonlocality without inequalities for almost all entangled multipartite states. Zbl 1353.81024 Abramsky, Samson; Constantin, Carmen M.; Ying, Shenggang 2016 Contextuality, cohomology and paradox. Zbl 1373.03048 Abramsky, Samson; Barbosa, Rui Soares; Kishida, Kohei; Lal, Raymond; Mansfield, Shane 2015 From Lawvere to Brandenburger-Keisler: interactive forms of diagonalization and self-reference. Zbl 1328.03013 Abramsky, Samson; Zvesper, Jonathan 2015 Games for dependent types. Zbl 1395.68177 2015 An operational interpretation of negative probabilities and no-signalling models. Zbl 1415.81009 2014 Intensionality, definability and computation. Zbl 1344.03003 Abramsky, Samson 2014 Semantic unification. A sheaf theoretic approach to natural language. Zbl 1285.03021 2014 Relational hidden variables and non-locality. Zbl 1278.81101 Abramsky, Samson 2013 Relational databases and Bell’s theorem. Zbl 1397.68041 Abramsky, Samson 2013 Coalgebras, Chu spaces, and representations of physical systems. Zbl 1270.81040 Abramsky, Samson 2013 $$H^\ast$$-algebras and nonunital Frobenius algebras: first steps in infinite-dimensional categorical quantum mechanics. Zbl 1267.18007 Abramsky, Samson; Heunen, Chris 2012 Big toy models. Representing physical systems as Chu spaces. Zbl 1275.81008 Abramsky, Samson 2012 The cohomology of non-locality and contextuality. Zbl 1464.81012 Abramsky, Samson; Mansfield, Shane; Soares Barbosa, Rui 2012 From Lawvere to Brandenburger-Keisler: interactive forms of diagonalization and self-reference. Zbl 1328.03012 Abramsky, Samson; Zvesper, Jonathan 2012 The sheaf-theoretic structure of non-locality and contextuality. Zbl 1448.81028 2011 Introduction to categories and categorical logic. Zbl 1217.18001 Abramsky, S.; Tzevelekos, N. 2011 No-cloning in categorical quantum mechanics. Zbl 1192.81013 Abramsky, Samson 2010 Categorical quantum mechanics. Zbl 1273.81014 Abramsky, Samson; Coecke, Bob 2009 From IF to BI. A tale of dependence and separation. Zbl 1175.03016 Abramsky, Samson; Väänänen, Jouko 2009 Game semantics for access control. Zbl 1337.68157 2009 Temperley-Lieb algebra: from knot theory to logic and computation via quantum mechanics. Zbl 1135.81006 Abramsky, Samson 2008 Petri nets, discrete physics, and distributed quantum computation. Zbl 1143.68468 Abramsky, Samson 2008 A compositional game semantics for multi-agent logics of partial information. Zbl 1196.68241 Abramsky, Samson 2007 A categorical quantum logic. Zbl 1099.03059 Abramsky, Samson; Duncan, Ross 2006 What are the fundamental structures of concurrency? We still don’t know! Zbl 1315.68188 Abramsky, Samson 2006 A Cook’s tour of the finitary non-well-founded sets. Zbl 1279.03073 Abramsky, Samson 2005 A structural approach to reversible computation. Zbl 1081.68019 Abramsky, Samson 2005 Abstract physical traces. Zbl 1065.18005 Abramsky, Samson; Coecke, Bob 2005 Abstract scalars, loops, and free traced and strongly compact closed categories. Zbl 1151.81002 Abramsky, Samson 2005 Linear realizability and full completeness for typed lambda-calculi. Zbl 1064.03012 Abramsky, Samson; Lenisa, Marina 2005 A game semantics for generic polymorphism. Zbl 1066.68074 2005 Applying game semantics to compositional software modeling and verification. Zbl 1126.68343 Abramsky, Samson; Ghica, Dan R.; Murawski, Andrzej S.; Ong, C.-H. Luke 2004 Sequentiality vs. concurrency in games and logic. Zbl 1129.03014 Abramsky, Samson 2003 A game semantics for generic polymorphism. Zbl 1029.68038 2003 Physical traces: quantum vs. classical information processing. Zbl 1270.68182 Abramsky, Samson; Coecke, Bob 2003 Geometry of interaction and linear combinatory algebras. Zbl 1014.03056 Abramsky, Samson; Haghverdi, Esfandiar; Scott, Philip 2002 Algorithmic game semantics. A tutorial introduction. Zbl 1097.68574 Abramsky, Samson 2002 A fully abstract denotational semantics for the calculus of higher-order communicating systems. Zbl 0974.68109 Thomsen, B.; Abramsky, S. 2001 Fully complete minimal PER models for the simply typed $$\lambda$$-calculus. Zbl 0999.03010 Abramsky, Samson; Lenisa, Marina 2001 Full abstraction for PCF. Zbl 1006.68028 2000 A fully complete PER model for ML polymorphic types. Zbl 0973.03015 Abramsky, Samson; Lenisa, Marina 2000 Axiomatizing fully complete models for ML polymorphic types. Zbl 0996.03041 Abramsky, Samson; Lenisa, Marina 2000 Process realizability. Zbl 0995.68064 Abramsky, Samson 2000 Game semantics. Zbl 0961.68080 Abramsky, Samson; McCusker, Guy 1999 Nuclear and trace ideals in tensored $$^$$-categories. Zbl 0946.18004 Abramsky, Samson; Blute, Richard; Panangaden, Prakash 1999 Full abstraction for idealized Algol with passive expressions. Zbl 0954.68028 Abramsky, Samson; McCusker, Guy 1999 A specification structure for deadlock-freedom of synchronous processes. Zbl 0932.68061 Abramsky, S.; Gay, S. J.; Nagarajan, R. 1999 Call-by-value games. Zbl 0908.03035 Abramsky, Samson; McCusker, Guy 1998 Semantics of interaction: An introduction to game semantics. Zbl 0938.91500 Abramsky, Samson 1997 Specifying interaction categories. Zbl 0884.18008 Pavlović, D.; Abramsky, S. 1997 A type-theoretic approach to deadlock-freedom of asynchronous systems. Zbl 0882.18002 Abramsky, Samson; Gay, Simon; Nagarajan, Rajagopal 1997 Linearity, sharing and state: A fully abstract games semantics for idealized Algol with active expressions. (Extended abstract). Zbl 0909.68029 Abramsky, Samson; McCusker, Guy 1996 Interaction categories and the foundations of typed concurrent programming. Zbl 0934.18007 Abramsky, Samson; Gay, Simon; Nagarajan, Rajagopal 1996 Games for recursive types. Zbl 0840.03054 Abramsky, Samson; McCusker, Guy 1995 Handbook of logic in computer science. Vol. 4: Semantic modelling. Zbl 0876.68001 1995 Games and full completeness for multiplicative linear logic. Zbl 0822.03007 1994 Full abstraction for PCF. Zbl 0942.68615 1994 Proofs as processes. Zbl 0850.68297 Abramsky, Samson 1994 New foundations for the geometry of interaction. Zbl 0803.03014 1994 Handbook of logic in computer science. Vol. 3: Semantic structures. Zbl 0829.68111 1994 Computational interpretations of linear logic. Zbl 0791.03003 Abramsky, Samson 1993 Quantales, observational logic and process semantics. Zbl 0823.06011 Abramsky, Samson; Vickers, Steven 1993 Full abstraction in the lazy lambda calculus. Zbl 0779.03003 Abramsky, Samson; Ong, C.-H. Luke 1993 An internal language for autonomous categories. Zbl 0806.03044 Mackie, Ian; Román, Leopoldo; Abramsky, Samson 1993 Handbook of logic in computer science. Vol. 2: Background: Computational structures. Zbl 0777.68001 1992 Games and full completeness for multiplicative linear logic. (Extended Abstract). Zbl 0925.03041 1992 Handbook of logic in computer science. Vol. 1: Background: Mathematical structures. Zbl 0806.68003 1992 Domain theory in logical form. Zbl 0737.03006 Abramsky, Samson 1991 A domain equation for bisimulation. Zbl 0718.68057 Abramsky, Samson 1991 Abstract interpretation, logical relations, and Kan extensions. Zbl 0727.03020 Abramsky, Samson 1990 Mixed computation of Prolog programs. Zbl 0654.68021 Fuller, David A.; Abramsky, Samson 1988 Observation equivalence as a testing equivalence. Zbl 0626.68016 Abramsky, Samson 1987 Strictness analysis for higher-order functions. Zbl 0603.68013 Burn, Geoffrey L.; Hankin, Chris; Abramsky, Samson 1986 The theory of strictness analysis for higher order functions. Zbl 0596.68009 Burn, G. L.; Hankin, C. L.; Abramsky, S. 1986 Strictness analysis and polymorphic invariance. Zbl 0624.68034 Abramsky, Samson 1986 Category theory and computer programming. Tutorial and Workshop, Guildford, U.K., September 16-20, 1985. Proceedings. Zbl 0607.00015 1986 On semantic foundations for applicative multiprogramming. Zbl 0538.68064 Abramsky, Samson 1983 Experiments, powerdomains and fully abstract models for applicative multiprogramming. Zbl 0586.68022 Abramsky, Samson 1983 all top 5 #### Cited by 936 Authors 34 Abramsky, Samson 18 Murawski, Andrzej S. 14 Coecke, Bob 13 Dezani-Ciancaglini, Mariangiola 11 Honsell, Furio 10 Heunen, Chris 10 Yoshida, Nobuko 9 Ghica, Dan R. 9 Lenisa, Marina 9 McCusker, Guy Andrew 9 Paolini, Luca 9 Scott, Philip J. 9 Solovyov, Sergey A. 9 Tzevelekos, Nikos 8 Gogioso, Stefano 8 Honda, Kohei 8 Jacobs, Bart 8 Jagadeesan, Radha 8 Mislove, Michael W. 8 Panangaden, Prakash 8 Vicary, Jamie 8 Vickers, Steven 7 Bezhanishvili, Nick 7 de’Liguoro, Ugo 7 Dzhafarov, Ehtibar N. 7 Galliani, Pietro 7 Ingólfsdóttir, Anna 7 Japaridze, Giorgi 7 Jung, Achim 7 Klop, Jan Willem 7 Winskel, Glynn 7 Zhao, Bin 6 Aceto, Luca 6 Alessi, Fabio 6 Blute, Richard F. 6 Clairambault, Pierre 6 Kurz, Alexander 6 Mackie, Ian 6 Melliès, Paul-André 6 Plotkin, Gordon D. 6 Rosenthal, Kimmo I. 6 Ulidowski, Irek 6 Yang, Fan 5 Berger, Martin J. 5 Bezhanishvili, Guram 5 Curien, Pierre-Louis 5 Doberkat, Ernst-Erich 5 Ehrhard, Thomas 5 Han, Shengwei 5 Hasegawa, Masahito 5 Laird, James D. 5 Li, Qingguo 5 Moshier, M. Andrew 5 Paseka, Jan 5 Pavlović, Duško 5 Phillips, Iain W. 5 Piccolo, Mauro 5 Resende, Pedro 5 Ronchi Della Rocca, Simona 5 Sadrzadeh, Mehrnoosh 5 Scedrov, Andre 5 van Benthem, Johan F. A. K. 4 Alves, Sandra 4 Barbanera, Franco 4 Carù, Giovanni 4 Cervesato, Iliano 4 Ciardelli, Ivano A. 4 de Paiva, Valeria 4 de Vries, Fer-Jan J. 4 Fiore, Marcelo P. 4 Florido, Mário 4 Fu, Yuxi 4 Gehrke, Mai 4 Haghverdi, Esfandiar 4 Hasuo, Ichiro 4 Hennessy, Matthew C. B. 4 Hötzel Escardó, Martín 4 Kissinger, Aleks 4 Kontinen, Juha 4 Kujala, Janne V. 4 Malacaria, Pasquale 4 Nielson, Flemming 4 Power, John 4 Pym, David J. 4 Santocanale, Luigi 4 Saurin, Alexis 4 Schmidt, David A. 4 Soares Barbosa, Rui 4 van Bakel, Steffen 4 Wang, Longchun 4 Worrell, James B. 3 Axelsen, Holger Bock 3 Baltag, Alexandru 3 Başkent, Can 3 Bernardo, Marco 3 Bloom, Bard 3 Bonsangue, Marcello Maria 3 Bucciarelli, Antonio 3 Chen, Yixiang 3 Cho, Kenta ...and 836 more Authors all top 5 #### Cited in 111 Serials 185 Theoretical Computer Science 68 Information and Computation 62 Annals of Pure and Applied Logic 46 MSCS. Mathematical Structures in Computer Science 17 Logical Methods in Computer Science 14 Fuzzy Sets and Systems 14 Journal of Pure and Applied Algebra 11 Foundations of Physics 10 Acta Informatica 10 Studia Logica 10 Synthese 10 Formal Aspects of Computing 10 Applied Categorical Structures 10 Philosophical Transactions of the Royal Society of London. A. Mathematical, Physical and Engineering Sciences 9 Journal of Functional Programming 9 The Journal of Logic and Algebraic Programming 7 International Journal of Theoretical Physics 7 Journal of Mathematical Psychology 7 Journal of Logic, Language and Information 7 Journal of Logical and Algebraic Methods in Programming 6 Information Processing Letters 6 Journal of Mathematical Physics 6 Journal of Computer and System Sciences 6 Journal of Philosophical Logic 6 The Journal of Symbolic Logic 6 Topology and its Applications 6 The Bulletin of Symbolic Logic 6 Quantum Information Processing 5 Cahiers de Topologie et Géométrie Différentielle Catégoriques 4 Semigroup Forum 4 Soft Computing 4 Journal of Applied Logic 3 Communications in Algebra 3 Communications in Mathematical Physics 3 Notre Dame Journal of Formal Logic 3 RAIRO. Informatique Théorique et Applications 3 Journal of Applied Non-Classical Logics 3 RAIRO. Theoretical Informatics and Applications 2 Computers & Mathematics with Applications 2 Advances in Mathematics 2 Journal of Algebra 2 New Generation Computing 2 International Journal of Approximate Reasoning 2 International Journal of Algebra and Computation 2 International Journal of Foundations of Computer Science 2 Journal of Knot Theory and its Ramifications 2 Cybernetics and Systems Analysis 2 Formal Methods in System Design 2 Theory and Applications of Categories 2 Annals of Mathematics and Artificial Intelligence 2 Theory of Computing Systems 2 Higher-Order and Symbolic Computation 2 LMS Journal of Computation and Mathematics 2 Computer Languages, Systems & Structures 2 RAIRO. Theoretical Informatics and Applications 1 Artificial Intelligence 1 Lithuanian Mathematical Journal 1 The Mathematical Gazette 1 Nuclear Physics. B 1 Journal of Geometry and Physics 1 The Mathematical Intelligencer 1 Information Sciences 1 Kybernetika 1 Mathematica Slovaca 1 Proceedings of the American Mathematical Society 1 Programming and Computer Software 1 Science of Computer Programming 1 Mathematical Social Sciences 1 History and Philosophy of Logic 1 Journal of Computer Science and Technology 1 International Journal of Parallel Programming 1 Journal of Automated Reasoning 1 International Journal of Computer Mathematics 1 Distributed Computing 1 Archive for Mathematical Logic 1 Indagationes Mathematicae. New Series 1 Topology Proceedings 1 Turkish Journal of Mathematics 1 Selecta Mathematica. New Series 1 Topoi 1 Journal of the ACM 1 New Journal of Physics 1 Communications in Nonlinear Science and Numerical Simulation 1 International Journal of Applied Mathematics and Computer Science 1 Fundamenta Informaticae 1 Journal of High Energy Physics 1 Annales Henri Poincaré 1 International Game Theory Review 1 Journal of the Australian Mathematical Society 1 Logic and Logical Philosophy 1 Journal of Applied Mathematics and Computing 1 Cahiers de Topologie et Géométrie Différentielle Catégoriques 1 Journal of Intelligent and Fuzzy Systems 1 Computational Intelligence 1 Journal of Algebra and its Applications 1 International Journal of Quantum Information 1 Iranian Journal of Fuzzy Systems 1 Logica Universalis 1 Journal of Homotopy and Related Structures 1 São Paulo Journal of Mathematical Sciences ...and 11 more Serials all top 5 #### Cited in 42 Fields 561 Computer science (68-XX) 428 Mathematical logic and foundations (03-XX) 169 Category theory; homological algebra (18-XX) 126 Quantum theory (81-XX) 107 Order, lattices, ordered algebraic structures (06-XX) 81 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 39 General topology (54-XX) 21 Functional analysis (46-XX) 16 General algebraic systems (08-XX) 14 Group theory and generalizations (20-XX) 13 Probability theory and stochastic processes (60-XX) 10 Information and communication theory, circuits (94-XX) 9 General and overarching topics; collections (00-XX) 9 Associative rings and algebras (16-XX) 8 Statistics (62-XX) 6 Combinatorics (05-XX) 6 Algebraic topology (55-XX) 5 Linear and multilinear algebra; matrix theory (15-XX) 5 Topological groups, Lie groups (22-XX) 4 Measure and integration (28-XX) 4 Operator theory (47-XX) 4 Manifolds and cell complexes (57-XX) 4 Relativity and gravitational theory (83-XX) 3 History and biography (01-XX) 3 Number theory (11-XX) 3 Commutative algebra (13-XX) 3 Partial differential equations (35-XX) 3 Dynamical systems and ergodic theory (37-XX) 3 Global analysis, analysis on manifolds (58-XX) 2 Nonassociative rings and algebras (17-XX) 2 $$K$$-theory (19-XX) 2 Convex and discrete geometry (52-XX) 2 Differential geometry (53-XX) 2 Biology and other natural sciences (92-XX) 1 Algebraic geometry (14-XX) 1 Real functions (26-XX) 1 Ordinary differential equations (34-XX) 1 Geometry (51-XX) 1 Numerical analysis (65-XX) 1 Mechanics of particles and systems (70-XX) 1 Statistical mechanics, structure of matter (82-XX) 1 Mathematics education (97-XX) #### Wikidata Timeline The data are displayed as stored in Wikidata under a Creative Commons CC0 License. Updates and corrections should be made in Wikidata.	2022-01-22 18:12:30	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4802730977535248, "perplexity": 10090.822895176541}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320303868.98/warc/CC-MAIN-20220122164421-20220122194421-00286.warc.gz"}
https://questions.examside.com/past-years/jee/question/the-boolean-expression-p-wedge-q-rightarrow-r-jee-main-mathematics-trigonometric-functions-and-equations-uy71tapm8wjk2qf7	1 JEE Main 2021 (Online) 27th August Evening Shift +4 -1 The Boolean expression (p $$\wedge$$ q) $$\Rightarrow$$ ((r $$\wedge$$ q) $$\wedge$$ p) is equivalent to : A (p $$\wedge$$ q) $$\Rightarrow$$ (r $$\wedge$$ q) B (q $$\wedge$$ r) $$\Rightarrow$$ (p $$\wedge$$ q) C (p $$\wedge$$ q) $$\Rightarrow$$ (r $$\vee$$ q) D (p $$\wedge$$ r) $$\Rightarrow$$ (p $$\wedge$$ q) 2 JEE Main 2021 (Online) 27th August Morning Shift +4 -1 The statement (p $$\wedge$$ (p $$\to$$ q) $$\wedge$$ (q $$\to$$ r)) $$\to$$ r is : A a tautology B equivalent to p $$\to$$ $$\sim$$ r C a fallacy D equivalent to q $$\to$$ $$\sim$$ r 3 JEE Main 2021 (Online) 26th August Evening Shift +4 -1 Consider the two statements : (S1) : (p $$\to$$ q) $$\vee$$ ($$\sim$$ q $$\to$$ p) is a tautology . (S2) : (p $$\wedge$$ $$\sim$$ q) $$\wedge$$ ($$\sim$$ p $$\wedge$$ q) is a fallacy. Then : A only (S1) is true. B both (S1) and (S2) are false. C both (S1) and (S2) are true. D only (S2) is true. 4 JEE Main 2021 (Online) 26th August Morning Shift +4 -1 If the truth value of the Boolean expression $$\left( {\left( {p \vee q} \right) \wedge \left( {q \to r} \right) \wedge \left( { \sim r} \right)} \right) \to \left( {p \wedge q} \right)$$ is false, then the truth values of the statements p, q, r respectively can be : A T F T B F F T C T F F D F T F JEE Main Subjects Physics Mechanics Electricity Optics Modern Physics Chemistry Physical Chemistry Inorganic Chemistry Organic Chemistry Mathematics Algebra Trigonometry Coordinate Geometry Calculus EXAM MAP Joint Entrance Examination	2023-03-26 00:28:16	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5927223563194275, "perplexity": 8167.999678477674}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945376.29/warc/CC-MAIN-20230325222822-20230326012822-00061.warc.gz"}
http://pldi17.sigplan.org/event/pldi-2017-papers-decomposition-instead-of-self-composition-for-k-safety	Tue 20 Jun 2017 10:50 - 11:15 at Aula Master - Static Analysis and Security Chair(s): Mayur Naik We describe a novel technique for proving $k$-safety properties (non-interference, determinism, etc.) via a decomposition that enables one to leverage non-relational reasoning techniques. The key is the inter-operation of the following principles. First, we observe that many $k$-safety properties of interest have a particular structure that we call $\psi$\emph{-quotient partitionability} where $\psi$ is a $k$-ary formula. Second, we develop a partitioning strategy of execution traces based on the $k$-safety property $\Phi$ of interest such that if $\psi$ holds for $k$ traces then they must be in the same partition. Finally, within a partition component $T_i$, we observe that we can prove $k$-safety by instead proving a universal property: all traces within the partition satisfy some common property $P_i$, chosen to be strong enough that it implies the $k$-safety property $\Phi$ of any $k$-tuple of traces in components $T_i$. We apply this strategy to the task of discovering timing side channels. A key feature of our approach is a demand-driven partitioning strategy that uses high/low-annotated regex-like \emph{trails} to reason about one partition component of execution traces at a time. We have applied our technique in a prototype implementation tool called {\sc Blazer}, based on WALA, PPL, Z3, and the brics automaton library. We have proved non-interference of (or synthesized an attack specification for) 25 programs written in Java bytecode, including 7 classic examples from the literature, and 6 examples extracted from the DARPA STAC challenge problems. Tue 20 Jun 10:50 - 12:30: PLDI Research Papers - Static Analysis and Security at Aula Master Chair(s): Mayur Naik 10:50 - 11:15Talk Decomposition Instead of Self-Composition for Proving the Absence of Timing Channels Media Attached 11:15 - 11:40Talk Automatic Program Inversion using Symbolic Transducers Media Attached 11:40 - 12:05Talk Control-Flow Recovery from Partial Failure Reports Pre-print Media Attached 12:05 - 12:30Talk Rigorous Analysis of Software Countermeasures against Cache Attacks Media Attached	2017-08-23 04:23:08	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5808961391448975, "perplexity": 2882.204274848489}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886117519.92/warc/CC-MAIN-20170823035753-20170823055753-00132.warc.gz"}
https://math.stackexchange.com/questions/1833668/riemann-zeta-function-functional-equation-proof-explanation	Riemann zeta function functional equation proof explanation In Riemann zeta function functional equation proof I arrived to a following equation $$\frac{\Gamma\left(\frac{s}2\right) \zeta(s)}{\pi^{\frac{s}2}}=\sum_{n=1}^\infty \int_0^\infty x^{\frac{s}2-1}e^{-n^{2}\pi x}dx,$$ where $s\in \mathbb{C}$ and Re$(s)>1$. Next step is to prove that $$\frac{\Gamma\left(\frac{s}2\right) \zeta(s)}{\pi^{\frac{s}2}}=\int_0^\infty x^{\frac{s}2-1}\sum_{n=1}^\infty e^{-n^{2}\pi x}dx.$$ Why exactly can we change the order of integration and summation? I'm having trouble with this equation, since $s\in \mathbb{C}$. What exactly should I use? For inverting $\sum$ and $\int$ from scratch the method is always the same (see this discussion), adapted to your problem it gives : • use that for $x >0$ $$\sum_{n=N}^\infty e^{- \pi n^2 x} < \sum_{n=N}^\infty e^{- \pi n x} = \frac{e^{-\pi N x}}{1-e^{-\pi x}}$$ so that for $Re(s) > 1$ : $$\lim_{N \to \infty} \left\|\int_0^\infty x^{s/2-1} \sum_{n=N}^\infty e^{- \pi n^2 x} dx\right\| < \lim_{N \to \infty} \int_0^\infty \|x^{s/2-1}\| \frac{e^{-\pi N x}}{1-e^{-\pi x}} dx = 0$$ • then using that $\int_0^\infty x^{s/2-1} e^{- \pi n^2 x} dx =n^{-s} \int_0^\infty y^{s/2-1} e^{- \pi y} dy$ we get that $$\lim_{N \to \infty} \sum_{n=N}^\infty \int_0^\infty x^{s/2-1} e^{- \pi n^2 x} dx = \lim_{N \to \infty} \sum_{n=N}^\infty n^{-s} \int_0^\infty y^{s/2-1} e^{-\pi y} dy = 0$$ • finally $$\int_0^\infty x^{s/2-1} \sum_{n=1}^\infty e^{- \pi n^2 x} dx = \int_0^\infty x^{s/2-1} \sum_{n=1}^N e^{- \pi n^2 x} dx + \int_0^\infty x^{s/2-1} \sum_{n=N+1}^\infty e^{- \pi n^2 x} dx$$ $$= \sum_{n=1}^N\int_0^\infty x^{s/2-1} e^{- \pi n^2 x} dx + \int_0^\infty x^{s/2-1} \sum_{n=N+1}^\infty e^{- \pi n^2 x} dx$$ $$=\sum_{n=1}^\infty\int_0^\infty x^{s/2-1} e^{- \pi n^2 x} dx - \sum_{n=N+1}^\infty\int_0^\infty x^{s/2-1} e^{- \pi n^2 x} dx + \int_0^\infty x^{s/2-1} \sum_{n=N+1}^\infty e^{- \pi n^2 x} dx$$ and let $N \to \infty$	2019-06-19 02:59:08	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9759612083435059, "perplexity": 132.84704534702706}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627998882.88/warc/CC-MAIN-20190619023613-20190619045613-00471.warc.gz"}
http://math.stackexchange.com/questions/328228/how-is-hausdorff-distance-sensitive-to-position	# How is Hausdorff Distance sensitive to position? I am working on my final year M.Tech. project. It involves using Hausdorff Distance to compare images. In the process of understanding it, I stumbled upon this website which states that Hausdorff Distance is sensitive to position. I have refered several papers and articles but unable to understand how this is possible. Can someone please explain it to me or atleast point me towards a link that will help me to understand this. Thank you - Note that the first definition given for Hausdorff distance is not actually even a distance; the real Hausdorff distance is the "generalized" version. Could you explain more in detail what you mean by the expression: sensitive to position? – Thomas E. Mar 12 '13 at 8:23 @Thomas E, "Sensitive to position" is exactly what I am not understanding. According to the link in my question, it says that Hausdorff Distance is sensitive to position and they show it by drawing circles enclosing the polygons with radius equal to the Hausdorff distance between the polygons. They show two figures where in the dimension of the polygons remain same but their positions are changed and the radius of the enclosing circles are different in the figures. This, they suggest is the proof for Hausdorff distance being sensitive to position, which i am unable to understand. – Yash Mar 13 '13 at 12:46 Compare figures 1 and 2. They depict two triangles such that the shortest distance between them is the same. However, qualitatively, the two triangles in figure 2 appear closer than the two triangles in figure 1. This is what the website calls the insensitivity of the shortest distance metric to position (the way the shapes are posed in the ambient space). Now look at figures 4 and 5. Figure 4 depicts the same triangles as in figure 1, and figure 5 depicts the same triangles as in figure 2. Now, what is computed is the Hausdorff distance instead of the shortest distance. It is now seen that the Hausdorff distance between the triangles in figure 4 is larger than for those in figure 5. Thus, it is seen that the Hausdorff distance captures the initial intuition that the triangles in figure 5 are indeed closer. This property, that the Hausdorff distance considers the position (again, the way the shapes are posed in the ambience), is what the website calls the sensitivity of the Hausdorff distance to position. I'm not sure this terminology is very illuminating. I would say something along the lines of "the Hausdorff distance is sensitive to the global positioning of the two sets, and not just to the local distances". A more straightforward example would be in $\mathbb R$. Consider $A=[0,1]$ and $B=[2,3]$. The shortest distance metric between $A$ and $B$ is $1$. The Hausdorff distance between $A$ and $B$ is 2. Now, consider $B'=[2,100]$. The shortest distance between $A$ and $B'$ did not change, it's still $1$. But the Hausdorff distance is between $A$ and $B'$ is now $99$. The shortest distance metric is blind to the global relative positioning of the two shapes, but the Hausdorff distance picks it up. - A small note to the second sentence in the last paragraph: the shortest distance is not actually a metric. It fails to satisfy e.g. the condition $d(A,B)=0\Leftrightarrow A=B$. – Thomas E. Mar 13 '13 at 12:55 that depends on the definition of metric space. More and more texts drop the demand that $d(x,y)=0$ implies $x=y$, calling metric spaces that do satisfy this property separated. There are plenty of reasons to do that, one is the enhance the analogy with topological spaces. Separated metric spaces correspond to Hausdorff spaces. – Ittay Weiss Mar 13 '13 at 21:11 I only mentioned that condition in order to avoid writing concrete counter-examples. However, the shortest distance also fails to satisfy the triangle-inequality. Take $A=[0,1]$, $B=[2,3]$, and $C=[4,5]$ as a simple example. By shortest distance, $d(A,C)>d(A,B)+d(B,C)$, since $3>2$. I don't think the triangle inequality is negotiable when defining metric spaces. What is there left, the symmetry? – Thomas E. Mar 13 '13 at 21:18 of course, no letting go of the triangle inequality in metric spaces. I thought you were referring to the Hausdorff distance limited to closed subsets in order to force separability. As you say, due to failure of the triangle inequality, the shortest distance does not make the set of subsets (closed or not) of a metric space into a metric space. – Ittay Weiss Mar 13 '13 at 21:24	2015-01-29 08:40:02	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8099813461303711, "perplexity": 251.84648648265178}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-06/segments/1422122022571.56/warc/CC-MAIN-20150124175342-00243-ip-10-180-212-252.ec2.internal.warc.gz"}
https://math.stackexchange.com/questions/3356883/harmonic-in-punctured-ball-and-bounded-implies-harmonic-in-ball	# Harmonic in punctured ball and bounded implies harmonic in ball. If $$u$$ is harmonic and bounded in $$B_1(0)\setminus\{0\}$$, then can we say that $$u$$ is harmonic in $$B_1(0)$$? I believe the answer is yes and I think the way to show it is by the Mean Value Property... but there is a problem. For $$\|x\|<1/2$$ and $$\|x\|, how do we know that $$\frac{1}{\|B_r\|}\int_{B_r(x)} u(y) dy =u(x)$$? Since $$u$$ is harmonic in the punctured ball, the MVP holds only when $$B_r(x)\subset B_1(0)\setminus\{0\}$$. How does one get around this problem? Note we are working in $$\mathbb{R}^n$$ for some $$n\geq 3$$ Define $$\newcommand{\dashint}{\mathchoice{\rlap{\,\,-}\int}{\rlap{\,-}\int}{\rlap{\ -}\int}{\rlap{\,-}\int}} \dashint_Af(x)\,\mathrm{d}x=\frac1{\|A\|}\int_Af(x)\,\mathrm{d}x$$ Using $$(1)$$ and $$(2)$$ from this answer and $$\|S(r,p)\|=\omega_{n-1}r^{n-1}$$, we get that \begin{align} &r_2^{n-1}\frac\partial{\partial r}\dashint_{S(r_2,p_2)}u(x)\,\mathrm{d}\sigma-r_1^{n-1}\frac\partial{\partial r}\dashint_{S(r_1,p_1)}u(x)\,\mathrm{d}\sigma\\ &=\frac1{\omega_{n-1}}\int_{B(r_2,p_2)\setminus B(r_1,p_1)}\Delta u(x)\,\mathrm{d}x\\[6pt] &=0\tag1 \end{align} where $$r_k\gt\|p_k\|$$ (the origin is inside both spheres). Thus, for some constant $$C$$, independent of $$p$$ (as long as $$r\gt\|p\|$$), $$r^{n-1}\frac\partial{\partial r}\dashint_{S(r,p)}u(x)\,\mathrm{d}\sigma=C\tag2$$ Therefore, $$\dashint_{S(r,p)}u(x)\,\mathrm{d}\sigma =\left\{\begin{array}{} A(p)-\frac{C}{(n-2)\,r^{n-2}}&\text{if }n\ge3\\ A(p)+C\log(r)&\text{if }n=2 \end{array}\right. \tag3$$ If $$u$$ is bounded, then by considering $$p=0$$, we get $$C=0$$, and therefore, for $$r\gt\|p\|$$, $$\dashint_{S(r,p)}u(x)\,\mathrm{d}\sigma=A(p)\tag4$$ Since $$u$$ is harmonic away from $$0$$, the Mean Value Property says that for $$r\lt\|p\|$$, $$\dashint_{S(r,p)}u(x)\,\mathrm{d}\sigma=u(p)\tag5$$ However, because $$u$$ is smooth away from $$0$$, and bounded, $$\dashint_{S(r,p)}u(x)\,\mathrm{d}\sigma$$ is a continuous function of $$r$$. That is, $$A(p)=u(p)$$ and we have $$(5)$$ for all $$r$$. The function $$u$$ is harmonic on a neighborhood of $$0$$ with $$u(x) \|x\|^{n-2} \to 0$$ as $$x\to 0$$, so it extends to a harmonic function across $$0$$. (This is a standard result about harmonic functions, but the proof is basically solving the Dirichlet problem on a small punctured ball around $$0$$ with boundary value $$u$$, then using the maximum principle and the condition above to ensure that this new solution must just be $$u$$ itself.)	2019-10-24 00:16:53	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 46, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9759963154792786, "perplexity": 1808.157399625835}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570987836368.96/warc/CC-MAIN-20191023225038-20191024012538-00380.warc.gz"}
https://www.hpmuseum.org/forum/showthread.php?tid=4645&pid=62107&mode=threaded	10-04-2016, 12:16 PM (This post was last modified: 10-04-2016 12:53 PM by Bruno.) Post: #401 Bruno Member Posts: 74 Joined: Sep 2014 Thank you Claudio, now I can try and learn funny stuffs This is very impressive all the work you've done !!! Congrats. Let's talk about the code from now, and here is my first question (Don't worry, I'll try to not flood you !) : Why did you use a slow O(n) switch/case structure in the library handlers instead of a fast O(1) structure like an array of function pointers ? (remember the linktable from our old and so loved RPL) An dirty example to be sure you understand me : Code: #define LIBRARY_NUMBER 1234 #define COMMAND_LIST \ CMD(CMD1,MKTOKENINFO(4,TITYPE_NOTALLOWED,1,2)), \ ... CMD(CMDN,MKTOKENINFO(4,TITYPE_NOTALLOWED,1,2)) ... // One function per exported command void l1234_cmd1 (void) { ... } ... void l1234_cmdN (void) { ... } void LIB_HANDLER() { ... // Here is the command's array of function pointers void (*cmds[N]) (void) = { l1234_cmd1, ..., l1234_cmdN } if (OPCODE(CurOpcode) < N) { // Direct call to the command's function cmds[OPCODE(CurOpcode)](void); return; } // Add here the switch/case to handle system opcodes rplError(ERR_INVALIDOPCODE); return; } I do not find any 'bad' reason for that : - the enum will generate appropriate index for an array - the system opcodes could be handled outside the array - ??	2021-05-07 07:22:37	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4360508322715759, "perplexity": 2304.368766630041}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243988775.25/warc/CC-MAIN-20210507060253-20210507090253-00186.warc.gz"}
https://stacks.math.columbia.edu/tag/0661	Lemma 15.60.2. The comparison map (15.60.0.1) is an isomorphism if $A' = A \otimes _ R R'$ and $A$ and $R'$ are Tor independent over $R$. Proof. To prove this we choose a free resolution $F^\bullet \to R'$ of $R'$ as an $R$-module. Because $A$ and $R'$ are Tor independent over $R$ we see that $F^\bullet \otimes _ R A$ is a free $A$-module resolution of $A'$ over $A$. By our general construction of the derived tensor product above we see that $K^\bullet \otimes _ A A' \cong \text{Tot}(K^\bullet \otimes _ A (F^\bullet \otimes _ R A)) = \text{Tot}(K^\bullet \otimes _ R F^\bullet ) \cong \text{Tot}(E^\bullet \otimes _ R F^\bullet ) \cong E^\bullet \otimes _ R R'$ as desired. $\square$ In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).	2021-01-16 11:47:15	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.9736420512199402, "perplexity": 216.12966115361306}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703506640.22/warc/CC-MAIN-20210116104719-20210116134719-00040.warc.gz"}
https://datascience.stackexchange.com/questions/53399/compute-gradients-in-parallel	Here is part of my code: class SimpleNet(nn.Module): def __init__(self): super().__init__() self.linear1 = nn.Linear(2, 1, bias=False) self.linear2 = nn.Linear(1, 2, bias=False) def forward(self, x): z = self.linear1(x) y_pred = self.linear2(z) return y_pred, z model = SimpleNet().cuda() for epoch in range(1): model.train() for i, dt in enumerate(data.trn_dl): output = model(dt[0]) loss2 = 0 for j in range(0,len(output[0])): loss2 = loss2 + abs(torch.sqrt(l12+l22)-1) loss1 = F.mse_loss(output[0], dt[1]) loss = loss1+loss2 loss.backward() optimizer.step() if epoch%100==0: print(loss1,loss2,loss) So I need the gradient of the output layer with respect to some node (this is a simple example, the real one has more layers in between), which I calculate using torch.autograd.grad(output[0][j][0], output[1], create_graph=True)[0][j]. However the way I do it now requires that for loop, over each element of the batch which is very slow. Is there a way to take this gradient all at once for a batch? Thank you! l1 = torch.autograd.grad(output[0][0][j], ...) • So I tried this: outp = torch.t(output[0]) l1 = torch.autograd.grad(outp[0], output[1], create_graph=True)[0] But I still get the grad can be implicitly created only for scalar outputs error unless I use the loop over j. How exactly should I do what you suggested? Jun 9, 2019 at 1:52	2023-03-23 10:16:40	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8301399946212769, "perplexity": 3683.4313917382583}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945144.17/warc/CC-MAIN-20230323100829-20230323130829-00190.warc.gz"}
https://my-assignmentexpert.com/2022/02/25/%E7%BB%9F%E8%AE%A1%E4%BB%A3%E5%86%99naive-definition-of-probability-stat-%E4%BB%A3%E5%86%99/	19th Ave New York, NY 95822, USA # 统计代写\|Naive definition of probability stat 代写 ## 统计代考 Historically, the earliest definition of the probability of an event was to count the number of ways the event could happen and divide by the total number of possible outcomes for the experiment. We call this the naive definition since it is restrictive and relies on strong assumptions; nevertheless, it is important to understand, and useful when not misused. Definition 1.3.1 (Naive definition of probability). Let $A$ be an event for an experiment with a finite sample space $S$. The naive probability of $A$ is $$P_{\text {naive }}(A)=\frac{\|A\|}{\|S\|}=\frac{\text { number of outcomes favorable to } A}{\text { total number of outcomes in } S} .$$ (We use $\|A\|$ to denote the size of $A$; see Section A. $1.5$ of the math appendix.) In terms of Pebble World, the naive definition just says that the probability of $A$ is the fraction of pebbles that are in $A$. For example, in Figure $1.1$ it says $$P_{\text {naive }}(A)=\frac{5}{9}, P_{\text {naive }}(B)=\frac{4}{9}, P_{\text {naive }}(A \cup B)=\frac{8}{9}, P_{\text {naive }}(A \cap B)=\frac{1}{9} .$$ For the complements of the events just considered, $$P_{\text {naive }}\left(A^{c}\right)=\frac{4}{9}, P_{\text {naive }}\left(B^{c}\right)=\frac{5}{9}, P_{\text {naive }}\left((A \cup B)^{c}\right)=\frac{1}{9}, P_{\text {naive }}\left((A \cap B)^{c}\right)=\frac{8}{9} .$$ In general, $$P_{\text {naive }}\left(A^{c}\right)=\frac{\left\|A^{c}\right\|}{\|S\|}=\frac{\|S\|-\|A\|}{\|S\|}=1-\frac{\|A\|}{\|S\|}=1-P_{\text {naive }}(A)$$ In Section 1.6, we will see that this result about complements always holds for probability, even when we go beyond the naive definition. A good strategy when trying to find the probability of an event is to start by thinking about whether it will be easier to find the probability of the event or the probability of its complement. De Morgan’s laws are especially useful in this context, since it may be easier to work with an intersection than a union, or vice versa. The naive definition is very restrictive in that it requires $S$ to be finite, with equal mass for each pebble. It has often been misapplied by people who assume equally likely outcomes without justification and make arguments to the effect of “either it will happen or it won’t, and we don’t know which, so it’s 50-50”. In addition to sometimes giving absurd probabilities, this type of reasoning isn’t even internally consistent. For example, it would say that the probability of life on Mars is $1 / 2$ (“either there is or there isn’t life there”), but it would also say that the probability of intelligent life on Mars is $1 / 2$, and it is clear intuitively-and by the properties of probability developed in Section $1.6$-that the latter should have strictly lower probability than the former. But there are several important types of problems where the naive definition is applicable: • when there is symmetry in the problem that makes outcomes equally likely. It is common to assume that a coin has a $50 \%$ chance of landing Heads when tossed, due to the physical symmetry of the coin. ${ }^{1}$ For a standard, well-shuffled deck of cards, it is reasonable to assume that all orders are equally likely. There aren’t certain overeager cards that especially like to be near the top of the deck; any particular location in the deck is equally likely to house any of the 52 cards. • when the outcomes are equally likely by design. For example, consider conducting a survey of $n$ people in a population of $N$ people. A common goal is to obtain a simple random sample, which means that the $n$ people are chosen randomly with all subsets of size $n$ being equally likely. If successful, this ensures that the naive definition is applicable, but in practice this may be hard to accomplish because of various complications, such as not having a complete, accurate list of contact information for everyone in the population. ## 统计代考 $$P_{\text {naive }}(A)=\frac{\|A\|}{\|S\|}=\frac{\text { 有利于 } A}{\text { } S 中的结果总数} 。$$ （我们使用 $\|A\|$ 来表示 $A$ 的大小；参见数学附录的 A 节。$1.5$。）就 Pebble World 而言，朴素的定义只是说 $A$ 的概率是分数以 $A$ 为单位的鹅卵石。例如，在图 $1.1$ 中它说 $$P_{\text {naive }}(A)=\frac{5}{9}, P_{\text {naive }}(B)=\frac{4}{9}, P_{\text {naive }} (A \cup B)=\frac{8}{9}, P_{\text {naive }}(A \cap B)=\frac{1}{9} 。$$ $$P_{\text {naive }}\left(A^{c}\right)=\frac{4}{9}, P_{\text {naive }}\left(B^{c}\right)=\ frac{5}{9}, P_{\text {naive }}\left((A \cup B)^{c}\right)=\frac{1}{9}, P_{\text {naive }} \left((A \cap B)^{c}\right)=\frac{8}{9} 。$$ $$P_{\text {naive }}\left(A^{c}\right)=\frac{\left\|A^{c}\right\|}{\|S\|}=\frac{\|S\|-\|A \|}{\|S\|}=1-\frac{\|A\|}{\|S\|}=1-P_{\text {天真}}(A)$$ • 当问题中存在使结果同样可能的对称性时。由于硬币的物理对称性，通常假设硬币在被抛掷时有 50 美元 \%$的机会正面朝上。${ }^{1}$对于一副标准的、经过良好洗牌的纸牌，可以合理地假设所有订单的可能性相同。没有某些过度渴望的牌特别喜欢靠近牌组的顶部；牌组中的任何特定位置都有可能放置 52 张牌中的任何一张。 • 当结果在设计上同样可能时。例如，考虑在$N$人的人口中对$n$人进行调查。一个共同的目标是获得一个简单的随机样本，这意味着$n$人是随机选择的，所有大小为$n\$ 的子集的可能性相同。如果成功，这将确保幼稚的定义适用，但在实践中，由于各种复杂情况，这可能难以实现，例如没有针对人群中每个人的完整、准确的联系信息列表。	2023-01-30 08:52:57	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8394056558609009, "perplexity": 284.48120327659717}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499804.60/warc/CC-MAIN-20230130070411-20230130100411-00638.warc.gz"}
https://www.futurelearn.com/courses/maths-linear-quadratic/0/steps/12143	## Want to keep learning? This content is taken from the UNSW Sydney's online course, Maths for Humans: Linear, Quadratic & Inverse Relations. Join the course to learn more. 3.12 ## UNSW Sydney Hooke's law tells us how much a spring is extended if a weight is hung from it. # Hooke's law and the stiffness of springs Hooke’s law is a fundamental relation that explains how a weight on a spring stretches that spring. There is a fundamental direct proportionality here, with a constant of proportionality called the spring constant $$\normalsize{k}$$. However when we inquire as to the relation between $$\normalsize{k}$$ and the length of the spring, we find an inverse proportionality! Hooke’s law explains the oscillation of a spring and the connection with circular functions. In this step you will learn • about the mathematical form of Hooke’s law • how the spring constant changes with the length of the spring • how the oscillation of a spring is a model for harmonic motion. ## Hooke’s law: Restoring force is proportional to displacement In physics, the motion of a oscillating spring is an important oscillation, that yields the familiar cosine and sine circular functions as amplitudes depending on time. The origins of this motion are in a particularly simple linear relationship that springs have, up to a point, when you displace them. © “Hookes-law-springs” Svjo/Wikimedia Commons CC BY SA In the diagram, we see that doubling the force doubles the displacement. This is another way of stating Hooke’s law. This was discovered by the British physicist Robert Hooke in 1660, and stated as $\Large{F=kx}$ where $$\normalsize{F}$$ is the force required to create a displacement of $$\normalsize{x}$$ in the position of a spring. (Sometimes this law appears with a negative sign and the meaning of the force reversed.) It is only a valid law for relatively small values of $$\normalsize{x}$$, but in this range if you want to double the displacement, you need to double the force on the spring. In this case the constant of proportionality $$\normalsize{k}$$ depends on the stiffness of the spring. ## An example Suppose we have a spring which is naturally $$\normalsize 6$$ cm long when hanging freely. If we place a small weight of $$\normalsize 30$$ gm on the spring we notice that it stretches by $$\normalsize 2$$ mm. From this we can use a linear relation to predict what kind of extensions we get with other masses. If we double the mass to $$\normalsize 60$$ gm we expect the string to stretch twice as far, to $$\normalsize 4$$ mm. If on the other hand we replace the mass with $$\normalsize 10$$ gm, then we expect the extension to also reduce by to a third of what it was, namely to $$\normalsize 2/3$$ mm. Q1 (E): If a mass of $$\normalsize 10$$ kg on a heavy industrial spring of $$\normalsize 100$$ cm creates an extension of $$\normalsize 3$$ mm, then how much extension would be caused by $$\normalsize 15$$ kg? How much mass would we have to load onto the original spring to get an extension of $$\normalsize 5$$ mm? ## How does the spring constant depend on the length? But aren’t we discussing inverse relations? Hooke’s law is an example of a direct proportionality, as we discussed in week 1. But behind this law is another law which is a fine example of an inverse proportionality. Suppose we have a given spring with a given spring constant $$\normalsize{k}$$. What happens if we cut this spring into two equally sized pieces? One of these shorter springs will have a new spring constant, which will be $$\normalsize{2k}$$. More generally, the spring constant of a spring is inversely proportional to the length of the spring, assuming we are talking about a spring of a particular material and thickness. So suppose we cut the spring in the example above exactly in two, creating two shorter springs each of length $$\normalsize 3$$ cm. One of the smaller springs will have a spring constant which is twice the original. That is because the spring constant and the length of the spring are inversely proportional. That means that the original mass of $$\normalsize 30$$ gm will only yield a stretch of $$\normalsize 1$$ mm on the shorter spring. The larger the spring constant, the smaller the extension that a given force creates. Hopefully this makes intuitive sense — it should not be a surprise. If we think of the original spring as being two shorter springs attached together, then the mass of $$\normalsize 30$$ is stretching both smaller springs by $$\normalsize 1$$ mm, giving a total stretch of $$\normalsize 2$$ mm. For those who are physically inclined, with proper units in this case the force is $$\normalsize{F=0.3}$$ Newtons, and the original displacement is $$\normalsize{x=0.002}$$ m. So since $$\normalsize{F=kx}$$, in these units $$\normalsize k=\frac{0.3}{0.002}=1500$$ N/m. And a Newton/metre is really the same as a $$\normalsize \text{kilogram}/\text{sec}^2$$. Q2 (M): If a given force $$\normalsize{F}$$ on a spring creates a displacement of $$\normalsize{8}$$ cm, and then we cut this spring into three equal pieces, how much force should be applied to one of the pieces to create a displacement of $$\normalsize{4}$$ cm? ## Spring oscillation and harmonic motion The particularly simple relation between the restoring force and displacement in Hooke’s law has a lovely consequence for the motion of an oscillating spring. If you pull a weight on a spring down and let it go, then it will oscillate around its mean position in what is called harmonic motion. Rather surprisingly, this is exactly identical to another fundamental motion: the $$\normalsize{y}$$-coordinate of a particle which moves around a unit circle uniformly at a constant speed. If you look at these two examples, hopefully you will see the similarities in motions. The reason for this seeming coincidence has to do with differential equations, but it comes down to the relatively simple relation of Hooke’s law. A1. Since by Hooke’s law the extension is directly proportional to the force, which for a hanging weight is directly proportional to the mass, we argue that if a mass of $$\normalsize 10$$ kg creates an extension of $$\normalsize 3$$ mm, then a mass of $$\normalsize 15$$ kg would cause an extension of $$\normalsize 4.5$$ mm. To get an extension of $$\normalsize 5$$ mm, we would need a mass of m, where $\Large \frac{10}{m}=\frac{3}{5}.$ Solving we get $$\normalsize m=50/3=16.7$$ kg. A2. If the spring constant is originally $$\normalsize{k}$$ then by Hooke’s law the force applied is $$\normalsize{F=8k}$$. Now for one of the smaller pieces, we know the spring constant has tripled, to $$\normalsize k_1=3k$$. If the new force required is $$\normalsize{F_1}$$, then $$\normalsize{F_1=4k_1=4(3k)}$$. It follows that $\Large{\frac{F}{F_1}=\frac{8k}{12k}=\frac{2}{3}}$ and so $$\normalsize{F_1}$$ must be $$\normalsize{\frac{3}{2}}=1.5$$ times as large as $$\normalsize{F}$$.	2020-09-24 01:17:47	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7297284603118896, "perplexity": 312.65348094888134}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400213006.47/warc/CC-MAIN-20200924002749-20200924032749-00228.warc.gz"}
https://mathspp.com/blog/til/012	## TIL #012 – At operator for matrix multiplication Today I learned that Python 3.5+ supports the operator @ for matrix multiplication. # At operator @ Since Python 3.5, Python has the infix operator @. This operator was introduced with PEP 465 to be used in matrix multiplication. You can try to use it with just vanilla Python, but no vanilla Python types define their behaviour with @: >>> 3 @ 5 Traceback (most recent call last): File "<stdin>", line 1, in <module> TypeError: unsupported operand type(s) for @: 'int' and 'int' However, just looking at the error above, you see that the error is in @ not knowing what to do with integers. The error is not the fact that @ is an invalid operator! So cool! # Matrix multiplication in numpy with @ If you have numpy at hand, you can check @ works, because numpy arrays added support to be used with @: >>> import numpy as np >>> np.random.rand(3, 3) @ np.random.rand(3, 3) array([[0.89431673, 0.57949659, 0.59470797], [0.47364302, 0.29837518, 0.33552972], [1.12634752, 0.75218169, 0.78876082]]) >>> _ @ np.eye(3) # The identity (eye) matrix leaves the other matrix unchanged. array([[0.89431673, 0.57949659, 0.59470797], [0.47364302, 0.29837518, 0.33552972], [1.12634752, 0.75218169, 0.78876082]]) # Using @ with custom classes/types If you want your own objects to add support for @, all you have to do is implement the dunder methods __matmul__ and __rmatmul__: >>> class Dummy: ... def __matmul__(self, other): ... print("Works!") ... return 42 ... def __rmatmul__(self, other): ... print("Also works!") ... return 73 ... >>> d = Dummy() >>> d @ 1 Works! 42 >>> 1 @ d Also works! 73 There's also the __imatmul__ method for in-place matrix multiplication: >>> class Dummy: ... def __imatmul__(self, other): ... print("In-place!") ... >>> d = Dummy() >>> d @= 1 In-place! Of course, this silly example above doesn't show you the proper semantics of the __matmul__, __rmatmul__, and __imatmul__ methods. It just shows you they exist and they interact with the operator @! By the way, for reference, here is the tweet that showed me this: That's it for now! Stay tuned and I'll see you around! I hope you learned something new! If you did, consider following the footsteps of the readers who bought me a slice of pizza 🍕. Your small contribution helps me produce this content for free and without spamming you with annoying ads.	2022-12-08 02:56:15	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.437516987323761, "perplexity": 8218.176445420006}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711232.54/warc/CC-MAIN-20221208014204-20221208044204-00573.warc.gz"}
https://baincapitalcrypto.com/insights/?authors=667	### Abstract The rise of Ethereum and other blockchains that support smart contracts has led to the creation of decentralized exchanges (DEXs), such as Uniswap, Balancer, Curve, mStable, and SushiSwap, which enable agents to trade cryptocurrencies without trusting a centralized authority. While traditional exchanges use order books to match and execute trades, DEXs are typically organized as constant function market makers (CFMMs). CFMMs accept and reject proposed trades based on the evaluation of a function that depends on the proposed trade and the current reserves of the DEX. For trades that involve only two assets, CFMMs are easy to understand, via two functions that give the quantity of one asset that must be tendered to receive a given quantity of the other, and vice versa. When more than two assets are being exchanged, it is harder to understand the landscape of possible trades. We observe that various problems of choosing a multi-asset trade can be formulated as convex optimization problems, and can therefore be reliably and efficiently solved. ## 1. Introduction In the past few years, several new financial exchanges have been implemented on blockchains, which are distributed and permissionless ledgers replicated across networks of computers. These decentralized exchanges (DEXs) enable agents to trade cryptocurrencies, i.e., digital currencies with account balances stored on a blockchain, without relying on a trusted third party to facilitate the exchange. DEXs have significant capital flowing through them; the four largest DEXs on the Ethereum blockchain (Curve Finance , Uniswap , SushiSwap , and Balancer) have a collective trading volume of several billion dollars per day. Unlike traditional exchanges, DEXs typically do not use order books. Instead, most DEXs (including Curve, Uniswap, SushiSwap, and Balancer) are organized as constant function market makers (CFMMs). A CFMM holds reserves of assets (cryptocurrencies), contributed by liquidity providers. Agents can offer or tender baskets of assets to the CFMM, in exchange for another basket of assets. If the trade is accepted, the tendered basket is added to the reserves, while the basket received by the agent is subtracted from the reserves. Each accepted trade incurs a small fee, which is distributed pro-rata among the liquidity providers. CFMMs use a single rule that determines whether or not a proposed trade is accepted. The rule is based on evaluating a trading function, which depends on the proposed trade and the current reserves of the CFMM. A proposed trade is accepted if the value of the trading function at the post-trade reserves (with a small correction for the trading fee) equals the value at the current reserves, i.e., the function is held constant. This condition is what gives CFMMs their name. One simple example of a trading function is the product , implemented by Uniswap and SushiSwap ; this CFMM accepts a trade only if it leaves the product of the reserves unchanged. Several other functions can be used, such as the sum or the geometric mean (which is used by Balancer ). For trades involving just two assets, CFMMs are very simple to understand, via a scalar function that relates how much of one asset is required to receive an amount of the other, and vice versa. Thus the choice of a two-asset trade involves only one scalar quantity: how much you propose to tender (or, equivalently, how much you propose to receive). In all practical cases, including the ones mentioned above, the trading function is concave . In this paper we make use of this fact to formulate various multi-asset trading problems as convex optimization problems. Because convex optimization problems can be solved reliably and efficiently (in theory and in practice) , we can solve the formulated trading problems exactly. This gives a practical solution to the problem of choosing among many possible multi-asset trades: the trader articulates their objective and constraints, and a solution to this problem determines the baskets of assets to be tendered and received. Outline. We start by surveying related work in §1.1. In §2, we give a complete description of CFMMs, describing how agents may trade with a CFMM, as well as add or remove liquidity. In §3 we study some basic properties of CFMMs, many of which rely on the concavity of the trading function. In §4 we examine trades involving just two assets, and show how to understand them via two functions that give the amount of asset received for a given quantity of the tendered asset. Finally, in §5 we formulate the general multi-asset trading problem as a convex optimization problem, and give some specific examples. Blockchain. CFMMs are typically implemented on a blockchain: a decentralized, permissionless, and public ledger. The blockchain stores accounts, represented by cryptographic public keys, and associated balances of one or more cryptocurrencies. A blockchain allows any two accounts to securely transact with each other without the need for a trusted third party or central institution, using public-key cryptography to verify their identities. Executing a transaction, which alters the state of the blockchain, costs the issuer a fee, typically paid out to the individuals providing computational power to the network. (This network fee depends on the amount of computation a transaction requires and is paid in addition to the CFMM trading fee mentioned above and described below.) Blockchains are highly tamper resistant: they are replicated across a network of computers, and kept in consensus via simple protocols that prevent invalid transactions such as double-spending of a coin. The consensus protocol operates on the level of blocks (bundles of transactions), which are verified by the network and chained together to form the ledger. Because the ledger is public, anyone in the world can view and verify all account balances and the entire record of transactions. The idea of a blockchain originated with a pseudonymously authored whitepaper that proposed Bitcoin, widely considered to be the first cryptocurrency . Cryptocurrencies. A cryptocurrency is a digital currency implemented on a blockchain. Every blockchain has its own native cryptocurrency, which is used to pay the network transaction fees (and can also be used as a standalone currency). A given blockchain may have several other cryptocurrencies implemented on it. These additional currencies are sometimes called tokens, to distinguish them from the base currency. There are thousands of tokens in circulation today, across various blockchains. Some, like the Uniswap token UNI, give holders rights over the governance of a protocol, while others, like USDC, are stablecoins, pegged to the market value of some external or real-world currency or commodity. Smart contracts. Modern blockchains, such as Ethereum , Polkadot , and Solana , allow anyone to deploy arbitrary stateful programs called smart contracts. A contract’s public functions can be invoked by anyone, via a transaction sent through the network and addressed to the contract. (The term ‘smart contract’ was coined in the 1990s, to refer to a set of promises between agents codified in a computer program .) Because creators are free to compose deployed contracts or remix them in their own applications, software ecosystems on these blockchains have developed rapidly. CFMMs are implemented using smart contracts, with functions for trading, adding liquidity, and removing liquidity. Their implementations are usually simple. For example, Uniswap v2 is implemented in just 200 lines of code. In addition to DEXs, many other financial applications have been deployed on blockchains, including lending protocols (e.g., ) and various derivatives (e.g., ). The collection of financial applications running on blockchains is known as decentralized finance, or DeFi for short. CFMMs have some similarities to exchange-traded funds (ETFs). A CFMM’s liquidity providers are analogous to an ETF’s authorized participants; adding liquidity to a CFMM is analogous to the creation of an ETF share, and subsequently removing liquidity is analogous to redemption. But while the list of authorized participants for an ETF is typically very small, anyone in the world can provide liquidity to a CFMM or trade with it. Comparison to order books. Another advantage of CFMMs over order book exchanges is their efficiency of storage, since they do not need to store and maintain a limit order book, and their computational efficiency, since they only need to evaluate the trading function. Because users must pay for computation costs for each transaction, and these costs can often be nonnegligible in some blockchains, exchanges implementing CFMMs can often be much cheaper for users to interact with than those implementing order books. Previous work. Academic work on automated market makers began with the study of scoring rules within the statistics literature, e.g., . Scoring rules furnish probabilities for baskets of events, which can be viewed as assets or tokens in a prediction market. The output probability from a scoring rule was first proposed as a pricing mechanism for a binary option (such as a prediction market) in . Unlike CFMMs, these early automated market makers were shown to be computationally complicated for users to interact with. For example. Chen demonstrated that computing optimal arbitrage portfolios in logarithmic scoring rules (the most popular class of scoring rules) is #P-hard. The first CFMM on Ethereum (the most commonly used blockchain for smart contracts) was Uniswap . The first formal analysis of Uniswap was first done in and extended to general concave trading functions in . Evans first proved that constant mean market makers could replicate a large set of portfolio value functions. The converse result was later proven, providing a mechanism for constructing a trading function that replicates a given portfolio value function . Analyses of how fees and trading function curvature affect liquidity provider returns are also common in the literature. Finally, we note that there exist investigations of privacy in CFMMs , suitability of liquidity provider shares as a collateral asset , and the question of triangular arbitrage in CFMMs. ### 1.2. Convex analysis and optimization Convex analysis. A function f : D \to {{\bf R}}, with D \subseteq {{\bf R}}^n, is convex if D is a convex set and f(\theta x + (1 - \theta)y) \leq \theta f(x) + (1 - \theta)f(y), for 0 \le \theta \le 1 and all x,y \in D. It is common to extend a convex function to an extended-valued function that maps {{\bf R}}^n to {{\bf R}}\cup \{\infty\}, with f(x)=+\infty for x \not\in D. A function f is concave if -f is convex , Chap. 3]. When f is differentiable, an equivalent characterization of convexity is f(z) \geq f(x)+ \nabla f(x)^T(z-x), for all z,x \in D. A differentiable function f is concave if and only if for all z,x\in D we have (1)f(z) \leq f(x)+ \nabla f(x)^T(z-x). The right hand side of this inequality is the first-order Taylor approximation of the function f at x, so this inequality states that for a concave function, the Taylor approximation is a global upper bound on the function. By adding (1) and the same inequality with x and z swapped, we obtain the inequality (2)(\nabla f(z)-\nabla f(x))^T (z-x) \leq 0, valid for any concave f and z,x\in D. This inequality states that for a concave function f, -\nabla f is a monotone operator . Convex optimization. A convex optimization problem has the form \begin{array}{ll} {minimize} & f_0(x) \\ {subject to} & f_i(x) \leq 0, \quad i=1, \ldots, m\\ & g_i(x) = 0, \quad i=1, \ldots, p, \end{array} where x \in {{\bf R}}^n is the optimization variable, the objective function f_0 : D \to {{\bf R}} and inequality constraint functions f_i : D \to {{\bf R}} are convex, and the equality constraint functions g_i : {{\bf R}}^n \to {{\bf R}} are affine, i.e., have the form g_i(x) = a_i^T x + b_i for some a_i \in {{\bf R}}^n and b_i \in {{\bf R}}. (We assume the domains of the objective and inequality functions are the same for simplicity.) The goal is to find a solution of the problem, which is a value of x that minimizes the objective function, among all x satisfying the constraints f_i(x) \leq 0, i=1, \ldots, m, and g_i(x) = 0, i=1, \ldots, p , Chap. 4]. In the sequel we will refer to the problem of maximizing a concave function, subject to convex inequality constraints and affine equality constraints, as a convex optimization problem, since this problem is equivalent to minimizing -f_0 subject to the constraints. Convex optimization problems are notable because they have many applications, in a wide variety of fields, and because they can be solved reliably and efficiently . The list of applications of convex optimization is large and still growing. It has applications in vehicle control , finance , dynamic energy management , resource allocation , machine learning , inverse design of physical systems , circuit design , and many other fields. In practice, once a problem is formulated as a convex optimization problem, we can use off-the-shelf solvers (software implementations of numerical algorithms) to obtain solutions. Several solvers, such as OSQP , SCS , ECOS , and COSMO , are free and open source, while others, like MOSEK , are commercial. These solvers can handle problems with thousands of variables in seconds or less, and millions of variables in minutes. Small to medium-size problems can be solved extremely quickly using embedded solvers or code generation tools . For example, the aerospace and space transportation company SpaceX uses CVXGEN to solve convex optimization problems in real-time when landing the first stages of its rockets . Domain-specific languages for convex optimization. Convex optimization problems are often specified using domain-specific languages (DSLs) for convex optimization, such as CVXPY or JuMP , which compile high-level descriptions of problems into low-level standard forms required by solvers. The DSL then invokes a solver and retrieves a solution on the user’s behalf. DSLs vastly reduce the engineering effort required to get started with convex optimization, and in many cases are fast enough to be used in production. Using such DSLs, the convex optimization problems that we describe later can all be implemented in just a few lines of code that very closely parallel the mathematical specification of the problems. ## 2. Constant function market makers In this section we describe how CFMMs work. We consider a DEX with n>1 assets, labeled 1, \ldots, n, that implements a CFMM. Asset n is our numeraire, the asset we use to value and assign prices to the others. ### 2.1. CFMM state Reserve or pool. The DEX has some reserves of available assets, given by the vector R \in {{\bf R}}_+^n, where R_i is the quantity of asset i in the reserves. Liquidity provider share weights. The DEX maintains a table of all the liquidity providers, agents who have contributed assets to the reserves. The table includes weights representing the fraction of the reserves each liquidity provider has a claim to. We denote these weights as v_1, \ldots, v_N, where N is the number of liquidity providers. The weights are nonnegative and sum to one, i.e., v \geq 0, and \sum_{i=1}^N v_i =1. The weights v_i and the number of liquidity providers N can change over time, with addition of new liquidity providers, or the deletion from the table of any liquidity provider whose weight is zero. State of the CFMM. The reserves R and liquidity provider weights v constitute the state of the DEX. The DEX state changes over time due to any of the three possible transactions: a trade (or exchange), adding liquidity, or removing liquidity. These transactions are described in §2.2 and §2.6. The proposed trade can either be rejected by the DEX, in which case its state does not change, or accepted, in which case the basket \Delta is transfered from the trader to the DEX, and the basket \Lambda is transfered from the DEX to the trader. The DEX reserves are updated as (3)R^+ = R + \Delta - \Lambda, where R^+ denotes the new reserves. A proposed trade is accepted or rejected based on a simple condition described in §2.3, which always ensures that R^+ \geq 0. Intuition suggests that a trade would not include an asset in both the proposed tender and receive baskets, i.e., we should not have \Delta_i and \Lambda_i both positive. We will see later that while it is possible to include an asset in both baskets, it never makes sense to do so. This means that \Delta and \Lambda can be assumed to have disjoint support, i.e., we