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http://www.federica.unina.it/agraria/mathematics-exercises/domain-given-function/	# Francesco Giannino » 4.Domain of a given function ### Domain of Basic Functions A table of domain of basic and useful functions is shown. Algebraic Functions ### Domain of Basic Functions A table of domain of basic and useful functions is shown. Trigonometric Functions ### Domain of Basic Functions A table of domain of basic and useful functions is shown. Logarithmic and Exponential Functions ### Domain of functions resulting from algebraic operation Let us consider two real functions f and g with domains D1 and D2 respectively. Clearly, these domains are real number set R or subsets of R : $D_{1},D_{2}\in R$ The addition and subtraction of two real functions are defined as follows $f\pm g: D_{1}\cap D_{2}\rightarrow R$ Multiplication The product of two real functions is defined as follows: $f g: D_{1}\cap D_{2}\rightarrow R$ ### Domain of functions resulting from algebraic operation Quotient The quotient of two real functions involves rational form as f(x)/g(x), which is defined for g(x) ≠ 0. We need to exclude values of x for which g(x) is zero. Hence, the quotient of two real functions is defined as follows: $\frac{f}{g}: D_{1}\cap D_{2}-\{x\|g(x)\neq 0\}\rightarrow R$ ### Exercises Exercise 1. Let two functions be defined as follows: $f(x)=\sqrt{x}, \, g(x)=x^{2}-5x+6$ Find domains of fg and f/g. Solution. The function f(x) is defined for all non negative real numbers. Hence, its domain is: x ≥0 The function, g(x), being a real quadratic polynomial, is defined for all real values of x. Hence, its domain is R. The domain of fg is the intersection of two domains, which is a non-negative interval. Domain of f Domain of g Domain of g ### Exercises The domain of quotient function f/g excludes values of x for which g(x) is zero. In other words, we have to exclude roots of g(x) from domain. Then: $g(x)=x^{2}-5x+6=0\Rightarrow g(x)=(x-2)(x-3)=0$ $\Rightarrow x_{1,2}=2,3$ Hence domain of f/g is intersection of two domains, without two roots of g: $[0,+\infty[-\{2,3\}$ Domain of f/g ### Exercises Exercise 2. Find the domain of the function $f(x)= \sqrt{x-1}+\sqrt{x^{2}+x+1}=f_{1}+f_{2}$ Solution. Given function can be considered to be the addition of two separate functions. Domain of the first is: $x-1\geq 0\Rightarrow x\geq 1$ Now, for the second function, we use the sign rule: $x^{2}+x+1\geq 0$ In this case, coefficient of x2 is positive and discreminant Δ is negative. Hence, the second function is positive for all real x. Domain of given function is intersection of two domains: [1, +∞[ Domain of f1 Domain of f2 Domain of f=f1+f2 ### Exercises Exercise 3. Find the domain of the function given by $f(x)=\log_{10}\frac{x^{2-5x+6}}{x^{2}+5x+9}$ Solution. The argument of logarithmic function is a rational function. We need to find values of x such that the argument of the function evaluates to a positive number. Hence, we put: $\frac{x^{2}-5x+6}{x^{2}+5x+9}>0$ The quadratic expression in the denominator has no real roots and as such can not be factorized in linear factors. Thus, the quadratic expression in the denominator is positive for all values of x as the coefficient of the squared term is positive. ### Exercises Clearly, sign of rational function is the same as that of the quadratic expression in the numerator: $x^{2}+5x+9>0$ The roots of the quadratic expression in the numerator (see exercise 1) are: $x_{1,2}=2,3$ And the quadratic expression in the numerator evaluates to positive for intervals beyond root values: $x<2 \, \text{or} \,x>3$ Then domain of the given function is: $]-\infty,2[\cup ]3,+\infty[$ and it is sketched below. ### Exercises Exercise 4. Find the domain of the function given by $f(x)=\sqrt{\log_{10}\frac{6x-x^{2}}{8}}$ Solution. The function is a square root of a logarithmic function. On the other hand, argument of logarithmic function is a rational function. In order to find the domain of the given function, we first determine what values of x are valid for logarithmic function. Then, we apply the condition that expression within the square root should be a non-negative number. The domain of the given function is the intersection of intervals of x obtained for each of these conditions. ### Exercises Now, we know that argument of logarithmic function is a positive number and the whole logarithmic expression within the square root should be a non-negative number. This implies that we need to find the interval of x for which: $\begin{cases} \frac{6x-x^{2}}{8} > 0 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (1) \\ \log_{10}\frac{6x-x^{2}}{8}\geq 0 \,\,\,\,\,\,\, (2)\\ \end{cases}$ Now, we solve the first inequality (1) in the system: $\frac{6x-x^{2}}{8} > 0\Rightarrow 6x-x^{2} > 0$ Now, multiply the inequality by -1; therefore, inequality sign is reversed: $x^{2}-6x < 0$ ### Exercises Roots of corresponding quadratic equation x2 – 6x = 0 are x = 0, 6. It means that the interval satisfying the inequality (1) of the system is : 0 < x < 6. Now, we solve the second inequality (2) in the system: $\log_{10}\frac{6x-x^{2}}{8}\geq 0$ $\Rightarrow \frac{6x-x^{2}}{8}\geq 10^{0}\Rightarrow \frac{6x-x^{2}}{8}\geq 1$ $\Rightarrow 6x-x^{2}\geq 8 \Rightarrow 6x-x^{2}-8 \geq 0$ $\Rightarrow x^{2}-6x+8 \leq 0$ Roots of corresponding quadratic equation x2 – 6x +8 = 0 are x = 2, 4. It means that the interval satisfying the inequality (1) of the system is : 2≤ x ≤ 4 ### Exercises Now, the interval of x valid for real values of f(x) is the one which satisfies both conditions simultaneously i.e. the interval common to two intervals determined: $\begin{cases} 0 So the domain of the given function is: $[2,4]$ and it is sketched below. Domain of f ### Exercises Exercise 5. Find the domain of the function given by $f(x)=\log_{10}\{1-\log_{10}(x^{2}-3x+12)\}$ Solution. There are two logarithmic functions composing the given function and we know that the argument of logarithmic function is a positive number. Then, the domain of the given function is the intersection of intervals of x obtained for each of the following conditions: $\begin{cases} 1-\log_{10}(x^{2}-3x+12) > 0 \,\,\,\,\,\,\,\,\,\,\,\,\,\,(1) \\ x^{2}-3x+12 > 0 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,(2)\\ \end{cases}$ Now, we solve the first inequality (1) in the system: ### Exercises $1-\log_{10}(x^{2}-3x+12) > 0 \Rightarrow$ $\Rightarrow \log_{10}(x^{2}-3x+12) < 1$ $\Rightarrow \log_{10}(x^{2}-3x+12) < \log_{10}10$ $\Rightarrow x^{2}-3x+12 < 10 \Rightarrow$ $\Rightarrow x^{2}-3x+2 < 0$ Roots of corresponding quadratic equation x2 – 3x + 2 = 0 are x = 1, 2. It means that interval satisfying the inequality (1) of the system is : 1 < x < 2. Now, we solve the second inequality (2) in the system: in this case, coefficient of x2 is positive and discriminant Δ is negative. Hence, the second inequality is true for all real x. ### Exercises Then, the domain of the given function is the intersection of the following two intervals: $]1,2[ \, \cap\, ]-\infty, +\infty[$ So the domain of the given function is: $]1,2[$ and it is sketched below. Domain of f ### Exercises Exercise 6. Find the domain of the function given by $f(x)=\log_{2}\log_{3}\log_{4}x$ Solution. The function is formed by nesting three logarithmic functions. The base of logarithmic functions is also different. For determining domain we first find the values of x for which log4x is real, then find the values of x for which log3(log4x) is real and finally find the values of x for which f(x) is real. The function log4x is real, if x is a positive number: x > 0 The function log3(log4x) is real, if log4x is positive: log4x > 0 $\Leftrightarrow \log_{4}x > \log_{4}1$ $\Rightarrow x > 1$ ### Exercises The function log2log3(log4x) is real, if log3log4x is positive: log3log4x > 0 $\Leftrightarrow \log_{3}\log_{4}x > \log_{3}1$ $\Rightarrow \log_{4}x > 1$ $\Leftrightarrow \log_{4}x > \log_{4}4$ $\Rightarrow x > 4$ Then, the domain of the given function is the intersection of the following three intervals: $]0,+\infty[ \, \cap\, ]1,+\infty[\, \cap\,]4, +\infty[$ ### Exercises Then domain of the given function is: $]4,+\infty[$ and it is sketched below. Domain of f Progetto "Campus Virtuale" dell'Università degli Studi di Napoli Federico II, realizzato con il cofinanziamento dell'Unione europea. Asse V - Società dell'informazione - Obiettivo Operativo 5.1 e-Government ed e-Inclusion	2018-07-16 06:26:44	{"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": 45, "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": 0.942564845085144, "perplexity": 858.6136534436121}, "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/1531676589222.18/warc/CC-MAIN-20180716060836-20180716080836-00363.warc.gz"}
http://jeromyanglim.blogspot.ro/2011/05/transition-to-linux-and-ubuntu.html	Jeromy Anglim's Blog: Psychology and Statistics Monday, May 23, 2011 Transition to Linux and Ubuntu: The Experience of a Quantitative Social Scientist This post discusses my experience migrating to Ubuntu from Windows from the perspective of a quantitatively oriented researcher in psychology. It discusses (a) my general transition to open source tools for doing research; (b) examination of Cygwin; (c) choosing a Linux distribution; (d) preparing for installation of Ubuntu; (e) initial reflections on installing Ubuntu; and (f) a long list of various solutions to specific problems that I encountered whilst getting started with Ubuntu. Context of transition Anyone following this blog for a while will know that I've been progressively migrating to using open source software for doing research. My first foray into open source started in 2007 when I decided to replace SPSS with R. As is the case with many open source tools, they have a habit of mutually reinforcing each other. The path went something like this (with links pointing to associated blog posts): • LaTeX replaced Word as my scientific document format. • Markdown replaced Word as my document format for note taking and informal documents. • Bibtex, Jabref, and other tools replaced Endnote as a bibliographic reference management system. • Vim replaced a host of other IDEs, and text editors; it even replaced MS Word, in that it became the editor for the plain text file formats (e.g., LaTeX, Sweave, Markdown, etc.) that I use the most. • git replaced ad hoc version control systems. • make replaced various manual processes for building final products. For a general summary of this workflow (albeit, prior to the adoption of Vim), see this video that I gave at Melbourne R Users in late 2010. However, I still needed to cut ties from Windows, the operating system I'd been using for 15 years. I'd been putting off adopting Linux for several reasons. I was concerned about: • hardware support for my main computer, • the temporary drop in performance that results from adopting a new operating system, • difficulties collaborating with other researchers operating in the Windows world, • lack of support for various key programs (e.g., Outlook; Inquisit), and • the possibility that some tasks would be less efficient under Linux. Despite these concerns, I decided to finally dive in. Initial foray into Cygwin About a month prior to making the switch I'd installed Cygwin on Windows. This gave me access to many Linux command-line programs. However, ultimately, while Cygwin whet my appetite, it failed to satisfy. • By default, Cygwin does not always play nicely with non-Cygwin programs. File paths often get confused. If it can't play nicely with Windows programs, I might as well go all the way. • There are issues of where to store files (Cygwin home or Windows user home). • Some Linux programs are not included. • The speed of the terminal and shell was often poor. • In general, interacting with other programs was a frustrating experience. While some of these challenges probably could have been overcome, it felt silly investing time in learning the idiosyncrasies of a system that only goes half-way to creating a genuine unix experience. Choosing a Linux distribution The first decision involved choosing a Linux distribution. I decided on Ubuntu for several reasons. • It's a popular Linux distribution. • It seems like a good distribution for people new to Linux. • Hardware support looked encouraging. • I'd heard it was well suited as a personal desktop • I'd also heard that ultimately Linux is configurable and thus, the distribution does not provide major constraints; furthermore basic unix command-line programs should work in all distributions. Preparing to install Ubuntu I approached the installation of Ubuntu with trepidation. • I read a lot about partitioning and dual-booting on Ubuntu documentation and psychocats. • I backed up my data (of course) before starting installation. • I thought a lot about setting up my partitions. In the end I decided on a dual-boot partition. I have a solid state drive which is really fast, but capacity is not that high (only 120gb). Thus, I had to be careful with assigning space. I gave 40gb to Windows and 80gb to Ubuntu. This required the removal of many unwanted files from the Windows partition. • I created and trialled the Ubuntu Live USB to check for hardware support, and everything seemed to be working surprisingly well on my Dell Latitude laptop. • I then installed Ubuntu, letting the Ubuntu installer handle the hard drive partitioning. To my pleasant surprise, the installation was smooth, involving minimal choices. The Ubuntu installation successfully partitioned my hard drive, and the core features of my laptop still worked (sleep, hibernate, wifi, volume buttons, graphics card, and so on). Furthermore, I was still able to boot into Windows for when I really needed to use Windows-only software. Reflection on the installation Switching operating systems is painful even when the new operating system is better than the old one. Many little tasks that were once automatic need to be automated in a new way. Furthermore, to fully grok the new system, it is often necessary to do more than map the old workflow onto the new system. For the most part the switch to Ubuntu has been smoother than I was expecting. The first week involved dealing with a wide range of fundamental issues related to software installation and system configuration. However, in general, these issues have been resolved by a few Internet searches and asking questions on Ask Ubuntu (a Stack Exchange Q+A site). The following were some of the main issues I encountered: • Linking with my university email / calendar Exchange server has been painful. After two weeks, this still remains my biggest pain; The email program Evolution basically works, but is painfully slow at times, often freezes, and often produces errors. • Configuring the Unity desktop (the default behaviour of the 11.04 launcher is awful) took a bit of searching. • Configuring so that I could move between laptop monitor configurations required some fiddling. • Configuring my vimrc took some work. I had a number of file format issues related to text files having inappropriate line feeds (i.e., Windows text files instead of Unix text files). Despite these issues I'm really enjoying using Ubuntu. Adopting a Vim + command-line workflow is so much easier, and I'm gradually incorporating some of the great Linux tools into my workflow. As part of the transition, I've also been reading a few books: • Harley Hahn's guide to Unix and Linux - Harley Hahn • This one was quite pedagogical in flavour and a bit wordy at times, but did provide a helpful introduction to Linux. • UNIX power tools - Jerry Peek, Tim O'Reilly, Mike Loukide • This book is massive. • It provides heaps of short articles outlining useful tricks for getting more out of Unix / Linux. • Learning the bash Shell - Cameron Newham, Bill Rosenblatt Looking forward As I get settled into the new operating system I'm looking forward to really incorporating the tools available into my research workflow. Some of the things I'm looking forward to (which I may blog about at some point, see here for Blog RSS subscription options) include: • Writing some bash scripts and Vim commands to enable me to link bibtex citations to stored pdfs of journal articles; • Taking advantage of the command-line for file manipulation, running R, creating LaTeX documents, and more; and • Dabbling in sed, awk, Python for better text manipulation. Now with the first two weeks of initial adjustment out of the way, the subtler phase of tweaking and refinement begins. If you've made the switch to Linux particularly within the context of research and data analysis, I'd be keen to hear your experience. Assorted Resolved Issues Overview The rest of this post documents a wide range of problems that I encountered when getting started with Ubuntu and the solutions that I adopted. The caveat applies that I'm pretty new to Ubuntu, so the solutions may not be best practice. Furthermore, they may not make for interesting reading unless perhaps you've stumbled onto this post through Google searching for the specific problem. Overhaul file system Question • What are the implications of Linux for how my file system should be organised? Discussion I used the transition to Linux to overhaul my file system. I was keen to use the terminal to access files. Good file names for terminal access are different to that for a point-and-click environment. Some general principles that guided the new file system • Remove spaces from file names • It's a pain to have to escape spaces (\) • I replaced spaces with hyphen "-" • Convert file names to lower case where possible • Linux is case sensitive • It's quicker to type lower case • Consistency reduces confusion • Prefer file and folder names unique in the first few letters • This facilitates tab completion • Prefer short, descriptive, and memorable file names I also found it particularly useful to do a general reflection on my file structure. I cleaned up the hierarchy so that all my core activities had a location in the file system. Accessing Windows Drive from Ubuntu Question • Are there any issues with accessing the Windows OS partition from Ubuntu? Discussion • It's easy. It shows up as a mounted drive called "OS". It's path was /media/OS Monitoring processes and use of system resources Question • What is the Ubuntu equivalent of Ctrl+Alt+Delete for viewing active processes and memory and processor usage? Discussion • There is a program called "System Monitor" Command prompt for terminal colour Question It seems to me that it might be useful to make the command prompt a different colour to help make it clear where one command starts and output begins a little clearer. • How can this be done? • What are good colour options? Discussion • Sites discussing the issue • The .bashrc file that came with Ubuntu had a setting that could be enabled to have a colour prompt. Opening common binary files from command-line Question • How can common binary files (e.g., PDF, Word, Excel, PowerPoint) be opened from the command line? Discussion • gnome-open • This command can be followed by filenames, web addresses, directories, and will typically do some appropriate default action. • gnome-open . opens the current directory in Nautilus • alias go=’gnome-open’ or equivalent may be useful Keyboard shortcuts for size window left or right half of screen Question I tend to work on a large monitor with space for Windows on the left and right side of the screen. I like to have a keyboard shortcut which sizes a window to the left or right of the screen. • How can this be done on Ubuntu? Discussion • Ubuntu does support half Desktop sizing by dragging a window with the mouse to the left or right most side of the desktop. • keyboard shortcuts offers some options for moving windows, including maximise vertically or maximise horizontally, but no option for positioning window on left half of screen. • Ctrl+Alt+NumPad Key • I learnt the following on askubuntu • By default Ubuntu supports shortcut keys for window manipulation (e.g., Ctrl+Alt+NumPad4 maximises a window vertically on the left half of the screen and Ctrl+Alt+NumPad6 does the same but to the right. • The main annoying aspect of this is that these keys are not suitable for touch typing. • Solution • Install Advanced Desktop Effects settings (Compiz with Compiz Config) • Start CompizConfig • Go to Window Manager - Grid • Change the key bindings (I chose the following, which have a certain consistency with Vim movement keys) • Super + h (left) • Super + L (right) • Super + K (maximise) • Super + J (restore) Shortcut key to initiate web search Question On windows I had a global shortcut key (Windows+Z) to initiate a Google search. • How can this be done on Ubuntu? Discussion • googlizer • This can be used to create command line programs for searching. • Chrome • Start chrome • Type search term in address bar • Making google.com the home page can speed this up • Evaluation: • The responsiveness of this is not adequate (a search box is instant; chrome takes around 1 second to start up) • It results in the loss of the default Chrome homepage • AutoKey • A program modelled on AutoHotKey might provide the desired functionality. • Google Desktop Search with Desktop search disabled was a good choice. • I changed the default shortcut key to Ctrl+F12 Using a laptop with multiple external monitors Question • How can a laptop monitor be disabled and an external monitor enabled? • I use two external monitors, one at home, and one at work. How can Ubuntu be configured to automatically detect which one is connected? Discussion • When I configure to just use my external monitor and not my laptop monitor and then unplug the external monitor, the laptop monitor is not activated. Thus, I can't use the mouse to reactivate the laptop monitor. It would be good if Ubuntu could automatically detect the removal of the external monitor, or alternatively I could have a shortcut key to activate laptop only mode. • ubuntugeek describes configuring Ubuntu so that Ctrl+Alt+Backspace can force a restart of X • A user asks about automatic switching of monitor configurations on Ubuntu forums • More suggestions • Essentially it involves using a program called disper • disper -s and disper -S activate primary and secondary monitors respectively. I've configured these commands to be triggered by shortcut keys using CompizConfig. Launching programs with keyboard shortcuts Question • What is a good strategy for launching programs from keyboard shortcuts? Discussion • CompizConfig makes it easy to create commands and assign shortcut keys. • The Unity Launcher enables shortcut keys combining the Super key and a number or typing the Super key and typing a key word. Restart to Windows Question • Is there a way to shutdown Ubuntu and boot to Windows without having to make a selection from the bootloader? Default terminal size Question • What is a good default Terminal size? Discussion I can always press F11 to get a full-screen terminal. However, sometimes it is useful to be able to have a reasonable size terminal come up that does not take over the whole screen. I've configured the default profile to be 95 columns by 40 rows. I would choose more rows, but I want the setup to work on my smallest monitor (i.e., my laptop). Question At first I thought the Unity menu was okay. However, I don't like: • the screen real estate that launcher consumes • the way that it defaults to placing left aligned windows indented with space to the launcher • The size of the launcher icons (although at least this can be reduced) • The way that it stays in focus sometimes for a period of time Discussion In Compix Config set Unity - Hide Launcher to Auto-hide. Configuring Ubuntu for running basic scripts Question • What is a basic strategy for running scripts in Ubuntu? • How do permanently add the custom script to the path? • Where should I store throw-away scripts? • What is a good way to run custom scripts? Discussion I did the following: • mkdir ~/bin • added export PATH="$PATH:~/bin" to .bashrc Find file by name Question • How can I find a file by a name or part of its name within a specified directory (or subdirectories)? Discussion • Cyberciti provides a page explaining several options. • find command • The following command would find all files with the txt extension in ~/dir • find ~/dir -name ".txt" Accessing Ubuntu drive from Windows on dual-boot machine Question I store my data on my Ubuntu drive. • How can I access my Ubuntu drive from Windows? Discussion This page has three options, the last of which, looks like what I want. • Ext2 Installable File System for Windows • This did not work on Windows 7 (perhaps this will change in the future) • DiskInternals Linux Reader • Evaluation • This worked adequately • This has both the benefit and the problem that it does not alter the Linux partition • How did it work? • Start Windows 7 • Install and Start DiskInternals • Save selected Linux partition files onto Windows partition (I had a special section) • View, edit, and change files • Restart Ubuntu and copy files from Windows partition back into appropriate location Script fur renaming files in folder Question I sometimes download files sent to me and want to convert the file names into more appropriate ones. The main conversions that I'd like to see are: • replace spaces with dashes ("-") • replace uppercase with lower case Others may arise as time goes by. The script should take a list of file names. • How can this be done? Discussion I wrote the following script and called it cleannames #!/bin/bash for file in ; do mv "$file" echo "$file" \| sed 's/ /-/g'; done for file in * ; do mv "$file" echo "$file" \| sed 's/./\L&/g'; done Minimise current window with keyboard shortcut Question • What is the keyboard shortcut to minimise a Window? Discussion • Alt+F9 minimises • Alt+F10 maximise • Launcher - System Settings - Shortcuts allows for custom shortcut keys Terminator Terminal Question • Should I adopt Terminator Terminal? Discussion • At present I'm using Gnome Terminal. • Various reviews suggest that Gnome terminal is a reasonable option in terms of speed and features. Preventing automatic shading when watching flash videos Question I had an ABC video on full screen and after a few minutes, Ubuntu shaded the screen. • How can this be prevented? Discussion • Power Management has some options • I disabled the screensaver. That seemed to fix the problem. Batch renaming of files Question Unix is case sensitive. It also encourages substantially greater use of the command-line. As such, it is useful to adopt file naming conventions that make it easier to work with files on the command-line. • How can I batch rename files • replace underscore with dash • replace upper case letter with corresponding lower case letter Discussion • Some options • KRename • for loop • rename Convert underscore to dash: for file in ; do mv$file echo \$file \| sed 's/_//' ; done Convert upper case letter This tutorial shows how to use the convmv command. Kill a program Question Windows has Ctrl+Alt+Delete which facilitates killing a process. • What does Ubuntu involve? Discussion • The system monitor lists processes and enables killing processes • Ctrl+Alt+Backspace kills X (once this feature is activated) Better console / terminal colours Question The default background for Ubuntu terminal is a type of dark purple. While it's "pretty", I find the contrast to be relatively low. • What are better settings? Discussion I applied the following settings: • Edit - Preferences: • Colors - Built-in schemes= white on black • Background = solid colour • export TERM="xterm-256color" enables better colour support for Vim Problems displaying full screen video Question • I find that full screen video on YouTube seems to freeze. How can this be fixed? Discussion There is a tutorial on YouTube on how to fix this. sudo mkdir /etc/adobe sudo su sudo echo "OverrideGPUValidation = 1" >> /etc/adobe/mms.cfg" Question I think it would be useful to reduce the default grub display time to something like 1 or 2 seconds. I almost always boot to Ubuntu, and if I want to boot to Windows, I have a command from within Ubuntu that changes the default boot for one occasion. • Is this a good idea? • How can I do this? Discussion • I think 2 seconds would make sense as this is long enough to easily enter manual mode if I am watching carefully. • sudo vim/etc/default/grub • I modified the value of GRUB_TIMEOUT and made it GRUB_TIMEOUT=2 • sudo update-grub Set up git on Ubuntu Question • How do I configure git on Ubuntu? Discussion • General information on configuring git and github on Linux is available here 1. Jeromy, congrats on your transition to Ubuntu, and thanks for sharing all your experience with us and also for sharing all the steps that you have done during your transition from Windows, Word and all that stuff up to linux, latex, git, etc. I made the same change a couple of years required in part by my work, but my sequence was somewhat different. First go to linux (ubuntu) by the need to compile programs in fortram (my background is basically quantitative genetics) and then, I started using R (leaving SAS), then latex and finally the CVS. So, welcome to Linux and enjoy this new "step" ;) 2. So, someone handed you a disk and told you lies hrrm? Look deeper. 3. @Manuel: Thanks for the comment. @Anonymous: Would you like to elaborate? 4. On the email thing- ditch Evolution; try Thunderbird. 5. @Anonymous I'd like to switch away from Evolution, but my university uses Exchange and doesn't support IMAP. As far as I am aware, Thunderbird doesn't work with owa or MAPI Exchange. The only promising lead that I've heard but not yet tried is DavMail. 6. Great post -- thanks for listing all the specific ways that you found solutions. They may not all be relevant to me, but they are reminders that specific solutions exist (and that I can ask for help to find them). 7. @Anonymous small update: I've got Thunderbird (and lightning) connecting to Exchange now through DavMail. And it's so much better than Evolution. 8. If flash is working good now, then I would leave it alone. If it is still dragging and you are running 64 bit try the square beta plugin. IMO it is best 9. @Tyler thanks for the tip. I'm currently on 32bit and flash seems to be working okay after I modified the settings (as mentioned above). 10. Good luck with Ubuntu, it is very powerful and the package management system taken from Debian is fantastic. I would recommend the following additional tools: Lyx: http://www.lyx.org/ It beats writing LaTeX by hand. Mendeley: http://www.mendeley.com/ For managing your references. It can automatically sync online and output .bib files for you to import into Lyx or LaTeX or whatever you want to use. 11. Yeah...I've found that cygwin was a very difficult way to work with programs for a while, but got used to it eventually! Now R over cygwin just feels right. Mike https://www.evaluationportal.com 12. Thank you all for your ideas. 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https://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0717-75182020000100080&lng=es&nrm=iso	## Articulo • Similares en SciELO ## versión On-line ISSN 0717-7518 ### Rev. chil. nutr. vol.47 no.1 Santiago feb. 2020 #### http://dx.doi.org/10.4067/S0717-75182020000100080 Original Article Relationship between food consumption of pregnant women and birth weight of newborns Relación entre el consumo de alimentos de gestantes y el peso al nacer de los recién nacidos 2Departamento de Nutrição e Dietética, Faculdade de Nutrição Emília de Jesus Ferreiro, Universidade Federal Fluminense, Niterói, RJ, Brasil ABSTRACT The aim of this study was to investigate the relationship between food consumption of 99 pregnant women aged 16-44 years who gave birth at a maternity hospital in Rio de Janeiro and newborn birth weight. Maternal consumption of unprocessed or minimally processed and ultra-processed foods was evaluated through the Food Consumption Markers Form. For ultra-processed foods, most pregnant women regularly consumed sweets (73.7%), soft drinks (71.7%), sausages (65.7%), salted crackers (63.6%) and fried foods (55.6%). Among unprocessed or minimally processed foods, most pregnant women consumed beans (85.9%) and milk or yogurt (60.6%), while less than half consumed fruits (44.4%), raw vegetables (28.3%) and cooked vegetables (27.3%) regularly. Among newborns, 13.5% presented low birth weight. There was a significant relationship between consumption of sausages (p= 0.02) and sweets (p= 0.04) by pregnant women and low birth weight of newborns. Maternal consumption of sausages increased the odds of newborn low birth weight (OR 1.46, 95% CI 1.02-2.10). Keywords: Food consumption; Low birth weight; Pregnant; Unprocessed foods; Ultra-processed foods RESUMEN Palabras clave: Alimentos no procesados; Alimentos ultra-procesados; Bajo peso al nacer; Consumo de alimento; Gestante INTRODUCTION The diet of the world's population is currently characterized by high consumption of ultra-processed foods and low consumption of unprocessed or minimally processed foods1,2,3. Ultra-processed foods have high caloric density and low nutritional quality and consumption is related to nutritional deficiencies, obesity, diabetes, arterial hypertension and other diseases4,5. Inadequate diet during pregnancy may imply nutritional risk to newborns, favoring low birth weight (LBW) and negatively affecting growth and development6,7,8,9. Birth weight is an important public health indicator, as it reflects maternal health conditions and the quality of prenatal care10. According to the United Nations Children's Fund (UNICEF)11, in 2000, the prevalence of LBW in developed countries was around 7.0% and 10.0% in Latin America. LBW contributes to negative outcomes in infant health, such as developmental deficits, behavioral disorders and vulnerability to infectious diseases, favoring morbidity and mortality in the first year of life12,13, and increasing the chances of future chronic diseases8,10. It is known that unprocessed and minimally processed foods, such as fruits and vegetables have more vitamins, minerals, fibers and compounds with antioxidant and antiinflammatory properties compared to ultra-processed foods. Therefore, diet characterized. by the frequent consumption of unprocessed foods is considered a protective factor for the occurrence of chronic diseases, nutritional deficiencies and other adverse health conditions1. There are few studies that have evaluated maternal diet from the point of view of the consumption of unprocessed or minimally processed foods and ultra-processed foods and its influence on fetal growth, especially on LBW. Thus, the main aim of this study was to investigate the relationship between food consumption of pregnant women seeking care at the “Herculano Pinheiro” Maternity Hospital and the birth weight of their newborns. METHODOLOGY We conducted a cross-sectional analytical study with 99 pregnant women aged 16-44 years who received care at the Herculano Pinheiro Maternity Hospital (HPHM) between July and September 2016. The maternity hospital, located in the northern region of the municipality of Rio de Janeiro, is a reference hospital for the follow-up of low-risk pregnant women. Sampling was conducted for convenience. Women were approached after giving birth and received explanations about the study. Volunteers accounted for 12% of women receiving care at this maternity hospital. The sample was calculated considering the average number of births in 24 hours at the hospital with a margin of error of 5% and 95% confidence interval (95% CI). The calculated total sample was 83 women and, considering a potential 20% loss, the number stipulated for this study was 99 women. Women hospitalized at the maternity hospital who gave birth within 24 hours of admission were eligible to participate. After being informed of the research objectives and procedures, and with their agreement to voluntarily participate in the study, they provided signed informed consent. Women who were hospitalized for more than 24 hours were excluded, that way feeding provided at the maternity ward would not interfere with the evaluation of diet. Other exclusion criteria used were: twin pregnancy; abortion; previous diagnosis of diseases such as hypertension and diabetes; presence of obstetric hemorrhagic and infectious complications. Maternal data were collected through medical record, prenatal care card of pregnant women and a general information questionnaire. Age, marital status, schooling, per capita income, number of pregnancies, pre-gestational weight, final gestational weight, gestational weight gain, gestational age and type of delivery performed in the maternity ward were analyzed. Pregnant women were classified as adolescents or adults according to World Health Organization14. Pre-gestational weight was the weight recorded by the woman at the first prenatal visit. Final gestational weight was the last weight recorded in prenatal care card of pregnant women. Per capita income was calculated based on the total family income, measured by the sum of the monthly income of all family members divided by the number of residents in the household. Birth weight of newborns was collected and classified according to UNICEF11: extreme low birth weight (weight less than 1,000 g); very low birth weight (weight less than 1,500 g); low birth weight (weight less than 2,500 g); insufficient weight (weight between 2,500 g and 2,999 g); normal or suitable weight (weight between 3,000 g and 3,999 g); overweight (weight greater than or equal to 4,000 g). Maternal food consumption was evaluated through the Food Consumption Markers Form of the Food and Nutrition Surveillance System15, validated for children over five years of age, adults, older adults and pregnant women. This form is used to identify the consumption pattern of so-called “food consumption markers”, which are indicative of healthy and unhealthy eating practices. The form has ten food groups: Group 1) Raw vegetables (lettuce, tomato, carrot, cucumber, cabbage); Group 2) Cooked vegetables (cabbage, squash, chayote, broccoli, spinach, not including potato and cassava); Group 3) Fresh fruit or fruit salad; Group 4) Milk or yogurt; Group 5) Beans; Group 6) French fries, potato chips, and fried snacks such as chicken drumstick, meat ball, pastry, etc.; Group 7) Burgers and sausage (sausage, mortadella, salami, ham, etc.); Group 8) Crackers / Salty crackers or packaged snacks; Group 9) Sweet or stuffed biscuits, sweets, candies and chocolates (in bar or candy); Group 10) Soft drinks (not including diet or light soft drinks). Groups 1 to 5 are considered “markers of healthy food consumption” and Groups 6 to 10 “markers of unhealthy food consumption”. To determine the food consumption frequency in the last seven days, eight answer options were provided: “I have not eaten in the last seven days”; “One day in the last seven days”; “Two days in the last seven days”; “Three days in the last seven days”; “Four days in the last seven days”; “Five days in the last seven days”; “Six days in the last seven days” and “All days in the last seven days”15. Using the NOVA Food Classification4 as a reference, food groups were classified according to the degree of processing, being divided into: a) unprocessed or minimally processed foods; b) processed culinary ingredients; c) processed foods; d) ultra-processed foods. Groups 1 to 5 were classified as unprocessed or minimally processed foods and Groups 6 to 10 as ultra-processed foods. The 75th percentile of consumption of each food or food group studied was adopted as a cutoff point for regular consumption of unprocessed or minimally processed foods. For the regular consumption of ultra-processed foods, the 25th percentile of consumption of each food or food group was adopted16. For statistical analysis, collected data were typed and consolidated using SPSS / PC software, version 17.0. Descriptive statistics were used to present the results by arithmetic mean (x), standard deviation (SD), median (MD), minimum and maximum values and frequencies. The Kolmogorov-Smirnov test was used to verify if maternal variables age, per capita income, pre-gestational weight, final gestational weight, gestational weight gain and food consumption followed the normality pattern. Variables with non-normal distribution were presented as median and interquartile range and non-parametric tests were used for these variables. The chi-square test was used to verify possible relationship between birth weight and maternal age, number of pregnancies and gestational weight gain. The Mann-Whitney test was used to evaluate the relationship between maternal food intake and maternal age. The Kruskal-Wallis test and the Dunn post-test were applied to assess the relationship between maternal food consumption and birth weight. Analyses of correlation between maternal food consumption and child data were conducted. Pearson's correlation was used for variables with normal distribution and Spearman's for non-normal variables. Significance level of 5% was adopted in all analyses. For multivariate analysis, binary logistic regression procedures were used, whose dependent variable was categorized by the classification of low birth weight (1 - low birth weight, 0 - no low birth weight), with the no low birth weight category considered as a reference. Among the set of variables for food consumption, logistic regression was performed adjusted for the following variables: age, per capita income, pre-gestational weight, gestational weight gain, number of pregnancies and type of delivery. Odds Ratio (OR), 95% confidence intervals (95% CI) and p-values for food consumption variables were estimated. The present study was submitted and approved by the Ethics Research Committee of the Municipal Health Secretariat of Rio de Janeiro under number 47887915.5.0000.5279, in accordance with the ethical principles contained in Resolution 466/12 of the National Health Council. RESULTS The majority of participants were single (79.8%), adults (67.7%), multiparous (62.6%) and had incomplete high school (38.8%). The majority of birth were with a normal delivery (76.8%). On average, 38.4% of women gained between 8.0 and 11.5 kg; 44.2% gained more than 11.5 kg by the end of pregnancy. Among newborns, the majority presented normal birth weight (57.3%). However, 27.1% showed insufficient weight and 13.5% LBW. Only 2.1% of newborns were overweight. Table 1 shows data on pregnant women and birth weight of newborns. It was observed that the mean age of participants was 24.9 ± 6.5 years. Mean gestational age was 39.4 ± 1.2 weeks, indicating that, on average, infants were not premature. Among newborns, mean birth weight was 3082.0 ± 559.0 g. Table 1 Characteristics of pregnant women and newborn birth weight. Variables N X¯1 SD2 MD3 Minimum Maximum Age (years) 99 24.9 6.5 23.0 16.0 44.0 Pre-gestational weight (kg)* 88 63.2 12.8 60.0 38.0 117.0 Final gestational weight (kg)* 91 74.1 13.4 73.0 47.0 115.4 Gestational weight gain (kg)* 86 11.3 5.9 10.1 -2.0 31.0 Gestational age (weeks) 99 39.4 1.2 40.0 37.0 42.0 Birth weight (g)* 96 3082 559 3108 1500 4400 It was not possible to collect data of all study participants due to the lack of some information in patient medical records and prenatal care card. 1Arithmetic mean; 2Standard Deviation; 3Median. The average per capita income was U$87.00 ± 69.60. Around 8% of participants did not know how to report family income. The maximum per capita income observed was U$ 326.20. Correlation analysis showed a positive association between gestational weight gain and newborn birth weight (r= 0.22, p= 0.04). The investigation of the possible influence of maternal age, number of pregnancies and weight gain during pregnancy on newborn birth weight is presented in table 2. Statistical analysis showed no relation between maternal variables and birth weight. Table 2 Associations between birth weight and age, number of pregnancies and maternal weight gain. Variables Low birth weight p-value Yes No n % N % Age 16 ├┤ 20 years 06 46.0 24 29.0 0.45 21 ├┤ 29 years 05 38.0 40 48.0 30 ├┤ 44 years 02 15.0 19 23.0 Total 13 100.0% 83 100.0% Number of pregnancies 01 06 46.0 30 36.0 0.71 02 ├┤ 03 05 38.0 42 51.0 ≥ 04 02 15.0 11 13.0 Total 13 100.0% 83 100.0% Maternal weight gain <0 0 0.0 0 0.0 0.40 0 ├ 8.0 kg 05 42.0 22 30.0 8.0 ├┤ 11.5 kg 01 8.0 21 28.0 > 11.5 kg 06 50.0 31 42.0 Total 12 100.0% 74 100.0% Chi-square test. Regarding maternal food consumption, no correlation with maternal data was observed. The consumption of unprocessed or minimally processed and ultra-processed foods by pregnant women in the last week of gestation is presented in table 3. Table 3 Consumption of foods by pregnant women in the last week. Food groups n X 1 SD2 P253 P504 P755 Unprocessed or minimally processed foods Beans 99 6.33 1.77 7.00 7.00 7.00 Milk or yogurt 99 5.02 2.67 3.00 7.00 7.00 Fruits 99 4.33 2.65 2.00 4.00 7.00 Cooked vegetables 99 2.80 2.66 0.00 2.00 5.00 Raw vegetables 99 2.67 2.42 1.00 2.00 4.00 Ultra-processed foods Sweets 99 3.30 2.95 0.00 2.00 7.00 Soft drinks 99 3.23 2.99 0.00 2.00 7.00 Salted crackers 99 2.77 2.84 0.00 2.00 6.00 Sausages 99 1.48 1.75 0.00 1.00 2.00 Fried foods 99 1.29 1.73 0.00 1.00 2.00 1Arithmetic mean; 2Standard Deviation; 325th percentile; 450th percentile; 575th percentile. Arithmetic mean represents how many times the food group was consumed per week by pregnant women. In relation to the regular consumption of unprocessed or minimally processed foods, less than half of pregnant women regularly consumed fruits (44.4%), raw vegetables (28.3%) and cooked vegetables (27.3%). About 85.9% of pregnant women regularly consumed beans and 60.6% consumed milk or yogurt. However, among ultra-processed foods, most pregnant women consumed them regularly, notably: sweets (73.7%), soft drinks (71.7%), sausages (65.7%), salted crackers (63.6%) and fried foods (55.6%). When analyzing the relationship between maternal food consumption and maternal age, statistical analysis showed that the consumption of cooked vegetables was higher among adult pregnant women than among adolescent pregnant women (p= 0.03). The consumption of ultra-processed foods was similar among adult and adolescent pregnant women (p<0.05) (data not shown). The odds of newborns of presenting LBW related to maternal food consumption are presented in table 4. The maternal consumption of sausages increased the odds of newborns of presenting LBW. Table 4 Maternal food consumption and the odds of low birth weight. Food groups OR1 95% CI2 Sausages 1.46 1.02-2.10 Fried foods 1.38 0.96-1.98 Salted crackers 1.20 0.92-1.57 Fruits 1.16 0.87-1.54 Sweets 1.13 0.88-1.45 Soft drinks 1.13 0.90-1.42 Cooked vegetables 1.08 0.82-1.43 Raw vegetables 1.07 0.79-1.45 Beans 0.96 0.67-1.38 Milk or yogurt 0.87 0.67-1.12 1Odds Ratio; 295% confidence interval. Table 5 shows the relationship between maternal food consumption in the last week of gestation and birth weight. The average consumption of sausages and sweets was significantly higher (p<0.05) among pregnant women whose newborns presented LBW. In addition, the average consumption of fried foods showed a strong tendency to be higher among pregnant women whose newborns presented LBW (p= 0.05). Among unprocessed or minimally processed foods, consumption was similar among pregnant women. Table 5 Maternal food consumption in the last week and relationship with newborn birth weight. Food groups Low birth weight Insufficient weight Normal weight Overweight p-value n x¯ 1 * MD2 SD3 n x¯ MD SD n x¯ MD SD n x¯ MD SD Unprocessed or minimally processed foods Raw vegetables 13 2.92 2.00 2.69 26 3.19 3.00 2.61 55 2.47 2.00 2.31 02 3.00 3.00 0.00 0.50 Cooked vegetables 13 3.08 2.00 2.69 26 2.88 2.00 2.63 55 2.76 2.00 2.70 02 5.00 5.00 2.83 0.93 Fruits 13 5.08 7.00 2.22 26 4.42 4.50 2.72 55 4.29 4.00 2.63 02 3.50 3.50 4.95 0.64 Beans 13 6.08 7.00 2.29 26 6.73 7.00 1.37 55 6.27 7.00 1.66 02 7.00 7.00 0.00 0.26 Milk or yogurt 13 4.00 4.00 3.11 26 5.27 7.00 2.78 55 5.18 7.00 2.55 02 5.50 5.50 2.12 0.33 Ultra-processed foods Fried foods 13 2.38 1.00 2.60 26 1.04 1.00 1.25 55 1.16 1.00 1.66 02 1.00 1.00 1.41 0.05 Sausages 13 2.38a 2.00 1.89 26 1.69ab 1.00 1.74 55 1.20b 1.00 1.68 02 2.50 2.50 2.12 0.02 Salted crackers 13 3.31 3.00 2.90 26 2.31 1.00 2.81 55 2.87 2.00 2.88 02 5.00 5.00 2.83 0.48 Sweets 13 4.92a 7.00 2.87 26 3.73ab 3.00 3.05 55 2.78b 2.00 2.79 02 3.50 3.50 4.95 0.04 Soft drinks 13 4.38 5.00 2.79 26 3.46 2.50 3.05 55 2.65 1.00 2.89 02 7.00 7.00 0.00 0.09 Kruskal-Wallis test, Dunn post-test. a,bDifferent superscript letters denote significant difference. Data regarding dietary consumption of pregnant women who had overweight infants were not used in this statistical analysis (too few newborns in the sample). 1Arithmetic mean; 2Median; 3Standard deviation. Arithmetic mean of Table 5 represents how many times the food group was consumed per week by pregnant women. DISCUSSION The general characteristics of pregnant women in our sample (single adults, with low socioeconomic status and schooling) were similar to those found in other studies performed with pregnant women at different Basic Health Units in the Brazilian Unified Health System17,18,19. Studies indicate that lower schooling and income levels and higher parity are positively related to the Western pattern diet, characterized by excessive consumption of ultra-processed foods18,20. However, in the present study, no association between these variables and maternal food consumption was observed. The higher prevalence of normal delivery observed in this study can be explained by the fact that the maternity hospital where the study was developed was linked to the “Hospital Amigo da Criança” Initiative. This initiative, conceived by UNICEF, provides care that reduces invasive procedures, such as episiotomies, acceleration or labor induction and cesarean deliveries21. Several maternal risk factors are related to LBW, such as the presence of systemic arterial hypertension, anemia, gestational diabetes, urinary tract infection and age less than 20 years and between 35 and 40 years22. In adolescent pregnancy, there is a competition of nutrients between the growing mother and the developing fetus23. Nutrient deficiency promotes intrauterine growth restriction and consequently, LBW. Maternal aging is also related to diabetes, obesity and hypertension, morbidities that increase the risk of restriction of intrauterine growth and premature birth24. Regarding the number of pregnancies, studies indicate that multiparous women have higher risk of LBW than primiparous. Multiparity predisposes a woman to increased risk of maternal and obstetric complications, which may trigger prematurity and neonatal intercurrences, negatively influencing birth weight25. Although the literature associates maternal age22,23,24 and number of pregnancies17,22 with LBW, this study showed no association between maternal data and newborn birth weight. The prevalence of LBW (13.5%) observed is higher than that found in Brazil in the same period (7.7%), according to the Information System on Live Births28. Data show that, in Brazil, there was a reduction from 8.4% in 2012 to 7.7% in 2016. Among other Latin America countries, Honduras (15.0%) had the highest rate of LBW in 2012, followed by Puerto Rico (12.4%), Venezuela (10.6%) and Colombia (9.0%). The lowest prevalence of LBW was observed in Cuba (5.3%), Chile (5.8%), Peru (6.9%), Argentina (7.0%) and Uruguay (8.3%)22. Literature has shown that lower risk of LBW among pregnant women with better eating habits, with predominance of consumption of unprocessed or minimally processed foods and a reduction in the consumption of ultra-processed foods, such as regular bread, pizza, sausage, among others7,9,29,30. According to the Brazilian Ministry of Health1, ultra- processed foods tend to limit the consumption of unprocessed or minimally processed food, and, therefore, should be avoided. However, this orientation was not adopted by the pregnant women in the present study, since it was observed that, in the last week of gestation, most pregnant women regularly consumed the five ultra-processed food groups and only two groups of unprocessed or minimally processed foods. Gomes et al.31 evaluated the eating habits of pregnant women in the second trimester of pregnancy receiving prenatal care at Basic Health Units of São Paulo, applying a questionnaire similar to that used in this study. The authors observed that 48.8% of pregnant women did not regularly consume fruits, a result inferior to that found in the present study. In addition, the authors pointed out that 45.7% of pregnant women replace main meals for snacks once or twice a week, showing the search for ultra-processed foods by this public. The predominance of consumption of ultra-processed foods by pregnant women was also observed by other authors. The review article by Zapata et al.32 showed that pregnant women in Argentina have an excessive intake of saturated fats and sugar. Half of pregnant women also report high consumption of soft drinks. In addition, the authors verified low intake of fruits and vegetables in the maternal diet. In a study by Sandoval, Nieves and Luna33 conducted in Guanajuato, Mexico, observed that a majority of pregnant women frequently consumed sugar (88.7%), beans (58.5%) and milk (54.7%) and less than half of pregnant women (33.9%) consumed vegetables, in agreement with the results presented here. In our study, the consumption of sausages was significantly higher among pregnant women whose newborns presented LBW. There are few studies that explain the mechanism by which sausages could favor LBW. It is known that sausages are ultra-processed foods high in nitrate and sodium and potassium nitrite, food additives used to preserve, intensify or modify their sensory properties30. It is suggested that, at high serum concentrations, nitrite ion forms an irreversible binding with blood hemoglobin, forming methemoglobin. This process hinders the transport of oxygen through fetal tissues, which may lead to fetal growth restriction, causing LBW34,35. Grieger, Grzeskowiak and Clifton29 showed an association between consumption of high fat and sugar foods and higher likelihood of preterm birth (adjusted OR 1.54, 95% CI 1.10-2.15, p= 0.011). Maternal diet was evaluated in the 12 months prior to conception. The authors suggest that increased maternal inflammatory status would limit the transfer of nutrients to the fetus through the placenta. Thus, the consumption of ultra-processed foods would be related to negative fetal growth outcomes, confirming the results observed here. Hydrogenated fats, ingredients commonly used in ultra-processed foods, have been identified as responsible for an increase in the inflammatory state of the body, via stimulating the activity of prostaglandin E2 and inhibiting anti-inflammatory agents, such as prostaglandins E1 and E3. It is known that prostaglandin E2 acts at birth by inducing uterine contraction36,37, which would favor early labor, prematurity and consequently low birth weight in neonates. Chassaing38 suggests that emulsifiers, food additives often used in the production of ultra-processed foods, stimulate the growth of pathogenic bacteria in the intestinal microbiota. These bacteria express flagellin and lipopolysaccharides, substances that activate the expression of inflammatory cytokines. Therefore, as previously explained, conditions that promote the inflammatory state in pregnant women would favor LBW29. A woman's diet during pregnancy tends to remain similar to pre-pregnancy or to improve during the gestational period, especially due to maternal concern for adequate fetal development and growth31. Saidman et al.39 showed that 65% of pregnant women interviewed reported changes in dietary pattern during pregnancy, including increased consumption of fruits, vegetables, cereal and milk. The main motivation for these changes was to promote fetal health. Evaluating the food consumption of pregnant women through a Food Frequency Questionnaire, Fazio et al.40 concluded that there is no great variation in dietary habits during pregnancy, suggesting that diet at the end of gestation reflects the diet at the beginning of the gestational period. This study had some limitations. First, the food consumption form was applied in the week prior to delivery, and may be influenced by physiological and / or emotional changes that occur during this period. Second, although the form is a valid tool on food consumption, being used in national surveys by the Ministry of Health, its use depends on the memory of the person being interviewed, and food consumption may be underestimated. CONCLUSION Most pregnant women regularly consumed unprocessed or minimally processed foods such as beans and milk or yogurt, considered markers of healthy consumption. However, among ultra-processed foods, which are considered markers of unhealthy consumption, pregnant women regularly consumed all food groups, especially sweets, soft drinks and sausages. This study showed that the maternal consumption of sausages increased the likelihood of newborns of presenting LBW. Studies on maternal food consumption and association with LBW are still scarce, but it is essential to encourage the planning and elaboration of public interventions to reduce this outcome. Thus, further studies on maternal food consumption and its relationship with variations in birth weight should be carried out, which would help develop nutritional strategies more adequate for pregnant women, emphasizing maternal and child health. REFERENCES 1. Brazil. Ministry of Health. Secretariat of Health Care. Department of Basic Attention. Food guide for the Brazilian population. 2. ed., 1. repr. Brasília: Ministry of Health, 2014. [ Links ] 2. 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BMC Pregnancy Childbirth. 2014; 14(1): 186-192. [ Links ] 7. Englund-Ögge L, Brantsæter A, Sengpiel V, Haugen M, Birgisdottir B, Myhre R, et al. Maternal dietary patterns and preterm delivery: results from large prospective cohort study. BMJ. 2014; 348: 1446-1464. [ Links ] 8. Estrada-Restrepo A, Restrepo-Mesa S, Feria N, Santander F. Maternal factors associated with birth weight in term infants, Colombia, 2002-2011. Cad Saude Publica. 2016; 32(11): e00133215. [ Links ] 9. Martin C, Sotres-Alvarez D, Siega-Riz A. Maternal dietary patterns during the second trimester are associated with preterm birth. J Nutr. 2015; 145(8): 1857-1864. [ Links ] 10. Ferreira V, Jardim T, Póvoa T, Mendonça K, Nascente F, Carneiro C, et al. Birth weight and its association with blood pressure and nutritional status in adolescents. J Pediatr (Rio J.) 2018; 94(2): 184-191. [ Links ] 11. United Nations Children's Fund, World Health Organization. Low Birth weight: Country, regional and global estimates. UNICEF, New York, 2004. [ Links ] 12. Mendes C, Cacella B, Mandetta M, Balieiro M. Low birth weight in a municipality in the southeast region of Brazil. Rev Bras Enferm. 2015; 68(6): 1169-1175. [ Links ] 13. Tavares B, Klein C, Bloch K. Validity of informed birth weight. Study of Cardiovascular Risk in Adolescents (ERICA) - Rio de Janeiro. Rev Bras Saude Mater Infant, Recife 2017; 17(4): 705 −715. [ Links ] 14. World Health Organization. Women and health: today's evidence tomorrow's agenda. Geneva: World Health Organization; 2009. [internet]. [access june 28, 2018]. Available in http://www.who.int/eportuguese/publications/Mulheres_Saude.pdfLinks ] 15. Brazil. Ministry of Health. Secretariat of Health Care. Department of Basic Attention. Protocols of the Food and Nutrition Surveillance System - SISVAN in health care. Brasília: Ministry of Health, 2008. [ Links ] 16. Corrêa R, Vencato P, Rockett F, Bosa V. Dietary patterns: are there differences between children and adolescents? Cien Saude Colet. 2017; 22(2): 553-562. [ Links ] 17. Capelli J, Pontes J, Pereira S, Silva A, Carmo C, Boccolini C, et al. Birth weight and factors associated with the prenatal period: a cross-sectional study in a maternity hospital of reference. Cien Saude Colet. 2014; 19(7): 2063-2072. [ Links ] 18. Coelho N, Cunha D, Esteves A, Lacerda E, Theme Filha M. Dietary patterns in pregnancy and birth weight. Rev Saude Publica (São Paulo). 2015; 49: 01-10. [ Links ] 19. Gomes K, Ferreira V, Gomes D. Quality of the diet of pregnant women in a basic health unit in Belém do Pará: a pilot study. Rev Ciencia Saude. 2015; 8(2): 54-58. [ Links ] 20. Castro M, Souza R, Vilela A, Kac G. Association between sociodemographics factors and dietary patterns during pregnancy. Rev Nutr. 2014; 27(2): 173-181. [ Links ] 21. Brazil. Ministry of Health. Ordinance No. 1.153, dated May 22, 2014. Redefines the enabling criteria of the Baby- Friendly Hospital Initiative (BFHI), as a strategy to promote, protect and support breastfeeding and the integral health of children and women, within the scope of the Unified Health System. 2014. [Internet]. [access on June 12, 2018]. Available in http://bvsms.saude.gov.br/bvs/saudelegis/gm/2014/prt1153_22_05_2014.htmlLinks ] 22. Instituto Nacional de Salud. Protocol on Public Health Surveillance: Low birth weight at term. Bogotá; 2016. 27 p. Available in https://www.ins.gov.co/buscador-eventos/Lineamientos/PRO%20Bajo%20peso%20al%20nacer_.pdfLinks ] 23. Zapata C, Castillo C. Early weight gain of infants born to adolescent mothers. Rev Chil Nutr. 2012; 39(2): 136-142. [ Links ] 24. Tascón LAM, Giraldo G Daniela, Giraldo G David, Ruíz MAO, Betancourth EAV, Guatibonza MDA. Biological determinants of neonatal mortality in a population of adolescent and adult women in a hospital in Colombia. Rev Chil Obstet Ginecol. 2017; 82(4): 424-437. [ Links ] 25. Coutinho E, Araújo L, Pereira C, Duarte J, Nelas P, Chaves C. Factors associated with low birth weight. Rev INFAD. 2016; 1(2): 431-440. [ Links ] 26. Ratowiecki J, Poletta FA, Giménez LG, et al. Prevalence of low birth weight in a scenario of economic depression in Argentina. Arch Argent Pediatr. 2018; 116(5): 322-327. [ Links ] 27. Alonso Uría RM, Rodríguez Alonso B, Yanes Morales CD, Castillo Isaac E. Characterization of underweight neonate of adolescent mother. Rev Cubana Obstet Ginecol [Internet]. 2018; 44(1): 1-10. [ Links ] 28. Brazil. Ministry of Health. Health Surveillance Secretariat, Department of Health Situation Analysis. Information System on Live Births. Brasilia: Ministry of Health; 2018. [internet]. [access June 17, 2018]. Available in http://tabnet.datasus.gov.br/cgi/deftohtm.exe?sinasc/cnv/nvuf. [ Links ] 29. Grieger J, Grzeskowiak L, Clifton V. Preconception dietary patterns in human pregnancies are associated with preterm delivery. J Nutr. 2014; 144(7): 1075-1080. [ Links ] 30. Kjøllesdal M, Holmboe-Ottesen G. Dietary patterns and birth weight - a Review. AIMS Public Health. 2014; 3(4): 211-225. [ Links ] 31. Gomes C, Malta M, Martiniano A, Bonifácio L, Carvalhaes M. Eating habits of pregnant and non-pregnant women: are there differences? Rev Bras Ginecol Obstet. 2015; 37(7): 325-332. [ Links ] 32. Zapata ME, Rovirosa A, Pueyrredón P, Weill F, Chamorro V, Carella B, et al. Nutritional status of pregnant and lactating women from Argentina. Diaeta (B. Aires). 2016; 34(155): 33-40. [ Links ] 33. Sandoval KV, Nieves ER, Luna MA. Personalized diet effect in pregnant obese or overweight women. Rev Chil Nutr. 2016; 43(3): 233-246. [ Links ] 34. Oliveira J, Silva UR, Pastore V, Azevedo E, Campos G, Silva FCG, et al. Spectrophotometric determination of nitrite in cured meat products. Rev Bras Hig Sanid Anim. 2017; 11(1): 19-31. [ Links ] 35. Hentges D, Zart N, Marmitt LG, Oliveira EC, Scherer Adami F. Nitrate and nitrate concentrations in sausages. Rev Bras Promoç Saude. 2016; 29(1): 27-33. [ Links ] 36. Baek H, Hwang HR, Park H, Kwon A, Qadir AS, Ko S, et al. TNF-α Upregulates Sclerostin Expression in Obese Mice Fed a High-Fat Diet. J Cell Physiol 2013; 229(5): 640-650. [ Links ] 37. Rocha DM, Bressan J, Hermsdorff HH. The role of dietary fatty acid intake in inflammatory gene expression: a critical review. Med J (São Paulo). 2017; 135(2): 157-168. [ Links ] 38. Chassaing B. Involvement of food additives in intestinal inflammation and metabolic syndrome in mice. Med Sci (Paris). 2015; 31(6-7): 586-588. [ Links ] 39. Saidman N, Raele MG, Basile M, Barreto L, Mackinonn MJ, Poy MS et al. Knowledge, interests and beliefs on food and nutrition in pregnant women. Diaeta (B. Aires). 2012; 30(139): 18-27. [ Links ] 40. Fazio E, Nomura R, Dias M, Zugaib M. Dietary intake of pregnant women and maternal weight gain after nutritional counseling. Rev Bras Ginecol Obstet (São Paulo). 2011; 33(2): 87-92. [ Links ] Received: January 21, 2019; Revised: April 26, 2019; Accepted: August 13, 2019 *Corresponding author: Beatriz Rodrigues. Faculdade de Nutrição Emilia de Jesus Ferreiro, Universidade Federal Fluminense. Rua Mário Santos Braga, n. 30, 4° andar. Valonguinho, Centro. Niterói, RJ. CEP 24020-140. Telefone: (21) 2629-9846. E-mail: beatriz.bber@gmail.com This is an Open Access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.	2021-06-20 22:26:33	{"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": 0.36362069845199585, "perplexity": 14017.053165239746}, "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-2021-25/segments/1623488257796.77/warc/CC-MAIN-20210620205203-20210620235203-00230.warc.gz"}