have \Delta_i \Lambda_i = 0 for each i. This allows us to define two disjoint sets of assets associated with a proposed or accepted trade: \mathcal T = \{ i \mid \Delta_i >0 \}, \qquad \mathcal R = \{ i \mid \Lambda_i >0 \}. Thus \mathcal T are the indices of assets the trader proposes to give to the DEX, in exchange for the assets with indices in \mathcal R. If j \not\in \mathcal T \cup \mathcal R, it means that the proposed trade does not involve asset j, i.e., \Delta_j = \Lambda_j =0. A very common type of proposed trade involves only two assets, one that is tendered and one that is received, i.e., \|\mathcal T\| =\|\mathcal R\|=1. Suppose \mathcal T= \{i\} and \mathcal R=\{j\}, with i\neq j. Then we have \Delta = \delta e_i and \Lambda = \lambda e_j, where e_i denotes the ith unit vector, and \lambda \geq 0 is the quantity of asset j the trader wishes to receive in exchange for the quantity \delta \geq 0 of asset i. (This is referred to as exchanging asset i for asset j.) When a trade involves more than two assets, it is called a multi-asset trade. We will study two-asset and multi-asset trades in §4 and §5, respectively. Trade acceptance depends on both the proposed trade and the current reserves. A proposed trade (\Delta,\Lambda) is accepted only if (4)\varphi(R+\gamma\Delta-\Lambda) = \varphi(R), where \varphi: {{\bf R}}_+^n \to {{\bf R}} is the trading function associated with the CFMM, and the parameter \gamma \in (0,1] introduces a trading fee (when \gamma<1). The ‘constant function’ in the name CFMM refers to the acceptance condition (4). We can interpret the trade acceptance condition as follows. If \gamma =1, a proposed trade is accepted only if the quantity \varphi(R) does not change, i.e., \varphi(R^+)=\varphi(R). When \gamma <1 (with typical values being very close to one), the proposed trade is accepted based on the devalued tendered basket \gamma \Delta. The reserves, however, are updated based on the full tendered basket \Delta as in (3). Properties. We will assume that the trading function \varphi is concave, increasing, and differentiable. Many existing CFMMs are associated with functions that satisfy the additional property of homogeneity, i.e., \varphi(\alpha R) = \alpha \varphi(R) for \alpha>0. We mention some trading functions that are used in existing CFMMs. Linear and sum. The simplest trading function is linear, \varphi(R) = p^TR = p_1R_1+ \cdots + p_nR_n, with p>0, where p_i can be interpreted as the price of asset i. The trading condition (4) simplifies to \gamma p^T \Delta = p^T \Lambda. We interpret the righthand side as the total value of received basket, at the prices given by p, and the lefthand side as the value of the tendered basket, discounted by the factor \gamma. A CFMM with p= \mathbf 1, i.e., all asset prices equal to one, is called a constant sum market maker. The CFMM mStable, which held assets that were each pegged to the same currency, was one of the earliest constant sum market makers. Geometric mean. Another choice of trading function is the (weighted) geometric mean, \varphi(R) = \prod_{i=1}^n R_i^{w_i}, where total w>0 and \mathbf 1^T w=1. Like the linear and sum trading functions, the geometric mean is homogeneous. CFMMs that use the geometric mean are called constant mean market makers. The CFMMs Balancer , Uniswap and SushiSwap are examples of constant mean market makers. (Uniswap and SushiSwap use weights w_i = 1/n, and are sometimes called constant product market makers .) Other examples. Another example combines the sum and geometric mean functions, \varphi(R) = (1 - \alpha) \mathbf 1^TR + \alpha \prod_{i=1}^n R_i^{w_i}, where \alpha \in [0,1] is a parameter, w \geq 0, and \mathbf 1^T w =1. This trading function yields a CFMM that interpolates between a constant sum market (when \alpha=0) and a constant geometric mean market (when \alpha=1). Because it is a convex combination of the sum and geometric mean functions, which are themselves homogeneous, the resulting function is also homogeneous. The CFMM known as Curve uses the closely related trading function \varphi(R) = \mathbf 1^TR - \alpha \prod_{i=1}^n R_i^{-1}, where \alpha > 0. Unlike the previous examples, this trading function is not homogeneous. ### 2.5. Prices and exchange rates In this section we introduce the concept of asset (reported) prices, based on a first order approximation of the trade acceptance condition (4). These prices inform how liquidity can be added and removed from the CFMM, as we will see in §2.6. Unscaled prices. We denote the gradient of the trading function as P=\nabla \varphi(R). We refer to P, which has positive entries since \varphi is increasing, as the vector of unscaled prices, (5)P_i = \nabla \varphi(R)_i = \frac{\partial \varphi}{\partial R_i}(R), \quad i=1,\ldots, n. To see why these numbers can be interpreted as prices, we approximate the exchange acceptance condition (4) using its first order Taylor approximation to get 0 = \varphi(R+\gamma \Delta - \Lambda) - \varphi(R) \approx \nabla \varphi(R) ^T (\gamma \Delta - \Lambda)= P^T(\gamma \Delta - \Lambda), when \gamma \Delta - \Lambda is small, relative to R. We can express this approximation as (6)\gamma \sum_{i \in \mathcal T} P_i \Delta_i \approx \sum_{i \in \mathcal R} P_i \Lambda_i. The righthand side is the value of the received basket using the unscaled prices P_i. The lefthand side is the value of the tendered basket using the unscaled prices P_i, discounted by the factor \gamma. Prices. The condition (6) is homogeneous in the prices, i.e., it is the same condition if we scale all prices by any positive constant. The reported prices (or just prices) of the assets are the prices relative to the price of the numeraire, which is asset n. The prices are p_i = \frac{P_i}{P_n}, \quad i=1, \dots, n. (The price of the numeraire is always 1.) In general the prices depend on the reserves R. (The one exception is with a linear trading function, in which the prices are constant.) In terms of prices, the condition (6) is (7)\gamma \sum_{i \in \mathcal T} p_i \Delta_i \approx \sum_{i \in \mathcal R} p_i \Lambda_i. We observe for future use that the prices for two values of the reserves R and \tilde R are the same if and only if (8)\nabla \varphi(\tilde R) = \alpha \nabla \varphi(R), for some \alpha >0. For the special case \varphi(R)=\prod_{i=1}^n R_i^{w_i}, with w_i >0 and \sum_{i=1}^nw_i=1, the unscaled prices are P = \nabla\varphi(R) = \varphi(R) (w_1R_1^{-1}, w_2R_2^{-1}, \dots, w_nR_n^{-1}), and the prices are (9)p_i = \frac{w_iR_n}{w_nR_i}, \quad i=1, \ldots, n. Exchange rates. In a two-asset trade with \Delta= \delta e_i and \Lambda = \lambda e_j, i.e., we are exchanging asset i for asset j, the exchange rate is E_{ij} = \gamma \frac{\nabla \varphi(R)_i}{\nabla \varphi(R)_j} = \gamma \frac{P_i}{P_j} = \gamma \frac{p_i}{p_j}. This is approximately how much asset j you get for each unit of asset i, for a small trade. Note that E_{ij}E_{ji} = \gamma^2 < 1, when \gamma<1, i.e., round-trip trades lose value. These are first order approximations. We remind the reader that the various conditions described above are based on a first order Taylor approximation of the trade acceptance condition. A proposed trade that satisfies (7) is not (quite) valid; it is merely close to valid when the proposed trade baskets are small compared to the reserves. This is similar to the midpoint price (average of bid and ask prices) in an order book; you cannot trade in either direction exactly at this price. Reserve value. The value of the reserves (using the prices p) is given by (10)V = p^T R = \frac{\nabla \varphi(R)^TR}{\nabla \varphi(R)_n}. When \varphi is homogeneous we can use the identity \nabla\varphi(R)^T R =\varphi(R) to express the reserves value as (11)V = p^T R = \frac{\varphi(R)}{\nabla \varphi(R)_n}. ### 2.6. Adding and removing liquidity In this section we describe how agents called liquidity providers can add or remove liquidity from the reserves. When an agent adds liquidity, she adds a basket \Psi \in {{\bf R}}_+^n to the reserves, resulting in the updated reserves R^+ = R+\Psi. When an agent removes liquidity, she removes a basket \Psi \in {{\bf R}}_+^n from the reserves, resulting in the updated reserves R^+ = R-\Psi. (We will see below that the condition for removing liquidity ensures that R^+ \geq 0.) Adding or removing liquidity also updates the liquidity provider share weights, as described below. Liquidity change condition. Adding or removing liquidity must be done in a way that preserves the asset prices. Using (8), this means we must have (12)\nabla \varphi(R^+) = \alpha \nabla \varphi(R), for some \alpha>0. (We will see later that \alpha>1 corresponds to removing liquidity, and \alpha<1 corresponds to adding liquidity.) This liquidity change condition is analogous to the trade exchange condition (4). We refer to \Psi as a valid liquidity change if this condition holds. The liquidity change condition (12) simplifies in some cases. For example, with a linear trading function the prices are constant, so any basket can be used to add liquidity, and any basket with \Psi \leq R can be removed. (The constraint comes from the requirement R^+ \geq 0, the domain of \varphi.) Liquidity change condition for homogeneous trading function. Another simplification occurs when the trading function is homogeneous. For this case we have, for any \alpha >0, \nabla \varphi(\alpha R)= \nabla \varphi(R), (by taking the gradient of \varphi(\alpha R) = \alpha \varphi(R) with respect to R). This means that \Psi = \nu R, for \nu >0, is a valid liquidity change (provided \nu \leq 1 for liquidity removal). In words: you can add or remove liquidity by adding or removing a basket proportional to the current reserves. Liquidity provider share update. Let V = p^T R denote the value of the reserves before the liquidity change, and V^+=(p^+)^T R^+ = p^T R^+ the value after. The change in reserve value is V^+-V= p^T \Psi when adding liquidity, and V^+- V= -p^T\Psi when removing liquidity. Equivalently, p^T\Psi is the value of the basket a liquidity provider gives, when adding liquidity, or receives when removing liquidity. The fractional change in reserve value is (V^+-V)/V^+. When liquidity provider j adds or removes liquidity, all the share weights are adjusted pro-rata based on the change of value of the reserves, which is the value of the basket she adds or removes. The weights are adjusted to (13) v_i^+ = \begin{cases} v_i V/V^+ + (V^+ - V)/V^+ & i= j\\ v_iV/V^+ & i \ne j. \end{cases} Thus the weight of liquidity provider j is increased (decreased) by the fractional change in reserve value when she adds (removes) liquidity. These new weights are also nonnegative and sum to one. When \varphi is homogeneous and we add liquidity with the basket \Psi = \nu R, with \nu >0, we have V_+ = (1+\nu)p^TR, so V/V^+ = 1/(1+\nu), \qquad (V^+-V)/V^+ = \nu / (1+\nu). The weight updates for adding liquidity \Psi= \nu R are then v_i^+ = \begin{cases} (v_i+\nu)/(1+\nu) & i=j \\ v_i/(1+\nu) & i\neq j. \end{cases} For removing liquidity with the basket \Psi = \nu R, we replace \nu with -\nu in the formulas above, along with the constraint \nu \le v_j. ### 2.7. Agents interacting with CFMMs Agents seeking to trade or add or remove liquidity make proposals. These proposals are accepted or not, depending on the acceptance conditions given above. A proposal can be rejected if another agent’s proposed action is accepted (processed) before their proposed action, thus changing R and invalidating the acceptance condition. Slippage thresholds. One practical and common approach to mitigating this problem during trading is to allow agents to set a slippage threshold on the received basket. This slippage threshold, represented as some percentage 0 \le \eta \le 1, is simply a parameter that specifies how much slippage the agent is willing to tolerate without their trade failing. In this case, the agent presents some trade (\Delta, \Lambda) along with a threshold \eta, and the contract accepts the trade if there is some number \alpha satisfying \eta \le \alpha such that the trade (\Delta, \alpha\Lambda) can be accepted. In other words, the agent allows the contract to devalue the output basket by at most a factor of \eta. If no such value of \alpha exists, the trade fails. Maximal liquidity amounts. While setting slippage thresholds can help with reducing the risk of trades failing, another possible failure mode can occur during the addition of liquidity. A simple solution to this problem is that the liquidity provider specifies some basket \Psi to the CFMM contract, and the contract accepts the largest possible basket \Psi^- such that \Psi^- \le \Psi, returning the remaining amount, \Psi - \Psi^-, to the liquidity provider. In other words, \Psi can be seen as the maximal amount of liquidity a user is willing to provide. ## 3. Properties In this section we present some basic properties of CFMMs. Non-uniqueness. If we replace the trading function \varphi with \tilde \varphi= h\circ \varphi, where h is concave, increasing, and differentiable, we obtain another concave increasing differentiable function. The associated CFMM has the same trade acceptance condition, the same prices, the same liquidity change condition, and the same liquidity provider share updates as the original CFMM. Any valid trade satisfies \varphi(R+\gamma \Delta-\Lambda) = \varphi(R), so in particular R+\gamma \Delta -\Lambda \geq 0. Since we assume \Delta and \Lambda have non-overlapping support, it follows that \Lambda \le R. A valid trade cannot ask to receive more than is in the reserves. Here we show why a valid proposed trade with \Delta_k>0 and \Lambda_k>0 for some k does not make sense when \gamma<1, justifying our assumption that this never happens. Let (\tilde \Delta,\tilde \Lambda) be a proposed trade which coincides with (\Delta,\Lambda) except in the kth components, which we set to \tilde \Delta_k = \Delta_k - \tau/\gamma, \qquad \tilde \Lambda_k = \Lambda_k - \tau, where \tau = \min\{\gamma \Delta_k,\Lambda_k\} >0. Evidently \tilde \Delta \geq 0, \tilde \Lambda \geq 0, and R+\gamma \Delta - \Lambda = R+\gamma \tilde \Delta - \tilde \Lambda, so the proposed trade (\tilde \Delta,\tilde \Lambda) is also valid. If the trader proposes this trade instead of (\Delta,\Lambda), the net change in her assets is \tilde \Lambda- \tilde \Delta = \Lambda -\Delta + \left(\frac{1}{\gamma}-1\right) \tau e_k. The last vector on the right is zero in all entries except k, and positive in that entry. Thus the valid proposed trade (\tilde \Delta,\tilde \Lambda) has the same net effect as the trade (\Delta,\Lambda), except that the trader ends up with a positive amount more of the kth asset. Assuming the kth asset has value, we would always prefer this. For an accepted nonzero trade, we have \varphi(R^+) =\varphi(R+\Delta-\Lambda) > \varphi(R+\gamma\Delta- \Lambda) = \varphi(R), since \varphi is increasing and R+\Delta-\Lambda \ge R+\gamma\Delta- \Lambda, with at least one entry being strictly greater, whenever \gamma < 1. We can derive a stronger inequality using concavity of \varphi, which implies that \varphi(R+\gamma \Delta-\Lambda) \leq \varphi(R+\Delta-\Lambda) + (\gamma-1) \nabla \varphi(R+\Delta-\Lambda)^T \Delta. This can be re-arranged as \varphi(R^+) \geq \varphi(R) + (1-\gamma) (P^+)^T \Delta, where P^+ = \nabla \varphi(R^+) are the unscaled prices at the reserves R^+. This tells us the function value increases at least by (1-\gamma) times the value of tendered basket at the unscaled prices. Suppose (\Delta, \Lambda) is a valid trade. The net change in the trader’s holdings is \Lambda-\Delta. We can interpret \delta = p^T (\Delta-\Lambda) as the decrease in value of the trader’s holdings due to the proposed trade, evaluated at the current prices. We can interpret \delta as a trading cost, evaluated at the pre-trade prices, and now show it is positive. Since \varphi is concave, we have \varphi(R+\gamma \Delta -\Lambda) \leq \varphi(R) + \nabla \varphi(R)^T(\gamma \Delta - \Lambda). Using \varphi(R+\gamma \Delta -\Lambda) = \varphi(R), this implies 0 \leq \nabla \varphi(R)^T (\gamma \Delta - \Lambda) = P^T(\gamma \Delta-\Lambda). %\sum_{i=1}^n P_i (\gamma \Delta_i - \Lambda_i). From this we obtain P^T(\Delta-\Lambda) = P^T(\gamma \Delta-\Lambda) + (1-\gamma) P^T \Delta \geq (1-\gamma) P^T \Delta . Dividing by P_n gives \delta \geq (1-\gamma) p^T \Delta. Thus the trading cost is always at least a factor (1-\gamma) of p^T\Delta, the total value of the tendered basket. The trading cost \delta is also the increase in the total reserve value, at the current prices. So we can say that each trade increases the total reserve value, at the current prices, by at least (1-\gamma) times the value of the tendered basket. ### 3.2. Properties of liquidity changes Liquidity change condition interpretation. One natural interpretation of the liquidity change condition (12) is in terms of a simple optimization problem. We seek a basket \Psi that maximizes the post-change trading function value subject to a given total value of the basket at the current prices, (14)\begin{aligned} & \text{maximize} && \varphi(R^+)\\ & \text{subject to} && p^T (R^+-R) \le M. \end{aligned} Here the optimization variable is R^+\in {{\bf R}}_+^n, and M is the desired value of the basket \Psi at the current prices, for adding liquidity, or its negative, for removing liquidity. The optimality conditions for this convex optimization problem are p^T(R^+-R)\le M, \qquad \nabla \varphi(R^+) - \nu p =0, where \nu \ge 0 is a Lagrange multiplier. Using p = \nabla \varphi(R)/\nabla \varphi(R)_n, the second condition is \nabla \varphi(R^+) = \frac{\nu}{\nabla \varphi(R)_n} \nabla \varphi(R), which is (12) with \alpha = \nu/ \nabla \varphi(R)_n. We can easily recover the trading basket \Psi from R^+ since \Psi = R^+ - R. Liquidity provision problem. When the trading function is homogeneous, it is easy to understand what baskets can be used to add or remove liquidity: they must be proportional to the current reserves. In other cases, it can be difficult to find an R^+ that satisfies (12). In the general case, however, the convex optimization problem (14) can be solved to find the basket \Psi that gives a valid liquidity change, with M denoting the total value of the added basket (when M>0) or removed basket (when M<0). Liquidity change and the gradient scale factor \alpha. Suppose that we add or remove liquidity. Since \varphi is concave (2) tells us that (\nabla \varphi(R^+)-\nabla \varphi(R))^T (R^+-R) \leq 0. Using \nabla \varphi(R^+) = \alpha \nabla \varphi(R), this becomes (\alpha-1) \nabla \varphi(R)^T (R^+-R) \leq 0. We have \nabla \varphi(R)>0. If we add liquidity, we have R^+ - R \geq 0 and R^+-R \neq 0, so \nabla \varphi(R)^T (R^+-R) >0. From the inequality above we conclude that \alpha<1. If we remove liquidity, a similar arguments tells us that \alpha>1. Two-asset trades, sometimes called swaps, are some of the most common types of trades performed on DEXs. In this section, we show a number of interesting properties of trades in this common special case. ### 4.1. Exchange functions Suppose we exchange asset i for asset j, so \Delta = \delta e_i and \Lambda = \lambda e_j, with \delta\geq 0, \lambda \geq 0. The trade acceptance condition (4) is (15)\varphi(R+\gamma \delta e_i - \lambda e_j)=\varphi(R). The lefthand side is increasing in \delta and decreasing in \lambda, so for each value of \delta there is at most one valid value of \lambda, and for each value of \lambda, there is at most one valid value of \delta. In other words, the relation (15) between \lambda and \gamma defines a one-to-one function. This means that two-asset trades are characterized by a single parameter, either \delta (how much is tendered) or \lambda (how much is received). Forward exchange function. Define F: {{\bf R}}_+ \to {{\bf R}}, where F(\delta) is the unique \lambda that satisfies (15). The function F is called the forward exchange function, since F(\delta) is how much of asset j you get if you exchange \delta of asset i. The forward exchange function F is increasing since \varphi is componentwise increasing and nonnegative since F(0) = 0. We will now show that the function F is concave. Concavity. Using the implicit function theorem on (15) with \lambda = F(\delta), we obtain (16)F'(\delta) = \gamma\frac{\nabla\varphi(R')_i}{\nabla\varphi(R')_j}, where we use R' = R + \gamma\delta e_i - F(\delta) e_j to simplify notation. To show that F is concave, we will show that, for any nonnegative trade amounts \delta, \delta' \ge 0, the function F satisfies (17)F(\delta') \le F'(\delta)(\delta' - \delta) + F(\delta), which establishes that F is concave. We write R'' = R + \gamma \delta' e_i - F(\delta')e_j, and note that \varphi(R) = \varphi(R') = \varphi(R'') from the definition of F. Since \varphi is concave it satisfies \varphi(R'') \le \nabla \varphi(R')^T(R'' - R') + \varphi(R'), so \nabla \varphi(R')^T(R'' - R') \ge 0. Using the definitions of R'' and R', we have 0 \le \gamma(\delta' - \delta)\nabla \varphi(R')_i - (F(\delta') - F(\delta))\nabla\varphi(R')_j. Dividing by \nabla\varphi(R')_j and using (16), we obtain (17). Reverse exchange function. Define G: {{\bf R}}_+ \to {{\bf R}}\cup \{\infty\}, where G(\lambda) is the unique \delta that satisfies (15), or G(\lambda)=\infty is there is no such \delta. The function F is called the reverse exchange function, since F(\lambda) is how much of asset i you must exchange, to receive \lambda of asset j. In a similar way to the forward trade function, the reverse exchange function is nonnegative and increasing, but this function is convex rather than concave. (This follows from a nearly identical proof.) Forward and reverse exchange functions are inverses. The forward and reverse exchange functions are inverses of each other, i.e., they satisfy G(F(\delta)) = \delta, \qquad F(G(\lambda)) = \lambda, when both functions are finite. Analogous functions for a limit order book market. There are analogous functions in a market that uses a limit order book. They are piecewise linear, where the slopes are the different prices of each order, while the distance between the kink points is equal to the size of each order. The associated functions have the same properties, i.e., they are increasing, inverses of each other, F is concave, and G is convex. Evaluating F and G. In some important special cases, we can express the functions F and G in a closed form. For example, when the trading function is the sum function, they are F(\delta) = \min\{\gamma\delta, R_j\}, \qquad G(\lambda) = \begin{cases} \lambda/\gamma & \lambda/\gamma \le R_j\\ +\infty & \text{otherwise}. \end{cases} When the trading function is the geometric mean, the functions are F(\delta) = R_j \left(1-\frac{R_i^{w_i/w_j}}{(R_i + \gamma \delta)^{w_i/w_j}}\right), \qquad G(\lambda) = \frac{R_i}{\gamma}\left(\frac{R_j^{w_j/w_i}}{(R_j - \lambda)^{w_j/w_i}} - 1\right), whenever \lambda < R_j, and G(\lambda) = \infty otherwise. On the other hand, when the forward and reverse trading functions F and G cannot be expressed analytically, we can use several methods to evaluate them numerically . To evaluate F(\delta), we fix \delta and solve for \lambda in (15). The lefthand side is a decreasing function of \lambda, so we can use simple bisection to solve this nonlinear equation. Newton’s method can be used to achieve higher accuracy with fewer steps. Exploiting the concavity of \varphi, it can be shown an undamped Newton iteration always converges to the solution. With superscripts denoting iteration, this is \lambda^{k+1} = \lambda^k + \frac{\varphi(R+\gamma \delta e_i - \lambda^k e_j) - \varphi(R)} {\nabla \varphi(R+\gamma \delta e_i - \lambda^k e_j)_j}, with starting point based on the exchange rate, \lambda^0 = \delta E_{ij} = \delta\frac{\gamma p_i}{p_j}. (It can be shown that the convergence is monotone decreasing.) We note that one of the largest CFMMs, Curve, uses a trading function that is not homogeneous and uses this method in production . Figure 1: Left. Forward exchange functions for two values of the reserves. Right. Reverse exchange functions for the same two values of the reserves. Slope at zero. Using (16), we see that F'(0^+) = E_{ij}, i.e., the one-sided derivative at 0 is exactly the exchange rate for assets i and j. Since F is concave, we have (18)F(\delta) \leq F'(0^+)\delta = E_{ij} \delta. This tells us that the amount of asset j you will receive for trading \delta of asset i is no more than the amount predicted by the exchange rate. The one-sided derivative of the reverse exchange function G at 0 is G'(0^+) = E_{ji}. The analog of the inequality (18) is (19)G(\lambda) \geq G'(0^+)\lambda = \gamma^{-2} E_{ji} \lambda, which states that the amount of asset i you need to tender to receive an amount of asset j is at least the amount predicted by the exchange rate. Examples. Figure 1 shows the forward and reverse exchange functions for a constant geometric mean market with two assets and weights w_1 = .2 and w_2 = .8, and \gamma =0.997. We show the functions for two values of the reserves: R=(1,100) and R=(0.1,10). The exchange rate is the same for both values of the reserves and equal to E_{12} = \gamma w_1R_2/w_2R_1 = 25. ### 4.2. Exchanging multiples of two baskets Here we discuss a simple generalization of two-asset trade, in which we tender and receive a multiple of fixed baskets. Thus, we have \Delta = \delta \tilde \Delta and \Lambda = \lambda \tilde \Lambda, where \lambda\geq 0 and \delta \geq 0 scale the fixed baskets \tilde \Delta and \tilde \Lambda. When \tilde \Delta = e_i and \tilde \Lambda= e_j, this reduces to the two-asset trade discussed above. The same analysis holds in this case as in the simple two-asset trade. We can introduce the forward and reverse functions F and G, which are inverses of each other. They are increasing, F is concave, G is convex, and they satisfy F(0)=G(0)=0. We have the inequality F(\delta) \leq E \delta, where E is the exchange rate for exchanging the basket \tilde \Delta for the basket \tilde \Lambda, given by E = \gamma \frac{\nabla \varphi(R)^T \tilde \Delta}{\nabla \varphi(R)^T \tilde \Lambda}. There is also an inequality analogous to (19), using this definition of the exchange rate. We mention two specific important examples in what follows. Liquidating assets. Let \Delta\in {{\bf R}}_+^n denote a basket of assets we wish to liquidate, i.e., exchange for the numeraire. We can assume that \Delta_n=0. We then find the \alpha>0 for which (\Delta,\alpha e_n) is a valid trade, i.e., (20)\varphi(R+\gamma \Delta - \alpha e_n) = \varphi(R). We can interpret \alpha as the liquidation value of the basket \Delta . We can also show that the liquidation value is at most as large as the discounted value of the basket; i.e., \alpha \le \gamma p^T\Delta. To see this, apply (1) to the left hand side of (20), which gives, after cancelling \varphi(R) on both sides, \nabla\varphi(R)^T(\gamma \Delta - \alpha e_n) \ge 0. Rearranging, we find: \alpha \le \frac{\gamma\nabla\varphi(R)^T\Delta}{\nabla\varphi(R)_n} = \gamma p^T\Delta. Let \Lambda \in {{\bf R}}_+^n denote a basket we wish to purchase using the numeraire. We find \alpha >0 for which (\alpha e_n,\Lambda) is a valid trade, i.e., \varphi(R+\gamma \alpha e_n - \Lambda) = \varphi(R). We interpret \alpha as the purchase cost of the basket \Lambda. It can be shown that \alpha \geq (1/\gamma) p^T \Lambda, i.e., the purchase cost is at least a factor 1/\gamma more than the value of the basket, at the current prices. This follows from a nearly identical argument to that of the liquidation value. Example. We consider an example with n=4, geometric mean trading function with weights w_i = 1/4 and fee \gamma = .997, with reserves R = (4, 5, 6, 7). We fix the received basket to be \Lambda = (2,4,0,0). There are many valid tendered baskets, which are shown in figure 2. The plot shows valid values of (\Delta_3, \Delta_4), since the first two components of \Delta are zero. ### 5.1. The general trade choice problem We formulate the problem of choosing (\Delta,\Lambda) as an optimization problem. The net change in holdings of the trader is \Lambda - \Delta. The trader judges a net change in holdings using a utility function U:{{\bf R}}^n \to {{\bf R}}\cup\{-\infty\}, where she prefers (\Delta,\Lambda) to (\tilde \Delta, \tilde \Lambda) if U(\Lambda - \Delta)> U(\tilde \Lambda - \tilde \Delta). The value -\infty is used to indicate that a change in holdings is unacceptable. We will assume that U is increasing and concave. (Increasing means that the trader would always prefer to have a larger net change than a smaller one, which comes from our assumption that all assets have value.) To choose a valid trade that maximizes utility, we solve the problem (21)\begin{aligned} & \text{maximize} && U(\Lambda - \Delta) \\ & \text{subject to} && \varphi(R+\gamma \Delta - \Lambda) = \varphi(R), \quad \Delta \geq 0, \quad \Lambda \geq 0, \end{aligned} with variables \Delta and \Lambda. Unfortunately the constraint \varphi(R+\gamma \Delta - \Lambda) = \varphi(R) is not convex (unless the trading function is linear), so this problem is not in general convex. Instead we will solve its convex relaxation, where we change the equality constraint to an inequality to obtain the convex problem (22)\begin{aligned} & \text{maximize} && U(\Lambda - \Delta) \\ & \text{subject to} && \varphi(R+\gamma \Delta - \Lambda) \geq \varphi(R), \quad \Delta \geq 0, \quad \Lambda \geq 0, \end{aligned} which is readily solved. It is easy to show that any solution of (22) satisfies \varphi(R+\gamma \Delta - \Lambda) = \varphi(R), and so is also a solution of the problem (21). (If a solution satisfies \varphi(R+\gamma \Delta - \Lambda) > \varphi(R), we can decrease \Delta or increase \Lambda a bit, so as to remain feasible and increase the objective, a contradiction.) Thus we can (globally and efficiently) solve the non-convex problem (21) by solving the convex problem (22). Assuming U(0) > - \infty, the solution to the problem (22) can be \Delta = \Lambda = 0, which means that trading does not increase the trader’s utility, i.e., the trader should not propose any trade. We can give simple conditions under which this happens for the case when U is differentiable. They are (23)\gamma p \leq \alpha\nabla U(0) \leq p, for some \alpha>0. We can interpret the set of prices p for which this is true, i.e., K = \{p \in {{\bf R}}^n_+ \mid \gamma p \le \alpha \nabla U(0) \le p ~ \text{for some} ~ \alpha > 0\}, as the no-trade cone for the utility function U. (It is easy to see that K is a convex polyhedral cone.) We interpret \nabla U(0) as the vector of marginal utilities to the trader, and p as the prices of the assets in the CFMM. For \gamma =1, the condition says that we do not trade when the marginal utility is a positive multiple of the current asset prices; if this does not hold, then the solution of the trading problem (22) is nonzero, i.e., the trader should trade to increase her utility. When \gamma<1, the trader will not trade when the prices are in K. To derive condition (23), we first derive the optimality conditions for the problem (22). We introduce the Lagrangian L(\Delta, \Lambda, \lambda, \omega, \kappa) = U(\Lambda -\Delta) + \lambda(\varphi(R+\gamma \Delta-\Lambda)-\varphi(R)) + \omega^T\Delta + \kappa^T\Lambda, where \lambda \in {{\bf R}}_+, \omega\in {{\bf R}}_+^n, and \kappa \in {{\bf R}}_+^n are dual variables or Lagrange multipliers for the constraints. The optimality conditions for (22) are feasibility, along with \nabla_\Delta L = 0, \qquad \nabla _\Lambda L = 0. The choice \Delta=0, \Lambda =0 is feasible, and satisfies this condition if \nabla_\Delta L(0,0,\lambda,\omega,\kappa) = 0, \qquad \nabla_\Lambda L(0,0,\lambda,\omega,\kappa) = 0. These are - \nabla U(0) + \lambda \gamma \nabla \varphi(R) + \omega =0, \qquad \nabla U(0) - \lambda \nabla \varphi(R) + \kappa=0, which we can write as \nabla U(0) \geq \lambda \gamma \nabla \varphi(R), \qquad \nabla U(0) \leq \lambda \nabla \varphi(R). Dividing these by \lambda P_n, we obtain (23) with \alpha = 1/(\lambda P_n). ### 5.2. Special cases Linear utility. When U(z) = \pi^T z, with \pi \geq 0, we can interpret \pi as the trader’s private prices of the assets, i.e., the prices she values the assets at. From (23) we see that the trader will not trade if her private asset prices satisfy (24)\gamma p \le \alpha\pi \le p for some \alpha > 0. In the special case where \pi satisfies (\pi_2, \dots, \pi_n) = \lambda (p_2, \dots, p_n), for \lambda \ge 0, i.e., \pi is collinear with p except in the first entry, (24) is satisfied if and only if \lambda \gamma p_1 \le \pi_1 \le \lambda \gamma^{-1}p_1. If \lambda = 1, then this simplifies to the condition \gamma p_1 \le \pi_1 \le \gamma^{-1}p_1. (This will arise in an example we present below.) Suppose the trader models the return r\in {{\bf R}}^n on the assets over some period of time as a random vector with mean \mathop{\bf E{}}r = \mu \in {{\bf R}}^n and covariance matrix \mathop{\bf E{}}(r-\mu)(r-\mu)^T =\Sigma \in {{\bf R}}^{n\times n}. If the trader holds a portfolio of assets z \in {{\bf R}}^n_+, the return is r^T z; the expected portfolio return is \mu^Tz and the variance of the portfolio return is z^T\Sigma z. In Markowitz trading, the trader maximizes the risk-adjusted return, defined as \mu^T z - \kappa z^T\Sigma z, where \kappa>0 is the risk-aversion parameter . This leads to the Markowitz trading problem (25)\begin{aligned} & {maximize} && \mu^T z - \kappa z^T \Sigma z\\ &{subject to} && z = z^\text{curr} - \Delta + \Lambda \\ &&&\varphi(R+\gamma \Delta - \Lambda) \geq \varphi(R)\\ &&& \Delta \geq 0, \quad \Lambda \geq 0, \end{aligned} with variables z, \Delta, \Lambda, where z^\text{curr} is the trader’s current holdings of assets. This is the general problem (22) with concave utility function U(Z) = \mu^T(z^\text{curr} + Z) - \kappa (z^\text{curr} + Z)^T \Sigma (z^\text{curr} + Z). A well-known limitation of the Markowitz quadratic utility function U, i.e., the risk-adjusted return, is that it is not increasing for all Z, which implies that the trading function relaxation need not be tight. However, for any sensible choice of the parameters \mu and \Sigma, it is increasing for the values of Z found by solving the Markowitz problem (25), and the relaxation is tight. As a practical matter, if a solution of (25) does not satisfy the trading constraint, then the parameters are inappropriate. Here the trader models the returns r\in {{\bf R}}^m on the assets over some time interval as random, with some known distribution. The trader seeks to maximize the expected utility of the portfolio return, using a concave increasing utility function \psi: {{\bf R}}\to {{\bf R}} to introduce risk aversion. (Thus we use the term utility function to refer to both the trading utility function U: {{\bf R}}_+^n \to {{\bf R}} and the portfolio return utility function \psi:{{\bf R}}\to {{\bf R}}, but the context should make it clear which is meant.) This leads to the problem (26)\begin{aligned} & {maximize} && \mathop{\bf E{}}\psi(r^Tz)\\ &{subject to} && z = z^\text{curr} - \Delta + \Lambda \\ &&&\varphi(R+\gamma \Delta - \Lambda) \geq \varphi(R)\\ &&& \Delta \geq 0, \quad \Lambda \geq 0, \end{aligned} where the expectation is over r. This is the general problem (22), with utility U(Z) = \mathop{\bf E{}}\psi (r^T (z^\text{curr}+Z)), which is concave and increasing. This problem can be solved using several methods. One simple approach is to replace the expectation with an empirical or sample average over some Monte Carlo samples of r, which leads to an approximate solution of (26). The problem can also be solved using standard methods for convex stochastic optimization, such as projected stochastic gradient methods. ### 5.3. Numerical examples In this section we give two numerical examples. Figure 3: Solutions \Lambda - \Delta for the linear utility maximization problem, as the private price for asset 1 is varied by the factor t from the CFMM price. The blue curve shows asset 1. Linear utility. Our first example involves a CFMM with 6 assets, geometric mean trading function with equal weights w_i = 1/6, and trading fee parameter \gamma = .9. (We intentionally use an unrealistically small value of \gamma so the no-trade condition is more evident.) We take reserves R = (1, 3, 2, 5, 7, 6). The corresponding prices are given by (9), p = (R_6/R_1, R_6/R_2, \ldots, 1) = (6, 2, 3, 6/5, 6/7, 1). We consider linear utility, with the trader’s private prices given by \pi = (tp_1, p_2, \dots, p_n), where t is a parameter that we vary over the interval t \in [1/2,2]. For t=1, we have \pi=p, i.e., the CFMM prices and the trader’s private prices are the same (and not surprisingly, the trader does not trade). As we vary t, we vary the trader’s private price for asset 1 by up to a factor of two from the CFMM price. The family of optimal trades are shown in figure 3, as a function of the parameter t. We plot \Lambda - \Delta versus t, which shows assets in the tender basket as negative and the received basket as positive. The blue curve shows asset 1, which we tender when t is small, and receive when t is large. The no-trade region is clearly seen as the interval t\in [0.9,1.1]. Figure 4: Solutions \Lambda - \Delta for instances of an example Markowitz trading problem as the risk-aversion parameter \kappa is varied. Our second example uses nearly the same CFMM and reserves as the previous example, but with a more realistic trading fee parameter \gamma = .997. (This is a common choice of trading fee for many CFMMs.) We solve the Markowitz trading problem (25), with current holdings z^\text{curr} = (2.5, 1, .5, 2.5, 3, 1), mean return \mu = (-.01, .01, .03, .05, -.02, .02), and covariance \Sigma = V^TV/100, where the entries of V\in {{\bf R}}^{6\times 6} are drawn from the standard normal distribution. We solve the optimal trading problem for values of the risk aversion parameter \kappa varying between 10^{-2} and 10^{1}. (For all of these values, the trading constraint is tight.) These optimal trades are shown in figure 4. It is interesting to note that depending on the risk aversion, we either tender or receive assets 2 and 3. The CVXPY code for the Markowitz optimal trading problem is given below. In this snippet we assume that mu, sigma, gamma, kappa, R, and z_curr have been previously defined. Note that the code closely follows the mathematical description of the problem given in (25).	2022-10-01 06:05:45	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9679831266403198, "perplexity": 1935.2199713941177}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335530.56/warc/CC-MAIN-20221001035148-20221001065148-00752.warc.gz"}
http://physics.stackexchange.com/questions/29628/understanding-dynamic-light-scattering/29689	# Understanding Dynamic light scattering I'd like to understand the physics of dynamic light scattering experiment. In particular I want to understand the basic relation between relaxation time $\tau_q$ and the diffusion coefficient $D$: $\tau_q=(q^2 D)^{-1}.$ I suppose that more or less the following steps are necessary: 1) Calculate the scattered wave from Maxwell equations. 2) Find a relation between the scattered intensity autocorrelation function and the brownian motion of the particles. I found this book by Berne and Pecora, but at least for point 1 I got lost because there seem to a strange version (or an approximation) of maxwell equations that I don't understand (it is citing Landau but looking at this book was not such an help). Is there any book were I can find what I need or at least point 1 written with some detail? thank you - Yes, Berne and Pecora is typically used as the canonical reference for Dynamic Light Scattering. I want to make sure I understand your question correctly. You're asking about the derivation in Appendix 3.A, and specifically equations 3.A.3-6, correct? The only thing that looks weird to me is that they look like they are written in Heaviside-Lorentz units, but I don't see that fact specified anywhere in the book. I've never tried to read Landau and Lifshitz, so I don't know what units they use; if they are using SI or Gaussian, that might be the source of your confusion. One other source to check is Jackson. Most of 3rd Edition Jackson uses SI, but an appendix has a chart to convert the Maxwell equations between the different unit systems. Jackson also has a derivation of the scattering equations that you might find helpful. I can't find it in the copy in my office, but my home copy is more marked up and it might be easier to find that when I get home. One other possible source of confusion is that Berne and Pecora are assuming no charges and currents, so they are setting $\mathbf{J} = 0$ and $\rho = 0$. They at least make that explicit. - Yes, correct. Also if I don't care about units, there is a point I don't understand. Let us take as a reference for the maxwell equation this page (en.wikipedia.org/wiki/Maxwell_equations), than eq. 3.A.3 reads $\nabla \times E = - \partial B/\partial t$, and in fact in note 14 of B.P. H is the magnetic field. But then $\nabla \times B = \partial E / \partial t$, not $D$ as in eq. 3.A.4. A hint comes from Landau and Lifshitz which puts the magmetization $M=0$. in this case $B=H$. Now the question is: which is the physical reason for this? – user7669 Jun 8 '12 at 7:41 (I'm posting a new Answer because editing my previous Answer would be an enormous re-write. I hope that this is not too bad a violation of SE etiquette. If it is, please correct me.) Physical assumptions • $\rho = 0$. There are no charges in the system. Yes, there are charges on the protons and electrons in the molecules, but those aren't relevant on the scale of the scattering event. • $\mathbf{J} = 0$. There are no currents in the system. • $\mathbf{P} = \chi_e\mathbf{E}$. The medium is isotropic, so the induced polarization is parallel to the $\mathbf{E}$ field from the light wave. This allows you to write $\mathbf{D} = \epsilon\mathbf{E}$. [EDIT: this is now correct.] • $\mathbf{M} = 0$. The light wave's $\mathbf{B}$ field doesn't magnetize anything in the system. This allows you to write $\mathbf{H} = \frac{1}{\mu}\mathbf{B}$. • $\mu = \mu_0$, (or $\mu = 1$ if you prefer). This is approximately true for most materials. In the DLS experiment at least, all of the scattering comes from variations in $\epsilon$ rather than $\mu$. Those physical assumptions give you the form of Maxwell's Equations that B&P uses. the Scattered Wave As I said, one way of understanding the scattering is that the light wave scatters as a result of a patch of volume with a different dielectric constant $\epsilon$. That in turn comes from the solute you have in the solution. For a given solute/solvent pair, you can relate the concentration of the solute to the change in the local dielectric constant. That gives you the total scattering. That total scattering is useful, and it is used in Static Light Scattering (SLS). But in DLS you're not directly interested in the total scattering; you're interested in how the total scattering fluctuates with time. Autocorrelation and Einstein Relations Your scattering detector sees the concentration of solute in a particular volume in your sample. The signal changes with changes in the local concentration at that volume. Those changes in local concentration are due to diffusion. The other post that @Ron linked to has a good description of how the diffusion constant $D$ relates to the relaxation time $\tau$ (he also provides a lot of background on how that idea relates to other systems). This is a very general process, and (as Ron says) is completely independent of how you probe your system. All you need is a probe that correlates with local concentration. The local concentration that you measure is random (about some mean value) over "long" times. Over short times, the local concentration at one time is highly correlated with the local concentration at another time. The autocorrelation function is the formal definition of this idea. The relaxation time constant $\tau$ is a parameter in the autocorrelation function. In the DLS experiment, there will be a fall off in the value of the autocorellation at a particular value of $\tau$. That tells you the diffusion coefficient of your solute. References • Jackson, Classical Electrodynamics (3 ed). Section 10.2 in particular does the EM derivation, starting from the same point that B&P starts from. • Sun, Physical Chemistry of Macromolecules (2 ed). Chapter 10 discusses diffusion. Section 16.3 discusses DLS, calling it "Laser Light Scattering," and discusses the autocorrlation function. • Teraoka, Polymer Solutions. Ch 3 discusses the dynamics of polymer solutions, autocorelation, and the dynamic light scattering experiment. Any of those should be a useful supplement to B&P. -	2014-04-20 16:28:27	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7870201468467712, "perplexity": 343.25102514628145}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609538824.34/warc/CC-MAIN-20140416005218-00472-ip-10-147-4-33.ec2.internal.warc.gz"}
https://www.gamedev.net/forums/topic/673591-pygame-adjust-character-speed-upon-landing-from-a-jump/	Public Group # Pygame: Adjust character speed upon landing from a jump? This topic is 1017 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic. ## Recommended Posts Hey all, I am working on a crude 2D platformer-brawler (think Smash Bros.) prototype in Pygame. I have mostly been following tutorials at programarcadegames.com and referencing the Pygame docs more and more as I get a better idea of what I'm doing. Right now I'm just working on a very basic physics/input handling sandbox while I learn what constitutes good architecture (proper encapsulation, best practices, avoiding short-term fixes that will provide errors and/or poor performance in a more complex project, etc). I know that this is a fairly ambitious project for a beginning game developer, let alone as a solo project, so I'm not trying to rush development -- I'm introducing features very, very slowly so that I can clearly see what works and what doesn't work when I try to add a desired feature. Movement, input, and animation are pretty much my only focuses for the present. So far I have been able to make the test character collide with/land on platforms, run right and left on platforms, jump, air jump, and come to a stop without trouble. However, I'm having trouble implementing what I thought would be a simple movement mechanic. Here's what I'm trying to do (relevant code will follow): When the character jumps, they can move left and right in the air, but at a lower speed than if they were standing on a platform. They also can't change their directional facing (left or right) while in the air. (This would be an odd design choice for a single-player/co-op platformer, but makes more sense for a fighting game -- jumping needs to be a bigger commitment for the player, with less room to "change your mind" about where you are and where/when you'll land.) Upon landing on a platform, I want the character's left/right movement speed to be immediately re-adjusted to its (higher) ground value, and I also want the character to be able to change directional facing on the 1st frame possible if the player is holding the opposite direction as they land (land and immediately run full speed in that direction, a la Castlevania, Mega Man, etc). I thought this would be as easy as just changing member variable values whenever a platform landing collision is detected in the character's update() method, but apparently I was wrong. What's happening instead is this: if I move backwards in the air (relative to the direction I was facing when I jumped), my slower air movement speed is properly applied, but it's never readjusted when I land. This doesn't happen if I hold forward when I jump, or if I release forward and press it again during a jump that's already moving forward. So if I move backwards in the air from a forward jump or a neutral jump (jump while not holding left or right), when my character lands, it moves slowly (air steer speed) and cannot change its directional facing. If I release left/right to trigger my keyup event handling, the character's stop() method gets called, which resets deltaX to 0, and the speed/facing variables are assigned the new values that I wanted. Here's my input handling in the main loop: if event.type == pygame.KEYDOWN: # Keydown left: Move character left if event.key == pygame.K_LEFT: player.go_left() # Keydown right: Move character right if event.key == pygame.K_RIGHT: player.go_right() # Keydown up: Jump / air jump if event.key == pygame.K_UP: if player.airborne: player.air_jump() else: player.jump() if event.type == pygame.KEYUP: # Stop character if left is released while moving # left if event.key == pygame.K_LEFT and player.deltaX < 0: player.stop() # Stop character if right is released while moving # right if event.key == pygame.K_RIGHT and player.deltaX > 0: player.stop() Here are the movement methods called from the character class: # Player-controlled movement: def go_left(self): """ Called when the user hits the left arrow. """ self.deltaX = -1 * self.movement_speed if not self.airborne: self.direction = "L" def go_right(self): """ Called when the user hits the right arrow. """ self.deltaX = self.movement_speed if not self.airborne: self.direction = "R" def jump(self): """ Called when user hits 'jump' button. """ # move down a bit and see if there is a platform below us. # Move down 2 pixels because it doesn't work well if we only move down # 1 when working with a platform moving down. self.rect.y += 2 platform_hit_list = pygame.sprite.spritecollide(self, self.level.platform_list, False) self.rect.y -= 2 # If it is ok to jump, apply jump force and # declare the character airborne if len(platform_hit_list) > 0 or self.rect.bottom >= constants.SCREEN_HEIGHT: self.deltaY = self.jump_force self.airborne = True self.movement_speed = self.air_steer_speed def air_jump(self): """ Called when user hits 'jump' button while airborne """ if self.airborne == True and self.air_jumped == False: self.air_jumped = True self.deltaY = self.air_jump_force def stop(self): """ Called when the user releases left or right """ self.deltaX = 0 def land(self): """ Called when player lands on a platform """ # Stop player's vertical movement and declare that # character is not airborne self.deltaY = 0 # Set horizontal movement speed back to run speed self.movement_speed = self.run_speed # Reset jumping state values self.airborne = False self.air_jumped = False And here is the order of operations in the character's update() method: def update(self): # Calculate and apply gravity self.calc_grav() # Move left/right (apply deltaX) self.rect.x += self.deltaX pos = self.rect.x + self.level.world_shift # Set running frame based on direction, screen position, # and frame rate if self.direction == "R": frame = (pos // 30) % len(self.running_frames_R) self.image = self.running_frames_R[frame] else: frame = (pos // 30) % len(self.running_frames_L) self.image = self.running_frames_L[frame] # Check for platform collisions (x-axis) block_hit_list = pygame.sprite.spritecollide(self, self.level.platform_list, False) for block in block_hit_list: # If we are moving right, # set our right side to the left side of the item we hit if self.deltaX > 0: self.rect.right = block.rect.left elif self.deltaY < 0: # Otherwise if we are moving left, do the opposite. self.rect.left = block.rect.right # Move up/down (apply deltaY) self.rect.y += self.deltaY # Check for platform collisions (y-axis) block_hit_list = pygame.sprite.spritecollide(self, self.level.platform_list, False) for block in block_hit_list: # Reset rect position based on the top/bottom of the object. if self.deltaY > 0: self.rect.bottom = block.rect.top elif self.deltaY < 0: self.rect.top = block.rect.bottom # Call platform landing method self.land() I used print statements to see exactly when and how frequently these methods were called. I know that for every left/right KEYDOWN event, go_left() and go_right() are only called once. I know I won't get a fresh call unless I release the key (for a KEYUP event that triggers a stop() call) and then press again (for a new KEYDOWN event and thus a new go_right() or go_left() call). I am not entirely sure how Pygame event handling works. I'm still going through tutorials and poring over the docs to understand where they come from, what members they have, what they can responsibly be used to accomplish in one's project, those kinds of things. So one suspicion I have about my problem is that I don't have enough information (or current enough information) about the key state passed to my character's methods -- I need more/better info in order to keep the data state updated. My impression is that in update(), when land() is called (as it is every frame while the character bottom collides with the top of a platform, logically reaffirming that it is standing on a platform), I need to somehow verify if a key is still pressedor perhaps has not been released, since the last time go_left() or go_right() was called. That may give me enough information to drive an if block that adjusts the character's speed and direction on the 1st frame the character is determined to have landed. (I am confused, however, as to why my boolean members for air jump and airborne get set properly when land() is called in update(), but not my movement speed and directional facing members.) How correct or incorrect are any of these assumptions I make? If you need to see any more of my code, let me know and I'll edit this post. Thank you in advance for any help you can provide. ##### Share on other sites You don't re-adjust the player direction nor deltaX when he lands, so, if you jump facing right, then press left, when the player lands the facing variable will still be 'R' and deltaX will still be the slow left since it too hasn't changed when he lands. You need a check in Land to see if airborne == true, and, if so, you may need to adjust the facing and deltaX variables. Edited by BeerNutts ##### Share on other sites Hey BeerNutts, Thank you very much for your help. I had to think about it a bit, but I realized what I was doing wrong in this case. Here is my new code in the character's stop() method: def land(self): """ Called when player lands on a platform """ # Stop player's vertical movement and declare that # character is not airborne self.deltaY = 0 # Reset jumping state values self.airborne = False self.air_jumped = False self.movement_speed = self.run_speed # Re-adjust deltaX to ground movement speed, # based on directional facing if self.deltaX < 0 and self.direction != "L": self.deltaX = -1 * self.movement_speed self.direction = "L" elif self.deltaX < 0 and self.direction == "L": self.deltaX = -1 * self.movement_speed if self.deltaX > 0 and self.direction != "R": self.deltaX = self.movement_speed self.direction = "R" elif self.deltaX > 0 and self.direction == "R": self.deltaX = self.movement_speed This corrects both the movement speed issue and the character's unfortunate tendency to "moonwalk" upon landing after a backward air steer while holding that direction. The problem I had, I assume, was that I was treating a lot of the landing logic as implicit from update(), when I needed to be more explicit about boolean tests in land() in order to make sure values got adjusted that very same frame upon landing. I was surprised that this problem turned out to be so subtle. Definitely something to think about as I move forward. I think I'm going to review a bit more about event handling and animation next, since I feel my grasp on how those work in Pygame is still pretty tenuous. Thank you again for your help! ##### Share on other sites I wonder why you call "self.land()" for every block in the block_hit_list in your first post. In your second post, the code block at lines 16 to 20 can be simplified (and 22 to 26 too). If you compare 16 vs 19, the only difference is the self.direction test. Lines 17 and 20 are the same, and then line 18 you correct self.direction if it is not what it should be. Since assigning a value to a variable that it already has is not a problem, you can remove the self.direction test, and simply always assign, as in if self.deltaX < 0: self.deltaX = -self.movement_speed self.direction = "L" It does not matter what "self.direction" old value was, afterwards, it's always "L" I also replaced "-1 * self.movement_speed" by "-self.movement_speed". Both are equivalent, just use what you like best. ##### Share on other sites Alberth, Apologies for the late reply. Busy at work as of late and I stopped to review and clean up my code a bit before I replied to this post. I agree with you about self.land(), I was calling it too generically. I changed the call to only happen if the player collides with a platform along the y-axis and their deltaY is greater than 0. This should hopefully make self.update() a little less busy, although I realize the collision logic is still a bit broad (more on that below). Here's the new condition under which I call self.land(): def update(self): # (more code above) # Check for collisions (y-axis) block_hit_list = pygame.sprite.spritecollide(self, self.level.platform_list, False) for block in block_hit_list: # Reset rect position based on the top/bottom of the object. if self.deltaY > 0: self.rect.bottom = block.rect.top # DEBUG: Land only if character was falling when # collision occurred self.land() elif self.deltaY < 0: self.rect.top = block.rect.bottom # Cause character to start falling if they bump # their head on the bottom of a platform self.stop_rising() def stop_rising(self): # Apply half of jump force in the opposite # y direction to deltaY to accelerate # the effect of gravity, resulting in # a short jump or a fall after bumping # against the bottom of a platform self.deltaY += -0.5 * self.jump_force stop_rising() was a method I wrote for use with my short jump logic, which was a surprisingly straightforward implementation -- I call player.jump(), count frames in main loop since jump keydown event happened, and call player.stop_rising() if jump keyup event gets polled within a certain number of frames elapsed since the keydown event(I semi-arbitrarily chose 10). I noticed my jump against the underside of low-height platforms was unintentionally floaty, so I added a stop_rising() call in the top->bottom collision logic to cause a sharp drop when the player "bumps their head". I also changed the syntax in self.land() like you suggested. I was in a hurry the night I fixed it and wasn't sure how much of my logic was being redundant until I looked at it with fresh eyes. For adjusting the movement speed, I also think your syntax (-foo instead of -1 * foo) is better, because for switching between left and right movement, I will always just be adjusting that value by 1foo or -1foo instead of a different multiplier (like when I apply or adjust a proportion of my character's jump force to deltaY in self.stop_rising()). Closer to plain human language that way. Much of the collision detection logic is a holdover from the tutorial code I was following (specifically, this tutorial by Paul Vincent Craven at http://programarcadegames.com/python_examples/en/sprite_sheets/). I'm not sure how/if I can improve on the for block in block_hit_list method (having gotten block_hit_list from a spritecollide() on the player and the level member's list of Sprite-extending platforms) he uses. The most obvious problem I can see at a glance is that colliding with overlapping/tiled platforms could result in a bunch of redundant (and expensive?) self.land() calls. However, I'm not sure yet how/if I might change the collision logic to limit self.land() calls beyond what I already have in place. I'll have to take another pass through collision logic to see exactly what I need and what I don't need for my purposes. If I have more questions about different problems with my project (not directly related to the topic in the top post), is it better to post in this same thread, or make a new thread for a new problem? I want to avoid forum spam as much as possible. (And of course I'll do a forum + Google search for similar problems before I make my own thread.) Thank you again for your help. EDIT: I see a potential solution to redundant collision generally suggested in the MDN collision detection tutorial here: https://developer.mozilla.org/en-US/docs/Games/Techniques/2D_collision_detection I'll start looking at how to accomplish broad vs narrow phases in Pygame. Edited by benreed 1. 1 2. 2 Rutin 21 3. 3 4. 4 frob 17 5. 5 • 9 • 12 • 9 • 33 • 13 • ### Forum Statistics • Total Topics 632595 • Total Posts 3007266 ×	2018-09-20 23:57:29	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2426861673593521, "perplexity": 3246.3005058590284}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267156690.32/warc/CC-MAIN-20180920234305-20180921014705-00348.warc.gz"}
https://www.vedantu.com/physics/reactance-and-impedance	# Reactance and Impedance Impedance meaning – It is the measure of overall opposition of a circuit to current denoted by Z. In simple words, it gives the amount of circuit that impedes the flow of change. Impedance is like resistance, which also takes into account the effects of inductance and capacitance. The measurement unit for Impedance is ohms. As impedance considers the effects of inductance and capacitance and varies with the frequency of current passing through the circuit, it is more complex than resistance. As compared to resistance, which is constant regardless of frequency, impedance varies with frequency. When it comes to define reactance, it is the measure of opposition of inductance and capacitance to current. Let’s learn more about these two terms in brief. ### Impedance Formula The mathematical symbol of impedance is Z, and the unit of measurement is ohm. It is the superset of both resistance and reactance combined. In phasor terms, impedance Z is characterized as a summation of resistance R and reactance X as: X = R + j X Where reactance X is the summation of Inductive reactance XL and capacitive Xc X = XL + Xc Impedance, Z=V/I Resistance, R= V/I V = voltage in volts (V) I = current in amps (A) Z= impedance in ohms (Ω) R= resistance in ohms (Ω) Impedance can be splitted into two parts: • Resistance R (a part which is constant regardless of frequency) • Reactance X (a part which varies with frequency due to capacitance and inductance) The alternating current lags or leads the voltage depends upon the nature of reactance component of impedance (whether predominantly inductive or capacitance) The inductance and capacitance causes phase shift between current and voltage, which means the resistance and reactance cannot be simply summed up to give impedance. Instead, they must be summed up as vectors with reactance at right angles to resistance, as shown in the figure below. Impedance Z = $\sqrt{R^{2} + X^{2}}$ There are four electrical quantities which determine the impedance (Z) of a circuit: These are: resistance (R), capacitance (C), inductance (L) and frequency (f). ### What Is Reactance? The measure of opposition of inductance and capacitance to current is known as reactance, denoted by X. It varies with the frequency of electrical signals and is measured in ohms. Reactance is of two types: • Capacitive reactance (Xc), and • Inductive reactance (XL). ### Reactance formula The total reactance (X) is equal to the difference between the two: Total Reactance, X =XL – Xc a. Capacitive Reactance Xc The reactance, which is large at low frequencies and small at high frequencies is known as capacitive reactance (Xc). Xc is infinite for steady DC, at zero frequency (f=0Hz). This means that the capacitor passes AC but blocks DC. Capacitive Reactance, Xc = 1/2fC Where, Xc=reactance in ohms (Ω) f =frequency in hertz (Hz) For eg: 1µF capacitor has a reactance of 3.2k for 50Hz signal, but when a frequency is higher at 10kHz, the reactance is only 16. b. Inductive Reactance, XL The reactance which is small at low frequencies and large at high frequencies is known as inductive reactance. XL is zero for steady DC, at zero frequency (f=0Hz). This means that the inductor passes DC but blocks AC. The formula for calculating inductive reactance of a coil is: Inductive reactance, or XL, is a product of 2 times p (pi), or 6.28, frequency of the ac current in hertz, and the inductance of the coil, in henries. XL =2p x f x L L= the inductance value of coil in henries. Inductive Reactance, XL=2fL Where, XL =reactance in ohms (Ω) f =frequency in hertz (Hz) L =inductance in henrys (H) For example, a 1mH inductor has a reactance of only 0.3 for a 50Hz signal, but when frequency is higher at 10 kHz, its reactance is 63. ### Do you know? When current and voltage are out of step with each other, it means there is a phase shift. For example, when you change a capacitor, the voltage across it is zero. However the current is maximum. When the capacitor is charged, the voltage will be maximum, and current will be at minimum. The charging and discharging occur continually with AC, where the current reaches maximum shortly before the voltage reaches maximum, so it is called current leading voltage. 1. How do you calculate reactance from impedance? Ans- To calculate impedance, calculate resistance and reactance of a circuit, label resistance as R and reactance as X. Square both R and X, and sum the two products together. Take the square root of the sum of the squares of R and X to get impedance. Display the answer in ohms. Impedance Z = V/I Z = √R2 + X2 Z2= (R2+X2) 2. What is the difference between resistance, reactance, and impedance? Ans- The measure of opposition to current flow offered by a material is known as resistance, denoted by R. The resistance offered to ac currents only by inductors and capacitors is known as reactance, denoted by X. For capacitor X=1/(2πfC), where f is frequency, and C is capacitance. For inductor X= 2πfL, where f is frequency, L is inductance. Impedance is the summation of resistance and reactance of circuit Symbolized for primarily inductive circuit Z=R+jX or for primarily capacitive circuit Z=R-jX. Where j=√(-1). 3. When should I use impedance instead of reactance vice versa? Ans- There is a phase shift between voltage and current, and Impedance is sensitive to frequency. Reluctance is a measure of resistance of an ideal coil at a specific frequency. Reactance is a measure of resistance of an ideal capacitor at a specific frequency. As an engineer: Coils are imperfect, and the wire has an intrinsic resistance and loops form stay capacitance. Thus, a coil has resistance, reactance, and reluctance; this combination is a measure of its impedance. Resistors are also imperfect, wire wound resistors have inductance to give small reluctance, and some surface mount resistors have tiny capacitance. Normally speaking, if there is a dc circuit, you need to use resistance, and if the circuit is ac, you need impedance. 4. What is the difference between reactance and resistance? Ans- The opposition of the inductor to change the current is usually translated as the alternating current, which by definition, always varies in instantaneous size and direction. This is similar to resistance, but different in that it always results in phase shift between current and voltage and it disperse zero power. Due to the difference, it has a different name reactance. Just like resistance, reactance to AC is measured in ohms, but its mathematical symbol is X instead of R. Some more difference between Reactance and Resistance is: Parameter Resistance Reactance Variation in current It is the property of the electrical component which opposes the flow of current. It is the property of the electrical component which opposes the change in current. Power Dissipation It leads to power dissipation It does not lead to power dissipation. Denoted by It is denoted by R It is denoted by X	2020-08-06 22:26:23	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7866172790527344, "perplexity": 1557.9024168239193}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439737039.58/warc/CC-MAIN-20200806210649-20200807000649-00221.warc.gz"}
https://math.stackexchange.com/questions/599126/continuous-bounded-function-f-mathbbr-rightarrow-mathbbr	# Continuous bounded function $f:\mathbb{R}\rightarrow \mathbb{R}$ Question is to check which of the following holds (only one option is correct) for a continuous bounded function $f:\mathbb{R}\rightarrow \mathbb{R}$. • $f$ has to be uniformly continuous. • there exists a $x\in \mathbb{R}$ such that $f(x)=x$. • $f$ can not be increasing. • $\lim_{x\rightarrow \infty}f(x)$ exists. What all i have done is : • $f(x)=\sin(x^3)$ is a continuous function which is bounded by $1$ which is not uniformly continuous. • suppose $f$ is bounded by $M>0$ then restrict $f: [-M,M]\rightarrow [-M,M]$ this function is bounded ad continuous so has fixed point. • I could not say much about the third option "$f$ can not be increasing". I think this is also true as for an increasing function $f$ can not be bounded but i am not sure. • I also believe that $\lim_{x\rightarrow \infty}f(x)$ exists as $f$ is bounded it should have limit at infinity.But then I feel the function can be so fluctuating so limit need not exists. I am not so sure. So, I am sure second option is correct and fourth option may probably wrong but i am not so sure about third option. Thank You. :) • Have you studied any theorems related to fix point of functions. – Mhenni Benghorbal Dec 9 '13 at 3:18 • I only know that continuous bounded function on compact set has a fixed point.. – user87543 Dec 9 '13 at 4:51 For the third point, consider $f(x) = \arctan(x)$. For the fourth point, you've already found a counterexample in one of your other points! • I am sorry.. I did not recognize counterexample for fourth option in what i have done... could you please explain a bit more. – user87543 Dec 9 '13 at 2:38 • @PraphullaKoushik: Sure! Does $\lim_{x\to\infty}\sin(x^3)$ exist? – Dan Dec 9 '13 at 2:40 • This is quite interesting... :) I do not have limit at $\infty$ for $\sin (x^3)$.. This is very beautiful.. :) – user87543 Dec 9 '13 at 2:42 $\tan^{-1}x$ is increasing. $\sin (x^3)$ has no limit at infinity. • yes yes.. $\tan^{-1}x$ is increasing but i do not know if it is bounded :( – user87543 Dec 9 '13 at 2:41 • @PraphullaKoushik It is. $-\pi/2 < \tan^{-1}(x) < \pi/2$ – Eric Auld Dec 9 '13 at 2:42 • Oh my bad... I got it... I am sorry for that dumb question... I was thinking of something else... Thank you so much... – user87543 Dec 9 '13 at 2:43 • @PraphullaKoushik No problem, it happens to everyone. – Eric Auld Dec 9 '13 at 2:44 Well, for $x$ really, really large, what can you say about $f(x) - x?$ For $x$ really, really small, what can you say about $f(x) - x?$ • I am sorry, I could not understand your idea... please explain a bit more. – user87543 Dec 9 '13 at 2:31 Here is an incredibly non-interesting trivial example: $f(x)=a$ for $a$ being some real number. • This is a uniformly continuous, non-increasing function that has a fixed point (namely, $x=a$) with the property that the limit at infinity exists. – Hayden Dec 9 '13 at 2:35	2019-09-18 15:21:45	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9030653834342957, "perplexity": 618.1665739155296}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514573309.22/warc/CC-MAIN-20190918151927-20190918173927-00081.warc.gz"}
http://mathhelpforum.com/math-topics/126525-converting-questions-how-find-isotopes-protons-neutrons.html	# Math Help - Converting questions, and how to find isotopes protons neutrons. 1. ## Converting questions, and how to find isotopes protons neutrons. I have a few questions i need some help on. its converting (my weakest category). 1)What is the price of 1 gallon of gasoline in U.S. dollars in France? (The value of the Euro is currently $1.33 U.S. and the price of 1 liter of gasoline in France is 0.90 Euro.) 2)What is its density in pounds per cubic inch (lb/inch^3). (The density of lead is 11.4 .) 3)How many kilometers can the Insight travel on the amount of gasoline that would fit in a soda pop can? The volume of a soda pop can is 355 . (The Honda Insight, a hybrid electric vehicle, has an EPA gas mileage rating of 57 in the city.) and last how do i solve for protons and neutrons if something says , ,, etc? thxs for teh help. 2. If you are just starting out, it pays to do things one step at a time. For now, it takes a bit longer but makes it really clear. With practice you should be able to skip quite a few steps. 1)What is the price of 1 gallon of gasoline in U.S. dollars in France? (The value of the Euro is currently$1.33 U.S. and the price of 1 liter of gasoline in France is 0.90 Euro.) Let x = the price of a gallon of gas in USD. I will refer to gallon as G in the following equation. First off we have $\frac{0.9 \,Euro}{1 L}$ 1 euro = 1.33 USD $0.9 \,Euro = 1 \times \frac{9}{10} \, Euro = 1.33 \times \frac{9}{10} \, USD = 1.197 \, USD$ So now we have $\frac{0.9 \,Euro}{1 L} = \frac{1.197 \, USD }{1 L}$ 1 Litre = 0.264 liquid Gallons (US) so we have $\frac{0.9 \,Euro}{1 L} = \frac{1.197 \, USD }{1 L} = \frac{1.197 \, USD}{0.264 G} = \frac{4.534 \, USD }{1G}$ 2)What is its density in pounds per cubic inch (lb/inch^3). (The density of lead is 11.4 .) Please try to work this out yourself using question 1 as a guide. 1g = 453.6 lbs and 1 inch = 2.54 cm. Ask for more help and show your working so far if you really can't do it. 3)How many kilometers can the Insight travel on the amount of gasoline that would fit in a soda pop can? The volume of a soda pop can is 355 . (The Honda Insight, a hybrid electric vehicle, has an EPA gas mileage rating of 57 in the city.) Same as question 2. 1 mile = 1.61 km and last how do i solve for protons and neutrons if something says , ,, etc? You'll need a periodic table for this. Most periodic tables have two numbers - one is an integer, which ascends consecutively from left to right, continuing on the next row. The other one is a larger number with decimal places. The integer is called the atomic number. It is defined as the number of protons in the element. In a non-ionised state, each element has the same number of protons and electrons. However, once ionised, electrons are given away or shared. Lets take your example of $V^{3+}$. V by itself has an atomic number of 23 on the periodic table. That means it has 23 protons and 23 electrons. However, once ionised, V gives away 3 electrons and becomes $V^{3+}$ It now has a charge of 3 and has 23 protons and 3 electrons. Now for the second number, the bigger number called the mass number. Because electrons have much smaller masses than protons and neutrons, their mass is negligeble when calculating the mass of individual atoms. The mass number is defined as: Mass number = # of protons + # of neutrons. Since # of protons = atomic number, another way of saying this is: Mass number = Atomic Number + # of Neutrons. It is thus very straightforward to find the number of neutrons. The mass number is in decimals because it is an average masses of all isotopes weighted by abundance. Because we are most likely dealing with the most abundant isotope, just round the mass number to the nearest integer and proceed with your calculation.	2015-04-25 00:38:25	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 6, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5472875833511353, "perplexity": 815.0343743240292}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246645538.5/warc/CC-MAIN-20150417045725-00011-ip-10-235-10-82.ec2.internal.warc.gz"}
https://math.stackexchange.com/questions/595750/properties-of-convergence-in-distribution	# Properties of convergence in distribution? If you have a sequence of random variables $(W_n)$ which converges in distribution to N, and another sequence of random variables $(X_n)$ which converges in distribution to $B$: i) Will $(W_nX_n)$ converge in distribution to N multiplied by B? ii) Similarly, will $(W_n + X_n)$ converge in distribution to $N + B$? I know this is probably a very basic question, but I can't find proofs/discussions properties of convergence in my text books :( If it's not too complicated, could you let me know whether i and ii happen to be true (or false) for the other basic types of convergence (almost-sure, L^2, pointwise) ? • Yes, that's what I meant to say. Ooops. – Wanda1989 Dec 6 '13 at 19:33 If $X_n\to X$ in distribution and $Y_n\to Y$ in distribution, we may not have $X_nY_n\to XY$ in distribution. For example, assume that $X$ takes the values $-1$ and $1$ with probability $1/2$, $X_n=X$ and $Y_n=X$ for $n$ even and $Y_n=-X$ for $n$ odd. We have that $X_n\to X$ in distribution and $Y_n\to X$ in distribution but $X_nY_n$ is equal to $1$ for $n$ even and $-1$ for $n$ odd.	2021-07-27 05:14:05	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9343783855438232, "perplexity": 60.94958753284488}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046152236.64/warc/CC-MAIN-20210727041254-20210727071254-00015.warc.gz"}
http://1156466043.srv040084.webreus.net/gw-wvw-tplt/article.php?id=system-of-linear-equations-calculator-646efe	## system of linear equations calculator december 1, 2020 This website uses cookies to ensure you get the best experience. System of Linear Equations Calculator This calculator will solve the system of linear equations of any kind, with steps shown, using either the Gauss-Jordan Elimination method or the Cramer's Rule. Download free on Google Play. You can input only integer numbers or fractions in this online calculator. Step 2: Click the blue arrow to submit. Free linear equation calculator - solve linear equations step-by-step. In mathematics, a system of linear equations is a set of one or more linear equations with the same number of variables (or unknowns). Change the names of the variables in the system, For example, the linear equation x1 - 7x2 - x4 = 2. 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Previous post Ringpootbuizerd	2021-01-20 19:10:58	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4114914834499359, "perplexity": 879.5513607898811}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703521987.71/warc/CC-MAIN-20210120182259-20210120212259-00504.warc.gz"}
http://mathoverflow.net/feeds/user/7595	User daizhuo - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T07:22:18Z http://mathoverflow.net/feeds/user/7595 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/44777/how-to-characterize-real-square-matrices-a-such-that-vav-0-for-all-real-vec How to characterize real square matrices A, such that v'Av >= 0, for all real vectors v with 1'v=0 (1 is the vector of all ones)? daizhuo 2010-11-04T03:21:47Z 2013-01-01T22:19:30Z <p>I derive this question while trying to prove the monotonicity of a differentiable vector function $f(x)$ that maps from $X\subset R^n$ to $R^n$ (Here function $f(x)$ is called monotone if $(x-y)'(f(x)-f(y))\geq 0$, $\forall x,y\in X$). The domain $X$ only consists of vectors $x$ such that $1'x=0$, here $1$ is the vector of all ones.</p> <p>Using the mean-value theorem, we have that $f(x)$ is locally monotone at $x$ (namely $(y-x)'(f(y)-f(x))\geq 0$, $\forall y\in X$) if its Jacobian matrix evaluated at $x$, which we label as $A$, satisfies the following condition:</p> <p>$$v'Av\geq 0,\quad \forall v \text{ such that } 1'v=0.$$</p> <p>This is a weaker condition than positive semidefiniteness. However, while there are a number of ways to characterize positive semidefinite matrices, for example, see <a href="http://en.wikipedia.org/wiki/Positive-semidefinite_matrix#Characterizations" rel="nofollow">this Wikipedia page</a>, how can I characterize the above defined matrices?</p> http://mathoverflow.net/questions/32515/how-to-write-matlabs-dot-operators-in-mathematical-expressions How to write Matlab's dot operators in mathematical expressions? daizhuo 2010-07-19T17:20:11Z 2011-04-26T05:27:13Z <p>Matlab has a set of dot operators, such as .*, ./, .^. Each of these operators consists of a dot and a normal algebraic operator. They perform element-wise algebraic operations on a matrix. For example, consider the following codes</p> <pre><code>A = [1 2 3; 3 2 1]; x = [1 2 4]; B = A.^2 y = 1./x </code></pre> <p>The result is</p> <pre><code>B = 1 4 9 9 4 1 y = 1.0000 0.5000 0.2500 </code></pre> <p>I find these dot operators very convenient. My question is, how to write these dot operators in mathematical expressions? (By mathematical expressions, I mean the expressions used in proofs.)</p> <p>EDIT - One obvious way is to define the result matrix element-wise. But is there a way to write this result in a more compact manner?</p> http://mathoverflow.net/questions/47888/expectation-multinomial-distribution-and-monotonicity-a-conjecture Expectation, multinomial distribution, and monotonicity (A conjecture) daizhuo 2010-12-01T10:40:22Z 2010-12-01T16:26:37Z <p>Let $n$ and $k$ be two positive integers. Let $S = \{ \mathbf{p} \in \mathbb{R}^k : \mathbf{p} \geq 0, \sum_{i=1}^k p_i = 1 \}$ (i.e., a simplex). Consider a function $\mathbf{f}:\mathbb{Z}^k \rightarrow \mathbb{R}^k$. And from $\mathbf{f}$ we may define function $\mathbf{g}: S\subset \mathbb{R}^k \rightarrow \mathbb{R}^k$ as follows $$\mathbf{g}(\mathbf{p}) = \mathbb{E}[\mathbf{f}(\mathbf{X})], \quad \mathbf{X}\sim \text{Multinomial}(n,\mathbf{p}).$$ Here $\mathbf{X}$ is a random vector that follows the multinomial distribution determined by the number of trials $n$ and probabilities $p_1,p_2,\dots,p_k$.</p> <p><strong>Conjecture</strong>: If $\mathbf{f}$ is monotone, that is, $$(\mathbf{x}-\tilde{\mathbf{x}})^T (\mathbf{f}(\mathbf{x})-\mathbf{f}(\tilde{\mathbf{x}})) \geq 0\quad \forall \mathbf{x}, \tilde{\mathbf{x}} \in \mathbb{Z}^k,$$ then $\mathbf{g}$ is also monotone, i.e., $$(\mathbf{p}-\tilde{\mathbf{p}})^T (\mathbf{g}(\mathbf{p})-\mathbf{g}(\tilde{\mathbf{p}})) \geq 0\quad \forall \mathbf{p}, \tilde{\mathbf{p}} \in S.$$</p> <p><strong>Remark</strong>: The above result should hold in the following two special cases (I omit the proofs) -</p> <ol> <li><p>The function $\mathbf{f}$ is affine, namely, $\mathbf{f}(\mathbf{x})=G\mathbf{x}+\mathbf{b}$ for some matrix $G$ and vector $\mathbf{b}$.</p></li> <li><p>The function $\mathbf{f}$ is "separable", meaning that $$\mathbf{f}(\mathbf{x}) = (f_1(x_1), \dots, f_k(x_k))^T$$ for some non-decreasing scalar functions $f_1(\cdot), \dots, f_k(\cdot)$.</p></li> </ol> <p>But does it hold in the general case? </p> http://mathoverflow.net/questions/47630/for-what-k-matrix-k-a-b-is-positive-semidefinite For what k, matrix (k A - B) is positive semidefinite? daizhuo 2010-11-29T02:39:35Z 2010-11-30T07:58:22Z <p>Suppose $A$ and $B$ are two $n\times n$ real symmetric matrices. $A$ is positive semidefinite. Then for what values of real number $k$, matrix $(kA-B)$ is positive semidefinite (we write as $kA-B\succeq0$)?</p> <p>If $A$ is positive definite, we may find an $n\times n$ nonsingular matrix $D$ such that $A=D^T D$. As a result, $kA-B\succeq0$ is equivalent to $$kI\succeq (D^{-1})^TBD^{-1},$$ or $k\geq \lambda_{\max}((D^{-1})^TBD^{-1}))$.</p> <p>But how to deal with the situation when $A$ is singular (but still positive semidefinite)? I know for certain that in this case we must impose additional constraint on matrix $B$. In particular, let the columns of matrix $N$ consist of a basis of the null space of $A$, then we must have that $N^T B N \preceq 0$ (i.e., $N^T B N$ is negative semidefinite). But what is the lower bound on $k$?</p> <p>Thanks.</p> http://mathoverflow.net/questions/47630/for-what-k-matrix-k-a-b-is-positive-semidefinite/47739#47739 Comment by daizhuo daizhuo 2010-11-30T17:35:26Z 2010-11-30T17:35:26Z Thanks. In fact, $B_{13}$ must be 0. Let $P_1$ be a matrix consisting of columns that form a basis of the range of $A$. Let $P_2$ be a matrix consisting of columns that form a basis of the intersection of the kernel of $A$ and the range of $B$. Let $P_3$ be a matrix consisting of columns that form a basis of the intersection of the kernel of $A$ and the kernel of $B$. Then columns of $P=(P_1, P_2, P_3)$ form a basis of $\mathbb{R}^n$ in which $A$ and $B$ took the form you showed above. In particular, $B_{13} = P_1^T B P_3 = 0$ because $B P_3 = 0$. http://mathoverflow.net/questions/47630/for-what-k-matrix-k-a-b-is-positive-semidefinite/47642#47642 Comment by daizhuo daizhuo 2010-11-29T09:56:51Z 2010-11-29T09:56:51Z Actually $kA - B \succeq 0$ for $k \geq -1$. http://mathoverflow.net/questions/47630/for-what-k-matrix-k-a-b-is-positive-semidefinite/47642#47642 Comment by daizhuo daizhuo 2010-11-29T09:52:11Z 2010-11-29T09:52:11Z Can you be more specific on your last sentence? In particular, why do I need $\text{ker}A\subseteq\text{ker}B$? A counter example would be the following. $A = \left(\begin{array}{cc}1 & 0\\0 & 0\end{array}\right)$ and $B = -I$. In this case $k A - B\succeq 0$ for $k\geq 0$, but $\text{ker}A$ is larger than $\text{ker} B$. http://mathoverflow.net/questions/44777/how-to-characterize-real-square-matrices-a-such-that-vav-0-for-all-real-vec/44851#44851 Comment by daizhuo daizhuo 2010-11-05T03:34:00Z 2010-11-05T03:34:00Z Thanks for your comment. I added to my question the definition of vector function monotonicity. http://mathoverflow.net/questions/32515/how-to-write-matlabs-dot-operators-in-mathematical-expressions Comment by daizhuo daizhuo 2010-07-20T02:06:01Z 2010-07-20T02:06:01Z Thanks for the comment. I followed the link to APL; it is interesting! http://mathoverflow.net/questions/32515/how-to-write-matlabs-dot-operators-in-mathematical-expressions/32518#32518 Comment by daizhuo daizhuo 2010-07-19T17:57:25Z 2010-07-19T17:57:25Z Oh thanks for your response! But are there more compact ways to write these facts?	2013-05-22 07:22:28	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9678531289100647, "perplexity": 482.1727239892368}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368701459211/warc/CC-MAIN-20130516105059-00083-ip-10-60-113-184.ec2.internal.warc.gz"}
http://www.scl.csd.uwo.ca/~bill/Papers/omVmml/omVmml.html	# On the relationship between OpenMath and MathML Bill Naylor Stephen Watt Ontario Research Center for Computer Algebra University of Western Ontario {bill,watt}@orcca.on.ca ### Abstract: MathML provides a rather complete set of primitives for notational representation of mathematics but only a limited set of primitives to describe the meaning of mathematical expressions. OpenMath provides an extensible mechanism to describe mathematical semantics and is not concerned with notational nuances. Both MathML and OpenMath give XML representations for mathematical objects. This paper describes how OpenMath may be used to extend MathML and vice versa: Two mechanisms are described whereby OpenMath content dictionaries may be used as an extension mechanism for MathML, overcoming the limitations in MathML's content model. Further, a description is given of a possible use of MathML elements within OpenMath objects. # Introduction MathML [1] is an emerging standard from the World Wide Web Consortium, W3C [2]. Two of its main design goals are: 1. [1)] to provide a means of representing publication quality'' layout (presentation MathML) of mathematics on the internet, 2. [2)] to provide a means of representing the semantic meaning of mathematics (content MathML), again with a major concern that this should be transferable between applications on a single computer or over a network. MathML is an XML application, and as such MathML data may readily be converted into other document styles using XSLT [4] style sheets. It has been designed primarily with the aim of representing mathematics to the level of K-12. Since mathematical notation consist of a small number of relative positioning primitives, it is fairly well covered by MathML. Problems arise, however, in the representation of the meaning of general mathematical objects. One of the limitations with MathML as it stands is that the meaning of an element resides in the name of the tag. Therefore, to represent mathematical concepts not pre-defined in MathML it is necessary to refer to some external definitions. A resource which may be able to provide a definition server may exist in the form of OpenMath [5] Content Dictionaries. OpenMath is an XML application whose primary goal is to serve as a communication medium for the semantics of mathematical objects. One of its primary concerns is that it should be extensible, as it is a huge task to represent the objects of all of known mathematics, and indeed we may need to represent mathematics which is not yet defined! Extensibility is provided via content dictionaries which are XML repositories of mathematical information. The OpenMath concept can provide a definition resource which may be used for extending MathML, whilst XSLT style-sheets enable a simple way of performing a translation between a document which contains new elements and a document which contains the required semantic elements containing pointers to Symbols in OpenMath Content Dictionaries [10]. In this paper we describe how this translation takes place and give an example of an original document together with the transformed document and the content dictionary it points to. # Using OpenMath to extend MathML ## The technology available The need to refer to externally defined concepts was anticipated by the authors of the MathML specification, and it was designed so that these mechanisms would be compatible with OpenMath. There are two seperate ways in which such an external source of mathematical knowledge may be accessed from within a MathML document, these are: via semantics elements and using the definitionURL attribute. A semantics'' element has a number of children. The first child is the element which is having semantic information attached to it. The second and subsequent children must be either annotation-xml or annotation elements. In the first case it contains XML data which specifies the semantics of the first child, for example a section of OpenMath. In the second case it may be non-XML data for example a Maple command. Example The MathML segment below attaches the meaning of the BesselJ symbol from the specialFunctions content dictionary to the element <ci>J</ci>. <semantics> <ci>J</ci> <annotation-xml encoding="OpenMath"> <OMS cd="specialFunctions" name="BesselJ" xmlns="http://www.openmath.org/OpenMath"/> </annotation-xml> </semantics> A definitionURL'' attribute may be used with the elements csymbol, declare, semantics and any operator element. We may set the definitionURL to point at a symbol in an OpenMath content dictionary using the element csymbol, in the following way: If the new symbol we wish to define is denoted by <mi>new</mi> and the OpenMath symbol is defined at the URL http://www.openmath.org/cdName#newSymbol then we could use the following XML fragment: <csymbol encoding="OpenMath" definitionURL="http://www.openmath.org/cdName#newSymbol"> <mi>new</mi> </csymbol> Example The MathML element below could usefully be taken to have the same meaning as the BesselJ symbol in the specialFunctions content dictionary. <csymbol encoding="OpenMath" definitionURL="http://www.openmath.org/specialFunctions#BesselJ"> <mi>J</mi> </csymbol> An OpenMath Content Dictionary specifies meanings for a collection of names, such as the Bessel function J of the examples above. In an OpenMath fragment, each symbol appearing indicates both the name of the symbol and the content dictionary in which it occurs. That is, in order to specify an OpenMath symbol (this will be represented by an OMS element, with attributes set as follows) with name symbol_name from a content dictionary called cd_name, one could use the XML fragment: <OMS name="symbol_name" cd="cd_name"/> In the most common use of a symbol (to represent the application of a function to a number of arguments) this would be enclosed in an OMA element (standing for OpenMath Application), and the following children would be its arguments in a specified order. We use this to provide either an appropriate OpenMath XML fragment for a semantics annotation or URI for a definitionURL attribute. # Using semantics elements Given a document containing some new symbol, we may apply an XSLT style-sheet to that document to give a document which has equivalent but now semantically meaningful semantics elements. The semantics elements could have an annotation part which referenced a symbol in an OpenMath content dictionary thus endowing the element with semantic meaning. In the following example we replace a new element by the element <m:semantics>, as was done in [9]. Assuming that the required content dictionary (see Appendix [1]) were available this element would be semantically meaningful. Example Suppose we were to be given the following section of extended MathML, which contains the element <xm:BesselJ/>, undefined in the standard MathML: (we assume that m is the MathML namespace [7], xsl is the namespace for the XSLT elements, xm is a namespace which contains the names of the new elements and om is the namespace containing the names of the OpenMath elements). <m:apply> <xm:BesselJ/><m:cn>1</m:cn><m:cn type="rational">1<m:sep/>2</m:cn> </m:apply> If we apply an XSLT style sheet containing the following rule, then we shall replace all applications of <xm:BesselJ/> with appropriate <m:semantics> markup. <xsl:template match="m:apply[m:BesselJ]"> <m:apply> <m:semantics> <m:ci><m:mo>J</m:mo></m:ci> <m:annotation-xml encoding="OpenMath"> <om:OMS cd="specialFunctions" name="BesselJ"/> </m:annotation-xml> </m:semantics> <xsl:apply-templates select="[2]"/> <xsl:apply-templates select="[3]"/> </m:apply> </xsl:template>} These templates'' are effectively re-write rules which transform the input and, on application, may activate further re-write rules. Assuming that there were templates available which copied through (verbatim) <m:cn> elements, then this would transform the preceding extended MathML into the semantically meaningful element (assuming that the required content dictionary see Appendix [1] was available): <m:apply> <m:semantics> <m:ci><m:mo>J</m:mo></m:ci> <m:annotation-xml encoding="OpenMath"> <om:OMS cd="specialFunctions" name="BesselJ"/> </m:annotation-xml> </m:semantics> <m:cn>1</m:cn><m:cn type="rational">1<m:sep/>2</m:cn> </m:apply> We may have intended different behavior, for example, that the presentation information should render as . This is possible, and only requires that an alternative style-sheet is given, in this case: <xsl:template match="m:apply[m:BesselJ]"> <m:apply> <m:semantics> <m:mrow> <m:msub><m:mi>J</m:mi><xsl:apply-templates select="[2]"/></m:msub> <m:mfenced> <xsl:apply-templates select="[3]"/> </m:mfenced> </m:mrow> <m:annotation-xml encoding="OpenMath"> <om:OMS cd="specialFunctions" name="BesselJ"/> </m:annotation-xml> </m:semantics> <xsl:apply-templates select="[2]"/> <xsl:apply-templates select="[3]"/> </m:apply> </xsl:template> # Using definitionURL attributes We continue with the example of the previous section: <m:apply> <xm:BesselJ/><m:cn>1</m:cn><m:cn type="rational">1<m:sep/>2</m:cn> </m:apply> We add semantics using the definitionURL attribute in the manner discussed earlier. Valid semantically meaningful code for this example using this technique is: <m:apply> <m:csymbol definitionURL="http://www.openmath.org/specialFunctions#BesselJ"/> <m:cn>1</m:cn><m:cn type="rational">1<m:sep/>2</m:cn> </m:apply> We could achieve this using the following style sheet: <xsl:template match="m:apply[m:BesselJ]"> <m:csymbol definitionURL="http://www.openmath.org/specialFunctions#BesselJ"/> <xsl:apply-templates select="[2]"/> <xsl:apply-templates select="[3]"/> </xsl:template> We see that given a library of XSLT style sheets together with some resource which allows the storage of mathematical knowledge, for example a library of OpenMath Content Dictionaries, extension and indeed personalisation (relative to a set of XSLT stylesheets) of MathML is indeed possible. ## Macro-aware translation One of the uses made of MathML shall be the generation of TeX. If the semantic extentions introducing new elements, e.g. <mx:BesselJ>, are expanded prior to TeX generation, then the result will potentially lose usefull information. It is not unlikely that an author shall have TeX definitions corresponding to these functions, e.g. \BesselJ. The translation between TeX and MathML, with an automated correspondence between TeX definitions and XSLT rules has been investigated in [9]. We investigate how information may be stored in order to allow this translation in our paper [8]. # MathML in OpenMath There are two places in which MathML elements should occur naturally in OpenMath Content Dictionaries. First, we propose that the <Description> element be extended to contain two new subelements, <CommonName>" and \verb"<Notation>. The contents of the notation element(s) would be expressed as MathML. Secondly, we note that each definition element may have Commented Mathematical Properties (<CMP> elements). These commented mathematical properties are intended for human readers, not proof systems, and could be expressed naturally in MathML to facilitate browsing.1 Example Here we show MathML in use in the proposed <Notation> element. <CD> <CDName>Bessel</CDName> <CDURL>http://www.openmath.org/CDs/bessel.ocd</CDURL> <CDReviewDate>2001-07-22</CDReviewDate> <CDStatus>experimental</CDStatus> <CDDate>2001-07-22</CDDate> <CDVersion>1</CDVersion> <CDRevision>0</CDRevision> <CDUses> </CDUses> <Description> This content dictionary contains symbols to describe the bessel functions and associated functions. </Description> <CDDefinition> <Name>BesselJ</Name> <Description> <CommonName>Bessel Functions of Integer Order</CommonName> <Notation> <version> <img src="besseleqn.jpg"/> <tex>J_\xref{arg1}{\nu}(\xref{arg2}{z})</tex> $<mrow> <msub> <mi>J<mi> <mi xref="arg1">ν</mi> </msub> <mo>⁡</mo> <mfenced> <mi xref="arg2">z</mi> </mfenced> </mrow>$ </version> <FMP> <OMOBJ> <OMA> <OMS name="BesselJ" cd="bessel"/> <OMV id="arg1" name="nu"/> <OMV id="arg2" name="z"/> </OMA> </OMOBJ> </FMP> </Notation> The solutions to the differential equation ... </Description> ... </CDDefinition> ... </CD> These elements give various possibilities for how a symbol is intended to appear textually. Within a Notation element we shall allow 1. [1)] version elements There will be one of these for each notation used for this symbol. Each version element has some or all of the following children: 1. [i)] img element, a URI specifying an image (for example a gif file) giving some sort of generic form of the notation (to be used in pull-down menus and the like). 2. [ii)] tex element, This will contain a string which is easily translateable to TeX, this would be usefull for a tool which translates between extended MathML and Tex as suggested in section . 3. [iii)] math element, these contain presentation MathML, this specifies the notation in a browser understandable fashion. 2. [3)] FMP, these will normally be an application of the symbol to its arguments, the arguments being cross-linked with the relevant arguments in the corresponding math element. This is intended to be a template which specifies the ordering of the arguments. It should be possible for a computer algebra system to accurately comprehend extended presentation MathML, with no ambiguity assuming that the relevant Content Dictionaries were present and that the specified notational styles are followed. We discuss the use of <Notation> elements in more detail in our paper [8]. # Conclusion We have shown how extended tag sets can be useful in MathML. Expressions including new elements may be rewritten using XSLT stylesheets to produce either parallel markup with OpenMath annotations in <semantics> elements or to produce definitionURL attributes in <csymbol> elements. This allows semantically meaningful expressions to use extended mathematical concepts as though they were in the base MathML definition. We expect that writing expressions in this way shall be more readily adopted by human and machine authors of MathML than explicit, detailed <semantics> or <csymbol> markup. This may in fact be the only way that a significant proportion of MathML documents contain meaningful content in some years' time. Secondly, MathML notations may be usefully incorporated in OpenMath Content Dictionaries in such a way as to allow OpenMath objects to be rendered in different notational variants using MathML. ## Bibliography 1 Mathematical Markup Language (MathML) Version 2.0, R.Aubrooks, S.Buswell, S.Dalmass, S.Devitt, A.Diaz, R.Hunter, B.Smith, N.Soiffer, R.Sutor and S.Watt, http://www.w3.org/TR/2001/REC-MathML2-20010221/. 2 World Wide Web Consortium, http://www.w3.org. 3 W3Cs Editor/Browser, Amaya:http://www.w3.org/Amaya. 4 XSL Transformations (XSLT) Version 1.0, J.Clark, http://www.w3.org/TR/1999/REC-xslt-19991116/. 5 OpenMath: http://www.openmath.org 6 OMDoc: http://www.mathweb.org/omdoc/ 7 Namespaces in XML, T.Bray, D.Hollander, A.Laymen, http://www.w3.org/TR/1999/REC-xml-names-19990114/ 8 B.Naylor, S.Watt: Meta stylesheets for the conversion of Mathematical documents into multiple forms. To appear. 9 Igor Rodionov: Tools for MathML, MSc thesis, U Western Ontario, 2001. 10 Xuehong Li: XML and the Commnunication of Mathematical Objects, MSc Thesis, U Western Ontario, 1998. # 1 Appendix In this Appendix we contain a html form of a content dictionary which contains the definition of the symbol BesselJ. The content dictionary has been written as an XML file and then translated into the html style using an XSLT style-sheet designed for the task. In order that the MathML parts of the content dictionary render satisfactorily a MathML knowledgeable browser eg. Amaya [3], must be used. On the relationship between OpenMath and MathML This document was generated using the LaTeX2HTML translator Version 2K.1beta (1.47) Copyright © 1993, 1994, 1995, 1996, Nikos Drakos, Computer Based Learning Unit, University of Leeds. Copyright © 1997, 1998, 1999, Ross Moore, Mathematics Department, Macquarie University, Sydney. The translation was initiated by Bill Naylor on 2002-04-02 #### Footnotes ... browsing.1 Here we refer both to OpenMath CDDefinition elements and OMDoc definition elements	2014-03-08 11:44:11	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6197898387908936, "perplexity": 3007.3706074740644}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1393999654396/warc/CC-MAIN-20140305060734-00068-ip-10-183-142-35.ec2.internal.warc.gz"}
https://mathoverflow.net/questions/158276/kernel-of-the-induced-map-of-the-wedge-product	# Kernel of the induced map of the wedge product Let $A$ be a noetherian ring and let $M$ be a finitely generated $A$-module. Let $F$ be a free $A$-module and let $d: F \to M$ be a homomorphism which maps a basis of $F$ to a minimal set of generators of $M$. Consider the submodule $N$ of $\bigwedge^2 F$ which is generated by the elements $x \wedge y$ where $x \in \ker (d)$. It is clear that $N$ lies the kernel of the induced map $\hat{d}:\bigwedge^2 F \to \bigwedge^2M$. I am interested in the quotient $\ker(\hat{d}) / N$. My question: Are there some results concerning this object? For example in the case where $A$ is a local ring? Is there perhaps some geometric interpretation of it in the case where $A$ is a finitely generated algebra over some field $k$? You don't need most of your assumptions to ensure that your quotient is $0$. More generally, we have: (1) If $A$ is any commutative ring, and $f : M \to N$ is a surjective homomorphism of $A$-modules, then the kernel of the induced map $\wedge f : \wedge M \to \wedge N$ is the left ideal generated by $\mathrm{ker} f$ in the exterior algebra $\wedge M$. This ideal, of course, is a graded two-sided ideal, and its $2$-nd graded component (i.e., its intersection with $\wedge^2 M$) is precisely the $A$-linear span of all $x \wedge y$ with $x \in \mathrm{ker} f$ and $y \in M$.	2020-10-27 03:41:50	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9797854423522949, "perplexity": 35.471834380012204}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107893011.54/warc/CC-MAIN-20201027023251-20201027053251-00338.warc.gz"}
https://formulasearchengine.com/wiki/Conditional_entropy	# Conditional entropy Individual (H(X),H(Y)), joint (H(X,Y)), and conditional entropies for a pair of correlated subsystems X,Y with mutual information I(X; Y). In information theory, the conditional entropy (or equivocation) quantifies the amount of information needed to describe the outcome of a random variable ${\displaystyle Y}$ given that the value of another random variable ${\displaystyle X}$ is known. Here, information is measured in shannons, nats, or hartleys. The entropy of ${\displaystyle Y}$ conditioned on ${\displaystyle X}$ is written as ${\displaystyle H(Y\|X)}$. ## Definition If ${\displaystyle H(Y\|X=x)}$ is the entropy of the variable ${\displaystyle Y}$ conditioned on the variable ${\displaystyle X}$ taking a certain value ${\displaystyle x}$, then ${\displaystyle H(Y\|X)}$ is the result of averaging ${\displaystyle H(Y\|X=x)}$ over all possible values ${\displaystyle x}$ that ${\displaystyle X}$ may take. Given discrete random variables ${\displaystyle X}$ with domain ${\displaystyle {\mathcal {X}}}$ and ${\displaystyle Y}$ with domain ${\displaystyle {\mathcal {Y}}}$, the conditional entropy of ${\displaystyle Y}$ given ${\displaystyle X}$ is defined as:[1] {\displaystyle {\begin{aligned}H(Y\|X)\ &\equiv \sum _{x\in {\mathcal {X}}}\,p(x)\,H(Y\|X=x)\\&=-\sum _{x\in {\mathcal {X}}}p(x)\sum _{y\in {\mathcal {Y}}}\,p(y\|x)\,\log \,p(y\|x)\\&=-\sum _{x\in {\mathcal {X}}}\sum _{y\in {\mathcal {Y}}}\,p(x,y)\,\log \,p(y\|x)\\&=-\sum _{x\in {\mathcal {X}},y\in {\mathcal {Y}}}p(x,y)\log \,p(y\|x)\\&=-\sum _{x\in {\mathcal {X}},y\in {\mathcal {Y}}}p(x,y)\log {\frac {p(x,y)}{p(x)}}.\\&=\sum _{x\in {\mathcal {X}},y\in {\mathcal {Y}}}p(x,y)\log {\frac {p(x)}{p(x,y)}}.\\\end{aligned}}} Note: It is understood that the expressions 0 log 0 and 0 log (c/0) for fixed c>0 should be treated as being equal to zero. ${\displaystyle H(Y\|X)=0}$ if and only if the value of ${\displaystyle Y}$ is completely determined by the value of ${\displaystyle X}$. Conversely, ${\displaystyle H(Y\|X)=H(Y)}$ if and only if ${\displaystyle Y}$ and ${\displaystyle X}$ are independent random variables. ## Chain rule Assume that the combined system determined by two random variables X and Y has joint entropy ${\displaystyle H(X,Y)}$, that is, we need ${\displaystyle H(X,Y)}$ bits of information to describe its exact state. Now if we first learn the value of ${\displaystyle X}$, we have gained ${\displaystyle H(X)}$ bits of information. Once ${\displaystyle X}$ is known, we only need ${\displaystyle H(X,Y)-H(X)}$ bits to describe the state of the whole system. This quantity is exactly ${\displaystyle H(Y\|X)}$, which gives the chain rule of conditional entropy: ${\displaystyle H(Y\|X)\,=\,H(X,Y)-H(X)\,.}$ The chain rule follows from the above definition of conditional entropy: ${\displaystyle H(Y\|X)=\sum _{x\in {\mathcal {X}},y\in {\mathcal {Y}}}p(x,y)\log {\frac {p(x)}{p(x,y)}}}$ ${\displaystyle =-\sum _{x\in {\mathcal {X}},y\in {\mathcal {Y}}}p(x,y)\log \,p(x,y)+\sum _{x\in {\mathcal {X}},y\in {\mathcal {Y}}}p(x,y)\log \,p(x)}$ ${\displaystyle =H(X,Y)+\sum _{x\in {\mathcal {X}}}p(x)\log \,p(x)}$ ${\displaystyle =H(X,Y)-H(X).}$ ## Bayes' rule Bayes' rule for conditional entropy states ${\displaystyle H(Y\|X)\,=\,H(X\|Y)-H(X)+H(Y)\,.}$ Proof. ${\displaystyle H(Y\|X)=H(X,Y)-H(X)}$ and ${\displaystyle H(X\|Y)=H(Y,X)-H(Y)}$. Symmetry implies ${\displaystyle H(X,Y)=H(Y,X)}$. Subtracting the two equations implies Bayes' rule. QED. ## Generalization to quantum theory In quantum information theory, the conditional entropy is generalized to the conditional quantum entropy. The latter can take negative values, unlike its classical counterpart. Bayes' rule does not hold for conditional quantum entropy, since ${\displaystyle H(X,Y)\neq H(Y,X)}$.{{ safesubst:#invoke:Unsubst\|\|date=__DATE__ \|\$B= {{#invoke:Category handler\|main}}{{#invoke:Category handler\|main}}[citation needed] }} ## Other properties ${\displaystyle H(Y\|X)\leq H(Y)\,}$ ${\displaystyle H(X,Y)=H(X\|Y)+H(Y\|X)+I(X;Y),\qquad }$ ${\displaystyle H(X,Y)=H(X)+H(Y)-I(X;Y),\,}$ ${\displaystyle I(X;Y)\leq H(X),\,}$ where ${\displaystyle I(X;Y)}$ is the mutual information between ${\displaystyle X}$ and ${\displaystyle Y}$. ${\displaystyle H(Y\|X)=H(Y){\text{ and }}H(X\|Y)=H(X)\,}$ Although the specific-conditional entropy, ${\displaystyle H(X\|Y=y)}$, can be either less or greater than ${\displaystyle H(X\|Y)}$, ${\displaystyle H(X\|Y=y)}$ can never exceed ${\displaystyle H(X)}$. ## References 1. {{#invoke:citation/CS1\|citation \|CitationClass=book }}	2023-03-23 14:01:43	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 59, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.93864506483078, "perplexity": 685.1575064922209}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945168.36/warc/CC-MAIN-20230323132026-20230323162026-00693.warc.gz"}
https://www.futureschool.com/australian-curriculum/south-australia/mathematics-year-11-14-periodic-phenomena/	### SA Year 11 – 14: Periodic Phenomena # TOPIC TITLE 1 Study Plan Study plan – Year 11 – 14: Periodic Phenomena Objective: On completion of the course formative assessment a tailored study plan is created identifying the lessons requiring revision. 2 Trig complementary angles Complementary angle results. Objective: On completion of the lesson the student will understand how to establish the complementary angle results for the sine and cosine ratios and then how to use these results to solve trig equations. 3 Trig identities Trigonometric identities Objective: On completion of the lesson the student will be able to simplify trigonometrical expressions and solve trigonometry equations using the knowledge of trig identities. 4 Trig larger angles Angles of any magnitude Objective: On completion of the lesson the student will be able to find the trigonometric values of angles of any magnitude by assigning angles to the four quadrants of the circle. 5 Trig larger angles Trigonometric ratios of 0°, 90°, 180°, 270° and 360° Objective: On completion of the lesson the student will learn how to find the Trigonometric Ratios of 0, 90, 180, 270 and 360 degrees. 6 Graph sine Graphing the trigonometric ratios – I Sine curve. Objective: On completion of the lesson the student will recognise and draw the sine curve exploring changes in amplitude and period. 7 Graph cosine Graphing the trigonometric ratios – II Cosine curve. Objective: On completion of the lesson the student will know how to recognise and draw the cosine curve exploring changes in amplitude and period. 8 Graphs tan curve Graphing the trigonometric ratios – III Tangent curve. Objective: On completion of the lesson the student will know how to recognise and draw the tan curve. 9 Trig larger angles Using one ratio to find another. Objective: On completion of the lesson the student will find other trig ratios given one trig ratio and to work with angles of any magnitude. 10 Trig equations Solving trigonometric equations – Type I. Objective: On completion of the lesson the student will solve simple trig equations with restricted domains. 11 Trig equations Solving trigonometric equations – Type II. Objective: On completion of the lesson the student will solve trig equations with multiples of theta and restricted domains. 12 Trig equations Solving trigonometric equations – Type III. Objective: On completion of the lesson the student will solve trig equations with two trig ratios and restricted domains. 13 Trigonometry Sin(A+B) etc sum and difference identities (Stage 2) Objective: On completion of the lesson the student will be using the reference triangles for 30, 45 and 60 degrees with the sum and difference of angles to find additional exact values of trigonometric ratios. 14 Trigonometry Double angle formulas (Stage 2) Objective: On completion of the lesson the student will derive and use the double angle trig identities. 15 Trigonometry Half angle identities (Stage 2) Objective: On completion of the lesson the student will derive and use the power reducing formulas and the half angle trig identities. 16 Trigonometry t Formulas (Stage 2) Objective: On completion of the lesson the student will solve trig equations using the t substitution. 17 Polar coordinates Plotting polar coordinates and converting polar to rectangular Objective: On completion of the lesson the student will understand the polar coordinate system and relate this to the rectangular coordinate system. 18 Polar coordinates Converting rectangular coordinates to polar form Objective: On completion of the lesson the student will understand the polar coordinate system and report these from rectangular coordinates. 19 Exam Exam – Year 11 – 14: Periodic Phenomena Objective: Exam	2020-08-06 07:40:16	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8993027210235596, "perplexity": 1701.1752337449957}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439736883.40/warc/CC-MAIN-20200806061804-20200806091804-00188.warc.gz"}
http://cpr-condmat-statmech.blogspot.com/2013/06/13063806-tadeusz-kosztoowicz.html	## Cattaneo--type subdiffusion--reaction equation [PDF] Subdiffusion in a system in which mobile particles $A$ can chemically react with static particles $B$ according to the rule $A+B\rightarrow B$ is considered within a persistent random walk model. This model, which assumes a correlation between successive steps of particles, provides hyperbolic Cattaneo normal diffusion or fractional subdiffusion equations for a system without chemical reactions. Starting with the difference equation, which describes a persistent random walk in a system with chemical reactions, using the generating function method and the continuous time random walk formalism, we will derive the Cattaneo--type subdiffusion differential equation with fractional time derivatives in which the chemical reactions mentioned above are taken into account. We will also find its solution over a long time limit. Based on the obtained results, we will find the Cattaneo--type subdiffusion--reaction equation in the case in which mobile particles of species $A$ and $B$ can chemically react according to a more complicated rule. View original: http://arxiv.org/abs/1306.3806	2017-08-16 13:19:03	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6636251211166382, "perplexity": 574.0760735736781}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886101966.48/warc/CC-MAIN-20170816125013-20170816145013-00643.warc.gz"}
https://zbmath.org/authors/?q=ai%3Ahale.jack-k	zbMATH — the first resource for mathematics Hale, Jack Kenneth Compute Distance To: Author ID: hale.jack-k Published as: Hale, J.; Hale, J. K.; Hale, Jack; Hale, Jack K. External Links: MGP · Wikidata · dblp · GND Documents Indexed: 242 Publications since 1954, including 26 Books Reviewing Activity: 3 Reviews Biographic References: 7 Publications all top 5 Co-Authors 108 single-authored 17 Raugel, Geneviève 10 Chow, Shui-Nee 8 Verduyn Lunel, Sjoerd M. 7 Huang, Wenzhang 5 LaSalle, Joseph Pierre 5 Lin, Xiaobiao 4 Cruz, Marianito A. 4 Rocha, Carlos 4 Stokes, Arnold P. 3 Cesari, Lamberto 3 Fusco, Giorgio 3 Magalhães, Luis T. 3 Mallet-Paret, John Joseph 3 Muniz Oliva, Waldyr 3 Sakamoto, Kunimochi 3 Táboas, Plácido Zoega 2 Arrieta, José M. 2 Chen, Xu-Yan 2 Cholewa, Jan W. 2 De Oliveira, Jose Carlos Fernandes 2 Fan, Haitao 2 Infante, Ettore F. 2 Ladeira, Luiz A. C. 2 Lopes, Orlando Francisco 2 Martinez-Amores, Pedro 2 Nolasco de Carvalho, Alexandre 2 Peletier, Lambertus Adrianus 2 Seifert, George 2 Troy, William C. 2 Wiener, Joseph 1 Afraimovich, Valentin S. 1 Bancroft, Stephen 1 Carr, Jack 1 Chen, Mingxiang 1 Chipot, Michel 1 Galves, A. P. T. 1 Gambill, Robert A. 1 González, José Domingo Salazar 1 Graef, John R. 1 Han, Qing 1 Imaz, Carlos 1 Ivanov, Anatoli F. 1 Izé, A. F. 1 Kato, Junji 1 Kocak, Huseyin 1 Massatt, Paul 1 Mawhin, Jean L. 1 Meyer, Kenneth R. 1 Onuchic, Nelson 1 Pavlu, Luiz Carlos 1 Perelló, Carles 1 Perissinotto, Anizio jun. 1 Rodringues, Hildebrando M. 1 Rodriques, Hildebrando M. 1 Rybakowski, Krzysztof P. 1 Salazar, Domingo 1 Scheurle, Jürgen 1 Slemrod, Marshall 1 Somolinos, Alfredo S. 1 Spezamiglio, Adalberto 1 Stavrakakis, Nikolaos M. 1 Sternberg, Natalia 1 Sweet, Daniel 1 Tan, Bin 1 Tanaka, Sueli M. 1 Tsen, Fu-Shiang Peter 1 Vegas, José Manuel 1 Waltman, Paul Elvis 1 Wang, Xiaodong 1 Weedermann, Marion 1 Xun, Jianping 1 Zhang, Weinian all top 5 Serials 26 Journal of Differential Equations 13 Journal of Mathematical Analysis and Applications 9 Archive for Rational Mechanics and Analysis 9 Nonlinear Analysis. Theory, Methods & Applications 7 Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 7 Journal of Dynamics and Differential Equations 5 Applied Mathematical Sciences 4 Boletín de la Sociedad Matemática Mexicana. Segunda Serie 4 SIAM Journal on Mathematical Analysis 4 Rivista di Matematica della Università di Parma 3 Applicable Analysis 3 Annali di Matematica Pura ed Applicata. Serie Quarta 3 Journal de Mathématiques Pures et Appliquées. Neuvième Série 3 Proceedings of the National Academy of Sciences of the United States of America 3 Resenhas do Instituto do Matemática e Estatística da Universidade de São Paulo 2 ZAMP. Zeitschrift für angewandte Mathematik und Physik 2 Funkcialaj Ekvacioj. Serio Internacia 2 Illinois Journal of Mathematics 2 Proceedings of the American Mathematical Society 2 Tohoku Mathematical Journal. Second Series 2 Transactions of the American Mathematical Society 2 Dynamic Systems and Applications 2 Annals of Mathematics Studies 1 Computer Methods in Applied Mechanics and Engineering 1 International Journal of Systems Science 1 Journal of Computational Physics 1 Journal of Mathematical Biology 1 Journal of Mathematical Physics 1 Revue Roumaine de Mathématiques Pures et Appliquées 1 Ukraïns’kyĭ Matematychnyĭ Zhurnal 1 Mathematics of Computation 1 Acta Mexicana de Ciencia y Tecnologia 1 Applied Mathematics and Computation 1 Hiroshima Mathematical Journal 1 Memoirs of the American Mathematical Society 1 Quarterly of Applied Mathematics 1 Rendiconti del Circolo Matemàtico di Palermo. Serie II 1 Physica D 1 RAIRO. Modélisation Mathématique et Analyse Numérique 1 Japan Journal of Applied Mathematics 1 IMA Journal of Mathematical Control and Information 1 Applied Mathematics Letters 1 CWI Quarterly 1 Journal of Integral Equations and Applications 1 Communications in Partial Differential Equations 1 SIAM Journal on Applied Mathematics 1 SIAM Review 1 The Canadian Applied Mathematics Quarterly 1 Zeszyty Naukowe Uniwersytetu Jagiellońskiego. Universitatis Iagellonicae Acta Mathematica 1 Dynamics of Continuous, Discrete and Impulsive Systems 1 Revista Matemática Complutense 1 Annals of Mathematics. Second Series 1 Dynamics of Continuous, Discrete & Impulsive Systems. Series A. Mathematical Analysis 1 Bollettino della Unione Matematica Italiana. Series IV 1 Journal of Mathematics and Mechanics 1 Contemporary Mathematics 1 Fields Institute Communications 1 Grundlehren der Mathematischen Wissenschaften 1 Mathematical Surveys and Monographs 1 NATO ASI Series. Series F. Computer and Systems Sciences 1 Regional Conference Series in Mathematics 1 Texts in Applied Mathematics 1 Journal of Fixed Point Theory and Applications 1 Journal of Rational Mechanics and Analysis 1 Anais da Academia Brasileira de Ciências 1 Abhandlungen der Deutschen Akademie der Wissenschaften zu Berlin, Klasse für Mathematik, Physik und Technik all top 5 Fields 121 Ordinary differential equations (34-XX) 70 Partial differential equations (35-XX) 47 Dynamical systems and ergodic theory (37-XX) 22 Operator theory (47-XX) 13 Global analysis, analysis on manifolds (58-XX) 10 General and overarching topics; collections (00-XX) 8 Numerical analysis (65-XX) 8 Systems theory; control (93-XX) 5 Mechanics of particles and systems (70-XX) 4 Biology and other natural sciences (92-XX) 3 History and biography (01-XX) 3 Difference and functional equations (39-XX) 3 Integral equations (45-XX) 3 Mechanics of deformable solids (74-XX) 2 Calculus of variations and optimal control; optimization (49-XX) 2 General topology (54-XX) 1 Functional analysis (46-XX) 1 Computer science (68-XX) 1 Information and communication theory, circuits (94-XX) Citations contained in zbMATH 223 Publications have been cited 14,572 times in 10,717 Documents Cited by Year Introduction to functional differential equations. Zbl 0787.34002 Hale, Jack K.; Verduyn Lunel, Sjoerd M. 1993 Theory of functional differential equations. 2nd ed. Zbl 0352.34001 Hale, Jack 1977 Asymptotic behavior of dissipative systems. Zbl 0642.58013 Hale, Jack K. 1988 Ordinary differential equations. Zbl 0186.40901 Hale, Jack K. 1969 Methods of bifurcation theory. Zbl 0487.47039 Chow, Shui-Nee; Hale, Jack K. 1982 Ordinary differential equations. 2nd ed. Zbl 0433.34003 Hale, Jack K. 1980 Some remarks on averaging and integral manifolds in evolutionary equations. Zbl 0425.34048 Hale, J. K. 1977 Phase space for retarded equations with infinite delay. Zbl 0383.34055 Hale, Jack K.; Kato, Junji 1978 Functional differential equations. Zbl 0222.34003 Hale, J. 1971 Persistence in infinite-dimensional systems. Zbl 0692.34053 Hale, Jack K.; Waltman, Paul 1989 Dynamics and bifurcations. Zbl 0745.58002 Hale, Jack K.; Koçak, Hüseyin 1991 Oscillations in nonlinear systems. Zbl 0115.07401 Hale, J. K. 1963 Dynamical systems and stability. Zbl 0179.13303 Hale, J. K. 1969 An example of bifurcation to homoclinic orbits. Zbl 0439.34035 Chow, Shui-Nee; Hale, Jack K.; Mallet-Paret, John 1980 Slow-motion manifolds, dormant instability, and singular perturbations. Zbl 0684.34055 Fusco, G.; Hale, J. K. 1989 Competition for fluctuating nutrient. Zbl 0525.92024 Hale, J. K.; Somolinos, A. S. 1983 Forward and backward continuation for neutral functional differential equations. Zbl 0213.36901 Hale, J. K. 1971 Geometric theory of functional-differential equations. Zbl 0189.39904 Hale, J. K. 1967 Existence, uniqueness and continuous dependence for hereditary systems. Zbl 0194.41002 Hale, J. K.; Cruz, M. A. 1970 A damped hyperbolic equation with critical exponent. Zbl 0815.35067 Arrieta, José; Carvalho, Alexandre N.; Hale, Jack K. 1992 Reaction-diffusion equation on thin domains. Zbl 0840.35044 Hale, Jack K.; Raugel, Geneviève 1992 A class of functional equations of neutral type. Zbl 0179.20501 Hale, Jack K.; Meyer, Kenneth R. 1967 Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation. Zbl 0666.35012 Hale, Jack K.; Raugel, Geneviève 1988 Stability in linear delay equations. Zbl 0569.34061 Hale, Jack K.; Infante, Ettore F.; Tsen, Fu-Shiang Peter 1985 Stability of functional differential equations of neutral type. Zbl 0191.38901 Cruz, Marianito A.; Hale, J. K. 1970 Theory of functional differential equations. Transl. from the English. Zbl 1092.34500 Hale, Jack 1984 Sufficient conditions for stability and instability of autonomous functional-differential equations. Zbl 0135.30301 Hale, J. K. 1965 Partial neutral functional differential equations. Zbl 0817.35119 Hale, Jack K. 1994 Strong stabilization of neutral functional differential equations. Zbl 1005.93026 Hale, Jack K.; Verduyn Lunel, Sjoerd M. 2002 Integral manifolds of perturbed differential systems. Zbl 0163.32804 Hale, J. K. 1961 Convergence in gradient-like systems with applications to PDE. Zbl 0751.58033 Hale, Jack K.; Raugel, Geneviève 1992 Coincidence degree and periodic solutions of neutral equations. Zbl 0274.34070 Hale, Jack K.; Mawhin, J. 1974 Upper semicontinuity of attractors for approximations of semigroups and partial differential equations. Zbl 0666.35013 Hale, Jack K.; Lin, Xiao-Biao; Raugel, Geneviève 1988 An introduction to infinite dimensional dynamical systems - geometric theory. With an append. by Krzysztof P. Rybakowski. Zbl 0533.58001 Hale, Jack K.; Magalhães, Luis T.; Oliva, Waldyr M. 1984 Lower semicontinuity of attractors of gradient systems and applications. Zbl 0712.47053 Hale, Jack K.; Raugel, Geneviève 1989 Alternative problems for nonlinear functional equations. Zbl 0159.20001 Bancroft, Stephen; Hale, Jack K.; Sweet, Daniel 1968 Global geometry of the stable regions for two delay differential equations. Zbl 0787.34062 Hale, Jack K.; Huang, Wenzhang 1993 Fixed point theorems and dissipative processes. Zbl 0256.34069 Hale, Jack K.; Lopes, Orlando 1973 Attracting manifolds for evolutionary equations. Zbl 1098.34552 Hale, Jack K. 1997 Dynamics in infinite dimensions. Appendix by Krzysztof P. Rybakowski. 2nd ed. Zbl 1002.37002 Hale, Jack K.; Magalhães, Luis T.; Oliva, Waldyr M. 2002 Diffusive coupling, dissipation, and synchronization. Zbl 1091.34532 Hale, Jack K. 1997 Large diffusivity and asymptotic behavior in parabolic systems. Zbl 0602.35059 Hale, Jack K. 1986 Periodic solutions of a class of hyperbolic equations containing a small parameter. Zbl 0152.10002 Hale, J. K. 1967 Averaging methods for differential equations with retarded arguments and a small parameter. Zbl 0151.10302 Hale, J. K. 1966 Asymptotic behaviour and dynamics in infinite dimensions. Zbl 0653.35006 Hale, J. K. 1985 A damped hyperbolic equation on thin domains. Zbl 0761.35052 Hale, Jack K.; Raugel, Geneviève 1992 Heteroclinic orbits for retarded functional differential equations. Zbl 0611.34074 Hale, Jack K.; Lin, X.-B. 1986 A nonlinear parabolic equation with varying domain. Zbl 0569.35048 Hale, Jack K.; Vegas, Jose 1984 Synchronization in lattices of coupled oscillators. Zbl 1194.34056 Afraimovich, V. S.; Chow, S.-N.; Hale, J. K. 1997 Applications of generic bifurcation. I. Zbl 0328.47036 Chow, Shui-Nee; Hale, Jack K.; Mallet-Paret, John 1975 Functional differential equations with infinite delays. Zbl 0289.34107 Hale, Jack K. 1974 Existence and stability of transition layers. Zbl 0669.34027 Hale, Jack K.; Sakamoto, Kunimochi 1988 On the behavior of the solutions of linear periodic differential systems near resonance points. Zbl 0117.30403 Hale, J. K. 1960 Invariant foliations for $$C^1$$ semigroups in Banach spaces. Zbl 0994.34047 Chen, Xu-Yan; Hale, Jack K.; Tan, Bin 1997 Abelian integrals and bifurcation theory. Zbl 0587.34033 Carr, Jack; Chow, Shui-Nee; Hale, Jack K. 1985 Exact homoclinic and heteroclinic solutions of the Gray-Scott model for autocatalysis. Zbl 0965.34037 Hale, J. K.; Peletier, L. A.; Troy, W. C. 2000 Generic bifurcation with applications. Zbl 0382.34013 Hale, J. K. 1977 Periodic and almost periodic solutions of functional-differential equations. Zbl 0129.06006 Hale, J. K. 1964 Shadow systems and attractors in reaction-diffusion equations. Zbl 0667.34072 Hale, Jack K.; Sakamoto, K. 1989 Interaction of damping and forcing in a second order equation. Zbl 0369.34014 Hale, Jack K.; Taboas, Placido Z. 1978 Linear functional-differential equations with constant coefficients. Zbl 0143.30702 Hale, J. K. 1963 Stability and control of feedback systems with time delays. Zbl 1052.93028 Hale, J. K.; Verduyn Lunel, S. M. 2003 Averaging in infinite dimensions. Zbl 0755.45012 Hale, Jack K.; Verduyn Lunel, Sjoerd M. 1990 Symbolic dynamics and nonlinear semiflows. Zbl 0632.58027 Hale, Jack K.; Lin, Xiao-Biao 1986 Stable equilibria in a scalar parabolic equation with variable diffusion. Zbl 0597.35040 Fusco, G.; Hale, J. K. 1985 Applications of generic bifurcation. II. Zbl 0346.47050 Chow, Shui-Nee; Hale, Jack K.; Mallet-Paret, John 1976 Critical cases for neutral functional differential equations. Zbl 0223.34057 Hale, Jack K. 1971 The neighborhood of a singular point of functional differential equations. Zbl 0136.07901 Hale, J. K.; Perello, C. 1964 Coupled oscillators on a circle. Zbl 0857.35127 Hale, Jack K. 1994 Structural stability for time-periodic one-dimensional parabolic equations. Zbl 0779.35061 Chen, Mingxiang; Chen, Xu-Yan; Hale, Jack K. 1992 Partial differential equations on thin domains. Zbl 0785.35050 Hale, J. K.; Raugel, G. 1992 Lower semicontinuity of the attractor for a singularly perturbed hyperbolic equation. Zbl 0752.35034 Hale, Jack K.; Raugel, Geneviève 1990 Onset of chaos in differential delay equations. Zbl 0644.65050 Hale, Jack K.; Sternberg, Natalia 1988 Flows on centre manifolds for scalar functional differential equations. Zbl 0582.34058 Hale, Jack K. 1985 Theory of a general class of dissipative processes. Zbl 0238.34098 Hale, J. K.; LaSalle, J. P.; Slemrod, Marshall 1972 Asymptotic behavior of neutral functional differential equations. Zbl 0211.12301 Hale, J. K.; Cruz, M. A. 1969 Attractors for dissipative evolutionary equations. Zbl 0938.34536 Hale, Jack K.; Raugel, Geneviève 1993 On the asymptotic behavior of solutions of a class of differential equations. Zbl 0135.30002 Hale, J. K.; Onuchic, N. 1963 Strongly limit-compact maps. Zbl 0297.47048 Chow, Shui-Nee; Hale, Jack K. 1974 Large diffusion with dispersion. Zbl 0781.35028 Carvalho, Alexandre N.; Hale, Jack K. 1991 Stability in neutral equations. Zbl 0359.34070 Hale, Jack K.; Martinez-Amores, Pedro 1977 Extended dynamical systems and stability theory. Zbl 0155.42301 Hale, J. K.; Infante, E. F. 1967 Regularity, determining modes and Galerkin methods. Zbl 1043.35048 Hale, Jack K.; Raugel, Geneviève 2003 Effects of small delays on stability and control. Zbl 0983.34070 Hale, Jack K.; Verduyn Lunel, Sjoerd M. 2001 Stability and instability in the Gray-Scott model: The case of equal diffusivities. Zbl 0936.92034 Hale, J. K.; Peletier, L. A.; Troy, W. C. 1999 A reaction-diffusion equation on a thin $$L$$-shaped domain. Zbl 0828.35055 Hale, Jack K.; Raugel, Geneviève 1995 The effect of rapid oscillations in the dynamics of delay equations. Zbl 0751.34037 Hale, Jack K.; Verduyn Lunel, Sjoerd M. 1991 Eigenvalue problems for nonsmoothly perturbed domains. Zbl 0736.35073 Arrieta, José M.; Hale, Jack K.; Han, Qing 1991 Bifurcation from simple eigenvalues for several parameter families. Zbl 0383.34050 Hale, Jack K. 1978 Period doubling in singularly perturbed delay equations. Zbl 0817.34040 Hale, Jack K.; Huang, Wenzhang 1994 Large-time behavior in inhomogeneous conservation laws. Zbl 0807.35085 Fan, Haitao; Hale, Jack K. 1993 From sine waves to square waves in delay equations. Zbl 0764.34048 Chow, S.-N.; Hale, J. K.; Huang, W. 1992 Functional differential equations. Zbl 0222.34063 Hale, Jack K. 1971 Varying boundary conditions with large diffusivity. Zbl 0557.35078 Hale, Jack K.; Rocha, Carlos 1987 Dynamic behavior from bifurcation equations. Zbl 0454.34035 De Oliveira, Jose C. Fernandes; Hale, Jack K. 1980 Periodic solutions of autonomous equations. Zbl 0397.34091 Chow, Shui-Nee; Hale, Jack K. 1978 Behavior near constant solutions of functional differential equations. Zbl 0273.34049 Hale, Jack K. 1974 Behavior of solutions near integral manifolds. Zbl 0093.08903 Hale, J. K.; Stokes, A. P. 1960 Interaction of diffusion and boundary conditions. Zbl 0661.35047 Hale, Jack K.; Rocha, Carlos 1987 Asymptotic behavior of gradient-like systems. Zbl 0542.34027 Hale, Jack K.; Massatt, Paul 1982 Persistence of periodic orbits for perturbed dissipative dynamical systems. Zbl 1263.35016 Hale, Jack K.; Raugel, Geneviève 2012 A modified Poincaré method for the persistence of periodic orbits and applications. Zbl 1189.35018 Hale, Jack K.; Raugel, Geneviève 2010 Some results in asymptotic fixed point theory. Zbl 1176.47039 Hale, Jack K.; Lopes, Orlando 2008 From point dissipative to compact dissipative. Addendum to “Some counterexamples in dissipative systems”. Zbl 1110.37059 Cholewa, J. W.; Hale, J. K. 2007 History of delay equations. Zbl 1130.34037 Hale, J. K. 2006 Dissipation and compact attractors. Zbl 1119.37046 Hale, Jack K. 2006 Eigenvalues and perturbed domains. Zbl 1099.35068 Hale, J. K. 2005 A Lyapunov-Schmidt method for transition layers in reaction-diffusion systems. Zbl 1093.35006 Hale, Jack K.; Sakamoto, Kunimochi 2005 Stability and gradient dynamical systems. Zbl 1070.37055 Hale, Jack K. 2004 On perturbations of delay-differential equations with periodic orbits. Zbl 1071.34074 Hale, Jack K.; Weedermann, Marion 2004 On uniformity of exponential dichotomies for delay equations. Zbl 1063.34051 Hale, Jack K.; Zhang, Weinian 2004 Stability and control of feedback systems with time delays. Zbl 1052.93028 Hale, J. K.; Verduyn Lunel, S. M. 2003 Regularity, determining modes and Galerkin methods. Zbl 1043.35048 Hale, Jack K.; Raugel, Geneviève 2003 Strong stabilization of neutral functional differential equations. Zbl 1005.93026 Hale, Jack K.; Verduyn Lunel, Sjoerd M. 2002 Dynamics in infinite dimensions. Appendix by Krzysztof P. Rybakowski. 