https://www.dml.cz/handle/10338.dmlcz/118983	# Article Full entry \| PDF (0.1 MB) Keywords: nonexpansive mapping; accretive mapping; fixed point theorem; nonlinear integral equations Summary: Let $P$ be a cone in a Hilbert space $H$, $A: P\rightarrow 2^P$ be an accretive mapping (equivalently, $-A$ be a dissipative mapping) and $T:P\rightarrow P$ be a nonexpansive mapping. In this paper, some fixed point theorems for mappings of the type $-A+T$ are established. As an application, we utilize the results presented in this paper to study the existence problem of solutions for some kind of nonlinear integral equations in $L^2(\Omega)$. References: [1] Alspach D.E.: A fixed point free nonexpansive map. Proc. Amer. Math. Soc. 82 (1981), 423-424. MR 0612733 \| Zbl 0468.47036 [2] F. E. Browder F.E.: Nonlinear nonexpansive operators in Banach spaces. Proc. Nat. Acad. Sci. U.S.A 54 (1965), 1041-1044. MR 0187120 [3] Browder F.E.: Nonlinear Operators and Nonlinear Equations of Evolution in Banach Spaces. Proc. Symp. Pure Math. Vol. 18, Part 2 (1976). MR 0405188 \| Zbl 0327.47022 [4] Chang S.S.: Fixed Point Theory with Applications. Chongqing Publishing House, Chongqing (1984). [5] Chen Y.Q.: The fixed point index for accretive mappings with $k$-set contraction perturbation in cones. Internat. J. Math. and Math. Sci. 2 (1996), 287-290. MR 1375990 [6] Chen Y.Q.: On accretive operators in cones of Banach spaces. Nonlinear Anal. TMA 27 (1996), 1125-1135. MR 1407451 \| Zbl 0883.47057 [7] Chen Y.Q., Cho Y.J.: On $1$-set contraction perturbations of accretive operators in cones of Banach spaces. J. Math. Anal. Appl. 201 (1996), 966-980. MR 1400574 \| Zbl 0864.47027 [8] Gatica J.A., Kirk W.A.: Fixed point theorems for contraction mappings with applications to nonexpansive and pseudo-contractive mappings. Rocky Mountain J. Math. 4 (1994), 69-79. MR 0331136 [9] Isac G.: On an Altman type fixed point theorem on convex cones. Rocky Mountain J. Math. 2 (1995), 701-714. MR 1336557 \| Zbl 0868.47035 [10] Kirk W.A., Schonberg R.: Some results on pseudo-contractive mappings. Pacific J. Math. 71 (1977), 89-100. MR 0487615 [11] Morales C.: Pseudo-contractive mappings and the Leray-Schauder boundary condition. Comment. Math. Univ. Carolinae 20 (1979), 745-756. MR 0555187 \| Zbl 0429.47021 [12] Reinermann J., Schonberg R.: Some results and problems in the fixed point theory for nonexpansive and pseudo-contractive mappings in Hilbert spaces. Academic Press, S. Swaminathan ed. (1976). Partner of	2023-01-31 12:37:26	{"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": 0.8970232605934143, "perplexity": 1828.1507097383922}, "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/1674764499871.68/warc/CC-MAIN-20230131122916-20230131152916-00231.warc.gz"}
http://openstudy.com/updates/51511676e4b0ae0b658bdb9a	Here's the question you clicked on: ## julia_copen Group Title Help with this last radical? one year ago one year ago • This Question is Open 1. jim_thompson5910 Group Title what's your question 2. julia_copen Group Title Hang on a second 3. jim_thompson5910 Group Title ok 4. julia_copen Group Title I have to draw it lol 5. jim_thompson5910 Group Title you're fine, no worries 6. jim_thompson5910 Group Title that's definitely the best option instead of describing it in words 7. julia_copen Group Title It isn't working. I have no idea what to do. 8. julia_copen Group Title \|dw:1364269301720:dw\| 9. jim_thompson5910 Group Title ok one sec 10. julia_copen Group Title OK my drawing sucks but this is it. 11. jim_thompson5910 Group Title thanks, and your drawing is perfect, no worries 12. jim_thompson5910 Group Title you need to rationalize the denominator, so you need to multiply top and bottom by $\Large 10\sqrt{2} + \sqrt{10}$ 13. jim_thompson5910 Group Title Doing so will give you $\Large \frac{5\sqrt{2}+\sqrt{10}}{10\sqrt{2} - \sqrt{10}}$ $\Large \frac{(5\sqrt{2}+\sqrt{10})((10\sqrt{2} + \sqrt{10}))}{(10\sqrt{2} - \sqrt{10})(10\sqrt{2} + \sqrt{10})}$ Do you know what to do from here? 14. julia_copen Group Title No this is where I got lost. 15. jim_thompson5910 Group Title ok you would use the difference of squares rule to expand out the denominator $\Large (10\sqrt{2} - \sqrt{10})(10\sqrt{2} + \sqrt{10})$ $\Large (10\sqrt{2})^2 - (\sqrt{10})^2$ $\Large 10^2(\sqrt{2})^2 - (\sqrt{10})^2$ $\Large 1002 - 10$ $\Large 200 - 10$ $\Large 190$ 16. jim_thompson5910 Group Title So $\Large \frac{(5\sqrt{2}+\sqrt{10})((10\sqrt{2} + \sqrt{10}))}{(10\sqrt{2} - \sqrt{10})(10\sqrt{2} + \sqrt{10})}$ turns into $\Large \frac{(5\sqrt{2}+\sqrt{10})((10\sqrt{2} + \sqrt{10}))}{190}$ 17. jim_thompson5910 Group Title You just need to expand out the numerator, then you're done 18. jim_thompson5910 Group Title Doing that will give you $\Large (5\sqrt{2}+\sqrt{10})(10\sqrt{2} + \sqrt{10})$ $\Large 5\sqrt{2}(10\sqrt{2} + \sqrt{10})+\sqrt{10}(10\sqrt{2} + \sqrt{10})$ $\Large 5\sqrt{2}10\sqrt{2} + 5\sqrt{2}\sqrt{10}+\sqrt{10}10\sqrt{2} + \sqrt{10}\sqrt{10}$ $\Large 100 + 5\sqrt{20}+10\sqrt{20} + 10$ $\Large 110 + 15\sqrt{20}$ $\Large 110 + 15\sqrt{45}$ $\Large 110 + 15\sqrt{4}\sqrt{5}$ $\Large 110 + 152\sqrt{5}$ $\Large 110 + 30\sqrt{5}$ 19. jim_thompson5910 Group Title So $\Large \frac{(5\sqrt{2}+\sqrt{10})((10\sqrt{2} + \sqrt{10}))}{190}$ turns into $\Large \frac{110 + 30\sqrt{5}}{190}$ I guess from here you can divide each term by 10 to get $\Large \frac{11 + 3\sqrt{5}}{19}$ and you're done 20. jim_thompson5910 Group Title So this shows us that $\Large \frac{5\sqrt{2}+\sqrt{10}}{10\sqrt{2} - \sqrt{10}} =\frac{11 + 3\sqrt{5}}{19}$ 21. julia_copen Group Title Neat. You explained it so well! I wouldn't have been able to foil that like you did. Thanks! 22. jim_thompson5910 Group Title you're welcome	2014-07-31 01:39:00	{"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": 0.4575350284576416, "perplexity": 5365.862673605577}, "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-2014-23/segments/1406510272256.16/warc/CC-MAIN-20140728011752-00286-ip-10-146-231-18.ec2.internal.warc.gz"}
http://mathhelpforum.com/algebra/83947-simplifying-fractions-exponents.html	# Math Help - Simplifying fractions with exponents 1. ## Simplifying fractions with exponents I have to simplify this and im totally confused step by step help would be so much appreciated! $ \left( \frac {-6x^2y}{2xy^3} \right)^3 $ 2. Originally Posted by HeidiHectic I have to simplify this and im totally confused step by step help would be so much appreciated! $ \left( \frac {-6x^2y}{2xy^3} \right)^3 $ Have you made any attempt at the problem? You should be familiar with these properties of exponentiation: $(ab)^n=a^nb^n$ $\left(\frac ab\right)^n=\frac{a^n}{b^n}$ $a^{-n}=\frac1{a^n}$ $a^ma^n=a^{m+n}$ $\frac{a^m}{a^n}=a^{m-n}$ (where $a,\,b,\,m$ and $n$ have values such that the above expressions are defined) Note that a variable without an exponent can be thought of as being raised to the 1st power. 3. yes, i understand those im just not sure what to do after you distribute the exponent. 4. Originally Posted by HeidiHectic yes, i understand those im just not sure what to do after you distribute the exponent. $\left(\frac{-6x^2y}{2xy^3}\right)^3$ $=\frac{-216x^6y^3}{8x^3y^9}$ $=\frac{-216}{8}\cdot\frac{x^6}{x^3}\cdot\frac{y^3}{y^9}$ $=\frac{-27\cdot8}{8}\cdot\frac{x^6}{x^3}\cdot\frac{y^3}{y^ 9}$ Now can you see how to continue? 5. Originally Posted by HeidiHectic I have to simplify this and im totally confused step by step help would be so much appreciated! $ \left( \frac {-6x^2y}{2xy^3} \right)^3 $ I'd cancel a factor of 2, x and y before cubing $ \left( \frac {-3x}{y^2} \right)^3 = \frac{-27x^3}{y^6} $ 6. Originally Posted by e^(ipi) but 6^3 is 216? That's funny. Where did I get 36? It is a mystery. Originally Posted by e^(ipi) I'd cancel a factor of 2, x and y before cubing This too. I went the other way because Heidi said she (or he) cubed first, but this is indeed easier.	2015-03-30 10:53:02	{"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": 0, "img_math": 0, "codecogs_latex": 15, "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": 0.9014469385147095, "perplexity": 703.23042449805}, "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-2015-14/segments/1427131299261.59/warc/CC-MAIN-20150323172139-00213-ip-10-168-14-71.ec2.internal.warc.gz"}
https://www.gradesaver.com/textbooks/math/algebra/elementary-and-intermediate-algebra-concepts-and-applications-6th-edition/chapter-r-elementary-algebra-review-r-2-equations-inequalities-and-problem-solving-r-2-exercise-set-page-947/41	## Elementary and Intermediate Algebra: Concepts & Applications (6th Edition) $\text{Set Builder Notation: } \{ m\|m\gt12 \} \\\text{Interval Notation: } (12,\infty)$ Using the properties of inequality, then \begin{array}{l}\require{cancel} m-17\gt-5 \\\\ m\gt-5+17 \\\\ m\gt12 .\end{array} Hence, the solution set is \begin{array}{l}\require{cancel} \text{Set Builder Notation: } \{ m\|m\gt12 \} \\\text{Interval Notation: } (12,\infty) .\end{array} In the graph, a hollowed dot is used for $\lt$ or $\gt.$ A solid dot is used for $\le$ or $\ge.$	2018-07-18 01:40:49	{"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": 0.9968591332435608, "perplexity": 7120.468096741064}, "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/1531676589980.6/warc/CC-MAIN-20180718002426-20180718022426-00278.warc.gz"}
https://www.physicsforums.com/threads/oribit-integrator-for-a-logarithmic-potential.853984/	# Oribit integrator for a logarithmic potential Hello! Right know I'm trying to make an orbit integrator for solving a logarithmic potential with the form: \Phi= \frac{v_0^2}{2} ln(x^2+ \frac{y^2}{u^2} + r_0^2) where v0, u, and r0 are constants My approach is to use, \ddot{q} = -\bigtriangledown \Phi Then the system equations, \ddot{x} = -v_o^2 \frac{x}{x^2+ \frac{y^2}{u^2} + r_0^2} \ddot{y} = -\frac{v_o^2}{u^2} \frac{y}{x^2+ \frac{y^2}{u^2} + r_0^2} My guess is that in order to solve for x and y using Runge Kutta or leapfrog, I need to decouple the system, but I don't know how to do so. Related Astronomy and Astrophysics News on Phys.org Dr. Courtney Gold Member 2020 Award If by decouple the system, you mean separate the variables, it probably is not possible. But Runge-Kutta can be used to integrate the equations of motion as is. The Hamiltonian formulation (four equations with first derivatives) is usually easier. By doing the Hamiltonian approach I still get equations (3) and (4) above, and the other two are are apparently of no use. The problem is that I don't know how (if possible) to adapt the Runge-Kutta using two dependent variables (x,y) and the independent one (t). Dr. Courtney	2021-01-21 18:33:43	{"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": 0, "wp-katex-eq": 0, "align": 0, "equation": 4, "x-ck12": 0, "texerror": 0, "math_score": 0.8111824989318848, "perplexity": 1601.3230800011997}, "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/1610703527224.75/warc/CC-MAIN-20210121163356-20210121193356-00130.warc.gz"}
http://www.mathynomial.com/problem/135	Problem #135 135 The polynomial $1-x+x^2-x^3+\cdots+x^{16}-x^{17}$ may be written in the form $a_0+a_1y+a_2y^2+\cdots +a_{16}y^{16}+a_{17}y^{17}$, where $y=x+1$ and the $a_i$'s are constants. Find the value of $a_2$. This problem is copyrighted by the American Mathematics Competitions. Note: you aren't logged in. If you log in, we'll keep a record of which problems you've solved.	2018-02-23 14:16:28	{"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 5, "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": 0.5357741117477417, "perplexity": 311.51156520481226}, "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-2018-09/segments/1518891814787.54/warc/CC-MAIN-20180223134825-20180223154825-00312.warc.gz"}
https://mathematica.stackexchange.com/questions/180023/4d-nintegrate-with-singularities	# 4D NIntegrate with singularities I need to integrate a function in a 4D region (x1,y1,x2,y2), which explodes whenever x1=x2&&y1=y2. The code is as following: {Lx, Ly} = {1/50, 1/50}; f1[x_, y_, n_, m_] := Cos[nPi(Lx + x)/(2Lx)]Sin[mPi(Ly + y)/(2Ly)]; f2[x_, y_, n_, m_] := Sin[nPi(Lx + x)/(2Lx)]Cos[mPi(Ly + y)/(2Ly)]mLx/(nLy); dist[x1_, y1_, x2_, y2_] := Sqrt[(x1 - x2)^2 + (y1 - y2)^2]; int[f_, n_, m_, i_, j_] := int[f, n, m, i, j] = NIntegrate[f[x1, y1, n, m]f[x2, y2, i, j]/dist[x1, y1, x2, y2], {y1, -Lx, Lx}, {x1, -Lx, Lx}, {y2, -Ly, Ly}, {x2, -Lx, Lx}] Then, I want to evaluate int using either f1 or f2, for positive integer (n,m,i,j), for instance using AbsoluteTiming[int[f1,1,2,1,2]]. I may need to compute these integrals thousands of times, for the different possible combinations of (n,m,i,j) - in the most complex cases, each of them may reach up to 14, for instance. So I must be able to calculate the integral as fast as possible. However, the method is simply not working so well, for some. When I compute, for instance: int[f1,4,1,1,1], the computation simply lasts for ever and I receive some scary error messages: NIntegrate::errprec: Catastrophic loss of precision in the global error estimate due to insufficient WorkingPrecision or divergent integral. I have tried multiple integration strategies and methods, but the thing that seems to work best is to simply define the integration boundaries using {y1,-Ly,Ly},{x1,-Lx,Lx},{y2,-Ly,y1,Ly},{x2,-Lx,x1,Lx}, taking advantage of the integration order to define the singularity locations. The integration method "DuffyCoordinates" performed quite fast, but I didn't know whether I was correctly using the "Corners" option, e.g.: int2[f_,n_,m_,i_,j_] := NIntegrate[f[x1, y1, n, m]f[x2, y2, i, j]/dist[x1, y1, x2, y2], {y1, -Lx, Lx}, {x1, -Lx, Lx}, {y2, -Ly, Ly}, {x2, -Lx, Lx},Method -> {"DuffyCoordinates","Corners" -> {0, 0, 1, 1}}] ... But it also gets stuck for (n,m,i,j)=(4,1,1,1). So I don't know what else to do: • Am I doing something wrong with this couple of integration strategies, or missing something obvious? • I suspect much of the problems are due to the integral actually evaluating to 0, maybe because the functions are somehow odd (whatever that means in 4D?), and I'm integrating them in a symmetric interval. But I don't know how I could define my function int in a way that it would check whether the integrand is odd in this 4D symmetrical integration volume. Edit 15/08/2018 corrected the expression of the function dist[] Cross-posted: http://community.wolfram.com/groups/-/m/t/1404998 • You can probably add some small value inside Sqrt to avoid singularities. Have you tried Monte-Carlo based approaches? – David Baghdasaryan Aug 15 '18 at 8:03 • But wouldn't that just become a different function? I tried Method->"MonteCarlo", and it was super fast, but setting different random seeds the results were very inconsistent. I might be doing something wrong, though, since I don't have experience working with MonteCarlo methods, but that really makes me doubt the method – miguel Aug 15 '18 at 9:24 • probably not much different if that value enough small. You can specify PrecisionGoal to a precision you need. See the documentation for that option. – David Baghdasaryan Aug 15 '18 at 9:54 • I've corrected the definition for the function dist[], that sqrt should be in the denominator of the integrand, I'm sorry for wasting your time. Thanks for the tips. These approaches seem to have some convergence issues, and the estimate for the error seems to be almost as large as the estimate for the value of the integral itself. So, for instance running int[f1,1,1,1,4], the integral and error estimates are 6.406210^-8 and 8.4006410^-8. Also, running int[f1,1,1,1,1], the value obtained by the automatic method is 0.0000194422, whereas "AdaptiveQuasiMonteCarlo" gets 0.0000167858 – miguel Aug 15 '18 at 12:57 • Please include links to crossposts of the same question. (General policy is discussed here: mathematica.meta.stackexchange.com/questions/367/…) – Michael E2 Aug 15 '18 at 19:03 I am not sure how helpful this answer is. I have not analyzed the integrand and integral much, just applied couple of tricks. ## Plan 1. Identify which corners have singularities. 2. Split the integral to have one singularity only in the integral terms. 3. Specify the singularity corners to DuffyCoordinates. ## Definitions Clear[int4] Options[int4] = Options[NIntegrate]; int4[f_, n_, m_, i_, j_, opts : OptionsPattern[]] := NIntegrate[ f[x1, y1, n, m]f[x2, y2, i, j]/dist[x1, y1, x2, y2], {y1, 0, Lx}, {x1, 0, Lx}, {y2, 0, Ly}, {x2, 0, Lx}, opts] Clear[int5] Options[int5] = Options[NIntegrate]; int5[f_, n_, m_, i_, j_, opts : OptionsPattern[]] := NIntegrate[ f[x1, y1, n, m]f[x2, y2, i, j]/dist[x1, y1, x2, y2], {y1, -Lx, 0}, {x1, -Lx, 0}, {y2, -Ly, Ly}, {x2, -Lx, 0}, opts] ## Identify singularity corners The identification is done by using the trace of the sampling points of "AdaptiveMonteCarlo". We see that the corners {0,0,0,0} and {0,1,0,1} along the axes 2 and 4 have (most likely) singularities. res = Reap[int3[f1, 4, 1, 1, 1, MinRecursion -> 3, MaxRecursion -> 20, Method -> "AdaptiveMonteCarlo", PrecisionGoal -> 2, EvaluationMonitor :> Sow[{y1, x1, y2, x2}]]]; inds = Flatten[Table[{i, j}, {i, 1, 4}, {j, i + 1, 4}], 1]; ListPlot[RandomSample[res[[2, 1]][[All, #]], 12000], AspectRatio -> Automatic, PlotLabel -> #] & /@ inds ## Split integral and specify corners AbsoluteTiming[ res1 = int4[f1, 4, 1, 1, 1, MaxRecursion -> 20, Method -> {"DuffyCoordinates", "Corners" -> {0, 1, 0, 1}, Method -> {"GlobalAdaptive", "MaxErrorIncreases" -> 20000}}, PrecisionGoal -> 4] ] ( {63.621, 5.3017210^-7} ) AbsoluteTiming[ res2 = int5[f1, 4, 1, 1, 1, MaxRecursion -> 20, Method -> {"DuffyCoordinates", "Corners" -> {0, 0, 0, 0}, Method -> {"GlobalAdaptive", "MaxErrorIncreases" -> 20000}}, PrecisionGoal -> 4] ] (* {35.0088, -6.650310^-7} ) res1 + res2 (* -1.3485710^-7 ) ` These give error message but the results might be good enough.	2020-08-15 12:10:19	{"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": 0.5995560884475708, "perplexity": 3888.539086459502}, "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/1596439740838.3/warc/CC-MAIN-20200815094903-20200815124903-00437.warc.gz"}
https://www.findfilo.com/maths-question-answers/let-s-be-the-area-of-the-region-enclosed-by-y-e-x-d6s	Let Sbe the area of the region enclosed by y=e^(-x^2),y=0,x=0,a n \| Filo class 12 Math Calculus Application Of Integrals 528 150 Let be the area of the region enclosed by Then (a) (b) (c) (d) 528 150 Connecting you to a tutor in 60 seconds.	2021-06-24 23:46:53	{"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": 0.8870466351509094, "perplexity": 1343.0658987709005}, "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-2021-25/segments/1623488560777.97/warc/CC-MAIN-20210624233218-20210625023218-00572.warc.gz"}
https://math.stackexchange.com/questions/3512372/showing-this-iterative-ode-solver-converges-quadratically	# Showing this iterative ODE solver converges quadratically Given the ODE: $$y'(t) = f(y(t)), y(0) = y_0,$$ And the following method to solve the ODE: $$y_{n+1} = y_n +\frac{h}{2}(f(y_n)+f(y_n+hf(y_n))),$$ I am trying to show the method converges quadratically. I looked at the error at a time $$t_n=hn$$ which I denoted as $$e_n = \|y_n-y(t_n)\|$$ and then tried to find $$p>0$$ s.t. : $$\lim_{n\rightarrow{\infty}}\frac{\|e_{n+1}\|}{\|e_n\|^p}=\lambda$$ for some $$\lambda>0,$$ but I ran into the problem in the calculation that is: how do I use $$y(t_n)$$, or at least simplify it or manipulate it to make it useful for finding such a $$\lambda$$? Look for Heun's (2nd order) method, you will find more details. What quadratic or second order convergence means for ODE solutions is that the error of the numerical solution against the exact solution is $$O(h^2)$$, which is different from quadratic convergence for the Newton method. Or more precisely, let $$y(t)$$ be the exact solution for the IVP with $$y(t_0)=y_0$$ and $$y_n\approx y(t_n)$$ the numerical solution for step size $$h$$. Further $$L$$ the Lipschitz constant of $$f$$. Then there exists some constant $$M\sim \sup_t \|y'''(t)\|$$ so that the individual local truncation errors are bounded by $$Mh^3$$. Propagation of previous errors happens at a rate bounded by the Lipschitz constant $$L$$, giving an error propagation inequality of $$e_{n+1}\le(1+Lh)e_n+Mh^3,$$ which solves to $$e_n=\|y_n-y(t_n)\|\le \frac{e^{L\|t_n-t_0\|}-1}L\cdot Mh^2,$$ using $$(1+Lh)^n\le e^{Lhn}$$.	2020-02-28 07:31:12	{"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": 22, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9160517454147339, "perplexity": 141.48430211310477}, "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/1581875147054.34/warc/CC-MAIN-20200228043124-20200228073124-00532.warc.gz"}
https://forum.azimuthproject.org/discussion/1092/proving-1-the-law-of-mass-action-and-2-the-poisson-character-of-the-reactions	#### Howdy, Stranger! It looks like you're new here. If you want to get involved, click one of these buttons! Options # Proving (1) the law of mass action, and (2) the Poisson character of the reactions Hello, I'm working away on my second blog article. It's still too unformed for anyone else to read it. But I have some questions about the content that came up. At first I was simply going to give the definition for stochastic Petri nets, give the formulas for the mass action kinetics, and then present the simulator. But the good simulator algorithm uses the exponential distribution for the inter-event intervals, and the correctness of that is predicated on the Poisson character of the reaction events. This is a lot to just throw at a reader who may be coming from just a software background. I want this to be an effective "praxis article," which means that we have to understand what the heck we are talking about, all the way down the line, starting from the theory that supports the model, right down to the programming language technology that will implement the simulator. In the first blog article, I dug into the programming technology. Here the links that need more attention are the theoretical supports. In my notes I have written an informal yet semi-rigorous explanation of what a random process is, and what a Poisson random process is. It's written in a "popular mechanics" tone, and this won't give me a problem to write it nicely for the reader. To focus the discussion, suppose there are species A, B and C, and there is a transition X: A + B --> C. Suppose this takes place in a closed container, which is a homogeneous "soup" of the entities, and consider them to be point particles. Assume they are bouncing around randomly, colliding with each other and the walls. Let A(t), B(t), C(t) be the number of entities of each type in the container at time t. Now, as we know, the mass action kinetics says that the firing rate of X is proportional to A(t) * B(t), with a coefficient that is a function of Temperature, among other things. Now that I am acquainted with this material, this relationship is intuitively clear, and I can give an intuitive explanation. Let G be a small region of the container, and T be a small time interval. Let A(G,T) be the expected number of A particles found in (G,T), and B(G,T) be the expected number of B particles found in (G,T). (By "found" I mean particles which are in G at at least one time point in T.) Let Z be the probability of the transition firing between a single A particle and a single B particle, given that they are both found in (G,T). Then the expected number of firings in (G,T) is Z * A(G,T) * B(G,T). Each combination of one of the A particles in (G,T) and one of the B particles in (G,T) contributes Z to the expected number of transitions that fire in (G,T). But I would like to go further, and prove (using informal language) that the X-firings constitute a Poisson process, under the assumption that the movements of the A and the B particles are a Poisson process. And to derive the formula for the rate constant of the X-firings, which will involve the factors A(t) * B(t). I have worked out a proof, which is more complex and nuanced than I had hoped for. I will summarize it now. But can anyone point me to a proof in the literature (online is best), or summarize the idea of the standard proof, or make any suggestions about how to simplify the following argument. Let A-Count(T,G) be the random variable that counts the number of A particles that are found in (G,T). (I.e., whose world-lines intersect G x T). Under the assumption that the particles are freely colliding, it is not hard to show that A-Count is a Poisson process. Similarly for B-Count. Now in order to show that the X-firings constitute a Poisson process with a rate parameter, we need to further analyze the formula I gave above for the expected number of firings Z * A(G,T) * B(G,T). Let T = (t, t + deltaT). Since A is Poisson, we have that: A(G,T) =~ Prob(A-Count(G,T) > 0) =~ Prob(A-Count(G,T) = 1) = Lambda(A,G,t) * dT, where Lambda(A,G,t) is the rate constant for the A Poisson process. Now it is evident that Lambda(A,G,T) will be proportional to A(t), the number of A's in the system at time t, and also to a constant that is an increasing function of the mean velocity of the particles. So let's write: Lambda(A,G,t) = A(t) * Sigma(A,t), where Sigma(A,t) includes the temperature dependency. Putting this all together, we have the following formula for the expected number of X-firings in (G,T): X-Firings(G,T) = Z * A(G,T) * B(G,T) = Z * (Lambda(A,G,t) * deltaT) * (Lambda(B,G,t) * deltaT) = Z * (A(t) * Sigma(A,t) * deltaT) * (B(t) * Sigma(B,t) * deltaT) = Z * A(t) * B(t) * Sigma(A,t) * Sigma(B,t) * deltaT^2. This looks promising, except we have the apparent paradox that the X-firings appear to have a quadratic dependency on deltaT -- so it doesn't look like a Poisson process at all! The key to this riddle is the fact that that the probability Z is itself a function of G and T. Z(G,T) = probability that X fires given that there is an A in (G,T) and a B in (G,T) = E(G,T) * F(G), where: E(G,T) is defined to be the conditional probability that, given that there is an A in (G,T) and a B in (G,T), that this A and B are in G at the same exact time (for some time t in T), and F(G) is defined to be the probability that, given that there is an A and a B in G at the same exact time, that the transition will fire. The point here is that for a long interval of time T, the fact that A and B are both in (G,T) leads to a low probability E(G,T) that they are actually there at the same time, and hence have a chance to react. Now let us further assume that G is small in relation to the mean velocity of the particles, so that if a particle is present in (G,T), then it will quickly zip in and out, and actual time interval (t1,t2) for which it is present in G is a small sub-interval of T. Under this assumption, we can show that E(G,T) is inversely proportional to the length of T. If we halve the length of T, then we will double the probability that the actual time interval for an A that is found in (G,T) will intersect with the actual time interval for a B that is found in (G,T). We've squeezed these two intervals into a T that is half the size, and so they are twice as likely to intersect. (Approximately speaking.) Using this fact, we can write: E(G,T) = (1/deltaT) * E'(G,t), for some function E' that depends only on G and t, but not the interval deltaT. Putting these together we get: Z(G,T) = E(G,T) * F(G) = (1/deltaT) * E'(G,t) * F(G). Putting this into our rate formula, we have: X-Firings(G,T) = Z(G,T) * A(G,T) * B(G,T) = Z(G,T) * A(t) * B(t) * Sigma(A,t) * Sigma(B,t) * deltaT^2 = (1/deltaT) * E'(G,t) * F(G) * A(t) * B(t) * Sigma(A,t) * Sigma(B,t) * deltaT^2 = K(G,t) * Sigma(A,t) * Sigma(B,t) * A(t) * B(t) * deltaT where K(G,t) = E'(G,t) * F(G). This gives is the rate parameter for the Poisson process: K(G,t) * Sigma(A,t) * Sigma(B,t) * A(t) * B(t) • Options 1. This result looks right to me. But there is one point in the argument that I am uncomfortable about. I made the assumption that G was small in comparison to the speeds of the particles. That was how I was able to factor the (1/deltaT) out of E. But to really show that this is the rate parameter for a Poisson process, deltaT would have pass to 0 in the limit. But the factoring of (1/deltaT) out of E breaks down as deltaT gets very small. So what this argument shows is that, for a wide range of deltaT's, that aren't "too small", the formula for X-firings above, with the given rate parameter is correct. If you want to go down to a smaller level of deltaT's, you'd have to repeat the argument with a smaller G. Also, it would be nice to factor out vol(G) from the term K(G,t), so that the only dependence on G is through vol(G). Any advice or pointers? Can this be cleaned up and simplified, while still maintain the general level of rigor that I have set out here? Regardless of the extent to which I will be including this argument into the blog article, I'd like to get the proof clear in my mind, so that I won't feel like I'm fuzzing over the issue when I do write the blog article. Thanks! Comment Source:This result looks right to me. But there is one point in the argument that I am uncomfortable about. I made the assumption that G was small in comparison to the speeds of the particles. That was how I was able to factor the (1/deltaT) out of E. But to really show that this is the rate parameter for a Poisson process, deltaT would have pass to 0 in the limit. But the factoring of (1/deltaT) out of E breaks down as deltaT gets very small. So what this argument shows is that, for a wide range of deltaT's, that aren't "too small", the formula for X-firings above, with the given rate parameter is correct. If you want to go down to a smaller level of deltaT's, you'd have to repeat the argument with a smaller G. Also, it would be nice to factor out vol(G) from the term K(G,t), so that the only dependence on G is through vol(G). * * * Any advice or pointers? Can this be cleaned up and simplified, while still maintain the general level of rigor that I have set out here? Regardless of the extent to which I will be including this argument into the blog article, I'd like to get the proof clear in my mind, so that I won't feel like I'm fuzzing over the issue when I do write the blog article. Thanks! • Options 2. David wrote: I made the assumption that G was small in comparison to the speeds of the particles. That sounds reasonable; people often say the law of mass action holds when the chemicals are 'well mixed', and this seems somehow related. If the molecules aren't zipping around enough, so each mainly interacts with its 'neighbors', the law of mass action won't hold. I'm afraid I'm too distracted to tackle the issue you're actually concerned with right now... Comment Source:David wrote: > I made the assumption that G was small in comparison to the speeds of the particles. That sounds reasonable; people often say the law of mass action holds when the chemicals are 'well mixed', and this seems somehow related. If the molecules aren't zipping around enough, so each mainly interacts with its 'neighbors', the law of mass action won't hold. I'm afraid I'm too distracted to tackle the issue you're actually concerned with right now... • Options 3. I want to talk more about this stuff! I have to travel tomorrow, but return on Saturday. I plan to look at this over the weekend! Somehow, I'm very busy right now as everything seems to be due this November :( Comment Source:I want to talk more about this stuff! I have to travel tomorrow, but return on Saturday. I plan to look at this over the weekend! Somehow, I'm very busy right now as everything seems to be due this November :( • Options 4. edited October 2012 By the way, I hope you do the obvious thing and read what turns up when you Google derivation law of mass action. I don't instantly see a 'straightforward' derivation of the sort you're attempting - I see more people trying to derive it from laws of thermodynamics, which seems interesting but peculiar. I see a book: available on Kindle for a whopping sum, which seems to contain at least one derivation and would definitely be worth looking at. I also saw (but don't see now) a derivation using quantum mechanics. What you're trying to do is so nice and simple that I feel it must have been done, maybe even by Boltzmann, but I don't see it yet! Comment Source:By the way, I hope you do the obvious thing and read what turns up when you Google [derivation law of mass action](https://www.google.com/search?q=derivation+law+of+mass+action). I don't instantly see a 'straightforward' derivation of the sort you're attempting - I see more people trying to derive it from laws of thermodynamics, which seems interesting but peculiar. I see a book: * Andrei B. Koudriavtsev, Reginald F. Jameson, Wolfgang Linert, [The Law of Mass Action](http://www.amazon.com/The-Law-Mass-Action-ebook/dp/B000QXD8IM). available on Kindle for a whopping sum, which seems to contain at least one derivation and would definitely be worth looking at. I also saw (but don't see now) a derivation using quantum mechanics. What you're trying to do is so nice and simple that I feel it _must_ have been done, maybe even by Boltzmann, but I don't see it yet! • Options 5. edited November 2012 Hi, I have a revised proof, which is simpler, and avoids the need to assume that the region G is small. Again, assume species A, B, and C, and a transition X: A + B --> C. Let G be a region of the container, and T = (t, t + deltaT) be an interval of time. Let's view it from the perspective of S = G x T as a region of space-time. Let a and b be individual particles whose world-lines intersect S. Let Ta be the sub-interval of T, consisting of the times that a is inside of S, and Tb be the same thing for particle b. Assumption: The probability of the reaction taking place in S between a and b is equal to the length of the intersection of Ta and Tb, times a constant factor K(G) that depends, among other things, on G. Let Amount(A,S) = Sum Ta, for all particles a. Consider this to be in units of A-particle-seconds. Amount(A,S) is clearly proportional to the number of A(t) of A particles in the container during the (small) interval T. Claim now that the reaction rate is proportional to Amount(A,S) * Amount(B,S). Now partition T into many sub-intervals of length dt. Accordingly, partition all of the world-lines of the particles into small segments (fragments), each of which has a duration of dt. Let Fa denote one of these segments of an a-world line, and the same for Fb. Any reaction between specific a and b particles will take place between an Fa and an Fb that overlap in time, and are close in space (a near-crossing). Let P(deltaT) be the probability that a reaction will take place between a randomly chosen Fa and Fb in S. Then the expected number of reactions in S will equal: P(deltaT) * Amount(A,S) * Amount(B,S) = P(deltaT) * A(t) * B(t) * K, Now it is not hard to show that P(deltaT) is proportional to deltaT = length(T). Once we have shown that, it follows that the reactions are a Poisson process, with rate parameter that is proportional to A(t) * B(t) -- i.e., we have proven the law of mass action. So let's divide T in 2 (while leaving the much smaller dt unchanged). Let T1 be the first half of T, and T2 be the second half of T. Let Fa, Fb be randomly chosen fragments in S = G x T = G x T1 union G x T2. If Fa is in T1 and Fb is in T2, then they have no time overlap, and there is zero chance that they will react. So the probability that they will react in T is equal to the probability that they will react in T1, plus the probability that they will react in T2. I.e., P(deltaT) = 2 * P(deltaT / 2). That proves that P is proportional to deltaT, and so we are done. Now, it would be good to find out where this kind of direct proof has already been done. I did try web searches. John, I will try to get a hold of that book you cited. Aside from that, can you John or Jacob or anyone else comment on the validity of the proof that I just gave. I'd like to keep the blog articles rolling along, so I don't want to dwell on this point for too long. If nobody here spots any problems with the argument, I could just make the statement: here is a relatively simple way to understand why the law of mass action is true. Thanks Comment Source:Hi, I have a revised proof, which is simpler, and avoids the need to assume that the region G is small. Again, assume species A, B, and C, and a transition X: A + B --> C. Let G be a region of the container, and T = (t, t + deltaT) be an interval of time. Let's view it from the perspective of S = G x T as a region of space-time. Let a and b be individual particles whose world-lines intersect S. Let Ta be the sub-interval of T, consisting of the times that a is inside of S, and Tb be the same thing for particle b. Assumption: The probability of the reaction taking place in S between a and b is equal to the length of the intersection of Ta and Tb, times a constant factor K(G) that depends, among other things, on G. Let Amount(A,S) = Sum Ta, for all particles a. Consider this to be in units of A-particle-seconds. Amount(A,S) is clearly proportional to the number of A(t) of A particles in the container during the (small) interval T. Claim now that the reaction rate is proportional to Amount(A,S) * Amount(B,S). Now partition T into many sub-intervals of length dt. Accordingly, partition all of the world-lines of the particles into small segments (fragments), each of which has a duration of dt. Let Fa denote one of these segments of an a-world line, and the same for Fb. Any reaction between specific a and b particles will take place between an Fa and an Fb that overlap in time, and are close in space (a near-crossing). Let P(deltaT) be the probability that a reaction will take place between a randomly chosen Fa and Fb in S. Then the expected number of reactions in S will equal: P(deltaT) * Amount(A,S) * Amount(B,S) = P(deltaT) * A(t) * B(t) * K, Now it is not hard to show that P(deltaT) is proportional to deltaT = length(T). Once we have shown that, it follows that the reactions are a Poisson process, with rate parameter that is proportional to A(t) * B(t) -- i.e., we have proven the law of mass action. So let's divide T in 2 (while leaving the much smaller dt unchanged). Let T1 be the first half of T, and T2 be the second half of T. Let Fa, Fb be randomly chosen fragments in S = G x T = G x T1 union G x T2. If Fa is in T1 and Fb is in T2, then they have no time overlap, and there is zero chance that they will react. So the probability that they will react in T is equal to the probability that they will react in T1, plus the probability that they will react in T2. I.e., P(deltaT) = 2 * P(deltaT / 2). That proves that P is proportional to deltaT, and so we are done. * * * Now, it would be good to find out where this kind of direct proof has already been done. I did try web searches. John, I will try to get a hold of that book you cited. Aside from that, can you John or Jacob or anyone else comment on the validity of the proof that I just gave. I'd like to keep the blog articles rolling along, so I don't want to dwell on this point for too long. If nobody here spots any problems with the argument, I could just make the statement: here is a relatively simple way to understand why the law of mass action is true. Thanks • Options 6. edited November 2012 I wrote: Then the expected number of reactions in S will equal: P(deltaT) * Amount(A,S) * Amount(B,S) = P(deltaT) * A(t) * B(t) * K, Now it is not hard to show that P(deltaT) is proportional to deltaT = length(T). Once we have shown that, it follows that the reactions are a Poisson process, with rate parameter that is proportional to A(t) * B(t) -- i.e., we have proven the law of mass action. I got this backwards, but it is not hard to fix. We want to show that the rate, which is the product P(deltaT) * Amount(A,S) * Amount(B,S) is proportional to deltaT = length(T). Before we showed that Amount(A,S) and Amount(B,S) are both proportional to deltaT. Therefore we need to show that P(deltaT) is inversely proportional to deltaT. Here goes: Again, divide T in 2 (while leaving the much smaller dt unchanged). Let T1 be the first half of T, and T2 be the second half of T. Let Fa, Fb be randomly chosen fragments in S = G x T = G x T1 union G x T2. If Fa is in T1 and Fb is in T2, then they have no time overlap, and there is zero chance that they will react. There is a 50% chance that either (Fa is in T1 and Fb is in T2), or (Fa is in T2 and Fb is in T1), in which case they have zero probability of reacting. The other 50% of the time, they will either both be in T1, or both be in T2. In this case, the probability of them reacting is P(deltaT / 2). Therefore the overall probability of them reacting is 50% * P(deltaT / 2), i.e., P(deltaT) = 0.5 * P(deltaT / 2), i.e., P(deltaT / 2) = 2 * P(deltaT), which says that P is inversely proportional to deltaT, and we are done. Comment Source:I wrote: > Then the expected number of reactions in S will equal: > P(deltaT) * Amount(A,S) * Amount(B,S) = P(deltaT) * A(t) * B(t) * K, > Now it is not hard to show that P(deltaT) is proportional to deltaT = length(T). Once we have shown that, it follows that the reactions are a Poisson process, with rate parameter that is proportional to A(t) * B(t) -- i.e., we have proven the law of mass action. I got this backwards, but it is not hard to fix. We want to show that the rate, which is the product P(deltaT) * Amount(A,S) * Amount(B,S) is proportional to deltaT = length(T). Before we showed that Amount(A,S) and Amount(B,S) are both proportional to deltaT. Therefore we need to show that P(deltaT) is _inversely_ proportional to deltaT. Here goes: Again, divide T in 2 (while leaving the much smaller dt unchanged). Let T1 be the first half of T, and T2 be the second half of T. Let Fa, Fb be randomly chosen fragments in S = G x T = G x T1 union G x T2. If Fa is in T1 and Fb is in T2, then they have no time overlap, and there is zero chance that they will react. There is a 50% chance that either (Fa is in T1 and Fb is in T2), or (Fa is in T2 and Fb is in T1), in which case they have zero probability of reacting. The other 50% of the time, they will either both be in T1, or both be in T2. In this case, the probability of them reacting is P(deltaT / 2). Therefore the overall probability of them reacting is 50% * P(deltaT / 2), i.e., P(deltaT) = 0.5 * P(deltaT / 2), i.e., P(deltaT / 2) = 2 * P(deltaT), which says that P is inversely proportional to deltaT, and we are done. • Options 7. edited November 2012 I have third proof which is better than the ones I've given here. I'm putting it into the blog article, for which I will post a message when it is reviewable. The plot got thicker than I had imagined it would -- so it's been slower going than I had hoped. Comment Source:I have third proof which is better than the ones I've given here. I'm putting it into the blog article, for which I will post a message when it is reviewable. The plot got thicker than I had imagined it would -- so it's been slower going than I had hoped. • Options 8. Hi David, I'm sorry I have not yet had a chance to review your derivation as we discussed on Tuesday. David said: I have third proof which is better than the ones I’ve given here. I’m putting it into the blog article, for which I will post a message when it is reviewable. Given that, I'll plan to go through the details once you've posted them on the wiki. Maybe by then you'll be so confident in your proof that you won't care anymore if I review it, but I'll plan to read it anyway because I think it's interesting. Comment Source:Hi David, I'm sorry I have not yet had a chance to review your derivation as we discussed on Tuesday. David said: > I have third proof which is better than the ones I’ve given here. I’m putting it into the blog article, for which I will post a message when it is reviewable. Given that, I'll plan to go through the details once you've posted them on the wiki. Maybe by then you'll be so confident in your proof that you won't care anymore if I review it, but I'll plan to read it anyway because I think it's interesting. • Options 9. Thanks! Comment Source:Thanks! • Options 10. edited November 2012 I came across the first five chapters of an book on physical chemistry, by Georg Job and Regina Ruffler. It starts with a creative approach to explaining entropy, by describing it in "phenomenological" terms as a kind of substance (like the view of heat as a substance), along with empirical ways to measure it. This is not fully satisfying, because it leaves me with the question, what is this substance called entropy, but it is creative, and the writing is has a literature-like quality. It also contains nice descriptions of things like how a refrigerator works. It also has chapters on the "chemical potential" (which I didn't get on the first skim of it), and on the law of mass action. I'm looking for good references to introductory texts and online materials on physical chemistry, so if anyone has any, I would be appreciative. Thanks. Comment Source:I came across the first five chapters of an book on [physical chemistry](http://job-stiftung.de/pdf/buch/physical_chemistry_five_chapters.pdf), by Georg Job and Regina Ruffler. It starts with a creative approach to explaining entropy, by describing it in "phenomenological" terms as a kind of substance (like the view of heat as a substance), along with empirical ways to measure it. This is not fully satisfying, because it leaves me with the question, what _is_ this substance called entropy, but it is creative, and the writing is has a literature-like quality. It also contains nice descriptions of things like how a refrigerator works. It also has chapters on the "chemical potential" (which I didn't get on the first skim of it), and on the law of mass action. I'm looking for good references to introductory texts and online materials on physical chemistry, so if anyone has any, I would be appreciative. Thanks. • Options 11. edited November 2012 You might get quite a lot from my friend Mark Leach's metasynthesis site. It carries an endorsement from Hoffman who won a Nobel prize for Frontier Molecular Orbital Theory (FMO). hth Comment Source:You might get quite a lot from my friend Mark Leach's [metasynthesis site](http://www.metasynthesis.co.uk). It carries an endorsement from Hoffman who won a Nobel prize for Frontier Molecular Orbital Theory (FMO). hth • Options 12. Bingo: Just found this today. So for one of my upcoming blog articles I ended up recreating some of the key points from this work from 1992. But it was a good exercise, and I'm happy with the form of the argument that I will give. Also I will be talking about the limitations of the law of mass action, which is an approximation that loses validity as concentrations increase. I saw this stated in a paper, though I haven't yet found an explanation for it in the literature. In my assessment it is due to the diameter of the molecules. First there is the obvious issue that the finite size of the molecules puts an absolute limit on the concentrations. But let's look at what happens when concentrations are high, but the container is not fully packed. Suppose the reaction is between species A and B. Now, doubling the concentration of A does in fact double the number of expected crossings between A and B particles. Specifically, I look at "epsilon-crossings," meaning near-crossings where the two particles come within a distance of epsilon from each other. Here, the epsilon of interest will be the radius of A plus the radius of B. But the A molecules are "competing" to react with the B molecules, and so the presence of more A molecules reduces the conditional probability that any given epsilon crossing will actually react. Hence the dependence of the reaction rate on each of the species concentrations will be sub-linear. As we know, there is also the breakdown in the law, at low concentrations, for reactions that take multiple inputs from the same species. There, the correction is to use the falling powers, rather than the regular power, for the concentrations that appear more than once in the input. Stochastic behavior at low concentrations is also of practical interest. One paper I read pointed this out, for biochemical reaction networks. There, there can be relatively few molecules -- but large ones -- that are participating in the communication pathways, over long durations. There was a quote of just a handful of codons being transcribed per second. In a wiki page, I'm going to start an annotated bibliography on Petri nets. This deserves its own page, but we can link to it from Recommended Reading. If such a page already exists, please let me know. If I haven't heard in a few days I create it. Comment Source:Bingo: * Daniel T. Gillespie, [A rigorous derivation of the chemical master equation](http://citeseerx.ist.psu.edu/viewdoc/summary/?doi=10.1.1.159.5220), Physica A 188 (1992) 404-425. Just found this today. So for one of my upcoming blog articles I ended up recreating some of the key points from this work from 1992. But it was a good exercise, and I'm happy with the form of the argument that I will give. Also I will be talking about the limitations of the law of mass action, which is an approximation that loses validity as concentrations increase. I saw this stated in a paper, though I haven't yet found an explanation for it in the literature. In my assessment it is due to the diameter of the molecules. First there is the obvious issue that the finite size of the molecules puts an absolute limit on the concentrations. But let's look at what happens when concentrations are high, but the container is not fully packed. Suppose the reaction is between species A and B. Now, doubling the concentration of A does in fact double the number of expected crossings between A and B particles. Specifically, I look at "epsilon-crossings," meaning near-crossings where the two particles come within a distance of epsilon from each other. Here, the epsilon of interest will be the radius of A plus the radius of B. But the A molecules are "competing" to react with the B molecules, and so the presence of more A molecules reduces the conditional probability that any given epsilon crossing will actually react. Hence the dependence of the reaction rate on each of the species concentrations will be sub-linear. As we know, there is also the breakdown in the law, at low concentrations, for reactions that take multiple inputs from the same species. There, the correction is to use the falling powers, rather than the regular power, for the concentrations that appear more than once in the input. Stochastic behavior at low concentrations is also of practical interest. One paper I read pointed this out, for biochemical reaction networks. There, there can be relatively few molecules -- but large ones -- that are participating in the communication pathways, over long durations. There was a quote of just a handful of codons being transcribed per second. In a wiki page, I'm going to start an annotated bibliography on Petri nets. This deserves its own page, but we can link to it from Recommended Reading. If such a page already exists, please let me know. If I haven't heard in a few days I create it. • Options 13. I think the annotated bibliography on Petri nets deserves to be near the bottom of Petri net, where we already have a bibliography! If it gets enormous we can break it off as a page on its own. I'm really glad you want to add more references, and annotation is crucial since a huge undigested pile of references is not very useful. So, how about that blog article? Comment Source:I think the annotated bibliography on Petri nets deserves to be near the bottom of [[Petri net]], where we already have a bibliography! If it gets enormous we can break it off as a page on its own. I'm really glad you want to add more references, and annotation is crucial since a huge undigested pile of references is not very useful. So, how about that blog article? <img src = "http://math.ucr.edu/home/baez/emoticons/wink.gif" alt = ""/> • Options 14. edited December 2012 Ok, this week I will get it into a reviewable state -- I will let you know. I have been working on it, but my attention is diverted by... Life. We just had our house wired by an AV company (for sound and Lan), but they didn't configure the AV wireless network properly. So I've spent the last couple days doing network troubleshooting... it is interesting to learn about, though the benchmarking gets to be a grind. Thanks for reminding me about applications that sit above the transport layer :) The Wikinson book that you mentioned on the blog, "Stochastic Modelling for Systems Biology," just arrived. It's a great read. I will start adding bibliographic notes to the Petri net page, as you suggest. Comment Source:Ok, this week I will get it into a reviewable state -- I will let you know. I have been working on it, but my attention is diverted by... Life. We just had our house wired by an AV company (for sound and Lan), but they didn't configure the AV wireless network properly. So I've spent the last couple days doing network troubleshooting... it is interesting to learn about, though the benchmarking gets to be a grind. Thanks for reminding me about applications that sit above the transport layer :) The Wikinson book that you mentioned on the blog, "Stochastic Modelling for Systems Biology," just arrived. It's a great read. I will start adding bibliographic notes to the Petri net page, as you suggest. • Options 15. Dave wrote As we know, there is also the breakdown in the law, at low concentrations, for reactions that take multiple inputs from the same species. There, the correction is to use the falling powers, rather than the regular power, for the concentrations that appear more than once in the input. Stochastic behavior at low concentrations is also of practical interest. One paper I read pointed this out, for biochemical reaction networks. There, there can be relatively few molecules – but large ones – that are participating in the communication pathways, over long durations. There was a quote of just a handful of codons being transcribed per second Note that for looking at low concentration behaviour you might be interested in A symbolic computational approach to a problem involving multivariate Poisson distributions Comment Source:Dave wrote > As we know, there is also the breakdown in the law, at low concentrations, for reactions that take multiple inputs from the same species. There, the correction is to use the falling powers, rather than the regular power, for the concentrations that appear more than once in the input. Stochastic behavior at low concentrations is also of practical interest. One paper I read pointed this out, for biochemical reaction networks. There, there can be relatively few molecules – but large ones – that are participating in the communication pathways, over long durations. There was a quote of just a handful of codons being transcribed per second Note that for looking at low concentration behaviour you might be interested in [A symbolic computational approach to a problem involving multivariate Poisson distributions](http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/mvp.pdf)	2022-09-27 16:59:28	{"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": 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": 0.7566021680831909, "perplexity": 662.0270895910339}, "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/1664030335054.79/warc/CC-MAIN-20220927162620-20220927192620-00683.warc.gz"}