2nd ed. Zbl 1002.37002 Hale, Jack K.; Magalhães, Luis T.; Oliva, Waldyr M. 2002 Bifurcation from families of periodic solutions. Zbl 1071.70014 Hale, Jack K.; Táboas, Plácido 2002 Effects of small delays on stability and control. Zbl 0983.34070 Hale, Jack K.; Verduyn Lunel, Sjoerd M. 2001 On monodromy matrix computation. Zbl 0982.70004 Wang, Xiaodong; Hale, Jack K. 2001 Some problems in FDE. Zbl 0984.34061 Hale, Jack K. 2001 Attractors and dynamics in partial differential equations. Zbl 1004.37062 Hale, Jack K. 2001 Exact homoclinic and heteroclinic solutions of the Gray-Scott model for autocatalysis. Zbl 0965.34037 Hale, J. K.; Peletier, L. A.; Troy, W. C. 2000 Square and pulse waves with two delays. Zbl 0949.34062 Hale, J. K.; Tanaka, S. M. 2000 Some counterexamples in dissipative systems. Zbl 0983.37094 Cholewa, J. W.; Hale, J. K. 2000 Effects of time delays on the dynamics of feedback systems. Zbl 0979.34054 Hale, Jack K.; Verduyn Lunel, Sjoerd M. 2000 Stability and instability in the Gray-Scott model: The case of equal diffusivities. Zbl 0936.92034 Hale, J. K.; Peletier, L. A.; Troy, W. C. 1999 Multiple internal layer solutions generated by spatially oscillatory perturbations. Zbl 0978.35019 Hale, Jack K.; Lin, Xiao-Biao 1999 Attractors of some reaction diffusion problems. Zbl 0937.35091 Hale, Jack K.; González, José Domingo Salazar 1999 Boundary layers in a semilinear parabolic problem. Zbl 0954.35086 Hale, Jack K.; Salazar, Domingo 1999 Dynamics of a scalar parabolic equation. Zbl 0979.35069 Hale, Jack K. 1999 Synchronization by diffusive coupling. Zbl 1002.34035 Hale, Jack K. 1998 Dynamics of numerical approximations. Zbl 0911.65063 Hale, Jack K. 1998 Attracting manifolds for evolutionary equations. Zbl 1098.34552 Hale, Jack K. 1997 Diffusive coupling, dissipation, and synchronization. Zbl 1091.34532 Hale, Jack K. 1997 Synchronization in lattices of coupled oscillators. Zbl 1194.34056 Afraimovich, V. S.; Chow, S.-N.; Hale, J. K. 1997 Invariant foliations for $$C^1$$ semigroups in Banach spaces. Zbl 0994.34047 Chen, Xu-Yan; Hale, Jack K.; Tan, Bin 1997 Dynamics of a scalar parabolic equation. Zbl 0924.35002 Hale, Jack K. 1997 Periodic solutions of singularly perturbed delay equations. Zbl 0841.34080 Hale, Jack K.; Huang, Wenzhang 1996 Traveling waves as limits of solutions on bounded domains. Zbl 0874.35008 Fusco, Giorgio; Hale, Jack K.; Xun, Jianping 1996 Hopf bifurcation analysis for hybrid systems. Zbl 0928.34050 Hale, Jack K.; Huang, Wenzhang 1996 A reaction-diffusion equation on a thin $$L$$-shaped domain. Zbl 0828.35055 Hale, Jack K.; Raugel, Geneviève 1995 Variation of constants for hybrid systems of functional differential equations. Zbl 0830.34055 Hale, Jack K.; Huang, Wenzhang 1995 Attractors in inhomogeneous conservation laws and parabolic regularizations. Zbl 0831.35103 Fan, Haitao; Hale, Jack K. 1995 Effects of delays on dynamics. Zbl 0834.34084 Hale, Jack K. 1995 Partial neutral functional differential equations. Zbl 0817.35119 Hale, Jack K. 1994 Coupled oscillators on a circle. Zbl 0857.35127 Hale, Jack K. 1994 Period doubling in singularly perturbed delay equations. Zbl 0817.34040 Hale, Jack K.; Huang, Wenzhang 1994 Numerical dynamics. Zbl 0808.34061 Hale, Jack K. 1994 Introduction to functional differential equations. Zbl 0787.34002 Hale, Jack K.; Verduyn Lunel, Sjoerd M. 1993 Global geometry of the stable regions for two delay differential equations. Zbl 0787.34062 Hale, Jack K.; Huang, Wenzhang 1993 Attractors for dissipative evolutionary equations. Zbl 0938.34536 Hale, Jack K.; Raugel, Geneviève 1993 Large-time behavior in inhomogeneous conservation laws. Zbl 0807.35085 Fan, Haitao; Hale, Jack K. 1993 On a high order differential delay equation. Zbl 0774.34053 Hale, Jack K.; Ivanov, Anatoli F. 1993 Global attractor and convergence for one-dimensional semilinear thermoelasticity. Zbl 0776.35004 Hale, Jack K.; Perissinotto, Anizio jun. 1993 Attractors and convergence of PDE on thin L-shaped domains. Zbl 0806.35076 Hale, J. K.; Raugel, G. 1993 Differentiability with respect to delays for a retarded reaction-diffusion equation. Zbl 0811.35060 Hale, Jack K.; Ladeira, Luiz A. C. 1993 Limits of semigroups depending on parameters. Zbl 0863.58046 Hale, Jack K.; Raugel, Geneviève 1993 Ordinary and delay differential equations 1. Proceedings of the international conference on theory and applications of differential equations, held at the University of Texas-Pan American, Edinburg, TX, USA, on May 15-18, 1991. Zbl 0780.00043 Wiener, Joseph (ed.); Hale, Jack K. (ed.) 1993 A damped hyperbolic equation with critical exponent. Zbl 0815.35067 Arrieta, José; Carvalho, Alexandre N.; Hale, Jack K. 1992 Reaction-diffusion equation on thin domains. Zbl 0840.35044 Hale, Jack K.; Raugel, Geneviève 1992 Convergence in gradient-like systems with applications to PDE. Zbl 0751.58033 Hale, Jack K.; Raugel, Geneviève 1992 A damped hyperbolic equation on thin domains. Zbl 0761.35052 Hale, Jack K.; Raugel, Geneviève 1992 Structural stability for time-periodic one-dimensional parabolic equations. Zbl 0779.35061 Chen, Mingxiang; Chen, Xu-Yan; Hale, Jack K. 1992 Partial differential equations on thin domains. Zbl 0785.35050 Hale, J. K.; Raugel, G. 1992 From sine waves to square waves in delay equations. Zbl 0764.34048 Chow, S.-N.; Hale, J. K.; Huang, W. 1992 Square and pulse waves in matrix delay differential equations. Zbl 0753.34048 Hale, J. K.; Huang, W. 1992 Dynamics and numerics. Zbl 0769.58053 Hale, J. K. 1992 Partial differential equations. Proceedings of the international conference on theory and applications of differential equations held at the University of Texas-Pan American, Edinburg, TX (USA), May 15-18, 1991. Zbl 0785.00039 Wiener, Joseph (ed.); Hale, Jack K. (ed.) 1992 In memoriam Lamberto Cesari (1910-1990). Zbl 0755.01032 Hale, Jack K. 1992 Oscillation and dynamics in delay equations. Proceedings of an AMS special session, held at the 863rd meeting of the American Mathematical Society in San Francisco, CA, USA, on January 16-19, 1991. Zbl 0745.00045 Graef, John R. (ed.); Hale, Jack K. (ed.) 1992 Dynamics and bifurcations. Zbl 0745.58002 Hale, Jack K.; Koçak, Hüseyin 1991 Large diffusion with dispersion. Zbl 0781.35028 Carvalho, Alexandre N.; Hale, Jack K. 1991 The effect of rapid oscillations in the dynamics of delay equations. Zbl 0751.34037 Hale, Jack K.; Verduyn Lunel, Sjoerd M. 1991 Eigenvalue problems for nonsmoothly perturbed domains. Zbl 0736.35073 Arrieta, José M.; Hale, Jack K.; Han, Qing 1991 Differentiability with respect to delays. Zbl 0735.34045 Hale, Jack K.; Ladeira, Luiz A. C. 1991 Dynamics and delays. Zbl 0735.34051 Hale, Jack K. 1991 Averaging in FDE and PDE. Zbl 0796.34030 Hale, Jack K.; Verduyn Lunel, Sjoerd M. 1991 Averaging in infinite dimensions. Zbl 0755.45012 Hale, Jack K.; Verduyn Lunel, Sjoerd M. 1990 Lower semicontinuity of the attractor for a singularly perturbed hyperbolic equation. Zbl 0752.35034 Hale, Jack K.; Raugel, Geneviève 1990 Persistence in infinite-dimensional systems. Zbl 0692.34053 Hale, Jack K.; Waltman, Paul 1989 Slow-motion manifolds, dormant instability, and singular perturbations. Zbl 0684.34055 Fusco, G.; Hale, J. K. 1989 Lower semicontinuity of attractors of gradient systems and applications. Zbl 0712.47053 Hale, Jack K.; Raugel, Geneviève 1989 Shadow systems and attractors in reaction-diffusion equations. Zbl 0667.34072 Hale, Jack K.; Sakamoto, K. 1989 Shadow systems for evolutionary equations. Zbl 0692.34052 Hale, Jack K. 1989 Continuity of attractors. Zbl 0687.58021 Raugel, Geneviève; Hale, Jack K. 1989 Asymptotic behavior of dissipative systems. Zbl 0642.58013 Hale, Jack K. 1988 Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation. Zbl 0666.35012 Hale, Jack K.; Raugel, Geneviève 1988 Upper semicontinuity of attractors for approximations of semigroups and partial differential equations. Zbl 0666.35013 Hale, Jack K.; Lin, Xiao-Biao; Raugel, Geneviève 1988 Existence and stability of transition layers. Zbl 0669.34027 Hale, Jack K.; Sakamoto, Kunimochi 1988 Onset of chaos in differential delay equations. Zbl 0644.65050 Hale, Jack K.; Sternberg, Natalia 1988 Compact attractors for weak dynamical systems. Zbl 0593.58018 Hale, J. K.; Stavrakakis, N. 1988 Varying boundary conditions with large diffusivity. Zbl 0557.35078 Hale, Jack K.; Rocha, Carlos 1987 Interaction of diffusion and boundary conditions. Zbl 0661.35047 Hale, Jack K.; Rocha, Carlos 1987 Asymptotic behavior of gradient dissipative systems. Zbl 0657.35014 Hale, Jack K. 1987 Some examples of infinite dimensional systems. Zbl 0611.34040 Hale, Jack K. 1987 Dynamics of infinite dimensional systems. (Proceedings of the NATO Advanced Study Institute on Dynamics of Infinite Dimensional Systems, held in Lisbon, Portugal, May 19-24, 1986). Zbl 0623.00009 Chow, Shui-Nee (ed.); Hale, Jack K. (ed.) 1987 Large diffusivity and asymptotic behavior in parabolic systems. Zbl 0602.35059 Hale, Jack K. 1986 Heteroclinic orbits for retarded functional differential equations. Zbl 0611.34074 Hale, Jack K.; Lin, X.-B. 1986 Symbolic dynamics and nonlinear semiflows. Zbl 0632.58027 Hale, Jack K.; Lin, Xiao-Biao 1986 Examples of transverse homoclinic orbits in delay equations. Zbl 0613.34060 Hale, Jack K.; Lin, Xiao-Biao 1986 Local flows for functional differential equations. Zbl 0612.34066 Hale, Jack K. 1986 ...and 123 more Documents all top 5 Cited by 9,526 Authors 86 Hale, Jack Kenneth 69 Park, Juhyun (Jessie) 62 Xu, Rui 59 Wei, Junjie 54 Wu, Jianhong 53 Ezzinbi, Khalil 52 Zhang, Weinian 49 Zhao, Xiao-Qiang 48 Chen, Lansun 48 Nolasco de Carvalho, Alexandre 47 Zhou, Shengfan 44 Barreira, Luis Manuel 43 Cao, Jinde 43 Valls Anglés, Cláudia 41 Teng, Zhi-dong 40 Liu, Xinzhi 40 Smith, Hal Leslie 39 Huang, Lihong 39 Kloeden, Peter Eris 38 Benchohra, Mouffak 38 Ruan, Shigui 37 Caraballo Garrido, Tomás 37 Ma, Wanbiao 36 Han, Maoan 35 Zanolin, Fabio 33 Yuan, Rong 32 Wang, Bixiang 31 Lu, Kening 31 Zhong, Chengkui 31 Zou, Xingfu 30 Diblík, Josef 29 Chow, Shui-Nee 29 Henríquez, Hernán R. 29 Mao, Xuerong 29 Robinson, James Cooper 28 Elaiw, Ahmed M. 28 Fiedler, Bernold 28 Liu, Bingwen 28 Niculescu, Silviu-Iulian 28 Rodríguez-Bernal, Aníbal 27 Berezansky, Leonid M. 27 Hernández, Eduardo M. 27 Kwon, O. 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Updates and corrections should be made in Wikidata.	2021-04-11 02:15:29	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.47663602232933044, "perplexity": 10058.899345325524}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038060603.10/warc/CC-MAIN-20210411000036-20210411030036-00057.warc.gz"}
https://stats.stackexchange.com/questions/74240/gradient-boosting-in-r-uses-only-a-single-variable	# Gradient boosting in R uses only a single variable I am trying to build a boosting model using the package gbm in R. I have the following code: gb = gbm(aaa_target ~ ., data=myDdata, n.trees=100, verbose=TRUE) and when I have trained the model, I can get a summary like this: summary(gb) The issue I am having, is that only a single variable (out of around 30) is selected and is given 100% predictive power. I know for a fact that many of the variables carry information (although the selected one is the most significant one), and using the randomForest package gives me a model which assigns significance to many of the variables. Does anybody have a clue to why this might be the case? Because the overworked maintainers of the gbm package have not had time to implement random feature sampling at each split calculation yet. I submitted a bad patch that did this as a proof of concept, but:	2021-01-27 20:47:54	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6621127128601074, "perplexity": 745.4517639950212}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610704832583.88/warc/CC-MAIN-20210127183317-20210127213317-00691.warc.gz"}
https://aimsciences.org/article/doi/10.3934/jimo.2011.7.849	# American Institute of Mathematical Sciences October 2011, 7(4): 849-874. doi: 10.3934/jimo.2011.7.849 ## Uniform estimates for ruin probabilities in the renewal risk model with upper-tail independent claims and premiums 1 Department of Mathematics, Soochow University, Suzhou, 215006, China Received May 2010 Revised May 2011 Published August 2011 In this paper, we discuss a nonstandard renewal risk model, where the price process of the investment portfolio is modelled as a geometric Lévy process, the claim sizes and premium sizes form sequences of identically distributed and upper-tail independent random variables, respectively, the claim size and its corresponding inter-claim time satisfy a certain dependence structure described via a conditional tail probability of the claim size given the inter-claim time before the claim occurs, and there is a similar dependence structure between the premium size and the inter-arrival time before the premium is paid. When the claim-size distribution belongs to the extended-regular-varying class, we obtain a uniform tail asymptotics for stochastically discounted aggregate claims. Furthermore, assuming that the tail of the premium-size distribution is lighter than that of the claim-size distribution, the uniform estimates for the finite- and infinite-time ruin probabilities are presented respectively. Citation: Yinghua Dong, Yuebao Wang. Uniform estimates for ruin probabilities in the renewal risk model with upper-tail independent claims and premiums. Journal of Industrial & Management Optimization, 2011, 7 (4) : 849-874. doi: 10.3934/jimo.2011.7.849 ##### References: [1] H. Albrecher and O. J. Boxma, A ruin model with dependence between claim sizes and claim intervals,, Insurance Math. Econom., 35 (2004), 245. doi: 10.1016/j.insmatheco.2003.09.009. Google Scholar [2] H. Albrecher and J. L. Teugels, Exponential behavior in the presence of dependence in risk theory,, J. 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Google Scholar show all references ##### References: [1] H. Albrecher and O. J. Boxma, A ruin model with dependence between claim sizes and claim intervals,, Insurance Math. Econom., 35 (2004), 245. doi: 10.1016/j.insmatheco.2003.09.009. Google Scholar [2] H. Albrecher and J. L. Teugels, Exponential behavior in the presence of dependence in risk theory,, J. App. Probab., 43 (2006), 257. doi: 10.1239/jap/1143936258. Google Scholar [3] A. V. Asimit and A. L. Badescu, Extremes on the discounted aggregate claims in a time dependent risk model,, Scand. Actuar. J., 2010 (): 93. doi: 10.1080/03461230802700897. Google Scholar [4] A. L. Badescu, E. C. K. Cheung and D. Landriault, Dependent risk models with bivariate phase-type distributions,, J. Appl. Probab., 46 (2009), 113. doi: 10.1239/jap/1238592120. Google Scholar [5] R. Biard, C. Lefévre and S. Loisel, Impact of correlation crises in risk theory: Asymptotics of finite-time ruin probabilities for heavy-tailed claim amounts when some independence and stationary assumptions are relaxed,, Insurance Math. Econom., 43 (2008), 412. doi: 10.1016/j.insmatheco.2008.08.004. Google Scholar [6] N. H. Bingham, C. M. Goldie and J. L. Teugels, "Regular Variation,", Encyclopedia of Mathematics and its Applications, 27 (1987). Google Scholar [7] A. V. Boĭkov, The Cramer-Lundberg model with stochastic premiums,, Theory Probab. Appl., 47 (2003), 489. doi: 10.1137/S0040585X9797987. Google Scholar [8] M. Boudreault, H. Cossette, D. Landriault and E. Marceau, On a risk model with dependence between interclaim arrivals and claim sizes,, Scand. Actuar. J., 5 (2006), 265. doi: 10.1080/03461230600992266. Google Scholar [9] R. J. Boucherie, O. J. Boxma and K. Sigman, A note on negative customers, GI/G/I workload, and risk processes,, Prob. Eng. Inf. Sci., 11 (1997), 305. doi: 10.1017/S0269964800004848. 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Journal of Industrial & Management Optimization, 2018, 14 (1) : 35-63. doi: 10.3934/jimo.2017036 2018 Impact Factor: 1.025	2019-08-20 03:20:33	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.646832287311554, "perplexity": 7619.986135705886}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027315222.14/warc/CC-MAIN-20190820024110-20190820050110-00171.warc.gz"}
https://www.ias.ac.in/listing/articles/pmsc/090/03	• Volume 90, Issue 3 July 1981, pages 167-286 • EOL and ETOL array languages In this paper we study the families of ETOL and EOL array languages. Standard forms for ETOL and EOL array systems are defined and closure properties of the families are studied. Relations of these families with other developmental array languages and other array languages are studied. • On the relation of generalized Valiron summability to Cesàro summability A family (Vak) of summability methods, called generalized Valiron summability, is defined. The well-known summability methods (Bα,γ), (Eρ, (Tα), (Sβ) and (Va) are members of this family. In §3 some properties of the (Bα,γ) and (Vak) transforms are established. Following Satz II of Faulhaber (1956) it is proved that the members of the (Vak) family are all equivalent for sequences of finite order. This paper is a good illustration of the use of generalized Boral summability. The following theorem is established: Theorem.If sn (n ≥ 0) isa real sequence satisfying$$\mathop {lim}\limits_{ \in \to 0 + } \mathop {lim inf}\limits_{m \to \infty } \mathop {min}\limits_{m \leqslant n \leqslant m \in \sqrt m } \left( {\frac{{S_n - S_m }}{{m^p }}} \right) \geqslant 0(\rho \geqslant 0)$$, and if sns (Vak) thensn → s (C, 2ρ). • On the mean square value of Hurwitz zeta function R Balasubramanian has shown that$$\mathop \smallint \limits_1^{\rm T} \|\zeta (\tfrac{1}{2} + it)\|^2 dt = T\log \tfrac{T}{{2\pi }} + (2\gamma - 1)T + O(T^{\theta + \in } )$$ with θ = 1/3. In this paper we develop a hybrid analogue for the mean square value of the Hurwitz zeta function ζ (s, a) and show that (i) new asymptotic terms arise in the expression for ζ (s, a) which are not present in the above expression for the ordinary zeta function and (ii) the corresponding error term is given by$$O(T^{5/12} log^2 T) + O\left( {\frac{{logT}}{{\left\\| {2a} \right\\|}}} \right)$$ for 0 <a < 1. • On a generalization of the class of functions with bounded Mocanu variation The object of this paper is to generalise the well-known class of functions analytic in the unit disc having bounded Mocanu variation. Certain properties of this more general class are investigated using convolution techniques. • Analyse asymptotique des équations de Transport dans le cas d’évolution Nous démontrons la convergence de la solution d’une équation de Transport vers la solution d’une équation de Diffusion quand Je libre parcours moyen tend vers zéro, pour des équations d’évolution et un domaine borné dans deux cas: flux incident nul et réflexion spéculaire. • Application of Newton’s method to a homogenization problem The homogenization of a family (Pε) of uniformly elliptic semilinear partial differential equations of second order is studied. The main result is that any non-singular solutionu of the homogenized problem (P) is the limit of non-singular solutions of (Pε). The method consists of specifying a functionwε starting from which the Newton iterates converge to a solutionuε ofPε. These solutionsuε converge to the given solutionu of (P). • Homogenization of eigenvalue problems in perforated domains In this paper, we treat some eigenvalue problems in periodically perforated domains and study the asymptotic behaviour of the eigenvalues and the eigenvectors when the number of holes in the domain increases to infinity Using the method of asymptotic expansion, we give explicit formula for the homogenized coefficients and expansion for eigenvalues and eigenvectors. If we denote by ε the size of each hole in the domain, then we obtain the following aysmptotic expansion for the eigenvalues: Dirichlet: λε = ε−2 λ + λ0 +O (ε), Stekloff: λε = ελ1 +O2), Neumann: λε = λ0 + ελ1 +O2). Using the method of energy, we prove a theorem of convergence in each case considered here. We briefly study correctors in the case of Neumann eigenvalue problem. • A note on the mean value of L-series Using Hilbert’s inequality, we give a new asymptotic formula (uniform inq andT) for$$\mathop \Sigma \limits_{\begin{array}{*{20}c} {\chi (mod q)} \hfill \\ {\chi primitive} \hfill \\ \end{array} } \smallint _T^{2T} \|L(\tfrac{1}{2} + it,\chi )^4 \|dt$$ • # Proceedings – Mathematical Sciences Current Issue Volume 129 \| Issue 3 June 2019	2019-05-19 18:39:16	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8096109628677368, "perplexity": 1792.8652730956383}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232255092.55/warc/CC-MAIN-20190519181530-20190519203530-00302.warc.gz"}
https://codegolf.stackexchange.com/questions/258469/generate-a-permutation-from-the-high-water-marks	# Generate a permutation from the high-water marks Given a permutation, we can define its high-water marks as the indices in which its cumulative maximum increases, or, equivalently, indices with values bigger than all previous values. For example, the permutation $$\ 4, 2, 3, 7, 8, 5, 6, 1 \$$, with a cumulative maximum of $$\ 4, 4, 4, 7, 8, 8, 8, 8 \$$ has $$\1, 4, 5\$$ as (1-indexed) high-water marks, corresponding to the values $$\4, 7, 8\$$. Given a list of indices and a number $$\n\$$, generate some permutation of size $$\n\$$ with the list as its high-water marks in time polynomial in $$\n\$$. This is code golf, so the shortest solution in each language wins. # Test Cases In all of those except the first, more than one valid permutation exist, and you may output any one of them. 1-based high-water mark indices, n -> a valid permutation [1, 2, 3], 3 -> [1, 2, 3] [1, 2, 3], 5 -> [2, 3, 5, 4, 1] [1, 4, 5], 8 -> [4, 2, 3, 7, 8, 5, 6, 1] [1], 5 -> [5, 4, 3, 2, 1] [1, 5], 6 -> [4, 1, 3, 2, 6, 5] # Rules • You can use any reasonable I/O format. In particular, you can choose: • To take the input list as a bitmask of size $$\n\$$, and if you do that whether to take the size at all. • Whether the input is 0-indexed or 1-indexed. • Whether to have the first index (which is always a high-water mark) in the input. When taking a bitmask, you may take a bitmask of size $$\n-1\$$ without the first value. When taking a list of indices, you can have it indexed ignoring the first value. • Whether the output is a permutation of the values $$\0,1,...,n-1\$$ or $$\1,2,...,n\$$. • To output any non-empty set of the permutations, as long as the first permutation is outputted in polynomial time. • To have the cumulative minimum decrease in the given points instead. • To have the cumulative operation work right-to-left, instead of left-to-right. • It is allowed for your algorithm to be non-deterministic, as long as it outputs a valid permutation with probability 1 and there's a polynomial $$\p(n)\$$ such that the probability it takes more than $$\p(n)\$$ time to run is negligible. The particular distribution doesn't matter. • Standard loopholes are disallowed. • I'm genuinely confused by this... what does it mean by "in time polynomial in n" and why is the second test case different from the first (other than the length) Feb 27 at 16:34 • @12944qwerty "in time polynomial in n" means that there must exist some polynomial such that the time your program takes to execute is bounded by $p(n)$ for any input with size $n$. Practically, it means you can't do stuff like test all possible permutations. What do you mean by why is the second test case different form the first? [1, 2, 3, 4, 5] wouldn't be a valid answer to it, because it has all indices as high-water marks, and not only the first 3. Feb 27 at 16:37 • Brownie points for an answer that actually makes use of the probability clause. Feb 27 at 17:51 • Will the list contains duplicate numbers? – tsh Feb 28 at 2:29 • @tsh no, and you can assume whether it's increasing/decreasing Feb 28 at 3:22 # Nibbles, 5 bytes .,@?:-,_@@ Attempt This Online! Puts the $$\k\$$ highest numbers at the specified indices in ascending order and the remaining numbers at the remaining indices, where $$\k\$$ is the number of marks. [1, 4, 5], 8 -> [6, 1, 2, 7, 8, 3, 4, 5] ## Explanation .,@?:-,_@@ . Map , range @ second input ? find index in : join - list difference , range _ second input @ first input @ first input If I'm reasoning correctly, the time complexity should be $$\\mathcal O\left(n^2\right)\$$. # R, 23 bytes Edit: -9 bytes thanks to Command Master by switching to use a bitmask as input function(h)rank(h,,"f") Try it online! Input is simply a bitmask h, no need to provide n as this is implicitly the length of the bitmask. The "f" (short for ties="first") argument to rank() indicates that ties with the same rank should be broken using "increasing values at each index set of ties". # R, 32 bytes function(h,n)rank(1:n%in%h,,"f") Try it online! Same approach as above, except inputting h as the indices of high-water marks, together with n. 1:n %in% h returns a vector of mainly zeros, with ones where the high-water marks should be (in other words, converting the indices to a bitmask). • function(h,n){x=1:n;x[h]=n2:n;rank(x)} seems to work for -1 Feb 27 at 17:06 • 38 bytes Feb 27 at 17:09 • If you take a bitmask couldn't you replace 1:n%in%h with h and not take n? Feb 28 at 9:42 • @CommandMaster - yes, thanks! I hadn't noticed the generous input allowances! Feb 28 at 10:39 # Vyxal, 6 bytes ɾ$ÞṖRf Try it Online! Takes n and then a list of indices. ɾ # range$ # swap top two items on the stack ÞṖ # split before indices R # reverse each f # flatten # 05AB1E, 8 bytes LDŠåÅ¡í˜ Explanation: L # Push a list in the range [1, first (implicit) input n] D # Duplicate this list Š # Triple-swap it with the second (implicit) input: [1,n], input-list, [1,n] å # Check for each value of [1,n] whether it's in the input-list Å¡ # Split the second [1,n]-list before the truthy indices í # Reverse each inner list ˜ # Flatten it to a single list # (after which the result is output implicitly) • @xigoi Woops, thanks for noticing. Should be fixed now. Feb 28 at 7:37 # Jelly, 4 2 bytes ỤỤ Try it online! Or see the test-suite. #### How? ỤỤ - Link: bitmask e.g. [1,0,0,1,0,0] Ụ - grade-up (indices of sorted values) [2,3,5,6,1,4] Ụ - grade-up (indices of sorted values) [5,1,2,6,3,4] ### Previous at 4 bytes (not using a bitmask): ReÞỤ A dyadic Link that accepts the number on the left and the high watermark indices on the right and yields a permutation. Try it online! Or see the test-suite. #### How? ReÞỤ - Link: n; I e.g. n=6; I=[1,4] R - range (n) [1,2,3,4,5,6] Þ - sort by: e - exists in (I)? [2,3,5,6,1,4] Ụ - grade-up (indices of sorted values) [5,1,2,6,3,4] # Pyth, 10 bytes XKSE+-KQQK Try it online! Takes the indices then the length, runs in $$\\mathcal{O}(n^2)\$$. Uses the same general strategy as most others, putting the $$\n\$$ highest values at the given indices. ### Explanation # implicitly assign Q = eval(input()) to the indices KSE # assign K = [1, 2, ..., eval(input())] -KQ # elements of K not in Q + Q # concatenated to Q XK K # translate K from this to K # R, 16 bytes Not enough reputation to comment on Dominic's answer, but using the new anonymous function syntax in R, a few bytes can be shaved off: \(h)rank(h,,"f") • Welcome to Code Golf! I'm sure Dominic is aware of the new function syntax, but Try It Online (TIO) is still using 3.6, and its maintainer has, unfortunately, taken a leave of absence from CGCC, so it's not likely to get updated. There are alternatives, e.g., Attempt This Online (aka ATO) with 4.0+ installed, but sometimes I still find myself defaulting to use TIO instead. At any rate, I've upvoted and hope you enjoy your time golfing! Feb 28 at 15:58 • +1 from me, too. Giuseppe is right that I knew the \ syntax, but it's a valid improvement to use it. Mar 7 at 20:34 # Python 3, 64 bytes Uses zero-based indexing f=lambda m,n:sum(([range(t)][::-1]for t in zip(m,m[1:]+[n])),[]) • You can drop the f= since it's not recursive. Also, list(range(t)) can be [range(t)] Feb 27 at 17:27 • 58 bytes Feb 27 at 17:54 • Your output contains $n-1$ elements, not $n$ elements. Maybe you want f=lambda m,n:sum(([range(t)][::-1]for t in zip([0]+m,m+[n])),[]) instead. – tsh Feb 28 at 3:11 • Based on comment by @97.100.97.109, this is 55 bytes in Python 2: lambda m,n:sum(map(range,m+[n],[0]+m,[-1]-~len(m)),[]) – tsh Feb 28 at 3:27 • @tsh I think it's fine. Zero must be in the input anyway. Feb 28 at 9:43 # JavaScript (Node.js), 45 bytes m=>n=>m.map(i=>i?++n-eval(m.join+):++x,x=0) Attempt This Online! Takes a bit mask and length. Runs in $$\\mathcal O(n)\$$. This is probably my first time golfing in JavaScript, please leave suggestions how to improve the code. -7 thanks to Kevin Cruijssen, Shaggy and Arnauld • m.reduce((a,b)=>a+b) can be eval(m.join+) for -5 bytes. Feb 27 at 17:54 • @KevinCruijssen So dirty, I love it! Thanks! Feb 27 at 17:58 • Save a byte with currying Feb 27 at 19:00 • 45 bytes (including currying syntax, as already suggested by Shaggy) Feb 27 at 20:29 # Retina, 38 bytes ^.+ _ $.¶ ^D G. .+$&,$:& N .+, Try it online! Takes n as the first input and then the element of the 0-indexed list. Explanation: Inspired by @DominicVanEssen's R answer. ^.+ * Convert n to unary. _$.¶ Get a range from 0 to n. ^D Remove all but the last instance of all duplicates. G. Remove all blank lines. .+ $&,$:& Append the line index to each value. N Sort by value. .+, Delete the values, leaving their original indices. # Python NumPy, 56 bytes def f(m,n):x=m[:1]range(n);_,x[m-1]=*x[m-1],n;return x Attempt This Online! 1-based. Expects a numpy array and an integer. ## How? Populates the output with a 0-based (!) range. Then shifts the values at the marked indices one position to the left ignoring non marked indices and filling the right most position with n. As the left most marked index always is 0, a 0 was lost and an n gained, generating a 1-based permutation. # Charcoal, 12 bytes ＩＥη⌕⁺⁻…⁰ηθθι Try it online! Link is to verbose version of code. 0-indexed. Explanation: Port of @xigoi's Nibbles answer. η Input n Ｅ Map over implicit range ⌕ Find index of ι Current value in … Range from ⁰ Literal integer 0 η To input n ⁻ Remove elements from θ Input list ⁺ Concatenated with θ Input list Ｉ Cast to string Implicitly print # Japt, 11 10 bytes õ ò@VøYÃcÔ Try it -1 by Shaggy (crossed out 11 is still regular 11) õ ò@VøYÃcÔ õ # range [1, input integer] ò@ Ã # partitioned between pairs X, Y where VøY # Y is contained in input array cÔ # reverse each and flatten • ò1 can be õ here. Mar 6 at 12:48 • @Shaggy I’ll never be good enough for you, will I? Thanks, I realized what the difference between the different range built-ins was the day after posting this but forgot to update the answer Mar 6 at 12:51	2023-03-26 10:50:02	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 19, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3323184847831726, "perplexity": 3482.1116683656}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945440.67/warc/CC-MAIN-20230326075911-20230326105911-00644.warc.gz"}
https://mathoverflow.net/questions/439095/automorphism-of-moduli-space-of-stable-vector-bundles-over-a-curve	# Automorphism of moduli space of stable vector bundles over a curve Let $$C$$ be a smooth genus two hyperelliptic curve and $$\mathcal{M}_C$$ be a moduli space of stable rank two vector bundles of fixed degree(or fixed determinant). Then is $$\mathrm{Aut}(\mathcal{M}_C)\subset\mathrm{Aut}(C)?$$. The motivation is following. Let $$Y$$ be an index two prime Fano threefold of degree $$4$$, one can show that such threefold is isomorphic to a moduli space of stable vector bundle of rank two fixed determinant over a genus two curve $$C$$ such that its degree is odd(say degree=1). Then, I am wondering if $$\mathrm{Aut}(Y)\subset\mathrm{Aut}(C)?$$. In the paper Hilbert scheme of lines and conics and automorphism groups of Fano threefolds, the authors show that $$\mathrm{Aut}(Y)\subset\mathrm{Aut}(J(C))$$. But I think $$\mathrm{Aut}(J(C))$$ is not contained in $$\mathrm{Aut}(C)$$ since $$J(C)$$ has some translation which is not coming from $$\mathrm{Aut}(C)$$. On the other hand, for index one degree 12 prime Fano threefold $$X$$. The authors show that $$\mathrm{Aut}(X)\subset\mathrm{Aut}(C)$$, in this case $$X$$ is also purely determined by $$C$$. The moduli space of rank 2 vector bundles on $$C$$ with fixed determinant of odd degree is a smooth complete intersection of two quadrics in $$\ \mathbb{P}^5\qquad$$ (P. Newstead, Topology 7 (1968), 205-215); in an appropriate system of coordinates, it is given by $$\sum X_i^2=\sum \lambda _iX_i^2=0$$, for $$\lambda_0,\ldots ,\lambda_5\in\mathbb{C}$$ distinct. It always admits a group $$(\mathbb{Z}/2)^5$$ of automorphisms (acting by changing the sign of the coordinates). On the other hand, $$\operatorname{Aut}(C)=\mathbb{Z}/2$$ for $$C$$ general.	2023-03-27 00:54:30	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 23, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9634460806846619, "perplexity": 90.8516662334368}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296946584.94/warc/CC-MAIN-20230326235016-20230327025016-00120.warc.gz"}
https://history.jes.su/s207987840001324-5-1/	Shakespeare in Russian Cultural Consciousness in the 20th Century Share Metrics Shakespeare in Russian Cultural Consciousness in the 20th Century Annotation PII S207987840001324-5-1 DOI 10.18254/S0001324-5-1 Publication type Article Status Published Authors Igor Shaytanov Edition Abstract The essay outlines how Shakespeare’s work came to be known, translated and interpreted in Russia. The first acquaintance with his plays in the 18th — early 19th century required an intermediary, the role acted by French translations and much altered versions of the original. It was in the time of Alexander Pushkin that adequate Russian translations were published, played on stage, while Shakespeare was first viewed as a productive artistic model in whose tragedies and histories a political and moral standard was put up. Since then Shakespeare has been present in Russian cultural consciousness as an ‘‘eternal fellow-traveller’’. His plays in new translations and theatrical interpretations accompanied the most crucial and catastrophic events in the 20th century Russia: revolution, Stalin’s terror and the so-called ‘‘thaw period’’ that followed it. The foremost Russian poets reflected on Shakespeare in their original verse, critical essays, or participated in his translation. The most important part in translating Shakespeare belongs to Boris Pasternak for whom his Shakespearian work was not only a means to survive and to earn his living but his message to people here and his attempt to be heard beyond the ‘‘iron curtain”. Marshak’s translation of the sonnets, in the last Stalin’s decade when new lyrical poetry was practically banned in print, served to transmit eternal feelings and enjoyed popularity unparalleled for translated verse. Keywords Shakespeare in Russia, translation, Pushkin, political and moral standard, Stalin, Pasternak, Marshak, sonnet, nation 25.10.2015 Publication date 25.12.2015 Number of characters 20224 Number of purchasers 18 Views 10076 1 Shakespeare in Russia — and not in Russia alone — for the last two centuries has served as a true model for humanism established since the time of European Renaissance, which he closed to open up a new historical perspective that extends to our time. 2 Besides that, there are, at least, three good formal reasons to talk now on Shakespeare and on Shakespeare in Russia. Firstly, Shakespeare’s jubilee, widely celebrated in 2014. We Russians are great jubilators and choose for such occasions practically all dates with a zero at the end but prefer births to deaths. This is why 450 in 2014 is preferable for our cultural mentality than 400 since his death two years later. 3 Among different events and editions (some of them still in progress) I want to mention the first Shakespeare encyclopaedia in Russia (there were but small reference books hitherto) coming out in 2015. The block of materials ‘‘Shakespeare in Russia’’ is, if not the most important for us, then undoubtedly the most informative for the rest of the world, the fact proved by a recent publication of articles from the forthcoming encyclopaedia in the American Russian studies in Literature. But besides these formal reasons there exists one more which prompts the choice of Shakespeare for my talk at the conference: his unique importance in Russian cultural consciousness. 4 Some ten years ago, a friend told me about a public speech given by the British Ambassador in Moscow Tony Brenton titled ‘‘Shakespeare the Russian’’. Next time at the embassy, I asked the Ambassador whether the text of his speech was available, and he sent me a fine essay. I liked his joke about serious matters and received a permission from him to have it translated for the foremost Russian journal in literary criticism Problems of Literature. The translation was published in 2007 (issue 4). 5 Tony Brenton began his retrospective reminding us that it was in Shakespeare’s time that the first Russians had reached Britain, a dozen of them — under Boris Godunov — were even sent to Oxford to study there (none of them came back) — why not to omit one of these recusants as a candidate to have written all we know as Shakespeare’s: 6 Maybe I even have a Russian candidate in mind — an exile from the brutal court of Ivan the Terrible. Perhaps a clerk or a skoromokh [vagrant actor] who finds his way to England in some Ambassadorial suite, and whose fascination with the strange and insular ways of the English gives that clarity and concreteness of vision which is one of Shakespeare’s hallmarks. In London he is beguiled by some British beauty to settle, and brings Russian eloquence and intensity to the hitherto rather provincial English language and stage. So in the fusion of our two cultures he creates an example of Anglo/Russian co-operation which I can extol as the magnificent model for how relations between our two countries ought to be today (Brenton 216). 7 I will concentrate on some facts from the 20th century but first ought to sketch the way of Shakespeare into Russia1. He came, of course, not in his own time and not by a direct way but a century and a half later through France and Germany. First, Hamlet was remodeled by our leading classical tragedian Sumarokov (1748), then Merry Wives of Windsor — with their traditional appeal to a royal taste — adapted by the empress Catherine the Great. The reformer of the Russian language, historian and poet, Nikolai Karamzin translated Julius Caesar if not from the original then looking into it. In many European countries Julius Caesar was the first or among the first to become popular with its classical background in the age of classicism. But the time for translations was not ripe yet and on stage they played adaptations borrowed from French. 1. The history of Shakespeare in Russia, especially its earlier stages in the 18th and 19th centuries, has been substantially documented and outlined. The most fundamental study is a collection of chronological materials Shakespeare and Russian Culture / ed. M. P. Alekseev. M.; Leningrad, 1965. 8 It was Pushkin’s time and with his own admiration for Shakespeare when his fame spread wide and his experience was found important as a guide for contemporary thought setting up a moral and political standard. Pushkin ‘‘shakesperized in a broad sense’’ (as he put it) in his national tragedy Boris Godunov, wrote a parody of Shakespeare and history in his mock poem Count Nulin (what might have happened if a modern Lucrece would put an end to a rape by slapping the face of an offender?), and the most enigmatic Pushkin’s sally into Shakespeare’s ground was with Measure for Measure2. He first translated odd 20 lines opening the comedy, then gave it up and wrote a poem Angelo, appreciated by critics in the same vein as Shakespeare’s play — a failure. Pushkin was very much displeased and told a friend that Angelo was the best he ever wrote. His opinion, at least, is worth consideration. 2. This comparative theme had provoked not a few comparative studies. My own article titled ‘‘Two failures’’ came out first in the periodical Problems of literature 1 (2003) to be reprinted in several collections and in my book: Shaytanov, Igor. Comparative Literature or/and Poetics. Moscow, 2010. 9 Through the 19th century Shakespeare’s popularity and impact on the Russian mind grew steadily. There were short periods in 1860—1870s when a new nihilistic and utilitarian turn of mind made young people prefer Henry Buckle and Herbert Spencer to Shakespeare, but both in the capitals and provinces his plays were performed and admired. 10 The first years of the 20th century (1902—1904) record the first important edition of Shakespeare’s works of an academic type — a 5-volume collection brought out by the famous publishing house Brokhaus-Efron (edited by Semyon Vengerov). It was both a summing up and a step taken into the new epoch. The Best old translations and some new specially commissioned ones were provided with prefaces and commentary based on German and English criticism and textological work. All books were lavishly illustrated. Only 40 years later would a similar ambitious project be attempted. Shakespeare, though not a domineering influence, became an important figure among ‘‘eternal fellow-travellers’’ into the domain of culture (Merezhkovsky’s metaphor) for the Russian ‘‘silver age’’. Many of them were doing Shakespearian work: Alexander Blok played Hamlet on a home stage and reflected on him in poems: ‘‘I am Hamlet, and blood grows cold in my veins / When perfidy is at work, conspiring…’’ 11 Valery Briusov translated sonnets (their popularity in Russia is still half a century ahead); Marina Tzvetaeva wrote in defense of Ophelia addressing bitter words to Hamlet: ‘‘And you with your mixture of plaster / And decay… Gossip with bones, / Prince Hamlet! It is none of your business to judge an inflated blood…’’ 12 А Вы с Вашей примесью мела И тлена… С костями злословь, Принц Гамлет! Не Вашего разума дело Судить воспаленную кровь. 13 Boris Pasternak… However, his great Shakespearean achievement belongs not to his early pre-Revolutionary years but to a later Soviet time, when Shakespeare became for many persons from the ‘‘silver age’’ a source to revive and speak up: Mikhail Kuzmin, Anne Radlova, Mikhail Lozinsky and Boris Pasternak created a new Russian Shakespeare in their translations. At the same time banned from the field of philosophy Mikahail Bakhtin and Gustav Spet wrote on Shakespeare: Bakhtin not extensively for his book on Rabelais (Shakespeare’s pages published in 1990s), and Spet had great plans to be one of the commentators, editors of translation for the new collection of Shakespeare’s works. He was arrested and shot dead before the first volume came out, and his collected papers in the field of Shakespeare were brought out only in 2013. 14 In the first revolutionary years Shakespeare was enlisted to serve the great cause and even preferred to avant-garde art. Thus Mayakovskii was very much annoyed when in the autumn of 1918 his revolutionary play Mystery-buff (staged by Meyerhold) after three nights was (as he believed) ousted by Macbeth. His information was wrong — Macbeth was performed at another theatre in Petrograd (Cinizelli circus) but the futurist was infuriated against Shakespeare. 15 The authorities would attend to this revolutionary protest against old classics only some years later. The first Soviet theatre in Petrograd (Bolshoi Drama Theater initiated by Maxim Gorky and Alexander Blok) in the first 4 seasons (1919—1922) lived on Shakespeare: Much Ado, Othello, The Merchant of Venice, The Twelfth Night, Julius Caesar… But then they were told, or realized, that Friedrich Schiller better corresponded to the revolutionary spirit, and by mid-1920s all classical names had to give place to the new art. 16 In Moscow Shakespeare was also on stage attracting some new talented actors. Mikhail Chekhov (Anton Chekhov’s nephew) began his Shakespearean career as Malvolio in Moscow Art Theatre (1920), and 4 years later, when a group of young actors rehearsing Hamlet was institutionalized as MAT-2, he played his famous Prince of Denmark. His Hamlet had little to do with either revolution or heroism and much with melancholy and the transcendental anticipation of universal evil. The famous theatre director and teacher of dramatic art Maria Knebel much later wrote of Chekhov’s Hamlet: ‘‘Theoretically I know what catharsis is but I experienced it only once at the moment when Chekhov’s Hamlet died’’. 17 This art was no longer in demand. It must have looked like a leave-taking gesture from the ‘‘silver age’’. And those who practiced it were suspected or called enemies. Mikhail Chekhov, afraid of arrest, left Russia in 1928. At that time a threatening question, addressed to every citizen — ‘‘What did you do before 1917’’, was relevant towards classics too. Their social position and origin were scrutinized, and the first Soviet dispute about Shakespeare related to whether he belonged to the archaic feudal society or could be considered as a herald of a new bourgeoise age. The second was much preferable, of course, because this social diagnosis would allow Shakespeare to claim a more progressive role and to be positioned as not completely alien to the proletariat. 18 But before all these uncertainties were cleared up Shakespeare in late 1920s was not much staged or translated. Only when in the totalitarian state its ideological leaders came to a decision that in art and literature a hierarchy should be as rigid as it was in political reality they rehabilitated classics. Shakespeare was among the first and greatest. It was in mid-30s that new translations were commissioned and performed all over the Soviet Union. Shakespeare arrived at places were he had never been heard of before. According to the reviewers fantastic performances were produced in Asian republics, in the Caucuses, on the stage of national theatres. There was not much variety in choice from Shakespeare’s canon — only Richard III among histories, some comedies, among tragedies — Romeo and Juliet, Othello, MacbethKing Lear was too complex and mystical, Antony and Cleopatra — staged by Tairov seemed too aesthetical. Hamlet was viewed on with a suspicious eye by Stalin 19 The role of the Prince was rehearsed in Moscow art theatre by Boris Livanov (his son Vasilii is famous as Sherlock Holmes in the late Soviet film version). Livanov belonged to the artistic elite who were invited to Kremlin for receptions there. At one of such events, Stalin came up to him to ask about his plans, probably, aware of his work on Hamlet. Livanov heard: ‘‘Why Hamlet, he is a weak hero… Not the one we need’’. This, of course, put an end to a long rehearsal process in MAT. and to Hamlet in the USSR, but it served as a good reason for Hamlet to herald the political ‘‘thaw’’ after Stalin’s death. 20 In 1955 Peter Brook brought Hamlet with Paul Scofield to Moscow. It was not just theatre and Shakespeare, it was a break through when the iron curtain was raised. Scofield played Hamlet 12 times in 10 days. 21 A year earlier in Leningrad Grigory Kozintzev staged Hamlet in Pasternak’s translation. In 1964 he filmed the tragedy with Innokentii Smoktunovsii. This production coincided with the end of Khrushchev’s era, whose stay in power thus was bracketed with spectacular Shakespearean events and foremost with Hamlet. 22 Many eventful turns in Russian history in the 20th century took place to Shakespearean accompaniment which did not look either coincidental nor occasional but provided an important commentary, supplied a heavily censored mind with an opportunity to speak up. It concerned not the political side of life only. 23 When Kozintsev chose Pasternak’s translation of Hamlet in 1954 he decided to close a performance with the sonnet 74 (‘‘But be contented: when that fell arrest / Without all bail shall carry me away…’’) Pasternak was not happy with the idea but made a translation, that he did not consider a final version. The director seemed not to appreciate his work and made use of another translation — by Samuil Marshak. Pasternak was indignant and protested. But Kozintzev’s choice must have been dictated by the unprecedented popularity poetry in translation omit that Marshak’s sonnets enjoyed. 24 The picture of Shakespeare in the 20th century Russia would not be complete without Marshak and his translation of the sonnets. They have won popularity, that seldom befalls a translated text — they became a literary fact in the target language. This is largely due to the time and circumstances of Marshak’s work started in the wartime: first of the sonnets were published at the end of the war in journals and brought out as a book in 1948. 25 Before then Shakespeare’s sonnets had never enjoyed much popularity in Russia. Translated by different hands since mid-19th century, they were twice completed as a full sequence — once in the 19th century and then at the beginning of the 20th (the second go was by Chaikovsky’s brother Modest). Marshak set to work at the time when love lyric was practically ruled out — first because of the war and then because of the censor’s strictures in the last Stalin’s decade. Lyrics were written by the best poets (Pasternak, Akmatova, Zabolotsky) and by the young generation but to have them published (even if it could be imagined) would be to expose oneself to a scathing criticism. 26 The only possibility allowed was to have lyric translated through a classic original, and this is exactly what Marshak had done with Shakespeare. He wrote texts that were demanded by the common reader and could pass through censorship. Since then his translations are widely remembered, sung (though not many, of course) as pop songs. They were adapted to this popular usage. Marshak did exactly what he intended and from this point of view his job was perfectly done. But how close to the original? 27 When I have to illustrate Marshak’s interpretation / deformation with one example I prefer the first lines from the sonnet 129: 28 Th’expense of spirit in a waste of shame / Is lust in action… If we had Marshak translated back into English we would have: Th’expense of spirit AND a waste of shame Is VOLUPTIOUSNESS in action… 29 Style has been smoothed and bowdlerized (this is one of Marshak’s tendencies). Another tendency concerns the metaphor dissected in the first line that is against both rhetoric in this particular sonnet and the verbal nature of the renaissance sonnet in general. To retain metaphors in Shakespeare’s sonnet means to follow his generic instruction and the translator is not in his right to choose in this case but has to submit. 30 ‘‘All this the world well knows…’’ Yet none knows well how to translate better than Marshak did. By now generations of translators have challenged his achievement, as it seems fully aware what they should do and failing to win in this competition. They know that sonnets should sound in a lower style, rough and manly, and have a more elaborate, sophisticated imagery, but translators after Marshak persistently fail in their attempt to retain poetry, alive in Marshak’s translation, though in a different genre, for different public taste. The highly reflective renaissance sonnet in Marshak’s version had been transformed into a romantic song — жестокий романс, and survives in this quality in Russian poetry. 31 But it is not with poetry that I would wish to close my talk. I would rather come back to Shakespeare as an acknowledged political commentator that has so often been called on as a contemporary to most crucial events in Russian history in the 20th century too. 32 I had a personal chance to witness how powerful he is in this capacity and how earnestly his word is expected as an epitome of political wisdom. In mid-90s I was invited to talk on some literary matter on the radio station Echo of Moscow. When I finished and was about to leave the editor rushed into the room with the latest news that the Chechen war began or was renewed and with a request to me to provide an immediate comment… in Shakespeare’s words or images. I protested but was pushed back into a close room, found myself with earphones and a line running before my eyes: “Speak” which meant that I was put on the air. 33 I am not sure what my choice would be had I ten minutes to think, but on the spur of the moment I started talking on what seemed most obvious to me as an illustration of how the order of state is always in opposition with an ever “wavering multitude” — the final episode in Part II of Henry IV. The new king proves his right to become an ideal monarch when he expels the companion and master of his youth sir John Falstaff. Much has been said in this connection, and apology of the king acting in the interest of state does not look more convincing than the voices of rebuke addressed to the one inhumanely ungrateful to old friendship. 34 This is an everlasting problem, growing not less but more and more urgent with a worldwide migration processes dressed up as multiculturalism, absolutely bewitching some ten years ago and met with more and apprehension voiced by the leading politicians in the West: ‘‘We thought they are coming to us to embrace our principles” But what made you think so? 35 Sir John Falstaff is not a multicultural newcomer. He belongs to carnivalesque outlaws, he represents Nature opposed to History as the author of the best Russian book on Shakespeare in the 20th century Leonid Pinsky had put it and titled the chapter on Falstaff: ‘‘History and Nature face to face” (67-84). 36 Significantly simultaneous to Henry IV is another Shakespeare’s play where — probably for the first time the nation is problematized. I mean The Merchant of Venice, of course, and not Shylock alone but Venice as a multicultural community. It is there and nowhere else, as we are told, that the Jew is in his right, as well as any other national representative, to appeal to the law and to be heard by the law because otherwise Venetian trade would be undermined. And, as it is traditional with Shakespeare, he does not leave an important motif to stay alone but doubles and trebles it as it happens in Belmont with a competiton for Portia’s hand. 37 There is a contemporaneous impression of an Elizabethan play: ‘‘On another occasion not far from our inn, I beheld a play in Bishopsgate …a play in which they presented divers nations and an Englishman struggling together for a maiden…’’ 38 Is there another Elizabethan plot, known to us, that fits into this description better than The Merchant? I have not heard of such. Thomas Platter, a Swiss traveler and student of medicine, is famous as practically a unique person who left a theatrical description from the period — that of Julius Caesar performed at the new opened Globe theatre, but I have never come across an attempt to attribute this above quoted impression to The Merchant of Venice. And if it relates to The Merchant (as I believe) it demonstrates what a contemporary remembered: not the Jew, capitalism or homosexuality, much favoured by the critics of our time, but ‘‘diverse nations’’ where Shylock grows not into a single problem of whether Shakespeare was antisemitic or vice versa, but stands out against a multicultural background. 39 Of course, Shakespeare is our contemporary, but it is not less important to understand that we are his, because he spoke from the starting point of the period in world history which with us, probably, comes to its end. And Shakespeare has much to recommend us about the world we still live in because it is the world he viewed opening up in a long perspective. ## References	2019-11-21 15:57:48	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.24300113320350647, "perplexity": 5311.961484221511}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496670921.20/warc/CC-MAIN-20191121153204-20191121181204-00048.warc.gz"}
https://ane.github.io/this-site/design.html	Antoine Kalmbach Design of this site This page describes the design of this website. If you come here often you might notice that every now and then something changes. That is normal, because I really enjoy tweaking the smallest of things here, be it adjusting margins, adding padding, changing fonts, and so on. It’s a neverending project, I can’t tell you if the site as you see now will be how it will look like in five years. But here’s a rough idea how it looks like at the time of writing. Most people will ask, “what design?”, as you’re greeted with the browser default font, probably a serif or sans-serif font, no colors beside the links, barely any borders or visual frills. This is by design, or the design: the design emphasizes this site’s focus on content. I want the text to be as readable and distraction-free as possible. To this end, • the content is center-aligned and constrained to a maximum width on large screens, • the font is black-on-white at about 16px of height, • the navigation is at the top of the page, and also aligned to the center, • there are no colours or background highlights unless absolutely necessary. Part of the reason why I don’t specify any extra fonts or the like is to make sure the site works on any device possible. I also happen to enjoy browsing “unstyled” pages because it puts emphasis on content over form, although some styling is necessary to make this enjoyable. There is no need to design for responsiveness when the site mostly just text. The stylesheet is a couple hundred lines of SCSS and mostly just deals with the formating of the navbar and images, and removing some built-in ugliness of the default browser styles, e.g., by thinning out horizontal rules to a single line. I plan for the central index at the front page to evolve in a tree-like structure of the whole site itself, but as the content outside the blog is rather thin, it’s just a link to, well, this section.	2023-01-27 15:04:01	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.293354332447052, "perplexity": 1413.699282587322}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764494986.94/warc/CC-MAIN-20230127132641-20230127162641-00366.warc.gz"}
https://www.gurobi.com/documentation/10.0/quickstart_mac/cs_setting_the_objective.html	Filter Content By Version Languages Setting the objective The next step in the example is to set the optimization objective: // Set objective: maximize x + y + 2 z model.SetObjective(x + y + 2 * z, GRB.MAXIMIZE); The objective is built here using overloaded operators. The C# API overloads the arithmetic operators to allow you to build linear and quadratic expressions involving Gurobi variables. The second argument indicates that the sense is maximization. Note that while this simple example builds the objective in a single statement using an explicit list of terms, more complex programs will typically build it incrementally. For example: GRBLinExpr obj = 0.0;	2023-01-27 01:01:02	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.23617249727249146, "perplexity": 1583.4107611974935}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764494852.95/warc/CC-MAIN-20230127001911-20230127031911-00776.warc.gz"}
https://www.physicsforums.com/threads/intergrating-factor.374368/	Homework Help: Intergrating factor 1. Feb 1, 2010 annoymage 1. The problem statement, all variables and given/known data (2y-6x)dx + (3x+4x2y-1)dy = 0 2. Relevant equations 3. The attempt at a solution (3xy+4x2)dy/dx = 6xy-2y2 i'm stuck, want to find the intergrating factor, it's xy2 when i reverse from the answer :D.. give me any hint please 2. Feb 1, 2010 annoymage or am i suppose to use special integrating factor?? 3. Feb 1, 2010 Dick Are you sure there is an integrating factor? Maybe I'm just not seeing it. But you could also try the substitution v=y/x. That should make it separable. 4. Feb 2, 2010 annoymage yey, i got it now, there's another technique which i don't really know the details.. But to find suitable intergrating factor, its something like e($$1/M$$)[($$N(x,y)/dx$$) - ($$M(x,y)/dy$$)] and is equal to xy2 and multiply both sides by the integrating factor, and will get Exact form of equation and solve it, hoho, but maybe i need to try substituting v=y/x 5. Feb 2, 2010 HallsofIvy Every first order differential equation has an integrating factor- it just may be hard to find! Are you required to find an integrating factor? this equation is "homogeneous". Write it as $$\frac{dy}{dx}= \frac{6x- 2uy}{3x+ 4x^2y^{-1}}= \frac{6- 2\frac{y}{x}}{3+ 4\frac{x}{y}}$$ Let u= y/x so that y= xu and dy/dx= u+ x du/dx and that will become a separable equation for u as a function of x. If you really want to find an integrating factor, I think that solving for u and then y and deriving the integrating factor from the solution is simplest.	2018-06-23 04:54:07	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8606406450271606, "perplexity": 1209.6418221707638}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267864940.31/warc/CC-MAIN-20180623035301-20180623055301-00128.warc.gz"}
https://stats.stackexchange.com/questions/365610/prediction-with-circular-variable	# Prediction with circular variable Suppose I have a set of $N$ observations for a circular variable $\theta_i$. Each trial for which an observation of $\theta_i$ is taken can lead to an outcome $x_i \in \{0,1\}$. Next, suppose that the set of trials for which $x_i = 1$ has a preferred direction, such that the mean resultant length $r$: $r = \frac{1}{N} \sum_i^{N} e^{i\phi_i}$ is nonzero ($0.1 < r < 0.2$, to be concrete). The set of trials for which $x_i = 0$ is uniform. Now, with the observations $\theta_i$, the outcomes $x_i$, and the mean resultant length $r$, intuitively it seems possible to predict the outcome for a new observation, either through a classifier, some closed-form expression in terms of $r$, or a parametric fitting approach. After searching, however, a reference clearly addressing this problem appears difficult to find. Could anyone suggest a potential start for thinking about this problem? A particularly simple approach would be to map angles to 2d Cartesian coordinates on the unit circle. That is, let $z_i = [\cos \theta_i, \sin \theta_i]$. Then, use any off-the-shelf classifier to predict $x$ as a function of $z$. For example, a linear classifier can separate the circle into two regions--one for each outcome. Nonlinear classifiers can fit more complicated functions. An alternative would be to work directly in angle space. For example, use a generative model: Fit a circular distribution (e.g. a von Mises distribution) to angles observed for each outcome: $p(\theta \mid x = 0)$ and $p(\theta \mid x=1)$. Estimate the prior class probabilities $p(x=0)$ and $p(x=1)$ as the relative frequency of each outcome in the dataset. Then, to predict the outcome for a new angle, use Bayes' rule: $$p(x=1 \mid \theta) = \frac{ p(\theta \mid x=1) p(x=1) }{ p(\theta \mid x=0) p(x=0) + p(\theta \mid x=1) p(x=1) }$$	2020-07-14 16:50:01	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9019924402236938, "perplexity": 327.1358926412484}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655897168.4/warc/CC-MAIN-20200714145953-20200714175953-00119.warc.gz"}
https://zbmath.org/?q=an:1105.14036	# zbMATH — the first resource for mathematics The patchworking construction in tropical enumerative geometry. (English) Zbl 1105.14036 Lossen, Christoph (ed.) et al., Singularities and computer algebra. Selected papers of the conference, Kaiserslautern, Germany, October 18–20, 2004 on the occasion of Gert-Martin Greuel’s 60th birthday. Cambridge: Cambridge University Press (ISBN 0-521-68309-2/pbk). London Mathematical Society Lecture Note Series 324, 273-300 (2006). In the early 1990’s the author of this article suggested to use the patchworking construction for tracing properties of objects, defined by polynomials, other than in the original method invented by O. Viro in 1979-80. In the paper under review the author restricts himself to families of surfaces, and curves in these surfaces, and he traces the property to possess a certain collection of singularities. As a result he states in section 2.3 a patchworking theorem for curves on toric surfaces which is proved in section 2.4. This theorem is sufficient for applications to a tropical calculation of Gromov-Witten and Welschinger invariants on toric surfaces. The general situation in the patchworking construction is a one-parametric flat family of surfaces with irreducible fibers $$Y_t$$, $$t\neq 0$$, and reduced central fiber $$Y_0$$ that contains the zero locus $$X_0$$ of a section of the restriction to $$Y_0$$ of a line bundle on $$Y$$. In most previous papers it is assumed that the components of $$X_0$$ are reduced and they meet transversally the singular locus of $$Y_0$$. A major novelty of the work under review is that $$X_0$$ is allowed to have nonreduced components that may be not transversal to $$\text{ Sing}(Y_0)$$. This covers the degenerations which appear in the tropical enumeration of curves. For the entire collection see [Zbl 1086.14001]. ##### MSC: 14H15 Families, moduli of curves (analytic) 14H20 Singularities of curves, local rings 14P05 Real algebraic sets 14M25 Toric varieties, Newton polyhedra, Okounkov bodies	2021-07-25 09:22:37	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5081010460853577, "perplexity": 586.471085653899}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046151641.83/warc/CC-MAIN-20210725080735-20210725110735-00194.warc.gz"}
https://vcc.codeplex.com/discussions/390815	# \exists quantifier Hare Aug 9, 2012 at 1:43 AM Edited Aug 9, 2012 at 1:47 AM Hi, I am tring to verify some case with \exists quantifier. But I always get the error infor. I even get the error message for the simplest case. What can I do to handle this? I tried to add some triggers like: \match_ulong(t) to get rid of the warnings. But it doesn't work for error message. Thanks very much! ```#include void test(){ _(assert \exists unsigned t; \true) } /* === VCC started. === Command Line: "C:\Program Files (x86)\Microsoft Research\Vcc\Binaries\vcc.exe" /bvd /loc:"c:\Users\Hare\documents\visual studio 2010\Projects\Vcc-test\Vcc-test\test.h":4 "c:\Users\Hare\documents\visual studio 2010\Projects\Vcc-test\Vcc-test\test.h" /p:/D_MBCS /clpath:"C:\PROGRAM FILES (X86)\MICROSOFT VISUAL STUDIO 10.0\VC\bin\cl.exe" c:\Users\Hare\documents\visual studio 2010\Projects\Vcc-test\Vcc-test\test.h(4,14) : warning VC9121: failed to infer triggers for '\exists unsigned t; \true)' Verification of test failed. [0.64] Prover warning: failed to find a pattern for quantifier (quantifier id: testh.4:14) c:\Users\Hare\documents\visual studio 2010\Projects\Vcc-test\Vcc-test\test.h(4,14) : error VC9500: Assertion '\exists unsigned t; \true)' did not verify. Verification errors in 1 function(s) Exiting with 3 (1 error(s).) === Verification failed. === /``` fdupress Editor Aug 9, 2012 at 1:24 PM Edited Aug 9, 2012 at 1:25 PM Hi Hare, I don't have any answers to offer you right now, but you (and any devs looking into this) may be interested in the follwing facts: The following succeeds. ```#include void test(){ _(assert \exists \integer t; \true) }``` The following fails. ```#include void test(){ _(assert \exists \natural t; \true) } ``` The following succeeds (with the usual trigger warning). ```#include void test(){ _(assert \exists \integer t; 0 <= t ==> \true) } ``` In the meantime, I would advise using \integer variables in everything spec, and casting where appropriate. If you need the fact that it's non-negative, you can easily switch between \natural and \integer with a side-condition, although you'll have to add explicit casts where appropriate. Cheers, Francois hare Aug 9, 2012 at 4:43 PM Hi Francois, Thanks very much! Shu Cheng hare Aug 9, 2012 at 5:58 PM Edited Aug 9, 2012 at 6:01 PM Hi Francois, When I go back to my case, it seems \integer also can't help. For instance, ```#include void test(){ _(assert \exists \integer t; t == 5) } void test1(){ int x = 25; _(assert \exists \integer t; t == 5 ==> x == t t) _(assert \exists \integer t; x == t * t) _(assert \exists \integer t; t == 5) } void test2(){ int x = 25; _(assert \forall \integer t; t == 5 ==> x == t * t) _(assert \exists \integer t; t == 5 ==> x == t * t) }``` In these cases, all assertion with \exists are fail to verify. It seems the prover can't solve quantifier \exists correctly. Is this an bug with vcc or prover? If not, what should I do to avoid these problems. Thanks very much! erniecohen Developer Sep 11, 2012 at 4:39 PM Edited Sep 11, 2012 at 4:43 PM It's important to understand how VCC (and more generally, SMT solvers) treat quantifiers. When VCC needs to prove a formula, it first negates and Skolemizes it, and tries to prove \false. It will have two kinds of formulas to use: ground formulas and formulas with quantifiers. It reasons only with the ground formulas, using (essentially) logic and linear arithmetic. A quantified formula is used only by matching its trigger(s) using ground subterms in its formulas to obtain a ground instance, and adding this instance to the collection of ground formulas. So, for example, if you want to prove (\exists \integer t; t == 5), it will actually have the formula (\forall \integer t; t != 5) which it will use to try to prove \false. Even if you had given the existential formula a trigger, to obtain a formula like (\forall \integer t; {\match_long(t)} t != 5) it will still fail to prove this because it has no ground term of the form \match_long(a) with which to instantiate the formula. To get it to trigger, you have to provide a ground instance, for example as follows (which verifies): ```#include _(def \bool match(\integer t) { return \true; }) void test(){ _(assert match(5)) _(assert \exists \integer t; {match(t)} t == 5) } void test1(){ int x = 25; _(assert match(5)) _(assert \exists \integer t; {match(t)} t == 5 ==> x == t * t) _(assert \exists \integer t; {match(t)} x == t * t) _(assert \exists \integer t; {match(t)} t == 5) } void test2(){ int x = 25; _(assert match(5)) _(assert \forall \integer t; {match(t)} t == 5 ==> x == t * t) _(assert \exists \integer t; {match(t)} t == 5 ==> x == t * t) }```	2017-08-18 10:34:18	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8863019943237305, "perplexity": 1518.9323319992243}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886104634.14/warc/CC-MAIN-20170818102246-20170818122246-00366.warc.gz"}
https://socratic.org/questions/what-are-the-absolute-extrema-of-f-x-2x-3-15x-2-in-2-10	# What are the absolute extrema of f(x)=2x^3-15x^2 in[-2,10]? Jan 6, 2016 Minimum: $- 125$, Maximum: $500$. #### Explanation: $f \left(x\right) = 2 {x}^{3} - 15 {x}^{2}$ is continuous on the closed interval $\left[- 2 , 10\right]$, so it has a minimum and a maximum on the interval. (Extreme Value Theorem) The extrema must occur at either an endpoint of the interval or at a critical number for $f$ in the interval. Find the critical numbers for $f$. $f ' \left(x\right) = 6 {x}^{2} - 30 x = 6 x \left(x - 5\right)$. $f '$ is never undefined and $f ' \left(x\right) = 0$ at $x = 0$ and $5$. Both $0$ and $5$ are in the interval of interest (the domain of this problem). Do the arithmetic to find $f \left(- 2\right) = - 16 - 25 \left(4\right) = - 16 - 60 = - 76$ $f \left(0\right) = 0$ $f \left(5\right) = 2 {\left(5\right)}^{3} - 15 {\left(5\right)}^{2} = 10 \left(25\right) - 15 \left(25\right) = - 5 \left(25\right) = - 125$ $f \left(10\right) = 2000 - 1500 = 500$ Minimum: $- 125$, Maximum: $500$.	2019-01-24 10:17:38	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 19, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8978861570358276, "perplexity": 427.5408227931875}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547584520525.90/warc/CC-MAIN-20190124100934-20190124122934-00559.warc.gz"}
https://maaxmarket.com/brazilian-constitution-omdc/angular-velocity-symbol-74d2a0	kg. (When this symbol is used, angular velocity is usually specified in radians per second.) The direction of the angular velocity vector is perpendicular to the plane of rotation, in a direction which is usually specified by the right hand rule. Related Symbolab blog posts. Calculating angular velocity requires understanding the rotational motion of an object, θ. We would have code something like: revolutions per minute), it is often called the rotational velocity and its magnitude the rotational speed. If the positive z - axis points up, then we choose θ to be increasing in the counterclockwise direction as shown in Figures 6.6. rad/s² \| deg/s²: Final Angular Velocity. -vector: m × vector cross product operator (see here for definition) So the rotation velocity of a point is not an absolute value, but it depends on which point that the … Angular velocity is usually represented by the symbol omega (Ω or ω). . When we attach a ball with a rope and spin it in a circle at a constant angular velocity. Angular velocity is a measurement of the rate of change of angular position of an object over a period of time. Choice Crossword … angular velocity synonyms, angular velocity pronunciation, angular velocity translation, English dictionary definition of angular velocity. Define angular velocity. Formula to calculate angular momentum (L) = mvr, where m = mass, v = velocity, and r = radius. 60 km/h to the north). Angular Velocity. Learn vocabulary, terms, and more with flashcards, games, and other study tools. Angular velocity is the rate of change of θ with respect to time. "Angular Velocity." \Angular velocity is usually represented by the symbol omega (ω, sometimes Ω). What is angular velocity measured in? For example, in async pipe source we see the usage of ɵisPromise(_) and ɵisObservable(_) functions. The rate at which a body or particle moves about a fixed axis, i.e. I have two suggestions: First \SI{42}{\degree\per\second} resulting in and second \num{42}\si{\degree\per\second} which looks like . And the symbol we used to represent angular velocity is the Greek letter omega, which looks like a w, but it's really the Greek letter omega. Enter the answer length or the answer pattern to get better results. The symbol used for angular velocity is usually a lower case Greek symbol omega, ω. Angular velocity is represented in units of radians per time or degrees per time (usually radians in physics), with relatively straightforward conversions allowing the scientist or … the rate of change of angular displacement. The SI unit is radians per second (rad/s). Name. Subject. Angular Velocity = Angular displacement × [Time]-1. . s \| min \| h: Go. Angular velocity is usually represented by the symbol omega (Ω or ω). They would have an angular velocity For a point object undergoing circular motion about the z -axis, the angular velocity vector $$\vec{\omega}$$ is directed along … In this angular velocity calculator, we use two different formulas of angular velocity, depending on what input parameters you have. Angular momentum is a vector quantity (more precisely, a pseudovector) that represents the product of a body's rotational inertia and rotational velocity (in radians/sec) about a particular axis. Angular velocity is a pseudovector, with the direction being perpendicular to the plane of rotation as dictated by the right-hand rule. radians per second. Equation 5 gives us the angular velocity, which is constant because the angular acceleration is zero. The SI unit of angular velocity is expressed as radians/sec with the radian having a dimensionless value of unity, thus the SI units of angular velocity are listed as 1/sec. What is Angular Velocity? . If you haven't solved the crossword clue Angular velocity symbol yet try to search our Crossword Dictionary by entering the letters you already know! . Angular velocity is a pseudovector, with the direction being perpendicular to the plane of rotation as dictated by the right-hand rule.It is equal to the angular momentum (L) divided by the moment of inertia (I) of an object. Retrieved from https://www.thoughtco.com/angular-velocity-2699103. (3) On substituting equation (2) and (3) in equation (1) we get, Angular Velocity = Angular displacement × [Time]-1. angular velocity = angular velocity + angular acceleration In fact, the above is actually simpler than what we started with because an angle is a scalar quantity—a single number, not a vector! But the angular velocity must have units of rad/s. It is expressed in radians per second. Also, it refers to the rate of change of an object’s position with respect to time. Angular acceleration is measured in units of angle per unit time squared (which in SI units is radians per second squared), and is usually represented by the symbol alpha (α).In two dimensions, angular acceleration is a pseudoscalar whose sign is taken to be positive if the angular speed increases counterclockwise or decreases clockwise, and is taken to be negative if the angular … Angular deformation = + + 6-2 Fluid Kinematics (cont’d) • Consider the following 2D, differential fluid element with corner A moving with a velocity of + . Linear Velocity. It is expressed in radians per second. By convention, positive angular velocity indicates counterclockwise rotation, while negative is clockwise. However, if the particle's trajectory lies in a single plane, it is sufficient to discard the vector nature of angular momentum, and treat it as a scalar (more precisely, a pseudoscalar). Thus angular velocity, ω, is related to tangential velocity, Vt through formula: Vt = ω r. Here r is the radius of the wheel. Verify Related. \Angular velocity is usually represented by the symbol omega (ω, sometimes Ω). v = $\frac{\Delta S}{\Delta t}$ = $\frac{2 \pi r}{t}$ ω = $\frac{\Delta \theta}{\Delta t}$ = $\frac{2 \pi r}{t}$ Quantity. It is equal to the angular momentum (L) divided by the moment of inertia (I) of an object. On this page you will find the solution to Symbol of angular velocity crossword clue crossword clue. Also, in this topic, we will discover the definition, angular velocity formula its derivation and solved example. Similarly, the units for torque (N m) are dimensionally equivalent to the units for work and energy (joules), but joules are strictly a unit of energy and we never use them as a unit of torque. Figure 6.6 Right handed coordinate system. https://www.thoughtco.com/angular-velocity-2699103 (accessed February 5, 2021). Angular velocity is the rate of velocity at which an object or a particle is rotating around a center or a specific point in a given time period. Physical Quantity. Also, it refers to the rate of change of an object’s position with respect to time. + A B D C + The angular velocity refers to how fast an object is rotating or revolving in a circular path relative to another point. From Simple English Wikipedia, the free encyclopedia, https://simple.wikipedia.org/w/index.php?title=Angular_velocity&oldid=6420252, Creative Commons Attribution/Share-Alike License. Initial Angular Velocity. The concept of angular velocity is for the velocity of an object which undergoes circular motion . Vector quantities (F, g, v) are written in a bold, serif font — including vector quantities written with Greek symbols (α, τ, ω).Scalar quantities (m, K, t) and … symbol. The SI unit of angular velocity is expressed as radians/sec with the radian having a dimensionless value of unity, thus the SI units of angular velocity are listed as 1/sec. Symbol of angular velocity -- Find potential answers to this crossword clue at crosswordnexus.com angular velocity The angle θ represents the angular displacement as body b orbits. the rate of change of angular displacement. If you encounter two or more answers look at the most recent one i.e the last item on the answers box. radians per second. Ipeak - maximal current (=amplitude of … Angular velocity = (final angle) - (initial angle) / time = change in position/time. Mechanics. en. m, M. kilogram. The angular velocity applies to the entire object that moves along a circular path. Verify Related. In symbols, this is $\boldsymbol{\omega\:=}\boldsymbol{\frac{\Delta\theta}{\Delta{t}}},$ where an angular rotation Δθ takes place in a time Δt. Related pages Edit The English Wiktionary has a dictionary definition (meanings of a word) for: angular velocity. But it may be measured in other units as well (such as degrees per second, degrees per hour, etc.). Angular velocity is usually represented by the symbol omega (ω, sometimes Ω). Angular velocity is a measurement of the rate of change of angular position of an object over a period of time. It is a vector quantity. The unit is degrees/square of time like s 2, min 2, hr … . Both look unfamiliar, but the output of the … Also, in this topic, we will discover the definition, angular velocity formula its derivation and solved example. Angular Momentum Formula The angular momentum of an object having mass (m) and linear velocity (v) with … Symbol: ω. The greater the rotation angle in a given amount of time, the greater the angular velocity. As the name suggests, tangential velocity describes the motion of an object along the edge of this circle … angular-acceleration-calculator. One-Dimensional Kinematics: Motion Along a Straight Line, Two-Dimensional Kinematics or Motion in a Plane, The Most Affordable Online Degree Programs, What Is Calculus? It is an axial vector. How fast is an object rotating? This page was last changed on 4 February 2019, at 22:31. Linear velocity applies to an object or particle that is moving in a straight line. (See our article on dimensional analysis for some tips on performing this sort of conversion.). Velocity is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of bodies. Angular Velocity is a measure of how quickly an object moves through an angle. Using the answer from the previous challenge, let’s say we wanted to rotate a baton in ProcessingJS by some angle. The angular velocity is defined as the change in angular position divided by a change in time—that means the slope of angle vs. time would actually be the … Angular velocity is the rate of velocity at which an object or a particle is rotating around a center or a specific point in a given time period. Linear velocity applies to an object or particle that is moving in a straight line. . Having an angular velocity in degree (unit: degree per seconds; I do not want to use rad/s here), what is the recommended way to have a nice (correct) output and what is the recommended way with siunitx? In physics, angular velocity refers to how fast an object rotates or revolves relative to another point, i.e. Practice Makes Perfect. ω = 2π f. ω - angular velocity measured in radians per second (rad/s) f - frequency measured in hertz (Hz). Now I need to convert this from centimeters-per-minute to kilometers-per-hour: The velocity will be the (linear, or equivalent straight-line) distance traveled in one second, divided by the one second. ω = v/r, where ω is the Greek letter omega. Angular acceleration is the rate of change of angular velocity. Symbol of angular velocity NYT Crossword Clue Answers are listed below and every time we find a new solution for this clue we add it on the answers list. rotational velocity, rotational speed: rad/s: radian per second: α, α: rotational … Copyright © 2020 Multiply Media, LLC. Also Known As: average angular velocity, instantaneous angular velocity, Andrew Zimmerman Jones is a science writer, educator, and researcher. where ν is the frequency of the wave, λ is the wavelength, ω = 2πν is the angular frequency of the wave, and v p is the phase velocity of the wave. The angular velocity applies to the entire object that moves along a circular path. 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Kitchen Floor Tiles Vinyl, New Math Curriculum Pdf, Vegetable Oil Price Philippines 2020, Rustoleum Kona Stain Gallon, Medical-surgical Nursing Textbook Lewis, Kalanchoe Fedtschenkoi 'variegata Care, Congratulations On Recovery From Covid-19, Acacia Cognata Mini Cog Care,	2021-05-07 10:16:11	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8649453520774841, "perplexity": 927.5127289428717}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243988775.80/warc/CC-MAIN-20210507090724-20210507120724-00242.warc.gz"}
http://libros.duhnnae.com/2017/jul6/150032803514-A-microlensing-study-of-the-accretion-disc-in-the-quasar-MG-0414-0534-Astrophysics.php	# A microlensing study of the accretion disc in the quasar MG 0414 0534 - Astrophysics A microlensing study of the accretion disc in the quasar MG 0414 0534 - Astrophysics - Descarga este documento en PDF. Documentación en PDF para descargar gratis. Disponible también para leer online. Abstract: Observations of gravitational microlensing in multiply imaged quasarscurrently provide the only direct probe of quasar emission region structure onsub-microarcsecond scales. Analyses of microlensing variability areobservationally expensive, requiring long-term monitoring of lensed systems.Here we demonstrate a technique for constraining the size of the quasarcontinuum emission region as a function of wavelength using single-epochmulti-band imaging. We have obtained images of the lensed quasar MG 0414+0534in five wavelength bands using the Magellan 6.5-metre Baade telescope at LasCampanas Observatory, Chile. These data, in combination with two existingepochs of Hubble Space Telescope data, are used to model the size of thecontinuum emission region $\sigma$ as a power-law in wavelength,$\sigma\propto\lambda^ u$. We place an upper limit on the Gaussian width ofthe $r^\prime$-band emission region of $1.80 \times 10^{16} h {70}^{-1-2}- mn{M} {\odot}^{1-2}$cm, and constrain the power-law index to$0.77\leq u\leq2.67$ 95 per cent confidence range. These results can be usedto constrain models of quasar accretion discs. As a example, we find that theaccretion disc in MG 0414+0534 is statistically consistent with aShakura-Sunyaev thin disc model. Autor: N. F. Bate, D. J. E. Floyd, R. L. Webster, J. S. B. Wyithe Fuente: https://arxiv.org/	2018-08-16 04:33:53	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8091819882392883, "perplexity": 9268.786317162932}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221210413.14/warc/CC-MAIN-20180816034902-20180816054902-00719.warc.gz"}
http://math.stackexchange.com/questions/113919/growth-condition-for-ito-diffusions	# Growth condition for Ito diffusions Given a one-dimensional SDE $$\begin{cases} dX_t &= b(t,X_t)dt+\sigma(t,X_t)dB_t, \\ X_0 &= Z \end{cases}$$ for $t\in[0,T]$ where $Z$ is square integrable: $\mathsf E[Z^2]<\infty$ the sufficient conditions for existence and uniqueness are: the growth condition $$\|b(t,x)\|+\|\sigma(t,x)\|\leq C(1+\|x\|)\quad\forall x\in \mathbb R, t\in [0,T]$$ and the Lipschitz condition $$\|b(t,x'')-b(t,x')\|+\|\sigma(t,x'')-\sigma(t,x')\|\leq D\|x''-x'\|\quad\forall x',x''\in \mathbb R.$$ These conditions are stated e.g. in Theorem 5.2.1, "Stochastic Differential Equations" (p. 68 here). However, in the definition of an Ito diffusion as a process satisfying $$dX_t = b(X_t)dt+\sigma(X_t)dB_t$$ for $t\geq s$, where $X_s = x$, in the same book (p. 114 on the linked webpage) it is written that conditions simplify to Lipschitz condition. Could you help me to understand the reasoning of that? - @sos440: thank you - would you put this as an answer? – Ilya Feb 27 '12 at 8:48 Let's consider a simplest case: Let $f : \mathbb{R} \to \mathbb{R}$ satisfy global Lipschitz condition $$\|f(x) - f(y)\| \leq L \|x - y\|, \quad x, y \in \mathbb{R}$$ with Lipschitz constant $L$. Then by triangle inequality, we have $$\|f(x)\| \leq \|f(x) - f(0)\| + \|f(0)\| \leq L\|x\| + \|f(0)\| \leq C(\|x\| + 1)$$ for large $C \geq \max(L, \|f(0)\|)$. Same argument applies to this case.	2014-12-21 00:24:53	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9514204859733582, "perplexity": 309.32942898561726}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1418802770433.122/warc/CC-MAIN-20141217075250-00154-ip-10-231-17-201.ec2.internal.warc.gz"}
http://www.docstoc.com/docs/39263219/The-jed-Editor	# The jed Editor Document Sample The jed Editor Contents 1 Introduction 3 2 Starting JED 3 3 Emulating Other Editors 4 3.1 Emacs Emulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.2 EDT Emulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.3 Wordstar Emulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 4 Backup and Auto-save Files 4 5 Status line and Windows 5 6 Mini-Buffer 5 6.1 Command Line Completion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 6.2 File Names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 6.3 Buffer Name and File Name Completion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 7 Basic Editing 7 7.1 Undo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 7.2 Marking Text (Point and Mark) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 7.3 Tab Issues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 7.4 Searching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 7.5 Rectangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 7.6 Sorting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 The JED Editor 2 8 Modes 10 8.1 Wrap Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Formatting paragraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 8.2 Smart Quotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 8.3 C Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 8.4 Fortran Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 9 Keyboard Macros 11 10 Shells and Shell Commands 12 11 Getting Help 12 12 Editing Binary Files 12 13 Dired— the Directory editor 13 14 Customization 13 14.1 Setting Keys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 14.2 Predeﬁned Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 15 Eight Bit Clean Issues 14 15.1 Displaying Characters with the High Bit Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 15.2 Inputting Characters with the hight bit Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 15.3 Upper Case - Lower Case Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 16 Miscellaneous 16 16.1 Abort Character . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 16.2 Input Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 16.3 Display Sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 17 xjed 16 17.1 Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 17.2 Mouse Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 17.3 EDT emulation under Linux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 18 Using a mouse with jed and xjed 18 18.1 Native Mouse Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Clicking in a window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Clicking on a window status line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Tips for using the mouse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 18.2 XTerm Event Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Mouse Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Cut/Paste Tips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 The JED Editor 3 How do I obtain jed? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 How do I disable jed’s C mode? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 What is C mode? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 How do I turn on wrap mode or turn it off? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 What is the difference between internal and intrinsic functions? . . . . . . . . . . . . . . . . . . . . . . . . . 22 Sometimes during screen updates, jed pauses. Why is this? . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 How do I get jed to recognize Control-S and Control-Q? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Can I bind the Alt keys on the PC? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 How do I ﬁnd out what characters a particular key generates? . . . . . . . . . . . . . . . . . . . . . . . . . . 23 jed scrolls slow on my WizBang-X-Super-Terminal. What can I do about it? . . . . . . . . . . . . . . . . . . 24 How do I get a list of functions? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 How can I use edt.sl with jed386.exe? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 How do I set custom tab stops in jed? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 The JED Editor 4 Questo documento è un’elaborazione della documentazione originale di jed, un editor emacs-like disponibile su piattaforme UNIX, MS-DOS e VMS (esiste anche xjed, una versione per X11). 1 Introduction This document presents some basic information that users should know in order to use jed effectively. Any questions, comments, or bug reports, should be email-ed to the author. Please be sure to include the version number. To be notiﬁed of davis@space.mit.edu jed is primarily a text editor; however, it can also edit binary ﬁles (see the section on editing binary ﬁles). As a result, jed may edit lines of arbitrary length (actually this depends upon the size of an integer). It is capable of editing arbitrarily large buffers as long as there is enough memory for the buffer as well as the overhead involved. This editor employs a linked list representation; hence, the overhead can be quite high. 2 Starting JED Normally, jed is started as jed <file name> However, jed also takes the switches deﬁned in the following table: -batch run JED in batch mode. This is a non-interactive mode -n do not load jed.rc (.jedrc) ﬁle -g <n> goto line <n> in buffer -l <file> load <ﬁle> as S-Lang code -f <function> execute S-Lang function named <function> -s <string> search forward for <string> -2 split window -i <file> insert <ﬁle> into current buffer For example, the command line: jed slang.c -g 1012 -s error -2 file.c -f eob will start up jed, read in the ﬁle slang.c, goto line 1012 of slang.c and start searching for the string error, split the window, read in file.c and goto the end of the ﬁle. If the -batch parameter is used, it must be the ﬁrst parameter. Similarly, if -n is used, it must also be the ﬁrst parameter unless used with the -batch parameter in which case it must the second. jed should only be run in batch mode when non–interactive operation is desired. For example, jed is distributed with a ﬁle, mkdoc.sl, that contains S–Lang code to produce a help ﬁle for functions and variables. In fact, the help ﬁle jed_funs.hlp was created by entering jed -batch -n -l mkdoc.sl at the command line. Now suppose that you want to read in a ﬁle with the name of one of the switches, say -2. How can this be done? The answer depends upon the operating system. For Unix, instead of jed -2, use jed ./-2; for VMS, use jed []-2. The case for MS-DOS is similar to Unix except that one must use the backslash. Once jed has loaded the startup ﬁle site.sl, it will try to load the user’s personal initialization ﬁle. It ﬁrst looks in the directory pointed to by the environment variable JED_HOME. If that fails, it then searches the HOME directory and upon failure simply loads the one supplied in JED_LIBRARY. The name of the user initialization ﬁle varies according to the operating system. On Unix systems this ﬁle must be called .jedrc while on VMS and MSDOS, it goes by the name jed.rc. For VMS systems, the HOME directory corresponds to the SYS$LOGIN logical name while for the other two systems, it corresponds to the HOME environment variable. The purpose of this ﬁle is to allow an individual user to tailor jed to his or her personal taste. Most likely, this will involve choosing an initial set of key-bindings, setting some variables, and so on. The JED Editor 5 3 Emulating Other Editors jed’s ability to create new functions using the S–Lang programming language as well as allowing the user to choose key bindings, makes the emulation of other editors possible. Currently, jed provides reasonable emulation of the Emacs, EDT, and Wordstar editors. 3.1 Emacs Emulation Emacs Emulation is provided by the S-Lang code in emacs.sl. The basic functionality of Emacs is emulated; most Emacs users should have no problem with jed. To enable Emacs emulation in jed, make sure that the line evalfile ("emacs"); pop (): is in your jed.rc (.jedrc) startup ﬁle. jed is distributed with this line already present in the default jed.rc ﬁle. 3.2 EDT Emulation For EDT emulation, edt.sl must be loaded. This is accomplished by ensuring that the line evalfile ("edt"); pop (); is in present in the jed.rc (.jedrc) Startup File. jed is distributed with EDT emulation enabled on VMS and Unix systems but the above line is commented out in the jed.rc ﬁle on MS-DOS systems. This emulation provides a near identical emulation of the EDT keypad key commands. In addition, the smaller keypad on the newer DEC terminals is also setup. It is possible to have both EDT and Emacs emulation at the same time. The only restriction is that emacs.sl must be loaded before edt.sl is loaded. One minor difference between jed’s EDT emulation and the real EDT concerns the Ctrl-H key. EDT normally binds this to move the cursor to the beginning of the line. However, jed uses it as a help key. Nevertheless, it is possible to re-bind it. See the section on re-binding keys as well as the ﬁle edt.sl for hints. Alternatively, simply put unsetkey ("^H"); setkey ("bol", "^H"); in the jed.rc startup ﬁle after edt.sl is loaded. Keep in mind that the Ctrl-H key will no longer function as a help key if this is done. EDT emulation for PCs only work with the enhanced keyboard. When edt.sl is loaded, a variable NUMLOCK_IS_GOLD is set which instructs jed to interpret the Num-Lock key on the square numeric keypad to function as the EDT GOLD key. In fact, this keypad should behave exactly like the keypad on VTxxx terminals. The only other problem that remains concerns the + key on the PC keypad. This key occupies two VTxxx key positions, the minus and the comma (delete word and character) keys. Thus a decision had to be made about which key to emulate. I chose the + key to return the characters Esc O l which jed maps to the delete character function. This may be changed to the delete word function if you prefer. See the ﬁle edt.sl for details. The GOLD–GOLD key combination toggles the keypad between application and numeric states. On the PC, this is not possible. Instead, the PC F1 key has been instructed to perform this task. 3.3 Wordstar Emulation wordstar.sl contains the S-Lang code for jed’s Wordstar emulation. Adding the line evalfile ("wordstar"); pop (); to your jed.rc (.jedrc) startup ﬁle will enable jed’s Wordstar emulation. 4 Backup and Auto-save Files On UNIX and MS-DOS systems, jed creates backup ﬁles by appending a ~ character to the ﬁlename. The VMS operating system handles backup ﬁles itself. jed periodically auto-saves its buffers. On UNIX and MS-DOS, auto-save ﬁles are preﬁxed with the pound sign #. On VMS, they are preﬁxed with \_$. The auto-save interval may be changed by setting the variable The JED Editor 6 MAX_HITS to the desired value. The default is 300 “hits” on the buffer. A “hit” is deﬁned as a key which MAY change the state of the buffer. Cursor movement keys do not cause hits on the buffer. Like many of jed’s features, the names of auto-save and backup ﬁles can be controlled by the user. The ﬁle site.sl deﬁnes two functions, make_backup_filename, and make_autosave_filename that generate the ﬁle names described in the previous paragraph. Like all S–Lang functions, these functions may be overloaded and replaced with different ones. See also information about find_file_hook in the section on hooks. On UNIX systems, jed catches most signals and tries to auto-save its buffers in the event of a crash or if the user accidently disconnects from the system (SIGHUP). If an auto-save ﬁle exists and you is desire to recover data from the auto-save ﬁle, use the function recover_file. Whenever jed ﬁnds a ﬁle, it checks to see if an auto-save ﬁle exists as well as the ﬁle’s date. If the dates are such that the auto-save ﬁle is more recent jed will display a message in the mini-buffer alerting the user of this fact and that the function recover_file should be considered. 5 Status line and Windows jed supports multiple windows. Each window may contain the same buffer or different buffers. A status line is displayed immediately below each window. The status line contains information such as the jed version number, the buffer name, “mode”, etc. Please beware of the following indicators: ** buffer has been modiﬁed since last save m Mark set indicator. This means a region is being deﬁned d File changed on disk indicator. This indicates that the ﬁle associated with the buffer is newer than the buffer itself s spot pushed indicator + Undo is enabled for the buffer [Macro] A macro is being deﬁned [Narrow] Buffer is narrowed to a region of LINES 6 Mini-Buffer The Mini-Buffer consists of a single line located at the bottom of the screen. Much of the dialog between the user and jed takes place in this buffer. For example, when you search for a string, jed will prompt you for the string in the Mini-Buffer. The Mini-Buffer also provides a direct link to the S-Lang interpreter. To access the interpreter, press Ctrl-X Esc and the S-Lang> prompt will appear in the Mini-Buffer. Enter any valid S-Lang expression for evaluation by the interpreter. It is possible to recall data previously entered into the Mini-Buffer by using the up and down arrow keys. This makes it possible to use and edit previous expressions in a convenient and efﬁcient manner. 6.1 Command Line Completion The jed editor has several hundred built–in functions as well as many more written in the S-Lang extension language. Many of these functions are bound to keys and many are not. It is simply unreasonable to require the user to remember if a function is bound to a key or not and, if it is, to remember the key to which it is bound. This is especially true of those functions that are bound but rarely used. More often than not, one simply forgets the exact name or spelling of a function and requires a little help. For this reason, jed supports command line completion in the mini-buffer. This function, called emacs_escape_x, is bound to the key Esc X. This is one binding that must be remembered! As an example, suppose that you are editing several buffers and you wish to insert the contents of one buffer into the current buffer. The function that does this is called insert_buffer and has no default key-binding. Pressing Esc X produces the prompt M-x. This prompt, borrowed from the Emacs editor, simply means that Esc X was pressed. Now type in and hit the space bar or the Tab key. In this context (completion context) the space bar and the Tab will expand the string in the Mini-Buffer up until it is no longer unique. In this case, insert_file and insert_buffer are only the two functions uniquely specify the desired function. However, in a completion context, the space bar also has a special property that enables The JED Editor 7 the user to cycle among the possible completions. For this example, hitting the space bar twice consecutively will produce the string insert_file and hitting it again produces the desired string insert_buffer. The role of the space bar in completion is a point where Emacs and jed differ. Emacs will pop up a buffer of possible completions but jed expects the user to press the space bar to cycle among them. Both have there pros and cons. Frequently, one sees messages on the Usenet newsgroup gnu.emacs.help from Emacs users asking for the kind of completion jed employs. 6.2 File Names jed takes every ﬁle name and “expands it” according to a set of rules which vary according to the Operating System. For concreteness, consider jed running under MS-DOS. Suppose the user reads a new ﬁle into the editor via the find_file command which emacs binds to Ctrl-X Ctrl-F. Then the following might be displayed in the mini-buffer: Find File: C:\JED\SLANG\ Here jed is prompting for a ﬁle name in the directory \JED\SLANG on disk C:. However, suppose the user wants to get the ﬁle C:\JED\SRC\VIDEO.C. Then the following responses produce equivalent ﬁlenames when jed expands them internally: Find File: C:\JED\src\video.c Find File: C:\JED\SLANG\..\src\video.c Find File: C:\JED\SLANG\../src/video.c Note that the on MS-DOS systems, jed replaces the / with a \ and that case is not important. Now suppose you wish to get the ﬁle VIDEO.C from disk A:. The following are also valid: Find File: A:\video.c Find File: A:video.c Find File: C:\JED\SLANG\a:\video.c In the last case, jed is smart enough to ﬁgure out what is really meant. Although the above examples are for MS-DOS systems, the rules also apply to Unix and VMS systems as well. The only change is the ﬁle name syntax. For example, on VMS sys$manager:[misc]dev$user:[davis.jed]vms.c dev$user:[davis.jed]vms.c become equivalent ﬁlenames upon expansion. For unix, the following are equivalent: /user1/users/davis/jed/unix.c /usr/local/src//user1/users/davis/jed/unix.c /usr/local/src/~/jed/unix.c Note the last example: the tilde character ~ always expands into the users HOME directory, in this case to /user1/users/davis. When jed writes a buffer out to a ﬁle, it usually prompts for a ﬁle name in the minibuffer displaying the directory associated with the current buffer. At this point a name can be appended to the directory string to form a valid ﬁle name or the user may simply hit the RET key. If the latter alternative is chosen, jed simply writes the buffer to the ﬁle already associated with the buffer. Once the buffer is written to a ﬁle, the buffer becomes attached to that ﬁle. 6.3 Buffer Name and File Name Completion When jed prompts for a ﬁle name or a buffer name, the space bar and the Tab keys are special. Hitting the Tab key will complete the name that is currently in the minibuffer up until it is no longer unique. At that point, you can either enter more characters to complete the name or hit the space bar to cycle among the possible completions. The spacebar must be pressed at least twice to cycle among the completions. On MSDOS and VMS, it is possible to use wildcard characters in the ﬁle name for completion purposes. For example, entering .c and hitting the space bar will cycle among ﬁle names matching .c. Unfortunately, this feature is not available on unix systems. The JED Editor 8 7 Basic Editing Editing with jed is pretty easy— most keys simply insert themselves. Movement around the buffer is usually done using the arrow keys or page up and page down keys. If edt.sl is loaded, the keypads on VTxxx terminals function as well. Here, only the highlights are touched upon (cut/paste operations are not considered “highlights”). In the following, any character preﬁxed by the ^ character denotes a Control character. On keyboards without an explicit Escape key, Ctrl-[ will most likely generate and Escape character. A “preﬁx argument” to a command may be generated by ﬁrst hitting the Esc key, then entering the number followed by pressing the desired key. Normally, the preﬁx argument is used simply for repetition. For example, to move to the right 40 characters, one would press Esc 4 0 followed immediately by the right arrow key. This illustrates the use of the repeat argument for repetition. However, the preﬁx argument may be used in other ways as well. For example, to begin deﬁning a region, one would press the Ctrl-@ key. This sets the mark and begins highlighting. Pressing the Ctrl-@ key with a preﬁx argument will abort the act of deﬁning the region and to pop the mark. The following list of useful keybindings assumes that emacs.sl has been loaded. Ctrl-L Redraw screen Ctrl-_ Undo (Control-underscore, also Ctrl-X u) Esc q Reformat paragraph (wrap mode). Used with a preﬁx argument. will justify the para- graph as well. Esc n narrow paragraph (wrap mode). Used with a preﬁx argument will justify the paragraph as well. Esc ; Make Language comment (Fortran and C) Esc \ Trim whitespace around point Esc ! Execute shell command Esc$ Ispell word (unix) Ctrl-X ? Show line/column information ‘ quoted_insert — insert next char as is (backquote key) Esc s Center line Esc u Upcase word Esc d Downcase word Esc c Capitalize word Esc x Get M-x minibuffer prompt with command completion Ctrl-X Ctrl-B pop up a list of buffers Ctrl-X Ctrl-C exit jed Ctrl-X 0 Delete Current Window Ctrl-X 1 One Window Ctrl-X 2 Split Window Ctrl-X o Other window Ctrl-X b switch to buffer Ctrl-X k kill buffer Ctrl-X s save some buffers Ctrl-X Esc Get S-Lang> prompt for interface to the S-Lang interpreter Ctrl-Z Sospende jed (vedi nota sotto) Esc . Find tag (unix ctags compatible) Ctrl-@ Set Mark (Begin deﬁning a region). Used with a preﬁx argument aborts the act of deﬁning the region and pops the Mark A differenza di XEmacs, Ctrl-E non termina la sessione di editing, in modo da consentire di accedere agevolmente alla funzione di end-of-line anche a chi non ha a disposizione un keypad come nelle tastiere Digital. Chi volesse uniformarsi alla deﬁnizione di XEmacs, dovrà aggiungere questa riga nel proprio .jedrc: setkey("exit_jed", "^E"); Sotto VMS il comando Ctrl-Z permette di sospendere temporaneamente l’esecuzione di jed. Alla successiva esecuzione del comando jed viene ripristinata la situazione precedente (in modo analogo a quanto avviene sotto unix, dove però bisogna usare il comando fg). The JED Editor 9 7.1 Undo One of jed’s nicest features is the ability to undo nearly any change that occurs within a buffer at the touch of a key. If you delete a word, you can undo it. If you delete 10 words in the middle of the buffer, move to the top of the buffer and randomly make changes, you can undo all of that too. By default, the undo function is bound to the key Ctrl-_ (Ascii 31). Since some terminals are not capable of generating this character, it is also bound to the key sequence Ctrl-X u. Due to the lack of virtual memory support on IBMPC systems, the undo function is not enabled on every buffer. In particular, it is not enabled for the scratch buffer. However, it is enabled for any buffer which is associated with a ﬁle. A + character on the left hand side of the status line indicates that undo is enabled for the buffer. It is possible to enable undo for any buffer by using the toggle_undo function. 7.2 Marking Text (Point and Mark) Many commands work on certain regions of text. A region is deﬁned by the Point and the Mark The Point is the location of the current editing point or cursor position. The Mark is the location of a mark. The mark is set using the set_mark_cmd which is bound to Ctrl-@ (Control-2 or Control-Space on some keyboards). When the mark is set, the m mark indicator will appear on the status line. This indicates that a region is being deﬁned. Moving the cursor (Point) deﬁnes the other end of a region. If the variable HIGHLIGHT is non-zero, jed will highlight the region as it is deﬁned. Even without highlighting, it is easy to see where the location of the mark is by using the exchange command which is bound to Ctrl-X Ctrl-X. This simply exchanges the Point and the Mark. The region is still intact since it is deﬁned only by the Point and Mark. Pressing Ctrl-X Ctrl-X again restores the mark and Point back to their original locations. Try it. 7.3 Tab Issues. Strictly speaking, jed uses only ﬁxed column tabs whose size is determined by the value of the TAB variable. Setting the TAB variable to 0 causes jed to not use tabs as whitespace and to display tabs as Ctrl-I. Please note that changing the tab settings on the terminal will have no effect as far as jed is concerned. The TAB variable is local to each buffer allowing every buffer to have its own tab setting. The variable TAB_DEFAULT is the tab setting that is given to all newly created buffers. The default value for this variable is 8 which corresponds to eight column tabs. jed is also able to “simulate” arbitrary tabs as well through the use of user deﬁned tab stops. One simply presses Esc X to get the M-x prompt and enters edit_tab_stops. A window will pop open displaying the current tab settings. To add a tab stop, simply place a T in the appropriate column. Use the space bar to remove a tab stop. Here an argument is presented in favor of simulated tabs over real tab stops. First, consider what a “tab” really is. A “tab” in a ﬁle is nothing more than a character whose ASCII value is 9. For this reason, one also denotes a tab as ^I (Ctrl-I). Unlike most other ASCII characters, the effect of the tab character is device dependent and is controlled through the device tab settings. Hence, a ﬁle which displays one way on one device may look totally different on another device if the tab settings do not correspond. For this reason, many people avoid tabs altogether and others the adopt “standard” of eight column tabs. Even though people always argue about what the correct tab settings should be, it must be kept in mind that this is primarily a human issue and not a machine issue. On a device employing tab stops, a tab will cause the cursor to jump to the position of the next tab stop. Now consider the effect of changing the tab settings. Assume that in one part of a document, text was entered using the ﬁrst setting and in another part, the second setting was used. When moving from the part of the document where the current tab setting is appropriate to the part where the other tab setting was used will cause the document to look unformatted unless the appropriate tab settings are restored. Wordprocessors store the tab settings in the ﬁle with the text so that the tabs may be dynamically changed to eliminate such unwanted behavior. However, text editors such as jed, vi, Emacs, EDT, EVE (TPU), etc, do not store this information in the ﬁle. jed avoids this problem by using simulated tabs. When using simulated tabs, tabs are not really used at all. Rather jed inserts the appropriate number of spaces to achieve the desired effect. This also has the advantage of one being able to cut and paste from the part of a document using one tab setting to another part with a different tab setting. This simple operation may lead to unwanted results on some wordprocessors as well as those text editors using real tab stops. The JED Editor 10 7.4 Searching jed currently has two kinds of searches: ordinary searches and incremental searches. Both types of searches have forward and backward versions. The actual functions for binding purposes are: isearch_forward Ctrl-F isearch_backward Ctrl-B Dato che jed può venire utilizzato su teminali non graﬁci che utilizzano il protocollo XON/XOFF, Ctrl-S e Ctrl-R non hanno il binding tradizionale di search_forward e search_backward. Chi volesse ripristinare questi binding deve aggiungere queste righe nel proprio .jedrc setkey("search_forward", "^S"); setkey("search_backward", "^R"); There is also the occur function which ﬁnds all occurrences of a single word (string). This function has no backwards version. By default it is not bound to any keys, so to use it, occur must be entered at the M-x prompt (Esc X) or one is always free to bind it to a key. In the following only the incremental search is discussed. As the name suggests, an incremental search performs a search incrementally. That is, as you enter the search string, the editor begins searching right away. For example, suppose you wish to search for the string apple. As soon as the letter a is entered into the incremental search prompt, jed will search for the ﬁrst occurrence of a. Then as soon as the p is entered, jed will search from the current point for the string ap, etc. This way, one is able to quickly locate the desired string with only a minimal amount of information. The search is terminated with the Enter key. Finally, the DEL key (Ctrl-?) is used to erase the last character entered at the search prompt. In addition to erasing the last character of the search string, jed will return back to the location of the previous match. Erasing all characters will cause the editor to return to the place where the search began. Like many things, this is one of those that is easier to do than explain. Feel free to play around with it. 7.5 Rectangles jed has built-in support for the editing of rectangular regions of text. One corner of rectangle is deﬁned by setting the mark somewhere in the text. The Point (cursor location) deﬁnes the opposite corner of the rectangle. Once a rectangle is deﬁned, one may use the following functions: kill_rect Delete text inside the rectangle saving the rectangle in the internal rectangle buffer. n_rect Push all text in the rectangle to the right outside the rectangle copy_rect Copy text inside the rectangle to the internal rectangle buffer blank_rect Replace all text inside the rectangle by spaces The function insert_rect inserts a previously killed or copied rectangle into the text at the Point. These functions have no default binding and must be entered into the MiniBuffer by pressing Esc X to produce the M-x prompt. 7.6 Sorting jed is capable of sorting a region of lines using the heapsort algorithm. The region is sorted alphabetically based upon the ASCII values of the characters located within a user deﬁned rectangle in the region. That is, the rectangle simply deﬁnes the characters upon what the sort is based. Simply move to the top line of the region and set the mark on the top left corner of the rectangle. Move to the bottom line and place the point at the position which deﬁnes the lower right corner of the rectangle. Press Esc X to get the M-x prompt and enter sort As as example, consider the following data: Fruit: Quantity: lemons 3 pears 37 peaches 175 The JED Editor 11 apples 200 oranges 56 To sort the data based upon the name, move the Point to the top left corner of the sorting rectangle. In this case, the Point should be moved to the l in the word lemons. Set the mark. Now move to the lower right corner of the rectangle which is immediately after the s in oranges. Pressing Esc X and entering sort yields: Fruit: Quantity: apples 200 lemons 3 oranges 56 peaches 175 pears 37 Suppose that it is desired to sort by quantity instead. Looking at the original (unsorted) data, move the Point to two spaces before the 3 on the line containing lemons. The cursor should be right under the u in Quantity. Set the mark. Now move the Point to immediately after 56 on the oranges line and again press Esc X and enter sort. This yields the desired sort: Fruit: Quantity: lemons 3 pears 37 oranges 56 peaches 175 apples 200 8 Modes jed supports two internal modes as well as user deﬁned modes. The two internal modes consist of a “C” mode for C Language programming and a “Wrap” mode for ordinary text editing. Examples of user deﬁned modes are Fortran mode and DCL mode. Online documentation is provided for nearly every mode jed deﬁnes. For help on the current mode, press Esc X and enter describe_mode. A window will appear with a short description of the special features of the mode as well as a description of the variables affecting the mode. 8.1 Wrap Mode In this mode, text is wrapped at the column given by the WRAP variable. The default is 78. The text does not wrap until the cursor goes beyond the wrap column and a space is inserted. Formatting paragraphs Paragraph delimiters are: blank lines, lines that begin with either a percent character, %, or a backslash character \. This deﬁnition is ideally suited for editing LTEX documents. However, it is possible for the user to change this deﬁnition. See the A discussion of the hook, is_paragraph_separator, in the section on hooks for explicit details on how to do this. The paragraph is formatted according to the indentation of the current line. If the current line is indented, the paragraph will be given the same indentation. The default binding for this function is Esc q. In addition, a paragraph may be “narrowed” by the narrow_paragraph function which is bound to Esc N by default. This differs from the ordinary format_paragraph function described above in that the right margin is reduced by an amount equal to the indentation of the current line. For example: This paragraph is the result of using the function ‘‘narrow_paragraph’’. Note how the right margin is less here than in the above paragraph. Finally, if either of these functions is called from the keyboard with a preﬁx argument, the paragraph will be justiﬁed as well. For example, pressing Esc 1 Esc N on the previous paragraph yields: The JED Editor 12 This paragraph is the result of using the function ‘‘narrow_paragraph’’. Note how the right margin is less here than in the above paragraph. See the discussion of format_paragraph_hook in the section on hooks for details on how this is implemented. 8.2 Smart Quotes You have probably noticed that many key words in this document are quoted in double quotes like “this is double quot- ed” and ‘this is single quoted’. By default, the double quote key (") and single quote key (’) are bound to the function text_smart_quote. With this binding and in wrap mode, the single quote key inserts a single quote with the “proper” orientation and the double quote key inserts two single quotes of the “proper” direction. To turn this off, rebind the keys to self_insert_cmd. Some modes already do this (e.g., EDT). This brings up the question: if the double quote key is bound to text_smart_quote then how does one insert the character (")? The most common way is to use the quoted_insert function which, by default, is bound to the single backquote (‘) key. This is the same mechanism that is used to insert control characters. The other method is to use the fact that if the preceding character is a backslash, \, the character simply self inserts. Again, this is ideal for writing TEX documents. 8.3 C Mode C Mode facilitates the editing of C ﬁles. Much of the latter part of the development of the jed editor was done using this mode. This mode may be customized by a judicious choice of the variables C_INDENT and C_BRACE as well as the bindings of the curly brace keys { and }. Experiment to ﬁnd what you like or write your own using the S-Lang interface. By default, the Enter key is bound to the function newline_and_indent. This does what its name suggests: inserts a newline and indents. Again, some modes may rebind this key. In addition, the keys {, }, and Tab are also special in this mode. The Tab key indents the current line and the { and } keys insert themselves and reindent. If you do not like any of these bindings, simply rebind the offending one to self_insert_cmd. Finally, the key sequence Esc ; is bound to a function called c_make_comment. This function makes and indents a C comment to the column speciﬁed by the value of the variable C_Comment_Column. If a comment is already present on the line, it is indented. 8.4 Fortran Mode Fortran Mode is written entirely in S-Lang and is designed to facilitate the writing of Fortran programs. It features automatic indentation of Fortran code as well as automatic placement of Fortran statement Labels. In this mode, the keys 0-9 are bound to a function for_elebel which does the following: 1. Inserts the calling character (0-9) into the buffer. 2. If the character is preceded by only other digit characters, it assumes the character is for a label and moves it to the appropriate position. 3. Reindents the line. This function is very similar to the one Emacs uses for labels. 9 Keyboard Macros jed is able to record a series of keystrokes from the terminal and replay them. The saved series of keystrokes is known as a keyboard macro. To begin a keyboard macro, simply enter the begin keyboard macro key sequence which is bound to Ctrl-X ( if emacs.sl is loaded. To stop recording the keystrokes, enter Ctrl-X ). Then to “execute” the macro, press Ctrl-X e. Please note that it is illegal to execute a macro while deﬁning one and doing so generates an error. A macro can be aborted at anytime by pressing the Ctrl-G key. The JED Editor 13 One nice feature jed includes is the macro_query function. That is, while deﬁning a macro, the key sequence Ctrl-X q will cause jed to issue the prompt Enter String: in the minibuffer. Any string that is entered will be inserted into the buffer and the process of deﬁning the macro continues. Every time the macro is executed, jed will prompt for a NEW string to be inserted. Any time an error is generated, the process of deﬁning the macro is aborted as well as execution of the macro. This is very useful and may be exploited often. For example, suppose you want to trim excess whitespace from the end of ALL lines in a buffer. Let us also suppose that the number of lines in the buffer is less than 32000. Then consider the following keystrokes: Ctrl-X ( (begin macro) Ctrl-E (goto end of line) ESC (trim whitespace) Down Arrow (go down one line) Ctrl-X ) (end macro) Now the macro has been deﬁned. So move to the top of the buffer and execute it 32000 times: ESC < (top of buffer) ESC 3 2 0 0 0 (repeat next command 32000 times Ctrl-X e (execute macro) If the buffer has less than 32000 lines, the end of the buffer will be reached and an error will be generated aborting the execution of the macro. 10 Shells and Shell Commands The default binding to execute a shell command and pump the output to a buffer is Esc !. jed will prompt for a command line and spawn a subprocess for its execution. Strictly speaking, jed does not support interactive subprocesses. However, jed includes S-Lang code that “emulates” such a subprocess. It may invoked by typing shell at the M-x minibuffer prompt. A window will be created with a buffer named shell attached to it. Any text entered at the system dependent shell prompt will be executed in a subprocess and the result stuffed back in the shell buffer. Don’t try to execute any commands which try to take over the keyboard or the screen or something undesirable may happen. Examples of types of stupid commands are spawning other editors, logging in to remote systems, et cetera. Even chdir is stupid since its effect is not permanent. That is, > cd .. > dir will not do what might naively be expected. That is, the two commands above are not equivalent to the single command dir ... 11 Getting Help jed’s help functions are bound to Ctrl-H by default. For example, Ctrl-H C will show what function a key carries out, Ctrl-H i will run jed’s info reader, Ctrl-H f will give help on a particular S-Lang function, etc. However, some modes may use the Ctrl-H key for something else. For example, if EDT mode is in effect, then Ctrl-H may be bound to bol which causes the cursor to move to the beginning of the line. See the section on EDT for more information. If jed is properly installed, this entire document is accessable from within the editor using jed’s info reader. Ctrl-H i will load info_mode allowing the user to browse the document as well as other “info” documents. 12 Editing Binary Files jed may edit binary ﬁles as long as the proper precautions are taken. On IBMPC systems, this involves calling the S-Lang function set_file_translation with an integer argument. If the argument is 0, ﬁles are opened as text ﬁles; otherwise, they are opened in binary mode. There is no need to call this function for other systems. However, beware of the user variable ADD_NEWLINE which if non zero, a newline character will be appended to the ﬁle if the last character is not a newline character. If you are going to edit binary ﬁles, it is probably a good idea to set this variable to zero. The JED Editor 14 13 Dired— the Directory editor In addition to editing ﬁles, jed is also able to rename and delete them as well. jed’s Dired mode allows one to do just this is a simple and safe manner. To run dired, simply press Esc X and enter dired at the prompt. jed will load dired.sl and prompt for a directory name. Once the directory is given, jed will display a list ﬁles in the directory in a buffer named dired. One may use normal buffer movement keys to move around this buffer. To delete one or more ﬁles, use the d key to “tag” the ﬁles. This in itself does not delete them; rather, it simply marks them for deleting. A capital ‘D’ will appear in the left margin to indicate that a ﬁle has been tagged. Simply hit the u key to untag a ﬁle. The delete key will also untag the previously tagged ﬁle. To actually delete the tagged ﬁles, press the ‘x’ key. This action causes jed to display a list of the tagged ﬁles in a separate window and prompt the user for conﬁrmation. Only when the proper conﬁrmation is given, will the ﬁle be deleted. Renaming a ﬁle is just as simple. Simply move to the line containg the name of the ﬁle that you wish to rename and hit the ‘r’ key. jed will prompt for a ﬁlename or a directory name. If a directory is given, the ﬁle will be moved to the new directory but will keep the name. However, for the operation to succeed, the ﬁle must be one the same ﬁle system. To rename tagged ﬁles to a different directory residing on the same ﬁle system, use the m key. This has the effect of moving the tagged ﬁle out of the current directory to the new one. One may also use the f key to read the ﬁle indicated by the cursor position into a buffer for editing. If the ﬁle is a directory, the directory will be used for dired operations. In addition, one may also use the v to simply “view” a ﬁle. Finally, the g key will re-read the current directory and the h and ? keys provide some help. 14 Customization To extend jed, it is necessary to become familiar with the S-Lang programming language. S-Lang not a standalone program- ming language like C, Pascal, etc. Rather it is meant to be embedded into a C program. The S-Lang programming language itself provides only arithmetic, looping, and branching constructs. In addition, it deﬁnes a few other primitive operations on its data structures. It is up to the application to deﬁne other built-in operations tailored to the application. That is what has been done for the jed editor. See the document slang.txt for S-Lang basics as well as the jed Programmer’s Manual for functions jed has added to the language. In any case, look at the .sl ﬁles for explicit examples. For the most part, the average user will simply want to rebind some keys and change some variables (e.g., tab width). Here I discuss setting keys and the predeﬁned global variables. 14.1 Setting Keys Deﬁning a key to invoke a certain function is accomplished using the setkey function. This function takes two arguments: the function to be executed and the key binding. For example, suppose that you want to bind the key Ctrl-A to cause the cursor to go to the beginning of the current line. The jed function that causes this is bol (See the jed Programmer’s Manual for a complete list of functions). Putting the line: setkey ("bol", "^A"); in the startup ﬁle jed.rc (.jedrc) ﬁle will perform the binding. Here ^A consists of the two characters ^ and A which jed will interpret as the single character Ctrl-A. For more examples, see either of the S-Lang ﬁles emacs.sl or edt.sl. The ﬁrst argument to the setkey function may be any S-Langexpression. Well, almost any. The only restriction is that the newline character cannot appear in the expression. For example, the line setkey ("bol();skip_white ();", "^A"); deﬁnes the Ctrl-A key such that when it is pressed, the editing point will move the beginning of the line and then skip whitespace up to the ﬁrst non-whitespace character on the line. In addition to being able to deﬁne keys to execute functions, it is also possible to deﬁne a key to directly insert a string of characters. For example, suppose that you want to deﬁne a key to insert the string int main(int argc, char argv) whenever you press the key Esc m. This may be accomplished as follows: setkey (" int main(int argc, char argv)", "\em"); The JED Editor 15 Notice two things. First of all, the key sequence Esc m has been written as "\em" where \e will be interpreted by jed as Esc. The other salient feature is that the ﬁrst argument to setkey, the “function” argument, begins with a space. This tells jed that it is not be interpreted as the name of a function; rather, the characters following the space are to be inserted into the buffer. Omitting the space character would cause jed to execute a function called int main(int argc, char argv) which would fail and generate an error. Finally, it is possible to deﬁne a key to execute a series of keystrokes similar to a keyboard macro. This is done by preﬁxing the “function” name with the @ character. This instructs jed to interpret the characters following the @ character as characters entered from the keyboard and execute any function that they are bound to. For example, consider the following key deﬁnition which will generate a C language comment to comment out the current line of text. In C, this may be achieved by inserting symbol "/" at the beginning of the line and inserting "/" at the end of the line. Hence, the sequence is clear (Emacs keybindings): 1. Goto the beginning of the line: Ctrl-A or decimal "\001". 2. Insert /. 3. Goto end of the line: Ctrl-E or decimal \005. 