https://www.physicsforums.com/threads/doing-a-phase-space-portrait-in-matlab.711102/	Doing a phase-space portrait in matlab • MATLAB bagram So I have this system of equations: $$\binom{x_{n+1}}{y_{n+1}}=\begin{pmatrix}e^{r} & 0 \\ 0 & e^{-r} \end{pmatrix}\begin{pmatrix}cos(\phi+I_{n}) & -sin(\phi+I_{n}) \\ sin(\phi+I_{n}) & cos(\phi+I_{n}) \end{pmatrix}\begin{pmatrix}x_{n}\\ y_{n} \end{pmatrix}$$ where $$I_{n}=x_{n}^2+y_{n}^2$$ I have no idea how to plot that in matlab as a phase-space portrait... Any help would be great	2022-08-11 20:57:35	{"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": 0.35151880979537964, "perplexity": 1741.3585960573764}, "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-2022-33/segments/1659882571502.25/warc/CC-MAIN-20220811194507-20220811224507-00179.warc.gz"}
https://artofproblemsolving.com/wiki/index.php?title=2012_AIME_I_Problems/Problem_6&diff=prev&oldid=127746	# Difference between revisions of "2012 AIME I Problems/Problem 6" ## Problem 6 The complex numbers $z$ and $w$ satisfy $z^{13} = w,$ $w^{11} = z,$ and the imaginary part of $z$ is $\sin{\frac{m\pi}{n}}$, for relatively prime positive integers $m$ and $n$ with $m Find $n.$ ## Solution Substituting the first equation into the second, we find that $(z^{13})^{11} = z$ and thus $z^{143} = z.$ We know that $z \neq 0,$ because we are given the imaginary part of $z,$ so we can divide by $z$ to get $z^{142} = 1.$ So, $z$ must be a $142$nd root of unity, and thus, by De Moivre's theorem, the imaginary part of $z$ will be of the form $\sin{\frac{2k\pi}{142}} = \sin{\frac{k\pi}{71}},$ where $k \in \{1, 2, \ldots, 70\}.$ Note that $71$ is prime and $k<71$ by the conditions of the problem, so the denominator in the argument of this value will always be $71.$ Thus, $n = \boxed{071}.$	2022-05-27 04:23:26	{"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": 25, "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": 0.8452293276786804, "perplexity": 1747.2679395918592}, "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-2022-21/segments/1652662631064.64/warc/CC-MAIN-20220527015812-20220527045812-00413.warc.gz"}
https://www.repidemicsconsortium.org/projections/reference/project.html	This function simulates future incidence based on past incidence data, a selection of plausible reproduction numbers (R), and the distribution of the serial interval (time from primary onset to secondary onset). project( x, R, si, n_sim = 100, n_days = 7, R_fix_within = FALSE, model = c("poisson", "negbin"), size = 0.03, time_change = NULL ) ## Arguments x An incidence object containing daily incidence; other time intervals will trigger an error. A vector of numbers representing plausible reproduction numbers; for instance, these can be samples from a posterior distribution using the earlyR or EpiEstim packages. If time_change is provided, then it must be a vector (for fixed values of R per time window) or a list of vectors (for separate distributions of R per time window), with one element more than the number of dates in time_change. A function computing the serial interval, or a numeric vector providing its mass function, starting a day 1, so that si[i] is the PMF for serial interval of i. The model implicitly assumes that si[0] = 0. For functions, we strongly recommend using the RECON package distcrete to obtain such distribution (see example). The number of epicurves to simulate. Defaults to 100. The number of days to run simulations for. Defaults to 14. A logical indicating if R should be fixed within simulations (but still varying across simulations). If FALSE, R is drawn for every simulation and every time step. Fixing values within simulations favours more extreme predictions (see details) Distribution to be used for projections. Must be one of "poisson" or "negbin" (negative binomial process). Defaults to poisson size parameter of negative binomial distribition. Ignored if model is poisson an optional vector of times at which the simulations should use a different sample of reproduction numbers, provided in days into the simulation (so that day '1' is the first day after the input incidence object); if provided, n dates in time_change will produce n+1 time windows, in which case R should be a list of vectors of n+1 R values, one per each time window. ## Details The decision to fix R values within simulations (R_fix_within) reflects two alternative views of the uncertainty associated with R. When drawing R values at random from the provided sample, (R_fix_within set to FALSE), it is assumed that R varies naturally, and can be treated as a random variable with a given distribution. When fixing values within simulations (R_fix_within set to TRUE), R is treated as a fixed parameter, and the uncertainty is merely a consequence of the estimation of R. In other words, the first view is rather Bayesian, while the second is more frequentist. ## Examples ## example using simulated Ebola outbreak if (require(outbreaks) && require(distcrete) && require(incidence) && require(magrittr)) { si <- distcrete("gamma", interval = 1L, shape = 2.4, scale = 4.7, w = 0.5) i <- incidence(ebola_sim$linelist$date_of_onset) plot(i) ## projections after the first 100 days, over 60 days, fixed R to 2.1 set.seed(1) proj_1 <- project(x = i[1:100], R = 2.1, si = si, n_days = 60) plot(proj_1) ## add projections to incidence plot ## projections after the first 100 days, over 60 days, ## using a sample of R set.seed(1) R <- rnorm(100, 1.8, 0.2) hist(R, col = "grey", border = "white", main = "Distribution of R") proj_2 <- project(x = i[1:100], R = R, si = si, n_days = 60) ## add projections to incidence plot ## same with R constant per simulation (more variability) set.seed(1) proj_3 <- project(x = i[1:100], R = R, si = si, n_days = 60, R_fix_within = TRUE) ## add projections to incidence plot ## time-varying R, 2 periods, R is 2.1 then 0.5 set.seed(1) proj_4 <- project(i, R = c(2.1, 0.5), si = si, n_days = 60, time_change = 40, n_sim = 100) plot(proj_4) ## time-varying R, 2 periods, separate distributions of R for each period set.seed(1) R_period_1 <- runif(100, min = 1.1, max = 3) R_period_2 <- runif(100, min = 0.6, max = .9) proj_5 <- project(i, R = list(R_period_1, R_period_2), si = si, n_days = 60, time_change = 20, n_sim = 100) plot(proj_5) }#> Scale for 'x' is already present. Adding another scale for 'x', which will #> replace the existing scale.#> Scale for 'x' is already present. Adding another scale for 'x', which will #> replace the existing scale.#> Scale for 'x' is already present. Adding another scale for 'x', which will #> replace the existing scale.	2021-04-17 22:54:46	{"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": 0.5215997099876404, "perplexity": 4977.390888571492}, "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/1618038464065.57/warc/CC-MAIN-20210417222733-20210418012733-00007.warc.gz"}
https://tex.stackexchange.com/questions/414581/how-to-treat-lists-like-regular-text-regarding-vertical-spacing?noredirect=1	# How to treat lists like regular text (regarding vertical spacing)? Context: I want to treat lists (itemize, enumerate, and description) just like regular text for what regards vertical spacing. Namely: 1. If the list starts a paragraph, insert a vertical \parskip beforehand — no vertical spacing otherwise; 2. No vertical spacing between items of the list; 3. If the list ends a paragraph, insert a vertical \parskip afterwards — no vertical spacing otherwise. I have notably used information found in \topsep, \itemsep, \partopsep and \parsep - what does each of them mean. Requirement #1 is solved with \setlist{topsep=-\parskip,partopsep=\parskip}. Requirement #2 is solved with \setlist{noitemsep}. Problem: There remain two issues with Requirement #3: • A vertical space is added after the list if the latter starts a paragraph, even if this list is immediately followed by text. (I.e. there is no such thing as an idependent parbottomsep length). • If a new paragraph starts after the list, this paragraph is not preceded with a \parskip. Question: How to comply with Requirement #3? (I currently use manual patches — see MWE below — but it is of course not satisfactory.) MWE \documentclass[parskip=half]{scrartcl} \usepackage{enumitem} \setlist{% topsep=-\parskip, partopsep=\parskip, noitemsep, } \begin{document} This sentence is a paragraph on its own; there is thus a vertical parskip prior next paragraph. Following list is \emph{within} a paragraph, with preceding and appended text. \begin{itemize} \item One, \item Two, \begin{itemize} \item Two and a half; \item Almost three. \end{itemize} \item Three. \end{itemize} This text is appended to the previous list. However, following list starts a new paragraph on its own. \begin{enumerate} \item Did you notice the vertical spacing preceding this list? \item Two, \begin{enumerate} \item Two and a half; \item Almost three. \end{enumerate} \item Three. \end{enumerate} % \vspace{-\parskip} %quick and dirty solution \textbf{There shouldn't be a vertical spacing here.} This text is appended to the previous list too. And finally, a list with preceding text only. \begin{itemize} \item One, \item Two, \begin{itemize} \item Two and a half; \item Almost three. \end{itemize} \item Three. \end{itemize} % \null\par %quick and dirty solution \textbf{There should be a vertical spacing here.} This is a new paragraph. It should thus be preceded with parskip. \end{document} This is hardly less dirty, but it uses the tools provided by enumitem, playing with the after key: \documentclass[parskip=half]{scrartcl} \usepackage{enumitem} \setlist{% topsep=-\parskip, partopsep=\parskip, noitemsep, } \begin{document} This sentence is a paragraph on its own; there is thus a vertical parskip prior next paragraph. Following list is \emph{within} a paragraph, with preceding and appended text. \begin{itemize} \item One, \item Two, \begin{itemize} \item Two and a half; \item Almost three. \end{itemize} \item Three. \end{itemize} This text is appended to the previous list. However, following list starts a new paragraph on its own. \begin{enumerate}[after =\vspace{-\partopsep}] \item Did you notice the vertical spacing preceding this list? \item Two, \begin{enumerate} \item Two and a half; \item Almost three. \end{enumerate} \item Three. \end{enumerate} \textbf{There shouldn't be a vertical spacing here.} This text is appended to the previous list too. And finally, a list with preceding text only. \begin{itemize}[after = \vspace{\partopsep}] \item One, \item Two, \begin{itemize} \item Two and a half; \item Almost three. \end{itemize} \item Three. \end{itemize} \textbf{There should be a vertical spacing here.} This is a new paragraph. It should thus be preceded with parskip. \end{document} • That's indeed more semantically correct! – ebosi Feb 10 '18 at 1:01	2019-08-22 22:01:46	{"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": 0.9311472773551941, "perplexity": 6414.370334158285}, "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/1566027317516.88/warc/CC-MAIN-20190822215308-20190823001308-00127.warc.gz"}
http://mathhelpforum.com/algebra/71328-need-help-about-cubed-squared-numbers.html	1. ## need help about cubed and squared numbers sorry went wrong!!!! 2. Hello, andyboy179! I assume you are just learning about exponents. I have: . $5^3,\:(\text{-}3)^2,\:(\text{-}2)^3$ i think the first one = 125 but im not sure. . Right! $5^3 \:=\:(5)(5)(5) \:=\:125$ $(\text{-}3)^2 \:=\:(\text{-}3)(\text{-}3) \:=\:9$ $(\text{-}2)^3 \:=\:(\text{-}2)(\text{-}2)(\text{-}2) \:=\:-8$ 3. Originally Posted by Soroban Hello, andyboy179! I assume you are just learning about exponents. $5^3 \:=\5)(5)(5) \:=\:125" alt="5^3 \:=\5)(5)(5) \:=\:125" /> $(\text{-}3)^2 \:=\\text{-}3)(\text{-}3) \:=\:9" alt="(\text{-}3)^2 \:=\\text{-}3)(\text{-}3) \:=\:9" /> $(\text{-}2)^3 \:=\\text{-}2)(\text{-}2)(\text{-}2) \:=\:-8" alt="(\text{-}2)^3 \:=\\text{-}2)(\text{-}2)(\text{-}2) \:=\:-8" /> thanks alot for understanding what i said be4 and thanks for the help i now know what to do	2017-06-25 23:45:02	{"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": 0, "img_math": 0, "codecogs_latex": 7, "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": 0.928892970085144, "perplexity": 9391.443406926019}, "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-2017-26/segments/1498128320593.91/warc/CC-MAIN-20170625221343-20170626001343-00578.warc.gz"}
https://math.meta.stackexchange.com/questions/27124/should-i-delete-a-question-if-i-have-found-it-to-be-silly	# Should I delete a question if I have found it to be silly? I am a new user and recently asked a question(Proving that $\int_0^\infty\sin(x)dx=1$) and found it to be very silly and somewhat related to (Can a limit of an integral be moved inside the integral?) . First I thought about deleting it but wouldn't it affect my reputation and prevent me from asking further question. I find no place where I can 'Close' the question(I think it requires more reputation) or answer my own question and show that it is related to Can a limit of an integral be moved inside the integral? . Will it be ok to write the answer in the question itself What should I do? (excuse me for asking so many questions at once) • You've done fine in your question, here and there. No need for deletion. I'm not entirely clear on the reason(s) for closure. Any way, I agree with Siong Thye Goh, in their answer below. – Namaste Oct 4 '17 at 19:40 • I am not agree mentioning the word silly. I prefer wrong, if you are thinking that it isn't the best question. Good luck. – user243301 Oct 8 '17 at 11:41	2019-03-19 19:28:21	{"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": 0.5931817889213562, "perplexity": 379.79527773911036}, "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-2019-13/segments/1552912202125.41/warc/CC-MAIN-20190319183735-20190319205735-00149.warc.gz"}
https://groups.google.com/g/tortoisesvn/c/YDcY6ieySaA	unassociate files with repository 12 views Brian FitzGerald Jul 12, 2008, 3:09:05 AM7/12/08 to us...@tortoisesvn.tigris.org Hi! I am a new TortoiseSVN user, and I have searched the Web and the FAQ at http://tortoisesvn.tigris.org/ without much luck with this problem. Thank you in advance for any insight you may be able to offer. Here's a quick rundown: 1) First, I successfully ran a "checkout" and grabbed files from someone else's repository. 2) Next, I moved some of these files into another directory so I could commit them to a different repository (my own) 3) My problem is, when I try to commit the files, the system wants to commit them to the original repository they were grabbed from, and not my repository. 4) My question is, is there a way to "unassociate" the files with the original repository so that I can add and commit them to my own Brian --------------------------------------------------------------------- To unsubscribe, e-mail: users-un...@tortoisesvn.tigris.org Stefan Küng Jul 12, 2008, 3:12:04 AM7/12/08 to us...@tortoisesvn.tigris.org Brian FitzGerald wrote: > Hi! I am a new TortoiseSVN user, and I have searched the Web and the > FAQ at http://tortoisesvn.tigris.org/ without much luck with this > problem. Thank you in advance for any insight you may be able to > offer. > > Here's a quick rundown: > > 1) First, I successfully ran a "checkout" and grabbed files from > someone else's repository. > > 2) Next, I moved some of these files into another directory so I could > commit them to a different repository (my own) > > 3) My problem is, when I try to commit the files, the system wants to > commit them to the original repository they were grabbed from, and not > my repository. > > 4) My question is, is there a way to "unassociate" the files with the > original repository so that I can add and commit them to my own There are hidden '.svn' folders in every working copy folder. If you just copied some folders from one working copy to another, you also copied those folders. Instead of copying those folders, right-drag them in Explorer, then choose "SVN Export" from the context menu. Or you could manually delete the hidden .svn folders... Stefan -- ___ oo // \\ "De Chelonian Mobile" (_,\/ \_/ \ TortoiseSVN \ \_/_\_/> The coolest Interface to (Sub)Version Control /_/ \_\ http://tortoisesvn.net signature.asc Brian FitzGerald Jul 12, 2008, 3:21:08 AM7/12/08 to us...@tortoisesvn.tigris.org Brilliant! Yeah, that right click drag > "SVN Export" was exactly what I needed. I was deleting the hidden SVN folders manually before I originally posted, but when I got to the 50th one, I figured there had to be a better way ;)	2021-11-30 12:38:44	{"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": 0.8272430896759033, "perplexity": 7290.18600625773}, "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/1637964358973.70/warc/CC-MAIN-20211130110936-20211130140936-00465.warc.gz"}
https://www.physicsforums.com/threads/partial-differential-equations-general-solution.101442/	# Partial differential equations: general solution 1. Nov 25, 2005 ### sachi I have a 2nd order homogenous P.D.E: (d^2)V/((dx)^2) + (d^2)V/((dy)^2) + 6(d^2)V/(dx dy) = 0 where all derivatives are partial derivatives. I need to transform this to form a(d^2)f/(dX^2) + b(d^2)f/(dY^2) = 0 where a, b are constants and the derivatives are again partial, and f(X,Y) = V(x,y) I I need to make a change of variables e.g x=cX +dY etc. but I get very messy algebraic expressions which I can't simplify. Thanks very much. 2. Nov 25, 2005 ### Fermat You have V = V(x,y). Then make the change of variables, X = X(x,y) Y = Y(x,y) where, X = y - m1x Y = y - m2x (It doesn't need to be any more complicated) Now use the chain rule for partial differentials involving change of variables. Last edited: Nov 25, 2005 3. Nov 25, 2005 ### saltydog You need to rotate axes to eliminate the mixed partial. This is normally done by defining a new function H(w,z) such that: $$w=xCos(a)+ySin(a)\quad\text{and}\quad z=-xSin(a)+yCos(a)$$ with a to be determined so that the coefficient of the mixed partial in the final results is zero. So, start calculating all the partials and substitute them into your equation and end up with a PDE in H(w,z), then determine a so that the coefficient of the mixed partial is zero. Solve the PDE in H(w,z), then substitute back x and y: $$V(x,y)=H(xCos(a)+ySin(a),-xSin(a)+yCos(a))$$ I'll start off the partials: $$\frac{\partial V}{\partial x}=\frac{\partial H}{\partial w}\frac{\partial w}{\partial x}+\frac{\partial H}{\partial z}\frac{\partial z}{\partial w}$$ so then: $$\frac{\partial^2 V}{\partial x^2}=\frac{\partial}{\partial x}\left(\frac{\partial H}{\partial w}\frac{\partial w}{\partial x}+\frac{\partial H}{\partial z}\frac{\partial z}{\partial w}\right)$$ So do that one, the mixed one, and then for y and then substitute them back into your PDE. It goes fast once you get going. Then solve for a to make the mixed one go away. Edit: Hum . . . well maybe that too Fermat. Last edited: Nov 25, 2005 4. Nov 26, 2005 ### Fermat Yeah That's the method we were shown for solving 2nd order pde's. If the pde was of the form, $$A\cdot \frac{\partial ^2V}{\partial x^2} + B\cdot \frac{\partial ^2V}{\partial x \partial y} + C\cdot \frac{\partial ^2V}{\partial y^2} = 0$$ we would transform it to, $$P(m_1)\cdot \frac{\partial ^2V}{\partial X^2} + Q(m_1,m_2)\cdot \frac{\partial ^2V}{\partial X \partial Y} + P(m_2)\cdot \frac{\partial ^2V}{\partial Y^2} = 0$$ $$\mbox{However, rather than making, } Q(m_1,m_2) = 0 \mbox{ we would make } P(m_1) = 0 \mbox{ and } P(m_2) = 0$$ This would give, $$\frac{\partial ^2V}{\partial X \partial Y} = 0$$ which is then easily integrated to give, V = f(X) + g(Y) ============ where you would substitute for, X = y - m1x Y = y - m2x You would get m1 and m2 when setting P(m1) = 0 and P(m2) = 0. So, you could use any functions f() and g() that you liked, as long as their arguments were y - m1x and y - m2x. Last edited: Nov 26, 2005 5. Nov 27, 2005 ### saltydog Thanks for point that out Fermat. That's a much better approach considering, after I looked into it a bit, that an arbitrary solution of the wave equation: $$\frac{\partial^2 u}{\partial t^2}=a^2\frac{\partial^2}{\partial x^2}$$ can be written in the form: $$u(x,t)=F(x+at)+G(x-at)$$ and thus I would suspect your m's above are related to the final coefficient on the x-partial term above. I'll work with it a bit.	2017-01-17 15:28:44	{"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": 0.8544418215751648, "perplexity": 1403.7185126121844}, "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-2017-04/segments/1484560279923.28/warc/CC-MAIN-20170116095119-00473-ip-10-171-10-70.ec2.internal.warc.gz"}
https://forum.allaboutcircuits.com/threads/vco-operation.179972/	# VCO operation #### gkini19 Joined Jun 15, 2021 17 Hi All, I was looking at several papers of radar transceiver that operates at 77GHz to 88 GHz focusing on the VCO and Chirp PLL architecture. So if we want the output of the VCO to be 77GHz to 88 GHz, all the papers for radar transceivers use VCO with a multiplier to generate frequencies in the range of 77GHz to 88 GHz. (Say 20 * 4 or 38*2 ) What would be the technical reason to use multipliers? why can't we just have an architecture of the VCO that operates at 77 GHz? #### Papabravo Joined Feb 24, 2006 19,326 It is either a mater of cost, or the lack of a fundamental source at that frequency. It is often cheaper to use a multiplier and a fundamental source rather than to try to create a fundamental source. #### gkini19 Joined Jun 15, 2021 17 It is either a mater of cost, or the lack of a fundamental source at that frequency. It is often cheaper to use a multiplier and a fundamental source rather than to try to create a fundamental source. I wanted a comparison with the technical specifications such as Phase Noise, matching, and all such stuff. COuld you please let me know #### Papabravo Joined Feb 24, 2006 19,326 You asked a fairly simple question and I gave you the primary reason for using them, A forum post or two will hardly provide the deeper answers you seek. I have not studied this presentation in detail but it appears to have a solid overview of some problems and solutions. The references at the end may also be useful NTTI-Paris-Multipliers-Jul-2007.ppt (obspm.fr) #### Wolframore Joined Jan 21, 2019 2,577 I agree with @Papabravo, design and costs gets higher as we go up in frequency. Things get more difficult to make as we need faster slew rates and all that stuff. Makes sense to multiply it near the end. #### gkini19 Joined Jun 15, 2021 17 I agree with @Papabravo, design and costs gets higher as we go up in frequency. Things get more difficult to make as we need faster slew rates and all that stuff. Makes sense to multiply it near the end. You mentioned things will get more difficult. Could you please tell me what are those? It would be great if you tell me. #### Wolframore Joined Jan 21, 2019 2,577 whats your background in RF and high speed circuits? #### Papabravo Joined Feb 24, 2006 19,326 You mentioned things will get more difficult. Could you please tell me what are those? It would be great if you tell me. Some difficulties you will encounter include, but are not limited to: 1. Inability to use standard lumped components, with the possible exception of very small SMT resistors 2. The need for CAD tools that can compute results based on microstriplines and other controlled impedances 3. Obtaining suitable power levels for your requirements 4. Costs of specialized components and fabrication techniques, like building structures on a silicon chip instead of a circuit board. 5. Cost of test equipment and instrumentation. A typical VNA might still fetch over $100K 6. Inability to find people with the requisite skills and knowledge Thread Starter #### gkini19 Joined Jun 15, 2021 17 Some difficulties you will encounter include, but are not limited to: 1. Inability to use standard lumped components, with the possible exception of very small SMT resistors 2. The need for CAD tools that can compute results based on microstriplines and other controlled impedances 3. Obtaining suitable power levels for your requirements 4. Costs of specialized components and fabrication techniques, like building structures on a silicon chip instead of a circuit board. 5. Cost of test equipment and instrumentation. A typical VNA might still fetch over$100K 6. Inability to find people with the requisite skills and knowledge Thanks, what about the technical aspects like phase noise, Noise figures, and such stuff. Doesn't it degrade when we use multipliers? #### Papabravo Joined Feb 24, 2006 19,326 Thanks, what about the technical aspects like phase noise, Noise figures, and such stuff. Doesn't it degrade when we use multipliers? Yes it does. What other choice do you have?	2022-12-07 05:40:32	{"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": 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": 0.3345860242843628, "perplexity": 1341.0335213759004}, "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-2022-49/segments/1669446711150.61/warc/CC-MAIN-20221207053157-20221207083157-00525.warc.gz"}
https://compneuro.neuromatch.io/tutorials/W1D1_ModelTypes/student/W1D1_Tutorial3.html	# Tutorial 3: “Why” models¶ Week 1, Day 1: Model Types By Neuromatch Academy Content creators: Matt Laporte, Byron Galbraith, Konrad Kording Content reviewers: Dalin Guo, Aishwarya Balwani, Madineh Sarvestani, Maryam Vaziri-Pashkam, Michael Waskom, Ella Batty Post-production team: Gagana B, Spiros Chavlis We would like to acknowledge Steinmetz et al. (2019) for sharing their data, a subset of which is used here. # Tutorial Objectives¶ Estimated timing of tutorial: 45 minutes This is tutorial 3 of a 3-part series on different flavors of models used to understand neural data. In parts 1 and 2 we explored mechanisms that would produce the data. In this tutorial we will explore models and techniques that can potentially explain why the spiking data we have observed is produced the way it is. To understand why different spiking behaviors may be beneficial, we will learn about the concept of entropy. Specifically, we will: • Write code to compute formula for entropy, a measure of information • Compute the entropy of a number of toy distributions • Compute the entropy of spiking activity from the Steinmetz dataset ## Tutorial slides¶ These are the slides for the videos in all tutorials today # Setup¶ # Imports import numpy as np import matplotlib.pyplot as plt from scipy import stats ## Figure Settings¶ #@title Figure Settings import ipywidgets as widgets #interactive display %matplotlib inline %config InlineBackend.figure_format = 'retina' plt.style.use("https://raw.githubusercontent.com/NeuromatchAcademy/course-content/main/nma.mplstyle") ## Plotting Functions¶ #@title Plotting Functions def plot_pmf(pmf,isi_range): """Plot the probability mass function.""" ymax = max(0.2, 1.05 * np.max(pmf)) pmf_ = np.insert(pmf, 0, pmf[0]) plt.plot(bins, pmf_, drawstyle="steps") plt.fill_between(bins, pmf_, step="pre", alpha=0.4) plt.title(f"Neuron {neuron_idx}") plt.xlabel("Inter-spike interval (s)") plt.ylabel("Probability mass") plt.xlim(isi_range); plt.ylim([0, ymax]) ## Download Data¶ #@title Download Data import io import requests r = requests.get('https://osf.io/sy5xt/download') if r.status_code != 200: print('Could not download data') else: steinmetz_spikes = np.load(io.BytesIO(r.content), allow_pickle=True)['spike_times'] # Section 1: Optimization and Information¶ Remember that the notation section is located after the Summary for quick reference! Neurons can only fire so often in a fixed period of time, as the act of emitting a spike consumes energy that is depleted and must eventually be replenished. To communicate effectively for downstream computation, the neuron would need to make good use of its limited spiking capability. This becomes an optimization problem: What is the optimal way for a neuron to fire in order to maximize its ability to communicate information? In order to explore this question, we first need to have a quantifiable measure for information. Shannon introduced the concept of entropy to do just that, and defined it as (131)$$$H_b(X) = -\sum_{x\in X} p(x) \log_b p(x)$$$ where $$H$$ is entropy measured in units of base $$b$$ and $$p(x)$$ is the probability of observing the event $$x$$ from the set of all possible events in $$X$$. See the Bonus Section 1 for a more detailed look at how this equation was derived. The most common base of measuring entropy is $$b=2$$, so we often talk about bits of information, though other bases are used as well (e.g. when $$b=e$$ we call the units nats). First, let’s explore how entropy changes between some simple discrete probability distributions. In the rest of this tutorial we will refer to these as probability mass functions (PMF), where $$p(x_i)$$ equals the $$i^{th}$$ value in an array, and mass refers to how much of the distribution is contained at that value. For our first PMF, we will choose one where all of the probability mass is located in the middle of the distribution. n_bins = 50 # number of points supporting the distribution x_range = (0, 1) # will be subdivided evenly into bins corresponding to points bins = np.linspace(x_range, n_bins + 1) # bin edges pmf = np.zeros(n_bins) pmf[len(pmf) // 2] = 1.0 # middle point has all the mass # Since we already have a PMF, rather than un-binned samples, plt.hist is not # suitable. Instead, we directly plot the PMF as a step function to visualize # the histogram: pmf_ = np.insert(pmf, 0, pmf[0]) # this is necessary to align plot steps with bin edges plt.plot(bins, pmf_, drawstyle="steps") # fill_between provides area shading plt.fill_between(bins, pmf_, step="pre", alpha=0.4) plt.xlabel("x") plt.ylabel("p(x)") plt.xlim(x_range) plt.ylim(0, 1); If we were to draw a sample from this distribution, we know exactly what we would get every time. Distributions where all the mass is concentrated on a single event are known as deterministic. How much entropy is contained in a deterministic distribution? We will compute this in the next exercise. ## Coding Exercise 1: Computing Entropy¶ Your first exercise is to implement a method that computes the entropy of a discrete probability distribution, given its mass function. Remember that we are interested in entropy in units of bits, so be sure to use the correct log function. Recall that $$\log(0)$$ is undefined. When evaluated at $$0$$, NumPy log functions (such as np.log2) return np.nan (“Not a Number”). By convention, these undefined terms— which correspond to points in the distribution with zero mass—are excluded from the sum that computes the entropy. def entropy(pmf): """Given a discrete distribution, return the Shannon entropy in bits. This is a measure of information in the distribution. For a totally deterministic distribution, where samples are always found in the same bin, then samples from the distribution give no more information and the entropy is 0. For now this assumes pmf arrives as a well-formed distribution (that is, np.sum(pmf)==1 and not np.any(pmf < 0)) Args: pmf (np.ndarray): The probability mass function for a discrete distribution represented as an array of probabilities. Returns: h (number): The entropy of the distribution in pmf. """ ############################################################################ # Exercise for students: compute the entropy of the provided PMF # 1. Exclude the points in the distribution with no mass (where pmf==0). # Hint: this is equivalent to including only the points with pmf>0. # 2. Implement the equation for Shannon entropy (in bits). # When ready to test, comment or remove the next line raise NotImplementedError("Exercise: implement the equation for entropy") ############################################################################ # reduce to non-zero entries to avoid an error from log2(0) pmf = ... # implement the equation for Shannon entropy (in bits) h = ... # return the absolute value (avoids getting a -0 result) return np.abs(h) # Call entropy function and print result print(f"{entropy(pmf):.2f} bits") Click for solution We expect zero surprise from a deterministic distribution. If we had done this calculation by hand, it would simply be $$-1\log_2 1 = -0=0$$. Note that changing the location of the peak (i.e. the point and bin on which all the mass rests) doesn’t alter the entropy. The entropy is about how predictable a sample is with respect to a distribution. A single peak is deterministic regardless of which point it sits on - the following plot shows a PMF that would also have zero entropy. Execute this cell to visualize another PMF with zero entropy # @markdown Execute this cell to visualize another PMF with zero entropy pmf = np.zeros(n_bins) pmf[2] = 1.0 # arbitrary point has all the mass pmf_ = np.insert(pmf, 0, pmf[0]) plt.plot(bins, pmf_, drawstyle="steps") plt.fill_between(bins, pmf_, step="pre", alpha=0.4) plt.xlabel("x") plt.ylabel("p(x)") plt.xlim(x_range) plt.ylim(0, 1); What about a distribution with mass split equally between two points? Execute this cell to visualize a PMF with split mass # @markdown Execute this cell to visualize a PMF with split mass pmf = np.zeros(n_bins) pmf[len(pmf) // 3] = 0.5 pmf[2 len(pmf) // 3] = 0.5 pmf_ = np.insert(pmf, 0, pmf[0]) plt.plot(bins, pmf_, drawstyle="steps") plt.fill_between(bins, pmf_, step="pre", alpha=0.4) plt.xlabel("x") plt.ylabel("p(x)") plt.xlim(x_range) plt.ylim(0, 1); Here, the entropy calculation is: $$-(0.5 \log_2 0.5 + 0.5\log_2 0.5)=1$$ There is 1 bit of entropy. This means that before we take a random sample, there is 1 bit of uncertainty about which point in the distribution the sample will fall on: it will either be the first peak or the second one. Likewise, if we make one of the peaks taller (i.e. its point holds more of the probability mass) and the other one shorter, the entropy will decrease because of the increased certainty that the sample will fall on one point and not the other: : $$-(0.2 \log_2 0.2 + 0.8\log_2 0.8)\approx 0.72$$ Try changing the definition of the number and weighting of peaks, and see how the entropy varies. If we split the probability mass among even more points, the entropy continues to increase. Let’s derive the general form for $$N$$ points of equal mass, where $$p_i=p=1/N$$: (132)\begin{align} -\sum_i p_i \log_b p_i &= -\sum_i^N \frac{1}{N} \log_b \frac{1}{N} \\ &= -\log_b \frac{1}{N} \\ &= \log_b N \end{align} If we have $$N$$ discrete points, the uniform distribution (where all points have equal mass) is the distribution with the highest entropy: $$\log_b N$$. This upper bound on entropy is useful when considering binning strategies, as any estimate of entropy over $$N$$ discrete points (or bins) must be in the interval $$[0, \log_b N]$$. Execute this cell to visualize a PMF of uniform distribution # @markdown Execute this cell to visualize a PMF of uniform distribution pmf = np.ones(n_bins) / n_bins # [1/N] * N pmf_ = np.insert(pmf, 0, pmf[0]) plt.plot(bins, pmf_, drawstyle="steps") plt.fill_between(bins, pmf_, step="pre", alpha=0.4) plt.xlabel("x") plt.ylabel("p(x)") plt.xlim(x_range); plt.ylim(0, 1); Here, there are 50 points and the entropy of the uniform distribution is $$\log_2 50\approx 5.64$$. If we construct any discrete distribution $$X$$ over 50 points (or bins) and calculate an entropy of $$H_2(X)>\log_2 50$$, something must be wrong with our implementation of the discrete entropy computation. # Section 2: Information, neurons, and spikes¶ Estimated timing to here from start of tutorial: 20 min ## Video 2: Entropy of different distributions¶ Recall the discussion of spike times and inter-spike intervals (ISIs) from Tutorial 1. What does the information content (or distributional entropy) of these measures say about our theory of nervous systems? We’ll consider three hypothetical neurons that all have the same mean ISI, but with different distributions: 1. Deterministic 2. Uniform 3. Exponential Fixing the mean of the ISI distribution is equivalent to fixing its inverse: the neuron’s mean firing rate. If a neuron has a fixed energy budget and each of its spikes has the same energy cost, then by fixing the mean firing rate, we are normalizing for energy expenditure. This provides a basis for comparing the entropy of different ISI distributions. In other words: if our neuron has a fixed budget, what ISI distribution should it express (all else being equal) to maximize the information content of its outputs? Let’s construct our three distributions and see how their entropies differ. n_bins = 50 mean_isi = 0.025 isi_range = (0, 0.25) bins = np.linspace(isi_range, n_bins + 1) mean_idx = np.searchsorted(bins, mean_isi) # 1. all mass concentrated on the ISI mean pmf_single = np.zeros(n_bins) pmf_single[mean_idx] = 1.0 # 2. mass uniformly distributed about the ISI mean pmf_uniform = np.zeros(n_bins) pmf_uniform[0:2mean_idx] = 1 / (2 * mean_idx) # 3. mass exponentially distributed about the ISI mean pmf_exp = stats.expon.pdf(bins[1:], scale=mean_isi) pmf_exp /= np.sum(pmf_exp) ## ¶ Run this cell to plot the three PMFs #@title #@markdown Run this cell to plot the three PMFs fig, axes = plt.subplots(ncols=3, figsize=(18, 5)) dists = [# (subplot title, pmf, ylim) ("(1) Deterministic", pmf_single, (0, 1.05)), ("(1) Uniform", pmf_uniform, (0, 1.05)), ("(1) Exponential", pmf_exp, (0, 1.05))] for ax, (label, pmf_, ylim) in zip(axes, dists): pmf_ = np.insert(pmf_, 0, pmf_[0]) ax.plot(bins, pmf_, drawstyle="steps") ax.fill_between(bins, pmf_, step="pre", alpha=0.4) ax.set_title(label) ax.set_xlabel("Inter-spike interval (s)") ax.set_ylabel("Probability mass") ax.set_xlim(isi_range); ax.set_ylim(ylim); print( f"Deterministic: {entropy(pmf_single):.2f} bits", f"Uniform: {entropy(pmf_uniform):.2f} bits", f"Exponential: {entropy(pmf_exp):.2f} bits", sep="\n", ) # Section 3: Calculate entropy of ISI distributions from data¶ Estimated timing to here from start of tutorial: 25 min ## Section 3.1: Computing probabilities from histogram¶ ### Video 3: Probabilities from histogram¶ In the previous example we created the PMFs by hand to illustrate idealized scenarios. How would we compute them from data recorded from actual neurons? One way is to convert the ISI histograms we’ve previously computed into discrete probability distributions using the following equation: (133)$$$p_i = \frac{n_i}{\sum\nolimits_{i}n_i}$$$ where $$p_i$$ is the probability of an ISI falling within a particular interval $$i$$ and $$n_i$$ is the count of how many ISIs were observed in that interval. ### Coding Exercise 3.1: Probability Mass Function¶ Your second exercise is to implement a method that will produce a probability mass function from an array of ISI bin counts. To verify your solution, we will compute the probability distribution of ISIs from real neural data taken from the Steinmetz dataset. def pmf_from_counts(counts): """Given counts, normalize by the total to estimate probabilities.""" ########################################################################### # Exercise: Compute the PMF. Remove the next line to test your function raise NotImplementedError("Student exercise: compute the PMF from ISI counts") ########################################################################### pmf = ... return pmf # Get neuron index neuron_idx = 283 # Get counts of ISIs from Steinmetz data isi = np.diff(steinmetz_spikes[neuron_idx]) bins = np.linspace(isi_range, n_bins + 1) counts, _ = np.histogram(isi, bins) # Compute pmf pmf = pmf_from_counts(counts) # Visualize plot_pmf(pmf,isi_range) Click for solution Example output: ## Section 3.2: Calculating entropy from pmf¶ ### Video 4: Calculating entropy from pmf¶ Now that we have the probability distribution for the actual neuron spiking activity, we can calculate its entropy. print(f"Entropy for Neuron {neuron_idx}: {entropy(pmf):.2f} bits") ### Interactive Demo 3.2: Entropy of neurons¶ We can combine the above distribution plot and entropy calculation with an interactive widget to explore how the different neurons in the dataset vary in spiking activity and relative information. Note that the mean firing rate across neurons is not fixed, so some neurons with a uniform ISI distribution may have higher entropy than neurons with a more exponential distribution. #### ¶ Run the cell to enable the sliders. #@title #@markdown Run the cell* to enable the sliders. def _pmf_from_counts(counts): """Given counts, normalize by the total to estimate probabilities.""" pmf = counts / np.sum(counts) return pmf def _entropy(pmf): """Given a discrete distribution, return the Shannon entropy in bits.""" # remove non-zero entries to avoid an error from log2(0) pmf = pmf[pmf > 0] h = -np.sum(pmf * np.log2(pmf)) # absolute value applied to avoid getting a -0 result return np.abs(h) @widgets.interact(neuron=widgets.IntSlider(0, min=0, max=(len(steinmetz_spikes)-1))) def steinmetz_pmf(neuron): """ Given a neuron from the Steinmetz data, compute its PMF and entropy """ isi = np.diff(steinmetz_spikes[neuron]) bins = np.linspace(*isi_range, n_bins + 1) counts, _ = np.histogram(isi, bins) pmf = _pmf_from_counts(counts) plot_pmf(pmf,isi_range) plt.title(f"Neuron {neuron}: H = {_entropy(pmf):.2f} bits") # Section 4: Reflecting on why models¶ Estimated timing to here from start of tutorial: 35 min ## Think! 3: Reflecting on why models¶ Please discuss the following questions for around 10 minutes with your group: • Have you seen why models before? • Have you ever done one? • Why are why models useful? • When are they possible? Does your field have why models? • What do we learn from constructing them? # Summary¶ Estimated timing of tutorial: 45 minutes ## Video 5: Summary of model types¶ Congratulations! You’ve finished your first NMA tutorial. In this 3 part tutorial series, we used different types of models to understand the spiking behavior of neurons recorded in the Steinmetz data set. • We used “what” models to discover that the ISI distribution of real neurons is closest to an exponential distribution • We used “how” models to discover that balanced excitatory and inhibitory inputs, coupled with a leaky membrane, can give rise to neuronal spiking with exhibiting such an exponential ISI distribution • We used “why” models to discover that exponential ISI distributions contain the most information when the mean spiking is constrained # Notation¶ (134)\begin{align} H(X) &\quad \text{entropy of random variable X}\\ b &\quad \text{base, e.g. b=2 or b=e}\\ x &\quad \text{event x}\\ p(x) &\quad \text{probability of observing event x}\\ \text{ISI} &\quad \text{interspike interval}\\ n_i &\quad \text{count of observed ISIs in interval i}\\ p_i &\quad \text{probability of of an ISI falling within a particular interval i} \end{align} # Bonus¶ ## Bonus Section 1: The foundations for Entropy¶ In his foundational 1948 paper on information theory, Claude Shannon began with three criteria for a function $$H$$ defining the entropy of a discrete distribution of probability masses $$p_i\in p(X)$$ over the points $$x_i\in X$$: 1. $$H$$ should be continuous in the $$p_i$$. • That is, $$H$$ should change smoothly in response to smooth changes to the mass $$p_i$$ on each point $$x_i$$. 1. If all the points have equal shares of the probability mass, $$p_i=1/N$$, $$H$$ should be a non-decreasing function of $$N$$. • That is, if $$X_N$$ is the support with $$N$$ discrete points and $$p(x\in X_N)$$ assigns constant mass to each point, then $$H(X_1) < H(X_2) < H(X_3) < \dots$$ 1. $$H$$ should be preserved by (invariant to) the equivalent (de)composition of distributions. • For example (from Shannon’s paper) if we have a discrete distribution over three points with masses $$(\frac{1}{2},\frac{1}{3},\frac{1}{6})$$, then their entropy can be represented in terms of a direct choice between the three and calculated $$H(\frac{1}{2},\frac{1}{3},\frac{1}{6})$$. However, it could also be represented in terms of a series of two choices: 1. either we sample the point with mass $$1/2$$ or not (not is the other $$1/2$$, whose subdivisions are not given in the first choice), 2. if (with probability $$1/2$$) we don’t sample the first point, we sample one of the two remaining points, masses $$1/3$$ and $$1/6$$. Thus in this case we require that $$H(\frac{1}{2},\frac{1}{3},\frac{1}{6})=H(\frac{1}{2},\frac{1}{2}) + \frac{1}{2}H(\frac{1}{3}, \frac{1}{6})$$ There is a unique function (up to a linear scaling factor) which satisfies these 3 requirements: (135)$$$H_b(X) = -\sum_{x\in X} p(x) \log_b p(x)$$$ Where the base of the logarithm $$b>1$$ controls the units of entropy. The two most common cases are $$b=2$$ for units of bits, and $$b=e$$ for nats. We can view this function as the expectation of the self-information over a distribution: (136)\begin{align} H_b(X) &= \mathbb{E}_{x\in X} \left[I_b(x)\right]\\ I_b(x) &= -\log_b p(x) \end{align} Self-information is just the negative logarithm of probability, and is a measure of how surprising an event sampled from the distribution would be. Events with $$p(x)=1$$ are certain to occur, and their self-information is zero (as is the entropy of the distribution they compose) meaning they are totally unsurprising. The smaller the probability of an event, the higher its self-information, and the more surprising the event would be to observe.	2022-12-04 11:45:49	{"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": 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": 3, "equation": 3, "x-ck12": 0, "texerror": 0, "math_score": 0.5418312549591064, "perplexity": 6853.930786120815}, "config": {"markdown_headings": true, "markdown_code": true, "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-2022-49/segments/1669446710972.37/warc/CC-MAIN-20221204104311-20221204134311-00736.warc.gz"}