4. Insert / To bind this sequence of steps to the key sequence Esc ;, simply use setkey("@\001/\005/", "\e;"); Again, the preﬁx @ lets jed know that the remaining characters will carry out the functions they are currently bound to. Also pay particular attention to the way Ctrl-A and Ctrl-E have been written. Do not attempt to use the ^ to represent “Ctrl”. It does not have the same meaning in the ﬁrst argument to the setkey function as it does in the second argument. To have control characters in the ﬁrst argument, you must enter them as \xyz where xyz is a three digit decimal number coinciding with the ASCII value of the character. In this notation, the Esc character could have been written as \027. See the S-Lang Programmer’s Reference Manual for further discussion of this notation. The setkey function sets a key in the global keymap from which all others are derived. It is also possible to use the function local_setkey which operates only upon the current keymap which may or may not be the global map. 14.2 Predeﬁned Variables jed includes some predeﬁned variables which the user may change. By convention, predeﬁned variables are in uppercase. The variables which effect all modes include: TAB_DEFAULT [8] sets default tab setting for newly created buffers to speciﬁed number of columns TAB Value of tab setting for current buffer ADD_NEWLINE [1] adds newline to end of ﬁle if needed when writing it out to the disk META_CHAR [-1] preﬁx for chars with high bit set (see section on eight bit clean issues for details) DISPLAY_EIGHT_BIT see section on eight bit clean issues COLOR [23] IBMPC background color (see jed.rc for meaning) LINENUMBERS [0] if 1, show current line number on status line WANT_EOB [0] if 1, [EOB] denotes end of buffer TERM_CANNOT_INSERT [0] if 1, do not put the terminal in insert mode when writing to the screen IGNORE_BEEP [0] do not beep the terminal when signalling errors In addition to the above, there are variables which affect only certain modes. See the section on modes for details. 15 Eight Bit Clean Issues 15.1 Displaying Characters with the High Bit Set There are several issues to consider here. The most important issue is how to get jed to display 8 bit characters in a “clean” way. By “clean” I mean any character with the high bit set is sent to the display device as is. This is achieved by putting the line: The JED Editor 16 DISPLAY_EIGHT_BIT = 1; in the jed.rc (.jedrc) startup ﬁle. European systems might want to put this in the ﬁle site.sl for all users. The default is 1 so unless its value has been changed, this step may not be necessary. There is another issue. Suppose you want to display 8 bit characters with extended Ascii codes greater than or equal to some value, say 160. This is done by putting DISPLAY_EIGHT_BIT = 160;. I believe that ISO Latin character sets assume this. This is the default value for Unix and VMS systems. 15.2 Inputting Characters with the hight bit Set Inputting characters with the high bit set into jed is another issue. How jed interprets this bit is controlled by the variable META_CHAR. What happens is this: When jed reads a character from the input device with the high bit set, it: 1. Checks the value of META_CHAR. If this value is -1, jed simply inserts the character into the buffer. 2. For any other value of META_CHAR in the range 0 to 255, jed returns two 7-bit characters. The ﬁrst character returned is META_CHAR itself. The next character returned is the original character but with the high bit stripped. The default value of META_CHAR is -1 which means that when jed sees a character with the high bit set, jed leaves it as is. Please note that a character with the high bit set it cannot be the preﬁx character of a keymap. It can be a part of the keymap but not the preﬁx. Some systems only handle 7-bit character sequences and as a result, jed will only see 7-bit characters. jed is still able to insert any character in the range 0-255 on a 7-bit system. This is done through the use of the quoted_insert function which, by default, is bound to the backquote key ‘. If the quoted_insert function is called with a digit argument (repeat argument), the character with the value of the argument is inserted into the buffer. Operationally, one hits Esc, enters the extended Ascii code and hits the backquote key. For example, to insert character 255 into the buffer, simply press the following ﬁve keys: Esc 2 5 5 ‘. 15.3 Upper Case - Lower Case Conversions The above discussion centers around input and output of characters with the high bit set. How jed treats them internally is another issue and new questions arise. For example, what is the uppercase equivalent of a character with ASCII code 231? This may vary from language to language. Some languages even have characters whose uppercase equivalent correspond to multiple characters. For jed, the following assumptions have been made: Each character is only 8 bits. Each character has a unique uppercase equivalent. Each character has a unique lowercase equivalent. It would be nice if a fourth assumption could be made: The value of the lowercase of a character is greater than or equal to its uppercase counterpart. However, apparently this is not possible since most IBMPC character sets violate this assumption. Hence, jed does not assume it. Suppose X is the upper case value of some character and suppose Y is its lower case value. Then to make jed aware of this fact and use it case conversions, it may be necessary to put a statement of the form: define_case (X, Y); in the startup ﬁle. For example, suppose 211 is the uppercase of 244. Then, the line define_case (211, 244); will make jed use this fact in operations involving the case of a character. This has already been done for the ISO Latin 1 character set. See the ﬁle iso-latin.sl for details. For MSDOS, this will not work. Instead use the ﬁles dos437.sl and dos850.sl. By default, jed’s internal lookup tables are initialized to the ISO Latin set for Unix and VMS systems and to the DOS 437 code page for the IBMPC. To change the defaults, it is only necessary to load the appropriate ﬁle. For example, to load dos850.sl deﬁnitions, put evalfile ("dos850"); pop (); The JED Editor 17 in the startup ﬁle (e.g., site.sl). In addition to uppercase/lowercase information, these ﬁles also contain word deﬁnitions, i.e., which characters constitute a “word”. 16 Miscellaneous 16.1 Abort Character The abort character (Ctrl-G by default) is special and should not be rebound. On the IBMPC, the keyboard interrupt 0x09 is hooked and a quit condition is signaled when it is pressed. For this reason, it should not be used in any keybindings. A similar statement holds for the other systems. This character may be changed using the function set_abort_char Using this function affects all keymaps. For example, putting the line set_abort_char (30); in your jed.rc ﬁle will change the abort character from its current value to 30 which is Ctrl-^. 16.2 Input Translation By using the function map_input the user is able to remap characters input from the terminal before jed’s keymap rou- tines have a chance to act upon them. This is useful when it is difﬁcult to get jed to see certain characters. For example, consider the Ctrl-S character. This character is especially notorious because many systems use it and Ctrl-Q for ﬂow control. Nevertheless Emacs uses Ctrl-S for searching. Short of rebinding all keys which involve a Ctrl-S how does one work with functions that are bound to key sequences using Ctrl-S? This is where map_input comes into play. The map_input function requires two integer arguments which deﬁne how a given ascii character is to be mapped. Suppose that you wish to substitute Ctrl-\ for Ctrl-S everywhere. The line map_input (28, 19); will do the trick. Here 28 is the ascii character of Ctrl-\ and 19 is the ascii character for the Ctrl-S. As another example, consider the case where the backspace key sends out a Ctrl-H instead of the DEL character (Ctrl-?). map_input (8, 127); will map the Ctrl-H (ascii 8) to the delete character (ascii 127). 16.3 Display Sizes On VMS and unix systems, the screen size may be changed to either 80 or 132 columns by using the functions w80 and w132 respectively. Simply enter the appropriate function name at the M-x prompt in the minibuffer. The default binding for access to the minibuffer is Esc X. Most window systems, e.g., DECWindows, allow the window size to be changed. When this is done, jed should automatically adapt to the new size. On the PC, at this time the screen size cannot be changed while jed is running. Instead it is necessary to exit jed ﬁrst then set the display size and rerun jed. 17 xjed These are some notes about using xjed, the X Window version of jed. It also mentions information about how to setup the EDT emulation under Linux. Suspending xjed is not allowed. If ^Z is pressed, the window is raised if it is obscured, or lowered if it is totally visible. The JED Editor 18 17.1 Resources xjed recognizes the following resources: Display [d] Display to run on Name Instance name Geometry Initial geometry speciﬁcations font Default font to use background [bg] Background color foreground [fg] Forground color Title name to be displayed on the title bar fgStatus [fgs] foreground color of the xjed buffer status line bgStatus [bgs] background color of the xjed buffer status line fgRegion [fgr] foreground color of a region as deﬁned by point and mark bgRegion [bgr] background color of a region as deﬁned by point and mark fgCursor [fgc] text cursor foreground color bgCursor [bgc] text cursor background color fgMouse [fgm] mouse cursor foreground color bgMouse [bgm] mouse cursor background color fgMessage [fgms] Foreground color for messages bgMessage [bgms] Background color for messages fgError [fger] Foreground color for error messages bgError [bger] Background color for messages These resources specify color syntax highlighting options: fgOperator [fgop] forground color for operators (+, -, etc...) bgOperator [bgop] background color for operators fgNumber [fgnm] foreground color for numbers bgNumber [bgnm] background color for numbers fgString [fgst] foreground color for strings bgString [bgst] background color for strings fgKeyword [fgkw] foreground color for keywords bgKeyword [bgkw] background color for keywords fgKeyword1 [fgkw1] foreground color for keywords1 bgKeyword1 [bgkw1] background color for keywords1 fgDelimiter [fgde] foreground color for delimeters bgDelimiter [bgde] background color for delimeters fgPreprocess [fgpr] foreground color for preprocessor lines bgPreprocess [bgpr] background color for preprocessor lines Any of the above items may be speciﬁed on the xjed command line. Quantities enclosed in square brackets may be used as a shortform of their longer counterparts. For example, xjed -d space:0.0 -font 9x15 -bg blue -fg white will start xjed using the server on amy using a white on blue 9x15 font. Once the X Window resource speciﬁcations have been parsed, any remaining command line arguments are parsed as normal jed command line arguments. The easiest way to specify the resources is to make use of a .Xdefaults in your $HOME directory. Here is an example entry for xjed: xjedGeometry: 80x36+100+100 xjedfont: 10x20 xjedbackground: white xjedforeground: black xjedfgNumber: blue The JED Editor 19 The ﬁrst line speciﬁes that the initial window size is 80 columns by 36 rows and that the top left corner of the window is to be positioned at (100, 100). The second line speciﬁes a ﬁxed 10x20 font. The other two lines specify the foreground and background colors of the window. 17.2 Mouse Usage x_set_window_name Set the name of the window (for title bar) x_warp_pointer Move mouse position to cursor position x_insert_cutbuffer insert contents of system cut buffer in current buffer x_copy_region_to_cutbuffer insert a region in system cutbuffer x_set_keysym deﬁne an equivalence string to be returned when a function key is pressed also, set_color() may be used to set colors of mouse, cursor, normal, region, and status line as well as the colors used by the syntax highlighting routines. For example, set_color ("mouse", "red", "blue"); gives the mouse cursor a red forground with a blue background. The color values must be recognizable by the X server. In addition to the usual keybindings, the X version binds: Control-UP goto top of buffer Control-DOWN goto end of buffer Shift-UP move to top of window Shift-DOWN move to bottom of window Control-RIGHT Pan the window to the right Control-LEFT Pan the window to the left Shift-RIGHT skip to next word Shift-LEFT skip to previous word 17.3 EDT emulation under Linux Angelo Pagan (pagan@astrpd.pd.astro.it) suggests putting keycode 22 = Delete keycode 77 = KP_F1 keycode 112 = KP_F2 keycode 63 = KP_F3 keycode 82 = KP_F4 keycode 86 = KP_Separator in the .Xmodmap ﬁle to enable EDT keypad emulation. Send comments and suggestions to davis@space.mit.edu 18 Using a mouse with jed and xjed jed provides native support for a mouse on the following systems: A Linux console running the GPM server. This server is a replacement for the ‘selection’ program. It is available from sunsite.unc.edu:/pub/Linux/system/Daemons/gpm-0.97.tar.gz MSDOS xjed Later, OS/2 support will be added. In addition to “native” mouse support, jed is able to interact with a mouse using the ‘XTerm Event Protocol’. Support for this protocol is available when running jed in an XTerm as well as interacting with jed from an MSDOS terminal emulator, e.g., MS-Kermit, using the PCMOUSE TSR. The JED Editor 20 This document is divided into two sections. The ﬁrst section describes native mouse support (Linux, MSDOS, xjed) and the second section describes the support for the XTerm Event Protocol. 18.1 Native Mouse Support The S-Lang ﬁle jed/lib/mouse.sl provides a user interface to the mouse. It can only be loaded for systems which provide native support for the mouse. Currently this includes MSDOS, Linux console, and xjed. This ﬁle is automatically loaded from os.sl when jed is started up. (See os.sl for how this is accomplished). Once this ﬁle has been loaded, the mouse buttons behave as described below. This interface assumes the presence of a three button mouse. Unfortunately, in the MSDOS world, two button mice are are quite common. Nevertheless, jed is able to emulate a three button mouse by using the ALT key. Any button pressed in combination with the ALT key is considered to be the middle mouse button. For example, to get the effect of pressing Ctrl-Middle, hold down on the ALT and Ctrl key while pressing any mouse button. Clicking in a window Left If a region is already marked, simply un-mark it. If one is not marked, move cursor to the mouse point crossing windows if necessary. If the button is held down and the mouse is dragged, a region will be highlighted and then copied to the cutbuffer when the button is released. Middle If a region is already marked, copy it to the mouse paste-buffer. Otherwise, paste text from the system cut buffer to current editing point. This may not be the position of the mouse. Right If a region is already marked, delete it and place a copy into the mouse paste-buffer. Otherwise, simply position the editing point at the position of the mouse. If the button is held down and the mouse is dragged, a new region will be marked. Shift Middle Insert contents of the last jed mouse copy or kill. This function may be identical to simply clicking on the middle button without using the shift key on non-X systems. Simply clicking the middle mouse button will insert the contents of the current selection which may not be owned by jed. Other buttons combinations are undeﬁned. Some modes may utilize the mouse in a slightly different manner. Clicking on a window status line Left Switch to next buffer Ctrl-Left Kill buffer described by status line Shift-Left Scroll window back one page Shift-Right Scroll window forward one page Middle Split the window Right Delete the window For example, one can quickly move from one buffer to the next by simply clicking on the status line with the left mouse button. Tips for using the mouse To quicky move the cursor to another location, simply point the mouse at that spot and click the LEFT mouse button. To copy a region for subsequent pasting, move the mouse to the beginning of the region and press the LEFT mouse button. While holding it down, “drag” the mouse to the end of the region and release it. To cut a region and put it in the paste buffer, deﬁne a region by dragging with the RIGHT mouse button. Now release the RIGHT button and then press and immediately release it. The JED Editor 21 18.2 XTerm Event Support Xterm event support is provided by not only Xterm but also the Linux console running the ‘selection’ program. Only versions 1.6 and greater of selection provide this support. In addition, one must be using a recent Linux kernel (1.1.35 or newer.) Please note that the selection program is considered obsolete and should be replaced by the GPM mouse server. Mouse Usage Left Button If the left button is clicked on the status line of a window, the window will switch to a different buffer. If the button is pressed anywhere else in the window, the cursor will be positioned at the location of the click. Middle Button On status line: split the window Anywhere else: If the region is highlighted, the region will be copied to the pastebuffer. This does not delete the region. Otherwise, the contents in the pastebuffer will be pasted to the current editing point. Right Button On status line: delete the window. Anywhere else: If a region is highlighted, the region will be extended to the position of the mouse. Otherwise, the mark is set and a region will be deﬁned. Cut/Paste Tips To mark and manipulate a region do: 1. Click the LEFT mouse button at the beginning of the region. 2. Move the mouse to the end of the region and click the RIGHT mouse button. The region should now be marked. 3. Click the MIDDLE button to copy the region to the pastebuffer. 4. To paste from the pastebuffer, move the cursor to where you want to paste and press the MIDDLE button. 19 Frequently Asked Questions How do I obtain jed? jed is available via anonymous ftp from space.mit.edu in the pub/davis/jed directory. jed comes in three forms: jedxxx.tar.Z unix distribution for version xxx jedxxx._of_n n part VMS share distribution for xxx jedxxx.zip PC distribution with precompiled jed.exe All distributions are identical except that the zip ﬁle also contains a precompiled executable for PC systems. jed may also be obtained by email for those without ftp access. To learn about how to ftp using email, send email to ftpmail@pa.dec.com with the single line help. A message will be returned with instructions. For those with VMS systems, Hunter Goatley has made jed available via anonymous ftp from ftp.spc.edu in [.MACRO32.SAVESETS]JE This distribution includes VMS .OBJs and a .EXE ﬁle that was linked under VMS V5.1. [Note that although this distribution is intended for VMS systems, it includes makeﬁles and sources for unix as well. However, you will need to get unzip for your unix system. –John] The JED Editor 22 How do I disable jed’s C mode? The startup ﬁle ‘site.sl’ contains the function mode_hook which is called whenever a ﬁle is loaded. This function is passed the ﬁlename extension. If a ﬁle with c or h extension is read, this function will turn on C-mode for that buffer. You could modify this function to not select C-mode. However, this is not recommended. Rather, it is recommended that you simply rebind the offending keybinding. These include: {, }, the TAB key, and the RETURN key. Simply put any or all of: "self_insert_cmd" "{" setkey "self_insert_cmd" "}" setkey "self_insert_cmd" "^I" setkey "newline" "^M" setkey in your personal startup ﬁle (jed.rc or .jedrc). Before you do this, are you sure that you really understand what C mode does? If not, please read on. What is C mode? In C mode, the TAB key does not insert tabs. Instead, it runs a command called indent_line. It is really the quickest way to edit C code. In this mode, the TAB, RETURN, {, and } keys are special. If you edit a ﬁle called x.c, jed will invoke its C mode. Entering the 28 characters (no newline, TAB, etc...) main (){if (x == 2){x = 4;}}’ should result in: main () { if (x == 2) { x = 4; } } which would take alot more time using the TAB and NEWLINE keys. If you do not like the indentation style, you can customize it by setting the appropriate variables in jed.rc. To see the use of the tab key, delete the whitespace in front of all the lines, move to any of the lines (anywhere on the line) and hit the TAB key. This should correctly indent the line to according to your preferences (i.e., the variables in jed.rc). Finally, move to one of the lines and enter ESC ;. This should produce a C comment. Using the C mode and the TAB key as indent_line also helps you avoid syntax errors. Basically, a line simply will not indent properly. This indicats that you left off a brace, mismatched parenthesis, etc... If you bind TAB away from indent_line, you lose some of this. Note that these same comments apply to Fortran mode. Get a ﬁle called x.for. Enter the characters: TABprogram mainRETinteger4 iRETdo 10 i=1,3RETcall f(i)RET10continueRETend Here TAB means hit TAB and RET means hit return. This will result in: program main integer4 i do 10 i=1,3 call f(i) 10 continue end The JED Editor 23 Again, the editor has done all the work. Once you get used to this style of editing, you will not want to go back. Also note that this will not work if EDT is loaded. To get this functionality back, you will need to do: setkey("indent_line_cmd", "\t"); setkey("newline_and_indent_cmd", "^M"); AFTER edt.sl is loaded. How do I turn on wrap mode or turn it off? Normally, this is done automatically when jed loads a ﬁle with extensions .txt, .doc, etc... See question 2 for a discussion of how this is done. To turn on wrap mode for the current buffer, simply press Escape-X and enter: text_mode at the prompt. To turn it off, you must change the mode to something else. A fairly generic choice is the no_mode mode. To do this, press Escape-X and enter: no_mode at the prompt. It is easy to write a function to toggle the mode for you that can be bound to a key. This one (toggle_wrapmode) will work: define toggle_wrapmode () { variable mode, modestr; (modestr, mode) = whatmode (); if (mode & 1) % test wrap bit mode = mode & ~(1); % wrap bit on so mask it off else mode = mode \| 1; % wrap bit off so set it. setmode (modestr, mode); } What is the difference between internal and intrinsic functions? An intrinsic function is a function that is directly callable from S-Lang while an internal function cannot. However, internal functions can be called indirectly through the use of the intrinsic function call. For example, consider the internal function self_insert_cmd. Most typing keys are bound to this function and cause the key to be directly inserted into the buffer. Consider the effect of this. After a character to be inserted is received by jed, the buffer is updated to reﬂect its insertion. Then the screen is updated. Here lies the essential difference between the two types of functions. If the screen was in sync before the insertion, jed can simply put the terminal in insert mode, send out the character and take the terminal back out of insert mode. However, this requires knowing the state of the screen. If called from a S-Lang routine, all bets are off. Since the screen update is not performed until after any S-Lang function has returned to jed, the buffer and the screen will almost always be out of sync with respect to one another and a full screen update will have to be performed. But this is very costly to have to do for every insertion. Hence, jed makes a distinction between the two types of functions by making the most common ones internal. The upshot is this: intrinsic functions will cause a full screen update while internal ones may not. Sometimes during screen updates, jed pauses. Why is this? Since version 0.91, jed checks the baud rate and tries to output characters based on reported rate. jed will literally sleep when outputting many characters if the reported baud rate is small. One should ﬁrst check to see that terminal driver has the baud rate set appropriately. On Unix, this is done by typing stty -a at the shell prompt. If setting the baud rate to the correct value does not help, set the internal global variable OUTPUT_RATE to zero. This is achived by uncommenting the line referring to OUTPUT_RATE in the jed.rc initialization ﬁle. If there is still a problem, contact me. The JED Editor 24 How do I get jed to recognize Control-S and Control-Q? Many systems use ^S/^Q for ﬂow control— the so-called XON/XOFF protocol which is probably the reason jed does not see either of these two characters. Perhaps the most portable solution to this problem is to simply avoid using ^S and ^Q altogether. This may require the user to rebind those those functions that have key bindings composed of these characters. jed is able to enable or disable ﬂow control on the system that it is running. This may be done by putting the line: enable_flow_control (0); % turn flow control off in your .jedrc ﬁle. Using a value of 1 turns ﬂow control on. Another solution is to use the map_input function to map a different key to ^S (and ^Q). For example, one might simply choose to map ^\ to ^S and ^^ (Control-^) to ^Q. To do this, simply put: map_input (28, 19); % ^\ --> ^S map_input (30, 17); % ^^ --> ^Q in your .jedrc (jed.rc) ﬁle. Can I bind the Alt keys on the PC? Yes. The ALT keys return a two character key sequence. The key sequence for a particular ALT key as well as other function keys are listed in the ﬁle pc-keys.txt. Many users simply want to use the ALT key as a Meta Character. To have jed interpret ALT-X as ESC-X, put ALT_CHAR = 27; int your jed.rc ﬁle. Here ‘X’ is any key. (Actually, this should not be necessary– the default value for ALT_CHAR is 27). How do I ﬁnd out what characters a particular key generates? The simpliest way is to start jed via the command: jed -l keycode -f keycode jed will then prompt for a key to be pressed and return the escape sequence that the key sends. If xjed is used, it will also return the keysym (See online help on the x_set_keysym function for more information). An alternative approach is to use the quoted insert function. By default, this is bound to the backquote ‘ key. Simply switch to the scratch* buffer, press the backquote key followed by the key in question. The key sequence will be inserted into the buffer. This exploits the fact that most multi-character key sequences begin with the ESC character followed one or more printable characters. If this fails, the following function will sufﬁce: define insert_this_key () { variable c; pop2buf ("scratch"); eob (); message ("Press key:"); update (1); forever { c = getkey (); if (c == 0) insert("^@"); else insert (char (c)); !if (input_pending (3)) break; } } The JED Editor 25 Simply type it into the scratch buffer, press ESC-X and type evalbuffer. Then, to use the function, press ESC-X again and enter insert_this_key. jed scrolls slow on my WizBang-X-Super-Terminal. What can I do about it? On Unix, jed uses termcap (terminfo) and the value of the TERM environment variable. Chance are, even though you are using an expansive state of the art terminal, you have told unix it is a vt100. Even if you have set the TERM variable to the appropriate value for you terminal, the termcap ﬁle may be missing entries for your “WizBang” features. This is particularly the case for Ultrix systems— the vt102, vt200, and vt300 termcap entries are missing the AL and DL termcap ﬂags. In fact, the Ultrix man page for termcap does not even mention these capabilities! jed is able to compensate for missing termcap entries only for vtxxx terminals. If your terminal is a fancy vtxxx terminal, put the line: set_term_vtxxx (0); in your .jedrc ﬁle. How do I get a list of functions? Help on any documented function is available by pressing ‘Ctrl-H f’ and entering the function name at the prompt. If you simply hit return, you will get the documentation for all functions. How can I use edt.sl with jed386.exe? The basic problem is the current generation of the 32 bit compiler (DJGPP) used to generate jed386.exe cannot handle the hardware keyboard interrupt used to remap the numeric keypad. Nevertheless, it is possible to use edt.sl with jed386. However, the function keys, F1 to F10 must be used for the EDT keypad. The remapping is as follows: VT100 Keys IBM Function On the Numeric Keypad ------------------------- ------------------------- \| F1 \| F2 \| F3 \| F4 \| \| PF1 \| PF2 \| PF3 \| PF4 \| \|-----+-----+-----+-----\| \|-----+-----+-----+-----\| \| F5 \| F6 \| F7 \| F8 \| \| 7 \| 8 \| 9 \| - \| \|-----+-----+-----+-----\| \|-----+-----+-----+-----\| \| F9 \| F10 \| F11 \| F12 \| \| 4 \| 5 \| 6 \| , \| \|-----+-----+-----+-----\| \|-----+-----+-----+-----\| \| SF1 \| SF2 \| SF3 \| SF4 \| \| 1 \| 2 \| 3 \| \| \|-----------+-----\|-----\| \|-----------+-----\|ENTER\| \| SF5 \| SF6 \| SF7 \| SF8 \| \| 0 \| . \| \| ------------------------- ------------------------- Here, SF1 means SHIFT-F1, etc... How do I set custom tab stops in jed? Put something like: variable Tab_Stops; Tab_Stops = create_array(’i’, 20, 1); %% The following defines the tab stops to be 8 column: _for (0, 19, 1) { =$1; Tab_Stops[$1] =$1 * 8 + 1; } The JED Editor 26 in your jed.rc. To individually set them, do: Tab_Stops[0] = 4; Tab_Stops[1] = 18; etc... 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https://lifewithdata.com/2022/04/08/python-regular-expression-lookbehind/	# Python Regular Expression – Lookbehind ## Positive Lookbehind – In regular expression, the positive lookbehind matches a string if there is a specific pattern before it. #### syntax – (?<=lookbehind_regex) Let’s say you have some data about stock prices and you want to find the prices of stocks but do not want to match with the number of stocks. For this you can use lookbehind in regex. In [1]: import re In [2]: re.findall('(?<=\$)\d+', 'The price of 1 stock of apple is$172') Out[2]: ['172'] The \d+ says that there is one or more digit characters and (?<=\$) says only match with the strings which has dollar sign before the digit characters. Since$ is a special character in regex we need to escape it with a backslash to match with regular $character. ## Negative lookbehind – In regular expression, Negative lookbehind only matches with a string if there is not a specific pattern before it. #### syntax – (?<!lookbehind_regex) Now, if you only want to match with numbers that does not contains dollar signs before it then you can negative lookbehind. In [3]: re.findall(r'\b(?<!\$)\d+\b', 'The price of 1 stock of apple is $172') Out[3]: ['1'] The \b matches if a word begins or ends with the given characters. We used it to isolate words. The pattern (?<!\$) says that the string does not starts with a dollar sign, and it contains one or more digit characters. Rating: 1 out of 5.	2023-02-06 03:37:54	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.18721963465213776, "perplexity": 1621.8413562183762}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500303.56/warc/CC-MAIN-20230206015710-20230206045710-00678.warc.gz"}
https://nl.mathworks.com/help/econ/share-results-of-econometric-modeler-session.html	## Share Results of Econometric Modeler App Session This example shows how to share the results of an Econometric Modeler app session by: • Exporting time series and model variables to the MATLAB® Workspace • Generating MATLAB plain text and live functions to use outside the app • Generating a report of your activities on time series and estimated models During the session, the example transforms and plots data, runs statistical tests, and estimates a multiplicative seasonal ARIMA model. The data set `Data_Airline.mat` contains monthly counts of airline passengers. ### Import Data into Econometric Modeler At the command line, load the `Data_Airline.mat` data set. `load Data_Airline` At the command line, open the Econometric Modeler app. `econometricModeler` Alternatively, open the app from the apps gallery (see Econometric Modeler). Import `DataTable` into the app: 1. On the Econometric Modeler tab, in the Import section, click . 2. In the Import Data dialog box, in the Import? column, select the check box for the `DataTable` variable. 3. Click . The variable `PSSG` appears in the Time Series pane, its value appears in the Preview pane, and its time series plot appears in the Time Series Plot(PSSG) figure window. The series exhibits a seasonal trend, serial correlation, and possible exponential growth. For an interactive analysis of serial correlation, see Detect Serial Correlation Using Econometric Modeler App. ### Stabilize Series Address the exponential trend by applying the log transform to `PSSG`. 1. In the Time Series pane, select `PSSG`. 2. On the Econometric Modeler tab, in the Transforms section, click . The transformed variable `PSSGLog` appears in the Time Series pane, and its time series plot appears in the Time Series Plot(PSSGLog) figure window. The exponential growth appears to be removed from the series. Address the seasonal trend by applying the 12th order seasonal difference. With `PSSGLog` selected in the Time Series pane, on the Econometric Modeler tab, in the Transforms section, set Seasonal to `12`. Then, click . The transformed variable `PSSGLogSeasonalDiff` appears in the Time Series pane, and its time series plot appears in the Time Series Plot(PSSGLogSeasonalDiff) figure window. The transformed series appears to have a unit root. Test the null hypothesis that `PSSGLogSeasonalDiff` has a unit root by using the Augmented Dickey-Fuller test. Specify that the alternative is an AR(0) model, then test again specifying an AR(1) model. Adjust the significance level to 0.025 to maintain a total significance level of 0.05. 1. With `PSSGLogSeasonalDiff` selected in the Time Series pane, on the Econometric Modeler tab, in the Tests section, click > Augmented Dickey-Fuller Test. 2. On the ADF tab, in the Parameters section, set Significance Level to `0.025`. 3. In the Tests section, click . 4. In the Parameters section, set Number of Lags to `1`. 5. In the Tests section, click . The test results appear in the Results table of the ADF(PSSGLogSeasonalDiff) document. Both tests fail to reject the null hypothesis that the series is a unit root process. Address the unit root by applying the first difference to `PSSGLogSeasonalDiff`. With `PSSGLogSeasonalDiff` selected in the Time Series pane, click the Econometric Modeler tab. Then, in the Transforms section, click . The transformed variable `PSSGLogSeasonalDiffDiff` appears in the Time Series pane, and its time series plot appears in the Time Series Plot(PSSGLogSeasonalDiffDiff) figure window. In the Time Series pane, rename the `PSSGLogSeasonalDiffDiff` variable by clicking it twice to select its name and `PSSGStable`. The app updates the names of all documents associated with the transformed series. ### Identify Model for Series Determine the lag structure for a conditional mean model of the data by plotting the sample autocorrelation function (ACF) and partial autocorrelation function (PACF). 1. With `PSSGStable` selected in the Time Series pane, click the Plots tab, then click . 2. Show the first 50 lags of the ACF. On the ACF tab, set Number of Lags to `50`. 3. Click the Plots tab, then click . 4. Show the first 50 lags of the PACF. On the PACF tab, set Number of Lags to `50`. 5. Drag the ACF(PSSGStable) figure window above the PACF(PSSGStable) figure window. According to [1], the autocorrelations in the ACF and PACF suggest that the following SARIMA(0,1,1)×(0,1,1)12 model is appropriate for `PSSGLog`. `$\left(1-L\right)\left(1-{L}^{12}\right){y}_{t}=\left(1+{\theta }_{1}L\right)\left(1+{\Theta }_{12}{L}^{12}\right){\epsilon }_{t}.$` Close all figure windows. ### Specify and Estimate SARIMA Model Specify the SARIMA(0,1,1)×(0,1,1)12 model. 1. In the Time Series pane, select the `PSSGLog` time series. 2. On the Econometric Modeler tab, in the Models section, click the arrow to display the models gallery. 3. In the models gallery, in the ARMA/ARIMA Models section, click . 4. In the SARIMA Model Parameters dialog box, on the Lag Order tab: • Nonseasonal section 1. Set Degrees of Integration to `1`. 2. Set Moving Average Order to `1`. 3. Clear the Include Constant Term check box. • Seasonal section 1. Set Period to `12` to indicate monthly data. 2. Set Moving Average Order to `1`. 3. Select the Include Seasonal Difference check box. 5. Click . The model variable `SARIMA_PSSGLog` appears in the Models pane, its value appears in the Preview pane, and its estimation summary appears in the Model Summary(SARIMA_PSSGLog) document. ### Export Variables to Workspace Export `PSSGLog`, `PSSGStable`, and `SARIMA_PSSGLog` to the MATLAB Workspace. 1. On the Econometric Modeler tab, in the Export section, click . 2. In the Export Variables dialog box, select the Select check boxes for the `PSSGLog` and `PSSGStable` time series, and the `SARIMA_PSSGLog` model (if necessary). The app automatically selects the check boxes for all variables that are highlighted in the Time Series and Models panes. 3. Click . At the command line, list all variables in the workspace. `whos` ``` Name Size Bytes Class Attributes Data 144x1 1152 double DataTable 144x1 3192 timetable Description 22x54 2376 char PSSGLog 144x1 1152 double PSSGStable 144x1 1152 double SARIMA_PSSGLog 1x1 7963 arima dates 144x1 1152 double series 1x1 162 cell ``` The contents of `Data_Airline.mat`, the numeric vectors `PSSGLog` and `PSSGStable`, and the estimated `arima` model object `SARIMA_PSSGLog` are variables in the workspace. Forecast the next three years (36 months) of log airline passenger counts using `SARIMA_PSSGLog`. Specify the `PSSGLog` as presample data. ```numObs = 36; fPSSG = forecast(SARIMA_PSSGLog,numObs,'Y0',PSSGLog);``` Plot the passenger counts and the forecasts. ```fh = DataTable.Time(end) + calmonths(1:numObs); figure; plot(DataTable.Time,exp(PSSGLog)); hold on plot(fh,exp(fPSSG)); legend('Airline Passenger Counts','Forecasted Counts',... 'Location','best') title('Monthly Airline Passenger Counts, 1949-1963') ylabel('Passenger counts') hold off``` ### Generate Plain Text Function from App Session Generate a MATLAB function for use outside the app. The function returns the estimated model `SARIMA_PSSGLog` given `DataTable`. 1. In the Models pane of the app, select the `SARIMA_PSSGLog` model. 2. On the Econometric Modeler tab, in the Export section, click > Generate Function. The MATLAB Editor opens and contains a function named `modelTimeSeries`. The function accepts `DataTable` (the variable you imported in this session), transforms data, and returns the estimated SARIMA(0,1,1)×(0,1,1)12 model `SARIMA_PSSGLog`. 3. On the Editor tab, click > Save. 4. Save the function to your current folder by clicking in the Select File for Save As dialog box. At the command line, estimate the SARIMA(0,1,1)×(0,1,1)12 model by passing `DataTable` to `modelTimeSeries.m`. Name the model `SARIMA_PSSGLog2`. Compare the estimated model to `SARIMA_PSSGLog`. ```SARIMA_PSSGLog2 = modelTimeSeries(DataTable); summarize(SARIMA_PSSGLog) summarize(SARIMA_PSSGLog2) ``` ``` ARIMA(0,1,1) Model Seasonally Integrated with Seasonal MA(12) (Gaussian Distribution) Effective Sample Size: 144 Number of Estimated Parameters: 3 LogLikelihood: 276.198 AIC: -546.397 BIC: -537.488 Value StandardError TStatistic PValue _________ _____________ __________ __________ Constant 0 0 NaN NaN MA{1} -0.37716 0.066794 -5.6466 1.6364e-08 SMA{12} -0.57238 0.085439 -6.6992 2.0952e-11 Variance 0.0012634 0.00012395 10.193 2.1406e-24 ARIMA(0,1,1) Model Seasonally Integrated with Seasonal MA(12) (Gaussian Distribution) Effective Sample Size: 144 Number of Estimated Parameters: 3 LogLikelihood: 276.198 AIC: -546.397 BIC: -537.488 Value StandardError TStatistic PValue _________ _____________ __________ __________ Constant 0 0 NaN NaN MA{1} -0.37716 0.066794 -5.6466 1.6364e-08 SMA{12} -0.57238 0.085439 -6.6992 2.0952e-11 Variance 0.0012634 0.00012395 10.193 2.1406e-24``` As expected, the models are identical. ### Generate Live Function from App Session Unlike a plain text function, a live function contains formatted text and equations that you can modify by using the Live Editor. Generate a live function for use outside the app. The function returns the estimated model `SARIMA_PSSGLog` given `DataTable`. 1. In the Models pane of the app, select the `SARIMA_PSSGLog` model. 2. On the Econometric Modeler tab, in the Export section, click > Generate Live Function. The Live Editor opens and contains a function named `modelTimeSeries`. The function accepts `DataTable` (the variable you imported in this session), transforms data, and returns the estimated SARIMA(0,1,1)×(0,1,1)12 model `SARIMA_PSSGLog`. 3. On the Live Editor tab, in the File section, click > Save. 4. Save the function to your current folder by clicking in the Select File for Save As dialog box. At the command line, estimate the SARIMA(0,1,1)×(0,1,1)12 model by passing `DataTable` to `modelTimeSeries.m`. Name the model `SARIMA_PSSGLog2`. Compare the estimated model to `SARIMA_PSSGLog`. ```SARIMA_PSSGLog2 = modelTimeSeries(DataTable); summarize(SARIMA_PSSGLog) summarize(SARIMA_PSSGLog2) ``` ``` ARIMA(0,1,1) Model Seasonally Integrated with Seasonal MA(12) (Gaussian Distribution) Effective Sample Size: 144 Number of Estimated Parameters: 3 LogLikelihood: 276.198 AIC: -546.397 BIC: -537.488 Value StandardError TStatistic PValue _________ _____________ __________ __________ Constant 0 0 NaN MA{1} -0.37716 0.066794 -5.6466 1.6364e-08 SMA{12} -0.57238 0.085439 -6.6992 2.0952e-11 Variance 0.0012634 0.00012395 10.193 2.1406e-24 ARIMA(0,1,1) Model Seasonally Integrated with Seasonal MA(12) (Gaussian Distribution) Effective Sample Size: 144 Number of Estimated Parameters: 3 LogLikelihood: 276.198 AIC: -546.397 BIC: -537.488 Value StandardError TStatistic PValue _________ _____________ __________ __________ Constant 0 0 NaN NaN MA{1} -0.37716 0.066794 -5.6466 1.6364e-08 SMA{12} -0.57238 0.085439 -6.6992 2.0952e-11 Variance 0.0012634 0.00012395 10.193 2.1406e-24``` As expected, the models are identical. ### Generate Report Generate a PDF report of all your actions on the `PSSGLog` and `PSSGStable` time series, and the `SARIMA_PSSGLog` model. 1. On the Econometric Modeler tab, in the Export section, click > Generate Report. 2. In the Select Variables for Report dialog box, select the Select check boxes for the `PSSGLog` and `PSSGStable` time series, and the `SARIMA_PSSGLog` model (if necessary). The app automatically selects the check boxes for all variables that are highlighted in the Time Series and Models panes. 3. Click . 4. In the Select File to Write dialog box, navigate to the `C:\MyData` folder. 5. In the File name box, type `SARIMAReport`. 6. Click . The app publishes the code required to create `PSSGLog`, `PSSGStable`, and `SARIMA_PSSGLog` in the PDF `C:\MyData\SARIMAReport.pdf`. The report includes: • Plots that include the selected time series • Descriptions of transformations applied to the selected time series • Results of statistical tests conducted on the selected time series • Estimation summaries of the selected models ## References [1] Box, George E. P., Gwilym M. Jenkins, and Gregory C. Reinsel. Time Series Analysis: Forecasting and Control. 3rd ed. Englewood Cliffs, NJ: Prentice Hall, 1994.	2022-06-30 07:15:38	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 1, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5098170638084412, "perplexity": 4779.197577157515}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103669266.42/warc/CC-MAIN-20220630062154-20220630092154-00368.warc.gz"}
https://www.embibe.com/exams/microbes-in-household-products/	Microbes in Household Products: Uses & Significance - Embibe • Written By Sagarika Swamy • Written By Sagarika Swamy # Microbes in Household Products – Definition, Uses & Significance Microbes in Household Products: Microbes are used extensively in household products, industrial products, medicine, etc. Microbes are small and tiny organisms, but they can do wonders like converting milk to curd, normal coconut water to a traditional drink, etc. Microbes can act as parasites and can cause a number of infectious diseases. Microbes and their products are also in the making of cheese, curd, dough, bread, vinegar, and other foods in everyday life. Lactic acid bacteria (LAB), also known as Lactobacillus acidophilus, Lactobacillus lactis, and Streptococcus lactis, are probiotic bacteria that inhibit the growth of harmful bacteria in the stomach and other regions of the digestive tract. Let’s explore more in detail the role of microbes in household products in the below article. ## What are Microbes? Microbes are tiny organisms, they are too tiny to see without a microscope, yet they are abundant on Earth. They live everywhere, in air, soil, rock and water. These microscopic organisms are found in plants and animals as well as in the human body. ### Role of Microbes in Household Products: Types Microbes are a vast community of vivid organisms. This group includes Bacteria, Viruses, unicellular Fungi, and unicellular Algae; all unicellular Protozoa members are the four major types of microbes. Learn Exam Concepts on Embibe Fig: Types of Microbes ### Microbes in Household Production Curd: Microorganisms such as Lactobacillus and others commonly called Lactic Acid Bacteria (LAB) grow in milk and convert it into curd. During growth, the Lactic acid bacteria produce acids that coagulate and partially digest the milk proteins called casein. 1. Lactobacillus acidophilus converts lactose sugar of milk into lactic acid at a temperature of about $${\rm{40^\circ C}}$$ or less. Streptococcus lactis and Streptococcus thermophilus are also used in the formation of curd. 2. The starter or inoculation used in the preparation of milk products is a small amount of curd (spoonful), which actually contains millions of lactic acid bacteria. 3. Curd is more nutritious than milk as it contains a number of organic acids and vitamin $${{\rm{B}}_{{\rm{12}}}}{\rm{.}}$$ 4. LAB present in curd also checks the growth of disease-causing microbes in the stomach and other parts of the digestive tract. 5. Curd also used to obtain buttermilk. Curd also possesses antimutagenic and anticancer biochemicals. Learn 11th CBSE Exam Concepts Paneer: Paneer (cottage cheese) is a fresh cheese common in South Asia, especially in India. It is made by curdling milk with lemon juice, vinegar and other edible acids. Large holes in Swiss cheese is due to the production of a large amount of carbon-di-oxide by the bacterium Propionibacterium shermanii. Dhokla: The dough of dhokla is prepared by mixing gram flour (Besan) with buttermilk. The Lactobacilli bring about the fermentation process. Jalebi and Nan: Jalebi and Nan are prepared from maida by the activity of many Lactobacilli. Idli and Dosa: The dough which is used for making dosa and idli is the fermented preparation of rice and black gram (Urad dal). 1. The dough is allowed to ferment overnight for $$10 – 12$$ hours. 2. The lactic acid bacteria from Leuconostoc and Streptococcus species bring about fermentation. Several other genera, like Bacillus, Candida, and Saccharomyces, are also used in this process. 3. During fermentation, $${\rm{C}}{{\rm{O}}_{\rm{2}}}$$ is evolved, which causes doughing of the raw material. The bubbles of $${\rm{C}}{{\rm{O}}_{\rm{2}}}$$ trapped in gluten makes the idli puffy. Practice Exam Questions Yoghurt: Yoghurt provides protein and calcium, and it may enhance healthy gut bacteria. 1. Yoghurt is produced by curdling milk with the help of Streptococcus thermophilus and Lactobacillus bulgaricus, Streptococcus lactis, etc. 2. Initially, milk is heated to $${\rm{80^\circ – 90^\circ C}}$$ for half an hour. It is cooled to $${\rm{40^\circ – 43^\circ C}}$$ and mixed with an inoculum containing the Streptococcus thermophilus bacteria. 3. After four hours, yoghurt becomes ready. It has an original flavour of lactic acid and acetaldehyde. 4. Since it is already acidic $$\left( {{\rm{pH \;3}}{\rm{.7 – 4}}{\rm{.3}}} \right){\rm{,}}$$ yoghurt is often sweetened and flavoured with fruits. 1. Yoghurt is actually Turkish! Many people think that it originated in Greece, but its root is from the Turkish word ‘Yog‘, meaning “to condense or intensity. 2. It is great for your skin, and in many parts of India and Asia was used as far back as ancient times to moisturize and revive skin. 3. It can be used to polish brass. Yoghurt contains lactic acid, which eats away the tarnish on your brass and leaves your ornaments shining. 4. It is good for your digestion. It contains good bacteria known as probiotics which aid your digestive system and help to keep you regular. 1. The dough that is used for making bread is fermented by using the microbe Saccharomyces cerevisiae which is commonly called Baker’s Yeast Toddy Drink: The toddy is a refreshing traditional drink in some parts of south India. The toddy is prepared by the fermentation process of coconut water and sap of palm trees. Toddy can be heated to produce jaggery or palm sugar. Process of Toddy: Toddy undergoes fermentation if left for a few hours with the help of naturally occurring yeast to form a beverage containing about $$6\%$$ alcohol. Cheese: Different varieties of cheese are known for their characteristic texture, flavour and taste. The specificity comes from the microbe that is used in it. For example, the large holes in ‘Swiss cheese’ are due to the production of a large amount of $${\rm{C}}{{\rm{O}}_{\rm{2}}}$$ by a bacterium named Propionibacterium shermanii. In contrast, the ‘Roquefort cheese’ are ripened by growing a specific fungus on them, which gives them a specific flavour. Attempt Mock Tests There are two types of cheese: (a). Ripened cheese: It is ripened only from the outside. It is soft in nature. (b). Unripened cheese: It is ripened externally as well as internally. It is hard in nature. Practice 11th CBSE Exam Questions Other Foods: Microbes are used in many other foods from the process of fermentation. Soyabean Preparations: (a) Tempeh- It is Indonesian food formed by fermenting soybean. It is an important source of vitamin $${{\rm{B}}_{{\rm{12}}}}{\rm{.}}$$ The preparation of this food also requires the involvement of microbes. A fungus, Rhizopus oligosporus, is used in the fermentation process and is also known as a tempeh starter. (b) Sofu (Chinese) and Tofu (Japanese)- They are cheese-like products of soybean obtained after fermentation with Mucor species. (c) Soya sauce- It is a flavoured, salted, brown sauce obtained by fermentation from a mash of soybean and wheat with the help of Aspergillus oryzae, Lactobacillus, Saccharomyces rouxii and Torulopsis species. Sausages: Sausages are prepared by fermentation and curing of fish and meat. They possess particular flavours and tastes depending upon the fermenting agent. Lactic acid bacteria Pediococcus cerevisiae are most commonly used. Bamboo Shoots: Tender bamboo shoots are used as vegetables directly as well as after fermentation. Sauerkraut: Sauerkraut is produced by Leuconostoc and Lactobacillus species within shredded cabbage. This is a fermented cabbage that is used as a delicate food. Pickles: Pickles prepared by lactic acid bacteria (LAB) fermentation have a unique flavour and beneficial health effects. These LABs are slightly different as they can comfortably tolerate higher concentrations of salt and sugar. Lactobacillus plantarum, L. brevis, Leuconostoc mesenteroides and Pediococcus cerevisiae, Pediococcus pentosaceus and Enterococcus faecalis are used in pickle making. ### Microbes in Industrial Products Microbes are grown in very large vessels called fermentors or bioreactors. In industries, microbes are very much used to synthesize different types of products which are valuable for us. These products are food additives, beverages (alcoholic and non-alcoholic), organic acids, enzymes, vitamins, biofuels, bio-fertilizers, metabolites, different types of antibiotics, and vaccines in the medical field for the treatment of diseases. Microbes play a vital role in the fermentation process to obtain several products. ### Microbes in Household Products: Summary Humans manipulate microbes to do work for them in industry, medicine, and caring for the environment. Microbiology is a complex discipline that includes many branches studying different aspects of microbes. There are many microbes that help humans in the household products like the formation of curd, yoghurt, cheese, paneer, traditional drink toddy, etc., that are rich in calcium and good for young children’s growth. Infectious diseases are caused by microbes acting as parasites. There are differences in infectious diseases affecting developing and industrialized countries. Attempt 11th CBSE Exam Mock Tests ### FAQs on Microbes in Household Products Frequently asked questions related to microbes in household products are listed below: Q.1: What is the role of microbes in household products? Ans: The role of microbes in household products includes the dough used to make idli or dosa, curd, and cheese under the process of the fermentation process. Q.2: What do you mean by microbes? Ans: It is an organism that can be seen only through a microscope. Microorganisms include bacteria, protozoa, algae, and fungi. Q.3: What products or services can be obtained from microbes? Ans: a. Food additives like curd, cheese, yogurt, paneer, etc. b. Alcoholic and non-alcoholic beverages for example toddy a traditional drink that consists of 6% alcohol. c. Biofuels, metabolites, and biofertilizers. d. Few Chemicals, Enzymes and other Bioactive Molecules. e. Vaccines and other Antibiotics kill or retard the growth of disease-causing microbes. Q.4: What are the microbes used in household products? Ans: a. Lactic acid bacteria (LAB) like Lactobacillus are added to milk for the formation of curd. b. Saccharomyces cerevisiae is commonly known as Baker’s Yeast which is used in the preparation of bread. Q.5: What are the different types of microbes present on our planet earth? Ans: The different types of microbes present on earth are algae, bacteria, viruses, protozoa, and fungi. We hope this detailed article on Microbes in Household Products helps you in your preparation. If you get stuck do let us know in the comments section below and we will get back to you at the earliest. Master Exam Concepts with 3D Videos	2022-06-25 19:15:01	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3041457533836365, "perplexity": 10258.49848655142}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103036099.6/warc/CC-MAIN-20220625190306-20220625220306-00184.warc.gz"}
https://www.shaalaa.com/question-bank-solutions/difference-between-energy-power-a-pump-raises-water-spending-4-105-j-energy-10-s-find-power-pump_30950	Share # A Pump Raises Water by Spending 4 × 105 J of Energy in 10 S. Find the Power of Pump. - Physics Course #### Question A pump raises water by spending 4 × 105 J of energy in 10 s. Find the power of pump. #### Solution Energy spent = w = 4 × 105 J Time× taken = 10 s Power = Energy / Time P = (4 × 10^5)/10 = 4 × 10^4 W Is there an error in this question or solution?	2020-08-14 11:57:56	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3335627317428589, "perplexity": 3104.996178729628}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439739211.34/warc/CC-MAIN-20200814100602-20200814130602-00075.warc.gz"}
http://amtoolbox.org/amt-1.0.0/doc/core/amt_disp.php	THE AUDITORY MODELING TOOLBOX Applies to version: 1.0.0 Go to function amt_disp - AMT-specific overload of the function 'disp' Usage amt_disp(X); amt_disp(X,'volatile'); amt_disp(X,'documentation'); amt_disp(); Description amt_disp(X); can be used to show message X in the command window. The output of amt_disp depends on the start-up configuration of the AMT. When the AMT is started in the default mode (verbose), amt_disp will display behaving as the Matlab/Octave funtion disp(), however, the output will not appear in the web documentation. When the AMT is started in the silent mode, amt_disp will never display. See amt_start for further explanation on the start-up configurations. amt_disp(X,'volatile'); can be used as volatile progress indicator. Any subsequent call of the amt_disp will delete the previous volatile message. This way a changing progress can be clearly shown even in loops. As the default messages, the volatile messages will not appear in the web documentation case. After the last usage of 'volatile' call amt_disp(); to fix the output and prevent subsequent deletions. amt_disp(X,'documentation'); can be used for information interesting to be displayed in the web documentation. amt_disp(X,'silent'); does not output at all. This is one of the available modes of the AMT. amt_disp(X,'no_debug'); does neither output at all. amt_disp(X,'debug'); does output and combined with no_debug, it can be used to implement a function in which the user defines the level of information to be displayed by passing a flag. For example,: definput.flags.disp = {'no_debug','debug'}; [flags,kv]=ltfatarghelper({},definput,varargin); ... amt_disp('Displayed only when flag debug provided',flags.disp);	2021-12-07 00:32:57	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3714662194252014, "perplexity": 2322.978929940165}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964363327.64/warc/CC-MAIN-20211206224536-20211207014536-00602.warc.gz"}
http://psychology.wikia.com/wiki/Birth_rate	# Birth rate 34,190pages on this wiki Crude Birth Rate is the natality or childbirths per 1,000 people per year. It can be mathematically represented by $CBR = \frac{n}{p}{1000}$ where n is the number of childbirths in that year, and p is the current population. This figure is combined with the crude death rate to produce the rate of natural population growth (natural in that it does not take into account net migration). As of 2007, the average birth rate for the whole world is 20.3 per year per 1000 total population, which for a world population of 6.6 billion comes to 134 million babies per year. Another indicator of fertility is frequently used: the total fertility rate — average number of children born to each woman over the course of her life. In general, the total fertility rate is a better indicator of (current) fertility rates because unlike the crude birth rate it is not affected by the age distribution of the population. Fertility rates tend to be higher in less economically developed countries and lower in more economically developed countries. ## Other methods of measuring birthEdit General fertility rate - Standardised birth rate (SBR) – This compares the age-sex structure to a hypothetical standard population. Total fertility rate (TFR) – The mean number of children a woman is expected to bear during her child-bearing years. It is also independent of the age-sex structure of the population. ## Factors affecting birth rateEdit • Pro-natalist policies and Antinatalist policies from government • Abortion rates • Birth control practices • Existing age-sex structure • Social and religious beliefs - especially in relation to contraception • Female literacy rates • Economic prosperity (although in theory when the economy is doing well families can afford to have more children in practice the lower the economic prosperity the lower the birth rate). • Poverty levels – children can be seen as an economic resource in developing countries as they can earn money. • Infant Mortality Rate – a family may have more children if a country's IMR is high as it is likely some of those children will die. • Urbanization • Typical age of marriage • Pension availability • Conflict ## Socioeconomic FactorsEdit In her 1994 book Pricing the Priceless Child (Princeton University Press), Princeton University sociology professor Viviana Zelizer describes how in the 19th century, children were "economic assets" that contributed to farm work and other important tasks. Then, during the early 20th century, the U.S. established laws removing many children from hard labor, sparking the "rise of the economically useless and emotionally priceless child," Zelizer says.	2015-07-05 06:26:23	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 1, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3903268575668335, "perplexity": 3341.4554775571683}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375097246.96/warc/CC-MAIN-20150627031817-00218-ip-10-179-60-89.ec2.internal.warc.gz"}
https://xmonad.github.io/xmonad-docs/xmonad-contrib/XMonad-Hooks-PositionStoreHooks.html	Contents Description This module contains two hooks for the PositionStore (see XMonad.Util.PositionStore) - a ManageHook and an EventHook. The ManageHook can be used to fill the PositionStore with position and size information about new windows. The advantage of using this hook is, that the information is recorded independent of the currently active layout. So the floating shape of the window can later be restored even if it was opened in a tiled layout initially. For windows, that do not request a particular position, a random position will be assigned. This prevents windows from piling up exactly on top of each other. The EventHook makes sure that windows are deleted from the PositionStore when they are closed. Synopsis # Usage You can use this module with the following in your ~/.xmonad/xmonad.hs: import XMonad.Hooks.PositionStoreHooks and adding positionStoreManageHook to your ManageHook as well as positionStoreEventHook to your event hooks. To be accurate about window sizes, the module needs to know if any decoration is in effect. This is specified with the first argument: Supply Nothing for no decoration, otherwise use 'Just def' or similar to inform the module about the decoration theme used. myManageHook = positionStoreManageHook Nothing <> manageHook def myHandleEventHook = positionStoreEventHook main = xmonad def { manageHook = myManageHook , handleEventHook = myHandleEventHook }	2022-10-03 19:05:08	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.21078678965568542, "perplexity": 3898.520687604762}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337428.0/warc/CC-MAIN-20221003164901-20221003194901-00759.warc.gz"}
https://puzzling.stackexchange.com/questions/5567/the-treasure-hunt/5590	# The treasure hunt An archaeologist who has just unearthed a long-sought-after pair of ancient treasure chests. One chest is plated with silver and the other is plated with gold. According to the legend one of the two chests is filled with a great treasure whereas the other chest houses a death cursing spell in the form of a dragon. Faced with a dilemma archaeologist then noticed that there are inscriptions on the chest. Based on these inscriptions which chest should he open? $$\mathrm{Gold\, Chest}- \mathrm{One\, of\, these\, inscriptions\, is\, true}$$ $$\mathrm{Silver\, Chest- \mathrm{This\, chest\, contains\, the\, dragon}}$$ Note: Thanks to his understanding of the culture from which these chests came, the archaeologist knows that the gold chest's inscription could also be translated as "One and only one of these inscriptions is true". I already know the answer, what I am seeking here is a nice explanation. • @generalcrispy Unless the question is purely: "Why is <this> the answer"? – d'alar'cop Nov 25 '14 at 16:40 • Correct, but everyone who posts a riddle here ostensibly knows the answer. @Enigmo didn't ask what you did. – generalcrispy Nov 25 '14 at 16:41 • Should the gold chest say "ONLY one of these..."? – Ste Nov 25 '14 at 16:57 • The ancients should've put a curse in both chests. The treasure could just be the removable plating. :) – Rob Watts Nov 25 '14 at 16:59 • Is the dragon good or bad? – user2357112 Nov 25 '14 at 18:12 He should open the Silver Chest Explanation of the logic (assumes the inscriptions are intended to be helpful): Assume the inscription on the gold chest is true. Because that inscription is true, we already have the one true inscription, so the other inscription must be false. Thus the silver chest does not contain the dragon, so it must have the treasure. Now assume the inscription on the gold chest is false. That means either both inscriptions are true, or both are false. Since the gold chest's inscription is false, the silver chest's inscription must also be false, so it must have the treasure. So regardless of whether the gold chest's inscription is true or false, the silver chest's inscription must be false and therefore must be the one with the treasure in it. I'm also assuming here that the dragon referred to by the silver chest's inscription is the same thing as the death curse. • All that we learn is that the silver chest does not contain a dragon. We are not given anything in the puzzle linking a dragon to either a treasure or the curse. Interpreting the inscriptions in this way does not provide any useful data about the actual contents of each chest. – eclipz905 Nov 25 '14 at 18:17 • @eclipz905 I forgot to mention that I'm assuming that "dragon" means the death curse – Rob Watts Nov 25 '14 at 19:46 • I figured that, but the puzzle does not make it clear. Someone else would be equally justified in assuming "dragon" is a code name for the treasure. – eclipz905 Nov 25 '14 at 19:51 • @DoubleDouble I agree that it is ambiguous, but Enigmo originally included which chest was correct as a part of the question. So the dragon is not the treasure. – Rob Watts Nov 25 '14 at 21:08 • @Cthulhu the gold chest says that only one of the inscriptions is true – Rob Watts Nov 26 '14 at 17:53 This is a more formal solution We have a small system of formulae to define the problem: If a chest $c_i$ is $true$ then it contains the treasure. If a chest $c_i$ is $false$ then it contains the dragon. We have 2 propositional variables, $c_1$ and $c_2$. The first formal assertion we can define is: $c_1 \oplus c_2$ We express the inscriptions as further assertions about each other, themselves and the propositional variables. $f_1$ expresses the inscription on the Gold chest and $f_2$ that of the Silver chest. Conjointly they are $F$. $f_1 = f_1 \oplus f_2$ $f_2 = (c_2 = false)$ We must deduce for which $i$, $c_i$ is true. We do this by observing contradictions in $F$. If $c_2 = false$, then that makes $f_2 = true$. Now, if $f_1$ is $false$ then $f_1$ is true. If $f_1$ is true then $f_1$ is false, this is a contradiction. Thus, $c_2 \not= false$. If $c_2 = true$, then that makes $f_2 = false$. Now, $f_1 \leftrightarrow f_1$. No problem. So, $c_2 = true$. The Silver chest contains the treasure. • Nice symbolism, but I think Rob's answer is more understandable. +1 anyway :-) – Rand al'Thor Nov 25 '14 at 16:56 • @randal'thor Yes, Rob's is probably more suitable here. But it's good to have this here as a record anyway :p – d'alar'cop Nov 25 '14 at 16:57 Most of the existing answers assume that you meant "Only one of these inscriptions is true", rather than "One of these inscriptions is true", so I will go ahead and assume that you meant what you said instead, for completeness, or just in case. In that case: He should pick the Silver chest, but he only has a 2/3 chance of this being the correct one. Let's see why: We know we only have 4 possibilities: 1. Gold is true, Silver is true: Definitely possible if we assume you didn't mean "Only one...". In this possibility, both statements are true and therefore Gold contains the treasure. 2. Gold is false, Silver is false: This is also possible since rendering the "one of these is true" statement false allows (and requires) for both to be false. The treasure chest would be Silver 3. Gold is true, Silver is false: Since "this contains death" is false, the Silver chest contains the treasure. 4. Gold is false, Silver is true: This is a contradiction and should be discarded. "One of these is true" being false (as a statement) would mean that neither inscription is true. Since by definition we are looking for a possibility where Silver is true, this possibility contradicts itself. • Excellent answer, instead of harping on "dragon" nailing the actual imprecise expression. – No. 7892142 Nov 26 '14 at 9:04 Assuming this is a bit of a trick question... Neither. There is not enough information to safely pick a chest. According to legend, one chest has treasure while the other has a death cursing spell. The fact that the silver chest must be lying and thus does not contain a dragon has no bearing on which chest has the curse and which has the treasure. A chest that lacks a dragon can instead contain a curse or a treasure. This assumes the question is correctly worded. I'm sure I've seen something similar in one of Ray Smullyan's books... If the position is exactly as stated, and we take the legend as true (no "I don't believe in curses" type solutions), the archaeologist should still not open either casket. If he does, he is making a big assumption: that the chest inscriptions have valid truth values. The standard solution is essentially that the dragon is in the gold chest, because if he's in the silver one then the silver inscription is true, and the gold inscription can neither be true nor false. Without additional information, though, it's perfectly possible for the gold chest to have a meaningless inscription - just as there's nothing physically stopping me carving "This statement is false" on to the chest. Compare this with the canonical knight/knave/joker type puzzle - there we know that we only ever deal with people who make true or false statements. • I agree. For it to work, the puzzle would need the appearance of some "guru" or "magician" that guarantees that the statements have valid truth values. – Daniel Daranas Nov 26 '14 at 13:06 • Found it - in "What Is The Name Of This Book?". The scene is set by several series of puzzles where the chest-makers are known to be knights/knaves (who inscribe the same way as they speak). Late in the book along comes one like this where you have no such context. – Julia Hayward Nov 26 '14 at 13:18 Hint: Consider four possibilities: (1) Both inscriptions are true, (2) both are false, (3) only the gold chest inscription is true, and (4) only the gold chest inscription is false. There are no other possibilities. Let's look at four possibilities: 1. Both inscriptions are true. This is a contradiction because the Gold inscription says "One of these inscriptions is true", so we can rule this out (see note below). 2. Both inscriptions are false. This means that the Silver inscription "This chest contains the dragon" is false and therefore, by elimination, the silver chest has the treasure. 3. The gold chest's inscription is true. If we assume the gold chest's inscription is true and the gold chest's inscription says that one of the inscriptions is true (see note below), then the silver chest's inscription is necessarily false and thus the silver chest has the treasure. 4. The gold chest's inscription is false. If we assume the gold chest's inscription is false, then either both inscriptions are false or both inscriptions are true. I've shown in #1 above that both inscriptions being true leads to a contradiction, so both inscriptions must be false. Therefore the silver chest has the treasure. These possibilities either are contradictory or point to the silver chest, so the silver chest is the answer. Note, I'm assuming "One of these inscriptions is true" means "only one" is true. • What reason do you have to believe that "the dragon" refers to the curse? – eclipz905 Nov 25 '14 at 21:03 • It seems pretty clear to me that the author made a mistake with this. I'm assuming "curse" is supposed to be "dragon". The logic puzzle would be underdetermined without that assumption. – user2023861 Nov 25 '14 at 21:10 • "There are no other possibilities" is demonstrably false - it is possible for the inscription on a container to be a logical paradox as I have just confirmed by writing "this statement is false" onto a shoebox with a Sharpie. – Joe Lee-Moyet Nov 26 '14 at 14:31 • @yjo I suppose that someone could put the treasure in the gold chest just to mess with us. Or they both have dragons. Or the dragon is actually a brand new car. Or whatever else you can dream up. Given the context of the question, there are only four possibilities. – user2023861 Nov 26 '14 at 14:54 • @user2023861 I disagree: the question gives evidence (albeit as 'legend') that one chest contains treasure and the other a dragon. On the other hand there is nothing to support the idea that the inscriptions are not paradoxical or self-contradictory. Furthermore it's not possible for both of the inscriptions to be true so we're already starting from a point of distrusting the inscriptions. – Joe Lee-Moyet Nov 26 '14 at 15:11 Three possibilities. 1. The archaeologist can CT scan both the chest. The one in which is finds treasure he can open it. 2. He can tap on both the treasure chest or do something that creates noise. If there is a dragon it will react 3. Ask someone else to open both the chest on his behalf It's most likely the... Gold Chest And here's why: Let's pretend for a while that I'm the person who made the treasure chests and was responsible for storing the treasure in them. (As well as making sure nobody made off with them.) A chest allows me to move treasure from point A to point B and provides a reasonable protection from external weather conditions, it does not however, provide a decent protection from thieves, raiders and other varieties of hoodlums. So what do I do? I make another chest, this time plated with silver and rig it with a deathtrap inside. Ok, what next? If I make a scary story about a curse residing in these chests, that should put off at least a good deal of people wanting to get their clutches on this treasure. But what about the people who do not believe in mystical curses? An idea springs into the mind: "Their rational way of thought will be their undoing." I will engrave inscriptions upon both of the chests portraying a conundrum. But what would initially seem to be a logical puzzle is actually a deception leading to their demise. A solution of which will not lead to the chest containing the treasure but instead to the chest with the deathtrap. For I have been given the task of protecting these chests and I am aware that the gold chest contains the treasure, and all staff who need access to the treasure know that the gold chest is where the treasure lies. For I have no reason to hand over the treasure to just about anyone who can solve a puzzle just as how the fair people of the future will not be able to solve a logical puzzle to access an account of a complete stranger. Some might argue that: - What if thieves anticipate such a cunning way and decide to pick the gold chest? Kudos to you, sir thief, you have bested my guards as you have bested my wit and are worthy of the treasure inside. For if I switch the chests around, my initial ploy becomes void and the measures taken described above would become ineffective. I think this fits quite nicely with the tag. I fully agree with Julia Hayward's answer. Apparent laws of logic cannot prevent anyone from putting a treasure into the gold chest, a dragon into the silver one, and then make arbitrary inscriptions on them. Valid conclusions about physical world require some preliminary assumptions that themselves are not purely logical, and here we are supposed to derive one from propositions of unknown logical value, based only on their logical structure. This is not possible. In Smullyan's book cited by Julia, there is also a statement which "proves" that unicorns exist. I don't remember it exactly, but I believe it was, "This statement is false, and unicorns do not exist," which seems to imply, on a purely logical basis, that unicorns exist indeed. (In fact, anything could be "proved" this way.) Self-referencing propositions have been a known problem in logic for over a century, leading to contradictions and antinomies. That's why contemporary mathematics became strictly formalized in a way that does not allow them as valid statements. I hate to be 'that guy' but since dragons and death curses aren't real, he can safely open either chest as long as he takes elementary precautions against explosives, booby-traps and poison. And yes, I fully appreciate that this answer sucks. Sorry. • You should change your profile pic to 'that guy' :-p – Rand al'Thor Nov 26 '14 at 13:08	2019-07-19 11:11:04	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.48308148980140686, "perplexity": 1422.0805697404583}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195526210.32/warc/CC-MAIN-20190719095313-20190719121313-00393.warc.gz"}
http://liroethenite.herokuapp.com/post/logistic-regression-cv	3.2.4.1.5. sklearn.linear_model.LogisticRegressionCV ... Logistic Regression CV (aka logit, MaxEnt) classifier. See glossary entry for cross validation estimator. This class implements logistic regression using liblinear, newton cg, sag of lbfgs optimizer. The newton cg, sag and lbfgs solvers support only L2 regularization with primal formulation. sklearn.linear_model.LogisticRegression — scikit learn 0 ... Logistic Regression (aka logit, MaxEnt) classifier. In the multiclass case, the training algorithm uses the one vs rest (OvR) scheme if the ‘multi_class’ option is set to ‘ovr’, and uses the cross entropy loss if the ‘multi_class’ option is set to ‘multinomial’. R Logistic Regression Tutorialspoint The Logistic Regression is a regression model in which the response variable (dependent variable) has categorical values such as True False or 0 1. It actually measures the probability of a binary response as the value of response variable based on the mathematical equation relating it with the predictor variables. What is Logistic Regression? A Beginner's Guide Regression analysis can be broadly classified into two types: Linear regression and logistic regression. In statistics, linear regression is usually used for predictive analysis. It essentially determines the extent to which there is a linear relationship between a dependent variable and one or more independent variables. Logit Regression \| R Data Analysis Examples The logistic regression coefficients give the change in the log odds of the outcome for a one unit increase in the predictor variable. For every one unit change in gre, the log odds of admission (versus non admission) increases by 0.002. How to optimize hyper parameters of a Logistic Regression ... Here, we are using Logistic Regression as a Machine Learning model to use GridSearchCV. So we have created an object Logistic_Reg. logistic_Reg = linear_model.LogisticRegression() Step 5 Using Pipeline for GridSearchCV. Pipeline will helps us by passing modules one by one through GridSearchCV for which we want to get the best parameters. Logistic Regression, Model Selection, and Cross Validation Logistic Regression, Model Selection, and Cross Validation GAO Zheng March 25, 2017. Classification problems. In this project we are trying to predict if a loan will be in good standing or go bad, given information about the loan and the borrower. Logistic regression Applications. Logistic regression is used in various fields, including machine learning, most medical fields, and social sciences. For example, the Trauma and Injury Severity Score (), which is widely used to predict mortality in injured patients, was originally developed by Boyd et al. using logistic regression.Many other medical scales used to assess severity of a patient have been developed ... Building A Logistic Regression in Python, Step by Step ... Logistic Regression is a Machine Learning classification algorithm that is used to predict the probability of a categorical dependent variable. In logistic regression, the dependent variable is a binary variable that contains data coded as 1 (yes, success, etc.) or 0 (no, failure, etc.). Logistic Regression Analysis of Quant’s Resume during His ... Logistic Regression Analysis of Quant’s Resume during His Job Interview April 17, 2018 by Pawel There are not too many creative opportunities to leave a person applying for a quant role dumbfounded with the interview question directly related to his CV. Grid Search with Logistic Regression \| Kaggle We use cookies on Kaggle to deliver our services, analyze web traffic, and improve your experience on the site. By using Kaggle, you agree to our use of cookies. Logistic regression with $$\ell_1$$ regularization — CVXPY ... Logistic regression with $$\ell_1$$ regularization¶. In this example, we use CVXPY to train a logistic regression classifier with $$\ell_1$$ regularization. We are ... Scikit learn: Logistic Regression CV \| Perspective What is Logistic Regression? A Logistic Regression is a regression model that uses the logistic sigmoid function to predict classification. The basic idea is to predict the feature vector sucht that it fits the Logistic_log function, . ... Fixing precompute and LogReg CV. Logistic Regression Model Tuning with scikit learn — Part ... Classifiers are a core component of machine learning models and can be applied widely across a variety of disciplines and problem statements. With all the packages available out there, running a logistic regression in Python is as easy as running a few lines of code and getting the accuracy of predictions on a test set. Logistic Regression A plete Tutorial with Examples in R Learn the concepts behind logistic regression, its purpose and how it works. This is a simplified tutorial with example codes in R. Logistic Regression Model or simply the logit model is a popular classification algorithm used when the Y variable is a binary categorical variable. What is Logistic Regression using Sklearn in Python ... Here, result is the dependent variable and gender is the independent variable. Since the result is of binary type—pass or fail—this is an example of logistic regression. Now that we have understood when to apply logistic regression, let us try and understand what logistic regression exactly is. prehensive Guide To Logistic Regression In R \| Edureka Logistic Regression is one of the most widely used Machine learning algorithms and in this blog on Logistic Regression In R you’ll understand it’s working and implementation using the R language. To get in depth knowledge on Data Science, you can enroll for live Data Science Certification Training by Edureka with 24 7 support and lifetime access. In k fold CV how use trained model (logistic regression ... In k fold CV how use trained model (logistic regression) to compare test data set and draw ROC Posted 11 16 2016 02:41 AM (2788 views) Hi I am new to SAS and need suggestions on my codes. I wanted to compare two logistic regression models. I tried to use 10 fold cross validation for both models. OpenCV: cv::ml::LogisticRegression Class Reference This function returns the trained parameters arranged across rows. For a two class classification problem, it returns a row matrix. It returns learnt parameters of the Logistic Regression as a matrix of type CV_32F. Logistic Regression in Python – Real Python Problem Formulation. In this tutorial, you’ll see an explanation for the common case of logistic regression applied to binary classification. When you’re implementing the logistic regression of some dependent variable 𝑦 on the set of independent variables 𝐱 = (𝑥₁, …, 𝑥ᵣ), where 𝑟 is the number of predictors ( or inputs), you start with the known values of the ... Practical Guide to Logistic Regression Analysis in R ... In Logistic Regression, we use the same equation but with some modifications made to Y. Let's reiterate a fact about Logistic Regression: we calculate probabilities. And, probabilities always lie between 0 and 1. In other words, we can say: The response value must be positive. It should be lower than 1. First, we'll meet the above two criteria. Example of Logistic Regression in Python Data to Fish In this guide, I’ll show you an example of Logistic Regression in Python. In general, a binary logistic regression describes the relationship between the dependent binary variable and one or more independent variable s.. The binary dependent variable has two possible outcomes: Chapter 21 The caret Package \| R for Statistical Learning trControl = trainControl(method = "cv", number = 5) specifies that we will be using 5 fold cross validation. method = glm specifies that we will fit a generalized linear model. The method essentially specifies both the model (and more specifically the function to fit said model in R ) and package that will be used. Logistic Regression in R Explained with Simple Examples Examples of Logistic Regression in R . Logistic Regression can easily be implemented using statistical languages such as R, which have many libraries to implement and evaluate the model. Following codes can allow a user to implement logistic regression in R easily: We first set the working directory to ease the importing and exporting of datasets. Python Examples of sklearn.linear_model.LogisticRegressionCV def logistic_regression_cv(): """Logistic regression with 5 folds cross validation.""" return LogisticRegressionCV(Cs=10, cv=KFold(n_splits=5)) Example 20 Project: pandas ml Author: pandas ml File: test_linear_model.py License: BSD 3 Clause "New" or "Revised" License logit.reg : Cyclic Coordinate Descent for Logistic regression CDLasso package: Coordinate descent algorithms for L1 and L2 regression cv.l1.reg: k fold Cross Validation cv.l2.reg: k fold Cross Validation cv.logit.reg: k fold Cross Validation l1.reg: Greedy Coordinate Descent for L1 regression l2.reg: Cyclic Coordinate Descent for L2 regression logit.reg: Cyclic Coordinate Descent for Logistic regression plot.cv.l1.reg: Cross validation plot 5.3.1 The Validation Set Approach Home Clark Science ... 5.3.2 Leave One Out Cross Validation. The LOOCV estimate can be automatically computed for any generalized linear model using the glm() and cv.glm() functions. In the lab for Chapter 4, we used the glm() function to perform logistic regression by passing in the family="binomial" argument. But if we use glm() to fit a model without passing in the family argument, then it performs linear ... Logistics Resume Sample \| Monster To be the successful job candidate in any field, it helps to have a comprehensive resume. To help guide your own resume efforts, check out our sample resume below for a logistics professional making the transition from military to civilian work, and download the sample resume for a logistics professional in Word. Jobs for logisticians are projected to grow by 7% (or 10,300 jobs) from 2016 ... Hyperparameter Tuning Using Grid Search Chris Albon Create Logistic Regression # Create logistic regression logistic = linear_model. LogisticRegression () ... # Create grid search using 5 fold cross validation clf = GridSearchCV (logistic, hyperparameters, cv = 5, verbose = 0) Conduct Grid Search # Fit grid search best_model = clf. fit (X, y) The logistic regression in python — how to prepare a data ... In first one, I will show my way of the data preparation and in the second you will see how to find the best logistic regression model. Every data science project you should observe as a maze. cv.clogitL1: Cross validation of conditional logistic ... The penalised conditional logistic regression model is fit to the non left out strata in turn and its deviance compared to an out of sample deviance computed on the left out strata. Fitting models to individual non left out strata proceeds using the cyclic coordinate descent warm start strong rule type algorithm used in clogitL1 , only with a prespecified sequence of λ .	2020-11-24 16:26:06	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4874889850616455, "perplexity": 1230.2195052887973}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141176864.5/warc/CC-MAIN-20201124140942-20201124170942-00023.warc.gz"}
https://eepower.com/new-industry-products/complete-airfuel-alliance-compatible-wireless-power-demo-kit/	New Industry Products # Complete AirFuel Alliance Compatible Wireless Power Demo Kit June 02, 2018 by Paul Shepard Efficient Power Conversion Corporation (EPC) today announces the availability of a complete class 4 wireless power kit, the EPC9129. The system can transmit up to 33W while operating at 6.78MHz (the lowest ISM band). The kit comes complete with two receivers, each with a regulated output − one capable of 5W capable and a second capable of delivering 27W at 19V. The purpose of this demonstration kit is to simplify the evaluation process of using eGaN FETs for highly efficient wireless power transfer. The EPC9129 utilizes the high frequency switching capability of EPC gallium-nitride transistors to facilitate wireless power systems with full power efficiency between 80% and 90% under various operating conditions. The popularity of highly resonant wireless power transfer is increasing rapidly, particularly for applications targeting large power transmitting surface areas, with the capability to place receiving devices anywhere on the surface, and the ability to simultaneously power (or charge) multiple devices placed on the surface. The end applications are varied and evolving quickly from cell phone charging to powering handheld tablets and laptop computers. Delivering up to 33 W supports all of these applications. The EPC9129 wireless power system consists of four boards: • Source Board (Transmitter or Power Amplifier) EPC9512 featuring the EPC8010, EPC2038 and EPC2019 • Class 4 Air Fuel compliant Source Coil (Transmit Coil) • Category 5 AirFuel compatible Receive Device EPC9514 featuring the EPC2016C • Category 3 AirFuel compatible Receive Device EPC9513 featuring the EPC2019 #### Source (Amplifier) Board The source board is a highly efficient Zero Voltage Switching (ZVS), Class-D amplifier configured in an optional half-bridge topology (for single-ended configuration) or default full-bridge topology (for differential configuration), and includes the gate driver(s), oscillator, and feedback controller for the pre-regulator. This allows for compliance testing operating to the AirFuel class 4 standard over a wide load range. The amplifier board is available separately as EPC9512 for evaluation in existing customer systems. #### Device (Receiving) Boards The Category 5 EPC9514 (19 V, 27 W) and Category 3 EPC9513 (5 V, 5 W) device or receiving boards included in the wireless power demonstration kits are also available separately for those that have their own source boards or who want to work with multiple receiving devices simultaneously. The efficiency of these first-generation systems is about 87% from input to the transmitter to the output of the receiver (end-to-end), and with future improvements in architecture and GaN IC technology this number can reach into the 95% range. As with the demonstration kits, these boards operate to the Airfuel standard, excluding Bluetooth Low Energy (BLE) communications. With the wide range of efficient receivers that can be used to power anything from lamps to laptops to tablets, while remaining compatible with cell phone charging, the system designer now has all the tools needed to create an entire wireless power, large area, efficient system. #### Price and Availability EPC9129 wireless power transfer demonstration systems are priced at $907.20 each. The EPC9512 amplifier board can also be purchased separately and is priced at$390.00 each The EPC9513 and EPC9514 receive boards can also be purchased separately and are priced at \$168.75 each. All products are available for immediate delivery from Digi-Key. The amplifier board (EPC9512) features the enhancement-mode, 100V-rated EPC8010 eGaN FET as the main power stage in a dual half-bridge configuration; the 100V-rated EPC2038 eGaN FET used as the synchronous bootstrap FET, and the 200V-rated EPC2019 eGaN FET used in the SEPIC pre-regulator. The amplifier can be set to operate in either differential mode or single-ended mode and includes the gate driver(s), oscillator, and feedback controller for the pre-regulator that ensures operation for wireless power control based on the AirFuel standard. Diagram of EPC9512 ZVS class-D amplifier circuit (click on image to enlarge) The EPC9512 can operate in either single-ended or differential mode by changing a jumper setting. This allows for high efficiency operation with load impedance ranges suitable for single ended operation. The circuits used to adjust the timing for the ZVS class-D amplifiers have been separated to further ensure highest possible efficiency setting. Each half bridge also includes separate ZVS tank circuits. Block diagram of the EPC9512 wireless power amplifier (click on image to enlarge) The amplifier is equipped with a pre-regulator controller that adjusts the voltage supplied to the ZVS class-D amplifier based on the limits of 3 parameters: coil current, DC power delivered to the ZVS class-D amplifier, and maximum operating voltage of the ZVS class-D amplifier. The coil current has the lowest priority followed by the power delivered and amplifier supply voltage having the highest priority. Changes in the device load power demand, physical placement of the device on the source coil and other factors, such as metal objects in proximity to the source coil, contribute to variations in coil current, DC power, and amplifier voltage requirements. Under any of these conditions, the controller will ensure the correct operating conditions for the ZVS class-D amplifier based on the AirFuel standard. The pre-regulator can be bypassed to allow testing with custom control hardware. The board further allows easy access to critical measurement nodes for accurate power measurement instrumentation hookup. The source and device coils are AirFuel compliant and have been pretuned to operate at 6.78 MHz with the EPC9512 amplifier. The source coil is AirFuel class 4 compliant and the device coils are AirFuel category 3 and category 5 compliant. The EPC9513 (Category 3) device board includes a high frequency Schottky diode based full-bridge rectifier, DC smoothing capacitor and 5 V regulator. The regulator is based on a SEPIC converter that features a 200 V EPC2019 eGaN FET. The power circuit is attached to the backside of the coil which is provided with a ferrite shield that prevents the circuit from shunting the coil’s magnetic field. Schematic diagram of the EPC9513 and EPC9514 demo board (click on image to enlarge) The EPC9514 (Category 5) device board includes a high frequency Schottky diode based full-bridge rectifier, DC smoothing capacitor and 19 V regulator. The regulator is based on a SEPIC converter that features a 100 V EPC2016C eGaN FET. The power circuit is attached to the side of the coil.	2022-08-10 14:14:13	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.18471382558345795, "perplexity": 6452.79169979683}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571190.0/warc/CC-MAIN-20220810131127-20220810161127-00736.warc.gz"}
http://mathhelpforum.com/calculus/224992-polar-rect-form-conversion.html	# Math Help - polar to rect form conversion 1. ## polar to rect form conversion Hi. I'm working on a practice problem which is stumping me. Even after looking up the answer in the back of the book, I still don't understand how it was arrived at. The question involves taking polar equation r = 2(hcos(t)+ksin(t)) And coverting it to rectangular form, and verifying that it is the equation of a circle. I keep playing around with the coord conversion equations and trig identities, but I can't seem to get it worked out. Help! 2. ## Re: polar to rect form conversion Originally Posted by infraRed Hi. I'm working on a practice problem which is stumping me. Even after looking up the answer in the back of the book, I still don't understand how it was arrived at. The question involves taking polar equation r = 2(hcos(t)+ksin(t)) And coverting it to rectangular form, and verifying that it is the equation of a circle. I keep playing around with the coord conversion equations and trig identities, but I can't seem to get it worked out. Help! $r = \sqrt{x^2 + y^2}$, $cos(t) = \frac{x}{\sqrt{x^2 + y^2}}$, and $sin(t) = \frac{y}{\sqrt{x^2 + y^2}}$ Hint: Plug these into your equation and multiply both sides by $\sqrt{x^2 + y^2}$. -Dan 3. ## Re: polar to rect form conversion Originally Posted by infraRed Hi. I'm working on a practice problem which is stumping me. Even after looking up the answer in the back of the book, I still don't understand how it was arrived at. The question involves taking polar equation r = 2(hcos(t)+ksin(t)) And coverting it to rectangular form, and verifying that it is the equation of a circle. I keep playing around with the coord conversion equations and trig identities, but I can't seem to get it worked out. Help! if (h $\neq$ k) then it's not a circle, it's an ellipse.	2015-01-27 09:32:04	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 5, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.887866199016571, "perplexity": 418.76348333945685}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-06/segments/1422115862141.23/warc/CC-MAIN-20150124161102-00154-ip-10-180-212-252.ec2.internal.warc.gz"}
https://www.rdocumentation.org/packages/stats/versions/3.6.1/topics/embed	# embed 0th Percentile ##### Embedding a Time Series Embeds the time series x into a low-dimensional Euclidean space. Keywords ts ##### Usage embed (x, dimension = 1) ##### Arguments x a numeric vector, matrix, or time series. dimension a scalar representing the embedding dimension. ##### Details Each row of the resulting matrix consists of sequences x[t], x[t-1], …, x[t-dimension+1], where t is the original index of x. If x is a matrix, i.e., x contains more than one variable, then x[t] consists of the tth observation on each variable. ##### Value A matrix containing the embedded time series x. • embed ##### Examples library(stats) # NOT RUN { x <- 1:10 embed (x, 3) # } Documentation reproduced from package stats, version 3.6.1, License: Part of R 3.6.1 ### Community examples Looks like there are no examples yet.	2019-09-17 20:05:23	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.36662545800209045, "perplexity": 6501.554895266225}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514573105.1/warc/CC-MAIN-20190917181046-20190917203046-00139.warc.gz"}
https://socratic.org/questions/how-do-you-simplify-sqrt-16393	# How do you simplify sqrt(16393)? $\sqrt{16393} = 13 \sqrt{97}$ If $a , b \ge 0$ then $\sqrt{a b} = \sqrt{a} \sqrt{b}$ $\sqrt{16393} = \sqrt{{13}^{2} \cdot 97} = \sqrt{{13}^{2}} \cdot \sqrt{97} = 13 \sqrt{97}$	2019-12-14 18:14:35	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 4, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9351511597633362, "perplexity": 1484.6398404425754}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575541288287.53/warc/CC-MAIN-20191214174719-20191214202719-00031.warc.gz"}
http://blog.amjith.com/?page=12	Memoization Decorator Recently I had the opportunity to give a short 10 min presentation on Memoization Decorator at our local UtahPython Users Group meeting. Memoization: • Everytime a function is called, save the results in a cache (map). • Next time the function is called with the exact same args, return the value from the cache instead of running the function. The code for memoization decorator for python is here: http://wiki.python.org/moin/PythonDecoratorLibrary#Memoize Example: The typical recursive implementation of fibonacci calculation is pretty inefficient O(2^n). ```def fibonacci(num): print 'fibonacci(%d)'%num if num in (0,1): return num return fibonacci(num-1) + fibonacci(num-2)>>> math_funcs.fibonacci(4) # 9 function calls fibonacci(4) fibonacci(3) fibonacci(2) fibonacci(1) fibonacci(0) fibonacci(1) fibonacci(2) fibonacci(1) fibonacci(0) 3``` But the memoized version makes it ridiculously efficient O(n) with very little effort. ```import memoized @memoized def fibonacci(num): print 'fibonacci(%d)'%num if num in (0,1): return num return fibonacci(num-1) + fibonacci(num-2) >>> math_funcs.mfibonacci(4) # 5 function calls fibonacci(4) fibonacci(3) fibonacci(2) fibonacci(1) fibonacci(0) 3``` We just converted an algorithm from Exponential Complexity to Linear Complexity by simply adding the memoization decorator. Slides: Presentation: I generated the slides using LaTeX Beamer. But instead of writing raw LaTeX code I used reStructured Text (rst) and used rst2beamer script to generate the .tex file. Source: The rst file and tex files are available in Github. https://github.com/amjith/User-Group-Presentations/tree/master/memoization_de... Posted Productive Meter A few weeks ago I decided that I should suck it up and start learning how to develop for the web. After asking around, my faithful community brethren, I decided to learn Django from its docs ::Django documentation is awesome:: Around this time I came across this post about Waking up at 5am to code. I tried it a few times and it worked wonders. I've been working on a small project that can keep track of my productivity on the computer. The concept is really simple, just log the window that is on top and find a way to display that data in a meaningful way. Today's 5am session got me to a milestone on my project. I am finally able to visaulize the time I spend using a decent looking graph. Which is a huge milestone for someone who learned how to display html tables 3 weeks ago. Tools: A huge thanks to my irc friends and random geeks who wrote awesome blog posts and SO answers on every problem I encountered. I will be open-sourcing the app pretty soon. Stay tuned. Posted Too Many Classes Too Little Time I'm taking a couple of the free online classes offered by Standford. One on Artifical Intelligence and one on Machine Learning I haven't had so much fun since kindergarten. Actually that's not fair, I didn't enjoy kindergarten this much. I'm listening to the classes during my lunch, after work, during weekends. I'm working on my assignment with so much enthusiasm, I dread the day when this class ends. Stanford just announced a slew of new online classes offered starting in Jan 2012. I was way too excited when I first read the description on them. Now I'm a little sad, becasue I want to take 8 out of the 11 courses that are being offered and I don't have enough time. :( Woe is me. ps: If you are not taking any of these classes you are missing out big time. Please do yourself a favor and sign up. Posted Picking 'k' items from a list of 'n' - Recursion Let me preface this post by saying I suck at recursion. But it never stopped me from trying to master it. Here is my latest (successful) attempt at an algorithm that required recursion. Background: You can safely skip this section if you're not interested in the back story behind why I decided to code this up. I was listening to KhanAcademy videos on probability. I was particularly intrigued by the combinatorics video. The formula to calculate the number of combinations of nCr was simple, but I wanted to print all the possible combinations of nCr. Problem Statement: Given 'ABCD' what are the possible outcomes if you pick 3 letters from it to form a combination without repetition (i.e. 'ABC' is the same as 'BAC'). At first I tried to solve this using an iterative method and gave up pretty quickly. It was clearly designed to be a recursive problem. After 4 hours of breaking my head I finally got a working algorithm using recursion. I was pretty adamant about not looking it up online but I seeked some help from IRC (Thanks jtolds). Code: ```def combo(w, l): lst = [] if l < 1: return lst for i in range(len(w)): if l == 1: lst.append(w[i]) for c in combo(w[i+1:], l-1): lst.append(w[i] + c) return lst``` Output: ```>>> combinations.combo('abcde',3) ['abc', 'abd', 'abe', 'acd', 'ace', 'ade', 'bcd', 'bce', 'bde', 'cde']``` Thoughts: • It helps to think about recursion with the assumption that an answer for step n-1 already exists. • If you are getting partial answers check the condition surrounding the return statement. • Recursion is still not clear (or easy). I have confirmed that this works for bigger data sets and am quite happy with this small victory. Posted Python Profiling I did a presentation at our local Python User Group meeting tonight. It was well received, but shorter than I had expected. I should've added a lot more code examples. We talked about usage of cProfile, pstats, runsnakerun and timeit. Here are the slides from the presentations: The slides were done using latex-beamer, but I wrote the slides in reStructuredText and used rst2beamer to create the tex file which was then converted to pdf using pdflatex. The source code for the slides are available on github. // Posted Programming - A Gateway Drug to Math I decided to try my hand at the Stanford's AI Class. The pre-requisites mentioned Probability and Linear Algebra. So I started watching Probability videos on KhanAcademy Sal Khan was teaching how to find the probability of 2 heads when you toss a coin 5 times. A classic nCk problem: $\small _nC_k = \frac{n!}{k!(n-k)!}$ The probability of getting 2 heads while tossing a coin 5 times is: $P(2) = \frac{{_5C_2}}{2^5}$ But I wanted to find out the probability of getting at least 2 heads when I toss 5 coins. Its really simple. All I had to do is P(2) + P(3) + P(4) + P(5). But then computing$_nC_k$by hand (or a calculator) was painfully slow, let alone do it 4 times. So I wrote two little functions in Python that will calculate factorial (yes I reinvented the wheel) and$_nC_k$ Nothing teaches you math faster than trying to write a program to do the math for you. Writing a program is the same as teaching the computer how to do a certain task. The only way you can teach someone to do a task is to become a master at doing that task yourself. Bonus: It also teaches you corner cases like 0! = 1 and $\small _5C_5 = 1$ that you wouldn't think of otherwise. Tags Posted When was the last time I vented about C++? The answer for that is always: "TOO LONG AGO". The initial friction to setup a substantial project using C++ is unfucking bearable. When we started code revamp at work recently, I decided to be a good citizen and decided to incorporate cpptest, a unit testing framework in our project. It made me realize how unreasonably complicated Makefiles can be. After 3 hours of peeling away at the complexity I managed to add cpptest to the build dependency of the project. Now time to write a few tests and check it out. I'm thinking "We are almost there". FALSE! Compilation gives me a gazillion error messages that make absolutely no sense. After about 30mins of StackOverflowing and Googling, I find out that its not enough to include string.h and map.h in my header files, but I also need to namespace it. Of course there is no indication (not even a hint) of that in the error messages. So I add 'using namespace std' and get past it. Awesome my first test is compiling successfully. Time to run this baby and declare victory. Close! But no cigar. The executable was unable to load the CppTest library during runtime. Argh! I set my LD_LIBRARY_PATH env variable and now it's running. But I can't ask everyone in my team to do that, so I have to figure out how to statically link that library. It's already 6pm and I'm hungry. That'll have to wait for another day. TL;DR - C++ and Makefile can burn in a fire of thousand suns. Posted Rapid Prototyping in Python I was recently assigned to a new project at work. Like any good software engineer I started writing the pseudocode for the modules. We use C++ at work to write our programs. I quickly realized it's not easy to translate programming ideas to English statements without a syntactic structure. When I was whining about it to Vijay, he told me to try prototyping it in Python instead of writing pseudocode. Intrigued by this, I decided to write a prototype in Python to test how various modules will come together. Surprisingly it took me a mere 2 hours to code up the prototype. I can't emphasize enough, how effortless it was in Python. What makes Python an ideal choice for prototyping: Dynamically typed language: Python doesn't require you to declare the datatype of a variable. This lets you write a function that is generic enough to handle any kind of data. For eg: ```def max_val(a,b): return a if a >b else b``` This function can take integers, floats, strings, a combination of any of those, or lists, dictionaries, tuples, whatever. A list in Python need not be homogenous. This is a perfectly good list: `[1, 'abc', [1,2,3]]` This lets you pack data in unique ways on the fly which can later be translated to a class or a struct in a statically typed language like C++. ```class newDataType { int i; String str; Vector vInts; };``` Rich Set to Data-Structures: Built-in support for lists, dictionaries, sets, etc reduces the time involved in hunting for a library that provides you those basic data-structures. Expressive and Succinct: The algorithms that operate on the data-structures are intuitive and simple to use. The final code is more readable than a pseudocode. For example: Lets check if a list has an element ```>>> lst = [1,2,3] # Create a list >>> res = 2 in lst # Check if 2 is in 'lst' True``` If we have to do it in C++. ```list lst; lst.push_back(3); lst.push_back(1); lst.push_back(7); list::iterator result = find(lst.begin(), lst.end(), 7); bool res = (result != lst.end())``` Python Interpreter and Help System: This is a huge plus. The presence of interpreter not only aids you in testing snippets of code, but it acts as an help system. Lets say we want to look up the functions that operate on a List. ```>>> dir([]) '__delslice__', '__doc__', '__eq__', '__format__', '__ge__', '__getattribute__', '__getitem__', '__getslice__', '__gt__', '__hash__', '__iadd__', '__imul__', '__init__', '__iter__', '__le__', '__len__', '__lt__', '__mul__', '__ne__', '__new__', '__reduce__', '__reduce_ex__', '__repr__', '__reversed__', '__rmul__', '__setattr__', '__setitem__', '__setslice__', '__sizeof__', '__str__', '__subclasshook__', 'append', 'count', 'extend', 'index', 'insert', 'pop', 'remove', 'reverse', 'sort'] >>> help([].sort) Help on built-in function sort: sort(...) L.sort(cmp=None, key=None, reverse=False) -- stable sort IN PLACE; cmp(x, y) -> -1, 0, 1``` • The type definition of the datastructures emerge as we code. • The edge cases start to emerge when you prototype. • A set of required supporting routines. • A better estimation of the time required to complete a task. Posted New Laptop I finally ordered a new Macbook air for myself. One of my friends remarked at the fact that this is the first brand new laptop that I've ordered for myself. Since I'm a bit of a Linux fanatic, I tend to restore old computers and install a linux distro and make them useable. So I always get old laptops for cheap for myself. But this time I decided it's time to checkout Mac OS X. So I'll be replacing my Netboook (yep!) with the Macbook air. Anyone need a Lenovo S10 netbook :). I will even do a clean-install of Ubuntu or your choice of Linux distro. This new Macbook air will be my primary development machine. Let's hope it can take the abuse. Posted Falsetto dude and the Fat man A typical conversation between my wife and I: Playing Ne Me Quitte Pas by Nina Simone Yoshi : Ow! What is that abomination? Amjith: It's a french song sung by the great Nina Simone. Yoshi : It's a woman? Sounded like a dude singing in falsetto. Yoshi : I'm leaving the room if you don't change the song. Amjith: FINE. You play something then. Yoshi puts on Ave Maria by Pavarotti. I wait for 2 mins. Amjith: Hey! Your fat man seems to be yelling at Maria. Yosh leaves the room and I sleep on the couch. The End Posted	2019-09-16 13:13:29	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 5, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3326587677001953, "perplexity": 1461.8753849470543}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514572556.54/warc/CC-MAIN-20190916120037-20190916142037-00114.warc.gz"}
http://physics.stackexchange.com/tags/waves/new	# Tag Info ## New answers tagged waves 2 In general, a mixed wave of the form $$f(x,t)=\cos(x-t)+r\cos(x+t)$$ will not have nodes, at least in the sense of points $x_0$ for which $f(x_0,t)\equiv 0$ for all times $t$. In general, there's relatively little to say beyond what the picture will convey: As you turn $r$ up from 0 to 1, you first start inducing modulations into the amplitude of the ... 6 Dispersion in waves arises from both material property variation with frequency and from the geometry of the fields in question. That wave dispersion will arise from material property variation is obvious. But wave geometry and boundary conditions also matter. Simple example: a conductive waveguide with rectangular cross-section with sidelengths $a$ and $b$... -2 A physical significance of a wave number would be the speed of a photons oscillation (frequency) times Planck's constant. Which can equate to a force strong enough to knock an electron loose or give you a sunburn. 4 Imagine something oscillating in space and time, for example a plane wave propagating across the axis $x$. This propagation is expressed via the so-called phase $$\phi(x,t)=\omega \cdot t - k\cdot x = \dfrac{2\pi}{T}\cdot t -\dfrac{2\pi}{\lambda}\cdot x \tag{01}$$ and the magnitude of the plane wave as $$E(x,t)=A\cos\phi(x,t) \tag{02}$$ As the ... 2 Basically it is whatever you need to multiply a distance by to find a phase difference (in radians). For a traveling wave, the wave number is the amount of phase difference per unit length. For a physical sine wave, it is the ratio between the maximal slope of the wave surface and the amplitude. In other words, it measures how dramatic the local ... 0 Actually both are analogous. When you quantize a single mode which is simply an oscillation associated with a single photon, you adopt the Lorentz model where you imagine your single photon as a mass on a spring, then you translate your quantum mechanical Hamiltonian that includes both position and momentum to phase space where the electric and magnetic ... 0 I think this is very commonly confused topic about this wave-particle thing. Let me clarify. Like the post above indicates, from Maxwell's Equations, we find evidence, in a vacuum, for the electric and magnetic fields to travel together at the speed of light, perpendicular to each other. This is simply the solution of the well known equations which govern ... 1 The difference between a fluid and a solid is the following: Fluid's have zero shear modulus, so they can't carry a shear force, but solids have non-zero shear modulus, so the can carry shear force. Fun little way to visualize this: Let's say we line up a bunch of second graders on rectangular grid. Now we push one of the students along one row. That ... 1 Treating a medium as continuous (which works for our purposes), for sound to travel through the medium the motion of one unit of it has to affect the motion of the next unit, and so on and so on. Sound in a gas or liquid does this by varying pressure: the air pressure next to a speaker suddenly increases, pushing a unit of air outwards towards the adjacent ... 0 According to this website: Sound is transmitted through gases, plasma, and liquids as longitudinal waves, also called compression waves...Through solids, however, it can be transmitted as both longitudinal waves and transverse waves. This means that through most forms of matter (gases, plasma, liquids), such as water, or the air, sound travels as ... 1 In the first case I get points where there is an oscillation in time, while in the second one, as stated, the points "do not oscillate": where there is a minimum, that minimum stays there in time and the same happens for maxima. This is not true. There is identical time dependence, of frequency $\omega$, in both situations; as I said in my answer to your ... 2 Yes, sound waves in a gas, liquid or solid can affect the light passing through it, as the motion of the atoms due to sound waves changes the atomic spacing, and this changes the index of refraction slightly. So the light would be diffracted and some amount of the light would experience a frequency shift up and a frequency shift down by the sound wave ... 6 Any physical phenomenon is potentially capable to cause some change to any other phenomenon, more or less directly. If it was not the case, the physical world could be divided into completely independent realms; there would not be the one single world we call Nature. Practically though, many if not most of the actually existing interactions between systems ... 5 I can answer half your question in that a sound can change the path of light. A change in the density of the air produces a change in the refractive index of the air and so a Schlieren photograph can make this visible. Here is a YouTube video to show a sound wave produced by clapping. 1 The answer has pretty much been given in the comments, but I think a nice pictorial representation might help. The mathematical form of a standing wave is $$y(x) = \sin \left(\frac{2 \pi}{ \lambda} x \right)$$ Here $y(x)$ is the displacement of the string at point $x$. If we plot the waves for the four wavelengths we obtain the following picture I will ... 6 My high school physics teacher was saying that “this is because of interference of sound waves. During the day, there are a lot of sounds and they cancel each other due to interference. But, during the night, there are few sounds and they can reach to our ears without canceling each other”. You need a better high school physics teacher. Temperatures tend ... -1 If we suppose that the phenomenon you describe is related with wave interference. A wave is a kind of mechanical disturbance in the medium through which it is travelling. A sound wave consists of areas of relatively high and low energy, in the form of relatively high and low pressure. To understand how sound is produced, consider a speaker. The cone or ... 3 I would tend to agree that background noise is a factor, but rather than reducing, adding to the sound you are trying to make sense of. So part of that may be how your brain is able to filter the information from the background noise. But at night the temperature is lower and according to this tutorial on sound propagation (which does cite reliable ... 22 You don't. You actually hear the high frequency notes from headphones. The bass really doesn't travel at all well, but the attack noise from the drum or bass guitar is what leaks from headphones. This is why on the tube you hear "tsss tsss tsss tsss" and very little else. From @leftaroundabout's answer on the post that valerio92 linked: Normal ... 3 The classical interference pattern is explained by the equations governing the behavior of light, and energy there is treated as a collective phenomenon, using the Pointing vector Energy transfer in a light beam can be best understood as an emergent phenomenon from the underlying quantum mechanical level. Innumerable photons create the visible interference ... 0 Helmholtz effect occurs when pressure outside the box (i.e. external pressure change). There will be a whistling at certain frequency. The controlling parameters are aperture size and volume of the box. When a loudspeaker inside the box radiates, it just radiates with wave bouncing on the walls and some emits out of the aperture. Resonance occurs at ... 0 Additionally, the fundamental frequency can be obtained by: $$f_0 = v_s/\lambda_0$$ Where $v_s$ is the speed of sound, and $\lambda_0 = 4L$ where $L$ is the length of the air column in the tube. so, $$f_0 = v_s/\lambda_0 = \frac{v_s}{4L}$$ and so, $$f_n = \frac{nf_0}{4L} = \frac{n}{4L}\frac{v_s}{4L} = \frac{nv_s}{16L^2}$$ 0 Water in a glass cup can be thought of as a one end closed organ pipe. The more volume of water you add, you effectively increase the length of the water column thereby reducing the length of the air column. The frequency is dependent on the length which can be understood from the following diagram. As you can see the length of the air column changes ... 0 By changing the amount of water in a tube you are changing the effective length of the tube. When you create a sound wave in the air inside the tube, the wave is reflected at the water surface and at the open end. With waves initially traveling in both directions inside the tube, they interfere with themselves and form standing waves. These standing waves ... 0 In a complex number $z=a+i b$,the real part is $a$ and abs is $\sqrt {a^2+b^2}$,. I am not an expert of acoustics but in general, when pressure is given in complex quantity and you want to measure its magnitude then take absolute. If you are comparing two complex numbers then equate real with real and imaginary with imaginary. 1 Difference between real and absolute value in general: Look at count_to_10 's answer. For acoustics and preasure measurement: Absolute pressure - pressure against perfect vacuum. Real pressure: Usually defined as the pressure against a reference-environment. Also called differential pressure. For example the pressure of the air inside a football against the ... 0 In a sense, the size of the universe limits the wavelength of a photon: any photon that has larger wavelength than the size of the universe, cannot exist entirely within this universe. It is not clear that this can ever be tested, however. In high energy (short wavelength) the lack of a limit to thermal radiated light was an important reason for the ... 0 Brian Dodson has posted a brief explanation to this at Quora. Basically it is suggested that "the largest energy that is sustainable as an electromagnetic wave is approximately 1 MeV, or a wavelength of 0.01 Angstrom." https://www.quora.com/Whats-the-longest-possible-wavelength-lowest-possible-frequency-lowest-possible-energy-of-electromagnetic-radiation ... 2 Let's back up. How do you know that a monochromatic wave of frequency $\omega$ doesn't contain any component at $\sqrt{2} \omega$? It's because these two frequencies are not commensurate: if you plot $\sin(\omega t)$ and $\sin(\sqrt{2} \omega t)$ they'll have no clear relation. The peaks of one look like random points in the other. Then it's clear their ... 2 To say that a wave, say with amplitude given by f(t), has a period $T$ means that not only $f(t+T) = f(t)$, but also that $T$ is the smallest value that has this property. Given that $f(t+T) = f(t)$ then it follows that $f(t + nT) = f(T)$ where $n$ is an integer (for example $f(t+2T) = f(t+T+T) = f(t+T) = f(t)$). 2 The assumptions under the statement are that A. the oscillation count in a wave is conserved and B. the passage of time is universal and uniform. Since the frequency of a wave is the count of oscillations measured within a given time interval by a stationary observer, it remains the same anywhere the wave can reach. On the other hand, the wavelength is how ... 0 You can visualize the situation like this. Lets say there is a source and an observer. They are not moving relative to each other. The source is emitting 10 peaks of wave per second (i.e. frequency is 10 Hz) and observer is observing 8 (only) peaks per time. i.e. due to medium property the frequency is changed. In this scenario 10 peaks enter the medium per ... 0 In a polaroid type of polarising material the molecules are aligned in the same direction throughout the sheet when the material is manufactured. Their electric dipoles are therefore also aligned and thus absorb photons whose electric field is parallel with the electric dipoles (to an extent that depends on cos(theta)^2 where theta is the angle between ... 1 I'm not really sure that my answer fits with your request of a "uniform" background pressure gradient, but anyway it's a related subject. Have you considered perturbations of a fluid parcel in a hydrostatically balanced atmosphere? Conservation of momentum here relates the pressure gradient to the gravitational acceleration. It can be shown that perturbed ... 2 Keep in mind that frequency is both the wave property that is preserved in a change of medium (both wavelength and velocity change) and the physical property of sound waves that we experience as pitch. So the frequencies you hear in both cases are ones produced by the vocal cords. Nor do we expect the gas environment of the vocal cords to have a large ... 3 To a very rough approximation we can say that frequencies of speech are selected by standing waves in the speakers mouth, larynx etc. If they breath helium the speed of these standing waves increases but their wave length, being constrained by dimensions of their body, remains the same. This results in higher frequency sounds produced. (think $f=v/\lambda$... 1 The formula you cite contains all you need to answer your question. The following is the procedure used to assess the suitability of a link for many microwave communication links defined by many RF link planning standards, indeed pretty much any terrestrial radio link at any frequency these days, since we no longer depend on ionospheric reflexions (used by "... 1 The sound wave is a property of the air, which does not care whether the object producing the sound was moving or not. Therefore in both cases the sound travels at the same velocity. That is the answer your book wanted, and while it is almost precisely correct, there is a slight complication. The point is that if an object in motion emits a sound which ... 1 For a travelling wave there is a phase difference between adjacent points in the medium. When two travelling waves superpose the resultant displacement of the medium is the vector sum of the displacements due to the individual travelling waves. At some points the two travelling waves arrive exactly in phase with one another and that is a position of ... 1 I think the Rayleigh-Taylor instability may be considered to be amplified under a pressure gradient. Due to the difference in densities, there is a pressure gradient across the interface which becomes unstable after a certain time. The instability grows exponentially in time according to an amplitude on the order of: $$a\propto\exp\left(\sqrt{A}\right)$$ ... 5 In general the wavenumber is a vector. That is, $e^{i(\vec{k}\cdot\vec{x}-\omega t)}$ is a solution to the wave equation in 3 (or any number) dimensions. We say this solution is a plane wave propagating in the $\hat{k}$ direction with wavenumber $\|\vec{k}\|$ or wavelength $\lambda = 2\pi/\|\vec{k}\|$. So properly the de Broglie relation is $\vec{p} = \hbar \... 0 According to huygens principle each point in the wave front acts as the source of secondary wavelet.By the time, the secondary wavelets from B travel a distance BC, the secondary wavelets from A on the reflecting surface would travel the same distance BC after reflection.Tking A as centre and BC as radius an arc is drawn.From C a tangent CD is drawn to ... 1 Remember that the air at the closed end can't move, so the amplitude at that location is always zero: it is a node for the standing wave. The displacement or amplitude will be maximal at the speaker, it is an antinode. So the length of the pipe determines the wavelength. The distance between node and antinode of a wave is$\lambda/4$, so$l=\lambda/4$. The ... 1 A simple harmonic motion is one where the acceleration (or restoring force) is directly proportional to the displacement and in the opposite direction of the displacement. For a mass$m$on a spring with spring constant$k$, the differential equation describing the motion becomes:$m\frac{\mathrm{d}^2 x}{\mathrm{d}t^2} = -kx$That equation has as solution: ... 1 I do not believe you can show it starting by the equation, but it becomes clear when you solve the differential equation that the two quantities are independent:$\omega$is an arbitrary parameter on the equation of motion, and A is an arbitrary constant that appears in the solution. 1 Yes. Let's consider the phase shifts to be random. Then we can consider each amplitude to be an independent random variable$A_i(t)$. Each one has a variance of $$\text{var}(A_i) = \langle A_i^2 \rangle - \langle A_i \rangle^2 = \langle A_i^2 \rangle \propto I_i$$ because the average amplitude should be zero, and intensity is proportional to amplitude ... 0 You cannot prove that because it is not generally true, or because it depends on your definition of "equal loudness". Consider the case where the sound sources are inside a flaring pipe with a standing wave (say, a modified trumpet). A standing wave is typically drawn as a standing wave of the air displacement in the pipe (illustrated), but note that the ... 1 I remember working this out in the opposite direction, hoping to get a paradox: that the amplitude of the sound produced by a choir of 100 singers is only 10 times the amplitude of the sounds produced by one of them. Each individual sound can be represented as a vector in 2-dimensional space, with length 1 (since all are equally loud) and random phase. The ... 4 They do cancel each other when the peaks of one coincide with the troughs of the other, but since they move in opposite directions, this only happens at specific moments in time (every half period). At other moments (one fourth period later), the peaks of both waves will coincide (and so will their troughs). In the figure above, the red and green wave move ... 0 The reason the waves do not cancel each other out at every position is that to do that the waves from the two sources must be exactly out of phase at every position and that is impossible. Suppose that you have two sources$A$and$B\$ emitting waves of exactly the same frequency and wavelength at each other. Further suppose that the waves are emitted from ... Top 50 recent answers are included	2016-06-29 23:36:04	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6817736625671387, "perplexity": 325.6798039120478}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783397864.87/warc/CC-MAIN-20160624154957-00176-ip-10-164-35-72.ec2.internal.warc.gz"}
https://socratic.org/questions/using-the-domain-values-1-0-4-how-do-you-find-the-range-values-for-relation-y-2x	Using the domain values {-1, 0, 4}, how do you find the range values for relation y=2x-7? Oct 16, 2017 See a solution process below: Explanation: To find the Range of the equation given the Domain in the problem we need to substitute each value in the Range for $x$ and calculate $y$: For $x = - 1$: $y = 2 x - 7$ becomes: $y = \left(2 \times - 1\right) - 7$ $y = - 2 - 7$ $y = - 9$ For $x = 0$: $y = 2 x - 7$ becomes: $y = \left(2 \times 0\right) - 7$ $y = 0 - 7$ $y = - 7$ For $x = 4$: $y = 2 x - 7$ becomes: $y = \left(2 \times 4\right) - 7$ $y = 8 - 7$ $y = 1$ Therefore the Domain is $\left\{- 9 , - 7 , 1\right\}$	2019-11-18 10:46:51	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 18, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5994381904602051, "perplexity": 844.9328781352087}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496669755.17/warc/CC-MAIN-20191118104047-20191118132047-00117.warc.gz"}
http://aas.org/archives/BAAS/v29n5/aas191/abs/S077004.html	Session 77 - Structures of Galaxies. Display session, Friday, January 09 Exhibit Hall, ## [77.04] The Frequency of Inner-truncated Disks W. E. Baggett (CSC/STScI), K. S. J. Anderson (NMSU), S. M. Baggett (STScI) We have performed bulge-disk decomposition on a sample of 659 spiral and lenticular galaxy brightness profiles from the Photometric Atlas of Northern Bright Galaxies (1990, Kodaira, et al.), and have used those fits to analyze the frequency of occurrence of inner-truncated disks. We selected a 76% complete sample containing 218 galaxies by choosing galaxies with profile fits having morphological types from S0 through Sdm (-3.5\leq T\leq 9.5), inclination \leq 60\deg (R_25\leq 0.3), declination north of -25\deg, and a total apparent magnitude brighter than 12.5 (B_T \leq 12.5); the complete sample of 286 galaxies was defined from the Third Reference Catalogue of Bright Galaxies (1991, deVaucouleurs, et al.) using the same selection criteria. Interacting galaxies were excluded from both samples. Excluding brightness profiles with the inner-truncation fitted to outer arms or rings and before completeness corrections, we find that approximately 28% of all galaxies exhibit inner-truncations, a rate that increases to almost 50% for the barred galaxy subsample.	2014-09-20 22:23:08	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8327637314796448, "perplexity": 9631.129439783082}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1410657133568.71/warc/CC-MAIN-20140914011213-00019-ip-10-196-40-205.us-west-1.compute.internal.warc.gz"}
https://www.physicsforums.com/threads/what-temperature-does-p-ge-translates-from-extrinsic-to-intrinsic.702932/	# What temperature does p-Ge translates from extrinsic to intrinsic ? 1. Jul 25, 2013 ### bushel Hi, I have expermentally measured the resistivity, Hall mobility concentration of a p-type germanium sample at the range of 300K-700K. The task I want to accomplish is, given the fact that I know NA= 4.5E17 and acceptor is boron. - How can I calculate the expected transition temperature from extrinsic to intrinsic? - How can I assume the expected resistivity, mobility and concentration values so I can compare them with the experimental values? Which scattering mechanisms will dominate?	2016-02-14 14:46:51	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8403136134147644, "perplexity": 2998.066577621663}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701987329.64/warc/CC-MAIN-20160205195307-00026-ip-10-236-182-209.ec2.internal.warc.gz"}