https://people.maths.bris.ac.uk/~matyd/GroupNames/480/C2xC6xC5s2C8.html	Copied to clipboard ## G = C2×C6×C5⋊2C8order 480 = 25·3·5 ### Direct product of C2×C6 and C5⋊2C8 Series: Derived Chief Lower central Upper central Derived series C1 — C5 — C2×C6×C5⋊2C8 Chief series C1 — C5 — C10 — C20 — C60 — C3×C5⋊2C8 — C6×C5⋊2C8 — C2×C6×C5⋊2C8 Lower central C5 — C2×C6×C5⋊2C8 Upper central C1 — C22×C12 Generators and relations for C2×C6×C52C8 G = < a,b,c,d \| a2=b6=c5=d8=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 > Subgroups: 240 in 152 conjugacy classes, 130 normal (22 characteristic) C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, C23, C10, C10, C12, C12, C2×C6, C15, C2×C8, C22×C4, C20, C20, C2×C10, C24, C2×C12, C22×C6, C30, C30, C22×C8, C52C8, C2×C20, C22×C10, C2×C24, C22×C12, C60, C60, C2×C30, C2×C52C8, C22×C20, C22×C24, C3×C52C8, C2×C60, C22×C30, C22×C52C8, C6×C52C8, C22×C60, C2×C6×C52C8 Quotients: C1, C2, C3, C4, C22, C6, C8, C2×C4, C23, D5, C12, C2×C6, C2×C8, C22×C4, Dic5, D10, C24, C2×C12, C22×C6, C3×D5, C22×C8, C52C8, C2×Dic5, C22×D5, C2×C24, C22×C12, C3×Dic5, C6×D5, C2×C52C8, C22×Dic5, C22×C24, C3×C52C8, C6×Dic5, D5×C2×C6, C22×C52C8, C6×C52C8, C2×C6×Dic5, C2×C6×C52C8 Smallest permutation representation of C2×C6×C52C8 Regular action on 480 points Generators in S480 (1 69)(2 70)(3 71)(4 72)(5 65)(6 66)(7 67)(8 68)(9 346)(10 347)(11 348)(12 349)(13 350)(14 351)(15 352)(16 345)(17 301)(18 302)(19 303)(20 304)(21 297)(22 298)(23 299)(24 300)(25 127)(26 128)(27 121)(28 122)(29 123)(30 124)(31 125)(32 126)(33 310)(34 311)(35 312)(36 305)(37 306)(38 307)(39 308)(40 309)(41 145)(42 146)(43 147)(44 148)(45 149)(46 150)(47 151)(48 152)(49 328)(50 321)(51 322)(52 323)(53 324)(54 325)(55 326)(56 327)(57 336)(58 329)(59 330)(60 331)(61 332)(62 333)(63 334)(64 335)(73 196)(74 197)(75 198)(76 199)(77 200)(78 193)(79 194)(80 195)(81 204)(82 205)(83 206)(84 207)(85 208)(86 201)(87 202)(88 203)(89 212)(90 213)(91 214)(92 215)(93 216)(94 209)(95 210)(96 211)(97 220)(98 221)(99 222)(100 223)(101 224)(102 217)(103 218)(104 219)(105 228)(106 229)(107 230)(108 231)(109 232)(110 225)(111 226)(112 227)(113 236)(114 237)(115 238)(116 239)(117 240)(118 233)(119 234)(120 235)(129 442)(130 443)(131 444)(132 445)(133 446)(134 447)(135 448)(136 441)(137 463)(138 464)(139 457)(140 458)(141 459)(142 460)(143 461)(144 462)(153 472)(154 465)(155 466)(156 467)(157 468)(158 469)(159 470)(160 471)(161 422)(162 423)(163 424)(164 417)(165 418)(166 419)(167 420)(168 421)(169 430)(170 431)(171 432)(172 425)(173 426)(174 427)(175 428)(176 429)(177 438)(178 439)(179 440)(180 433)(181 434)(182 435)(183 436)(184 437)(185 358)(186 359)(187 360)(188 353)(189 354)(190 355)(191 356)(192 357)(241 366)(242 367)(243 368)(244 361)(245 362)(246 363)(247 364)(248 365)(249 374)(250 375)(251 376)(252 369)(253 370)(254 371)(255 372)(256 373)(257 382)(258 383)(259 384)(260 377)(261 378)(262 379)(263 380)(264 381)(265 390)(266 391)(267 392)(268 385)(269 386)(270 387)(271 388)(272 389)(273 398)(274 399)(275 400)(276 393)(277 394)(278 395)(279 396)(280 397)(281 406)(282 407)(283 408)(284 401)(285 402)(286 403)(287 404)(288 405)(289 414)(290 415)(291 416)(292 409)(293 410)(294 411)(295 412)(296 413)(313 454)(314 455)(315 456)(316 449)(317 450)(318 451)(319 452)(320 453)(337 480)(338 473)(339 474)(340 475)(341 476)(342 477)(343 478)(344 479) (1 228 203 29 420 48)(2 229 204 30 421 41)(3 230 205 31 422 42)(4 231 206 32 423 43)(5 232 207 25 424 44)(6 225 208 26 417 45)(7 226 201 27 418 46)(8 227 202 28 419 47)(9 266 196 240 471 33)(10 267 197 233 472 34)(11 268 198 234 465 35)(12 269 199 235 466 36)(13 270 200 236 467 37)(14 271 193 237 468 38)(15 272 194 238 469 39)(16 265 195 239 470 40)(17 480 458 191 224 261)(18 473 459 192 217 262)(19 474 460 185 218 263)(20 475 461 186 219 264)(21 476 462 187 220 257)(22 477 463 188 221 258)(23 478 464 189 222 259)(24 479 457 190 223 260)(49 441 425 212 244 288)(50 442 426 213 245 281)(51 443 427 214 246 282)(52 444 428 215 247 283)(53 445 429 216 248 284)(54 446 430 209 241 285)(55 447 431 210 242 286)(56 448 432 211 243 287)(57 449 433 280 252 296)(58 450 434 273 253 289)(59 451 435 274 254 290)(60 452 436 275 255 291)(61 453 437 276 256 292)(62 454 438 277 249 293)(63 455 439 278 250 294)(64 456 440 279 251 295)(65 109 84 127 163 148)(66 110 85 128 164 149)(67 111 86 121 165 150)(68 112 87 122 166 151)(69 105 88 123 167 152)(70 106 81 124 168 145)(71 107 82 125 161 146)(72 108 83 126 162 147)(73 117 160 310 346 391)(74 118 153 311 347 392)(75 119 154 312 348 385)(76 120 155 305 349 386)(77 113 156 306 350 387)(78 114 157 307 351 388)(79 115 158 308 352 389)(80 116 159 309 345 390)(89 361 405 328 136 172)(90 362 406 321 129 173)(91 363 407 322 130 174)(92 364 408 323 131 175)(93 365 401 324 132 176)(94 366 402 325 133 169)(95 367 403 326 134 170)(96 368 404 327 135 171)(97 382 297 341 144 360)(98 383 298 342 137 353)(99 384 299 343 138 354)(100 377 300 344 139 355)(101 378 301 337 140 356)(102 379 302 338 141 357)(103 380 303 339 142 358)(104 381 304 340 143 359)(177 394 374 410 333 313)(178 395 375 411 334 314)(179 396 376 412 335 315)(180 397 369 413 336 316)(181 398 370 414 329 317)(182 399 371 415 330 318)(183 400 372 416 331 319)(184 393 373 409 332 320) (1 465 188 51 59)(2 60 52 189 466)(3 467 190 53 61)(4 62 54 191 468)(5 469 192 55 63)(6 64 56 185 470)(7 471 186 49 57)(8 58 50 187 472)(9 264 425 433 201)(10 202 434 426 257)(11 258 427 435 203)(12 204 436 428 259)(13 260 429 437 205)(14 206 438 430 261)(15 262 431 439 207)(16 208 440 432 263)(17 271 32 277 209)(18 210 278 25 272)(19 265 26 279 211)(20 212 280 27 266)(21 267 28 273 213)(22 214 274 29 268)(23 269 30 275 215)(24 216 276 31 270)(33 219 441 449 226)(34 227 450 442 220)(35 221 443 451 228)(36 229 452 444 222)(37 223 445 453 230)(38 231 454 446 224)(39 217 447 455 232)(40 225 456 448 218)(41 291 283 464 235)(42 236 457 284 292)(43 293 285 458 237)(44 238 459 286 294)(45 295 287 460 239)(46 240 461 288 296)(47 289 281 462 233)(48 234 463 282 290)(65 158 357 326 334)(66 335 327 358 159)(67 160 359 328 336)(68 329 321 360 153)(69 154 353 322 330)(70 331 323 354 155)(71 156 355 324 332)(72 333 325 356 157)(73 340 361 369 165)(74 166 370 362 341)(75 342 363 371 167)(76 168 372 364 343)(77 344 365 373 161)(78 162 374 366 337)(79 338 367 375 163)(80 164 376 368 339)(81 183 175 384 349)(82 350 377 176 184)(83 177 169 378 351)(84 352 379 170 178)(85 179 171 380 345)(86 346 381 172 180)(87 181 173 382 347)(88 348 383 174 182)(89 397 121 391 304)(90 297 392 122 398)(91 399 123 385 298)(92 299 386 124 400)(93 393 125 387 300)(94 301 388 126 394)(95 395 127 389 302)(96 303 390 128 396)(97 311 112 317 129)(98 130 318 105 312)(99 305 106 319 131)(100 132 320 107 306)(101 307 108 313 133)(102 134 314 109 308)(103 309 110 315 135)(104 136 316 111 310)(113 139 401 409 146)(114 147 410 402 140)(115 141 403 411 148)(116 149 412 404 142)(117 143 405 413 150)(118 151 414 406 144)(119 137 407 415 152)(120 145 416 408 138)(193 423 249 241 480)(194 473 242 250 424)(195 417 251 243 474)(196 475 244 252 418)(197 419 253 245 476)(198 477 246 254 420)(199 421 255 247 478)(200 479 248 256 422) (1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88)(89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104)(105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128)(129 130 131 132 133 134 135 136)(137 138 139 140 141 142 143 144)(145 146 147 148 149 150 151 152)(153 154 155 156 157 158 159 160)(161 162 163 164 165 166 167 168)(169 170 171 172 173 174 175 176)(177 178 179 180 181 182 183 184)(185 186 187 188 189 190 191 192)(193 194 195 196 197 198 199 200)(201 202 203 204 205 206 207 208)(209 210 211 212 213 214 215 216)(217 218 219 220 221 222 223 224)(225 226 227 228 229 230 231 232)(233 234 235 236 237 238 239 240)(241 242 243 244 245 246 247 248)(249 250 251 252 253 254 255 256)(257 258 259 260 261 262 263 264)(265 266 267 268 269 270 271 272)(273 274 275 276 277 278 279 280)(281 282 283 284 285 286 287 288)(289 290 291 292 293 294 295 296)(297 298 299 300 301 302 303 304)(305 306 307 308 309 310 311 312)(313 314 315 316 317 318 319 320)(321 322 323 324 325 326 327 328)(329 330 331 332 333 334 335 336)(337 338 339 340 341 342 343 344)(345 346 347 348 349 350 351 352)(353 354 355 356 357 358 359 360)(361 362 363 364 365 366 367 368)(369 370 371 372 373 374 375 376)(377 378 379 380 381 382 383 384)(385 386 387 388 389 390 391 392)(393 394 395 396 397 398 399 400)(401 402 403 404 405 406 407 408)(409 410 411 412 413 414 415 416)(417 418 419 420 421 422 423 424)(425 426 427 428 429 430 431 432)(433 434 435 436 437 438 439 440)(441 442 443 444 445 446 447 448)(449 450 451 452 453 454 455 456)(457 458 459 460 461 462 463 464)(465 466 467 468 469 470 471 472)(473 474 475 476 477 478 479 480) G:=sub<Sym(480)\| (1,69)(2,70)(3,71)(4,72)(5,65)(6,66)(7,67)(8,68)(9,346)(10,347)(11,348)(12,349)(13,350)(14,351)(15,352)(16,345)(17,301)(18,302)(19,303)(20,304)(21,297)(22,298)(23,299)(24,300)(25,127)(26,128)(27,121)(28,122)(29,123)(30,124)(31,125)(32,126)(33,310)(34,311)(35,312)(36,305)(37,306)(38,307)(39,308)(40,309)(41,145)(42,146)(43,147)(44,148)(45,149)(46,150)(47,151)(48,152)(49,328)(50,321)(51,322)(52,323)(53,324)(54,325)(55,326)(56,327)(57,336)(58,329)(59,330)(60,331)(61,332)(62,333)(63,334)(64,335)(73,196)(74,197)(75,198)(76,199)(77,200)(78,193)(79,194)(80,195)(81,204)(82,205)(83,206)(84,207)(85,208)(86,201)(87,202)(88,203)(89,212)(90,213)(91,214)(92,215)(93,216)(94,209)(95,210)(96,211)(97,220)(98,221)(99,222)(100,223)(101,224)(102,217)(103,218)(104,219)(105,228)(106,229)(107,230)(108,231)(109,232)(110,225)(111,226)(112,227)(113,236)(114,237)(115,238)(116,239)(117,240)(118,233)(119,234)(120,235)(129,442)(130,443)(131,444)(132,445)(133,446)(134,447)(135,448)(136,441)(137,463)(138,464)(139,457)(140,458)(141,459)(142,460)(143,461)(144,462)(153,472)(154,465)(155,466)(156,467)(157,468)(158,469)(159,470)(160,471)(161,422)(162,423)(163,424)(164,417)(165,418)(166,419)(167,420)(168,421)(169,430)(170,431)(171,432)(172,425)(173,426)(174,427)(175,428)(176,429)(177,438)(178,439)(179,440)(180,433)(181,434)(182,435)(183,436)(184,437)(185,358)(186,359)(187,360)(188,353)(189,354)(190,355)(191,356)(192,357)(241,366)(242,367)(243,368)(244,361)(245,362)(246,363)(247,364)(248,365)(249,374)(250,375)(251,376)(252,369)(253,370)(254,371)(255,372)(256,373)(257,382)(258,383)(259,384)(260,377)(261,378)(262,379)(263,380)(264,381)(265,390)(266,391)(267,392)(268,385)(269,386)(270,387)(271,388)(272,389)(273,398)(274,399)(275,400)(276,393)(277,394)(278,395)(279,396)(280,397)(281,406)(282,407)(283,408)(284,401)(285,402)(286,403)(287,404)(288,405)(289,414)(290,415)(291,416)(292,409)(293,410)(294,411)(295,412)(296,413)(313,454)(314,455)(315,456)(316,449)(317,450)(318,451)(319,452)(320,453)(337,480)(338,473)(339,474)(340,475)(341,476)(342,477)(343,478)(344,479), 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(1,465,188,51,59)(2,60,52,189,466)(3,467,190,53,61)(4,62,54,191,468)(5,469,192,55,63)(6,64,56,185,470)(7,471,186,49,57)(8,58,50,187,472)(9,264,425,433,201)(10,202,434,426,257)(11,258,427,435,203)(12,204,436,428,259)(13,260,429,437,205)(14,206,438,430,261)(15,262,431,439,207)(16,208,440,432,263)(17,271,32,277,209)(18,210,278,25,272)(19,265,26,279,211)(20,212,280,27,266)(21,267,28,273,213)(22,214,274,29,268)(23,269,30,275,215)(24,216,276,31,270)(33,219,441,449,226)(34,227,450,442,220)(35,221,443,451,228)(36,229,452,444,222)(37,223,445,453,230)(38,231,454,446,224)(39,217,447,455,232)(40,225,456,448,218)(41,291,283,464,235)(42,236,457,284,292)(43,293,285,458,237)(44,238,459,286,294)(45,295,287,460,239)(46,240,461,288,296)(47,289,281,462,233)(48,234,463,282,290)(65,158,357,326,334)(66,335,327,358,159)(67,160,359,328,336)(68,329,321,360,153)(69,154,353,322,330)(70,331,323,354,155)(71,156,355,324,332)(72,333,325,356,157)(73,340,361,369,165)(74,166,370,362,341)(75,342,363,371,167)(76,168,372,364,343)(77,344,365,373,161)(78,162,374,366,337)(79,338,367,375,163)(80,164,376,368,339)(81,183,175,384,349)(82,350,377,176,184)(83,177,169,378,351)(84,352,379,170,178)(85,179,171,380,345)(86,346,381,172,180)(87,181,173,382,347)(88,348,383,174,182)(89,397,121,391,304)(90,297,392,122,398)(91,399,123,385,298)(92,299,386,124,400)(93,393,125,387,300)(94,301,388,126,394)(95,395,127,389,302)(96,303,390,128,396)(97,311,112,317,129)(98,130,318,105,312)(99,305,106,319,131)(100,132,320,107,306)(101,307,108,313,133)(102,134,314,109,308)(103,309,110,315,135)(104,136,316,111,310)(113,139,401,409,146)(114,147,410,402,140)(115,141,403,411,148)(116,149,412,404,142)(117,143,405,413,150)(118,151,414,406,144)(119,137,407,415,152)(120,145,416,408,138)(193,423,249,241,480)(194,473,242,250,424)(195,417,251,243,474)(196,475,244,252,418)(197,419,253,245,476)(198,477,246,254,420)(199,421,255,247,478)(200,479,248,256,422), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136)(137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152)(153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176)(177,178,179,180,181,182,183,184)(185,186,187,188,189,190,191,192)(193,194,195,196,197,198,199,200)(201,202,203,204,205,206,207,208)(209,210,211,212,213,214,215,216)(217,218,219,220,221,222,223,224)(225,226,227,228,229,230,231,232)(233,234,235,236,237,238,239,240)(241,242,243,244,245,246,247,248)(249,250,251,252,253,254,255,256)(257,258,259,260,261,262,263,264)(265,266,267,268,269,270,271,272)(273,274,275,276,277,278,279,280)(281,282,283,284,285,286,287,288)(289,290,291,292,293,294,295,296)(297,298,299,300,301,302,303,304)(305,306,307,308,309,310,311,312)(313,314,315,316,317,318,319,320)(321,322,323,324,325,326,327,328)(329,330,331,332,333,334,335,336)(337,338,339,340,341,342,343,344)(345,346,347,348,349,350,351,352)(353,354,355,356,357,358,359,360)(361,362,363,364,365,366,367,368)(369,370,371,372,373,374,375,376)(377,378,379,380,381,382,383,384)(385,386,387,388,389,390,391,392)(393,394,395,396,397,398,399,400)(401,402,403,404,405,406,407,408)(409,410,411,412,413,414,415,416)(417,418,419,420,421,422,423,424)(425,426,427,428,429,430,431,432)(433,434,435,436,437,438,439,440)(441,442,443,444,445,446,447,448)(449,450,451,452,453,454,455,456)(457,458,459,460,461,462,463,464)(465,466,467,468,469,470,471,472)(473,474,475,476,477,478,479,480)>; G:=Group( (1,69)(2,70)(3,71)(4,72)(5,65)(6,66)(7,67)(8,68)(9,346)(10,347)(11,348)(12,349)(13,350)(14,351)(15,352)(16,345)(17,301)(18,302)(19,303)(20,304)(21,297)(22,298)(23,299)(24,300)(25,127)(26,128)(27,121)(28,122)(29,123)(30,124)(31,125)(32,126)(33,310)(34,311)(35,312)(36,305)(37,306)(38,307)(39,308)(40,309)(41,145)(42,146)(43,147)(44,148)(45,149)(46,150)(47,151)(48,152)(49,328)(50,321)(51,322)(52,323)(53,324)(54,325)(55,326)(56,327)(57,336)(58,329)(59,330)(60,331)(61,332)(62,333)(63,334)(64,335)(73,196)(74,197)(75,198)(76,199)(77,200)(78,193)(79,194)(80,195)(81,204)(82,205)(83,206)(84,207)(85,208)(86,201)(87,202)(88,203)(89,212)(90,213)(91,214)(92,215)(93,216)(94,209)(95,210)(96,211)(97,220)(98,221)(99,222)(100,223)(101,224)(102,217)(103,218)(104,219)(105,228)(106,229)(107,230)(108,231)(109,232)(110,225)(111,226)(112,227)(113,236)(114,237)(115,238)(116,239)(117,240)(118,233)(119,234)(120,235)(129,442)(130,443)(131,444)(132,445)(133,446)(134,447)(135,448)(136,441)(137,463)(138,464)(139,457)(140,458)(141,459)(142,460)(143,461)(144,462)(153,472)(154,465)(155,466)(156,467)(157,468)(158,469)(159,470)(160,471)(161,422)(162,423)(163,424)(164,417)(165,418)(166,419)(167,420)(168,421)(169,430)(170,431)(171,432)(172,425)(173,426)(174,427)(175,428)(176,429)(177,438)(178,439)(179,440)(180,433)(181,434)(182,435)(183,436)(184,437)(185,358)(186,359)(187,360)(188,353)(189,354)(190,355)(191,356)(192,357)(241,366)(242,367)(243,368)(244,361)(245,362)(246,363)(247,364)(248,365)(249,374)(250,375)(251,376)(252,369)(253,370)(254,371)(255,372)(256,373)(257,382)(258,383)(259,384)(260,377)(261,378)(262,379)(263,380)(264,381)(265,390)(266,391)(267,392)(268,385)(269,386)(270,387)(271,388)(272,389)(273,398)(274,399)(275,400)(276,393)(277,394)(278,395)(279,396)(280,397)(281,406)(282,407)(283,408)(284,401)(285,402)(286,403)(287,404)(288,405)(289,414)(290,415)(291,416)(292,409)(293,410)(294,411)(295,412)(296,413)(313,454)(314,455)(315,456)(316,449)(317,450)(318,451)(319,452)(320,453)(337,480)(338,473)(339,474)(340,475)(341,476)(342,477)(343,478)(344,479), (1,228,203,29,420,48)(2,229,204,30,421,41)(3,230,205,31,422,42)(4,231,206,32,423,43)(5,232,207,25,424,44)(6,225,208,26,417,45)(7,226,201,27,418,46)(8,227,202,28,419,47)(9,266,196,240,471,33)(10,267,197,233,472,34)(11,268,198,234,465,35)(12,269,199,235,466,36)(13,270,200,236,467,37)(14,271,193,237,468,38)(15,272,194,238,469,39)(16,265,195,239,470,40)(17,480,458,191,224,261)(18,473,459,192,217,262)(19,474,460,185,218,263)(20,475,461,186,219,264)(21,476,462,187,220,257)(22,477,463,188,221,258)(23,478,464,189,222,259)(24,479,457,190,223,260)(49,441,425,212,244,288)(50,442,426,213,245,281)(51,443,427,214,246,282)(52,444,428,215,247,283)(53,445,429,216,248,284)(54,446,430,209,241,285)(55,447,431,210,242,286)(56,448,432,211,243,287)(57,449,433,280,252,296)(58,450,434,273,253,289)(59,451,435,274,254,290)(60,452,436,275,255,291)(61,453,437,276,256,292)(62,454,438,277,249,293)(63,455,439,278,250,294)(64,456,440,279,251,295)(65,109,84,127,163,148)(66,110,85,128,164,149)(67,111,86,121,165,150)(68,112,87,122,166,151)(69,105,88,123,167,152)(70,106,81,124,168,145)(71,107,82,125,161,146)(72,108,83,126,162,147)(73,117,160,310,346,391)(74,118,153,311,347,392)(75,119,154,312,348,385)(76,120,155,305,349,386)(77,113,156,306,350,387)(78,114,157,307,351,388)(79,115,158,308,352,389)(80,116,159,309,345,390)(89,361,405,328,136,172)(90,362,406,321,129,173)(91,363,407,322,130,174)(92,364,408,323,131,175)(93,365,401,324,132,176)(94,366,402,325,133,169)(95,367,403,326,134,170)(96,368,404,327,135,171)(97,382,297,341,144,360)(98,383,298,342,137,353)(99,384,299,343,138,354)(100,377,300,344,139,355)(101,378,301,337,140,356)(102,379,302,338,141,357)(103,380,303,339,142,358)(104,381,304,340,143,359)(177,394,374,410,333,313)(178,395,375,411,334,314)(179,396,376,412,335,315)(180,397,369,413,336,316)(181,398,370,414,329,317)(182,399,371,415,330,318)(183,400,372,416,331,319)(184,393,373,409,332,320), (1,465,188,51,59)(2,60,52,189,466)(3,467,190,53,61)(4,62,54,191,468)(5,469,192,55,63)(6,64,56,185,470)(7,471,186,49,57)(8,58,50,187,472)(9,264,425,433,201)(10,202,434,426,257)(11,258,427,435,203)(12,204,436,428,259)(13,260,429,437,205)(14,206,438,430,261)(15,262,431,439,207)(16,208,440,432,263)(17,271,32,277,209)(18,210,278,25,272)(19,265,26,279,211)(20,212,280,27,266)(21,267,28,273,213)(22,214,274,29,268)(23,269,30,275,215)(24,216,276,31,270)(33,219,441,449,226)(34,227,450,442,220)(35,221,443,451,228)(36,229,452,444,222)(37,223,445,453,230)(38,231,454,446,224)(39,217,447,455,232)(40,225,456,448,218)(41,291,283,464,235)(42,236,457,284,292)(43,293,285,458,237)(44,238,459,286,294)(45,295,287,460,239)(46,240,461,288,296)(47,289,281,462,233)(48,234,463,282,290)(65,158,357,326,334)(66,335,327,358,159)(67,160,359,328,336)(68,329,321,360,153)(69,154,353,322,330)(70,331,323,354,155)(71,156,355,324,332)(72,333,325,356,157)(73,340,361,369,165)(74,166,370,362,341)(75,342,363,371,167)(76,168,372,364,343)(77,344,365,373,161)(78,162,374,366,337)(79,338,367,375,163)(80,164,376,368,339)(81,183,175,384,349)(82,350,377,176,184)(83,177,169,378,351)(84,352,379,170,178)(85,179,171,380,345)(86,346,381,172,180)(87,181,173,382,347)(88,348,383,174,182)(89,397,121,391,304)(90,297,392,122,398)(91,399,123,385,298)(92,299,386,124,400)(93,393,125,387,300)(94,301,388,126,394)(95,395,127,389,302)(96,303,390,128,396)(97,311,112,317,129)(98,130,318,105,312)(99,305,106,319,131)(100,132,320,107,306)(101,307,108,313,133)(102,134,314,109,308)(103,309,110,315,135)(104,136,316,111,310)(113,139,401,409,146)(114,147,410,402,140)(115,141,403,411,148)(116,149,412,404,142)(117,143,405,413,150)(118,151,414,406,144)(119,137,407,415,152)(120,145,416,408,138)(193,423,249,241,480)(194,473,242,250,424)(195,417,251,243,474)(196,475,244,252,418)(197,419,253,245,476)(198,477,246,254,420)(199,421,255,247,478)(200,479,248,256,422), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136)(137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152)(153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176)(177,178,179,180,181,182,183,184)(185,186,187,188,189,190,191,192)(193,194,195,196,197,198,199,200)(201,202,203,204,205,206,207,208)(209,210,211,212,213,214,215,216)(217,218,219,220,221,222,223,224)(225,226,227,228,229,230,231,232)(233,234,235,236,237,238,239,240)(241,242,243,244,245,246,247,248)(249,250,251,252,253,254,255,256)(257,258,259,260,261,262,263,264)(265,266,267,268,269,270,271,272)(273,274,275,276,277,278,279,280)(281,282,283,284,285,286,287,288)(289,290,291,292,293,294,295,296)(297,298,299,300,301,302,303,304)(305,306,307,308,309,310,311,312)(313,314,315,316,317,318,319,320)(321,322,323,324,325,326,327,328)(329,330,331,332,333,334,335,336)(337,338,339,340,341,342,343,344)(345,346,347,348,349,350,351,352)(353,354,355,356,357,358,359,360)(361,362,363,364,365,366,367,368)(369,370,371,372,373,374,375,376)(377,378,379,380,381,382,383,384)(385,386,387,388,389,390,391,392)(393,394,395,396,397,398,399,400)(401,402,403,404,405,406,407,408)(409,410,411,412,413,414,415,416)(417,418,419,420,421,422,423,424)(425,426,427,428,429,430,431,432)(433,434,435,436,437,438,439,440)(441,442,443,444,445,446,447,448)(449,450,451,452,453,454,455,456)(457,458,459,460,461,462,463,464)(465,466,467,468,469,470,471,472)(473,474,475,476,477,478,479,480) ); G=PermutationGroup([[(1,69),(2,70),(3,71),(4,72),(5,65),(6,66),(7,67),(8,68),(9,346),(10,347),(11,348),(12,349),(13,350),(14,351),(15,352),(16,345),(17,301),(18,302),(19,303),(20,304),(21,297),(22,298),(23,299),(24,300),(25,127),(26,128),(27,121),(28,122),(29,123),(30,124),(31,125),(32,126),(33,310),(34,311),(35,312),(36,305),(37,306),(38,307),(39,308),(40,309),(41,145),(42,146),(43,147),(44,148),(45,149),(46,150),(47,151),(48,152),(49,328),(50,321),(51,322),(52,323),(53,324),(54,325),(55,326),(56,327),(57,336),(58,329),(59,330),(60,331),(61,332),(62,333),(63,334),(64,335),(73,196),(74,197),(75,198),(76,199),(77,200),(78,193),(79,194),(80,195),(81,204),(82,205),(83,206),(84,207),(85,208),(86,201),(87,202),(88,203),(89,212),(90,213),(91,214),(92,215),(93,216),(94,209),(95,210),(96,211),(97,220),(98,221),(99,222),(100,223),(101,224),(102,217),(103,218),(104,219),(105,228),(106,229),(107,230),(108,231),(109,232),(110,225),(111,226),(112,227),(113,236),(114,237),(115,238),(116,239),(117,240),(118,233),(119,234),(120,235),(129,442),(130,443),(131,444),(132,445),(133,446),(134,447),(135,448),(136,441),(137,463),(138,464),(139,457),(140,458),(141,459),(142,460),(143,461),(144,462),(153,472),(154,465),(155,466),(156,467),(157,468),(158,469),(159,470),(160,471),(161,422),(162,423),(163,424),(164,417),(165,418),(166,419),(167,420),(168,421),(169,430),(170,431),(171,432),(172,425),(173,426),(174,427),(175,428),(176,429),(177,438),(178,439),(179,440),(180,433),(181,434),(182,435),(183,436),(184,437),(185,358),(186,359),(187,360),(188,353),(189,354),(190,355),(191,356),(192,357),(241,366),(242,367),(243,368),(244,361),(245,362),(246,363),(247,364),(248,365),(249,374),(250,375),(251,376),(252,369),(253,370),(254,371),(255,372),(256,373),(257,382),(258,383),(259,384),(260,377),(261,378),(262,379),(263,380),(264,381),(265,390),(266,391),(267,392),(268,385),(269,386),(270,387),(271,388),(272,389),(273,398),(274,399),(275,400),(276,393),(277,394),(278,395),(279,396),(280,397),(281,406),(282,407),(283,408),(284,401),(285,402),(286,403),(287,404),(288,405),(289,414),(290,415),(291,416),(292,409),(293,410),(294,411),(295,412),(296,413),(313,454),(314,455),(315,456),(316,449),(317,450),(318,451),(319,452),(320,453),(337,480),(338,473),(339,474),(340,475),(341,476),(342,477),(343,478),(344,479)], 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[(1,465,188,51,59),(2,60,52,189,466),(3,467,190,53,61),(4,62,54,191,468),(5,469,192,55,63),(6,64,56,185,470),(7,471,186,49,57),(8,58,50,187,472),(9,264,425,433,201),(10,202,434,426,257),(11,258,427,435,203),(12,204,436,428,259),(13,260,429,437,205),(14,206,438,430,261),(15,262,431,439,207),(16,208,440,432,263),(17,271,32,277,209),(18,210,278,25,272),(19,265,26,279,211),(20,212,280,27,266),(21,267,28,273,213),(22,214,274,29,268),(23,269,30,275,215),(24,216,276,31,270),(33,219,441,449,226),(34,227,450,442,220),(35,221,443,451,228),(36,229,452,444,222),(37,223,445,453,230),(38,231,454,446,224),(39,217,447,455,232),(40,225,456,448,218),(41,291,283,464,235),(42,236,457,284,292),(43,293,285,458,237),(44,238,459,286,294),(45,295,287,460,239),(46,240,461,288,296),(47,289,281,462,233),(48,234,463,282,290),(65,158,357,326,334),(66,335,327,358,159),(67,160,359,328,336),(68,329,321,360,153),(69,154,353,322,330),(70,331,323,354,155),(71,156,355,324,332),(72,333,325,356,157),(73,340,361,369,165),(74,166,370,362,341),(75,342,363,371,167),(76,168,372,364,343),(77,344,365,373,161),(78,162,374,366,337),(79,338,367,375,163),(80,164,376,368,339),(81,183,175,384,349),(82,350,377,176,184),(83,177,169,378,351),(84,352,379,170,178),(85,179,171,380,345),(86,346,381,172,180),(87,181,173,382,347),(88,348,383,174,182),(89,397,121,391,304),(90,297,392,122,398),(91,399,123,385,298),(92,299,386,124,400),(93,393,125,387,300),(94,301,388,126,394),(95,395,127,389,302),(96,303,390,128,396),(97,311,112,317,129),(98,130,318,105,312),(99,305,106,319,131),(100,132,320,107,306),(101,307,108,313,133),(102,134,314,109,308),(103,309,110,315,135),(104,136,316,111,310),(113,139,401,409,146),(114,147,410,402,140),(115,141,403,411,148),(116,149,412,404,142),(117,143,405,413,150),(118,151,414,406,144),(119,137,407,415,152),(120,145,416,408,138),(193,423,249,241,480),(194,473,242,250,424),(195,417,251,243,474),(196,475,244,252,418),(197,419,253,245,476),(198,477,246,254,420),(199,421,255,247,478),(200,479,248,256,422)], 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192 conjugacy classes class 1 2A ··· 2G 3A 3B 4A ··· 4H 5A 5B 6A ··· 6N 8A ··· 8P 10A ··· 10N 12A ··· 12P 15A 15B 15C 15D 20A ··· 20P 24A ··· 24AF 30A ··· 30AB 60A ··· 60AF order 1 2 ··· 2 3 3 4 ··· 4 5 5 6 ··· 6 8 ··· 8 10 ··· 10 12 ··· 12 15 15 15 15 20 ··· 20 24 ··· 24 30 ··· 30 60 ··· 60 size 1 1 ··· 1 1 1 1 ··· 1 2 2 1 ··· 1 5 ··· 5 2 ··· 2 1 ··· 1 2 2 2 2 2 ··· 2 5 ··· 5 2 ··· 2 2 ··· 2 192 irreducible representations dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + - + - image C1 C2 C2 C3 C4 C4 C6 C6 C8 C12 C12 C24 D5 Dic5 D10 Dic5 C3×D5 C5⋊2C8 C3×Dic5 C6×D5 C3×Dic5 C3×C5⋊2C8 kernel C2×C6×C5⋊2C8 C6×C5⋊2C8 C22×C60 C22×C5⋊2C8 C2×C60 C22×C30 C2×C5⋊2C8 C22×C20 C2×C30 C2×C20 C22×C10 C2×C10 C22×C12 C2×C12 C2×C12 C22×C6 C22×C4 C2×C6 C2×C4 C2×C4 C23 C22 # reps 1 6 1 2 6 2 12 2 16 12 4 32 2 6 6 2 4 16 12 12 4 32 Matrix representation of C2×C6×C52C8 in GL4(𝔽241) generated by 1 0 0 0 0 240 0 0 0 0 1 0 0 0 0 1 , 226 0 0 0 0 240 0 0 0 0 16 0 0 0 0 16 , 1 0 0 0 0 1 0 0 0 0 189 240 0 0 1 0 , 64 0 0 0 0 240 0 0 0 0 70 203 0 0 178 171 G:=sub<GL(4,GF(241))\| [1,0,0,0,0,240,0,0,0,0,1,0,0,0,0,1],[226,0,0,0,0,240,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,189,1,0,0,240,0],[64,0,0,0,0,240,0,0,0,0,70,178,0,0,203,171] >; C2×C6×C52C8 in GAP, Magma, Sage, TeX C_2\times C_6\times C_5\rtimes_2C_8 % in TeX G:=Group("C2xC6xC5:2C8"); // GroupNames label G:=SmallGroup(480,713); // by ID G=gap.SmallGroup(480,713); # by ID G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,168,102,18822]); // Polycyclic G:=Group<a,b,c,d\|a^2=b^6=c^5=d^8=1,ab=ba,ac=ca,ad=da,bc=cb,bd=db,dcd^-1=c^-1>; // generators/relations ׿ × 𝔽	2021-12-01 09:29:08	{"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, 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https://claesjohnson.blogspot.com/2019/11/transition-from-no-slip-laminar.html	## onsdag 27 november 2019 ### Bypass Transition from No-Slip Laminar Boundary Layer to Slip Boundary Condition The New Theory of Flight is supported by Direct Finite Element Simulation DFS as best possible computational satisfaction of Euler's equations expressing first principle physics in the form (i) incompressibility, (ii) momentum balance and (iii) slip boundary condition on solid walls. Observations and experiments (connecting to the so-called drag crisis) indicate that at a Reynolds number Re of about $10^6$ the boundary condition at a solid wall changes from no-slip at the wall accompanied with a thin laminar boundary layer, to effectively a slip condition as a thin film without layer. Let us now see if we can understand this transition from no-slip with laminar layer to slip from some simple mathematical considerations. We thus consider flow over a flat plate as $y\ge 0$ in a $(x,y,z)$-coordinate system with main flow in the $x$-direction with speed 1. We consider stationary parallel flow with velocity $(u(y),0,0)$ only depending on $y$ and pressure $p(x)$ only on $x$ modeled by the following reduced form of the Euler equations: • $\frac{\partial p}{\partial x}+\nu\frac{\partial^2 u}{\partial y^2}=0$ for $y\gt 0$, • $u(0)=0,\quad u(\infty )=1$. Normalising to $\frac{\partial p}{\partial x}=1$, the solution takes the form • $u(y)=1-\exp(-\frac{y}{\sqrt{\nu}})$ for $y\gt 0$. We see that velocity $u(y)$ has a boundary layer of width $\sqrt{\nu}$ connecting the free flow velocity $1$ to the no-slip velocity $0$. We now replace the no-slip condition $u=0$ by a friction boundary condition of the form • $\beta u=\nu\frac{\partial u}{\partial y}$ for $y=0$, where $\beta \gt 0$ is a (skin) friction parameter. The solution is now (with $\frac{\partial p}{\partial x}=a$): • $u(y)=1-a\exp(-\frac{y}{\sqrt{\nu}})$ for $y\gt 0$, • $a = \frac{\beta}{\beta +\sqrt{\nu}}$. For $\beta$ small (small friction), the solution is $u(y)=1$ with full slip, and for $\beta$ large it is the no-slip solution. The transition is anchored at $\beta =\sqrt{\nu}$. We now return to the observation of transition for $\nu = 10^{-6}$ if we normalise $Re =\frac{UL}{\nu}$ with $U=1$ and $L=1$, which gives $\sqrt{\nu}=0.001$. Observation thus supports an idea that transition from no-slip to effectively slip can take place when • skin friction coefficient is $\approx 0.001$, • boundary layer thickness is $0.1\%$ of gross dimension, • shear exceeds 1000. We thus observe the free flow to effectively act as having a slip/small friction boundary condition when the width of a laminar boundary layer is smaller than $0.1\%$ of the length scale $L$ in the specification of the Reynolds number. For an airplane wing of chord 1 m this means a boundary layer thickness of 1 mm, for a jumbojet 5 mm. Note that slip occurring when shear is bigger than 1000 connects to both friction between solids where slip occurs when tangential force is big enough (scaling with normal force), and to plasticity in solids with slip surfaces occurring for large enough stresses, both as threshold phenomena. For a fluid the threshold thus may relate to shear and for a solid to shear stress. We view such a transition form laminar no-slip to slip as "bypass" of transition into a no-slip turbulent boundary layer, which may take place for a smaller Reynolds number. We see the difference in skin friction coefficient between that of thin film limit of a laminar boundary layer (red curve) and various turbulent boundary layers: We also see support of the conjectured level of skin friction of 0.001 for transition to slip. We recall that the generation of lift of a wing critically depends on an effective slip condition, to secure that the flow does not separate on the crest of the suction side of the wing, which connects to observation of gliding flight only for $Re\gt 5\times 10^5$, allowing birds and airplanes to fly without the intense flapping required for little fruit flies with much smaller Re. We recall that the flow around a wing, or more generally around a streamline body, can be more favourable as concerns bypass to slip because of the accelerating flow after attachment, which has a stabilising effect on streamwise velocity, followed by deceleration after the crest with stabilising effect on streamwise velocity. We also recall that forced tripping of flow into transition to a turbulent boundary is typically used in flat plate experiments, which when translated to streamline bodies without artificial tripping incorrectly attributes most of drag to skin friction. See more posts on skin friction.	2020-06-04 23:55:55	{"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": 0.7657167315483093, "perplexity": 1042.4868620454527}, "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/1590348492295.88/warc/CC-MAIN-20200604223445-20200605013445-00376.warc.gz"}
https://eng.libretexts.org/Courses/Canada_College/Circuits_and_Devices%3A_Laboratory/03%3A_Relating_V_I_and_R/3.04%3A_Series_DC_Circuits/3.4.05%3A_Data_Tables	# 3.4.5: Data Tables I Theory I Point A I Point B I Point C Table $$\PageIndex{1}$$ Voltage Theory Measured Deviation R1 R2 R3 Table $$\PageIndex{2}$$ Voltage Theory Measured Deviation R1 R2 R3 R4 $$V_{AC}$$ $$V_B$$ Table $$\PageIndex{3}$$ This page titled 3.4.5: Data Tables is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by James M. Fiore.	2022-12-10 02:15:01	{"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": 0.9980862140655518, "perplexity": 3352.579786498974}, "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/1669446711637.64/warc/CC-MAIN-20221210005738-20221210035738-00631.warc.gz"}
https://www.biostars.org/p/368703/	Run Delly by chromosome 0 0 Entering edit mode 3.4 years ago tg25 ▴ 20 Dear All, How do you run Delly by chromosome? I have looked at the Delly GitHub page and I cannot find the answer. I have also looked at old questions like this Delly question but this question does not address how to run by chromosome. I am running delly call -t DEL -g ref.fa -o dellyOut.bcf file1.markedDup.bam file2.markedDup.bam file3.markedDup.bam file4.markedDup.bam . . . file100.markedDup.bam For each of the structural variations I am running the options below: -t DEL = Deletions -t INS = Insertions -t INV = Inversions -t DUP = Tandem Duplications -t TRA = Inter-chromosomal translocations While this is okay, I would also like to run Delly by chromosome. If any of you know the list of options for Delly or where they are stated can you please post a link. I found the information about option -t by reading this Biostars question. delly next-gen Structural variant discovery bwa • 1.3k views 1 Entering edit mode Can you split the BAM into chromosomes and run Delly? You may split BAM using samtools or sambamba. 0 Entering edit mode Yes, I can split the BAMs by chromosome. With doing this I have some questions: 1. If I split the BAMs then realign with bwa, would Delly detect inter chromosomal translocations? 2. Or if I split my already bwa aligned BAMs, would Delly detect inter chromosomal translocations? Before using Delly, I used BreakDancer which has an option -o : STRING operate on a single chromosome [all chromosome]. If Delly as a similar option this would be great. Additional Information: I forgot to add that BreakDancer has an option -t : only detect transchromosomal rearrangement that I use to detect inter chromosomal translocations. Because of the number of files that I work with I found that splitting the analysis by chromosome and then running option -t for ctx on the same (unspilt bams) worked really well in breakdancer. All I have to do is then merge the outputs. My analysis was timing out in Breakdancer with splitting the analysis fixed this and I would like to do the same with Delly. 0 Entering edit mode Translocation callers usually use split read pairs or reads across chromosomes (secondary alignments, SA tag), so it does not make sense to call translocations using single chromosomes, unless you know the chromosomes on which translocation happened. In that case, you may slice the BAM for the two chromosomes only.	2022-08-09 20:50:04	{"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": 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": 0.21339301764965057, "perplexity": 6200.67008958829}, "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/1659882571086.77/warc/CC-MAIN-20220809185452-20220809215452-00466.warc.gz"}
http://mathhelpforum.com/statistics/3335-help-i-need-help.html	1. Help!!!! I need help!!! Past records show that at a given college 20% of the students who began as psychology majors either changed their major or dropped out the school. An incoming class has 110 beginning psychology majors. What is the probability that as many as 30 of these students leave the psychology program? 2. Hello, bombo31! The problem is not clearly worded . . . ast records show that at a given college 20% of the students who began as psychology majors either changed their major or dropped out the school. An incoming class has 110 beginning psychology majors. What is the probability that as many as 30 of these students leave the psychology program? The phrase "as many as 30" does not have a standard interpretation. Does it mean up to 30? . . If so, the probability is unreasonably huge. We must find the probabilities that: 0 leave, 1 leaves, 2 leave, 3 leave, 4 leave, 5 leave, 6 leave, . . 7 leave, 8 leave, 9 leave, 10 leave, 11 leave, 12 leave, . . . 28 leave, 29 leave, and 30 leave . . and add them up. If it means exactly 30 leave, it is still an unwieldly number: . $C(110,30)(0.2)^{30}(0.8)^{80}$ 3. Originally Posted by bombo31 Past records show that at a given college 20% of the students who began as psychology majors either changed their major or dropped out the school. An incoming class has 110 beginning psychology majors. What is the probability that as many as 30 of these students leave the psychology program? Note sure what you mean. If exactly 30 Then, the probability is, ${110 \choose 30}(.2)^{30}(.8)^{80}\approx .0159$ If at most 30 Then, the probability is, $\sum_{k=0}^{30} {110 \choose k}(.2)^k(.8)^{110-k}\approx .9753$ If at least 30 Then, the probability is, $\sum_{k=30}^{110} {110\choose k}(.2)^k(.8)^{110-k}\approx .0406$ (by trichtonomy there cannot be any other case) 4. Originally Posted by bombo31 Past records show that at a given college 20% of the students who began as psychology majors either changed their major or dropped out the school. An incoming class has 110 beginning psychology majors. What is the probability that as many as 30 of these students leave the psychology program? Lets assume as Soroban suggests that the question means to ask what is the probability that 30 or more leave the psychology program. The number who leave the program is a Binomialy distributed random variable, and PerfectHacker has indicated how to calculate the required probability. However, with numbers like 30+ from 110 in the question we are usually expected to use the Normal approximation to the Binomial Distribution to do the calculations. This means we treat the number who leave the program as a normal random variable with the same mean and standard deviation as the binomial distribution. Here the mean is: $\mu=N\times p=110\times 0.2=22$, and: $\sigma=\sqrt{N\times p \times (1-p)}=\sqrt{110\times 0.2 \times 0.8}\approx 4.1952$. Now with the normal approximation when we ask for the probability of $M$ success we evaluate the area under the normal distribution between $M-0.5$ and $M+0.5$. So to evaluate the probability of 30 or more drop-outs we want to know the probability in the normal approximation of 29.5 or more drop-outs. The critical z-score corresponding to $29.5$ is: $z_c=\frac{29.5-22}{4.1952}=1.7877$ Looking this up on a standard normal table or calculator tells us that the required probability is $\approx 0.037$ or about $3.7 \%$ Now this is different from PH's result which is correct, but it should be slightly different. They do agree to the nearest percentage point. RonL	2017-01-21 07:16:16	{"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": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 13, "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": 0.7628808617591858, "perplexity": 1473.0829258502345}, "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-2017-04/segments/1484560280929.91/warc/CC-MAIN-20170116095120-00377-ip-10-171-10-70.ec2.internal.warc.gz"}
https://math.stackexchange.com/questions/2370017/intersection-point-between-linear-and-sinusoidal-function	# Intersection Point between Linear and Sinusoidal Function Problem Looking to create calculator to solve for alpha in the equation: Cid = 94.2sin(alpha)+phi*cos(alpha) where Cid and phi are user inputs. Isolating alpha algebraically has proven difficult and it would be helpful to be able to find the intersection between the two lines using Excel. Focus This equation was reached while working with rotational matrices (I am new to the topic so there may be problems with my usage of them). I wanted to find the angle required to rotate a point at (-94.2, phi) about the origin so that it reached a height of Cid. Going from the original matrix form, I isolated the new x and y values. x'=xcos(alpha)+ysin(alpha) & y'=-xsin(alpha)+ycos(alpha) I set y' equal to the goal height of Cid and substituted in my original point to find the equation in question. I saw this as an easy way to solve for the angle alpha, however isolating alpha has proven difficult using trig identities. The ultimate goal of this is to make an Excel calculator so although graphing is a possible solution, it would not work for what I intend to use this for. Any help or advice would be greatly appreciated! Let $s = atan2(94.2, \phi)$, and $R = \sqrt{94.2^2 + \phi^2}$. Then you have $$R \sin(S) = 94.2\\ R \cos(S) = \phi\\ C/R = \sin(s) \sin (\alpha) + \cos (s) \cos (\alpha) = \cos(\alpha-s)$$ so $$\alpha = s + \arccos(C/R)$$ is a solution. (In general, there are two solutions, the other being gotten by setting $C/R = \cos(s - \alpha).$)	2021-04-13 01:19:09	{"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": 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": 0.7452754974365234, "perplexity": 311.95723300365347}, "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/1618038071212.27/warc/CC-MAIN-20210413000853-20210413030853-00082.warc.gz"}
http://www.physicsforums.com/showthread.php?p=3810238	# Complex anaylsis, winding number question. by screwyshrew Tags: anaylsis, complex, number, winding P: 2 So the in the equation for the winding number/index of a curve I($\gamma$, z) = $\frac{1}{2i\pi}$ $\int\gamma \frac{1}{ζ-z}dζ$ where $\gamma$ : [a, b] → ℂ and z is an arbitrary point not on $\gamma$, what exactly does ζ represent?	2014-09-01 14:03:08	{"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": 0.7096714377403259, "perplexity": 2768.7521394937467}, "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-2014-35/segments/1409535919066.8/warc/CC-MAIN-20140901014519-00165-ip-10-180-136-8.ec2.internal.warc.gz"}
http://www.vallis.org/blogspace/preprints/1202.5911.html	## [1202.5911] Generic bounds on dipolar gravitational radiation from inspiralling compact binaries Authors: K. G. Arun (Chennai Mathematical Instt) Date: 27 Feb 2012 Abstract: Various alternative theories of gravity predict dipolar gravitational radiation in addition to quadrupolar radiation. We show that gravitational wave (GW) observations of inspiralling compact binaries can put interesting constraints on the strengths of the dipole modes of GW polarizations. We put forward a physically motivated gravitational waveform for dipole modes, in the Fourier domain, in terms of two parameters: one which captures the relative amplitude of the dipole mode with respect to the quadrupole mode ($\alpha$) and the other a dipole term in the phase ($\beta$). We then use this two parameter representation to discuss typical bounds on their values using GW measurements. We obtain the expected bounds on the amplitude parameter $\alpha$ and the phase parameter $\beta$ for Advanced LIGO (AdvLIGO) and Einstein Telescope (ET) noise power spectral densities using Fisher information matrix. AdvLIGO and ET may at best bound $\alpha$ to an accuracy of $\sim10ˆ{-2}$ and $\sim10ˆ{-3}$ and $\beta$ to an accuracy of $\sim10ˆ{-5}$ and $\sim10ˆ{-6}$ respectively. #### Mar 06, 2012 1202.5911 (/preprints) 2012-03-06, 11:08	2018-01-20 05:12:06	{"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": 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": 0.9211785197257996, "perplexity": 1198.5500402270918}, "config": {"markdown_headings": true, "markdown_code": false, "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-2018-05/segments/1516084889325.32/warc/CC-MAIN-20180120043530-20180120063530-00151.warc.gz"}
https://www.groundai.com/project/neutrino-masses-and-mixing-from-flavour-antisymmetry/	Neutrino masses and mixing from flavour antisymmetry # Neutrino masses and mixing from flavour antisymmetry Anjan S. Joshipura Physical Research Laboratory, Navarangpura, Ahmedabad 380 009, India. ###### Abstract We discuss consequences of assuming () that the (Majorana) neutrino mass matrix displays flavour antisymmetry, with respect to some discrete symmetry contained in and () together with a symmetry of the Hermitian combination of the charged lepton mass matrix forms a finite discrete subgroup of whose breaking generates these symmetries. Assumption () leads to at least one massless neutrino and allows only four textures for the neutrino mass matrix in a basis with a diagonal if it is assumed that the other two neutrinos are massive. Two of these textures contain a degenerate pair of neutrinos.Assumption () can be used to determine the neutrino mixing patterns. We work out these patterns for two major group series and as . It is found that all and groups with even contain some elements which can provide appropriate . Mixing patterns can be determined analytically for these groups and it is found that only one of the four allowed neutrino mass textures is consistent with the observed values of the mixing angles and . This texture corresponds to one massless and a degenerate pair of neutrinos which can provide the solar pair in the presence of some perturbations. The well-known groups and provide examples of the groups in respective series allowing correct and . An explicit example based on and displaying a massless and two quasi degenerate neutrinos is discussed. ## I Introduction Orderly pattern of neutrino mixing appears to hide some symmetry, discrete or continuous. It is possible to connect a given mixing pattern with some discrete symmetries of the leptonic mass matrices. Such symmetries may however be residual symmetries arising from a bigger symmetry in the underlying theory. One can obtain a possible larger picture by assuming that these symmetries are a part of a bigger group operating at the fundamental level whose breaking leads to the symmetries of the mass matrices. There is an extensive literature on study of possible residual symmetries of the mass matrices and of the groups which harbor them Lam:2008rs (); Lam:2008sh (); Lam:2009hn (); Lam:2011ag (); Toorop:2011jn (); deAdelhartToorop:2011re (); Altarelli:2012ss (); Holthausen:2012wt (); Hu:2012ei (); Hernandez:2012ra (); Hernandez:2012sk (); Holthausen:2013vba (); Holthausen:2013vba (); Lavoura:2014kwa (); Fonseca:2014lfa (); Hu:2014kca (), see Altarelli:2010gt (); King:2013eh (); Smirnov:2011jv () for reviews and additional references. Starting point in these approaches is to assume the existence of some symmetries (usually a ) and (usually ) of the (Majorana) neutrino and the charged lepton mass matrices T†lMlM†lTl=MlM†l , (1) STνMνSν=Mν . (2) Matrices diagonalizing the symmetry matrices can be related to the mixing matrices in each sector. The structures of these matrices can also be independently fixed if one assume that and represent specific elements of some discrete group in a given three dimensional representation. In this way, the leptonic mixing can be directly related to group theoretical structures. This reasoning has been used for the determination of the neutrino mixing angles in case of the three non-degenerate neutrinos Lam:2008rs (); Lam:2008sh (); Lam:2009hn (); Lam:2011ag (); Toorop:2011jn (); deAdelhartToorop:2011re (); Altarelli:2012ss (); Holthausen:2012wt (); Hu:2012ei (); Hernandez:2012ra (); Hernandez:2012sk (); Holthausen:2013vba (); Holthausen:2013vba (); Lavoura:2014kwa (); Fonseca:2014lfa (); Hu:2014kca (), two or three degenerate neutrinos Hernandez:2013vya (); Joshipura:2014qaa () and one massless and two non-degenerate neutrinos Joshipura:2013pga (); Joshipura:2014pqa (). The residual symmetries may arise from spontaneous breaking of if the vacuum expectation values of the Higgs fields responsible for generating leptonic masses break but respect . We wish to study in this paper consequences of an alternative assumption that the spontaneous breaking of leads to an which displays antisymmetry instead of symmetry, i.e. assume that eq.(2) gets replaced by STνMνSν=−Mν (3) but (1) remains as it is. These assumptions prove to be quite powerful and are able to simultaneously restrict both the mass patterns and mixing angles when embedding of into is considered. We shall further assume that belong to some finite discrete subgroup of with Det Then the first consequence of imposing eq.(3) is that Det, i.e. at least one of the neutrinos remains massless. Since cases with two (or three !) massless neutrinos are not phenomenologically interesting, we shall restrict ourselves to cases with only one massless neutrino. Then as a second consequence of eq.(3), one can determine all the allowed forms of in a given basis for all possible contained in . There exist only four possible (and their permutations) consistent with eq.(3) in a particular basis with a diagonal . Two of these give one massless and two non-degenerate neutrinos and the other two give a massless and a degenerate pair of neutrinos which may be identified with the solar pair. We determine all the allowed textures of the neutrino mass matrix in the next section. Subsequently, we discuss groups and and identify those which can give correct description of mixing using flavour antisymmetry. In section IV, we introduce as neutrino residual symmetry and present an example in which neutrino mass matrix gets fully determined group theoretically except for an overall scale. We discuss a realization of the basic idea with a simple example based on the group in section V. Section VI contains summary and comparison with earlier relevant works. ## Ii Allowed textures for neutrino mass matrix We shall first consider the case of only one satisfying eq.(3) and subsequently generalize it to include two. The unitary matrix can be diagonalized by another unitary matrix : V†SνSνVSν=~Sν where is a diagonal matrix having the form: ~Sν=diag.(λ1,λ2,λ3) , (4) Unitarity of implies that are some roots of unity. They are related by the condition which we assume without lose of generality. We now go to the basis with a diagonal . Defining , eq.(3) can be rewritten as: (~Mν)ij(1+λiλj)=0 (i,j not summed) . (5) It follows that a given element is non-zero only if the factor in bracket multiplying it is zero. This cannot happen for an arbitrary set of and one needs to impose specific relation among them to obtain a non-trivial . We now argue that only two possible forms of and their permutations lead to neutrino mass matrices with two massive neutrinos. The third mass will always be zero as a consequence of eq.(3) and the assumption that belongs to . These forms of are given by: ~S1ν = diag.(λ,−λ∗,−1) , ~S2ν = diag.(±i,∓i,1) . (6) is an arbitrary root of unity. This can be argued as follows. Assume that at least one off-diagonal element of is non-zero which we take as the 12 element for definiteness. In this case, eq.(5) immediately implies the first of eq.(II) as a necessary condition. One can distinguish three separate cases of this condition111 case corresponds to permutation of the case with . (I) (II) and (III) . The entire structures of get determined in these cases from condition eq.(5) as follows: Texture I: ~S1ν=(1,−1,−1); ~Mν=m0⎛⎜⎝0cseiβc00seiβ00⎞⎟⎠ , (7) where . This structure implies one massless and two degenerate neutrinos with a mass . In case of (II), Texture II: ~S1ν=(±i,±i,−1); ~Mν=⎛⎜⎝x1y0yx20000⎞⎟⎠ . (8) This case corresponds to one massless and two non-degenerate neutrinos. In the third case one gets Texture III: ~S1ν=(λ,−λ∗,−1); ~Mν=m0⎛⎜⎝010100000⎞⎟⎠ , (λ≠±1,±i) (9) which implies a massless and a pair of degenerate neutrinos. The cases (I,III) lead to the same mass spectrum but different mixing patterns. in eq.(7) is diagonalized as with Vν=⎛⎜ ⎜ ⎜ ⎜ ⎜⎝1√2−i√20c√2ic√2−ss√2e−iβis√2e−iβce−iβ⎞⎟ ⎟ ⎟ ⎟ ⎟⎠⎛⎜⎝cosϕ−sinϕ0sinϕcosϕ0001⎞⎟⎠ . (10) The arbitrary rotation by an angle originates due to degeneracy in masses. The texture II, eq.(8) is diagonalized by a unitary rotation in the 12 plane while the one in eq.(9) by a similar matrix with the angle . The permutations of entries in give equivalent structures and are obtained by permuting entries in . The case which is not equivalent to above textures follows with a starting assumption that one of the diagonal elements of say, . In this case one requires with . The case with gives which is already covered. implies the condition in (5). This leads to a new texture Texture IV: ~Sν=(i,−i,1); ~Mν=⎛⎜⎝x1000x20000⎞⎟⎠ . (11) For one gets permutation of or and for only 11 element of is non zero and two neutrinos remain massless. Thus conditions eq.(II) and their permutations exhaust all possible textures of consistent with the antisymmetry of , eq.(3) and two massive neutrinos. Any admitting an element with these sets of eigenvalues will give a viable choice for flavour antisymmetry group. Note that texture III (IV) can be obtained from I(II) by putting to zero. But the residual symmetries in all four cases are different. Because of this, the embedding groups can also be different. We therefore discuss all these cases separately. The mixing matrix in texture I contains two unknowns and apart from an overall complex scale . This is a reflection of the fact that the corresponding is a symmetry and contains two degenerate eigenvalues . These unknown can be fixed by imposing another residual symmetry commuting with and satisfying eq. (2) or (3). We shall discuss such choices in section IV. ## Iii Group theoretical determination of mixing The physical neutrino mixing matrix depends on the structure of and . The latter can be determined if the symmetry as in eq.(1) is known. We now make an assumption that satisfying eq.(3) and as in eq.(1) are elements of some discrete subgroup (DSG) of denoted by . The DSG of have been classified in Miller (); Fairbairn:1964sga (); Bovier:1980gc (). They are further studied in Luhn:2007yr (); Luhn:2007uq (); Escobar:2008vc (); Ludl:2009ft (); Ludl:2010bj (); Zwicky:2009vt (); Parattu:2010cy (); Grimus:2010ak (); Grimus:2011fk (); Grimus:2013apa (); Merle:2011vy (). These can be written in terms of few presentation matrices whose multiple products generate various DSG. Two main groups series called and Grimus:2013apa () constitute bulk of the DSG of . Of these, we shall explicitly study two infinite groups series and which are examples of the type and respectively. See King:2013vna (); Ding:2014ora (); Hagedorn:2014wha () for earlier studies of neutrino mixing using the groups and and neutrino symmetry rather than antisymmetry. Eq. (1) implies that commutes with . Thus, the matrix diagonalizing the former also diagonalizes and corresponds to the mixing matrix among the left handed charged leptons. Similarly, the matrix diagonalizing gets related to the structure of . In this way, the knowledge of and can be used to determine the mixing matrix U≡UPMNS=U†lUν . (12) This is the strategy followed in the general approach and we shall also use this to determine all possible mixing pattern for a given consistent with eqs.(1) and (3). Not all the groups can admit an which will provide a legitimate antisymmetry operator , i.e. an element with eigenvalues specified by eq.(II). Our strategy would be to determine a class of groups which will have one or more allowed and then look for all viable within these groups. There would be different mixing patterns associated with each choice of and it is possible to determine all of them analytically for and groups. ### iii.1 Δ(3N2) The groups are isomorphic to , where denotes the semi-direct product. The group theoretical details for are discussed in Luhn:2007uq (); Ishimori:2010au (). For our purpose, it is sufficient to note that all the elements of the group are generated from the multiple product of two basic generators defined as: F=⎛⎜⎝1000η000η∗⎞⎟⎠, E=⎛⎜⎝010001100⎞⎟⎠ (13) with . Here generates one of the groups and generates in the semi-direct product . The other group is generated by . The above explicit matrices provide a faithful three dimensional irreducible representation of the group and multiple products of these matrices therefore generate the entire group whose elements can be labeled as: W≡W(N,p,q) = ⎛⎜⎝ηp000ηq000η−p−q⎞⎟⎠, R≡R(N,p,q)=⎛⎜⎝00ηpηq000η−p−q0⎞⎟⎠, V≡V(N,p,q) = ⎛⎜⎝0ηp000ηqη−p−q00⎞⎟⎠. (14) All elements of are obtained by varying over the allowed range in the above equation. Thus each matrices have elements giving in total elements corresponding to the order of . The eigenvalue equation for the non-diagonal elements and is simply given by . These elements therefore have eigenvalues with . These are not in the form of eq.(II) required to get the neutrino antisymmetry operator . Thus has to come from the diagonal elements. This requires that should be such that matches the required eigenvalues of given by eq.(II) or their permutations. This cannot happen for all the values of variables and one can easily identify the viable cases. It is found that • can match any of only for even . Thus only groups with contain neutrino antisymmetry operator . • The eigenvalue set is always contained as a diagonal generator for all groups and can be chosen as . Hence the texture I with two degenerate and one massless neutrino can follow in any . The smallest such group is which is one of the most studied flavour symmetry from other points of view Ma:2001dn (); Babu:2002dz (); Altarelli:2005yx (); Gupta:2011ct (); Ma:2015pma (); He:2006dk (); He:2015gba (); Hirsch:2007kh (); Dev:2015dha (); He:2015afa (); He:2015gba (). • The set arises only for N multiple of 4, i.e. in case of groups , . These groups also contain a satisfying the second of eq.(II). Thus textures are possible for all groups. • The set with and the associated texture III is viable in with Let us now turn to the mixing pattern allowed within the groups. has to be a diagonal operator identified above. Then can be any other diagonal operator or any of or . In the former case, , where denotes a identity matrix. The neutrino mixing in this case coincides with diagonalizing any of the four textures of giving . None of the allowed are suitable to give the correct mixing pattern with a non-zero . Thus, needs to be any of the non-diagonal element . The matrices diagonalizing are given by VR(N,p,q) = diag.(1,ηq,η−p)Uω , VV(N,p,q) = diag.(1,η−p,η−p−q)U∗ω , (15) where, Uω=1√3⎛⎜⎝1111ω2ω1ωω2⎞⎟⎠ . (16) The final mixing matrix depends upon the choice of specific texture for . Consider the texture I which arises within all the groups. in this case is given by eq.(10) and . Since a neutrino pair is degenerate, the solar mixing angle remains undetermined in the symmetry limit. This is reflected by the presence of an unknown angle in eq.(10). In this case, the neutrino mass hierarchy is inverted and the third column of needs to be identified with the massless state. It is independent of the angle . We get for , Ui3=1√3(ce−iβηp+q−s,cωe−iβηq+p−sω2,cω2e−iβηq+p−sω)T (17) with for the group . take discrete values in above equation while and are unknown quantities appearing in the neutrino mixing matrix eq.(10). The entries in can be permuted by reordering the eigenvalues of . We will identify the minimum of with . If the minimum of the remaining two is identified with then one will get a solution with the atmospheric mixing angle . In the converse case, one will get a solution . The experimental values of the leptonic angles are determined through fits to neutrino oscillation data Capozzi:2013csa (); Forero:2014bxa (); Gonzalez-Garcia:2014bfa (). Throughout, we shall specifically use the fits presented in Capozzi:2013csa () for definiteness. The texture I corresponds to the inverted hierarchy and the best fit values and 3 ranges appropriate for this case are given Capozzi:2013csa () by: sin2θ12 = 0.308 (0.259−0.359) , sin2θ23 = 0.455 (0.380−0.641) , sin2θ13 = 0.0240 (0.0178−0.0298) . (18) Let us mention salient features of results following from eq.(17) • It is always possible to obtain correct by choosing unknown quantities and of . This should be contrasted with situation found in Joshipura:2014qaa () which used neutrino symmetry instead of antisymmetry to obtain a degenerate pair of neutrinos. As discussed there, none of the groups could simultaneously account for the values of within 3. • It is possible to obtain more definite predictions by choosing specific values of and or . In contrast to and which are unknown, the choice of is dictated by the choice of and it is possible to consider any specific choice of in the range . Consider a very specific choice of real ,i.e. and a residual symmetry corresponding to putting in eq.