https://lavelle.chem.ucla.edu/forum/viewtopic.php?f=130&t=44852	## free expansion $\Delta U=q+w$ paytonm1H Posts: 74 Joined: Fri Sep 28, 2018 12:18 am ### free expansion why, in free expansion, are work and heat transfer both equal to zero? skyeblee2F Posts: 62 Joined: Fri Sep 28, 2018 12:23 am ### Re: free expansion Free expansion is an irreversible process in which a gas expands into an insulated evacuated chamber. Because it's insulated, there is no change in temperature and q=0. Because there is no external pressure and gas is moving freely, w=0 too. Roberto Gonzalez 1L Posts: 39 Joined: Fri Sep 28, 2018 12:17 am ### Re: free expansion For free expansion there is no acting external pressure as it is in a vacuum and therefore no work being done. The insulation accounts for a lack of transfer of heat.	2020-08-09 21:02:08	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 1, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2950098514556885, "perplexity": 2130.095757849611}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439738573.99/warc/CC-MAIN-20200809192123-20200809222123-00060.warc.gz"}
https://binaryoptionsywq.web.app/coxisiheligu/fx-e-1-x8.html	Fx = e ^ 1 x Fx = e ^ 1 x 05.01.2021 given f(x)=2x and g(x+2)=3x+1 . Find the ... - Yahoo Answers May 26, 2019 Inverse Function for f(x) = 2 e^{x-2} - YouTube Dec 15, 2010 Equation Calculator & Solver \| Wyzant Resources For multiplication, use the * symbol. A * symbol is not necessary when multiplying a number by a variable. For instance: 2 * x can also be entered as 2x. Similarly, 2 * (x + 5) can also be entered as 2(x + 5); 2x * (5) can be entered as 2x(5). The * is also optional when multiplying with parentheses, example: (x + 1)(x - 1). Order of Operations Integral of the Type e^x[f(x) + f'(x)]dx: Integration ... 9 Nov 2015 Find the complete list of videos at http://www.prepanywhere.com Follow the video maker Min @mglMin for the latest updates. Apr 15, 2011 If g(x) = 3 + x + e^x, find g^-1(4) \| Physics Forums Dec 15, 2009 f(x) = x/(1 - ln(x - 1)) Differentiate f and find the ... Get an answer for 'f(x) = x/(1 - ln(x - 1)) Differentiate f and find the domain of f.' and find homework help for other Math questions at eNotes f(x) = (x-3)/(x+1) find f'(0) f(x) = (x-3)/(x+1) find f'(0 ... Free functions calculator - explore function domain, range, intercepts, extreme points and asymptotes step-by-step Precalculus. Find the Inverse Function f(x)=(1+e^x)/(1-e^x) Replace with . Interchange the variables. Solve for . Tap for more steps Rewrite the equation as . Solve for . Tap for more steps Multiply each term by and simplify. Tap for more steps Multiply each term in by . \frac{d}{dx}\left(\frac{1}{e^{x}}\right) en. image/svg+xml. Related Symbolab blog posts. Advanced Math Solutions – Derivative Calculator, Implicit Differentiation. We’ve covered methods and rules to differentiate functions of the form y=f(x), where y is explicitly defined as Integral of e^(1/x)/x^2 (substitution) - Duration: 1:09. Integrals ForYou 73,444 views. 1:09. Derivatives of Exponential Functions - Duration: 12:03. The Organic Chemistry Tutor 378,468 views. 1) If f(x) = ex, then f '(x) = ex 2) If f(x) = eg(x), then f '(x) = g'(x).eg(x) 3) If f(x) = ln x, then f '(x) = 1/x (x > 0). 4) If f(x) = ln g(x), then f '(x) = g'(x)/g(x) [g(x) > 0]. Sample Jul 19, 2018 Solutions of F(x)=x and F(F(x))=x - Alexander Bogomolny f(x)=x^2 - Wolfram\|Alpha f(x)= 1/(x^2) - Functions Calculator - Symbolab New Brokers Top Brokers	2021-10-21 21:51:41	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6014155745506287, "perplexity": 3915.925967478448}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585441.99/warc/CC-MAIN-20211021195527-20211021225527-00558.warc.gz"}
https://www.transtutors.com/questions/hello-this-is-as-far-as-i-can-get-eith-this-question-it-keeps-saying-incomplete-2816831.htm	# Hello. this is as far as I can get eith this question. it keeps saying incomplete Hello. this is as far as I can get eith this question. it keeps saying incomplete Required information Exercise 1-18 Preparing a statement of cash flows LO P2 Also assume the following: a. The owner's initial investment consists of $37,600 cash and$45.940 in land. b. The company's $17,530 equipment purchase is paid in cash. c. The accounts payable balance of$8,110 consists of the $2.850 office supplies purchase and$5,260 in employee salaries yet to be paid d. The compeny's rent, telephone, and miscellaneous expenses are peid in cash. e. No cash has been coliected on the \$13,520 consulting fees earned K Prev 8 of 8 Next	2020-02-25 11:51:19	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5262677073478699, "perplexity": 4824.100303818917}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875146066.89/warc/CC-MAIN-20200225110721-20200225140721-00368.warc.gz"}
http://vodnisystemy.cz/election-dvmc/31f7cb-most-important-math-for-economics	# most important math for economics Most economics Ph.D. programs expect applicants to have had advanced calculus, differential equations, linear algebra, and basic probability theory. This is a chronological list of some of the most important mathematicians in history and their major achievments, as well as some very early achievements in mathematics for which individual contributions can not be acknowledged.. Where the mathematicians have individual pages in this website, these pages are linked; otherwise more … Part of getting comfortable about using math to do economics is knowing how to go from graphs to the underlying equations, and part is going from equations to the appropriate graphs. So, if you want to be an informed citizen you have to know at least some of the most basic economic concepts. Abstract: This paper suggests that mathematics may have become so important in economicsforfour reasons: (1) to make use of existing human capital, (ii) to attain scientific respectability, (3) to help assure security with respect to claims of truth, and (4) because economics was created primarily by Western econo- Real analysis. The types of math used in economics are primarily algebra, calculus and statistics. Now that I am 41 this year, I am thinking of starting anew in a brand new area of applied mathematics. There is, of course, some flexibility about the order of courses. Its highest-ranked, Universität Mannheim , climbed up from 51-100 to place 44 th this year – while its second-highest ranked, Ludwig-Maximilians-Universität München , retains a spot in the global top 100. Math will be more represented in the future. So, I think the mathematics can be the most needed, important subject we should really know about. You Might Also Like to Read: Common Abbreviations in Economics, Your email address will not be published. You'd be surprised just how relevant multivariable calculus is to both micro and macro-economics. Although the above approach is perfectly fine, I personally feel there is another approach that is better especially for the people 1) who don’t have a solid quantitative background and 2) cannot afford the time to do all the prerequisite math courses. Downloadable as a PDF file, it has four chapters (Linear algebra, Calculus, Constrained Optimization and Dynamics) plus 14 pages of exercises. No prior coursework in economics is assumed, nor is necessarily helpful. When decisions are taken about the distribution of resources, it is beneficial to be able to articulate how a difference in one input will modify the result and conclusively change the utility of individuals. It seeks to understand what drives the accumulation and distribution of capital, the history of inequality, how wealth is concentrated, and prospects for economic growth. Economics Major. Math Preparation for Graduate School. Data Scientists are expert at solving complex problems in business and economics by collecting, managing and analyzing large data bases. By convention, these applied methods are beyond simple geometry, such as differential and integral calculus, difference and differential equations, matrix algebra, mathematical programming, and other computational methods. Through the language of mathematics, theoretical economic models have turned into useful tools for everyday economic policymaking. Future journalists and politicians will speak less and analyze more. This survey, therefore, partly reflects the perspective of one person at a point in time. Economics is an accurate language that is helpful in articulating causal associations between associated variables. what boas book is being referred to? That's why most lawyers, politicians, and wall street executives have extensive knowledge in this field. Required fields are marked *. 1 Two useful references on these topics are: We work hard to simplify and improve the course discovery experience. December 17, 2013. Economics. Mathematical economics is an approach to economic analysis where mathematical symbols and theorems are used. Mathematical Economics vs. Econometrics . economics, we outline the economic problem to be solved and then derive differential equation(s) for this problem. In addition, the selection of the most significant papers in a field will evolve over time, as certain lines of thinking become dominant while others die out. Whether we like that or not, math is becoming an increasingly important factor in a variety of industries. It will at least give you a flavor of the mathematics you might need, and then you can pursue it more with more detailed text if you so desire. JavaScript is disabled. If I want to explain Economics to a physicist say, what are considered to be the most important . If you're preparing for graduate school, the most important courses in your curriculum aren't in the economics department at all. Math is the most important. Economics is usually taught more as a social science. Modern economics is analytical and mathematical in structure. For instance, a great deal of pure mathematics, such as real analysis, appears in microeconomic theory. Check the prerequisites at your school to figure out the right sequencing. You don't know how the … ie; most schools don't require anything beyond calc and stat 1 or 2 unless your are an engineering/physics major. UC Berkeley. In response to my previous post offering advice to aspiring economists, a student emails me: Since the time allocation is limited, I can take only some math courses and the problem is that I am not sure which courses are most important for a successful economist and which course I … Economic literacy improves the competence of each individual for making personal and social decisions about the multitude of economic issues that will be encountered over a lifetime. MYTHS OF ECONOMICS: •Economics is ALL math •Economics is just a business degree, but business is more marketable •Economics only deals with money •It wasn't interesting in high school, so it won't be interesting now. linear algebra, calculus, and differential equations provide a good base for a lot of engineering and science though. This is obviously important and relevant information, but not decisive on its own. in economics, but you might even get a B.S. with just several math courses. Which math courses are the most useful for grad. Find out how and why mathematics are used in microeconomics, what its limitations are and the kinds of math skills that economics students should have. This amount of math should at the very least be sufficient to get a B.A. This course is an important part of the undergraduate stage in education for future economists. Differential Equations. Basically every field has stats and error analysis. Boas - Mathematical Methods in the Physical Sciences. Obviously economics requires a certain degree of competency in stat. This amount of math should at the very least be sufficient to get a B.A. As such, it is probably worth owning as a reference book. All of the other characteristics here depend upon you caring about your studies and your specific topic. A 1991 report by the American Economic Association presented economics Ph.D students with the following list of mathematical topics: high school mathematics only basic calculus and linear algebra Equity premium puzzle: The equity premium puzzle is thought to be one of the most important outstanding questions in neoclassical economics. Clarity is the rst priority in economics writing. I have found Fourier series and differential equations particularly useful as a physics undergraduate. Upper-division math and statistics courses for those who are adequately prepared (in order of importance) Math 110, Linear Algebra; Math 104, Introduction to Analysis; Stat 134, Concepts of Probability; Stat 150, Stochastic Processes Includes instruction in economic theory, micro- and macroeconomics, comparative economic systems, money and banking systems, international economics … Multivariable Calculus. Trig, pythagorean theorem, and arithmetic. What could be more useful than good old geometry and basic algebra? You'd be surprised just how relevant multivariable calculus is to both micro and macro-economics. In finance, probability and calculus are very important, as well as linear algebra (at points). Average\, Total\, Cost\, (ATC)\, =\, \frac {Total\, Cost} {Q} Average\, Variable\, Cost\, (AVC)\, =\, \frac {Total\, Variable\, Cost} {Average} Fixed Cost (AFC) = ATC – AVC. What fields of math progress the most continuously ? Going into astronomy/astrophysics after EE, Changing PhD/ collab PhD with a different department. Also Check: Accounting Formulas for Commerce Students. It is founded on the basis that over the last one hundred years or so the average real return to stocks in the US … Criticism of Math in Economics. uate courses economic arguments are often made using graphs. Which economy did better year-over-year (YOY) in the fourth quarter of 2018 compared to the fourth quarter of 2017? Mathematical Statistics. Most Important Topics in Math for Econ PhD. … Applying economics in everyday life. In order to be a successful economics student, or any kind of student for that matter, it's particularly important that you're interested in and engaged with your subject. Prerequisites The major has prerequisites in both mathematics and economics: MATH 120; ECON 110 or 115; and ECON 111 or 116. As most economics student will attest to, modern economic research certainly doesn't shy away from mathematical modeling, but its application of the math differs within the various subfields. Economics affects all parts of our daily lives. See: Applying economics in everyday life. Algebra is used to make computations such as total cost and total revenue. Which electrical engineering subdiscipline uses the most math? Most business/finance major requirements don't go too far beyond the lower level math courses. Behavioural economics examines the reasons why we make decisions. In graduate courses we tend to use equations. The economy significantly affects people's quality of life within a society, so completing a sociology minor may also be useful for economics majors. I was an average mathematics major as an undergraduate with a G.P.A. See: Behavioural economics. With the second-most entrants in the economics university ranking this year is Germany, which has 24 representatives in the economics university ranking this year. Future journalists and politicians will speak less and analyze more. New Zealand performed better. This is an interdisciplinary field of study that is rooted in statistics and computer coding, and data science is acknowledged to one of the most important new careers for the 21st century. R analysis is often cited as the most important math course for economics PhD preparation, according the California State University website. Future police and military personnel … Effective planning ahead protects fish and fisheries, Polarization increases with economic decline, becoming cripplingly contagious, https://www.amazon.com/dp/0471198269/?tag=pfamazon01-20&tag=pfamazon01-20, https://www.physicsforums.com/showthread.php?t=183900, https://www.physicsforums.com/showthread.php?t=185965, https://www.physicsforums.com/showthread.php?p=1416394, http://www.math.uni-hamburg.de/home/gunesch/Entropy/stat.html. Mathematical Economics majors are well suited for data science jobs as the major requires students to take introductory statistics and econometrics courses, which can then be su… 12. Any advanced physics students/academics that have failed? The "real" refers to the discipline's focus on real numbers, which excludes imaginary numbers and infinity. Numerical method approaches from applied mathematics are also used a great deal in most subfields of economics. Description: A general program that focuses on the systematic study of the production, conservation and allocation of resources in conditions of scarcity, together with the organizational frameworks related to these processes. However those probably fall under Calculus. well, if by real life you mean day to day life, or working in a non-technical field, algebra should do just fine. Economics is such an undervalued field that incorporates all disciplines such as math and sciences. Proponents of this approach claim that it allows the formulation of theoretical relationships with rigor, generality, and simplicity. What are the most math heavy subjects besides Math? LIST OF IMPORTANT MATHEMATICIANS – TIMELINE. The desirable style of writing is exempli ed by most … Having a good understanding of mathematics is crucial to success in economics. Whether we like that or not, math is becoming an increasingly important factor in a variety of industries. That is: Instead of going by the subjects, go by the topics. KNOWLEDGE CHECK Here is the most important economic data for New Zealand and Switzerland. It gives students skills for implementation of the mathematical knowledge and expertise to the problems of economics. berkeleytime. This means that undergraduates thinking about graduate school in economics should take 1-2 mathematics courses each semester. school? Use the two tables to investigate. Depends what you plan to do after your degree. Introduction to Economics. However, a … Math and science are part of a major contribution to children's education, so art isn't as important as math and science. Modern economists have examined economic forces behind everyday social issues. Probability Theory. Economics applications are given throughout the text. Mathematical economics is the application of mathematical methods to represent theories and analyze problems in economics. Fields like econometrics seek to analyze real-world economic scenarios and activity through statistical methods. Formulas for Economics plays an important role in all the students educational period. Journalistic writing is characterized by the lack of an analytical tone. Does anyone know which college math course uses conics the most? I have your basic single/multivariable calc, RA 1, matrix algebra, familiar with diff eq, trying to figure out what courses to prioritize. Different from most standard textbooks on mathematical economics, we use computer simulation to demonstrate motion of economic systems. I would recommend the Boas text without hesitation. It is simply not true that much of mathematics for economics could be learned by a competent physicist in two months. The book is dated 1999. For a better experience, please enable JavaScript in your browser before proceeding. Particularly in the current situation, using the technology many of us have all around us is a great way to stay on top of your economics research and remain knowledgeable while inside, staying safe during the coronavirus COVID-19 pandemic. Pick up an engineering maths book -- you'll find everything from calculus to linear algebra (as mathwonk suggested), through to differential equations and laplace/fourier transforms... without getting bogged down by proofs. I'm surprised no one's mentioned statistics yet! Yeah, imo pattern recognition and stats are probably the most useful. Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Most important, economics provides the tools to … (6) Students with a score of 4 should complete Math 1B. That said, if you are going to major in economics, you should consider learning a little calculus. Whether there is a case for economic literacy, however, is not the most important … The Economics and Mathematics major is intended for students with a strong interest in both mathematics and economics and for students who may pursue a graduate degree in economics. Math 1A - Calculus -- [4 units] Course Format: Three hours of lecture and three hours of discussion per week. We list five key topics below. An Example. and important articles. Citation counts are biased in favor of subfields of economics with the largest popu-lations. Offered by National Research University Higher School of Economics. Economic Education. Calculus is used to find the derivatives of utility curves, profit maximization curves and growth models. Although economics graduate programs have varying admissions requirements, graduate training in economics is highly mathematical. Behavioral economics looks specifically at the role of human psychology in economic theory, and completing a minor in psychology will provide you with a good foundation in this area. A 1991 report by the American Economic Association presented economics Ph.D students with the following list of mathematical topics: high school mathematics only Most undergraduate students, particularly those coming from North America, are often shocked by how mathematical graduate programs in economics are. The math goes beyond basic algebra and calculus, as it tends to be more proofs, such as "Let (x_n) be a Cauchy sequence. A little math ability goes a long way; the more advanced math you bring in, the less additional knowledge that will get you. Game Theory. Economist ee93. Economics is such an undervalued field that incorporates all disciplines such as math and sciences. In order to do this it is often desirable determine cause and effect relationships and to quantify variables. This book is somewhat basic by the standards of modern graduate economics courses, but is well written and organized, and touches on most of the important topics. You don't know how the world truly works until you study the economy. Economics is usually taught more as a social science. To know more, stay tuned to BYJU’S. That's why most lawyers, politicians, and wall street executives have extensive knowledge in this field. By useful, I mean math that is important in the real world. It is difficult to perform economic analysis without some rudimentary knowledge of fundamentals. The above mentioned is the concept, that is elucidated in detail about ‘Formulas for Economics’ for the Commerce students. Every David Rittenhouse Lab. Unless people intend to become artists - of which they should still be more focused on studies - then art would be equally, if not more, important than math and science. Statistics allows economists to make forecasts and determine the probability of an occurrence. in economics, but you might even get a B.S. Probability I know is very important. with just several math courses. Additional economics courses that emphasize theory and quantitative methods, such as Economics 103, 104, and 142. This is an interdisciplinary field of study that is rooted in statistics and computer coding, and data science is acknowledged to one of the most important new careers for the 21st century. Find out how and why mathematics are used in microeconomics, what its limitations are and the kinds of math skills that economics students should have. Real Analysis. As a starting point we used citation counts and numbers of searches in JSTOR. Why do suppliers sell at the price they do and what makes buyers buy at a particular price? the classic carus monograph by ralph boas, primer of real functions? Important Economics Formulas. By these reasons, I think math is the most important subject. I have about the average IQ for economists, but I know nothing about economics. There is not enough information to tell ali They performed identically. Useful does not mean interesting or worthy of a math paper. Economics Job Market Rumors » Economics » Questions from prospective grad students. Sociology. One of the most basic concepts in Economics is the study of Demand & Supply. What is the most efficient way to gain mathematical insight? Logic does play a strong role in economics, though, so you will need to be able to think along logical lines, just as you would in math classes. Real analysis is especially valued by economics PhD programs because of its emphasis on writing and understanding proofs. Most schools will offer some sort of "calculus for business majors" that generally satisfies the mathematical demands of undergraduate level economics. Plus it allows individuals to think critically. Real analysis builds on calculus and provides a foundation for the kinds of applied mathematics that are useful in economic study. Most Important Topics in Math for Econ PhD. Many applicants have completed a course in real analysis. Future police and military personnel will use technology that is certainly an invention of scientists. But is Real Analysis and Abstract Algebra really necessary to study the more "useful" applied math? What math is used the most in cosmology/astrophysics? Your phone is one of the most useful tools you’ve got, whether you’re studying or working in the field. Below, you will nd some notes about the economics style of writing. However, one of the significant responsibilities in economics is the assessment of options to decide which valid contents are given purposes or intentions. It is impossible to do economic… Highly regarded as one of the most important economics books, "Capital in the Twenty-First Century" by Thomas Piketty, a French economist, focuses on wealth and income inequality. In order to do this, it is usually beneficial to ascertain reason and influence associations and to quantify variables. most important papers will inevitably differ from another’s. More dramatically, the department would also be adding an entirely new major, Mathematical Economics (14-2), which would focus more on the abstract, mathy subjects in economics, incorporate a foundation of pure mathematics, and allow electives in Course 18 to count towards the degree. 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Undergraduates thinking about graduate school in economics, your email address will not published!	2021-04-12 22:59:33	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.34758809208869934, "perplexity": 1663.435549279453}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038069267.22/warc/CC-MAIN-20210412210312-20210413000312-00637.warc.gz"}
https://math.stackexchange.com/questions/57543/new-to-matlab-having-some-trouble-solving-systems-differential-equations	# New to MATLAB, having some trouble solving systems differential equations I need to numerically solve $$\frac{\mathrm dx}{\mathrm dt} = y - x^3$$ $$\frac{\mathrm dy}{\mathrm dt} = -x - y^3$$ with initial conditions $x_0,y_0 = 1,0$ for $t$ from $1$ to $100$. I don't know how to do this. I'd also like to plot this along with some other functions and don't know how to plot multiple functions on one plot. The specific functions are $$A = x^2 + y^2$$ and $$B = 1/t$$ I'm quite new to MATLAB, so I need some help. • You should be able to adapt the answer I gave you here to your current problem. Aug 15 '11 at 8:58 • As for plotting... Aug 15 '11 at 9:00 • I don't see how in the first case. I tried dy(1) = y(2) - y(1).^3, dy(2) = - y(1) - y(2).^3. That didn't work. Aug 15 '11 at 9:29 • "That didn't work." isn't very informative for me. What did you enter, and what did MATLAB spit out? Aug 16 '11 at 3:14 Let $z = [x,y]$, observe $z' = [x',y'] = [z(2)-z(1)^3,-z(1)-z(2)^3]$. Your initial condition is $z_0 = [x_0,y_0] = [1,0]$ The trick is now we have a vector $z$ and we know its derivative $z'$ and initial values, we can certainly solve it (numerically). Matlab solvers like ode45 or ode23 can easily tackle this, or you can write your own solver using Euler's method and so on. Example Code f = @(t,z)[z(2)-z(1)^3,-z(1)-z(2)^3]; [t z] = ode45(f,[1 100],[1 0]); Remember now your $z$ vector contains both $x$ and $y$. $z(:,1)$ will give you $x$ vector in correspondence to $t$ vector and $z(:,2)$ will give the $y$ vector You can for example plot(t,z(:,1)) to see how x evolve along t or plot(t,z(:,2)) to see how y evolve along t. Or plot(z(:,1),z(:,2)) to see the phase plot. Regarding plotting against $A$ and $B$, I think you might have misunderstood the problem. We have x(t) and y(t) and we solved them, this is different from solving z(x,y,t). Don't get them confused. Your A(x(t),y(t)) is basically a parametrized curve in 2 dimension (z against t). You can plot by A = @(x,y) x.^2+y.^2; plot(t,A(z(:,1),z(:,2)); And your B has nothing to do x(t) and y(t), it only depends on t, so again a 2 dimension plot. B = @(t) 1./t; plot(t,B(t)); To plot them on the same graph you can do either plot(t,A(z(:,1),z(:,2),t,B(t)); or plot(t,A(z(:,1),z(:,2)); hold on; plot(t,B(t)); hold off;	2022-01-23 08:44:07	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7418510317802429, "perplexity": 385.7444210741111}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320304217.55/warc/CC-MAIN-20220123081226-20220123111226-00183.warc.gz"}
http://math.stackexchange.com/questions/41200/trouble-finding-the-right-dtft-pair	Trouble finding the right DTFT pair i have a periodic signal $x[n] = \cos (\frac{2 \pi}{10}n)$. I found this DTFT pair: That's the only pair for a cosine function i found. But what is $\delta _{2\pi}$ ? Is it just a Dirac $\delta$ function for every period ? And the $\pi$ before the brackets is just for a higher amplitude of the impulse ? - Ultimately, the $\pi$ before the brackets is a convention. You can change how you define the Fourier transform, but then you'd also need to change the inverse transform. Normally, $\delta_{2\pi}$ denotes a periodic delta function, meaning that $\delta_{2\pi}(x) = \delta_{2\pi}(y) \Leftrightarrow x = y + 2 k \pi$ for some integer $k$. I can't really see if that should be the case here, but you should verify it. – Gerben May 25 '11 at 7:52 Well, it just says calculate the periodic signal $x[n]$. – madmax May 25 '11 at 8:35 The DTFT is always $2\pi$-periodic, since changing the frequency by $2\pi$ doesn't change the phase at the integers. The cosine is a sum of two exponentials and has frequency components at $\theta_0$ and $-\theta_0$. The $\delta_{2\pi}$ function replicates these with period $2\pi$. As was already stated in the comments, the factor $\pi$ is a matter of convention; it gives the right normalization for the definition used e.g. in the Wikipedia article.	2015-11-29 07:39:16	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9641711711883545, "perplexity": 159.62747605599534}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398456975.30/warc/CC-MAIN-20151124205416-00288-ip-10-71-132-137.ec2.internal.warc.gz"}
https://stat.ethz.ch/R-manual/R-devel/library/MASS/html/corresp.html	corresp {MASS} R Documentation Simple Correspondence Analysis Description Find the principal canonical correlation and corresponding row- and column-scores from a correspondence analysis of a two-way contingency table. Usage corresp(x, ...) ## S3 method for class 'matrix' corresp(x, nf = 1, ...) ## S3 method for class 'factor' corresp(x, y, ...) ## S3 method for class 'data.frame' corresp(x, ...) ## S3 method for class 'xtabs' corresp(x, ...) ## S3 method for class 'formula' corresp(formula, data, ...) Arguments x, formula The function is generic, accepting various forms of the principal argument for specifying a two-way frequency table. Currently accepted forms are matrices, data frames (coerced to frequency tables), objects of class "xtabs" and formulae of the form ~ F1 + F2, where F1 and F2 are factors. nf The number of factors to be computed. Note that although 1 is the most usual, one school of thought takes the first two singular vectors for a sort of biplot. y a second factor for a cross-classification. data an optional data frame, list or environment against which to preferentially resolve variables in the formula. ... If the principal argument is a formula, a data frame may be specified as well from which variables in the formula are preferentially satisfied. Details See Venables & Ripley (2002). The plot method produces a graphical representation of the table if nf=1, with the areas of circles representing the numbers of points. If nf is two or more the biplot method is called, which plots the second and third columns of the matrices A = Dr^(-1/2) U L and B = Dc^(-1/2) V L where the singular value decomposition is U L V. Thus the x-axis is the canonical correlation times the row and column scores. Although this is called a biplot, it does not have any useful inner product relationship between the row and column scores. Think of this as an equally-scaled plot with two unrelated sets of labels. The origin is marked on the plot with a cross. (For other versions of this plot see the book.) Value An list object of class "correspondence" for which print, plot and biplot methods are supplied. The main components are the canonical correlation(s) and the row and column scores. References Venables, W. N. and Ripley, B. D. (2002) Modern Applied Statistics with S. Fourth edition. Springer. Gower, J. C. and Hand, D. J. (1996) Biplots. Chapman & Hall. svd, princomp. Examples ## IGNORE_RDIFF_BEGIN ## The signs can vary by platform (ct <- corresp(~ Age + Eth, data = quine)) plot(ct) corresp(caith) biplot(corresp(caith, nf = 2)) ## IGNORE_RDIFF_END [Package MASS version 7.3-58.3 Index]	2023-03-23 14:23:19	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5832698345184326, "perplexity": 2823.4272354306445}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945168.36/warc/CC-MAIN-20230323132026-20230323162026-00661.warc.gz"}
https://www.groundai.com/project/two-level-discretization-techniques-for-ground-state-computations-of-bose-einstein-condensates/	Keywords ## Abstract This work presents a new methodology for computing ground states of Bose-Einstein condensates based on finite element discretizations on two different scales of numerical resolution. In a pre-processing step, a low-dimensional (coarse) generalized finite element space is constructed. It is based on a local orthogonal decomposition of the solution space and exhibits high approximation properties. The non-linear eigenvalue problem that characterizes the ground state is solved by some suitable iterative solver exclusively in this low-dimensional space, without significant loss of accuracy when compared with the solution of the full fine scale problem. The pre-processing step is independent of the types and numbers of bosons. A post-processing step further improves the accuracy of the method. We present rigorous a priori error estimates that predict convergence rates for the ground state eigenfunction and for the corresponding eigenvalue without pre-asymptotic effects; being the coarse scale discretization parameter. Numerical experiments indicate that these high rates may still be pessimistic. Two-Level discretization techniques for ground state computations of Bose-Einstein condensates [2em] Patrick Henning1, Axel Målqvist2 3, Daniel Peterseim4 5 [2em] October 9, 2018 Keywords eigenvalue, finite element, Gross-Pitaevskii equation, numerical upscaling, two-grid method, multiscale method AMS subject classifications 35Q55, 65N15, 65N25, 65N30, 81Q05 ## 1 Introduction Bose-Einstein condensates (BEC) are formed when a dilute gas of trapped bosons (of the same species) is cooled down to ultra-low temperatures close to absolute zero [10, 19, 22, 38]. In this case, nearly all bosons are in the same quantum mechanical state, which means that they loose their identity and become indistinguishable from each other. The BEC therefore behaves like one ’super particle’ where the quantum state can be described by a single collective wave function . The dynamics of a BEC can be modeled by the time-dependent Gross-Pitaevskii equation (GPE) [26, 31, 37], which is a nonlinear Schrödinger equation given by iℏ∂tΨ=−ℏ22m△Ψ+VeΨ+4πℏ2aNm\|Ψ\|2Ψ. (1) Here, denotes the atomic mass of a single boson, the number of bosons (typically in the span between and ), is the reduced Plank’s constant and is an external trapping potential that confines the system. The nonlinear term in the equation describes the effective two-body interaction between the particles. If the scattering length is positive, the interaction is repulsive, if it is negative the interaction is attractive. For there is no interaction and (1) becomes the Schrödinger equation. The parameter changes according to the considered species of bosons. We only consider the case in this paper. We are mainly interested in the ground state solution of the problem. This stationary state of the BEC is of practical relevance, e.g., in the context of atom lasers [35, 30, 41]. The ansatz , with the unknown chemical potential of the condensate and a proper nondimensionalization , reduces (1) to the time-independent GPE −12△u+Vu+β\|u\|2u=λu% withβ=4πaNxs, where denotes the dimensionless length unit and where denotes the accordingly rescaled potential (see, e.g., [8] for a derivation of the time-independent GPE). The ground state of the BEC is the lowest energy state of the system and is therefore stable. It minimizes the corresponding energy E(v)=∫Rd12\|∇v\|2+V\|v\|2+β2\|v\|4dx amongst all -normalized functions. For any -normalized minimizer , is the smallest eigenvalue of the GPE. In this paper, we shall focus on the computation of this ground state eigenvalue. Eigenfunctions whose energies are larger than the minimum energy are called excited states of the BEC and are not stable in general but may satisfy relaxed concepts of stability such as metastability (see [36]). Numerical approaches for the computation of ground states of a BEC typically involve an iterative algorithm that starts with a given initial value and diminishes the energy of the density functional in each iteration step. Different methodologies are possible: methods related to normalized gradient-flows [5, 3, 1, 2, 5, 7, 24, 6, 9, 20], methods based on a direct minimization of the energy functional [8, 11], explicit imaginary-time marching [32], the DIIS method (direct inversion in the iterated subspace) [40, 16], or the Optimal Damping Algorithm [14, 12]. We emphasize that, in any case, the dimensionality of the underlying space discretization is the crucial factor for computational complexity because it determines the cost per iteration step. The aim of this paper is to present a low-dimensional space discretization that reduces the cost per step and, hence, speeds up the iterative solution procedure considerably. In the literature, there are only a few contributions on rigorous numerical analysis of space discretizations of the GPE. In particular, explicit orders of convergence are widely missing. In [44, 17], Zhou and coworkers proved the convergence of general finite dimensional approximations that were obtained by minimizing the energy density in a finite dimensional subspace of . This justifies, e.g., the direct minimization approach proposed in [8]. The iteration scheme is not specified and not part of the analysis. The results of Zhou were generalized by Cancès, Chakir and Maday [13] allowing explicit convergence rates for finite element approximations and Fourier expansions. A-priori error estimates for a conservative Crank-Nicolson finite difference (CNFD) method and a semi-implicit finite difference (SIFD) method were derived by Bao and Cai [4]. In this work, we propose a new space discretization strategy that involves a pre-processing step and a post-processing step in standard finite element spaces. The pre-processing step is based on the numerical upscaling procedure suggested by two of the authors [33] for linear eigenvalue problems. In this step, a low-dimensional approximation space is assembled. The assembling is based on some local orthogonal decomposition that incorporates problem-specific information. The constructed space exhibits high approximation properties. The non-linear problem is then solved in this low-dimensional space by some standard iterative scheme (e.g., the ODA [14]) with very low cost per iteration step. The post-processing step is based on the two-grid method suggest by Xu and Zhou [42]. We emphasize that both, pre- and post-processing, involve only the solution of linear elliptic Poisson-type problems using standard finite elements. We give a rigorous error analysis for our strategy to show that we can achieve convergence orders of for the computed eigenvalue approximations without any pre-asymptotic effects. We do not focus on the iterative scheme that is used for solving the discrete minimization problem. The various choices previously mentioned, e.g., the ODA [14] are possible. Our new strategy is particularly beneficial in experimental setups with different types of bosons, because the results of the pre-processing step can be reused over and over again independent of . Similarly, the data gained by pre-processing can be recycled for the computation of excited states. Other applications include setups with potentials that oscillate at a very high frequency (e.g., to investigate Josephson effects [41, 43]). Here, normally very fine grids are required to resolve the oscillations, whereas our strategy still yields good approximations in low dimensional spaces and, hence, reduces the costs within the iteration procedure tremendously. ## 2 Model problem Consider the dimensionless Gross-Pitaevskii equation in some bounded Lipschitz domain where . Since ground state solutions show an extremely fast decay (typically exponential), the restriction to bounded domains and homogeneous Dirichlet condition are physically justified. We seek (in the sense of distributions) the minimal eigenvalue and corresponding -normalized eigenfunction with −divA∇u+bu+β\|u\|2u =λuin Ω, u =0on ∂Ω. The underlying data satisfies the following assumptions: • If , the domain is an interval. If (resp. ), has a polygonal (resp. polyhedral) boundary. • The diffusion coefficient is a symmetric matrix-valued function with uniform spectral bounds , σ(A(x))⊂[γmin,γmax]for almost all x∈Ω. (2) • is non-negative (almost everywhere). • is non-negative. The weak solution of the GPE minmizes the energy functional given by E(ϕ):=12∫ΩA∇ϕ⋅∇ϕdx+12∫Ωbϕ2dx+14∫Ωβ\|ϕ\|4dxfor ϕ∈H10(Ω). ###### Problem 2.1 (Weak formulation of the Gross-Pitaevskii equation). Find such that a.e. in , , and E(u)=infv∈H10(Ω)∥v∥L2(Ω)=1E(v). It is well-known (see, e.g., [31] and [13]) that there exists a unique solution of Problem 2.1. This solution is continuous in and positive in . The corresponding eigenvalue of the GPE is real, positive, and simple. Observe that the eigenpair satisfies ∫ΩA∇u⋅∇ϕdx+∫Ωbuϕdx+∫Ωβ\|u\|2uϕdx=λ∫Ωuϕdx for all . Moreover, is the smallest amongst all possible eigenvalues and satisfies the a priori bound . ## 3 Discretization This section recalls classical finite element discretizations and presents novel two-grid approaches for the numerical solution of Problem 2.1. The existence of a minimizer of the functional in discrete spaces is easily seen. However, uniqueness does not hold in general. We note that unlike claimed in [44] the uniqueness proof given in [31] does not generalize to arbitrary subspaces of the original solution space. ###### Remark 3.1 (Existence of discrete solutions [13]). Let denote a finite dimensional, non-empty subspace of , then there exists a minimizer with , , and E(uW)=infw∈W∥w∥L2(Ω)=1E(w). If represents a dense family of such subspaces, then any sequence of corresponding minimizers with converges to the unique solution of Problem 2.1. ### 3.1 Standard Finite Elements We consider two regular simplicial meshes and of . The finer mesh is obtained from the coarse mesh by regular mesh refinement. The discretization parameters represent the mesh size, i.e., (resp. ) for (resp. ) and (resp. ). For , let P1(T)={v∈L2(Ω)\|∀T∈T,v\|T is a polynomial of total degree≤1} denote the set of -piecewise affine functions. Classical -conforming finite element spaces are then given by Vh:=P1(Th)∩H10(Ω)% andVH:=P1(TH)∩H10(Ω)⊂Vh. Note that on the fine discretization scale, a different choice of polynomial degree, e.g., piecewise quadratic functions, is possible. This would be a better choice for smooth data that allows for a regular ground state. Our method and its analysis essentially require the inclusion . The discrete problem on the fine grid reads as follows. ###### Problem 3.2 (Reference finite element discretization on the fine mesh). Find with , and E(uh)=infvh∈Vh∥vh∥L2(Ω)=1E(vh). (3) The corresponding eigenvalue is given by . According to Remark 3.1, is not determined uniquely in general. Moreover, is not necessarily the smallest eigenvalue of the corresponding discrete eigenvalue problem. In what follows, refers to an arbitrary solution of Problem 3.2. It will serve as a reference to compare further (cheaper) numerical approximations with. The accuracy of has been studied in [13]. Under the assumption of sufficient regularity, optimal orders of convergence are obtained (cf. (14)). ### 3.2 Preprocessing motivated by numerical homogenization The aim of this paper is to accurately approximate the finescale reference solution of Problem 3.2 within some low-dimensional subspace of . For this purpose, we introduce a two-grid upscaling discretization that was initially proposed in [34] for the treatment of multiscale problems. The framework has been applied to non-linear problems in [27], to linear eigenvalue problems in [33] and in the context of Discontinuous Galerkin [23] and Partition of Unity Methods [28]. This contribution aims to generalize and analyze the methodology to the case of an eigenvalue problem with an additional nonlinearity in the eigenfunction. We emphasize that the co-existence of two difficulties, the nonlinear nature of the eigenproblem itself and the additional nonlinearity in the eigenfunction, requires new essential ideas far beyond simply plugging together existing theories for the isolated difficulties. Let denote the set of interior vertices in . For we let denote the corresponding nodal basis function with and for all . We define a weighted Clément-type interpolation operator (c.f. [15]) IH:H10(Ω)→VH,v↦IH(v):=∑z∈NHvzΦzwith vz:=(v,Φz)L2(Ω)(1,Φz)L2(Ω). (4) It is easily shown by Friedrichs’ inequality and the Sobolev embedding (for ) that a(v,ϕ):=∫ΩA∇v⋅∇ϕdx+∫Ωbvϕdxfor v,ϕ∈H10(Ω) defines a scalar product in and induces a norm on which is equivalent to the standard -norm. By means of the interpolation operator defined in (4), we construct an -orthogonal decomposition of the space into a low-dimensional coarse space (with favorable approximation properties) and a high-dimensional residual space . The residual or ’fine’ space is the kernel of the interpolation operator restricted to , VfH,h:=kernel(IH\|Vh). (5.a) The coarse space is simply defined as the orthogonal complement of in with respect to . It is characterized via the -orthogonal projection onto the fine space given by a(Pfv,ϕ)=a(v,ϕ)for all ϕ∈VfH,h. By defining , the coarse space is given by VcH,h:=PcVH. (5.b) A basis of is given by with . With this definition we obtain the splitting Vh=VcH,h⊕VfH,h. (5.c) Some favorable properties of the decomposition, in particular its -quasi-orthogonality, are discussed in Section 6.2. The minimization problem in the low-dimensional space reads as follows. ###### Problem 3.3 (Pre-processed approximation). Find with , and E(ucH)=infvc∈VcH,h∥vc∥L2(Ω)=1E(vc). The corresponding eigenvalue in is given by . ###### Remark 3.4 (Practical aspects of the decomposition). • The assembly of the corresponding finite element matrices requires only the evaluation of , i.e., the solution to one linear Poisson-type problem per coarse vertex. This can be done in parallel. Section 3.3 below will show that these linear problems may be restricted to local subdomains centered around the coarse vertices without loss of accuracy. Hence, even in a serial computing setup, the complexity of solving all corrector problems is equivalent (up to factor ) to the cost of solving one linear Poisson problem on the fine mesh. • The pre-processing step is independent of the parameter which characterizes the species of the bosons. Hence, the method becomes considerably cheaper when experiments need to be carried out for different types and numbers of bosons. A similar argument applies to variations on the trapping potential . Provided that this trapping potential is an element of (in practical applications it is usually even harmonic and admits the desired regularity) the bilinear form (and the associated constructions of and ) can be restricted to the second order term without a loss in the expected convergence rates stated in Theorems 4.1 and 4.2 below. The trapping potential may then be varied without affecting the pre-processed space . • Once the coarse space has been assembled it can also be re-used in computations of larger eigenvalues (i.e., not only in the ground state solution). ### 3.3 Sparse approximations of VcH,h The construction of the coarse space is based on fine scale equations formulated on the whole domain which makes them expensive to compute. However, [34] shows that decays exponentially fast away from . We specify this feature as follows. Let denote the localization parameter, i.e., a new discretization parameter. We define nodal patches of coarse grid layers centered around the node by ωz,1 :=suppΦz=∪{T∈TH\|z∈T}, (6) ωz,k :=∪{T∈TH\|T∩ωz,k−1≠∅}fork≥2. There exists depending on the contrast but not on mesh sizes and fast oscillations of such that for all for all vertices and for all , it holds ∥PfΦz∥H1(Ω∖ωz,k)≲θk∥PfΦz∥H1(Ω). (7) This result motivates the truncation of the computations of the basis functions to local patches . We approximate from (5.a)-(5.c) with such that a(Ψz,k,v)=a(Φz,v)for all v∈VfH,h(ωz,k). (8) This yields a modified coarse space with a local basis VcH,h,k=span{Φz−Ψz,k\|z∈NH}. (9) The number of non-zero entries of the corresponding finite element matrices is proportional to (note that we expect non-zero entries without the truncation). Due to the exponential decay, the very weak condition implies that the perturbation of the ideal method due to this truncation is of higher order and forthcoming error estimates in Theorems 4.1 and 4.2 remain valid. We refer to [34] for details and proofs. The modified localization procedure from [29] with improved accuracy and stability properties may also be applied. ### 3.4 Post-processing Although and will turn out to be highly accurate approximations of the unknown solution , the orders of convergence can be improved even further by a simple post-processing step on the fine grid. The post-processing applies the two-grid method originally introduced by Xu and Zhou [42] for linear elliptic eigenvalue problems to the present equation by using our upscaled coarse space on the coarse level. ###### Problem 3.5 (Post-processed approximation). Find with ∫ΩA∇uch⋅∇ϕhdx+∫Ωbuchϕhdx=λcH∫ΩucHϕhdx−∫Ωβ\|ucH\|2ucHϕhdx for all . Define . Let us emphasize that this approach is different from [18], where the post-precessing problem has a different structure and where classical finite element spaces are used on both scales. ## 4 A-priori error estimates This section presents the a-priori error estimates for the pre-processed/upscaled approximation with and without the post-processing step. Throughout this section, denotes the solution of Problem 2.1, the solution of reference Problem 3.2, the solution of Problem 3.3 and the post-processed solution of Problem 3.5. The notation abbreviates with some constant that may depend on the space dimension , , , , , , and interior angles of the triangulations, but not on the mesh sizes and . In particular it is robust against fast oscillations of and . ###### Theorem 4.1 (Error estimates for the pre-processed approximation). Assume that . For and as above, it holds ∥u−ucH∥H1(Ω) ≲H2+∥u−uh∥H1(Ω). (10) For sufficiently small (in the sense of Cancès et al. [13]), we also have \|λ−λcH\|+∥u−ucH∥L2(Ω) ≲H3+H∥u−uh∥H1(Ω). (11) ###### Proof. The proof is postponed to Section 6.3. ∎ The additional post-processing improves, roughly speaking, the order of accuracy by one. ###### Theorem 4.2 (Error estimates for the post-processed approximation). Assume that is sufficiently small. The post-processed approximation and the post-processed eigenvalue satisfy: ∥u−uch∥H1(Ω) ≲H3+∥u−uh∥H1(Ω), (12) \|λ−λch\|+∥u−uch∥L2(Ω) ≲H4+CL2(h,H). (13) The constant behaves roughly like and can be extracted from the proofs in Section 6.4.2. ###### Proof. The proof is postponed to Section 6.4. ∎ Let us emphasize that both theorems remain valid for replaced with its sparse approximation (cf. Section 3.3) for moderate localization parameter . We shall discuss the behavior of the finescale errors and . Recall from [13] that for a bounded domain with polygonal Lipschitz-boundary, , and sufficiently small , the fine scale error satisfies the optimal estimate ∥u−uh∥H1(Ω)+h−1∥u−uh∥L2(Ω)+h−1\|λ−λh\| ≲h. (14) The proof in [13] is for constant and hyperrectangle but it is easily checked that the estimates remain valid for any bounded domain with polygonal Lipschitz-boundary and . Under these assumptions our a priori estimates for the post-processed approximation of the ground state eigenvalue summarize as follows \|λ−λch\|≲H4+H2h. Hence, in this regular setting, the choice ensures that the loss of accuracy is negligible when compared to the accuracy of the expensive full fine scale approximation . However, with regard to the numerical experiment in Section 5.1 below, this choice might be pessimistic. Moreover, note that the fine scale error depends crucially on higher Sobolev regularity of the solution whereas our estimates for the coarse scale error require only minimal regularity that holds under the assumption (a)–(d) in Section 2. Thus, we believe that in a less regular setting, even coarser choices of relative to will balance the discretization errors on the coarse and the fine scale. ## 5 Numerical experiments Any numerical approach for the computation of ground states of a BEC involves an iterative algorithm that starts with a given initial value and diminishes the energy of the density functional in each iteration step. In this contribution, we use the Optimal Damping Algorithm (ODA) originally developed by Cancès and Le Bris [14, 12] for the Hartree-Fock equations, since it suits our pre-processing framework. The ODA involves solving a linear eigenvalue problem in each iteration step. However, after pre-processing these linear eigenvalue problems are very low dimensional and the precomputed basis of can be reused for each of these problems making the iterations extremely cheap. The approximations produced by the ODA are known to rapidly converge to a solution of the discrete minimization problem (see [21] and [12] for a proof in the setting of the Hartree-Fock equations). All subsequent numerical experiments have been performed using MATLAB. ### 5.1 Numerical results for harmonic potential In this section, we choose the smooth experimental setup of [13, Section 4, p. 109 and Fig. 2 (bottom)], i.e., , , , and with homogeneous Dirichlet boundary condition. Our method depends basically on three parameters, the coarse mesh size , the fine mesh size , and the localization parameter (cf. Section 3.3 and [29]). In all computations of this section we couple to the coarse mesh size by choosing . This choice is made such that the error of localization is negligible when compared with the errors committed be the fine scale discretization and the upscaling. All approximations are computed with the ODA method as presented in [21, Section 2] with accuracy parameter . #### Comparison with full fine scale approximation In the first experiment, we consider uniform coarse meshes with mesh width parameters of . The fine mesh for the pre- and post-processing has width and remains fixed. We study the error committed by coarsening from a fine scale to several coarse scales , i.e., we study the distance between the ground state of Problem 3.2 and either the coarse scale approximation of Problem 3.3 (with underlying finescale ) or its post-processed version of Problem 3.5. Our theoretical results do not allow predictions about the coarsening error. Most likely, this is an artifact of our theory and we conjecture that and its coarse approximations and are in fact super-close in the sense of H−1∥uh−ucH∥H1(Ω)+∥uh−ucH∥L2(Ω)+\|λh−λcH\| ≲H3, (15) H−1∥uh−uch∥H1(Ω)+∥uh−uch∥L2(Ω)+\|λh−λch\| ≲H4. This assertion is true in the limit . Section 5.1.2 supports numerically the assertion for positive . Figure 1 reports the numerical results. Observe that the experimental rates with respect to displayed in the figures are in fact better than the rates indicated by Theorems 4.14.2 and conjectured in (15). The reason could be the high regularity of the underlying (exact) solution . We do not exploit additional regularity in our error analysis. Similar observations have been made for the linear eigenvalue problem; see [33, Remark 3.3] for details and some justification of higher rates under additional regularity assumptions. Our implementation is not yet adequate for a fair comparison with regard to computational complexity and computing times between standard fine scale finite elements and our two-level techniques. However, to convince the reader of the potential savings in our new approach, let us mention that the number of iterations of the ODA were basically the same for both approaches in all numerical experiments. This statement applies as well to more challenging setups with larger values of (see, e.g., Section 5.2 below) where ODA needs many iterations to fall below some prescribed tolerance. We, hence, conclude that the actual speed-up of our approach is truly reflected by the dimension reduction from to up to the overhead induced by slightly denser (but still sparse) finite element matrices on the coarse level. #### Comparison with high-resolution numerical approximation In the second experiment we investigate the role of the fine scale parameter . We consider uniform coarse meshes with mesh width parameters and uniform fine meshes for for pre- and post-processing computations. The error between the exact eigenvalue and coarse approximations and is estimated via a high-resolution numerical solution on a mesh of width . The results are reported in Figure 2. For the sake of clarity, we show eigenvalue errors only. We conclude that it would have been sufficient to choose to achieve the accuracy of by our coarse approximation scheme with post-processing. ### 5.2 Numerical results for discontinuous periodic potential This section addresses the case of a BEC that is trapped in a periodic potential. Periodic potentials are of special interest since they can be used to explore physical phenomena such as Josephson oscillations and macroscopic quantum self-trapping of the condensate (c.f. [41, 43]). Here we use a potential that describes a periodic array of quantum wells that can be experimentally generated by the interference of overlapping laser beams (c.f. [39]). Let , , and . Given and , define b0(x1,x2):={0forx∈]14,34[2btelse and the potential . Consider the same numerical setup as in Section 5.1.1 (i.e., we draw our attention again to the coarsening error ) with the exception that we were able to reduce the localization parameter without affecting the best convergence rates possible. Figure 3 reports the errors between the finescale reference discretization and our coarse approximations. For the discontinuous potential, the experimental rates (with respect to ) are slightly worse than those ones observed in Section 5.1.1. However, they are still better than the rates indicated by Theorems 4.14.2 and conjectured in (15). ## 6 Proofs of the main results In this section we are concerned with proving the main theorems. ### 6.1 Auxiliary results An application of [13, Theorem 1] shows that and both converge to in , which guarantees stability. ###### Remark 6.1 (Stability of discrete approximations). For sufficiently small we have ∥uh∥H1(Ω) ≤√λh≲√λand (16) \|\|uh\|\|L4(Ω) ≤(λhβ)14≲(λβ)14. (17) The same results hold for replaced by and replaced by for and sufficiently small. The bound (16) is obvious using and the -convergence which guarantees . Estimate (17) directly follows from the definitions of and which gives us . ###### Remark 6.2 (L∞-bound). The solution of Problem 2.1 is in . This follows from the uniqueness of which shows that it is also the unique solution of the linear elliptic problem ∫ΩA∇u⋅∇ϕ+buϕdx=∫Ω~fϕdxfor all ϕ∈H10(Ω), where . Standard theory for linear elliptic problems (c.f. [25, Theorem 8.15, pp. 189–193]) then yields the existence of a constant only depending on , and such that ∥u∥L∞(Ω)≤c(∥u∥L2(Ω)+γ−1min∥~f∥L2(Ω))≲1+∥u∥3L6(Ω)≲1+∥u∥3H1(Ω). (18) ### 6.2 Properties of the coarse space VcH,h Recall the local approximation properties of the weighted Clément-type interpolation operator defined in (4), H−1T∥v−IH(v)∥L2(T)+∥∇(v−IH(v))∥L2(T)≤CIH∥∇v∥L2(ωT) (19) for all . Here, is a generic constant that depends only on interior angles of but not on the local mesh size and . Furthermore, for all and for all it holds ∫ωz(v−vz)2dx≤CIHH2∥∇v∥2L2(ωz), (20) where and is given by (4). ###### Lemma 6.3 (Properties of the decomposition). The decomposition of into and (stated in Section 3.2) is -orthogonal, i.e., Vh=VH⊕VfH,hand(vH,vf)L2(Ω)=0for allvH∈VH,vf∈VfH,h. (21) The decomposition of in and is -orthogonal Vh=VcH,h⊕VfH,handa(vc,vf)=0for allvc∈VcH,h,vf∈VfH,h (22) and -quasi-orthogonal in the sense that (vc,vf)L2(Ω)≲H2∥∇vc∥L2(Ω)∥∇vf∥L2(Ω). (23) ###### Proof. The proof is verbatim the same as in [33]. ∎ The following lemma estimates the error of the best-approximation in the modified coarse space . The lemma is also implicitly required each time that we use the abstract error estimates stated in [13, Theorem 1]. These estimates require a family of finite-dimensional spaces that is dense in . This density property is implied by the following lemma. ###### Lemma 6.4 (Approximation property of VcH,h). For any given with it holds infvcH∈VcH,h∥v−vcH∥H1(Ω)≲H∥divA∇v+bv∥L2(Ω)+infvh∈Vh∥v−vh∥H1(Ω). ###### Proof. Given , define (since ) and let denote the corresponding finite element approximation, i.e., a(vh,ϕh)=(fv,ϕh)L2(Ω)% for all ϕh∈Vh. With , Galerkin-orthogonality leads to ∥A1/2∇(vh−vcH)∥2L2(Ω) a(vh,Pfvh)=(fv,Pfvh)L2(Ω) γ−1/2min∥Hfv∥L2(Ω)∥A1/2∇(vh−vcH)∥L2(Ω). This, the triangle inequality, and norm equivalences readily yield the assertion.∎ Next, we show that there exists an element in the space that approximates in the energy-norm with an accuracy of order O. ###### Lemma 6.5 (Stability and approximability of the reference solution). Let solve Problem 3.2. Then it holds ∥Pcuh∥H1(Ω) ≤√λh, ∥Pcuh−uh∥H1(Ω)=∥Pfuh∥H1(Ω) ≲H2+H∥u−uh∥H1(Ω), (Pcuh,Pfuh)L2(Ω) ≲(H2+H∥u−uh∥H1(Ω))H2. ###### Proof. Recall . Since is a projection, we have ∥Pcuh∥2H1(Ω)≤∥uh∥2H1(Ω)=λh∥uh∥2L2(Ω)−β∥uh∥4L4(Ω)≤λh. The -orthogonality of (5.c) further yields ∥Pfuh∥2H1(Ω)=a(Pfuh,Pfuh)=a(uh,Pfuh)=λh(uh,(1−IH	2018-11-20 07:25:20	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8945632576942444, "perplexity": 609.8982018539959}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039746301.92/warc/CC-MAIN-20181120071442-20181120093442-00206.warc.gz"}
https://zbmath.org/?q=1024.57022	## Decomposing four-manifolds up to homotopy type.(English)Zbl 1024.57022 Let $$M$$ be a closed connected orientable topological $$4$$-manifold with fundamental group $$\pi_1=\pi_1(M)$$. This paper investigates the question of when $$M$$ is homotopy equivalent to a connected sum of the form $$P\#M'$$, with $$M'$$ a simply-connected closed $$4$$-manifold not homeomorphic to the $$4$$-sphere. The main Theorem 1 provides a characterization, given a degree one map $$f:M\to P$$ which induces an isomorphism on $$\pi_1$$. In this case, $$M$$ is homotopy equivalent to $$P\#M'$$ if and only if $\lambda_M^\Lambda \|_{K_2(f,\Lambda)}\cong \lambda_M^{\mathbb Z} \|_{K_2(f,{\mathbb Z})}\otimes_{\mathbb Z} \Lambda,$ where $$\Lambda={\mathbb Z}[\pi_1]$$, and $$\lambda_M^\Lambda \|_{K_2(f,\Lambda)}$$ and $$\lambda_M^{\mathbb Z} \|_{K_2(f,{\mathbb Z})}$$ are the intersection forms of $$M$$ with coefficients $$\Lambda$$ and $${\mathbb Z}$$, respectively, restricted to the kernels of the maps induced by $$f$$ on second homology. After giving two different proofs of Theorem 1, the authors list some consequences. They give various splitting theorems, providing algebraic conditions for $$M$$ to be identified, up to homotopy equivalence and in certain cases up to homeomorphism, as a connected sum of $$M'$$ with common manifolds such as $$S^1\times S^3$$, and $$F\times S^2$$ with $$F$$ a closed aspherical surface. A further consequence concerns $$4$$-manifolds $$M$$ with $$\pi_1$$ torsion free and infinite, and $$\pi_2(M)$$ trivial. Hillman has conjectured that any such $$M$$ is homeomorphic to a connected sum of aspherical closed $$4$$-manifolds and $$S^1\times S^3$$ factors. Theorem 1 implies that $$M$$ is homotopy equivalent to a connected sum of that form, verifying Hillman’s conjecture up to homotopy type. ### MSC: 57N65 Algebraic topology of manifolds 55P10 Homotopy equivalences in algebraic topology 57Q10 Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc. 57R67 Surgery obstructions, Wall groups 57N13 Topology of the Euclidean $$4$$-space, $$4$$-manifolds (MSC2010) ### Keywords: four-manifolds; homotopy type; decomposition; intersection form Full Text:	2022-07-06 19:17:39	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8565831780433655, "perplexity": 261.4694948200541}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104676086.90/warc/CC-MAIN-20220706182237-20220706212237-00595.warc.gz"}