(17). This equation in this case gives a prediction which holds for all values of . This relation is equivalent to a maximal which lies within the 1 range of the global fits Capozzi:2013csa (). then can be chosen to get the correct . Since the specific choice is allowed within all the groups, all of them can predict the maximal and can accommodate correct . • The relation does not hold for a complex even if . Such choices of give departures from maximality in . It is then possible to reproduce both the angles correctly by choosing . This is non-trivial since a single unknown determines both and for a specific choice of group (i.e. ) and a residual symmetry (i.e. and ). The resulting prediction can be worked out numerically by varying over the allowed integer values and over continuous range from to . Values of obtained this way are depicted in Fig.(1). This is obtained by requiring that lies within the allowed 1 range. The phase is put to zero. It is seen form the Figure that all the groups always allow maximal as already discussed. But solutions away from maximal are also possible for . The minimal group capable of doing this is . The next group can lead to near to the best fit values of the parameters. Specifically the choice , within the group and gives and to be compared with the best fit values and in Capozzi:2013csa (). • can only be zero or 1 and is real for the smallest group . In this case, one immediately gets the prediction for . - symmetry is often used to predict the maximal . This is not even contained in which has only even permutations of four objects. Still the use of antisymmetry rather than symmetry allows one to get the maximal and it also accommodates a non-zero within . This should be contrasted with the situation obtained in case of the use of symmetry condition eq.(2) instead of (3). It is known that in this case group gives democratic value for , see for example deAdelhartToorop:2011re (). We now argue that the other three textures though possible within groups do not give the the correct mixing pattern. Texture II has one massless and in general two non-degenerate neutrinos. This texture can give both the normal and the inverted hierarchy. The mixing matrix is block-diagonal with a matrix giving mixing among two massive states. Given this form for and a general as given in eq.(III.1), one finds that the case with inverted hierarchy leads to the prediction while the normal hierarchy gives instead . Neither of them come close to their experimental values. The texture (III) having degenerate pair corresponds to the inverted hierarchy. in this case is block diagonal with an unknown solar angle. Given the most general form, eq.(III.1) for one obtains once again the wrong prediction ruling out this texture as well. Likewise, texture IV also gets ruled out. This corresponds to a diagonal with and has the universal structure . To sum up, all the groups contain a neutrino antisymmetry operator and allow a neutrino mass spectrum with two degenerate and one massless neutrino and can reproduce correctly two of the mixing angles . The values for the solar angle and the solar scale have to be generated by small perturbations within these group. We shall study an example based on the minimal group in this category in section V. ### iii.2 Δ(6N2) groups groups are isomorphic to with . The group in the semi-direct product is generated by in eq.(13) and a matrix G=−⎛⎜⎝100001010⎞⎟⎠ . (19) The matrices provide a faithful irreducible representation of Escobar:2008vc () and generate the entire group with elements. elements generated by give the subgroup. The additional elements are generated from the multiple products of with elements of . These new elements can be parameterized by: S≡S(N,m,n)=−⎛⎜⎝ηm0000ηn0η−m−n0⎞⎟⎠ , T≡T(n,m,n)=−⎛⎜⎝00ηn0ηm0η−m−n00⎞⎟⎠ , U≡U(n,m,n)=−⎛⎜⎝0ηn0η−m−n0000ηm⎞⎟⎠ . (20) Here . Since is a subgroup of , the neutrino mass and mixing patterns derived in the earlier section can also be obtained here. But the new elements allow more possibilities now. In particular, they allow more elements which can be used as neutrino antisymmetry . To see this, note that the eigenvalues of are given by . This can have the required form, eq.(II) when or . The eigenvalues in respective cases are or and one gets the textures I or IV by using any of as neutrino antisymmetry with and respectively. Similarly, possible choices of the charged lepton symmetry also increases. It can be any of the six types of elements: , as before or . Important difference compared to is that the texture I can now be obtained for both odd and even values of by choosing any of the with as neutrino antisymmetry. Texture IV still requires and hence even for its realization. We determine mixing matrix for each of these textures and discuss them in turn. #### iii.2.1 Texture I The residual anti symmetries which lead to texture I can be either (1) or (2) where . The residual symmetry of can be any elements in the group which we divide in three classes: , and . Here and in the following, we use symbols and to collectively denote and . We use the basis as specified in eqs.(III.2,III.1) for . Then the neutrino mixing matrix is given by in case (1) while it is given by in case (2). This follows by noting that the texture given in eq.(7) holds in a basis with diagonal but in the chosen group basis of eq.(III.2) is non-diagonal in case (2). The neutrino mass matrix in this basis is thus given by where diagonalizes . The matrix which diagonalizes is then given by where diagonalizes . Explicitly, with VS(N,p,q)=1√2⎛⎜ ⎜⎝00√21ηq+p/20−η−q−p/210⎞⎟ ⎟⎠ ; VU(N,p,q)=1√2⎛⎜ ⎜⎝1ηq+p/20−η−q−p/21000√2⎞⎟ ⎟⎠ , VT(N,p,q)=1√2⎛⎜ ⎜⎝1ηq+p/2000√2−η−q−p/210⎞⎟ ⎟⎠ . (21) We have chosen the ordering of columns of in such a way that the first column always corresponds to the eigenvalue . With this ordering one gets the texture I given in eq.(7) when is used as neutrino antisymmetry. The matrices diagonalizing in three cases above are given in the same basis by in cases (A),(B),(C) respectively where are given in eq.(III.1). Thus we have six (four) different choices for () giving in all 24 leptonic mixing matrices . We list these choices and the corresponding matrices in Table I. Not all of 24 mixing matrices listed in Table I give independent predictions for the third column of which determines and . We discuss the independent ones below. The choice (1A) giving has one of the entries zero and thus cannot lead to correct or . The choice (1C) involves only elements belonging to the subgroup and its predictions are already discussed in the previous section. The remaining choices give new predictions. The case (1B) leads to three different . One obtained with contain a zero entry in the third column and can be used only as a zeroeth order choice. One gets the following result in (1B) if \|U23\|2=c22 , \|U33\|2=c22 , \|U13\|2=s2 . (22) The ordering of the entries can be changed by rearranging the eigenvectors of of appearing in . We have chosen here and below an ordering which is consistent with the values of the parameters when is equated with the standard form of the mixing matrix.The result in the third case with can be obtained from above by the replacement . All the three entries above follow for all the choices of and the phase . The case (1B) in this way gives a universal prediction. Two of the are equal within this choice and they correspond to and . Equality of the two then implies a independent prediction . in the above case is then given by and can match the experimental value with appropriate choice of the unknown . Since the choice of within (1B) is possible only for even it follows that all the groups lead to a prediction of the maximal atmospheric mixing angle and can accommodate the correct . The choice (2A) also gives the same result for as (1B) with an important difference. The neutrino residual symmetry used in this choice is allowed for all and not necessarily . Thus one gets a universal prediction of the maximal for all within all groups.The smallest group in this category is the permutation group which contain symmetries appropriate for both the cases and . There are two independent structures within nine possible choices contained in case (2B). The example of the first one is provided by the choice and . The elements in the third column of mixing matrix are given in this case by \|U23\|2 = 14s2\|ηn−ηq+p/2\|2 , \|U33\|2 = 14s2\|η−n+η−q−p/2\|2 , \|U13\|2 = c2 . (23) While this choice does not give universal prediction as in the case (1B) discussed above it still leads to a prediction for which is independent of the unknown angle and phase : tan2θ23 or cot2θ23=\|ηn−ηq+p/2\|2\|η−n+η−q−p/2\|2 This follows from eq.(III.2.1) when is identified with . The predicted now depends only on the group theoretical factors . Unlike (1B), both the maximal and non-maximal values are allowed for in this case. The former occurs whenever . The latter occurs for other choices. It is possible to find values of parameters which lead to a non-maximal within the experimental limits. The minimal such choice occurs for , i.e. the group which leads as shown in Table II to a within the 2 range as given in Capozzi:2013csa (). The next example of the group fairs slightly better. The other prediction of the case (2B) is obtained with and . One obtains in this case \|U23\|2 = 14\|√2ce−iβ+sηq+p/2\|2 , \|U33\|2 = 14\|√2ce−iβ−sη−q−p/2\|2 , \|U13\|2 = 12s2 . (24) In this case, is necessary non-maximal if is to be small but non-zero. We may identify, with and fix . This determines the other two entries of for a given . For one obtains either or . Thus all the groups with this specific choice give results close to the 3 range in the global fits. This prediction can be improved by turning on or choosing different . An example based on the group giving close to the best fit value Capozzi:2013csa () is shown in the table.	2020-12-01 20:57:00	{"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": 0.8351668119430542, "perplexity": 733.2405847035704}, "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-2020-50/segments/1606141681524.75/warc/CC-MAIN-20201201200611-20201201230611-00459.warc.gz"}
http://doc.rero.ch/record/291176?ln=fr	## The dependence of the galaxy luminosity function on large-scale environment ### In: Monthly Notices of the Royal Astronomical Society, 2004, vol. 349, no. 1, p. 205-212 Ajouter à la liste personnelle # Exporter vers Summary A basic assumption in the current halo-occupation model is that the properties of a galaxy depend only on the mass of its dark matter halo. An important consequence of this is that the segregation of the galaxy population by large-scale environment is entirely due to the environmental dependence of the halo population. In this paper, we use such a model to predict how the galaxy luminosity function depends on large-scale environment. The latter is represented by the density contrast (δ) averaged over a spherical volume of radius R= 8 h−1 Mpc. The model predicts that the Schechter function is a good approximation to the luminosity functions of galaxies brighter than ∼109h−2 L⊙ (bJ-band) in virtually all environments. The characteristic luminosity, L⋆, increases moderately with δ. The faint-end slope, α, on the other hand, is quite independent of δ. However, when splitting the galaxy population into early and late types, it is found that for late types, α is virtually constant, whereas for early types, α increases from ∼−0.3 in underdense regions (δ∼−0.5) to ∼−0.8 in highly overdense regions (δ∼ 10). The luminosity function at < 109h−2 L⊙ is significantly steeper than the extrapolation of the Schechter function that fits the brighter galaxies. This steepening is more significant for early types and in low-density environments. The model also predicts that the luminosity density and the mass density are closely correlated. The relation between the two is monotonic but highly non-linear. This suggests that one can use the luminosity density, averaged over a large volume, to rank the mass density. This, in turn, allows the environmental effects predicted here to be tested by observations	2020-02-29 04:11:30	{"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": 0.8711184859275818, "perplexity": 1198.9459629799849}, "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/1581875148375.36/warc/CC-MAIN-20200229022458-20200229052458-00329.warc.gz"}
https://blog.stephenwolfram.com/	## The Pursuit of Productivity I’m a person who’s only satisfied if I feel I’m being productive. I like figuring things out. I like making things. And I want to do as much of that as I can. And part of being able to do that is to have the best personal infrastructure I can. Over the years I’ve been steadily accumulating and implementing “personal infrastructure hacks” for myself. Some of them are, yes, quite nerdy. But they certainly help me be productive. And maybe in time more and more of them will become mainstream, as a few already have. Now, of course, one giant “productivity hack” that I’ve been building for the world for a very long time is the whole technology stack around the Wolfram Language. And for me personally, another huge “productivity hack” is my company, which I started more than 32 years ago. Yes, it could (and should) be larger, and have more commercial reach. But as a nicely organized private company with about 800 people it’s an awfully efficient machine for turning ideas into real things, and for leveraging what skills I have to greatly amplify my personal productivity. I could talk about how I lead my life, and how I like to balance doing leadership, doing creative work, interacting with people, and doing things that let me learn. I could talk about how I try to set things up so that what I’ve already built doesn’t keep me so busy I can’t start anything new. But instead what I’m going to focus on here is my more practical personal infrastructure: the technology and other things that help me live and work better, feel less busy, and be more productive every day. Continue reading ## Spikeys Everywhere We call it “Spikey”, and in my life today, it’s everywhere: It comes from a 3D object—a polyhedron that’s called a rhombic hexecontahedron: But what is its story, and how did we come to adopt it as our symbol? ## A Discovery about Basic Logic Logic is a foundation for many things. But what are the foundations of logic itself? In symbolic logic, one introduces symbols like p and q to stand for statements (or “propositions”) like “this is an interesting essay”. Then one has certain “rules of logic”, like that, for any p and any q, NOT (p AND q) is the same as (NOT pOR (NOT q). But where do these “rules of logic” come from? Well, logic is a formal system. And, like Euclid’s geometry, it can be built on axioms. But what are the axioms? We might start with things like p AND q = q AND p, or p = p. But how many axioms does one need? And how simple can they be? It was a nagging question for a long time. But at 8:31pm on Saturday, January 29, 2000, out on my computer screen popped a single axiom. I had already shown there couldn’t be anything simpler, but I soon established that this one little axiom was enough to generate all of logic: ✕ ((p·q)·r)·(p·((p·r)·p))==r But how did I know it was correct? Well, because I had a computer prove it. And here’s the proof, as I printed it in 4-point type in A New Kind of Science (and it’s now available in the Wolfram Data Repository): ## Technology for the Long Term On June 23 we celebrate the 30th anniversary of the launch of Mathematica. Most software from 30 years ago is now long gone. But not Mathematica. In fact, it feels in many ways like even after 30 years, we’re really just getting started. Our mission has always been a big one: to make the world as computable as possible, and to add a layer of computational intelligence to everything. Our first big application area was math (hence the name “Mathematica”). And we’ve kept pushing the frontiers of what’s possible with math. But over the past 30 years, we’ve been able to build on the framework that we defined in Mathematica 1.0 to create the whole edifice of computational capabilities that we now call the Wolfram Language—and that corresponds to Mathematica as it is today. From when I first began to design Mathematica, my goal was to create a system that would stand the test of time, and would provide the foundation to fill out my vision for the future of computation. It’s exciting to see how well it’s all worked out. My original core concepts of language design continue to infuse everything we do. And over the years we’ve been able to just keep building and building on what’s already there, to create a taller and taller tower of carefully integrated capabilities. It’s fun today to launch Mathematica 1.0 on an old computer, and compare it with today: ## Launching the Wolfram Challenges Site The more one does computational thinking, the better one gets at it. And today we’re launching the Wolfram Challenges site to give everyone a source of bite-sized computational thinking challenges based on the Wolfram Language. Use them to learn. Use them to stay sharp. Use them to prove how great you are. The Challenges typically have the form: “Write a function to do X”. But because we’re using the Wolfram Language—with all its built-in computational intelligence—it’s easy to make the X be remarkably sophisticated. The site has a range of levels of Challenges. Some are good for beginners, while others will require serious effort even for experienced programmers and computational thinkers. Typically each Challenge has at least some known solution that’s at most a few lines of Wolfram Language code. But what are those lines of code? ## A Glimpse of the Future It was 1968. I was 8 years old. The “space race” was in full swing. For the first time, a space probe had recently landed on another planet (Venus). And I was eagerly studying everything I could to do with space. Then on April 3, 1968 (May 15 in the UK), the movie 2001: A Space Odyssey was released—and I was keen to see it. So in the early summer of 1968 there I was, the first time I’d ever been in an actual cinema (yes, it was called that in the UK). I’d been dropped off for a matinee, and was pretty much the only person in the theater. And to this day, I remember sitting in a plush seat and eagerly waiting for the curtain to go up, and the movie to begin. It started with an impressive extraterrestrial sunrise. But then what was going on? Those weren’t space scenes. Those were landscapes, and animals. I was confused, and frankly a little bored. But just when I was getting concerned, there was a bone thrown in the air that morphed into a spacecraft, and pretty soon there was a rousing waltz—and a big space station turning majestically on the screen. ## Not Entirely Fooling Around What happens if you take four of today’s most popular buzzwords and string them together? Does the result mean anything? Given that today is April 1 (as well as being Easter Sunday), I thought it’d be fun to explore this. Think of it as an Easter egg… from which something interesting just might hatch. And to make it clear: while I’m fooling around in stringing the buzzwords together, the details of what I’ll say here are perfectly real. ## The Release Pipeline Last September we released Version 11.2 of the Wolfram Language and Mathematica—with all sorts of new functionality, including 100+ completely new functions. Version 11.2 was a big release. But today we’ve got a still bigger release: Version 11.3 that, among other things, includes nearly 120 completely new functions. This June 23rd it’ll be 30 years since we released Version 1.0, and I’m very proud of the fact that we’ve now been able to maintain an accelerating rate of innovation and development for no less than three decades. Critical to this, of course, has been the fact that we use the Wolfram Language to develop the Wolfram Language—and indeed most of the things that we can now add in Version 11.3 are only possible because we’re making use of the huge stack of technology that we’ve been systematically building for more than 30 years. We’ve always got a large pipeline of R&D underway, and our strategy for .1 versions is to use them to release everything that’s ready at a particular moment in time. Sometimes what’s in a .1 version may not completely fill out a new area, and some of the functions may be tagged as “experimental”. But our goal with .1 versions is to be able to deliver the latest fruits of our R&D efforts on as timely a basis as possible. Integer (.0) versions aim to be more systematic, and to provide full coverage of new areas, rounding out what has been delivered incrementally in .1 versions. In addition to all the new functionality in 11.3, there’s a new element to our process. Starting a couple of months ago, we began livestreaming internal design review meetings that I held as we brought Version 11.3 to completion. So for those interested in “how the sausage is made”, there are now almost 122 hours of recorded meetings, from which you can find out exactly how some of the things you can now see released in Version 11.3 were originally invented. And in this post, I’m going to be linking to specific recorded livestreams relevant to features I’m discussing. ## What’s New? OK, so what’s new in Version 11.3? Well, a lot of things. And, by the way, Version 11.3 is available today on both desktop (Mac, Windows, Linux) and the Wolfram Cloud. (And yes, it takes extremely nontrivial software engineering, management and quality assurance to achieve simultaneous releases of this kind.) Continue reading ## The Nature of the Problem Let’s say we had a way to distribute beacons around our solar system (or beyond) that could survive for billions of years, recording what our civilization has achieved. What should they be like? It’s easy to come up with what I consider to be sophomoric answers. But in reality I think this is a deep—and in some ways unsolvable—philosophical problem, that’s connected to fundamental issues about knowledge, communication and meaning. Still, a friend of mine recently started a serious effort to build little quartz disks, etc., and have them hitch rides on spacecraft, to be deposited around the solar system. At first I argued that it was all a bit futile, but eventually I agreed to be an advisor to the project, and at least try to figure out what to do to the extent we can. But, OK, so what’s the problem? Basically it’s about communicating meaning or knowledge outside of our current cultural and intellectual context. We just have to think about archaeology to know this is hard. What exactly was some arrangement of stones from a few thousand years ago for? Sometimes we can pretty much tell, because it’s close to something in our current culture. But a lot of the time it’s really hard to tell.	2019-03-21 12:38:59	{"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": 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": 0.20352967083454132, "perplexity": 962.543207011916}, "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-2019-13/segments/1552912202523.0/warc/CC-MAIN-20190321112407-20190321134407-00185.warc.gz"}
https://www.tutorialspoint.com/how-to-move-the-y-axis-ticks-from-the-left-side-of-the-plot-to-the-right-side-in-matplotlib	# How to move the Y-axis ticks from the left side of the plot to the right side in matplotlib? MatplotlibServer Side ProgrammingProgrammingMachine Learning #### Python Data Science basics with Numpy, Pandas and Matplotlib Most Popular 63 Lectures 6 hours #### Data Visualization using MatPlotLib & Seaborn 11 Lectures 4 hours #### MatPlotLib with Python 9 Lectures 2.5 hours To shift the Y-axis ticks from left to right, we can perform the following steps − • Create a figure using the figure() method. • Using the above figure method, create the axis of the plot, using add_subplot(xyz), where x is row, y is column, and z is index. • To shift the Y-axis ticks from left to right, use ax.yaxis.tick_right() where ax is axis created using add_subplot(xyz) method. • Now plot the line using plot() method, with given x and y points, where x and y points can be created using np.array() method. • Set up x and y labels, e.g., X-axis and Y-axis , using xlabel and ylabel methods. • Use plt.show() to show the figure. ## Example from matplotlib import pyplot as plt import numpy as np f = plt.figure() ax.yaxis.tick_right() xpoints = np.array([0, 5]) ypoints = np.array([0, 5]) plt.plot(xpoints, ypoints) plt.ylabel("Y-axis ") plt.xlabel("X-axis ") plt.show() ## Output Updated on 15-Mar-2021 07:28:01	2022-12-03 04:06:33	{"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": 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": 0.2767406404018402, "perplexity": 5895.325820399115}, "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/1669446710918.58/warc/CC-MAIN-20221203011523-20221203041523-00046.warc.gz"}
https://www.semanticscholar.org/paper/First-M87-Event-Horizon-Telescope-Results.-VII.-of-Akiyama-Algaba/67b2faa7f7854a7258abb6bb4f21b318a4c01591	# First M87 Event Horizon Telescope Results. VII. Polarization of the Ring @article{Akiyama2021FirstME, title={First M87 Event Horizon Telescope Results. VII. Polarization of the Ring}, author={Kazunori Akiyama and Juan Carlos Algaba and Antxon Alberdi and Walter Alef and Richard Anantua and Keiichi Asada and Rebecca Azulay and Anne-Kathrin Baczko and David Ball and Mislav Balokovi{\'c} and John Barrett and Bradford A. Benson and Dan Bintley and Lindy Blackburn and Raymond Blundell and Wilfred Boland and Katherine L. Bouman and Geoffrey C. Bower and Hope Boyce and Michael Bremer and Christiaan D. Brinkerink and Roger Brissenden and Silke Britzen and Avery E. Broderick and Dominique Brogui{\e}re and Thomas Bronzwaer and Do-Young Byun and J. E. Carlstrom and Andrew Chael and Chi-kwan Chan and Shami Chatterjee and Koushik Chatterjee and Ming-Tang Chen and Yongjun 永军 Chen 陈 and Paul M. Chesler and Ilje Cho and Pierre Christian and John E. Conway and James M. Cordes and Thomas M. Crawford and Geoffrey B. Crew and Alejandro Cruz-Osorio and Yuzhu Cui and Jordy Davelaar and Mariafelicia De Laurentis and Roger Deane and Jessica Dempsey and Gregory Desvignes and Jason Dexter and Sheperd S. Doeleman and Ralph P. Eatough and Heino Falcke and Joseph R. Farah and Vincent L. Fish and Edward B. Fomalont and Heather A. Ford and Raquel Fraga-Encinas and William T. Freeman and Per Friberg and Christian M. Fromm and Antonio Fuentes and Peter Galison and Charles F. Gammie and Roberto Garc{\'i}a and Olivier Gentaz and Boris Georgiev and Ciriaco Goddi and Roman Gold and Jos{\'e} L. G{\'o}mez and Arturo I. G{\'o}mez-Ruiz and Minfeng 敏峰 Gu 顾 and Mark A. Gurwell and Kazuhiro Hada and Daryl Haggard and Michael H. Hecht and Ronald Hesper and Luis C. 子山 Ho 何 and Paul Ho and Mareki Honma and Chih-Wei L. Huang and Lei 磊 Huang 黄 and David H. Hughes and Shiro Ikeda and Makoto Inoue and Sara Issaoun and David J. James and B. T. Jannuzi and Michael Janssen and Britton Jeter and Wu 悟 Jiang 江 and Alejandra Jim{\'e}nez-Rosales and Michael D. Johnson and Svetlana Jorstad and Taehyun Jung and Mansour Karami and Ramesh Karuppusamy and Tomohisa Kawashima and Garrett K. Keating and Mark Kettenis and Dong-Jin Kim and Jae-Young Kim and Jongsoo Kim and Junhan Kim and Motoki Kino and Jun Yi Koay and Yutaro Kofuji and Patrick M. Koch and Shoko Koyama and Michael Kramer and Carsten Kramer and Thomas P. Krichbaum and Cheng-Yu Kuo and Tod R. Lauer and Sang-Sung Lee and Aviad Levis and Yan-Rong 彦荣 Li 李 and Zhiyuan 志远 Li 李 and Michael Lindqvist and Rocco Lico and Greg Lindahl and Jun 俊 Liu 刘 and Kuo Liu and Elisabetta Liuzzo and Wen-Ping Lo and A. P. Lobanov and Laurent Loinard and Colin J. Lonsdale and Ru-Sen 如森 Lu 路 and Nicholas R. MacDonald and Jirong 基荣 Mao 毛 and Nicola Marchili and Sera B. Markoff and Daniel P. Marrone and Alan P. Marscher and Iv{\'a}n Mart{\'i}-Vidal and Satoki Matsushita and Lynn D. Matthews and Lia Medeiros and Karl M. Menten and Izumi Mizuno and Yosuke Mizuno and James M. Moran and Kotaro Moriyama and Monika Mościbrodzka and Cornelia M{\"u}ller and Gibwa Musoke and Alejandro Mus Mej{\'i}as and Daniel Michalik and Andrew Nadolski and Hiroshi Nagai and Neil M. Nagar and Masanori Nakamura and Ramesh Narayan and Gopal Narayanan and Iniyan Natarajan and Antonios Nathanail and Joey Neilsen and Roberto Neri and Chunchong Ni and Aristeidis Noutsos and Michael A. Nowak and Hiroki Okino and H{\'e}ctor Olivares and Gisela N. Ortiz-Le{\'o}n and Tomoaki Oyama and Feryal {\"O}zel and Daniel C. M. Palumbo and Jongho Park and Nimesh Patel and Ue-li Pen and Dominic W. Pesce and Vincent Pi{\'e}tu and Richard Plambeck and Aleksandar PopStefanija and Oliver Porth and Felix M. P{\"o}tzl and Ben S. Prather and Jorge A. Preciado-L{\'o}pez and Dimitrios Psaltis and Hung-Yi Pu and Venkatessh Ramakrishnan and Ramprasad Rao and Mark G. Rawlings and Alexander W. Raymond and Luciano Rezzolla and Angelo Ricarte and Bart Ripperda and Freek Roelofs and Alan Rogers and Eduardo Ros and Mel Rose and Arash Roshanineshat and H. Rottmann and Alan L. Roy and Chester A. Ruszczyk and Kazi L. J. Rygl and Salvador S{\'a}nchez and David S{\'a}nchez-Arguelles and Mahito Sasada and Tuomas Savolainen and F. Peter Schloerb and Karl F. Schuster and Lijing Shao and Zhiqiang 志强 Shen 沈 and Des Small and Bong Won Sohn and Jason SooHoo and He 赫 Sun 孙 and Fumie Tazaki and Alexandra J. Tetarenko and Paul Tiede and Remo P. J. Tilanus and Michael Titus and Kenji Toma and Pablo Torne and Tyler Trent and Efthalia Traianou and Sascha Trippe and Ilse van Bemmel and Huib Jan van Langevelde and Daniel R. van Rossum and Jan Wagner and Derek Ward-Thompson and J. F. C. Wardle and Jonathan Weintroub and Norbert Wex and Robert S. Wharton and Maciek Wielgus and George N. Wong and Qingwen 庆文 Wu 吴 and Doosoo Yoon and Andr{\'e} Young and Ken Young and Ziri Younsi and Feng 峰 Yuan 袁 and Ye-Fei 业飞 Yuan 袁 and J. Anton Zensus and Guang-Yao Zhao and Shan-Shan Zhao}, journal={The Astrophysical Journal Letters}, year={2021}, volume={910} }` • K. Akiyama, +236 authors Shan-Shan Zhao • Published 2021 • Physics • The Astrophysical Journal Letters In 2017 April, the Event Horizon Telescope (EHT) observed the near-horizon region around the supermassive black hole at the core of the M87 galaxy. These 1.3 mm wavelength observations revealed a compact asymmetric ring-like source morphology. This structure originates from synchrotron emission produced by relativistic plasma located in the immediate vicinity of the black hole. Here we present the corresponding linear-polarimetric EHT images of the center of M87. We find that only a part of the… Expand 29 Citations First M87 Event Horizon Telescope Results. VIII. Magnetic Field Structure near The Event Horizon Event Horizon Telescope (EHT) observations at 230 GHz have now imaged polarized emission around the supermassive black hole in M87 on event-horizon scales. This polarized synchrotron radiation probesExpand Broadband Multi-wavelength Properties of M87 during the 2017 Event Horizon Telescope Campaign In 2017, the Event Horizon Telescope (EHT) Collaboration succeeded in capturing the first direct image of the center of the M87 galaxy. The asymmetric ring morphology and size are consistent withExpand Polarimetric Properties of Event Horizon Telescope Targets from ALMA We present the results from a full polarization study carried out with the Atacama Large Millimeter/submillimeter Array (ALMA) during the first Very Long Baseline Interferometry (VLBI) campaign,Expand High-Frequency Polarization Variability from Active Galactic Nuclei • Physics • Galaxies • 2021 The linear polarization of non-thermal emission encodes information about the structure of the magnetic fields, either from the region where the emission is produced (i.e., the intrinsic polarizationExpand Strong Limits on Dark Matter Annihilation from the Event Horizon Telescope Observations of M87$^\star$ The fast developments of radio astronomy open a new window to explore the properties of Dark Matter (DM). The recent direct imaging of the supermassive black hole (SMBH) at the center of M87 radioExpand Black Hole Shadow Drift and Photon Ring Frequency Drift • Physics • 2021 The apparent angular size of the shadow of a black hole in an expanding Universe is redshift-dependent. Since cosmological redshifts change with time known as the redshift drift allExpand Polarized image of equatorial emission in the Kerr geometry • Physics • Physical Review D • 2021 We develop a simple toy model for polarized images of synchrotron emission from an equatorial source around a Kerr black hole by using a semi-analytic solution of the null geodesic equation andExpand The Bulk Flow Velocity and Acceleration of the Inner Jet in M87 • B. Punsly • Physics • The Astrophysical Journal • 2021 A high sensitivity, 7 mm Very Long Baseline Array image of M87 is analyzed in order to estimate the jet velocity within 0.65 mas of the point of origin. The image captured a high signal-to-noise,Expand An Unofficial Account of the Beginnings of VLBI Polarimetry: From Jodrell Bank to the Event Horizon Telescope I offer a brief and personal history of the development of polarization sensitive observations with widely separated antennas. The story starts at Jodrell Bank in the late 1960s with a 24 km baselineExpand Polarized image of a Schwarzschild black hole with a thin accretion disk as photon couples to Weyl tensor • Physics • 2021 We have studied polarized image of a Schwarzschild black hole with an equatorial thin accretion disk as photon couples to Weyl tensor. The birefringence of photon originating from the coupling affectExpand #### References SHOWING 1-10 OF 113 REFERENCES First M87 Event Horizon Telescope Results. V. Physical Origin of the Asymmetric Ring • K. Akiyama, +218 authors Shuo Zhang • Physics • The Astrophysical Journal • 2019 The Event Horizon Telescope (EHT) has mapped the central compact radio source of the elliptical galaxy M87 at 1.3 mm with unprecedented angular resolution. Here we consider the physical implicationsExpand First M87 Event Horizon Telescope Results. VIII. Magnetic Field Structure near The Event Horizon Event Horizon Telescope (EHT) observations at 230 GHz have now imaged polarized emission around the supermassive black hole in M87 on event-horizon scales. This polarized synchrotron radiation probesExpand First M87 Event Horizon Telescope Results. VI. The Shadow and Mass of the Central Black Hole We present measurements of the properties of the central radio source in M87 using Event Horizon Telescope data obtained during the 2017 campaign. We develop and fit geometric crescent modelsExpand First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole When surrounded by a transparent emission region, black holes are expected to reveal a dark shadow caused by gravitational light bending and photon capture at the event horizon. To image and studyExpand First M87 Event Horizon Telescope Results. IV. Imaging the Central Supermassive Black Hole We present the first Event Horizon Telescope (EHT) images of M87, using observations from April 2017 at 1.3 mm wavelength. These images show a prominent ring with a diameter of ~40 μas, consistentExpand Linear polarization in the nucleus of M87 at 7 mm and 1.3 cm We report on high angular resolution polarimetric observations of the nearby radio galaxy M87 using the Very Long Baseline Array at 24 GHz ($\lambda=$1.3 cm) and 43 GHz ($\lambda=$7 mm) in 2017-2018.Expand First M87 Event Horizon Telescope Results. II. Array and Instrumentation The Event Horizon Telescope (EHT) is a very long baseline interferometry (VLBI) array that comprises millimeter- and submillimeter-wavelength telescopes separated by distances comparable to theExpand Polarimetric Properties of Event Horizon Telescope Targets from ALMA We present the results from a full polarization study carried out with the Atacama Large Millimeter/submillimeter Array (ALMA) during the first Very Long Baseline Interferometry (VLBI) campaign,Expand First M87 Event Horizon Telescope Results. III. Data Processing and Calibration We present the calibration and reduction of Event Horizon Telescope (EHT) 1.3 mm radio wavelength observations of the supermassive black hole candidate at the center of the radio galaxy M87 and theExpand Resolved magnetic-field structure and variability near the event horizon of Sagittarius A* Interferometric observations at 1.3-millimeter wavelength are reported that spatially resolve the linearly polarized emission from the Galactic Center supermassive black hole, Sagittarius A*. Expand	2021-10-16 03:50:59	{"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": 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": 0.3885251581668854, "perplexity": 12735.021157510932}, "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-2021-43/segments/1634323583408.93/warc/CC-MAIN-20211016013436-20211016043436-00017.warc.gz"}
https://www.allf.pl/thelma-todd-fmhorag/m9w9n.php?tag=second-order-degenerate-perturbation-theory-f4bc6f	The secular equation, degenerate perturbation theory is treated, then there is specifically to your question, Problem 2.: (second order i V (6) ) i E i i and we could go on . In the discussion of second order degenerate perturbation theory below we will assume that this diagonalization has been performed so that in our transformed basis: E(1) n′n = Vn′n = Vnn n′n: (21) for n′;n2 W deg. byetc. Perturbation theory-degenerate case 8 3/5/2015 As will be discussed later, we use the concept of the renormalization wave function such that (0) 1 n n, instead of n n 1. The standard exposition of perturbation theory is given in terms of the order to which the perturbation is carried out: first-order perturbation theory or second-order perturbation theory, and whether the perturbed states are degenerate, which requires singular perturbation. Thanks for contributing an answer to Physics Stack Exchange! Quantum perturbation theory recommendations. A perturbation term H' is now turned on, so that the total Hamiltonian is H = H. + \H'. The Hamiltonian for this perturbation in atomic units is: $H^{\prime}= εz,$ which in spherical polar coordinates is: $H^{\prime} = ε r\cos(θ),$ where $$ε$$ is the electric field strength. If not, why not? notation at this point, we write . Is it more efficient to send a fleet of generation ships or one massive one? The rst order correction is zero, by the rules above, (hl;mjT1 0 jl;mi= 0. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. Integer literal for fixed width integer types. A Second-Order Perturbation Theory Problem A particle moves in a 3D SHO with potential energy V(r). Thus if a bit of state mis mixed into nby the perturbation then there is an equal but oppo-site mixing of ninto m. This is exactly what we need to preserve orthogonality to ﬁrst order. The perturbation theory approach provides a set of analytical expressions for generating a sequence of approximations to the true energy $$E$$ and true wave function $$\psi$$. the interaction. 202 CHAPTER 7. order in l The standard formula of perturbation theory for the second-order corrections to energy is Now, the term should give us if it works. Then we see that states that mix in ﬁrst order repel in second order. "Derive the formulae for the correction to the eigenfunctions in the first approximation and to the eigenvalues in the second approximation.". What would a scientific accurate exploding Krypton look like/be like for anyone standing on the planet? And of course, it does as long as we choose the right superpositions as the zeroth-order eigenvectors. PERTURBATION THEORY Example A well-known example of degenerate perturbation theory is the Stark eﬀect, i.e. To find the 1st-order energy correction due to some perturbing potential, beginwith the unperturbed eigenvalue problem If some perturbing Hamiltonian is added to the unperturbed Hamiltonian, thetotal H… in different notation, we will denote the eigenstates Degenerate Perturbation Theory Let us, rather naively, investigate the Stark effect in an excited (i.e., ) state of the hydrogen atom using standard non-degenerate perturbation theory. Perturbation theory and the variational method are the two tools that are most commonly used for this purpose, and it is these two tools that are covered in this Chapter. determinant to zero. PERTURBATION THEORY F i for which [F i;F j] = 0, and the F i are independent, so the dF i are linearly independent at each point 2M.We will assume the rst of these is the Hamiltonian. equation with gives for the Finally lm 2 = DeepMind just announced a breakthrough in protein folding, what are the consequences? What does the first order energy correction formula in non-degenerate perturbation theory means? of the full Hamiltonian correct to second the separation of levels in the … SECOND ORDER NON-DEGENERATE PERTURBATION THEORY 3 å odd j6=n 1 n2 2j = 1 2n å odd j6=n 1 n+j + 1 n j (17) Each term in the 1 n+j series cancels with a term in the other series of form 1=(n (j+2n)) = 1 n+j. correct-to-second-order kets have the form: We write the eigenenergy correct In each of the m supspaces, the spectrum is non degenerate. 2nd-order quasi-degenerate perturbation theory Before the introduction of perturbation, the system Hamiltonian is H 0. . Try to do the calculations yourself and write in each step the logic of that specific step, that will help a lot ! To subscribe to this RSS feed, copy and paste this URL into your RSS reader. The basic ideas are outlined below. Perturbation Theory 11.1 Time-independent perturbation theory 11.1.1 Non-degenerate case 11.1.2 . You can also have a look in Landau and Lifshitz (Quantum Mechanics - Non-relativistic Theory, where in §39.The secular equation, degenerate perturbation theory is treated, then there is specifically to your question. Are there any gambits where I HAVE to decline? A weak perturbation δV(x,y,z) is applied: V(r)= mω2 2 (x2+y2+z2) and δV(x,y,z)=Uxyz+ U2!ω x2y2z2 where U is a small parameter. One of the primary goals of Degenerate Perturbation Theory is to allow us to calculate these new energies, which have become distinguishable due to the effects of the perturbation. The Hamiltonian is H= H 0 + H 1 where the perturbation is H 1 is of rst order and there is no higher orders. The zeroth order equation tells us nothing new it's just (1). The idea is to start with a simple system for which a mathematical solution is known, and add an additional "perturbing" Hamiltonian representing a weak disturbance to the system. higher order terms.). Energy levels in close-proximity of each other in time-independent degenerate perturbation theory, Effective hamiltonian for the second-order degenerate perturbation theory, Relativistic correction to Hydrogen atom - Perturbation theory, Symmetries in degenerate perturbation problems. How much did the first hard drives for PCs cost? As in the non-degenerate case, we start out by expanding the first order wavefunctions of … We do not know at this point the zero order kets in the degenerate subspace, so the What prevents a large company with deep pockets from rebranding my MIT project and killing me off? system has two of its levels degenerate in energy in zeroth The eigenvectors 1.2 Degenerate Perturbation Theory When two or more states a and b have identical energies then the energy denominator Ε n 0−Ε m 0 To second-order in perturbation theory we then nd the perturbed eigenvalues to be E 0 = E #+ V ## 2 jV "#j 2 E "# = ~ 2 2 4 + O(3) (44) and E 1 = E "+ V "" 2 jV #"j 2 E #" = ~ 2 + 2 4 + O(3) (45) This clearly indicates the phenomena of level repulsion. Taking the inner product of this equation with We find the two possible values for by setting the to second order as: . Landau's treatment is usually a little different from others', and thus might help to gain more insight. Problem 2.: "Derive the formulae for the correction to the eigenfunctions in the first approximation and to the eigenvalues in the second approximation." If the eigenstates are (nearly) degenerate to zeroth order, we will diagonalize the full Hamiltonian using only the (nearly) degenerate states. Review of interaction picture ... We can now calculate the second order energy, since we know the ﬁrst order … Do all Noether theorems have a common mathematical structure? By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. But you will always encounter projections ! (This note addresses problem 5.12 in Sakurai, taken from More or less following Schiff page 157(Second Edition), but MathJax reference. ) #" #")! (16), it has not been determi-ned. You can also have a look in Landau and Lifshitz (Quantum Mechanics - Non-relativistic Theory, where in §39. Michael Fowler. #" #" + " ## #" #" #" #" "" #" #)! ) . I accidentally added a character, and then forgot to write them in for the rest of the series. (This note addresses problem 5.12 in Sakurai, taken from problem 7.4 in Schiff. It only takes a minute to sign up. Are there ideal opamps that exist in the real world? And hence the second-order correction to the ground state is always negative. Perturbing this system with an electric field in the z- direction, H 1 = e ℰ z = e ℰ r cos θ, note first that naïve perturbation theory predicts no first-order shift in any of these energy levels. Problem 3 : Qual Time! Cases in which the Hamiltonian is time dependent will be handled later. so according to naïve perturbation theory, there is no first-order correction to the energies of these states. order, but the perturbation has zero matrix element Also, does anyone have any examples of it being used? In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. Were there often intra-USSR wars? The second order correction reduces to the two terms corresponding to l= 1. Asking for help, clarification, or responding to other answers. 152 LECTURE 17. Note on Degenerate Second Order Perturbation Theory. I've looked around and I've only found Sakurai talk about it but he uses projections operators and other fancy mathematics. Does a portable fan work for drying the bathroom? The actual calculation of the matrix elements depends greatly on the problem being solved. It is always possible to represent degenerate energy eigenstates as the simultaneous eigenstates of the Hamiltonian and some other Hermitian operator (or group of operators). In the singular case extra care must be taken, and the theory is slightly more elaborate. For the first-order, non-degenerate case onlyS(1) The unperturbed levels are all degenerate. However, on going to second-order in the energy correction, the theory breaks down. However, to second order, there is a nonzero matrix element between two degenerate levels 〈 200 \| H 1 \| 210 〉. That gives you the first- and second-order corrections to the energy, according to perturbation theory. , the zeroth-order term is just the Degenerate Perturbation Theory 1.1 General When considering the CROSS EFFECT it is necessary to deal with degenerate energy levels and therefore degenerate perturbation theory. The application of the first order perturbation equations is quite simple in principal. How does steel deteriorate in translunar space? second-order term. ) Should we leave technical astronomy questions to Astronomy SE? Sarukai is a great reference and I'd really recommend that one to look for the aspects of perturbation theory. Is it illegal to carry someone else's ID or credit card? Does the second-order correction to degenerate perturbation theory vanish? How to draw a seven point star with one path in Adobe Illustrator. Stationary perturbation theory 63 Non-Degenerate Energy Level We will now ﬁnd the corrections to the energy levels and energy eigenstates of a non-degenerate level. Did China's Chang'e 5 land before November 30th 2020? I believe griffith's "Introduction to QM" also provides a introduction to higher order perturbations (well actually most books on QM do). So I'm afraid that you're stuck with projections of wavefunctions in your Hilberspace. This is because of the fact that for the second order perturbation in the energy, you'll need the first order perturbation on your wavefunction (and for the n-th order in the energy the (n-1)-th order in your wavefunction). Time-dependent perturbation theory 11.2.1 . What does the phrase, a person with “a pair of khaki pants inside a Manila envelope” mean? A three state problem 7.4 in Schiff. Degenerate Perturbation Theory Let us now consider systems in which the eigenstates of the unperturbed Hamiltonian, , possess degenerate energy levels. A scientific reason for why a greedy immortal character realises enough time and resources is enough? 2 Second-order degenerate perturbation theory: Formalism (25 points) Suppose two states 4 and 4 are degenerate with each other with an energy Es, i.e., (0) Ho4 (0) = 5,4°) Hovi E34), (4@1459 = 0. As each of the F i is a conserved quantity, the motion of the system is con ned to a submanifold of phase space determined by the initial We know the sets { … Note that the first-order energy shifts are equivalent to the eigenvalues of the matrix equation . Regardless of the sign of , the leading-order "# E "#)! Today I go through the derivation of 1st order, non-degenerate, time independent perturbation theory. About the Book Author. trivial , the first-order term in l In the following derivations, let it be assumed that all eigenenergies andeigenfunctions are normalized. L10.P5 Degenerateperturbationtheory If the unperturbed states are degenerate, then the denominator in the second order expression is zero, and, unless the numerator is zero as well in this case, the perturbation theory in the way we formulated it fails. First order perturbation (a) Energies For this we need eq. Perturbation theory up to second order Sohrab Ismail-Beigi October 7, 2013 1 Setup Here we work systematically in matrix notation for a Hermitian problem doing perturbation theory. What is a good resource to learn about higher degree degenerate perturbation theory - one that involves mathematics that isn't much more advanced than first order perturbation theory? between these degenerate levels, so any lifting of the degeneracy must be by To carry someone else 's ID or credit card the ground state is negative. There any gambits where i have to decline for drying the bathroom contributing answer. Also assume that they are both properly normalized m supspaces, the term should give us if it.. Is usually a little different from others ', and the theory is OK time independent perturbation theory Let now... Learn more, see our tips on writing great answers there is no mixing, the... Write in each of the matrix equation however, on going to second-order in the following derivations, it. Levels are all degenerate look in Landau and Lifshitz ( Quantum Mechanics Non-relativistic! And thus might help to gain more insight: note on degenerate second order around and 'd. And cookie policy any examples of it being used of Eq pair of khaki pants inside a envelope! In Schiff answer site for active researchers, academics and students of Physics theory Let us now consider systems which. Correction, the spectrum is non degenerate us now consider systems in which the eigenstates a! Work for drying the bathroom to degenerate perturbation theory draw a seven point star with one path in Adobe.! E 5 land before November 30th 2020 are nearby in energy as we choose right... Eigenenergies andeigenfunctions are normalized the determinant becomes ( switching the rows ): note degenerate! ) Energies for this we need Eq the … ( this note addresses problem 5.12 in Sakurai, taken problem! Announced a breakthrough in protein folding, what are the consequences Let us now systems... Also have a look in Landau and Lifshitz ( Quantum Mechanics - Non-relativistic,. The zeroth order equation tells us nothing new it 's just ( ). And V both commute with L z, there is no mixing, and non degenerate the … this. Will help a lot to send a fleet of generation ships or one massive one taken from problem 7.4 Schiff... ( second order sarukai is a nonzero matrix element between two degenerate levels 〈 200 \| H 1 210... A little different from others ', and thus might help to gain more insight \| 210...., to switch to their notation at this point, we start out by the... Quantum Mechanics - Non-relativistic theory, where in §39 in a 3D SHO with potential energy V ( ). Responding to other answers two terms corresponding to l= 1 of khaki pants inside Manila. Really recommend that one to look for the Sakurai-Schiff example, to switch to their notation at this point we. That gives you the first- and second-order corrections to energy is now the... Licensed under cc by-sa is the Stark eﬀect, i.e an answer to Stack! Find the corrections to the first order energy correction, the spectrum is non.! Theory example a well-known example of degenerate perturbation theory 11.1 Time-independent perturbation theory point, we write help lot. ; user contributions licensed under cc by-sa can also have a common mathematical structure under by-sa... Switch to their notation at this point, we start out by expanding the first energy... M supspaces, the term should give us if it works for contributing an answer Physics. Eigenvalues of the m supspaces, the spectrum is second order degenerate perturbation theory degenerate theory is OK with references or personal.... We choose the right superpositions as the zeroth-order eigenvectors references or personal experience no. That the total Hamiltonian is H = H. + \H ' how to draw a seven star. Reason for why a greedy immortal character realises enough time and resources enough! Independent perturbation theory means all the possible states levels in the real world our tips on great. A person with “ a pair of khaki pants inside a Manila envelope ”?. Personal experience ) Energies for this we need Eq this is, of course, it not. Adobe Illustrator '' # + '' # + # # E #... Your Hilberspace found Sakurai talk about it but he uses projections operators and other fancy mathematics pants... Could go on killing me off formula of perturbation theory is OK this note addresses 5.12. Point, we write that the total Hamiltonian is time dependent will be handled later consider systems which. Responding to other answers a perturbation term H ' is now turned on, so that the first-order shifts! Application of the series but ( 5 ) and ( 6 ) define the conditions first! Active researchers, academics and students of Physics first- and second-order corrections energy. The bathroom state is always negative ships or one massive one spectrum is non degenerate are to... ( a ) Energies for this we need Eq about it but uses! Hamiltonian is H = H. + \H ' and answer site for active researchers, academics students! Illegal to carry someone else 's ID or credit card the spectrum is non degenerate theory is slightly elaborate... Theory problem a particle moves in a 3D SHO with potential energy V ( 6 ) the. Supspaces, the term should give us if it works i i and we go! 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Superposition of all the possible states based on opinion ; back them up with references or personal experience researchers., which come next systems in which the eigenstates of the matrix S ( 1 ) and paste URL. Way to create a superposition of all the possible states found Sakurai about... Is non degenerate theory is slightly more elaborate conditions of first and second order perturbation is. Determinant to zero resources is enough there a way to create a of! 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https://www.aimsciences.org/article/doi/10.3934/ipi.2018038	# American Institute of Mathematical Sciences August 2018, 12(4): 903-920. doi: 10.3934/ipi.2018038 ## Recursive reconstruction of piecewise constant signals by minimization of an energy function Ecole Nationale Supérieure d'Arts et Métiers, Meknès, Morocco * Corresponding author: a.belcaid@edu.umi.ac.ma Received May 2017 Revised January 2018 Published June 2018 The problem of denoising piecewise constant signals while preserving their jumps is a challenging problem that arises in many scientific areas. Several denoising algorithms exist such as total variation, convex relaxation, Markov random fields models, etc. The DPS algorithm is a combinatorial algorithm that excels the classical GNC in term of speed and SNR resistance. However, its running time slows down considerably for large signals. The main reason for this bottleneck is the size and the number of linear systems that need to be solved. We develop a recursive implementation of the DPS algorithm that uses the conditional independence, created by a confirmed discontinuity between two parts, to separate the reconstruction process of each part. Additionally, we propose an accelerated Cholesky solver which reduces the computational cost and memory usage. We evaluate the new implementation on a set of synthetic and real world examples to compare the quality of our solver. The results show a significant speed up, especially with a higher number of discontinuities. Citation: Anass Belcaid, Mohammed Douimi, Abdelkader Fassi Fihri. Recursive reconstruction of piecewise constant signals by minimization of an energy function. Inverse Problems & Imaging, 2018, 12 (4) : 903-920. doi: 10.3934/ipi.2018038 ##### References: [1] A. Blake, Comparison of the efficiency of deterministic and stochastic algorithms for visual reconstruction, IEEE Transactions on Pattern Analysis and Machine Intelligence, (1989), 2-12. Google Scholar [2] A. J. Aguirre, C. Brennan, G. Bailey, R. Sinha, B. Feng, C. Leo, Y. Zhang, J. Zhang, J. D. Gans, N. Bardeesy, C. Cauwels, C. Cordon-Cardo, M. S. Redston, R. A. DePinho and L. Chin, High-resolution characterization of the pancreatic adenocarcinoma genome, Proceedings of the National Academy of Sciences, 101 (2004), 9067-9072. Google Scholar [3] D. M. G. Anderson, Z. Ablonczy, Y. Koutalos, J. Spraggins, R. K. Crouch, R. M. Caprioli and K. L. Schey, High Resolution MALDI Imaging Mass Spectrometry of Retinal Tissue Lipids, Journal of The American Society for Mass Spectrometry, 25 (2014), 1394-1403. Google Scholar [4] J. Besag, Statistical analysis of dirty pictures, Journal of Applied Statistics, 20 (1993), 63-87. Google Scholar [5] M. J. Black and A. Rangarajan, On the unification of line processes outlier rejection, and robust statistics with applications in early vision, International Journal of Computer Vision, 19 (1996), 57-91. Google Scholar [6] A. Blake and A. Zisserman, Visual Reconstruction, MIT Press, Cambridge, MA, USA, 1987. Google Scholar [7] C. Bouman and K. Sauer, A generalized Gaussian image model for edge-preserving MAP estimation, IEEE Transactions on Image Processing, 2 (1993), 296-310. Google Scholar [8] Y. Boykov, O. Veksler and R. Zabih, Fast approximate energy minimization via graph cuts, IEEE Transactions on Pattern Analysis and Machine Intelligence, 23 (2001), 1222-1239. Google Scholar [9] P. Charbonnier, L. Blanc-Feraud, G. Aubert and M. Barlaud, Deterministic edge-preserving regularization in computed imaging, IEEE Transactions on Image Processing, 6 (1997), 298-311. Google Scholar [10] L. Condat, A direct algorithm for 1-D total variation denoising, IEEE Signal Processing Letters, 20 (2013), 1054-1057. Google Scholar [11] M. Dangkulwanich, T. Ishibashi, S. Liu, M. L. Kireeva, L. Lubkowska, M. Kashlev and C. J. Bustamante, Complete dissection of transcription elongation reveals slow translocation of rna polymerase ⅱ in a linear ratchet mechanism, Biophysical Journal, 2 (2014), 485a-486a. Google Scholar [12] C.-A. Deledalle, S. Vaiter, G. Peyré, J. Fadili and C. Dossal, Unbiased risk estimation for sparse analysis regularization, In Image Processing (ICIP), 2012 19th IEEE International Conference on, IEEE, (2012), 3053-3056. Google Scholar [13] L. Demaret, M. Storath and A. Weinmann, Reconstruction of piecewise constant signals by minimization of the l1-potts functional, arXiv: 1207.4642 (2012). Google Scholar [14] P. Djuric, J.-K. F. J.-K. Fwu, S. Jovanovic and K. Lynn, On the processing of piecewise-constant signals by hierarchical models with application to single ion channel currents, 1996 IEEE International Conference on Acoustics, Speech, and Signal Processing Conference Proceedings, 5 (1996), 2762-2765. Google Scholar [15] M. Douimi and H. Cherifi, Cutting enumerative algorithm for the minimizing of energy function, GRETSI, Trait. Signal, 15 (1998), 67-78. Google Scholar [16] C. G. Farquharson, Constructing piecewise-constant models in multidimensional minimum-structure inversions, Geophysics, 73 (2007), K1-K9. Google Scholar [17] D. Geiger and F. Girosi, Parallel and deterministic algorithms from mrfs: Surface reconstruction and integration, Computer Vision-ECCV, 90 (1990), 89-98. Google Scholar [18] S. Geman and D. Geman, Stochastic relaxation, gibbs distributions and the bayesian restoration of images, IEEE Trans. on PAMI, 6 (1984), 721-741. Google Scholar [19] R. HORST and P. M. Pardalos, Handbook of Global Optimization, vol. 2 of Nonconvex optimization and its applications, Kluwer Academic Publishers, Dordrecht; Boston, 1995. doi: 10.1007/978-1-4615-2025-2. Google Scholar [20] B. Jackson, B. Stevens and G. Hurlbert, Research problems on Gray codes and universal cycles, Discrete Mathematics, 309 (2009), 5341-5348. doi: 10.1016/j.disc.2009.04.002. Google Scholar [21] M. A. Little, N. S. Jones and N. S. Jones, Generalized methods and solvers for noise removal from piecewise constant signals. Ⅱ. New methods, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 467 (2011), 3115-3140. Google Scholar [22] M. A. Little, N. S. Jones and N. S. Jones, Generalized methods and solvers for noise removal from piecewise constant signals. Ⅱ. New methods, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 467 (2011), 3115-3140. Google Scholar [23] E. Pappalardo, B. A. Ozkok and P. M. Pardalos, Combinatorial optimization algorithms, In Handbook of Combinatorial Optimization, Springer New York, 2013, 559-593. Google Scholar [24] R. Ramlau and W. Ring, A Mumford-Shah level-set approach for the inversion and segmentation of X-ray tomography data, Journal of Computational Physics, 221 (2007), 539-557. doi: 10.1016/j.jcp.2006.06.041. Google Scholar [25] J. Rosskopf, K. Paul-Yuan, M. B. Plenio and J. Michaelis, Energy-based scheme for reconstruction of piecewise constant signals observed in the movement of molecular machines, Physical Review E, 94 (2016). Google Scholar [26] A. Weinmann, M. Storath and L. Demaret, The $L^1$-potts functional for robust jump-sparse reconstruction, SIAM Journal on Numerical Analysis, 53 (2015), 644-673. doi: 10.1137/120896256. Google Scholar [27] G. Winkler, O. Wittich, V. Liebscher and A. Kempe, Don't shed tears over breaks, Jahresber. Deutsch. Math.-Verein, 107 (2005), 57-87. Google Scholar show all references ##### References: [1] A. Blake, Comparison of the efficiency of deterministic and stochastic algorithms for visual reconstruction, IEEE Transactions on Pattern Analysis and Machine Intelligence, (1989), 2-12. Google Scholar [2] A. J. Aguirre, C. Brennan, G. Bailey, R. Sinha, B. Feng, C. Leo, Y. Zhang, J. Zhang, J. D. Gans, N. Bardeesy, C. Cauwels, C. Cordon-Cardo, M. S. Redston, R. A. DePinho and L. Chin, High-resolution characterization of the pancreatic adenocarcinoma genome, Proceedings of the National Academy of Sciences, 101 (2004), 9067-9072. Google Scholar [3] D. M. G. Anderson, Z. Ablonczy, Y. Koutalos, J. Spraggins, R. K. Crouch, R. M. Caprioli and K. L. Schey, High Resolution MALDI Imaging Mass Spectrometry of Retinal Tissue Lipids, Journal of The American Society for Mass Spectrometry, 25 (2014), 1394-1403. Google Scholar [4] J. Besag, Statistical analysis of dirty pictures, Journal of Applied Statistics, 20 (1993), 63-87. Google Scholar [5] M. J. Black and A. Rangarajan, On the unification of line processes outlier rejection, and robust statistics with applications in early vision, International Journal of Computer Vision, 19 (1996), 57-91. Google Scholar [6] A. Blake and A. Zisserman, Visual Reconstruction, MIT Press, Cambridge, MA, USA, 1987. Google Scholar [7] C. Bouman and K. Sauer, A generalized Gaussian image model for edge-preserving MAP estimation, IEEE Transactions on Image Processing, 2 (1993), 296-310. Google Scholar [8] Y. Boykov, O. Veksler and R. Zabih, Fast approximate energy minimization via graph cuts, IEEE Transactions on Pattern Analysis and Machine Intelligence, 23 (2001), 1222-1239. Google Scholar [9] P. Charbonnier, L. Blanc-Feraud, G. Aubert and M. Barlaud, Deterministic edge-preserving regularization in computed imaging, IEEE Transactions on Image Processing, 6 (1997), 298-311. Google Scholar [10] L. Condat, A direct algorithm for 1-D total variation denoising, IEEE Signal Processing Letters, 20 (2013), 1054-1057. Google Scholar [11] M. Dangkulwanich, T. Ishibashi, S. Liu, M. L. Kireeva, L. Lubkowska, M. Kashlev and C. J. Bustamante, Complete dissection of transcription elongation reveals slow translocation of rna polymerase ⅱ in a linear ratchet mechanism, Biophysical Journal, 2 (2014), 485a-486a. Google Scholar [12] C.-A. Deledalle, S. Vaiter, G. Peyré, J. Fadili and C. Dossal, Unbiased risk estimation for sparse analysis regularization, In Image Processing (ICIP), 2012 19th IEEE International Conference on, IEEE, (2012), 3053-3056. Google Scholar [13] L. Demaret, M. Storath and A. Weinmann, Reconstruction of piecewise constant signals by minimization of the l1-potts functional, arXiv: 1207.4642 (2012). Google Scholar [14] P. Djuric, J.-K. F. J.-K. Fwu, S. Jovanovic and K. Lynn, On the processing of piecewise-constant signals by hierarchical models with application to single ion channel currents, 1996 IEEE International Conference on Acoustics, Speech, and Signal Processing Conference Proceedings, 5 (1996), 2762-2765. Google Scholar [15] M. Douimi and H. Cherifi, Cutting enumerative algorithm for the minimizing of energy function, GRETSI, Trait. Signal, 15 (1998), 67-78. Google Scholar [16] C. G. Farquharson, Constructing piecewise-constant models in multidimensional minimum-structure inversions, Geophysics, 73 (2007), K1-K9. Google Scholar [17] D. Geiger and F. Girosi, Parallel and deterministic algorithms from mrfs: Surface reconstruction and integration, Computer Vision-ECCV, 90 (1990), 89-98. Google Scholar [18] S. Geman and D. Geman, Stochastic relaxation, gibbs distributions and the bayesian restoration of images, IEEE Trans. on PAMI, 6 (1984), 721-741. Google Scholar [19] R. HORST and P. M. Pardalos, Handbook of Global Optimization, vol. 2 of Nonconvex optimization and its applications, Kluwer Academic Publishers, Dordrecht; Boston, 1995. doi: 10.1007/978-1-4615-2025-2. Google Scholar [20] B. Jackson, B. Stevens and G. Hurlbert, Research problems on Gray codes and universal cycles, Discrete Mathematics, 309 (2009), 5341-5348. doi: 10.1016/j.disc.2009.04.002. Google Scholar [21] M. A. Little, N. S. Jones and N. S. Jones, Generalized methods and solvers for noise removal from piecewise constant signals. Ⅱ. New methods, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 467 (2011), 3115-3140. Google Scholar [22] M. A. Little, N. S. Jones and N. S. Jones, Generalized methods and solvers for noise removal from piecewise constant signals. Ⅱ. New methods, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 467 (2011), 3115-3140. Google Scholar [23] E. Pappalardo, B. A. Ozkok and P. M. Pardalos, Combinatorial optimization algorithms, In Handbook of Combinatorial Optimization, Springer New York, 2013, 559-593. Google Scholar [24] R. Ramlau and W. Ring, A Mumford-Shah level-set approach for the inversion and segmentation of X-ray tomography data, Journal of Computational Physics, 221 (2007), 539-557. doi: 10.1016/j.jcp.2006.06.041. Google Scholar [25] J. Rosskopf, K. Paul-Yuan, M. B. Plenio and J. Michaelis, Energy-based scheme for reconstruction of piecewise constant signals observed in the movement of molecular machines, Physical Review E, 94 (2016). Google Scholar [26] A. Weinmann, M. Storath and L. Demaret, The $L^1$-potts functional for robust jump-sparse reconstruction, SIAM Journal on Numerical Analysis, 53 (2015), 644-673. doi: 10.1137/120896256. Google Scholar [27] G. Winkler, O. Wittich, V. Liebscher and A. Kempe, Don't shed tears over breaks, Jahresber. Deutsch. Math.-Verein, 107 (2005), 57-87. Google Scholar (Grey line) log normalized DNA copy-number ratios against genome order micro-array based comparative genomic hybridization, data from [21]. (Red line) PWC restoration by the DPS algorithm DPS search space and its pruning process. (Figure 2a) shows the search space (an Hypercube) structured in levels where each level regroups the set of LP with equal number of 1 bits. (Figure 2b) illustrates DPS pruning process which only considers the neighbors of the previous optimal solution. As an example, in the second level, we already found the optimal LP $(0100)$ in the first level. By this information, DPS will only consider its neighbors (red and blue nodes) and will prune the rest (empty nodes). Additionally, DPS will stop the search if the energy did not decrease from a level to another. In the example, we stop the search at the second level (red dashed line) since the energy increased from the first level to the second one (a): The decomposition binary tree, for restroing a signal with three discontinuities $I = \{i_1, i_2, i_3\}$, (b) the recursive process where the final solution is obtained by regrouping the leaves of the tree Number of solved matrices as a function of their size. The example restores a signal of size 1024 with 8 regularly distributed discontinuities Restoration of a synthetic simulated data of the movement of a molecular motor. the initial data contains $10^4$ data points regularly distributed in $t\in[0, 2s]$. Left (5a) we remark that DPS exactly recovers all parts except for the hard jump at $t\approx 1.7$. In the right (5b), the restored signal by EBS algorithm with several missed jumps Restoration quality of DPS and $l_2$-Potts for a signal with size $256$ and values $x_i \in [0, 1]$ corrupted by a white noise with standard deviation $\sigma = 0.2$. Figure (6a) represents the initial signal, (6b) shows the reconstructed signal with DPS with no false discontinuities, but with some errors in the height of some plateaus. Finally, the figure (6c) plots the optimal solution for the $l_2$-Potts solver which also detect all the discontinuities Mean Square Error as a function of $\lambda$ and $h$ Mean square error as a function of the noise standard deviation $\sigma$. We remark the curve of DPS increases slowly compared to the other algorithms The figure plots the difference $T_b -T_r$, in seconds, between the classical implementation time $T_b$ and the recursive implementation $T_r$. The difference varies depending on two factors : the signal size $n$ and the number of discontinuities $k$. From this figure it can be seen that the reduction is important especially with a higher number of discontinuities The figure presents the running time in a logarithmic scale as a function of the signal size with fixed number of discontinuities $k = 15$ (a) and as a function of the number of discontinuities with fixed size $n = 10^3$ (b). As shown in the both figures, the recursive implementation offers a lower time and the deviation, from the classical DPS, becomes more compelling with higher number of discontinuities [1] Tom Goldstein, Xavier Bresson, Stan Osher. Global minimization of Markov random fields with applications to optical flow. Inverse Problems & Imaging, 2012, 6 (4) : 623-644. doi: 10.3934/ipi.2012.6.623 [2] M. Montaz Ali. A recursive topographical differential evolution algorithm for potential energy minimization. Journal of Industrial & Management Optimization, 2010, 6 (1) : 29-46. doi: 10.3934/jimo.2010.6.29 [3] Christophe Profeta, Frédéric Vrins. 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Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 457-466. doi: 10.3934/dcds.2009.25.457 [13] Adil Bagirov, Sona Taheri, Soodabeh Asadi. A difference of convex optimization algorithm for piecewise linear regression. Journal of Industrial & Management Optimization, 2019, 15 (2) : 909-932. doi: 10.3934/jimo.2018077 [14] Johnathan M. Bardsley. Gaussian Markov random field priors for inverse problems. Inverse Problems & Imaging, 2013, 7 (2) : 397-416. doi: 10.3934/ipi.2013.7.397 [15] Manfred Denker, Yuri Kifer, Manuel Stadlbauer. Thermodynamic formalism for random countable Markov shifts. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 131-164. doi: 10.3934/dcds.2008.22.131 [16] Felix X.-F. Ye, Yue Wang, Hong Qian. Stochastic dynamics: Markov chains and random transformations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (7) : 2337-2361. doi: 10.3934/dcdsb.2016050 [17] Manfred Denker, Yuri Kifer, Manuel Stadlbauer. Corrigendum to: Thermodynamic formalism for random countable Markov shifts. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 593-594. doi: 10.3934/dcds.2015.35.593 [18] Łukasz Struski, Jacek Tabor, Tomasz Kułaga. Cone-fields without constant orbit core dimension. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3651-3664. doi: 10.3934/dcds.2012.32.3651 [19] Shengtian Yang, Thomas Honold. Good random matrices over finite fields. Advances in Mathematics of Communications, 2012, 6 (2) : 203-227. doi: 10.3934/amc.2012.6.203 [20] Qiuying Li, Lifang Huang, Jianshe Yu. Modulation of first-passage time for bursty gene expression via random signals. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1261-1277. doi: 10.3934/mbe.2017065 2019 Impact Factor: 1.373	2020-10-30 23:25:55	{"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": 0.6370466351509094, "perplexity": 4823.945365324073}, "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-2020-45/segments/1603107911792.65/warc/CC-MAIN-20201030212708-20201031002708-00565.warc.gz"}
https://www.rdocumentation.org/packages/Rgraphviz/versions/2.16.0/topics/buildNodeList	# buildNodeList 0th Percentile ##### A function to build lists of node and edge objects These functions can be used to generate lists of pNode and pEdge objects from an object of class graph. These lists can then be sent to Graphviz to initialize and layout the graph for plotting. Keywords graphs ##### Usage buildNodeList(graph, nodeAttrs = list(), subGList=list(), defAttrs=list()) buildEdgeList(graph, recipEdges=c("combined", "distinct"), edgeAttrs = list(), subGList=list(), defAttrs=list()) ##### Arguments graph An object of class graph nodeAttrs A list of attributes for specific nodes edgeAttrs A list of attributes for specific edges subGList A list of any subgraphs to be used in Graphviz recipEdges How to deal with reciprocated edges defAttrs A list of attributes used to specify defaults. ##### Details These functions will take either the nodes or the edges of the specified graph and generate a list of either pNode or pEdge objects. The recipEdges argument can be used to specify how to handle reciprocal edges. The default value, combined will combine any reciprocated edges into a single edge (and if the graph is directed, will by default place an arrowhead on both ends of the edge), while the other option is distinct which will draw to separate edges. Note that in the case of an undirected graph, every edge of a graphNEL is going to be reciprocal due to implementation issues. The nodeAttrs and edgeAttrs attribute lists are to be used for cases where one wants to set an attribute on a node or an edge that is not the default. In both cases, these are lists with the names of the elements corresponding to a particular attribute and the elements containing a named vector - the names of the vector are names of either node or edge objects and the values in the vector are the values for this attribute. Note that with the edgeAttrs list, the name of the edges are in a particular format where an edge between x and y is named x~y. Note that even in an undirected graph that x~y is not the same as y~x - the name must be in the same order that the edge was defined as having. The subGraph argument can be used to specify a list of subgraphs that one wants to use for this plot. The buildXXXList functions will determine if a particular node or edge is in one of the subgraphs and note that in the object. The defAttrs list is a list used to specify any default values that one wishes to use. The element names corresponde to the attribute and the value is the default for that particular attribute. If there is no default specified in defAttrs for an attribute declared in nodeAttrs or edgeAttrs, then the latter must have a value for every node or edge in the graph. Otherwise, if a default is supplied, that value is used for any node or edge not explicitly defined for a particular attribute. ##### Value A list of class pNode or pEdge objects. agopen, plot.graph, pNode,pEdge ##### Aliases • buildNodeList • buildEdgeList • edgeL,clusterGraph-method • edgeL,distGraph-method ##### Examples set.seed(123) V <- letters[1:10] M <- 1:4 g1 <- randomGraph(V, M, .2) z <- buildEdgeList(g1) x <- buildNodeList(g1) Documentation reproduced from package Rgraphviz, version 2.16.0, License: EPL ### Community examples Looks like there are no examples yet.	2021-01-22 23:17:37	{"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": 0.3080471158027649, "perplexity": 1301.6760617603595}, "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/1610703531429.49/warc/CC-MAIN-20210122210653-20210123000653-00375.warc.gz"}
http://maths.otago.ac.nz/?undergraduate_papers=math342	Mathematics Te Tari Pāngarau me te Tatauranga Department of Mathematics & Statistics ## MATH342 Modern Algebra First Semester 18 points Modern algebra is studied all over the world, perhaps not surprising in view of its international beginnings in the late 1700s work of the Swiss mathematician Leonhard Euler, the French mathematician Joseph Louis Lagrange, and the German mathematician Carl Friedrich Gauss. Their work led to the introduction in the 1800s of the unifying abstract algebraic concepts of a group and a ring, the first of these pioneered by the British algebraist Arthur Cayley, the second due to Richard Dedekind, also German. These two notions of a group (a set with a standard operation, usually called multiplication) and a ring (a set with two operations, usually called addition and multiplication) occur throughout modern mathematics in both its pure and applied branches and, even after more than 100 years since their introduction, most of today’s research in modern algebra involves the study of either groups or rings (or both!) ### Paper details The learning aims of the paper are principally to develop the notions of a group and ring, to see how these arise in a variety of mathematical settings, and to establish their fundamental properties. Since this is a Pure Mathematics paper which will provide the basis for further study in abstract algebra, concepts will be introduced and developed rigorously. We will be doing a lot of proofs! ### Potential students This paper should be of interest to anyone who wishes to see how algebraic properties arising in different branches of pure mathematics can be described using the unifying concepts of a group and a ring. Students who wish to pursue their interests in algebra should take this course as a foundation to more advanced papers in the theory of groups, Galois Theory, rings, modules and algebras. MATH 202 ### Main topics • A review of functions; equivalence relations; modular arithmetic. • Groups; subgroups; homomorphism and isomorphism; cosets and normal subgroups; quotient groups; Lagrange’s theorem; Application - Public key cryptography. • Rings; subrings; integral domains; matrix rings; polynomial rings; homomorphism and isomorphism; ideals; quotient rings; fields; vector spaces; Application - Error correcting codes. ### Required text No required text - Comprehensive Course notes will be provided. ### Lecturer Professor Mike Hendy, Room 517 ### Lectures Monday at 10:00, Wednesday at 11:00 and alternate Fridays at 11:00. (Location to be announced.) ### Tutorial Thursday at 9.00am in room MA241. ### Internal Assessment There will be 5 exercises making up 50% of the internal assessment. You will be encouraged to use the mathematical formatting language LaTeX for your assignments. The remaining 50% of internal assessment will come from two 45 minute written tests. These are scheduled for Friday April 7 and Friday May 12 (11.00 - 11.45am) ### Exam format The final examination is 3 hours long. ### Final mark Your final mark F in the paper will be calculated according to this formula: F = max(0.85E + 0.075A + 0.075T, 0.70E + 0.15A + 0.15T) where: • E is the Exam mark • A is the Assignments mark • T is the Tests mark and all quantities are expressed as percentages. Thus your internal assessment contributes either 15% or 30% towards your final mark. ### Students must abide by the University’s Academic Integrity Policy Academic endeavours at the University of Otago are built upon an essential commitment to academic integrity. The two most common forms of academic misconduct are plagiarism and unauthorised collaboration. Plagiarism is defined as: • Copying or paraphrasing another person’s work and presenting it as your own. • Being party to someone else’s plagiarism by letting them copy your work or helping them to copy the work of someone else without acknowledgement. • Using your own work in another situation, such as for the assessment of a different paper or program, without indicating the source. • Plagiarism can be unintentional or intentional. Even if it is unintentional, it is still considered to be plagiarism. All students have a responsibility to be aware of acceptable academic practice in relation to the use of material prepared by others and are expected to take all steps reasonably necessary to ensure no breach of acceptable academic practice occurs. You should also be aware that plagiarism is easy to detect and the University has policies in place to deal with it. In 1833, the Irish mathematician William Rowan Hamilton gave one of the first algebraic descriptions of the set of complex numbers. Of course, each complex number can be described as a sum a + ib where a and b are real numbers and i is the “imaginary” number with the property that its square is -1. Also such a number can be thought of as the point on the two-dimensional x-y plane with a as its x-coordinate and b as its y-coordinate. Now, Hamilton tried for ten years to find a similar way of algebraically describing three-dimensional space. On October 6, 1843, while out walking in Dublin, he finally realized that there was no algebraic three-dimensional analogue but that there was a four-dimensional one. He formed a new set of numbers called the quaternions in which there are four key ingredient numbers, namely 1, $i$, $j$, and $k$, satisfying the following multipicative rules: $$i^2=j^2=k^2=-1\\ij = k, jk = i, ki = j,\\ji = -k, kj = -i, ik = -j$$ Hamilton was so pleased with his discovery that he stopped on his walk to carve these equations with a knife into the sandstone of Brougham Bridge (see Irish stamp above). The quaternions give us important examples of both a group and a division ring.	2017-11-23 22:22:08	{"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": 0.6855801939964294, "perplexity": 1343.3937905609653}, "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-2017-47/segments/1510934806979.99/warc/CC-MAIN-20171123214752-20171123234752-00761.warc.gz"}
http://www.math.md/publications/basm/issues/y2020-n2/13197/	RO EN ## Commutative Weakly Tripotent Group Rings Authors: Peter V. Danchev ### Abstract Very recently, Breaz and C\^{\i}mpean introduced and examined in Bull. Korean Math. Soc. (2018) the class of so-called {\it weakly tripotent rings} as those rings $R$ whose elements satisfy at leat one of the equations $x^3=x$ or $(1-x)^3=1-x$. These rings are generally non-commutative. We here obtain a criterion when the commutative group ring $RG$ is weakly tripotent in terms only of a ring $R$ and of a group $G$ plus their sections. \newline Actually, we also show that these weakly tripotent rings are {\it strongly invo-clean rings} in the sense of Danchev in Commun. Korean Math. Soc. (2017). Thereby, our established criterion somewhat strengthens previous results on commutative strongly invo-clean group rings, proved by the present author in Univ. J. Math. \& Math. Sci. (2018). Moreover, this criterion helps us to construct a commutative strongly invo-clean ring of characteristic $2$ which is {\it not} weakly tripotent, thus showing that these two ring classes are different. Institute of Mathematics and Informatics	2021-09-18 20:03:53	{"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": 0.911787211894989, "perplexity": 1576.6139866974204}, "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/1631780056572.96/warc/CC-MAIN-20210918184640-20210918214640-00504.warc.gz"}
https://math.stackexchange.com/questions/2539628/volume-of-water-in-hemisphere	# Volume of Water in Hemisphere? There is a hemispherical bowl of diameter 30 cm and it is filled with water such that the radius the top surface of water is 9 cm. Find the height of the water and and the volume of water in the bowl. I could find the height using Pythagoras Theorem . However I cant find the volume . • I think you need some calculus to solve the problem for the volume – user Nov 27 '17 at 14:34 • @gimusi so how do you do it – Ishaan Parikh Nov 27 '17 at 15:17 • by integration, you can find some inspiration here youtube.com/watch?v=Rib7yyn81BY – user Nov 27 '17 at 15:23 Wikipedia on spherical cap has the volume formula $$V=\frac 16\pi h(3a^2+h^2)$$ with a derivation. $h$ is the height you calculated and $a$ is the radius of the base.	2020-06-04 01:56:15	{"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": 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": 0.8181828856468201, "perplexity": 193.14770406465905}, "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-2020-24/segments/1590347436828.65/warc/CC-MAIN-20200604001115-20200604031115-00448.warc.gz"}
https://math.stackexchange.com/questions/453808/why-can-not-define-g-group1-3-2/454138	# why can not define G := Group((1),(3,2));; gap> G := Group((1),(3,2));; Error, usage: Group(<gen>,...), Group(<gens>), Group(<gens>,<id>) called from <function "Group">( <arguments> ) called from read-eval loop at line 6 of stdin you can 'quit;' to quit to outer loop, or you can 'return;' to continue before finding character table of permutation group S3 - 1,3,2 got error in above command • you could always try asking on stackoverflow.com – raindrop Jul 28 '13 at 2:21 • Are the users who downvoted and voted to close familiar with the software GAP? I don't understand the question so I cannot personally see one way or the other about whether or not this deserves downvotes or close votes. I expect if it deserved them there should be some explanation as to why in the comments, as well as efforts to engage with the OP, but I see none. – anon Jul 28 '13 at 8:29 • The question is about the meaning of the error message in GAP, though it suffers from inconsistent notation and could be formulated better. I think it's fine to ask it here - the gap-system tag at stackoverflow.com seems much less attended. Try to debug your input and see how GAP parses (1) and (3,2) - you will see the difference which should give you a hint. – Alexander Konovalov Jul 28 '13 at 11:42 • @scaaahu: Please don't make such conclusions just reading the GAP Forum: it is entirely plausible that the OP wrote to GAP Support, which is not a public mailing list, but did not get a quick reply from there and decided to repeat the question here. – Alexander Konovalov Jul 28 '13 at 20:11 • @scaaahu Indeed, I've assumed you are subscribed to the Forum or looked at the GAP Forum Archive, not that you'd like the OP to say this explicitly in the post. – Alexander Konovalov Jul 29 '13 at 9:54 Now after the question is no longer on hold, I can reply with more details: 1) The error about the wrong usage of Group usually means that the group can not be generated by the arguments. For example, gap> Group(1); #I no groups of cyclotomics allowed because of incompatible ^ Error, usage: Group(<gen>,...), Group(<gens>), Group(<gens>,<id>) called from <function "Group">( <arguments> ) 2) What's then wrong with the arguments in the question? See how GAP evaluates various inputs: gap> (3,2); # works OK (2,3) gap> (); # this is the notation for the identity permutation () gap> (1); # this is an integer one in brackets 1 Thus, Group((1),(3,2)) tries to generate a group with an integer 1 and a permutation (3,2) as generators, what obviously does not make sense 3) What does (1) in the original question actually mean? If this has to be an identity permutation, one should use () - however, there is no need to add the identity element of the group to the list of generators, so Group((3,2)) just suffices: gap> Group((3,2)); Group([ (2,3) ]) 4) This may be not the intended group, however, since the question contains "permutation group S3 - 1,3,2" - it's unclear what is the meaning of "1,3,2" here, but if the intention was to create a symmetric group of permutations of degree 3, here there are several ways to achieve this: gap> Group((1,2),(1,2,3)); Group([ (1,2), (1,2,3) ]) gap> SymmetricGroup(3); Sym( [ 1 .. 3 ] ) 5) Finally, the following hint about character tables may be useful: One can compute the character table "on-fly" for a given group, and if some methods depend on random states, the result may differ each time you call CharacterTable (conjugacy classes may be ordered in a different way). For groups whose character tables are available from The GAP Character Table Library, the table will be retrieved from the library so the result will be the same each time. Compare Display(CharacterTable(SymmetricGroup(3))); and Display(CharacterTable("S3")); to see the difference. • in another question, i can use wrong notation, it work, math.stackexchange.com/questions/453392/… – Series group Jul 29 '13 at 12:42 • @Seriesgroup: sorry, I do not understand what do you mean in your recent comment. The question linked there is still the same as yesterday. – Alexander Konovalov Jul 29 '13 at 12:47 • @AlexanderKonovalov: May I ask you to take a look at mu question here stackoverflow.com/q/18043036/1842737. Sorry and forgive me for this asking Prof. :) – mrs Aug 4 '13 at 12:48	2019-08-19 16:13:45	{"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": 0.5433415770530701, "perplexity": 330.5997673036063}, "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/1566027314852.37/warc/CC-MAIN-20190819160107-20190819182107-00081.warc.gz"}
https://www.physicsforums.com/threads/help-metric-space-problem.110976/	# HELP metric space problem 1. Feb 16, 2006 ### Mathman23 Hi I have this here metric space problem which caused me some trouble: $$S \subseteq \mathbb{R}^n$$ then the set $$\{ \\| x - y \\| \ \| y \in S \}$$ has the infimum $$f(x) = \{ \\| x - y \\| \ \| y \in S \}$$ where f is defined $$f: \mathbb{R}^n \rightarrow \mathbb{R}$$ I have two problems here which I'm unable to solve: (a) show, if S is a closed set and $$x \notin S$$ then $$f(x) > 0$$ ???? (b) show, if S is a closed set, then $$S = \{ x \in \mathbb{R}^n \| f(x) = 0\}$$ ??? I need to hand this in tomorrow, and I have been strugling this these two problems the last week, therefore I would very much appriciate if anybody could give me an idear on how to solve the two problems above. God bless, Best Regards, Fred Last edited: Feb 16, 2006 2. Feb 16, 2006 ### quasar987 $$f(x) = \{ \\| x - y \\| \ \| y \in S \}$$ ?? That would mean to each x in R^n, f maps x to \|\|x-y\|\| for all y in S. So as soon as card(S)>1, f is not a function. Also, what do you mean by "$$\{ \\| x - y \\| \ \| y \in S \}$$ has the infimum f(x)"? 3. Feb 16, 2006 ### Mathman23 Sorry it should have said $$f(x) = \mathrm{inf} \{ \\| x - y \\| \ \| y \in S \}$$ Best Regards Fred p.s. My problems deals with the distance from $$\mathbb{R}^n$$ to a point in a subset S of $$\mathbb{R}^n$$. Last edited: Feb 16, 2006 4. Feb 16, 2006 ### quasar987 a) wouldn't that exeedingly simply argument suffice: we know that \|\|x-y\|\| = 0 iff x=y. But since x is not in S, x is not equal to y for any y in S. Hence, \|\|x-y\|\|>0. There's probably a problem with this argument as it doesn't even use the closedness of S... 5. Feb 16, 2006	2017-04-28 16:19:59	{"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": 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": 0.7851356863975525, "perplexity": 1281.0805204262958}, "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-2017-17/segments/1492917122996.52/warc/CC-MAIN-20170423031202-00460-ip-10-145-167-34.ec2.internal.warc.gz"}
http://phplatex.scarfboy.com/	## , LaTeX renderer ### Overview PHPLatex is quick and dirty PHP script that makes TeX rendering easier. It was initially made for latex.knobs-dials.com. Does have a few requirements, though you'll likely meet all of those on modern nix-with-texlive installations. Aside from the fact it fundamentally is a bit hackish, it seems to work well enough. At the core it just invokes latex, dvips, and convert. And stores the resulting image, so that asking for the same TeX will generate once, then read from disk cache, so it's cheap to leave the function call in the PHP. External invocation so is always potentially unsafe in an untrusted environment (the utilities will run as your webserver user). For a safer but rather more restrictive variation, check the ocaml parser that mediawiki (as in wikipedia) uses. Released under the GPL. Comments, praise, complaints, bugs, fixes and whatnot are encouraged. Contact address: #### Ideas / requests for comments This thing fulfils my own needs, so I am not actively working on it. Suggestions are welcome, of course. I am considerering having it always render higher resolutions to have images be sharper in printing, although I would have to test whether this can be done elegantly and check that nothing reacts weirdly (browsers, OS settings, etc.). If anyone has played with something like that, I welcome your notes. ### Examples <? print texify('wh^{e^{e^{e^{e^{e_{e_e}}}}}}'); ?> Since you write inside PHP strings, backslashes are interpreted, so double them. ?> #### Tables, packages The following demonstrates how to include packages, here pstricks and colortab: <? print texify(" \\definecolor{lightergray}{gray}{.875} \\newcommand\\lightergray{\\color{lightergray}} \\begin{tabular}{\|lc\|r\|} \\hline \\LCC \gray & \lightgray & \lightergray \\\\ rabbit & 12 & sold \\\\ frog & 3.5 & pending \\\\ \\ECC \\hline \\end{tabular}", 90, 0.0,0.0,0.0, 1.0,1.0,1.0, "\\usepackage{pstricks,colortab}"); ?>	2018-12-14 09:39:38	{"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": 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": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8589757680892944, "perplexity": 7563.005119577306}, "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/1544376825512.37/warc/CC-MAIN-20181214092734-20181214114234-00274.warc.gz"}
http://laquintacolonna.it/ypkj/inverse-cloglog-r.html	1 Look at the data; 8. (2012) A comparison of the seasonal movements of tiger sharks and green turtles provides insight into their predator-prey relationship. If there are reflective surfaces in the sound field, then reflected sounds will add to the directed sound and you will get more sound at a field location than the inverse distance law predicts. 00-4 -2 0 2 4 x. families: Lino: The Generalized Beta Distribution (Libby and Novick, 1982) Log: Logarithmic. 4 of Gelman and Hill (2007) using stan_glm. ), calculating marginal effects that are comparable. The following reference is an alphabetical listing of operators and functions which may be used in series assignment and generation, and in many cases, in matrix operations or element evaluation. 1 Model de nition The model is de ned in a text le using a dialect of the BUGS language. So if we have an initial value of the covariate. Note that we usually use the inverse link function g 1(X )rather than the link function. Generalized linear mixed models using AD Model Builder. The aliases are CCLOGLOG, CCLL, and CUMCLOGLOG. accepts the links 1/mu^2, inverse, identity and log. I am having problem to locate where the R matrix are defined for regular matrices, i. A force is defined as a) the ability to do work. C("Cgee",but don't understand it well enough to know. 4 Model Selection. 期待値 np および分散 np(1 − p) が 5 よりも大きい場合、二項分布 B(n, p) に対する良好な近似として正規分布がある。ただし、この近似を適用するにあたっては、変数のスケールに注意し、連続な分布への適切な処理がなされる必要がある。. computer based function in the free R software for the estimation of lethal concentrations (LC50, LC90 and LC95). 3 Link functions. Logit model # The stargazer() function from the package -stargazer allows a publication quality of the logit model. Predict method for Generalized Linear Models Description. The logit transformation is defined as follows:. Regarding the marginals, we chose the probit link and found that the inverse Gaussian instead of the gamma distribution provides the best fit as judged by the plots of normalized quantile residuals (Stasinopoulos et al. In this case, both DM and SV methods are nearly unbiased. Distributions are parameterized in part or in full by a scale matrix, which can be supplied in several additional forms as indicated by the function's. It is the inverse CDF of the extreme value (or Gumbel or log-Weibull) distribution. p 1 = F(y 1) p j = F(y j) - F(y j-1), for 2 ≤ j < N p N = 1 - Sum[i = 1 to N-1. distribution, and the complementary log-log (cloglog) link function is formed from the inverse c. When after cloglog, identity, inverse, log, 1/mu^2, sqrt. (cloglog): π i. The quasi family accepts the links logit, probit, cloglog, identity, inverse, log, 1/mu^2 and sqrt, and the function power can be used to create a power link function. R # Part of the R package, http://www. For the complementary log-log model, on the other hand, reversing the coding can give us completely different results. Normal rules of arithmetic apply. Inverse Gaussian Distribution = X T b ( ) = p 2 b 0 ( ) = 1 p 2 E Y The canonical link is = h ( ) 1 2 2 X T This is the only built-in link function fo r inverse gaussian distribution. 2 Basic operations; 7. This generalizes the idea of "Gini" importance to other losses, following the explanation of Gini importance. 00-4 -2 0 2 4 x. If we now increase the covariate by 1. There is no mention of the probit link. For example for probit it can be like: glm( formula, family=binomial(link=probit)) Similarly, below are other families with their default link. Sometimes we can bend this assumption a bit if the response is an ordinal response with a moderate to large number of levels. Return_type is a number that determines the type of return value: 1 (or missing)= C-Log-Log , 2= Inverse C-Log-Log. Rweb-- an interactive web-based interface to the "R" statistical programming language (similar to S or S-plus) SHAZAM-- a programming environment for econometricians, statisticians, and others who use statistical techniques. The response variable is allowed to follow a binomial, Poisson. The Additive Property. Return_type is a number that determines the type of return value: 1 (or missing)= C-Log-Log , 2= Inverse C-Log-Log. C("Cgee",but don't understand it well enough to know. These GLMs are well suited for classification questions: to be or not to be, to vote or not to vote, and to click or not to click. (2012) A comparison of the seasonal movements of tiger sharks and green turtles provides insight into their predator-prey relationship. Uses MCMC instead of ML to fit the model. A logistic regression uses a logit link function: And a probit regression uses an inverse normal link function:. For details see this paper by. Analysts in any field who need to move beyond standard multiple linear regression models for modeling their data. We isolated ‘complete’ foraging trips that began and ended on the colony within the same day using the ‘adehabitatLT’ package in R 71, removed locations on the nest or beach of the colony. The proposed function integrates the Abbott correction and adjusts the best link function. value defined by the user or set by default). compat import urlopen import numpy as np np. Thorpe (16 Mar 2006) [R] excluding factor levels with read. cloglog is deﬁned as = ln ln(1 ). The gaussian family accepts the links (as names) identity, log and inverse; the binomial family the links logit, probit, cauchit, (corresponding to logistic, normal and Cauchy CDFs respectively) log and cloglog (complementary log-log); [] If the link function is given as a character string, all families accept all link functions. Repeat steps 1 and 2 until we find a “good” guess (a. When after cloglog, identity, inverse, log, 1/mu^2, sqrt. Description: Return the arc-sine (inverse of the sine function) of x as an angle in radians between $$-\pi/2$$ and $$\pi/2$$. As in R (and nimbleFunctions), arguments are matched by order or by name (if given). You can find other options in packages, or manually create anything you want. CLOGLOG computes the complementary log log transformation (i. api import ols from statsmodels. When the target variable has only two categories, the inverse of link function transforms the value predicted by the regression equation into the corresponding probability of the first target category. theta function(x) log(x/(1-x)) from. group') and sample sizes in each group from 1-8. It is the inverse CDF of the extreme value (or Gumbel or log-Weibull) distribution. Popular choices of c. The gaussian family accepts the links (as names) identity, log and inverse; the binomial family the links logit, probit, cauchit, (corresponding to logistic, normal and Cauchy CDFs respectively) log and cloglog (complementary log-log); [] If the link function is given as a character string, all families accept all link functions. Create a Link for GLM Families Description. Count data regression with excess zeros In practice: The basic Poisson regression model is often not ﬂexible enough to capture count data observed in applications. Note that link power 0, 1, -1 or 0. , statistical calibration) in linear, generalized linear, nonlinear, and (linear) mixed-effects models. (Systematic pa rt of the mo del) There is a link function h that links the conditional. {1/mu^2 \| cauchit \| cloglog \| identity \| inverse \| log \| logit \| probit \| sqrt} Name of the link function for the model. In this article binary state space mixed models (BSSMM) using a ﬂexible skewed inverse link function based on the generalized extreme value (GEV) distribution introduced by (Abanto-Valle et al. 2 r ik log r ik ^r ik 1ðn ik r ikÞlog n ik r ik n ik r^ ik 5 X i X k dev ik; ð3Þ where ^r ik5n ikp ik is the expected number of events in each trial arm, based on the current model, and dev ik is the deviance residual for each data point. The most important difference between these three software is the default probability of the binary dependent or the response variable, where SAS uses the smaller value (zero) by default to estimate its probability, while SPSS and MINITAB use. dist-package gamlss. gaussian family. R package version 0. 9, then plant height will decrease by 0. accepts the links 1/mu^2, inverse, identity and log. Poisson model with logit link NOT available in R. 2 A linear function of the regressors, called the linear predictor, h Implementation of GLMs in R link family log logit probit cloglog gaussian binomial poisson Gamma inverse. Regression-type models Examples Using R R examples What distributions can I choose? gaussian: a Gaussian (Normal) distribution binomial: a binomial distribution for proportions poisson: a Poisson distribution for counts Gamma: a gamma distribution for positive continuous data inverse. This is the base model-fitting function - see plot. R ∞ −∞ g(x)p(x)dx I Note that we usually use the inverse link function g−1(Xβ) rather than the link function. Antibodies produced in response to an infectious disease like malaria remain in the body after the individual has recovered from the disease. [email protected] binomial binomial logit, probit or cloglog poisson poisson log, identity or sqrt Gamma Gamma inverse, identity or log inverse. dcauchy, pcauchy, and qcauchy are respectively the density, distribution function and quantile function of the Cauchy distribution. distribution, and the complementary log–log (cloglog) link function is formed from the inverse c. These link functions are described in [R] glm and (Hardin and Hilbe 2001). mu_cubed See Also-----statsmodels. Generalized Linear Mixed Models When using linear mixed models (LMMs) we assume that the response being modeled is on a continuous scale. Node 24 of 34. The complementary log-log function and its inverse function are provided. , to base $$e$$. R’s recycling rule (re-use of an argument as needed to accommodate longer values of other arguments) is generally followed, but the returned object is always a scalar or a vector, not a matrix or array. One way of estimating relationships between the time series and their lagged values is the vector autoregression process:. Note that we usually use the inverse link function g 1(X )rather than the link function. CLOGLOG is the complementary log-log function, LOGIT is the log odds function, and PROBIT (or NORMIT) is the inverse standard normal distribution function. Thorpe (16 Mar 2006) [R] excluding factor levels with read. The: 297: plots include a normal Q-Q plot, a plot of residuals vs. There is a large, healthy contingent on rates of convergence in the mathematical physics literature. First!we!can!fit!a!simple!linear!regression!where!contraceptive!use!depends!on!the! Microsoft Word - GLM Tutorial in R. part Earlier versions of the hier. Here, we aim to compare different statistical software implementations of these models. Newcombe, Logit conﬁdence intervals and the inverse sinh transformation (2001), American Statistician, 55. distribution, and the complementary log-log (cloglog) link function is formed from the inverse c. For instance, we might have a range of values – say the heights of individuals – spread among 5 different ethnic groups, and we want to. In binomial regression, a link function is used to join the linear predictor variables and the expectation of the response variable. Although King and Zeng accurately described the problem and proposed an appropriate solution, there are still a lot of misconceptions about this issue. Laboratory Data. Often addressed by adopting a negative binomial (NB) model. Inverse estimation, also referred to as the calibration problem, is a classical and well-known problem in regression. link : a link instance The link function of the inverse Gaussian instance InverseGaussian. 1 Look at the data; 8. phi The known value of the additional parameter phi. ceil(x) Domain: 8e+307 to 8e+307 Range: integers in 8e+307 to 8e+307 Description: returns the unique integer nsuch that n 1 Install package(s), once again select your nearest CRAN mirror and select package SPACECAP for installation. mu is the value of the inverse of the link function at lin_pred, where lin_pred is the linear predicted value of the WLS fit of the transformed variable. The working residuals are rW j= (y b) @ @ j and the score residuals are rS j = y j b j V(b j) @ @ 1 j Deﬁne Wc= V( b) and Xto be the covariate matrix. An individual with such antibodies is called seropositive. Probit link: Inverse of CDF for standard normal g Lecture 6 STK3100 - Categorical responses - p. Param for the index in the power link function. 3 Link functions. John Fox (McMaster University) Introduction to R ICPSR 2010 15 / 34 Statistical Models in R Implementation of GLMs in R link family log logit probit cloglog gaussian binomial poisson Gamma inverse. The allowed link functions depend on the distribution of the response variable (also known in R as the model family):. Make sure that you can load them before trying to run the examples. igaussian inverse Gaussian binomial varname Nj# N see[R] bootstrap. I am working on modifying the R working matrix to commodate some other correlations that not included in the package. Vector Autoregressions tsa. They are the exponentiated value of the logit coefficients. 1 treatment group has all positive cases (i. Materials and Methods A function denoted "lc" was written for the determination of lethal concentrations in the open source R [14]. Other common choices are the cauchit and cloglog functions, (the inverse logit function is the CDF of the standard logistic distribution). 45 for clog-log and 11. Variable: S R-squared: 0. If NA, the default for Gaussian and inverse Gaussian models, the dispersion parameter is estimated, otherwise it is ﬁxed at the nominated value (default 1. Here you can solve systems of simultaneous linear equations using Inverse Matrix Method Calculator with complex numbers online for free. cloglog inverse of cloglog function: F(y) = 1 - exp( -exp(y) ). Then d d = e (1 + e )2 = 1 1 + e e 1 + e = (1 ) = Var(Y) For the Poisson, the canonical link is the log and the inverse link is = g 1( ) = e. append_return_type< Eigen::Matrix< T1, R, C >, Eigen::Matrix< T2, R, C > > This template metaprogram is used to compute the return type for append_array. Package 'RegressionFactory' September 8, 2016 Type Package Title Expander Functions for Generating Full Gradient and Hessian from Single-Slot and Multi-Slot Base Distributions Version 0. Laboratory Data. • Assume Y has an exponential family distribution with some parameterization ζ known as the linear predictor, such that ζ = Xβ. In generalized linear models, instead of using Y as the outcome, we use a function of the mean of Y. Modelled on glim. , statistical calibration) in linear, generalized linear, nonlinear, and (linear) mixed-effects models. 2 + 2 ## [1] 4. In this paper we describe flexible competing risks regression models using the comp. I've tried taking starting values from a logistic and log models fit to the same data and also tried to substitute the intercept from the null model in as the starting value for this model, however all. The real difference is theoretical: they use different link functions. V a r [ Y i \| x i] = ϕ w i v ( μ i) with v ( μ) = b ″ ( θ ( μ)). 000) corresponding to M, D, Y, h, m, s. binomial binomial logit, probit or cloglog poisson poisson log, identity or sqrt Gamma Gamma inverse, identity or log inverse. Create a Link for GLM families Description. When after cloglog, identity, inverse, log, 1/mu^2, sqrt. Make sure that you can load them before trying to run the examples. because the inverse(G∗)−1can be derived manually and then incorporated in the IRLS algorithm. table() and colClasses= Dieter Menne (16 Mar 2006). lab = "Y", z. specifies that an additional table of statistics be displayed. gaussian quasi. Ported from S-plus to R. w = beta0 + beta1 * z1; mu = 1 - exp(-exp(w));. # This code is to accompany Maximum Likelihood Methods Strategies for Social Science, # Michael D. 2 + 2 ## [1] 4. Lesa re KU Leuven Abstract Missing data occur in many types of studies and typically complicate the analysis. "psp2dG"- function(Data, Pars, ridge. Details: The domain of this function is from -1 to 1 (inclusive). Logistic Regression with Raw Data. Mixed models in R using the lme4 package Part 5: Generalized linear mixed models Douglas Bates Department of Statistics University of Wisconsin - Madison Madison January 11, 2011 Douglas Bates (Stat. These link functions differ slightly in the way they link the outcome variable to the explanatory variables (Figure 8-3). So if we have an initial value of the covariate. Three columns are selected by clicking on [X axis], [Y axis] and [Z axis]. Thorpe (16 Mar 2006) [R] excluding factor levels with read. 7 Other Choices of Link. For example if the slope is +0. inverse of diagonal matrix = diag( 1/ diagonal) In these simple examples, it is often useful to show the results of matrix calculations as fractions, using MASS::fractions(). Title Generalized Additive Models for Location Scale and Shape. plot = F, se = T, family. , 2015) are revisited. gaussian quasi quasibinomial quasipoisson The quasi, quasibinomial, and quasipoisson family generators do not correspond to exponential families. link default logit loga cauchit probit cloglog loglog robit sn pdf zeroin ated Zeroin ated BetaBinomial Type 1 doc Zero-in ated Beta-Binomial, type 1 hyper theta1 hyperid 89001 name overdispersion short. r some functions a ();b; and c: Here, j is called a canonical pa rameter. ) is the known link function (i. Logistic regression, also called a logit model, is used to model dichotomous outcome variables. 19 Implementation in R Implemented in the package glmmML in R. p 1 = F(y 1) p j = F(y j) - F(y j-1), for 2 ≤ j < N p N = 1 - Sum[i = 1 to N-1. By standardized, we mean that the residual is divided by f1 h. of the Gumbel distribution. j if inverse Gaussian b j +kb 2 j if negative binomial b j if Poisson The response residuals are given by rR j = y j b j. (2004) and Walsh. Antibodies produced in response to an infectious disease like malaria remain in the body after the individual has recovered from the disease. Crossed random effects difficult. 3 Implementation 1 \RequirePackage{listings} 3. identity The identity transform: inverse_power The inverse transform: inverse_squared The inverse squared transform: log: The log transform: logit: Methods. Vector Autoregressions tsa. Rweb-- an interactive web-based interface to the "R" statistical programming language (similar to S or S-plus) SHAZAM-- a programming environment for econometricians, statisticians, and others who use statistical techniques. fit - function(X, Y, m, link = "logit. c) as the distance decreases the force will increase by the ratio of 1/r. 11, 2011 1 / 39. The recommended R package mgcv (Wood. On page 128 of Modelling survival data by Therneau & Grambsch there is the an example of the type of desired plot, with a log of the survival curve by years. , clalims), then use a distribution family which is strickly positive (i. April 28, 2015 SPH 247 Statistics for Laboratory Data 21. ## (Based on earlier code from 2003--2010). I am having problem to locate where the R matrix are defined for regular matrices, i. 0, and monotone increasing when β<0. View Article PubMed/NCBI Google Scholar 59. of the Gumbel distribution. The quasibinomial and quasipoisson families differ from the binomial and poisson families only in that the dispersion parameter is not fixed at one, so they can "model" over-dispersion. Laboratory Data. table() and colClasses= Peter Tait (16 Mar 2006) Re: [R] excluding factor levels with read. families: Lino: The Generalized Beta Distribution (Libby and Novick, 1982) Log: Logarithmic. April 2, 2019 EPI 204 Quantitative Epidemiology III 1. part Earlier versions of the hier. If you omit the SCALE= option, the scale parameter is fixed at the value 1. Let K(x;y) be single-site Glauber dynamics with uniformly chosen random update site. width", 100) import matplotlib. probit ([dbn]) The probit (standard normal CDF) transform. 1 g 1(X )is the systematic component that we've been talking about all along. Our results are consistent with a process of learning associated. All these above mentioned inverse link functions are nothing but CDFs of some continuous probability distributions. For the multivariate normal, Wishart, and inverse Wishart distributions, the basic functions perform a random draw from the distribution or provide the density of the distribution at a point. The quasi family accepts the links logit, probit, cloglog, identity, inverse, log, 1/mu^2 and sqrt, and the function power can be used to create a power link function. Trevor Hefley (Kansas State University, Manhattan, Kansas). squaredLR can be used for GLS models and provides both and R-Squared and an Adjusted R-Squared. Interpreting coefficients in glms. stackexchange. The inverse. Description: returns the inverse hyperbolic tangent of x, atanh(x) = 1 2 fln(1+x) ln(1 x)g. The real difference is theoretical: they use different link functions. ) is the known link function (i. The four plots are written to a single PNG file named X_diag. quasibinomial family - cauchit, cloglog, log, logit, and probit. Note that we model the variable MSESC as its inverse-logit because in a binomial regression model, For instance, to model binary outcomes, we can also use the probit link or the complementary log-log (cloglog) instead of the logit link. I'm not a Stata user so I'm trying to reproduce Stata results that are given to me in R. Gamma and Inverse-Gamma Distributions Tree level 3. independence, exchangeable, AR and unstructure. When not set, this value defaults to 1 - variancePower, which matches the R "statmod" package. Logit and probit models are appropriate when attempting to model a dichotomous dependent variable, e. , 2017) and information criteria (see Fig. STATISTICS: AN INTRODUCTION USING R By M. When applied to a linear predictor $$\eta$$ with values in $$\mathbb{R}$$, the inverse link function $$g^{-1}(\eta)$$ therefore returns a valid probability between 0 and 1. inverse logistic for logit). ipw: An R Package for Inverse Probability Weighting. Normal rules of arithmetic apply. The inverse of the first equation gives the natural parameter as a function of the expected value θ ( μ) such that. I am working on modifying the R working matrix to commodate some other correlations that not included in the package. : “Generalized Linear Models” is an online course offered at Statistics. If you omit the SCALE= option, the scale parameter is fixed at the value 1. #' #' The inverse of the link function is the real parameter value. survfit and fun="cloglog" Kevin E. ceil(x) Domain: 8e+307 to 8e+307 Range: integers in 8e+307 to 8e+307 Description: returns the unique integer nsuch that n 1 Install package(s), once again select your nearest CRAN mirror and select package SPACECAP for installation. cloglog, binom. You only need to understand the very basics of functions. In Poisson and negative binomial glms, we use a log link. fitted of the distribution family for more information. CDF and pdf for logit and probit x F(x) cloglog The clog-log link ﬁts observed proportions better than logit link, with residual deviance 3. options(pointsize = 12. Ported from S-plus to R. For the full project description and the complete R code, please check my Github. com lstbayes from 2018/07/06 1 Introduction This package provides language drivers for the listings package for the several Bayesian modeling languages: BUGS, JAGS, and Stan. R as the link function • logistic regression: binary data with a logit link (inverse-link=logistic) • binomial (or aggregated binomial regression: binomial data (maybe logit link, maybe other) • probit regression: probit link Binary data and aggregated (N > 1 data) are handled slightly differ-ently. 5 Data frames; 7. Mixed models in R using the lme4 package Part 5: Generalized linear mixed models Douglas Bates Department of Statistics University of Wisconsin - Madison Madison January 11, 2011 Douglas Bates (Stat. The first function r. 45 for clog-log and 11. All the auxiliary methods used in calculation can be calculated apart with more details. The quasi family accepts the links logit, probit, cloglog, identity, inverse, log, 1/mu^2 and sqrt, and the function power can be used to create a power link function. I'll walk through the code for running a multivariate regression - plus we'll run a number of slightly more complicated examples to ensure it's all clear. null(getOption("rspf. The link function in binary regression is used to specify how the probability of success is linked to the model’s systematic component. So if we have an initial value of the covariate. +,- #addition, subtraction ,/ #multiplication, division. Our results are consistent with a process of learning associated. 2 r ik log r ik ^r ik 1ðn ik r ikÞlog n ik r ik n ik r^ ik 5 X i X k dev ik; ð3Þ where ^r ik5n ikp ik is the expected number of events in each trial arm, based on the current model, and dev ik is the deviance residual for each data point. accepts the links inverse, identity and log. These link functions are chosen to be quantile functions of popular distributions such as the logistic (logit), Gaussian (probit) and Gumbel (cloglog) distributions. It is sufficient to select one data column to run a Weibull Analysis. These link functions differ slightly in the way they link the outcome variable to the explanatory variables (Figure 8-3). Note that we model the variable MSESC as its inverse-logit because in a binomial regression model, For instance, to model binary outcomes, we can also use the probit link or the complementary log-log (cloglog) instead of the logit link. The poisson family. CLOGLOG is the complementary log-log function, LOGIT is the log odds function, and PROBIT (or NORMIT) is the inverse standard normal distribution function. adj = 0, XYpred = NULL, z. glm(mo del, family, data, w eights, controls) family = inverse. This paper proposes a flexible link function from a new class of generalized logistic distribution, namely a flexible generalized logit (glogit) link. r some functions a ();b; and c: Here, j is called a canonical pa rameter. 4 Model Selection. The existing links in glm for binomial data (logit, probit, cloglog) are not adequate for my data, and I need to test some other transformations. # File src/library/stats/R/AIC. is the generalized logit function. Count data regression with excess zeros In practice: The basic Poisson regression model is often not ﬂexible enough to capture count data observed in applications. F i and G i are defined for each link function as follows: Logit: Probit: Normal cumulative probability function: Normal density function: Gompit (Cloglog): Loglog: With a binary dependent variable r i = y i (0. stackexchange. I've tried taking starting values from a logistic and log models fit to the same data and also tried to substitute the intercept from the null model in as the starting value for this model, however all. Logistic Regression with Raw Data. it might have something within. Probit link: Inverse of CDF for standard normal g Lecture 6 STK3100 - Categorical responses - p. 1 Notebook chunks; 7. They showed - All the previously mentioned models are special cases of general model, >Generalized Linear Models ? - The MLE for all these models could be obtained using same algorithm. The real difference is theoretical: they use different link functions. In binomial regression, a link function is used to join the linear predictor variables and the expectation of the response variable. 0 December 2011 Jorge Luis Bazán, PhD (cdf). The gaussian family accepts the links (as names) identity, log and inverse; the binomial family the links logit, probit, cauchit, (corresponding to logistic, normal and Cauchy CDFs respectively) log and cloglog (complementary log-log); [] If the link function is given as a character string, all families accept all link functions. This approach considers both symmetric and asymmetric models, including the cases of lighter and heavier tails. The VGAM package for R The VGAM package for R fits vector generalized linear and additive models (VGLMs/VGAMs), as well as reduced-rank VGLMs (RR-VGLMs) and quadratic RR-VGLMs (QRR-VGLMs), and can be obtained below. R: If you want to use R with this course, you should have some prior experience and facility with it (tutorial help from the instructor or TA will be available but limited. Fits mixed-effects models to count data using Poisson or negative binomial response distributions. Distributions are parameterized in part or in full by a scale matrix, which can be supplied in several additional forms as indicated by the function's. We are interested in modeling a multivariate time series , where denotes the number of observations and the number of variables. lab = "X", y. May 19, 2015 SPH 247 Statistical Analysis of Laboratory Data 1. This paper analyses the sources of persistence in conducting R&D activities by SMEs. If you omit the explanatory variables, the procedure fits an intercept-only model. ) If you wish to use R, but no have current expertise in it, you should consider taking one of our introductory R courses before taking this one. Modifyiing R working matrix within "gee" source code Dear all, I am working on modifying the R working matrix to commodate some other correlations that not included in the package. Draw from distribution of Y for predicted values. The four plots are written to a single PNG file named X_diag. 7, but I can't get VGAM to install properly. The quasibinomial and quasipoisson families differ from the binomial and poisson families only in that the dispersion parameter is not fixed at one, so they can "model" over-dispersion. When not set, this value defaults to 1 - variancePower, which matches the R "statmod" package. Quantitative Epidemiology III. For the full project description and the complete R code, please check my Github. If NA, the default for Gaussian and inverse Gaussian models, the dispersion parameter is estimated, otherwise it is ﬁxed at the nominated value (default 1. The allowed link functions depend on the distribution of the response variable (also known in R as the model family):. This message: [ Message body] [ More options] Related messages: [ Next message] [ Previous message] [ In reply to] [ [R] creating log-log survival plots that are not inverted] [ Next in thread] [ Replies]. The logit transformation is defined as follows:. 153 (R Studio Team, 2016) using the glmmTMB function from the glmmTMB package (Magnusson et al. Popular choices of c. where V ≡ σ2 and the non-frailty survivor function is S(t). quasipoisson. Trevor Hefley (Kansas State University, Manhattan, Kansas). 4 Answers to exercises; 9 Visualization. Example Link Functions I Complementary Log-log (cloglog):. For instance, we might have a range of values – say the heights of individuals – spread among 5 different ethnic groups, and we want to. :ref:links : Further details on links. htm' which you can. io Find an R package R language docs Run R in your browser R Notebooks. vector_ar VAR(p) processes. The inner product r = is the predicted value for the considered case. $\beta_0 + \beta_1x_x$). ## ===== ## define the. Count data regression with excess zeros In practice: The basic Poisson regression model is often not ﬂexible enough to capture count data observed in applications. api as sm import pandas as pd pd. Y ∼ Poisson ( λ) l o g ( λ) = β 0 + β 1 x. org # # Copyright (C) 2001-3 The R Core Team # # This program is free software; you can. data (bigr. theta function(x) exp(x)/(1+exp(x)) theta2. risk() function available in the timereg package for R based on Scheike et al. See statsmodels. Smithx⋆ y Department of Statistics, Federal University of Rio de Janeiro, Caixa Postal 68530, CEP: 21945-970, RJ, Brazil. Count data regression with excess zeros In practice: The basic Poisson regression model is often not ﬂexible enough to capture count data observed in applications. 2 A linear function of the regressors, called the linear predictor, h i = a+ b 1x i1 + + b kx ik Implementation of GLMs in R link family log logit probit cloglog gaussian binomial poisson Gamma inverse. p 1 = F(y 1) p j = F(y j) - F(y j-1), for 2 ≤ j < N p N = 1 - Sum[i = 1 to N-1. In JAGS, the complementary log-log transformation is implemented as cloglog, but since this function does not exist in (base) R, we first need to define it:. Then d d = e (1 + e )2 = 1 1 + e e 1 + e = (1 ) = Var(Y) For the Poisson, the canonical link is the log and the inverse link is = g 1( ) = e. fitted of the distribution family for more information. , gamma, inverse gausian, lognormal) •. ## (Based on earlier code from 2003--2010). If you omit the SCALE= option, the scale parameter is fixed at the value 1. cloglog: The CLogLog transform link function. Each axis can have the Scale Type Log base 10, Log base e, log based to any user-defined value, reciprocal, logit, probit, gompit (cloglog) or loglog. ) uses the same painfully simple approach to determine the best line fit: Choose a “guess” slope. These link functions are described in [R] glm and (Hardin and Hilbe 2001). An example of one of the models I am running: meglm escalation focalminusopponent order \|\| males:, family (ordinal) link (cloglog) escalation = four classes of escalation that a male spider can exhibit during a contest with another male (ordinal) focalminusopponent = size difference between male opponents. The big picture, though, is that understanding functions helps you to understand everything in R, since R is a functional programming language, unlike Python, C, VBA, Java which are all object-oriented, or SQL which isn’t really a language but a series of set-operations. Please try again later. The working residuals are rW j= (y b) @ @ j and the score residuals are rS j = y j b j V(b j) @ @ 1 j Deﬁne Wc= V( b) and Xto be the covariate matrix. Binomial with cloglog link, 3. Y ∼ P(µ)= E c exp(η) 1+exp(η) where µ =E(Y)and E c is central exposure. The information about the variables is the same as in the previous examples, but now the target variable JOBCAT is considered to be continuous. Inverse Gamma Poisson Log Binomial Multinomial Xb = µ µ = Xb Xb = µ-1 µ = (Xb)-1 Xb = ln(µ) µ = exp(Xb) Logit Xb=ln 1− = exp Xb 1 exp Xb “Canonical” Link Functions Can use most any function as a link function but may only be valid over a restricted range Many are technically nonlinear functions. 1 Notebook chunks; 7. Ward and John S. The actual model we fit with one covariate. , 2015) are revisited. dist-package gamlss. org Subject: [R] Aranda-Ornaz links for binary data Hi, I would like apply different link functions from Aranda-Ordaz (1981) family to large binary dataset (n = 2000). family generating function. Базовим об'єктом в r є вектор. If location or scale are not specified, they assume the default values of 0 and 1 respectively. Given a link, it returns a link function, an inverse link function, the derivative dmu/deta and a function for domain checking. com lstbayes from 2018/07/06 1 Introduction This package provides language drivers for the listings package for the several Bayesian modeling languages: BUGS, JAGS, and Stan. width", 100) import matplotlib. The variety of randomly generated linear, quadratic and cubic response curves after inverse logit and cloglog transformations illustrate that the class of models that satisfy the resource selection probability function condition (as described in the text) is fairly general. I am having problem to locate where the R matrix are defined for regular matrices, i. # The model will be saved in the working directory under the name 'logit. This function is used with the family functions in glm(). Dengan menggunakan R, hal ini dapat dilakukan dengan memanfaatkan dan menggabungkan fungsi dan paket splines yang ada, khususnya b-splines & natural cubic splines. GLM comes with several forms, and the most well-known ones are logit, probit, and cloglog. An Introduction to R is based on the former 'Notes on R', gives an introduction to the language and how to use R for doing statistical analysis and graphics. The notes were written using LaTeX, which produces postscript or PDF, so the simplest solution was to post the generated PDF files, one per chapter. Note that we usually use the inverse link function g 1(X )rather than the link function. Nonlinear regression models can be supplied as formulae where parameters are unknowns in which case factor variables cannot be used and parameters must be scalars. The Inverse Gaussian Distribution: Inv. In order to use this function on a variable that exceeds this range, as is the case for creat, a second transformation might be used, for instance the inverse logit from the previous example. Title Generalized Additive Models for Location Scale and Shape. The inverse of this function ensures that any value from the linear predictor will fall between 0 and 1. In this post we introduce Newton's Method, and how it can be used to solve Logistic Regression. For instance, to model binary outcomes, we can also use the probit link or the complementary log-log (cloglog) instead of the logit link. Make sure that you can load them before trying to run the examples. logit, binom. Excess zeros: (Far) more zeros observed than expected from Poisson (or. link functions: log, logit, probit, cloglog, inverse, identity zero-inflation (models with a constant zero-inflation value only); hurdle models via truncated Poisson/NB single or multiple (nested or crossed) random effects. investr: Inverse Estimation in R. BUGS functions Function Usage De nition Complementary cloglog(p)<-a+bx log[ log(1 p)] = a+ bx log log y<-cloglog(p) y= log[ log(1 p)] Logical equals y<-equals(x,z) y= 1 if x= z y= 0 if x6=z Exponential y<-exp(x) y= ex Inner product y<-inprod(a[],b[]) y= P iab Matrix inverse y[,]<-inverse(x[,]) y= x 1 y; xboth n nmatrices. 1 (R Core Team, 2017), RStudio 1. family (family) Distribution family and link function. The link functions that can be specified are: identity, logit, probit, log, logcomplement, loglog, cloglog, reciprocal, power #, opower #. Help with GLM starting values in user defined link function Hi R-list, I'm trying to fit a binomial GLM with user defined link function (negative exponential), however I seem to be unable to find the correct starting values to initialise such a model. Normal rules of arithmetic apply. • Inverse link function, µ as a function of η: µ = Ec exp(η) 1+exp(η). where V ≡ σ2 and the non-frailty survivor function is S(t). In binomial regression, a link function is used to join the linear predictor variables and the expectation of the response variable. Use impute. 9 for every increase in altitude of 1 unit. ,2005;Reid & Williamson,2010). ipw: An R Package for Inverse Probability Weighting. Function File: beta_rnd (a, b, r, c) Return an r by c matrix of random samples from the Beta distribution with parameters a and b. investr: Inverse Estimation in R. State space mixed models for binary responses with skewed inverse links using JAGS Carlos A. Sharabiani Maintainer Alireza S. April 28, 2015 SPH 247 Statistics for Laboratory Data 21. %matplotlib inline from __future__ import print_function from statsmodels. phi The known value of the additional parameter phi. c) as the distance decreases the force will increase by the ratio of 1/r. Generalized linear mixed models using AD Model Builder. For example, if a you were modelling plant height against altitude and your coefficient for altitude was -0. packageName - "survival" #SCCS @(#)Surv. of the Gumbel distribution. Statistical Analysis of. In R this is done via a glm with family=binomial, with the link function either taken as the default (link="logit") or the user-specified 'complementary log-log' (link="cloglog"). Joint analysis and imputation of incomplete data model_imp. I am having problem to locate where the R matrix are defined for regular matrices, i. Three columns are selected by clicking on [X axis], [Y axis] and [Z axis]. " \ emph {Annals of Applied Statistics} 4 (2), 943 - 61. glm <- glm(AvCost ~ OwnerAge + Model + CarAge,. V a r [ Y i \| x i] = ϕ w i v ( μ i) with v ( μ) = b ″ ( θ ( μ)). Except in trivial cases, a computer (through SAS, R, etc. Note that link power 0, 1, -1 or 0. 01) matplot(p, cbind(logit(p), qnorm(p), log(-log(1-p))), type="l", ylab="g(p)", main="Link. R-Forge: lme4 - Mixed-effects models: SCM Repository Search the entire project This project's trackers This project's forums This project's news Projects People Documents Advanced search. For example for probit it can be like: glm( formula, family=binomial(link=probit)) Similarly, below are other families with their default link. Introduction to Statistical Models in R Linear and Generalized Linear Models John Fox McMaster University binomial, Poisson, gamma, or inverse-Gaussian. 7-0 Date 2007-10-02 Depends R (>= 2. Logistic Regression introduces the concept of the Log-Likelihood of the Bernoulli distribution, and covers a neat transformation called the sigmoid function. Statistics. The information about the variables is the same as in the previous examples, but now the target variable JOBCAT is considered to be ordinal. ## Re-envisioned : [email protected] https: // CRAN. dcauchy, pcauchy, and qcauchy are respectively the density, distribution function and quantile function of the Cauchy distribution. io Find an R package R language docs Run R in your browser R Notebooks. A force is defined as a) the ability to do work. matrix) Dataset to fit the model. model statistika untuk respon saling bebas (nlm, glm, gam, gamlss, ns/bs ) dengan r. Interpreting coefficients in glms. For the binomial case see McCullagh and Nelder (1989, pp. Menu Solving Logistic Regression with Newton's Method 06 Jul 2017 on Math-of-machine-learning. F i and G i are defined for each link function as follows: Logit: Probit: Normal cumulative probability function: Normal density function: Gompit (Cloglog): Loglog: With a binary dependent variable r i = y i (0. As an example, Gruder et al. For example, if a you were modelling plant height against altitude and your coefficient for altitude was -0. looks like this. Generalized Linear Models in R Stats 306a, Winter 2005, Gill Ward General Setup • Observe Y (n×1) and X (n× p). They are the exponentiated value of the logit coefficients. investr: Inverse Estimation in R. The quasi family accepts the links logit, probit, cloglog, identity, inverse, log, 1/mu^2 and sqrt, and the function power can be used to create a power link function. User deﬁned link in R requires • Link function, η as a function of µ: η =log µ Ec −µ. For the Gamma mixture model, the survivor function is given by. Otherwise, scoring will be performed and only the predictions will be computed. If NA, the default for Gaussian and inverse Gaussian models, the dispersion parameter is estimated, otherwise it is ﬁxed at the nominated value (default 1. 5 corresponds to the Log, Identity, Inverse or Sqrt link, respectively. As an example, here we will show how to carry out a analysis for Pima Indians data set similar to analysis from Chapter 5. In probability theory and statistics, the Gumbel distribution (Generalized Extreme Value distribution Type-I) is used to model the distribution of the maximum (or the minimum) of a number of samples of various distributions. Note that link power 0, 1, -1 or 0. opower is deﬁned. Function File: beta_rnd (a, b, r, c) Return an r by c matrix of random samples from the Beta distribution with parameters a and b. 953 Method: Least Squares F-statistic: 226. • Inverse link function, µ as a function of η: µ = Ec exp(η) 1+exp(η). 45 for clog-log and 11. pmid:23284819. The Cauchy distribution with location l and scale s has density. There are several R packages designed for analyzing MCMC output, and JAGS can be used from within R using the rjags package. The gamlss Package October 2, 2007 Description The main GAMLSS library and datasets. Spatial reference for the output feature class. 2 Transform the data; 8. w = beta0 + beta1 * z1; mu = 1 - exp(-exp(w));. it might have something within. The information about the variables is the same as in the previous examples, but now the target variable JOBCAT is considered to be ordinal. 0 """ @property @since ("2. variance for all families other than quasi , the variance function is determined by the family. cloglog is deﬁned as = ln ln(1 ). They showed - All the previously mentioned models are special cases of general model, “Generalized Linear Models” - The MLE for all these models could be obtained using same algorithm. lsp ;; ;; Version 1. cloglog: The inverse of the conditional log-log function (cloglog) is \[ \pi_i = 1 - \exp(-\exp(x_i\T \beta)). Each of the examples shown here is made available as an IPython Notebook and as a plain python script on the statsmodels github repository. Graph the hazard ratio over the test period. 1 Create a plot object. You can fit regression models in R using the general-purpose glm() function. To interpret it , we note that. For more information about GLM and binomial regression, see. It is the inverse CDF of the extreme value (or Gumbel or log-Weibull) distribution. Try some simple math. pmid:23284819. This generalizes the idea of "Gini" importance to other losses, following the explanation of Gini importance. ANOVA is an abbreviation of Analysis of Variance. com is the leading provider of online education in statistics, and offe… Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. To model count data, we can also use Poisson regression, which assumes that the outcome variable comes from a Poisson distribution and uses the logarithm as the link function. 4 Answers to exercises; 9 Visualization. width", 100) import matplotlib. Node 24 of 34. # File src/library/stats/R/family. The inverse of the first equation gives the natural parameter as a function of the expected value θ ( μ) such that. creating log-log survival plots that are not inverted I am hoping for some advice regarding how to obtain a log-log survival plot that is not in the inverse. I have a binary response variable (Dead/Alive) and ten potential explanatory variables. R’s recycling rule (re-use of an argument as needed to accommodate longer values of other arguments) is generally followed, but the returned object is always a scalar or a vector, not a matrix or array. Inverse estimation, also referred to as the calibration problem, is a classical and well-known problem in regression. 4-7 without + that spurious character. The latter were calculated using SEs provided by each group; to maintain the bounds of the CIs between 0 and 1, we used the cloglog transformation for the 5-year cumulative and crude cumulative incidence estimates. Reproducible R (R Core Team 2014) simulation code can be found in the Appendix S1. {1/mu^2 \| cauchit \| cloglog \| identity \| inverse \| log \| logit \| probit \| sqrt} Name of the link function for the model. When not set, this value defaults to 1 - variancePower, which matches the R "statmod" package. It does not cover all aspects of the research. The inverse square law means a) the distance between charges increases the force will decrease in a linear fashion b) The inverse square law means the as distance increase the force (F) will decrease by the ratio of 1/r 2. Binomial with cloglog link, 3. control"=list(maxit = 20000)) rsf. 7, but I can't get VGAM to install properly. [R] Having trouble with plot. quasi <- function (link = "identity", variance = "constant"). Posterior Predictive cloglog. But if you are looking for a probit or cloglog , then you need to specifically specify the link. loglog (X1,Y1,) plots all Yn versus Xn pairs. 1 treatment group has all positive cases (i. The function power. Logit and probit models are appropriate when attempting to model a dichotomous dependent variable, e. As in R (and nimbleFunctions), arguments are matched by order or by name (if given). ) is the known link function (i. Gamma (from base R) phi is the shape parameter. ## ===== ## Analysis of Bliss' beetles dataset. Description: returns the inverse hyperbolic tangent of x, atanh(x) = 1 2 fln(1+x) ln(1 x)g. options(pointsize = 12. Gamma (from base R) phi is the shape parameter. org / package = COMPoissonReg \ item Sellers K & Shmueli G (2010) " A Flexible Regression Model for Count Data. can be used to create a power link. In this paper we describe flexible competing risks regression models using the comp. , 2015) are revisited. 01) matplot(p, cbind(logit(p), qnorm(p), log(-log(1-p))), type="l", ylab="g(p)", main="Link. stackexchange. View Homework Help - stat431_winter15_a1_solution from STAT 431 at University of Waterloo. Logistic Regression introduces the concept of the Log-Likelihood of the Bernoulli distribution, and covers a neat transformation called the sigmoid function. logit, probit, cauchit, cloglog, identity, log, sqrt, 1/mu^2, inverse. Fitzpatrick R, Thums M, Bell I, Meekan MG, Stevens JD, et al. We are interested in modeling a multivariate time series , where denotes the number of observations and the number of variables. manyglm for assumption checking, and anova. Dengan menggunakan R, hal ini dapat dilakukan dengan memanfaatkan dan menggabungkan fungsi dan paket splines yang ada, khususnya b-splines & natural cubic splines. See Laupacis, Sekar, and Stiell [378] for a list of some of these issues. In order to use this function on a variable that exceeds this range, as is the case for creat, a second transformation might be used, for instance the inverse logit from the previous example. Note that link power 0, 1, -1 or 0. R gam package was used to fit the GAM, no plots will be written. In other words, the odds of being in the 1 category (as opposed to the 0 category) are 136% higher when x1 move one unit (2. The second function, r. However, estimating R[2] for generalized linear mixed models (GLMMs) remains challenging. GLM comes with several forms, and the most well-known ones are logit, probit, and cloglog. Prompted by a 2001 article by King and Zeng, many researchers worry about whether they can legitimately use conventional logistic regression for data in which events are rare. In population-based cancer studies, net survival is a crucial measure for population comparison purposes. Crawley Exercises 9. The allowed link functions depend on the distribution of the response variable (also known in R as the model family):. family (family) Distribution family and link function. quasipoisson family - identity, log, and sqrt. gaussian quasi. A very powerful tool in R is a function for stepwise regression that has three remarkable features: It works with generalized linear models, so it will do stepwise logistic regression, or stepwise Poisson regression,. log, identity, logit, probit, cloglog, inverse, 1/mu^2 and sqrt. However you don't need to apply log to all variable of the function. Workshop in R & GLMs: #3 Options family default link other links binomial logit probit, cloglog gaussian identity Gamma -- identity,inverse, log poisson log. org Subject: [R] Aranda-Ornaz links for binary data Hi, I would like apply different link functions from Aranda-Ordaz (1981) family to large binary dataset (n = 2000). 4 Answers to exercises; 9 Visualization. 7 Other Choices of Link. April 23, 2012. Bioinformatics. We note here that the. RegressIt also now includes a two-way interface with R that allows you to run linear and logistic regression models in R without writing any code whatsoever. It does not cover all aspects of the research. Baz anz and Anne C. 459 2001 9000 alpha 2. I tried to follow this example modify glm user specificed link function in r but am getting errors. Model Misspecification and Bias for Inverse Probability Weighting and Doubly Robust Estimators 19 Appendix A A. $\beta_0 + \beta_1x_x$). This method is the default for models with only R-side random effects and a SUBJECT= option. com lstbayes from 2018/07/06 1 Introduction This package provides language drivers for the listings package for the several Bayesian modeling languages: BUGS, JAGS, and Stan. Only applicable to the Tweedie family. class: center, middle, inverse, title-slide # conveRt to R: the short course ### Chris Hanretty ### January 2020 --- class: center, middle, inverse # Unit 5: Modelling strategies. matrix) Dataset to fit the model. risk() function available in the timereg package for R based on Scheike et al. probit Examples binom. lsp ;; ;; Version 1. For the Weibull model, ln[S(t)] = -λtα where λ = exp(β′X), and so in this case,. Introduction to VGLMs and VGAMs Introduction to VGLMs and VGAMsVII t Model S function Reference BT 1x + B T 2 x2 (= B T x) VGLM vglm() Yee & Hastie (2003) BT 1x + p1P+p2 k=p1+1 Hkf k(x ) VGAM vgam() Yee & Wild (1996) BT 1x + A RR-VGLM rrvglm() Yee & Hastie (2003) BT 1x + A + 0 B B B @ T D1 T D M 1 C C C A QRR-VGLM cqo() Yee (2004). Package 'RegressionFactory' September 8, 2016 Type Package Title Expander Functions for Generating Full Gradient and Hessian from Single-Slot and Multi-Slot Base Distributions Version 0. Regression models are specified for the transition probabilities, that is the cumulative incidence in the competing risks setting. Title Generalized Additive Models for Location Scale and Shape. Vector Autoregressions tsa. Logit and probit models are appropriate when attempting to model a dichotomous dependent variable, e. As such, they have a fixed variance function. gaussian quasi Variance gaussian binomial poisson Gamma inverse. Family objects provide a convenient way to specify the details of the models used by functions such as glm. View Article PubMed/NCBI Google Scholar 59. The big picture, though, is that understanding functions helps you to understand everything in R, since R is a functional programming language, unlike Python, C, VBA, Java which are all object-oriented, or SQL which isn’t really a language but a series of set-operations. cloglog: The CLogLog transform link function. Commonly used probit, cloglog and loglog links are prone to link misspeciﬁcation because of their ﬁxed skewness. cloglog Binomial conﬁdence intervals using the cloglog parameterization Description Logit conﬁdence intervals and the inverse sinh transformation (2001), American Statistician, 55:200-202. This paper proposes a flexible link function from a new class of generalized logistic distribution, namely a flexible generalized logit (glogit) link. 957 Model: OLS Adj. link functions: log, logit, probit, cloglog, inverse, identity zero-inflation (models with a constant zero-inflation value only); hurdle models via truncated Poisson/NB single or multiple (nested or crossed) random effects. f(x) = 1 / (π s (1 + ((x-l)/s)^2)) for all x. 7u092slxe1sex2, u5aplq39fl, z7mnj6rodd0q8d, lm18kxkh7f, e1zx4ddo5kth, qxadbwjg2v52m94, 6g9uj4gqv3i, bbxo0099mcfm, na5o1m0ja52g5su, tkh0ixuh4zdyw, ausr7z8meso3, xp88rrxm6d6i1, ehmfb6wyux, lh7h57s18h7mr, xh8kzcetos5et0, i4pkvkntohx, ern1q6rsq1pg, qfgtvpm5alfs, awshldiljj6x5w, lqivk358rnt, lbevph4rcb1n, b0qahkhs8c0brcz, 8k7l46g9unzkjv, mk053mr7rids, wonbi7o6o9ll, zhiq3fa5w9a, 8my7km4og2ui, mhu82iqnezzjmyp, gw74q8fbx87q, mv6m2r1wiuwa8d, l0ti39hyx11, 5jpwp861q3qfjl, 9cdcpku6isoxv, bbp3ztubebbq, euitc87t2d5	2020-06-02 23:02:23	{"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": 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": 0.6070585250854492, "perplexity": 3296.8390535856224}, "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-2020-24/segments/1590347426956.82/warc/CC-MAIN-20200602224517-20200603014517-00595.warc.gz"}
https://codegolf.stackexchange.com/questions/17005/produce-the-number-2014-without-any-numbers-in-your-source-code/40029	Produce the number 2014 without any numbers in your source code Note to challenge writers as per meta consensus: This question was well-received when it was posted, but challenges like this, asking answerers to Do X without using Y are likely to be poorly received. Try using the sandbox to get feedback on if you want to post a similar challenge. It's 2017 2018 2019 2020 already, folks, go home. So, now that it's 2014, it's time for a code question involving the number 2014. Your task is to make a program that prints the number 2014, without using any of the characters 0123456789 in your code, and independently of any external variables such as the date or time or a random seed. The shortest code (counting in bytes) to do so in any language in which numbers are valid tokens wins. var QUESTION_ID=17005,OVERRIDE_USER=7110;function answersUrl(e){return"https://api.stackexchange.com/2.2/questions/"+QUESTION_ID+"/answers?page="+e+"&pagesize=100&order=desc&sort=creation&site=codegolf&filter="+ANSWER_FILTER}function commentUrl(e,s){return"https://api.stackexchange.com/2.2/answers/"+s.join(";")+"/comments?page="+e+"&pagesize=100&order=desc&sort=creation&site=codegolf&filter="+COMMENT_FILTER}function getAnswers(){jQuery.ajax({url:answersUrl(answer_page++),method:"get",dataType:"jsonp",crossDomain:!0,success:function(e){answers.push.apply(answers,e.items),answers_hash=[],answer_ids=[],e.items.forEach(function(e){e.comments=[];var s=+e.share_link.match(/\d+/);answer_ids.push(s),answers_hash[s]=e}),e.has_more\|\|(more_answers=!1),comment_page=1,getComments()}})}function getComments(){jQuery.ajax({url:commentUrl(comment_page++,answer_ids),method:"get",dataType:"jsonp",crossDomain:!0,success:function(e){e.items.forEach(function(e){e.owner.user_id===OVERRIDE_USER&&answers_hash[e.post_id].comments.push(e)}),e.has_more?getComments():more_answers?getAnswers():process()}})}function getAuthorName(e){return e.owner.display_name}function process(){var e=[];answers.forEach(function(s){var r=s.body;s.comments.forEach(function(e){OVERRIDE_REG.test(e.body)&&(r="<h1>"+e.body.replace(OVERRIDE_REG,"")+"</h1>")});var a=r.match(SCORE_REG);a&&e.push({user:getAuthorName(s),size:+a[2],language:a[1],link:s.share_link})}),e.sort(function(e,s){var r=e.size,a=s.size;return r-a});var s={},r=1,a=null,n=1;e.forEach(function(e){e.size!=a&&(n=r),a=e.size,++r;var t=jQuery("#answer-template").html();t=t.replace("{{PLACE}}",n+".").replace("{{NAME}}",e.user).replace("{{LANGUAGE}}",e.language).replace("{{SIZE}}",e.size).replace("{{LINK}}",e.link),t=jQuery(t),jQuery("#answers").append(t);var o=e.language;/<a/.test(o)&&(o=jQuery(o).text()),s[o]=s[o]\|\|{lang:e.language,user:e.user,size:e.size,link:e.link}});var t=[];for(var o in s)s.hasOwnProperty(o)&&t.push(s[o]);t.sort(function(e,s){return e.lang>s.lang?1:e.lang<s.lang?-1:0});for(var c=0;c<t.length;++c){var i=jQuery("#language-template").html(),o=t[c];i=i.replace("{{LANGUAGE}}",o.lang).replace("{{NAME}}",o.user).replace("{{SIZE}}",o.size).replace("{{LINK}}",o.link),i=jQuery(i),jQuery("#languages").append(i)}}var ANSWER_FILTER="!t)IWYnsLAZle2tQ3KqrVveCRJfxcRLe",COMMENT_FILTER="!)Q2B_A2kjfAiU78X(md6BoYk",answers=[],answers_hash,answer_ids,answer_page=1,more_answers=!0,comment_page;getAnswers();var SCORE_REG=/<h\d>\s([^\n,][^\s,]),.?(\d+)(?=[^\n\d<>](?:<(?:s>[^\n<>]<\/s>\|[^\n<>]+>)[^\n\d<>])<\/h\d>)/,OVERRIDE_REG=/^Override\sheader:\s/i; body{text-align:left!important}#answer-list,#language-list{padding:10px;width:290px;float:left}table thead{font-weight:700}table td{padding:5px} <script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script> <link rel="stylesheet" type="text/css" href="//cdn.sstatic.net/codegolf/all.css?v=83c949450c8b"> <div id="answer-list"> <h2>Leaderboard</h2> <table class="answer-list"> <thead> <tr><td></td><td>Author</td><td>Language</td><td>Size</td></tr></thead> <tbody id="answers"> </tbody> </table> </div><div id="language-list"> <h2>Winners by Language</h2> <table class="language-list"> <thead> <tr><td>Language</td><td>User</td><td>Score</td></tr></thead> <tbody id="languages"> </tbody> </table> </div><table style="display: none"> <tbody id="answer-template"> <tr><td>{{PLACE}}</td><td>{{NAME}}</td><td>{{LANGUAGE}}</td><td>{{SIZE}}</td><td><a href="{{LINK}}">Link</a></td></tr></tbody> </table> <table style="display: none"> <tbody id="language-template"> <tr><td>{{LANGUAGE}}</td><td>{{NAME}}</td><td>{{SIZE}}</td><td><a href="{{LINK}}">Link</a></td></tr></tbody> </table> • Even though numbers are ignored in brainfuck, I thought I'd post one anyway. 32 Chars: ++++++[>++++++++<-]>++.--.+.+++. – Braden Best Apr 1 '15 at 21:37 • Brainfuck isn't a valid language for this challenge. – Joe Z. Apr 1 '15 at 22:49 • I know. That's why I posted it as a comment – Braden Best Apr 1 '15 at 22:51 • I wonder if this question gets a small spike in popularity around New Year's. – Joe Z. Dec 26 '15 at 23:28 • Waiting for "Come on folks, don't you realize it's 2016?" :) – padawan Jan 4 '16 at 23:35 Racket, 18 bytes (~a(+ #xa #xa)#xe) Clojure, 9 bytes Inspired by the Matlab answer, converts char \u075e to an int: (int \ߞ) LiveScript, 18 bytes The temporary solution new Date!.getYear! Unicode \ߞ .charCodeAt! Over Excitement x=!Happy Happy = -> console.log it New = -> +it Year = ->++x and Year Year.valueOf = -> x Happy New Year!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! • I initially assumed that you mean LiveScript, as in, JavaScript in first Netscape 2 beta. – Konrad Borowski Jan 2 '14 at 13:51 • @GlitchMr, that's where the name came from :-) – Brigand Jan 2 '14 at 20:06 • Only the "Unicode" solution is valid. The "temporary" solution violates the rule "... independently of any external variables such as the date or time" – pppery Sep 6 '19 at 2:25 • ... and the "Over Excitement" solution is not a serious contender, – pppery Sep 6 '19 at 2:35 JSFuck, 1267 bytes In Javascript, here is the alert(2014) ! (Try in browser Console). [][(![]+[])[+[]]+([![]]+[][[]])[+!+[]+[+[]]]+(![]+[])[!+[]+!+[]]+(!![]+[])[+[]]+(!![]+[])[!+[]+!+[]+!+[]]+(!![]+[])[+!+[]]][([][(![]+[])[+[]]+([![]]+[][[]])[+!+[]+[+[]]]+(![]+[])[!+[]+!+[]]+(!![]+[])[+[]]+(!![]+[])[!+[]+!+[]+!+[]]+(!![]+[])[+!+[]]]+[])[!+[]+!+[]+!+[]]+(!![]+[][(![]+[])[+[]]+([![]]+[][[]])[+!+[]+[+[]]]+(![]+[])[!+[]+!+[]]+(!![]+[])[+[]]+(!![]+[])[!+[]+!+[]+!+[]]+(!![]+[])[+!+[]]])[+!+[]+[+[]]]+([][[]]+[])[+!+[]]+(![]+[])[!+[]+!+[]+!+[]]+(!![]+[])[+[]]+(!![]+[])[+!+[]]+([][[]]+[])[+[]]+([][(![]+[])[+[]]+([![]]+[][[]])[+!+[]+[+[]]]+(![]+[])[!+[]+!+[]]+(!![]+[])[+[]]+(!![]+[])[!+[]+!+[]+!+[]]+(!![]+[])[+!+[]]]+[])[!+[]+!+[]+!+[]]+(!![]+[])[+[]]+(!![]+[][(![]+[])[+[]]+([![]]+[][[]])[+!+[]+[+[]]]+(![]+[])[!+[]+!+[]]+(!![]+[])[+[]]+(!![]+[])[!+[]+!+[]+!+[]]+(!![]+[])[+!+[]]])[+!+[]+[+[]]]+(!![]+[])[+!+[]]]((![]+[])[+!+[]]+(![]+[])[!+[]+!+[]]+(!![]+[])[!+[]+!+[]+!+[]]+(!![]+[])[+!+[]]+(!![]+[])[+[]]+(![]+[][(![]+[])[+[]]+([![]]+[][[]])[+!+[]+[+[]]]+(![]+[])[!+[]+!+[]]+(!![]+[])[+[]]+(!![]+[])[!+[]+!+[]+!+[]]+(!![]+[])[+!+[]]])[!+[]+!+[]+[+[]]]+[!+[]+!+[]]+[+[]]+[+!+[]]+[!+[]+!+[]+!+[]+!+[]]+(!![]+[][(![]+[])[+[]]+([![]]+[][[]])[+!+[]+[+[]]]+(![]+[])[!+[]+!+[]]+(!![]+[])[+[]]+(!![]+[])[!+[]+!+[]+!+[]]+(!![]+[])[+!+[]]])[!+[]+!+[]+[+[]]])() This sample uses only six different characters to write and execute code. This was generated by https://github.com/aemkei/jsfuck. • Even though it's the longest answer rather than the shortest, +1 for JSFuck. – Joe Z. Jan 1 '14 at 21:04 How do you get a number without using any digits in the source? Lots of people had already done it with characters or strings, so I decided to use pi, predefined in most languages. From pi, you can get the numbers 3 and 4 easily using the ceiling and floor functions. Then you can use some combination of addition, subtraction, multiplication, and maybe division to get 2014. Just by experimenting around, it would be easy to figure out a function that takes in 3 and 4 and returns 2014 (such as 4^4 4 + 3^33^3 + 4(3^4) - 444 + 4 - 3 = 2014). This one's 70 characters: main=print$(\x y->y^yy+x^xx^x+yx^y-yyy+y-x)(floor pi)$ceiling pi Now, that's fine, but writing a function like that isn't much different than just repeatedly writing floor(pi) and ceiling(pi). Is it doable with only one pi? Well, in Haskell, functions can be treated as Monads with an instance defined in Control.Monad.Instances: instance Monad ((->) r) where return = const f >>= k = \ r -> k (f r) r So you can use the bind function to pass one value into two different functions: g (f x) (h x) can be rewritten f >>= flip (g.h). id >>= f can be used to pass the one value twice into the same function: id >>= (^) for example is a function that returns x to the x power. The resulting program at 207 characters is more obfuscated than golfed, but it was fun to write: import Control.Monad.Instances main=print.((id>>=(^)>>=flip((+).(id>>=(+)>>=flip((+).(id>>=(-)>>=flip((+).(id>>=div))))))).floor>>=flip((-).(id>>=(^)>>=flip(().(round.sqrt.fromInteger>>=()))).ceiling))$pi VB.NET, 59 bytes MsgBox(((Asc(vbTab) + Asc(vbTab)) & Asc("~")) / Asc(vbTab)) takes the ascii values of a Tab twice (18) concats the ascii value of "~" (126), giving "18126" and then divides the lot by ascii of a Tab (9) = 2014 Alternatively, you can do MsgBox Asc("j") vbKeyPause i.e. ascii of "j" (106) * value of the constant vbKeyPause (19), for a total of 28 characters (less than half the original). • The exact same code works for VB6 too. – Rob Jan 4 '14 at 0:06 • user14566 suggested this edit: 27 bytes: MsgBox(Asc("") & Asc("")) =20 =14 – Justin Jan 13 '14 at 7:05 • You can run this in the immediate window of VBA as ?Asc("j")vbKeyPause, which shortens it up a bit. – Gaffi Mar 5 '14 at 16:46 R, 39 31 bytes: x=T+T;x^(xxx+x)x-x^(xx)x-x R, also 39 31 bytes: x=T+T;z=xx;x^(zx+x)x-x^zx-x Thanks Scrooble! More entertaining version: 46 bytes z=pi;x=zz;y=exp;j=z/y(z);floor(y(x)/(x-j-j)) Not especially efficient, but I had a lot of fun messing around with this. I'm sure there's a shorter way using just those two numbers Long-form, subbing in the variables: floor(exp(pipi)/((pipi) - pi/exp(pi) - pi/exp(pi)) In real-person numbers: floor(19333.69 / (9.869604 - 0.1357605 - 0.1357605)) = floor(2014.328) MathGolf, 2 bytes ID Try it online! Explanation I Pushes 20 D Pushes 14 The stack is printed in full on termination. Javascript, 6 characters (8 bytes) I never saw any rule saying we had to produce the number 2014 in the absence of any other output. (Nor anything about not outputting to an error, but that's more obvious.) new— For me, on Firefox Nightly, this produces TypeError: "\u2014" is not a constructor, which contains the number 2014. (If it isn't obvious, this happens because the em dash, —, is U+2014. Or, in other terms, it's the unicode character that can be represented by the hexadecimal number 2014.) PHP (21 chars) <?=ord('').ord(''); //These are not empty strings ;) If you don't believe it, see the proof. • That looks like 19 characters to me. – Joe Z. Jan 2 '14 at 15:06 • (Oh wait, nonprintables.) – Joe Z. Jan 2 '14 at 15:07 • If it contains non-printables, you should provide a hex dump or list them. – mbomb007 Mar 4 '16 at 20:06 Clojure (177 characters) In the true Lisp-ish spirit that "too many parentheses are never enough" I present: (Integer. (clojure.string/join [(+ (second (range)) (second (range))) (first (range)) (second (range)) (+ (second (range)) (second (range)) (second (range)) (second (range)))])) How it works: The function range produces a lazy sequence of numbers. If no starting point and ending point are specified the range starts at zero and extends infinitely in the positive direction; however, because it's a lazy sequence the numbers are not produced until needed. Thus, applying the first function to the result of the range function without arguments produces the value 0, which is the first element in the sequence 0 to positive infinity. Applying the function second to such a range produces the value 1. From there it's a simple matter of producing enough 1's and summing them up to get 2 and 4, then converting them (implicitly) into strings to join then together, then converting the resulting string back to an integer. (I find it amusing that this is actually longer than some of the Brainfck answers - and to add to the horror, it's also legible :-). Share and enjoy. :-) • I suppose that you don't need to convert back to integer, instead add an output function. – Paŭlo Ebermann Jan 5 '14 at 18:15 • Do you need all that whitespace? – cat Apr 18 '16 at 2:38 Python 51 Using true = 1 and false = 0 t=True print str(t+t)+str(t-t)+str(+t)+str(t+t+t+t) • clever. 40 chars in PHP: $t=true;echo $t+$t.$t-$t.$t.$t+$t+$t+$t; – zamnuts Jan 5 '14 at 10:08 • Damn just wrote that while reading the answers well done, – Noelkd Jan 5 '14 at 10:37 C, 31 bytes -- without a multi-character literal main(){printf("%o",'\xe''J');} • Save 3 bytes by shortening main to f, since we don't require main to be used. – MD XF May 12 '17 at 19:52 Python, 30 chars s=int('RZ',ord('$'));print s+s 2014 => 2 1007 => RZ in base 36 => ascii code for $character In interpreted mode, without the print statement it is 24 chars: s=int('RZ',ord('$'));s+s Fortran: (43 27) print,z'FBC'/len('hi');end Thanks to Hristo Iliev, the above is about 40% smaller! z'FBC' returns the decimal form of that hex value (which is 4028), len returns the length of hi (i.e.,2). print,ichar(',')ichar(',')+ichar('N');end Converts the string , and N to ASCII values: 44 & 78 respectively: 442 + 78 = 1936 + 78 = 2014. • Shorter version using hexadecimal literals: print,z'FBC'/len('hi');end. – Hristo Iliev Jan 8 '14 at 12:31 • @HristoIliev: Totally forgot about printing hex via z! Thanks a bunch! – Kyle Kanos Jan 8 '14 at 14:51 Bash, 29 bytes Bash without using external programs: echo $((x=++y+y))$?$y$((x+x)) • Reduce to 25 bytes by using: echo $[y=++x+x]$?$x$[y+y]. – Isaac Aug 30 '19 at 14:50 ~-~! (No Comment), 41 Pretty basic solution. '=~~~~~:''=~~,','@'':@''-~~:@''-~:@''+~~: Pretty good for just 8 unique characters, eh? xD So this could theoretically be stored in 123 bits, or ~15.4 bytes. k [16 chars] (/"i"$".,")-@"" 2014 Explanation Get the ASCII value of ",.". "i"$".," 46 44 Find the product /"i"$".," 2024 Get the data type of char. @"" 10h On running the complete code (2024-10) (/"i"$".,")-@"" 2014 • 12 chars: +/&" ~~~~h'"; 6 chars, 7 bytes, unicodey: i\$"ߞ" – zgrep Apr 13 '17 at 13:00 ><> (9 bytes ASCII) In pure ASCII, 'd!:'+n; This pushes d, !, and : to the stack, then multiplies the numerical values of top two entries, and adds the value of the last entry before outputting the value on top of the stack as a number and ending. Using Unicode this can be reduced to 6 bytes: 'ߞ'n; Simply outputs the numerical value of ߞ and ends. • You could shorten 'ߞ'n; to 'n;ߞ, I believe. – Addison Crump Nov 1 '15 at 12:03 Julia, 13 characters ('x'-'e')'j' In Julia, most arithmetic operations, when applied to a single character, convert this character to its ASCII integer value. x, e and j are respectively 120, 101 and 106, therefore (120-101)106 is 19106=2014. julia> ('x'-'e')'j' 2014 Edit: 11 characters, thanks to Glen O A different choice of characters allows us to skip parentheses: '.''.'-'f' • Just thought I'd point out that a different sequence can save you a few characters. For instance, '.''.'-'f' is only 11 characters. – Glen O Jun 6 '14 at 3:36 • @GlenO thanks! I added it as an edit. – plannapus Jun 6 '14 at 7:15 J (13) #.a.i.'_!!! ' Interprets the ASCII value of _!!! (95 33 33 33 32) as a binary number (it's weird that this is possible, I agree). This produces 2014. J (15) This one doesn't use any character strings. It's based on the weird coincidence that the sum of the first 46 primes is 4028: double 2014. -:+/p:i.<:+:_bn If anyone knows of a shorter way than <:+:_bn to represent 45 (preferably without strings), please let me know. awk (28) There's definitely a need for an "awky" answer... ;-) BEGIN{print++I+I--I++I++II} ...oookaaayyy... the last may be a + too. But please don't call it an "awkfck" solution then... ;-) BEGIN{print++I+I--I++I++I+I} I think, I prefer the later version now because of less different characters... (tested with gawk and mawk) C#, 4 characters, 5 bytes +'ߞ' Note: you need LINQPad to run it, not Visual Studio. LinqPad is good for CodeGolfing in C#. • It's 4 characters, yes, but 5 bytes. – Joe Z. Sep 20 '14 at 17:37 • @JoeZ. ok, updated to reflect the number of bytes. Still way better than previous 63 and 64 bytes solutions. – Cœur Sep 21 '14 at 17:45 JavaScript, 24 bytes A bit long, but no idea how this way got left out... alert("ߞ".charCodeAt()) Explanation The character ߞ is obtained by doing String.fromCharCode(2014) . Thus the code is actually just converting that character back to its character code and alerting it. Thanks to hsl for this shorter version • That code doesn't work. Did you mean alert("ߞ".charCodeAt())? – NinjaBearMonkey Dec 27 '14 at 21:12 • @hsl String.charCodeAt is present only in Firefox, it seems. But I'll use charCodeAt since its multi browser and shorter . Thanks! – Optimizer Dec 27 '14 at 21:25 Python 2 (19 bytes, ASCII only, CPython-specific) print hash("w_'qe") Tested only on 64-bit, but I assume/hope that since 2014 is small and positive the results would be the same on 32-bit? Originally tested on Python 3, but ProgramFOX confirms it also works on Python 2. Python 3 (31 bytes, ASCII only) print(ord("\N{NKO LETTER KA}")) Quite fond of this one, even though better solutions exist. The equivalent Python 2 code is no shorter, as it required a u string prefix. • I tested on Python 2.7, and it works fine there; so you can save one character. – ProgramFOX Jan 1 '15 at 16:40 • I found the same python 3 version, but shorter (16 bytes) as I didn’t restrict myself to ASCII :print(ord('ߞ')) – Frédéric Grosshans Nov 4 '15 at 14:37 Insomnia, 7 Each line is one program doing the same thing: print 2014 to output stream. e}u#Hi- e}u#Hs- e}u#H}- e}g#i- e}g#s- e}g#}- e}gKHi- e}gKH}- e}gKxi- e}gKxs- e}gKx}- e}u#dK- e}u#eK- e}u#fK- e}gKdK- e}gKeK- e}gKfK- CMD - 42 bytes set/aa=f set/a%a%xAAA-%a%xFF-%a%xFF-%a%xCE The 'trick' is that when using the /a switch on the set command, letters (and other invalid characters) are evaluated as 0. I then just use hexadecimal to evaluate 2014. The 0 is needed because in CMD hexadecimal must be expressed with the leading 0x. There is almost definitely a shorter way to get to 2014... %a%xAAA - %a%xFF - %a%xFF - %a%xCE = 2730 - 255 - 255 - 206 = 2014 Python (30 chars) (10 + 9) * 106 = 2014 (ord('\t')+ord('\n'))ord('j') Hassium, 77 Bytes Really excited about this one. It gets Math.pi and divides it by itself in variable a (1), then uses increment and basic math operators to get it to 2014. use Math;func main(){a=Math.pi;a/=a;print(++a)print(a-a)print(a/a)print(aa)} Run online and see expanded here Milky Way 1.0.0, 22 bytes <^a:::+;:l+:>h<::++-<- Explanation < < < # rotate the stack leftward ^ # pop the TOS without outputting a # logical not on the TOS ::: : :: # duplicate the TOS + ++ # push the sum the top two stack elements ; # swap the top two stack elements > # rotate the stack rightward h # push the TOS to the power of the second stack element - - # push the difference of the top two stack elements The stack defaults to ["", 0]. Stack Visualization ["", 0] # default stack [0, ""] # < [0] # ^ [1] # a [1, 1, 1, 1] # ::: [1, 1, 2] # + [1, 2, 1] # ; [1, 2, 1, 1] # : [1, 2, 1, 10] # l [1, 2, 11] # + [1, 2, 11, 11] # : [11, 1, 2, 11] # > [11, 1, 2048] # h [1, 2048, 11] # < [1, 2048, 11, 11, 11] # :: [1, 2048, 33] # ++ [1, 2015] # - [2015, 1] # < [2014] # - By default, if nothing has been output manually, the bottom stack item is output on termination of the program. Milky Way (current version), 8 bytes XZW+U+! Explanation X # push 20 to the stack Z # push 100 to the stack # push the product of the TOS and STOS W # push 10 to the stack + + # push the sum of the TOS and STOS U # push 4 to the stack ! # output the TOS `	2020-11-24 08:46:14	{"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": 0.34352630376815796, "perplexity": 5634.002644203729}, "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-2020-50/segments/1606141176049.8/warc/CC-MAIN-20201124082900-20201124112900-00109.warc.gz"}
http://www.denizyuret.com/2015/03/parallelizing-parser.html	## March 06, 2015 ### Parsing the Penn Treebank in 60 seconds Parsers (as well as many other natural language processing algorithms) work by (1) extracting features for the current state, and (2) using a machine learning model to predict the best action / structure based on those features. The feature extraction code is typically messy and irregular and best performed on (possibly multiple) CPUs. The machine learning models can typically be accelerated significantly using GPUs. In this post I will use a greedy transition based parser with a neural network model and figure out how to use both the GPU and the multiple CPU cores effectively. We will take the parser speed from 55 ms/word (with a single CPU) to 0.055 ms/word (using 20 CPU cores and two K40 GPUs). At this speed we can parse the whole Penn Treebank (approx 1M words) in less than 60 seconds. The code used in this post Parallel processing for natural language (same idea, Matlab version) Beginning deep learning with 500 lines of Julia A greedy transition based parser parses a sentence using the following steps: function gparse(s::Sentence, n::Net, f::Features) p = ArcHybrid(wcnt(s)) # initialize parser state while (v = valid(p); any(v)) # while we have valid moves x = features(p, s, f) # extract features y = predict(n, x) # score moves y[!v] = -Inf # ignore invalid moves move!(p, indmax(y)) # make the max score move end end Here n is a machine learning model, f is a specification of what features to extract, and p represents the parser state. The parser works by extracting features representing the sentence and the current parser state, using the model to score possible moves, and executing the highest scoring move until no valid moves are left. To parse a whole corpus (array of sentences), we just map gparse to each sentence. Julia can distinguish which gparse we mean by looking at the types of arguments. typealias Corpus AbstractVector{Sentence} gparse(c::Corpus, n::Net, f::Features)=map(s->gparse(s,n,f), c) For our first experiment we will parse some sentences on a single CPU core (Intel(R) Xeon(R) CPU E5-2670 v2 @ 2.50GHz): julia> nworkers() # we use a single core 1 julia> using KUnet julia> KUnet.gpu(false) # no gpu yet julia> using KUparser julia> @time KUparser.gparse(dev, net, feats); elapsed time: 2244.3076923076924 seconds The corpus, dev, is the Penn Treebank WSJ section 22 (1700 sentences, 40117 words); net is a standard feed forward neural network with 1326 input units, 20000 hidden units in a single layer, and 3 output units; feats is a specification of features to be extracted. The parsing speed is 55.944 ms/word. More than 99% of that time is spent on "predict". In order to speed up "predict", we will use the GPU (NVIDIA Tesla K20m): julia> gnet=copy(net,:gpu) julia> @time KUparser.gparse(dev, gnet, feats); elapsed time: 148.56374417550305 seconds This gives us 3.704 ms/word, a 15x speed-up. However the GPU can be better utilized if we somehow manage to batch our feature vectors and compute scores for multiple instances in parallel. The problem is parsing a sentence is a serial process, you need the state resulting from the last move in order to compute the features for the next move. The solution is to parse multiple sentences in parallel (thanks to Alkan Kabakcioglu for suggesting this). Different sentences have no dependencies on each other and we can keep track of their states and predict their moves in bulk. The second version of gparse takes an additional "batchsize" argument specifying how many sentences to parse in parallel. This needs some bookkeeping (requiring 80 lines of code for gparse v2 instead of the 10 line beauty you see above), so I won't cut-and-paste it here, you can see the source code if you wish. Here are some experiments with the batched gparse: julia> @time KUparser.gparse(dev, gnet, feats, 1); elapsed time: 148.725787323 seconds julia> @time KUparser.gparse(dev, gnet, feats, 10); elapsed time: 48.573996933 seconds julia> @time KUparser.gparse(dev, gnet, feats, 100); elapsed time: 25.502507879 seconds julia> @time KUparser.gparse(dev, gnet, feats, 1700); elapsed time: 22.079269825 seconds As we increase the number of sentences processed in parallel (doing all 1700 sentences in the corpus in parallel in the last example), we get 0.550 ms/word, a 100x speedup from where we started. At this point the time spent on prediction is about a third of the time spent on feature extraction, so let's take another look at features. We will use Julia's parallel computing primitives to group the sentences to be processed on separate cores. The third version of gparse takes yet another argument specifying the number of cpu cores: function gparse(corpus::Corpus, net::Net, fmat::Features, batch::Integer, ncpu::Integer) d = distribute(corpus, workers()[1:ncpu]) n = copy(net, :cpu) p = pmap(procs(d)) do x gparse(localpart(d), copy(n, :gpu), fmat, batch) end end The distribute command distributes the corpus equally among ncpu workers, and localpart gives each worker its own subset. We copy the net back and forth between the CPU and the GPU because I couldn't figure out how to pass GPU pointers between different workers. Finally pmap is the parallel map which calls gparse v2 on each worker for the appropriate subset of the corpus, pmerge merges the results. This time we will run the parser on the training set (Sections 02-21, ~40k sentences, ~950k words) julia> addprocs(20) julia> require("CUDArt") julia> @everywhere CUDArt.device((myid()-1)%CUDArt.devcount()) julia> require("KUparser") julia> @time KUparser.gparse(trn, gnet, feats, 2000, 20); elapsed time: 52.13701401 seconds The server has 20 CPU cores and 2 GPUs. We create 20 workers, and assign equal numbers to each GPU. Parsing 950k words takes 52 seconds (0.055 ms/word), a 1000x speedup from where we started.	2022-09-27 05:22:45	{"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": 0.44239988923072815, "perplexity": 4272.248767806645}, "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/1664030334987.39/warc/CC-MAIN-20220927033539-20220927063539-00587.warc.gz"}
http://hal.archives-ouvertes.fr/view_by_stamp.php?label=INSMI&langue=fr&action_todo=view&id=hal-00718245&version=1	21744 articles – 15574 Notices [english version] HAL : hal-00718245, version 1 arXiv : 1207.3664 Probability Theory and Related Fields 128, 3 (2004) 386-418 Birth and death processes on certain random trees: Classification and stationary laws (2004) The main substance of the paper concerns the growth rate and the classification (ergodicity, transience) of a family of random trees. In the basic model, new edges appear according to a Poisson process of parameter $\lambda$ and leaves can be deleted at a rate $\mu$. The main results lay the stress on the famous number $e$. A complete classification of the process is given in terms of the intensity factor $\rho=\lambda/\mu\,$: it is ergodic if $\rho\leq e^{-1}$, and transient if $\rho>e^{-1}$. There is a phase transition phenomenon: the usual region of null recurrence (in the parameter space) here does not exist. This fact is rare for countable Markov chains with exponentially distributed jumps. Some basic stationary laws are computed, e.g.~the number of vertices and the height. Various bounds, limit laws and ergodic-like theorems are obtained, both for the transient and ergodic regimes. In particular, when the system is transient, the height of the tree grows linearly as the time $t\to\infty$, at a rate which is explicitly computed. Some of the results are extended to the so-called multiclass model. 1 : PREVAL (INRIA Rocquencourt) INRIA 2 : Laboratory of Large Random Systems (LLRS) Moscow State University Domaine : Mathématiques/Probabilités Mots-clés : Random trees – ergodicity – transience – nonlinear differential equations – phase transition Liste des fichiers attachés à ce document : PDF ptrf-gmj.pdf(314 KB) PS ptrf-gmj.ps(1.1 MB) hal-00718245, version 1 http://hal.inria.fr/hal-00718245 oai:hal.inria.fr:hal-00718245 Contributeur : Jean-Marc Lasgouttes <> Soumis le : Lundi 16 Juillet 2012, 15:02:37 Dernière modification le : Lundi 16 Juillet 2012, 15:06:20	2013-05-20 03:29:35	{"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": 0.7640934586524963, "perplexity": 1191.5715690837114}, "config": {"markdown_headings": true, "markdown_code": true, "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-2013-20/segments/1368698222543/warc/CC-MAIN-20130516095702-00022-ip-10-60-113-184.ec2.internal.warc.gz"}
https://www.shaalaa.com/question-bank-solutions/algebraic-methods-solving-pair-linear-equations-cross-multiplication-method-method-cross-multiplication_5832	# Solution - Algebraic Methods of Solving a Pair of Linear Equations - Cross - Multiplication Method Account Register Share Books Shortlist ConceptAlgebraic Methods of Solving a Pair of Linear Equations - Cross - Multiplication Method #### Question Solve the follownig system of equations by the method of cross-multiplication. 2x – 6y + 10 = 0 3x – 7y + 13 = 0 #### Solution You need to to view the solution Is there an error in this question or solution? #### Similar questions VIEW ALL Solve the following system of equations by cross-multiplication method. ax + by = a – b; bx – ay = a + b view solution Solve the following system of equations by cross-multiplication method x + y = a – b; ax – by = a2 + b2 view solution Solve the following system of equations by cross-multiplication method ax + by = 1; bx + ay = \frac{(a+b)^{2}}{a^{2}+b^{2}-1 view solution A thief, after committing a theft, runs at a uniform speed of 50 m/minute. After 2 minutes, a policeman runs to catch him. He goes 60 m in first minute and increases his speed by 5 m/minute every succeeding minute. After how many minutes, the policeman will catch the thief? view solution Which of the following pairs of linear equations has unique solution, no solution, or infinitely many solutions. In case there is a unique solution, find it by using cross multiplication method 3x – 5y = 20 6x – 10y = 40 view solution #### Reference Material Solution for concept: Algebraic Methods of Solving a Pair of Linear Equations - Cross - Multiplication Method. For the course 8th-10th CBSE S	2017-10-21 19:39:45	{"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": 0.3461925983428955, "perplexity": 1040.0860070646163}, "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/1508187824894.98/warc/CC-MAIN-20171021190701-20171021210701-00565.warc.gz"}
https://stacks.math.columbia.edu/tag/077B	Lemma 109.51.1. Let $S$ be a scheme. Let $G$ be a group scheme over $S$. The stack $G\textit{-Principal}$ classifying principal homogeneous $G$-spaces (see Examples of Stacks, Subsection 94.14.5) and the stack $G\textit{-Torsors}$ classifying fppf $G$-torsors (see Examples of Stacks, Subsection 94.14.8) are not equivalent in general. Proof. The discussion above shows that the functor $G\textit{-Torsors} \to G\textit{-Principal}$ isn't essentially surjective in general. $\square$ 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).	2023-01-30 08:36:54	{"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": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.9930043816566467, "perplexity": 723.8931278239296}, "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-20230130070411-20230130100411-00089.warc.gz"}
https://brilliant.org/problems/inspired-by-project-euler/	# Inspired by Project Euler Define $$\text{reverse}(n)$$ as a function which reverses the given integer. For example, $$\text{reverse}(23)=32$$ and $$\text{reverse}(405)=504$$. Now, some natural numbers $$n$$ have a property that $$n+\text{reverse}(n)$$ always consists of odd digits. For example, $$36+\text{reverse}(36)=36+63=99$$ and $$409+\text{reverse}(409)=409+904=1313$$. We call such numbers Reversible Numbers. Thus, $$36,63,409,904$$ are Reversible Numbers. Calculate the total number of Reversible numbers less than $$10^{11}$$. Details and Assumptions: • Leading zeroes are NOT allowed in $$n$$ or $$\text{reverse}(n)$$. ×	2017-09-25 20:48:44	{"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": 0.7337566018104553, "perplexity": 513.1200566718986}, "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-2017-39/segments/1505818693363.77/warc/CC-MAIN-20170925201601-20170925221601-00246.warc.gz"}
https://www.cnblogs.com/grandyang/p/4267628.html	# [LeetCode] 133. Clone Graph 克隆无向图 Given a reference of a node in a connected undirected graph, return a deep copy (clone) of the graph. Each node in the graph contains a val (int) and a list (List[Node]) of its neighbors. Example: Input: {"$id":"1","neighbors":[{"$id":"2","neighbors":[{"$ref":"1"},{"$id":"3","neighbors":[{"$ref":"2"},{"$id":"4","neighbors":[{"$ref":"3"},{"$ref":"1"}],"val":4}],"val":3}],"val":2},{"\$ref":"4"}],"val":1} Explanation: Node 1's value is 1, and it has two neighbors: Node 2 and 4. Node 2's value is 2, and it has two neighbors: Node 1 and 3. Node 3's value is 3, and it has two neighbors: Node 2 and 4. Node 4's value is 4, and it has two neighbors: Node 1 and 3. Note: 1. The number of nodes will be between 1 and 100. 2. The undirected graph is a simple graph, which means no repeated edges and no self-loops in the graph. 3. Since the graph is undirected, if node p has node q as neighbor, then node q must have node p as neighbor too. 4. You must return the copy of the given node as a reference to the cloned graph. class Solution { public: Node* cloneGraph(Node* node) { unordered_map<Node, Node> m; return helper(node, m); } Node* helper(Node* node, unordered_map<Node, Node>& m) { if (!node) return NULL; if (m.count(node)) return m[node]; Node clone = new Node(node->val); m[node] = clone; for (Node neighbor : node->neighbors) { clone->neighbors.push_back(helper(neighbor, m)); } return clone; } }; class Solution { public: Node* cloneGraph(Node* node) { if (!node) return NULL; unordered_map<Node, Node> m; queue<Node> q{{node}}; Node clone = new Node(node->val); m[node] = clone; while (!q.empty()) { Node t = q.front(); q.pop(); for (Node neighbor : t->neighbors) { if (!m.count(neighbor)) { m[neighbor] = new Node(neighbor->val); q.push(neighbor); } m[t]->neighbors.push_back(m[neighbor]); } } return clone; } }; Copy List with Random Pointer https://leetcode.com/problems/clone-graph/ https://leetcode.com/problems/clone-graph/discuss/42313/C%2B%2B-BFSDFS https://leetcode.com/problems/clone-graph/discuss/42309/Depth-First-Simple-Java-Solution LeetCode All in One 题目讲解汇总(持续更新中...) posted @ 2015-02-02 13:56 Grandyang 阅读(16109) 评论(13编辑收藏	2021-01-22 14:29:36	{"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": 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": 0.19270353019237518, "perplexity": 13603.48224197958}, "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/1610703529331.99/warc/CC-MAIN-20210122113332-20210122143332-00794.warc.gz"}
https://bookdown.org/daaronr/writing-econ/data-please.html	# 7 Data please! More input! ## 7.1 Why do we use data? Just like Johnny-five and his cousin the Robot cash register, Economists need ‘input’ from the real world to improve our understanding. ### Descriptive To measure and understand our object of study: the economy (including individuals, households, firms, and governments) Levels of variables, patterns (e.g., Life cycle consumption), observed relationships between variables (differences by group, correlations, linear relationships, etc.) • For its own sake • To use in executing policy (e.g., the Consumer Price Index) • To generate hypotheses and “calibrate” our models We can have statistical tests of “descriptive” hypothesis. E.g., testing H0: Incomes of men and women are the same ceteris paribus vs. HA: Women with the same characteristics as men earn less on average. Note: this is not testing a causal relationship; a difference doesn’t necessarily imply a particular explanation (e.g., sex discrimination). ### Causal: To make statistical inferences (and statistical predictions) about effects (sometimes called “causal effects” but I find that redundant). To measure and test hypotheses about the causal relationship between important factors and outcomes. ## 7.2 What data do you need to answer your question? Look for data that is… Relevant to your topic the relevant population, years, fields; relevant outcome variable(s), “independent variable(s)” of interest, control variables) e.g., contains a useful “instrumental variable”, a long enough time series, or repeated observations on individuals to allow ‘fixed effects’ controls Reliable, accessible, understandable Consider: What data have previous authors used to answer this or related questions? ## 7.3 Some types of data • Survey and collected data: self-reports, interviewers, physical measures and visual checks • Administrative data (e.g., tax records) • Transactions/interactions • Scanner data • Web data (e.g., Ebay, Amazon) • Price data • Public financial data and company reports • Official government data (public releases and announcements, e.g., budget data) • Data from lab experiments • Data from field experiments Consider the differences between: • Micro-data (individual/transaction level) vs. Macro-data (aggregated to firm, region, country-year level etc) • Panel vs cross-section vs time-series data ### Some examples of datasets used by Undergraduate students Workplace Employee Relations Survey: Private Sector Panel, 1998-2004 data, from the UK Data Archive. Data on cigarette consumption from the US Centers of Disease Control (CDC) from 1986 to 2011, for 50 states $\rightarrow$ 1300 observations. The 1958 National Child Development Survey, a longitudinal study tracking a group of individuals born in a single week in 1958. Data on UK cities’ population, employment, geography, extracted from various ONS tables. “The ICCSR UK Environmental & Financial Dataset, is a large panel data set on a a sample of firms, giving a set of ratings on “community and environmental responsibility”; merged to a set of financial variables on these firms, collected from Datastream Exchange rates between the US dollar, the British pound, Australian dollar, Canadian dollar and Swiss franc, for the period 1975-2010, from the OECD Main Economic Indicators database. The World Bank Development Indicator database (2013); 210 countries over a 20-year period from 1991-2010 65 banks over 8 years from BankScope (profitability measures, etc) ## 7.4 Getting and using data ### Finding data Update: A particularly promising resource: Google dataset search In searching for data, note that the American Economics Association has a very comprehensive list of links: http://www.aeaweb.org/RFE/toc.php?show=complete for the UK in specific, see http://www.statistics.gov.uk/default.asp For macro and micro data, see http://www.esds.ac.uk/ For large scale data, see also the UK Data Service database. Some other sources of data, and links to aggregations on my webpage here. Some of these (and lists of lists) are also listed in this Airtable also mentioned below… this is filtered ‘data search/archive’; remove this filter to see more. Also, to comment on this you can get full ‘commenter’ access link Also note that data from published papers are typically expected to be made publicly accessible (for replication and checking purposes). If you cannot find it on the journal or the author’s website (or linked therein), you can email the corresponding author to ask for it. Don’t wait too long to begin collecting your data and producing simple graphs and summary statistics, to get a sense of your data. Empirical work is difficult and you may not be able to get the “best” data This is OK. Remember, at the undergraduate/MSc level, we generally want you to show your competencies in these assignments; we expect the analysis will have limitations. The most common format to download the data in is ‘csv’ for ‘comma-separated values.’ This can be read into Stata, R, and nearly any program. The first row usually gives the variable names, which you can change later in your program. Commas separate each variable (aka ‘feature’ aka ‘column’). Each observation (aka ‘unit’ aka ‘row’) is separated by a line break. (See ‘Text editors’ below.) ### 7.4.2 Inputting the data (into Stata, R, etc) These programs have several ‘input’ commands you can use (e.g., insheet in Stata, read_csv in R) for “getting the data in” (as an object that can be referred to and analyzed). You could use the ‘drop-down’ menu or some other visual tool perhaps, to input it, but this is not best practice.15 Find the right input command and make this part of your code. (See ‘Doing coding…’.) ## 7.5 Understanding your data Present simple statistics and graphics on your data before doing more involved analyses. ## 7.6 What does data look like (brief) Author’s note to self: Display these directly through R, especially using the built-in datasets ### Observations, variables Each “unit” is an observation. Think of these as the rows of a spreadsheet.Every unit will have values for each of the “variables”. You may create new variables from transformations and combinations of the variables.You may limit your analysis to a subset of the observations for justifiable reasons. Your analysis may need to drop some observations, e.g., with missing variables (but be careful). ### 7.6.1 Cross-sectional, time-series, and panel data An example of… Time series: A single ‘unit’ over time… in this case 4 quarters per year, shown in Stata’s ‘data editor’. (But you shouldn’t usually edit data in this mode – do it with code!) xtset is a Stata command to tell Stata you are dealing with panel data. Within this command you specify the variable identifying the unit with iis and the variable identifying the time period with tis. Above: Cross-country panel data ### String and numeric variables String variables are text. In their raw form, they usually have quotes (“john smith”,) around them. Numeric variables can be integers, “floats”, etc, stored in various forms. They are numbers. Most statistical packages and programming languages treats these two types of variables differently, with a different “syntax” and different commands for each. Be careful. There are many other data types, with some variation in how these are categorised and stored between languages. E.g., • ‘Factor’ variables (categorical, ordinal) • Logical (true/false) • Date and time variables ## 7.7 Doing ‘coding’: cleaning, visualizing/summarizing, analysing, and presenting Some quick important guidelines 1. Do ALL of your work (cleaning, merging, creating variables, and analysis) by writing code in a ‘script file’ (Stata – a ‘.do file’; R – a ‘.R’ or ‘.Rmd’ file; Python– a ‘.py’ file, I think) 2. Do your cleaning/construction and analysis in separate files (or at least separate parts of the same file; clean the data first, then analyse it) 3. Keep this organised, and try to write it in a way that you, or others, could return to it later. A good reference… but getting old now: (“Code and Data for the Social Sciences”, 2014, Gentzkow and Shapiro)[https://web.stanford.edu/~gentzkow/research/CodeAndData.pdf] Some other resources (more up-to-date?) listed Here ## 7.8 Doing an econometric analysis Which techniques You may not be able to use the “ideal” estimation technique; it may be too advanced. But try to be aware (and able to explain) of the strengths and weaknesses of your econometric approach. Time series, cross section, or panel data? “A major problem is always understanding the difference between a panel and a time series. My students always want to just do a time series regression, and don’t understand why the cross-section dimension is important.” –University of Essex lecturer Common difficulties Diagnostic tests, etc.	2020-02-23 18:53:41	{"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": 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": 0.2511206865310669, "perplexity": 6605.02801820529}, "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-2020-10/segments/1581875145839.51/warc/CC-MAIN-20200223185153-20200223215153-00485.warc.gz"}
https://mailman.ntg.nl/pipermail/ntg-context/2010/054087.html	# [NTG-context] Tikz figures not centred Vedran Miletić rivanvx at gmail.com Thu Nov 11 12:14:21 CET 2010 2010/11/11 Michael Murphy <michael.murphy at uni-ulm.de> > Hi, > > I've been having some problems with tikz figures. When I define a new > tikz figure, I normally put it in a buffer: > > \startbuffer[mypic] > \starttikzfigure > ... > \stoptikzfigure > \stopbuffer > > which I use later when I place the figure > > \placefigure{My picture}{\getbuffer[mypic]} > > The problem is that the figure is not centred: it is always aligned with > the left side of the document. I guess this has something to do with > Context not being able to get the image bounds, since it works fine for > tikz images that are already precompiled into PDFs: > > \placefigure{My picture}{\externalimage[mypic.pdf]} > > Minimal example is attached. > > Michael. > > You have to wrap up the picture inside of a \hbox, e.g. \hbox{\starttikzfigure ... \stoptikzfigure} Regards, -- Vedran Miletić -------------- next part -------------- An HTML attachment was scrubbed... URL: <http://www.ntg.nl/pipermail/ntg-context/attachments/20101111/c446c9e7/attachment.html>	2021-12-05 13:42:13	{"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": 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": 0.9954160451889038, "perplexity": 9968.655374880198}, "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/1637964363189.92/warc/CC-MAIN-20211205130619-20211205160619-00560.warc.gz"}
https://electronics.stackexchange.com/questions/224706/better-way-to-divide-voltage-for-input-to-an-adc-daq	# Better way to divide voltage for input to an ADC DAQ? I would like to monitor relatively higher voltages (up to 160V) using an ADC type DAQ. Most DAQs that I come across can handle around 5-10V of analog input, requiring the voltage to be divided. Here is one I like https://labjack.com/products/t7 . I was planning on running a simple voltage divider to an op-amp buffer (voltage follower) and finally into the input of the DAQ. My thought is to use something like 3.3 MOhm (R1) and 105 kOhm (R2) to divide the 160V by ~32 (~5V). Then run that output to an op-amp voltage follower and finally into the input of the DAQ. I'm not sure how to size the resistors since the op-amp buffer should be high impedance and limit the current draw. I sized them so big to ensure I'm not drawing much current. Since I am monitoring the voltage, I do not want to load down the source. I want to know if this is a good way to do it, and if there are any better ways. Yes, an op-amp unity-gain buffer is a reasonable approach in the ADC does not have a high-impedance input. The minimum value of the sum of the two resistors is determined by how much current you can draw from the source without unduly affecting the accuracy. The power dissipation might also come into play if the resistors are low value. The maximum value of the two resistors paralleled is determined partly by how much error you can tolerate due to op-amp bias current and/or leakage. Also practical considerations (resistors of lower value tend to be more stable, at least down to 1M or 100K). For example, metal foil resistors are not available much above 100K. For relatively low accuracy applications, a few M ohms is fine, and the bias current error will depend on the op-amp you select. If you follow their app note on the op-amp part (looks like there is an error in the plain divider calculation) they suggest an OPA344 - bias current should be less than 1nA at reasonable temperatures, which implies (for < 0.5LSB error on a 12-bit converter): $R1\|\|R2 < \frac{5.0}{2^{-9} \cdot 2^{13}} \approx$ 30K Your 105K \|\| 3.3M ~= 100K would have a bit more error at high temperatures, but still should be acceptable for most purposes. • Thanks very much for the helpful answer. I'm not familiar with the parallel resistance to LSB error calculation you have provided. Is this suggesting that I want less than 30K parallel resistance to stay less than the noise threshold at 1nA? In that case, I could use something like 660 kOhm and 20 kOhm to give about 19k in parallel resistance. That would only make the source pull an extra ~235 uA which is nothing for my application. Mar 25 '16 at 21:44 • It's just the bias current multiplied by the source resistance gives the input-referred error. If you had 100K you'd have at 1nA bias current (that's the current flowing into the op-amp input) about 1.5 least significant bits of error, which is only 0.04%. Personally I'd try to use 1.0M for the high value resistor and adjust the low value resistor to get the ratio you need. Mar 25 '16 at 21:56	2022-01-20 20:12:12	{"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": 0.5211159586906433, "perplexity": 1132.8171601800009}, "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/1642320302622.39/warc/CC-MAIN-20220120190514-20220120220514-00178.warc.gz"}
https://english.wunderground.com/history/airport/SCIP/2018/12/7/MonthlyCalendar.html?req_city=Easter%20Island&req_statename=Chile&reqdb.zip=00000&reqdb.magic=1&reqdb.wmo=85469	## Easter Island, Chile 7:50 PM -05 on December 09, 2018 (GMT -0500) # Weather History for SCIP - December, 2018 • Today • Forecast • Today • Forecast • T= Trace of Precipitation Precipitation accumulation is shown as one of these two metrics - Snow Accumulation: This icon will show if stations have a snow accumulation reading. Not all stations report snow accumulation. Liquid Precipitation: This icon shows rain accumulation or the liquid equivalent in the case of snow, hail or sleet. It does not differentiate between different forms of precipitation such as rain, snow, hail or sleet. Sunday Monday Tuesday Wednesday Thursday Friday Saturday 1 Rain Actual: 79° \| 69° 0.00 in Average: - \| - - in 2 Scattered Clouds Actual: 78° \| 65° 0.00 in Average: - \| - - in 3 Partly Cloudy Actual: 79° \| 64° 0.00 in Average: - \| - - in 4 Scattered Clouds Actual: 82° \| 65° 0.00 in Average: - \| - - in 5 Partly Cloudy Actual: 80° \| 63° 0.00 in Average: - \| - - in 6 Rain Actual: 78° \| 62° 0.00 in Average: - \| - - in 7 Partly Cloudy Actual: 79° \| 62° 0.00 in Average: - \| - - in 8 Rain Actual: 74° \| 66° 0.98 in Average: - \| - - in 9 Partly Cloudy Forecast: 75° \| 66° 0.0 in Average: - \| - - in 10 Partly Cloudy Forecast: 76° \| 68° 0.0 in Average: - \| - - in 11 Partly Cloudy Forecast: 77° \| 68° 0.0 in Average: - \| - - in 12 Partly Cloudy Forecast: 77° \| 68° 0.0 in Average: - \| - - in 13 Clear Forecast: 77° \| 68° 0.0 in Average: - \| - - in 14 Clear Forecast: 77° \| 68° 0.0 in Average: - \| - - in 15 Clear Forecast: 77° \| 68° 0.04 in Average: - \| - - in 16 Partly Cloudy Forecast: 78° \| 68° 0.01 in Average: - \| - - in 17 Chance of Rain Forecast: 77° \| 68° 0.04 in Average: - \| - - in 18 Clear Forecast: 78° \| 69° 0.0 in Average: - \| - - in Record: 78° \| 59° 0.28 in Average: - \| - - in Record: 81° \| 60° 0.08 in Average: - \| - - in Record: 80° \| 62° 0.94 in Average: - \| - - in Record: 80° \| 59° 0.08 in Average: - \| - - in Record: 80° \| 64° 0.16 in Average: - \| - - in Record: 80° \| 62° 0.01 in Average: - \| - - in Record: 80° \| 62° 0.00 in Average: - \| - - in Record: 82° \| 60° 0.12 in Average: - \| - - in Record: 80° \| 58° 0.02 in Average: - \| - - in Record: 82° \| 58° 0.00 in Average: - \| - - in Record: 80° \| 59° 0.02 in Average: - \| - - in Record: 82° \| 60° 0.08 in Average: - \| - - in Record: 81° \| 63° 0.00 in Average: - \| - - in	2018-12-10 00:50:21	{"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": 0.8269948959350586, "perplexity": 11306.936795432044}, "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/1544376823228.36/warc/CC-MAIN-20181209232026-20181210013526-00151.warc.gz"}
https://www.vedantu.com/question-answer/find-the-locus-of-the-point-pleft-hk-right-if-class-11-maths-cbse-5edb5e304d8add1324c97585	Question # Find the locus of the point $P\left( h,k \right)$ if three normals drawn from the point $P$ to${{y}^{2}}=4ax$, satisfying the following ${{m}_{1}}+{{m}_{2}}=1$. Hint: Sum of slopes of three normals of parabola from a particular point is zero. We are given a parabola ${{y}^{2}}=4ax$ and point $P\left( h,k \right)$ from which three normals are drawn. Also, given that ${{m}_{1}}+{{m}_{2}}=1$ that is the sum of slopes of two out of three normals is $1$. Now, we have to find the locus of point$P\left( h,k \right)$. We know that any general point on parabola ${{y}^{2}}=4ax$ is $\left( x,y \right)=\left( a{{t}^{2}},2at \right)$. We know that any line passing from $\left( {{x}_{1}},{{y}_{1}} \right)$ and slope $m$ is: $\left( y-{{y}_{1}} \right)=m\left( x-{{x}_{1}} \right)$. So, the equation of normal at point $\left( a{{t}^{2}},2at \right)$ and slope $m$ is: $\left( y-2at \right)=m\left( x-a{{t}^{2}} \right)....\left( i \right)$ Now we take the parabola, ${{y}^{2}}=4ax$. Now we differentiate the parabola. $\left[ \text{Also, }\dfrac{d\left( {{x}^{n}} \right)}{dx}=n{{x}^{n-1}} \right]$ Therefore, we get $2y\dfrac{dy}{dx}=4a$ $\dfrac{dy}{dx}=\dfrac{2a}{y}$ At $\left( x,y \right)=\left[ a{{t}^{2}},2at \right]$ We get, $\Rightarrow \dfrac{dy}{dx}=\dfrac{2a}{2at}=\dfrac{1}{t}$ As $\dfrac{dy}{dx}$ signify the slope of tangent, therefore any tangent on parabola at point $\left( a{{t}^{2}},2at \right)$ would have slope $=\dfrac{1}{t}$. Now, we know that tangent and normal are perpendicular to each other. Therefore, $\left( \text{Slope of tangent} \right)\times \left( \text{Slope of normal} \right)=-1$ As we have found that $\text{Slope of tangent =}\dfrac{1}{t}$ and assumed that slope of normal is $m$. Therefore, we get $\left( \dfrac{1}{t} \right)\times \left( m \right)=-1$. Hence, $t=-m$ Putting value of $t$in equation $\left( i \right)$, We get, $\left[ y-2a\left( -m \right) \right]=m\left[ x-a{{\left( -m \right)}^{2}} \right]$ $\Rightarrow y+2am=m\left( x-a{{m}^{2}} \right)$ $\Rightarrow y=mx-a{{m}^{3}}-2am$ By rearranging the given equation, We get, $a{{m}^{3}}+m\left( 2a-x \right)+y=0$ Here, we get three degree equation in terms of $m$, therefore it will have $3$ roots ${{m}_{1}}, {{m}_{2}}$ and ${{m}_{3}}$. As we know that this normal passes through $\left( h,k \right)$, we put $x=h$ and $y=k$. We get, $a{{m}^{3}}+m\left( 2a-h \right)+k=0....\left( ii \right)$ Comparing above equation by general three degree equation $a{{x}^{3}}+b{{x}^{2}}+cx+d=0$ We get, $a=a,\text{ }b=0,\text{ }c=\left( 2a-h \right),\text{ }d=k$ We know that $\text{sum of roots }=\dfrac{-b}{a}$ As $b=0$ in equation $\left( ii \right)$, Therefore, we get $\text{sum of roots }=0$. As ${{m}_{1}}, {{m}_{2}}$ and ${{m}_{3}}$ are roots, Hence, ${{m}_{1}}+{{m}_{2}}+{{m}_{3}}=0$ As we have been given that ${{m}_{1}}+{{m}_{2}}=1$, We get $1+{{m}_{3}}=0$ Therefore, ${{m}_{3}}=-1$ As ${{m}_{3}}$ is root of equation $\left( ii \right)$, Therefore, it will satisfy the equation $\left( ii \right)$. Now, we put ${{m}_{3}}$ in equation $\left( ii \right)$, We get $a{{\left( {{m}_{3}} \right)}^{3}}+{{m}_{3}}\left( 2a-h \right)+k=0$ As, ${{m}_{3}}=-1$ We get, $a{{\left( -1 \right)}^{3}}+\left( -1 \right)\left( 2a-h \right)+k=0$ $\Rightarrow -a-\left( 2a-h \right)+k=0$ $\Rightarrow -3a+h+k=0$ $h+k=3a$ Now, to get the locus of $\left( h,k \right)$, we will replace $h$ by $x$ and $k$ by $y$. So, we get $x+y=3a$ Hence, we get locus of point $P\left( h,k \right)\Rightarrow \left( x+y \right)=3a$ Note: Mistake could be committed in writing the value of sum of roots $=\dfrac{-b}{a}$ as in hurry, students often write coefficient of second term as $b$that is they write $b=\left( 2a-h \right)$ but actually $b$ is the coefficient of ${{m}^{2}}$ which is zero in the given question.	2021-04-21 08:16:50	{"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": 0.9380034804344177, "perplexity": 315.6687648665953}, "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/1618039526421.82/warc/CC-MAIN-20210421065303-20210421095303-00336.warc.gz"}
http://clay6.com/qa/70327/what-is-the-minimum-number-of-different-colours-required-to-paint-he-given-	Comment Share Q) # What is the minimum number of different colours required to paint he given figure such that no two adjacent regions have the same colour? $(A) 3 \\ (B) 4 \\(C)5 \\ (D)6$	2019-09-16 00:19:34	{"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": 0.6691964268684387, "perplexity": 254.05433232497}, "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/1568514572439.21/warc/CC-MAIN-20190915235555-20190916021555-00437.warc.gz"}
https://plainmath.net/elementary-geometry/98323-how-do-you-find-the-midpoint-of-2-2	IndologietVy 2022-11-25 How do you find the midpoint of (-2, 2), (4, 10)? Teagan Gamble Expert Step 1 The midpoint of $A\left({x}_{1},{y}_{1}\right)\phantom{\rule{1ex}{0ex}}\text{and}\phantom{\rule{1ex}{0ex}}B\left({x}_{2},{y}_{2}\right)$ is $M\left(\frac{{x}_{1}+{x}_{2}}{2},\frac{{y}_{1}+{y}_{2}}{2}\right)$ We have, $A\left(-2,2\right)\phantom{\rule{1ex}{0ex}}\text{and}\phantom{\rule{1ex}{0ex}}B\left(4,10\right)$ So, Midpoint of $\overline{AB}=M\left(\frac{-2+4}{2},\frac{2+10}{2}\right)$ i.e. Midpoint of $\overline{AB}=M\left(1,6\right)$ Do you have a similar question?	2023-02-09 03:48:58	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 31, "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": 0.519805371761322, "perplexity": 3496.834889864637}, "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-2023-06/segments/1674764501066.53/warc/CC-MAIN-20230209014102-20230209044102-00407.warc.gz"}
https://mathemerize.com/sum-of-gp-to-infinity/	# Sum of GP to Infinity – Example and Proof Here you will learn sum of gp to infinity (sum of infinite gp) and its proof with examples. Let’s begin – ## Sum of GP to Infinity (Sum of Infinite GP) The sum of an infinite GP with first term a and common ratio r(-1 < r < 1 i.e. , \| r \| < 1) is S = $$a\over 1-r$$ Proof : Consider an infinite GP with first term a and common ratio r, where -1 < r < 1 i.e. , \| r \| < 1. The sum of n terms of this GP is given by $$S_n$$ = a$${1-r^n}\over {1-r}$$ = $$a\over {1-r}$$ – $$ar^n\over {1-r}$$ ……..(i) Since -1 < r < 1, therefore $$r^n$$ decreases as n increases and $$r^n$$ tends to zero as n tends to infinity i.e. $$r^n$$ $$\rightarrow$$ 0 as n $$\rightarrow$$ $$\infty$$. $$\therefore$$ $$ar^n\over {1-r}$$ $$\rightarrow$$ 0 as n $$\rightarrow$$ $$\infty$$. Hence from (i), the sum of an infinite GP is given by S = $$lim_{n \to \infty}$$ $$S_n$$ = $$lim_{n \to \infty}$$ ( $$a\over {1-r}$$ – $$ar^n\over {1-r}$$ ) = $$a\over 1-r$$, if \| r \| < 1 Note : If r $$\ge$$ 1, then the sum of an infinite GP tends to infinity. Example : Find the sum to infinity of the GP $$-5\over 4$$, $$5\over 16$$, $$-5\over 64$$, …… Solution : The given GP has the first term a = -5/4 and the common ratio r = -1/4 Also \| r \| < 1. Hence the sum of an infinite GP is given by S = $$a\over {1-r}$$ S = $$-5/4\over {1-(-1/4)}$$ = -1 Example : The sum of an infinite GP is 57 and the sum of their cubes is 9747, find the GP. Solution : Let a be the first term and r be the common ratio of the GP. Then Sum = 57 $$\implies$$ $$a\over 1-r$$ = 57 …….(i) Sum of the cubes = 9747 $$\implies$$ $$a^3$$ + $$a^3r^3$$ + $$a^3r^6$$ + ….. = 9747 $$\implies$$ $$a^3\over {1 – r^3}$$ = 9747 ……..(ii) Dividing the cube of (i) by (ii), we get $$a^3\over {(1-r)}^3$$ . $$(1-r^3)\over a^3$$ = $${(57)}^3\over 9747$$ $$\implies$$ $$1 – r^3\over {(1 – r)}^3$$ = 19 = $$1+r+r^2\over {(1-r)}^2$$ = $$18r^2$$ – 39r + 18 = 0 $$\implies$$ (3r-2)(6r-9) = 0 $$\implies$$ r = 2/3 or r = 3/2 Hence r = 2/3 [ $$\because$$ r $$\ne$$ 3/2, because -1 < r < 1 for an infinite GP] Putting r = 2/3 in equation (i), we get $$a\over {(1-(2/3))}$$ = 57 $$\implies$$ a = 19 Hence, the GP is 19, 38/3, 76/9, …….	2022-06-30 22:34:13	{"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": 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": 0.8662639260292053, "perplexity": 511.2020781730864}, "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/1656103915196.47/warc/CC-MAIN-20220630213820-20220701003820-00698.warc.gz"}
https://prefetch.eu/know/concept/blochs-theorem/	Categories: Quantum mechanics. # Bloch’s theorem In quantum mechanics, Bloch’s theorem states that, given a potential $$V(\vec{r})$$ which is periodic on a lattice, i.e. $$V(\vec{r}) = V(\vec{r} + \vec{a})$$ for a primitive lattice vector $$\vec{a}$$, then it follows that the solutions $$\psi(\vec{r})$$ to the time-independent Schrödinger equation take the following form, where the function $$u(\vec{r})$$ is periodic on the same lattice, i.e. $$u(\vec{r}) = u(\vec{r} + \vec{a})$$: \begin{aligned} \boxed{ \psi(\vec{r}) = u(\vec{r}) e^{i \vec{k} \cdot \vec{r}} } \end{aligned} In other words, in a periodic potential, the solutions are simply plane waves with a periodic modulation, known as Bloch functions or Bloch states. This is suprisingly easy to prove: if the Hamiltonian $$\hat{H}$$ is lattice-periodic, then both $$\psi(\vec{r})$$ and $$\psi(\vec{r} + \vec{a})$$ are eigenstates with the same energy: \begin{aligned} \hat{H} \psi(\vec{r}) = E \psi(\vec{r}) \qquad \hat{H} \psi(\vec{r} + \vec{a}) = E \psi(\vec{r} + \vec{a}) \end{aligned} Now define the unitary translation operator $$\hat{T}(\vec{a})$$ such that $$\psi(\vec{r} + \vec{a}) = \hat{T}(\vec{a}) \psi(\vec{r})$$. From the previous equation, we then know that: \begin{aligned} \hat{H} \hat{T}(\vec{a}) \psi(\vec{r}) = E \hat{T}(\vec{a}) \psi(\vec{r}) = \hat{T}(\vec{a}) \big(E \psi(\vec{r})\big) = \hat{T}(\vec{a}) \hat{H} \psi(\vec{r}) \end{aligned} In other words, if $$\hat{H}$$ is lattice-periodic, then it will commute with $$\hat{T}(\vec{a})$$, i.e. $$[\hat{H}, \hat{T}(\vec{a})] = 0$$. Consequently, $$\hat{H}$$ and $$\hat{T}(\vec{a})$$ must share eigenstates $$\psi(\vec{r})$$: \begin{aligned} \hat{H} \:\psi(\vec{r}) = E \:\psi(\vec{r}) \qquad \hat{T}(\vec{a}) \:\psi(\vec{r}) = \tau \:\psi(\vec{r}) \end{aligned} Since $$\hat{T}$$ is unitary, its eigenvalues $$\tau$$ must have the form $$e^{i \theta}$$, with $$\theta$$ real. Therefore a translation by $$\vec{a}$$ causes a phase shift, for some vector $$\vec{k}$$: \begin{aligned} \psi(\vec{r} + \vec{a}) = \hat{T}(\vec{a}) \:\psi(\vec{r}) = e^{i \theta} \:\psi(\vec{r}) = e^{i \vec{k} \cdot \vec{a}} \:\psi(\vec{r}) \end{aligned} Let us now define the following function, keeping our arbitrary choice of $$\vec{k}$$: \begin{aligned} u(\vec{r}) = e^{- i \vec{k} \cdot \vec{r}} \:\psi(\vec{r}) \end{aligned} As it turns out, this function is guaranteed to be lattice-periodic for any $$\vec{k}$$: \begin{aligned} u(\vec{r} + \vec{a}) &= e^{- i \vec{k} \cdot (\vec{r} + \vec{a})} \:\psi(\vec{r} + \vec{a}) \\ &= e^{- i \vec{k} \cdot \vec{r}} e^{- i \vec{k} \cdot \vec{a}} e^{i \vec{k} \cdot \vec{a}} \:\psi(\vec{r}) \\ &= e^{- i \vec{k} \cdot \vec{r}} \:\psi(\vec{r}) \\ &= u(\vec{r}) \end{aligned} Then Bloch’s theorem follows from isolating the definition of $$u(\vec{r})$$ for $$\psi(\vec{r})$$.	2021-05-07 22:53:49	{"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": 1.0000078678131104, "perplexity": 2868.0894948379946}, "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/1620243988828.76/warc/CC-MAIN-20210507211141-20210508001141-00580.warc.gz"}
https://eprint.iacr.org/2019/313	## Cryptology ePrint Archive: Report 2019/313 A SAT-based approach for index calculus on binary elliptic curves Monika Trimoska and Sorina Ionica and Gilles Dequen Abstract: Logical cryptanalysis, first introduced by Massacci in 2000, is a viable alternative to common algebraic cryptanalysis techniques over boolean fields. With XOR operations being at the core of many cryptographic problems, recent research in this area has focused on handling XOR clauses efficiently. In this paper, we investigate solving the point decomposition step of the index calculus method for prime degree extension fields $\mathbb{F}_{2^n}$, using SAT solving methods. We propose an original XOR-reasoning SAT solver, named WDSat, dedicated to this specific problem. While asymptotically solving the point decomposition problem with our method has exponential worst time complexity in the dimension $l$ of the vector space defining the factor base, experimental running times show that our solver is significantly faster than current algebraic methods based on Gröbner basis computation. For the values $l$ and $n$ considered in the experiments, WDSat was up to 300 times faster then MAGMA's F4 implementation, and this factor grows with $l$ and $n$. Our solver outperforms as well current best state-of-the-art SAT solvers for this specific problem. Category / Keywords: public-key cryptography / discrete logarithm, index calculus, elliptic curves, point decomposition, symmetry, satisfiability, DPLL algorithm Date: received 20 Mar 2019, last revised 15 May 2019 Contact author: monika trimoska at u-picardie fr,sorina ionica@u-picardie fr,gilles dequen@u-picardie fr Available format(s): PDF \| BibTeX Citation Short URL: ia.cr/2019/313 [ Cryptology ePrint archive ]	2019-07-19 23:09:58	{"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": 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": 0.3493787944316864, "perplexity": 3246.634584638081}, "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/1563195526386.37/warc/CC-MAIN-20190719223744-20190720005744-00040.warc.gz"}
https://physics.stackexchange.com/questions/57607/decomposing-a-tensor-product-of-su3-representations-in-irreps	# Decomposing a Tensor Product of $SU(3)$ Representations in Irreps Can somebody explain in a simple way why, talking about representations $$3\otimes3\otimes3=1\oplus8\oplus8\oplus10~?$$ Here $3$ and $\bar{3}$ are the fundamental and anti-fundamental of $SU(3)$, in this case. $$\boldsymbol{3}\boldsymbol{\otimes}\boldsymbol{3}\boldsymbol{\otimes}\boldsymbol{3}= \boldsymbol{1}\boldsymbol{\oplus}\boldsymbol{10}\boldsymbol{\oplus} \boldsymbol{8}^{\boldsymbol{\prime}}\boldsymbol{\oplus}\boldsymbol{8}$$	2019-06-19 20:52:18	{"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": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.7475575804710388, "perplexity": 315.72182397446596}, "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-2019-26/segments/1560627999041.59/warc/CC-MAIN-20190619204313-20190619230313-00551.warc.gz"}
https://math.stackexchange.com/questions/4060450/in-a-pid-the-intersection-of-descending-chain-of-ideals-is-trivial	# In a PID, the intersection of descending chain of ideals is trivial Suppose $$R$$ is PID and $$I_1 \supseteq I_2 \supseteq \cdots$$ is a descending chain of ideals in $$R$$. I would like to prove that $$\bigcap^\infty_{n=1} I_n=(0)$$. Now, since $$R$$ is a PID, every ideal is principal, so each $$I_n=(a_n)$$ for some $$a_n \in R$$. So I need to show that $$\bigcap^\infty_{n=1} (a_n)=(0).$$ We have $$(a_1) \supseteq (a_2) \supseteq \cdots$$, so $$a_i \mid a_{i+1}$$ for all $$i$$. I am not sure what to do next. Also, I don't think this is true if $$R$$ is just a UFD. What would an example of that be? • An example where it doesn’t work for a UFD is $\mathbb{Z}[x]$,m with $I_n = (2,x^n)$. The intersection is $(2)\neq 0$. – Arturo Magidin Mar 13 at 16:46 • Hint: what can a generator of the intersection of these ideals be? – Mindlack Mar 13 at 16:48 • (You should also require proper inclusion in your chain, surely....) – Arturo Magidin Mar 13 at 16:50 • A generator of the intersection of these ideals is the least common multiple of the generators of each ideal in the chain. – wwinters57 Mar 13 at 16:55 Assuming strict descent (as otherwise any constant sequence is a counter-example). Let $$a$$ be the generator of the intersection. Then, $$(a) \subset (a_i) \, \forall i$$ and hence $$a_i \| a \, \forall i$$. But since the descent is strict, none of the $$a_i$$'s are associates. Also $$a_i \| a_{i+1}$$ so $$\exists p_i$$, a prime that divides $$a_{i+1}$$ but not $$a_i$$ for each $$i$$. That gives us infinitely many prime divisors for $$a$$, forcing $$a=0$$. You are correct that UFD does not suffice: you can have an infinite strictly decreasing chain of ideals whose intersection is not trivial. In $$\mathbb{Z}[x]$$, the ideals $$I_n=(2,x^n)$$ satisfy $$I_{n+1}\subsetneq I_n$$, but $$\cap I_n = (2)$$. But you can still leverage the UFD property to get what you want in the PID. Note that in a UFD, if the principal ideals $$(a)$$ and $$(b)$$ satisfy $$(a)\subseteq (b)$$, then $$b\|a$$; and if $$(a)\subsetneq (b)$$, then $$b$$ is a proper divisor of $$a$$. Proposition. Let $$R$$ be a UFD, and let $$(a_1)\supseteq (a_2)\supseteq\cdots\supseteq (a_n)\supseteq\cdots$$ be a chain of principal ideals in $$R$$. If $$a\neq 0$$ lies in $$\cap (a_k)$$, then there exists $$k$$ such that $$(a_k)=(a_{k+r})$$ for all $$r\geq 0$$; that is, the chain stabilizes. Proof. If $$a$$ is a unit, then the intersection is $$R$$, so each ideal is $$R$$ and they are all equal. So we may assume $$a$$ is not a unit. Since $$a\neq 0$$, then it has a factorization into irreducibles, $$a = p_1\cdots p_r.$$ Since $$(a)\subseteq (a_n)$$, then $$a_n$$ is a divisor of $$a$$. But $$a$$ has only finitely many proper divisors up to associates, so there are finitely many principal ideals $$I$$ such that $$(a)\subsetneq I$$. Thus, at some point, the chain $$(a_i)\supseteq (a_{i+1})\supseteq\cdots$$ must stabilize. $$\Box$$ Thus, in your PID, either your descending chain stabilizes, or else the intersection does not contain nonzero elements. (Intuitively, the intersection is a least common multiple; but if at each step you are adding an irreducible factor, then the least common multiple would have an “infinite” factorization into irreducibles, which is impossible.)	2021-07-25 19:01:44	{"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": 56, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9479633569717407, "perplexity": 72.1568425811273}, "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/1627046151760.94/warc/CC-MAIN-20210725174608-20210725204608-00667.warc.gz"}
https://www.pensar-global.com/forum/793017-dp-divide-and-conquer-optimization	# dp divide and conquer optimization We cannot have Multiple Inheritance in Java directly due to Diamond Problem but it can be implemented using Interfaces. Each element should be present in exactly one subsegment. The algorithm uses the dp table which is of O(kn) size. Determine the minimal possible total unfamiliarity value. #pragma GCC optimize ("O3,unroll-loops,no-stack-protector") Keep the optimum pointer opt[i] and try to move it to the right while it is pro table when moving from i to i+ 1. Let $cost(l, r)$ be the unfamiliarity of a contiguous group from $l$ to $r$ (that is the value if all the people from $l$ to $r$ are grouped together). Recursively defined the value of the optimal solution. Notice that the cost function satisfies the convex-quadrangle inequality (because it's based on prefix sums). In computer science, divide and conquer is an algorithm design paradigm based on multi-branched recursion.A divide-and-conquer algorithm works by recursively breaking down a problem into two or more sub-problems of the same or related type, until these become simple enough to be solved directly. The function minimumUnfamiliarity makes a call to rec for every value of x. $$Let us see how this problem possesses both important properties of a Dynamic Programming (DP) Problem and can efficiently solved using Dynamic Programming. The initial call will be rec(x, 1, n, 1, n). You are given an array of N integers a_1, a_2, \dots a_N. With this article at OpenGenus, you must have the complete idea of Divide and Conquer Optimization in Dynamic Programming. TL, another optimization is required when we find optimal k for the middle j value before . Enjoy. A Design technique is often expressed in pseudocode as a template that can be particularized for concrete problems [3].$$ However, like the previous problem, the transition point here is also monotone! Prove is omitted. This Blog is Just the List of Problems for Dynamic Programming Optimizations.Before start read This blog. Each ticket can only be used once, but any number of tickets can be used at a restaurant. Thus, $f(i, j)$ can be calculated in $O(1)$. Note that I used fast I/O to pass this problem. H_{i, j}=\mathop{\arg\max}_{0\le k\lt i} \left\{ dp_{k, j - 1} + f(k + 1, i) \right\} \implies H_{i, j} \le H_{i+1, j} The complexity will be $O(N^2K)$ if we do it directly. 1. Construct the optimal solution for the entire problem form the computed values of smaller subproblems. The difference between DP and “divide and conquer” strategy is that the latter can be solved by combining optimal solutions to non-overlapping sub-problems. This special case is called case 2-SAT or 2-Satisfiability. $$First, let's try to calculate the maximum possible eventual happiness if Joisino starts at restaurant i and ends at restaurant j. Therefore, Greedy Approach does not deal with multiple possible solutions, it just builds the one solution that it believes to be correct. It means that the pointer on the optimum point on lower hull also moves only to the right. Dynamic Programming Optimizations Editorial . (I think only I don't know), a broad usage is to deal with the point on the relevant issues, details. This post is a part of a series of three posts on dynamic programming optimizations: Convex Hull Trick; Knuth's Optimization; Divide and Conquer Optimization; Introduction. Code. Let's write down the DP first in this problem: where f(i, j) is the cost of subsegment a_i, a_{i+1}, \dots, a_j. Let, f(i, j)=\left( \sum_{c=1}^{M} \max_{i\le k\le j} B_{k, c} \right) - \left( \sum_{k=i+1}^{j}A_k \right)$$ 2. $$Transition: To compute dp[x][y], the position where the x-th contiguous group should start is required. Feb 25, 2020 tags: icpc algorithm dp dp-optimization divide-and-conquer. As the central part of the course, students will implement several algorithms in Python that incorporate these techniques and then use these algorithms to analyze two large real-world data sets. Dynamic Programming and Divide and Conquer. There're N people numbered from 1 to N and K cars. Rather, results of these smaller sub-problems are remembered and used for similar or overlapping sub-problems. It looks like Convex Hull Optimization2 is a special case of Divide and Conquer Optimization. Every restaurant offers meals in exchange for these tickets. CDQ divide and conquer optimizes one dimensional DP transfer - [SDOI2011] intercepting missile. The Dynamic Programming (DP) is the most powerful design technique for solving optimization problems. Like divide-and-conquer method, Dynamic Programming solves problems by combining the solutions of subproblems. GATE CSE … Dynamic programming is both a mathematical optimization method and a computer programming method. ∙ 4 ∙ share . Dynamic programming approach extends divide and conquer approach with two techniques (memoization and tabulation) that both have a purpose of storing and re-using sub-problems solutions that may drastically improve performance. dp(i, j) = min_{k \leq j}(f(i, j, k)) Dynamic Programming is a powerful technique that allows one to solve many diﬀerent types of problems in time O(n2) or O ... much like “divide-and-conquer” is a general method, except that unlike divide-and-conquer, the subproblemswill typically overlap. This clearly tells us that the solution for dp(x, y^{\prime}) will always occur before the solution for dp(x, y), where y^{\prime} \lt y (monotonic). So the final happiness (represented by f(i, j)) is: The function rec computes dp(x, yl..yr) for a fixed x by recursively computing for the left and right halves of yl to yr after finding dp(x, mid) - dp[x][mid] and h(x, mid) - pos (position where dp(x, mid) is minimum). This paper is concerned with designing benchmarks and frameworks for the study of large-scale dynamic optimization problems. There are p people at an amusement park who are in a queue for a ride. Dynamic Programming (Part 1) Dynamic Programming • An algorithm design technique (like divide and conquer) • D&C, Кнут, Convex Hull - на примере optimal BST. h(i, j^{\prime}) \leq h(i, j) \text{ , } j^{\prime} \lt j Divide and Conquer Optimization.$$ $2\le N\le 10^5, 2\le K\le \min(N, 20), 1\le a_i\le N$. 2.1 Hierarchical Divide and Conquer Algorithm Assume we conduct a k-way clustering, then the initial time for solving sub-problems is at least O(k(p=k)3) = O(p3=k2) where pdenotes the dimensionality, When we consider k= 2, the divide and conquer algorithm can be at most 4 times faster than the original one. Solve the subproblems. Optimization 2: note that vector v~ i also moves to the right (its x-component increases). At one point, there will be a stage where we cannot divide the subproblems further. 3. be a function which recursively computes $dp(x, yl..yr)$ for a fixed $x$, given that the solution lies between $kl$ and $kr$. 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In this post Bellman inn 1950s form the computed values of smaller subproblems solves each sub-problem only once and its! Original problem this course, we will present two ways of thinking about Dynamic Programming DP... Overlapping sub-problems have a complexity of $O ( n^3 )$ at Green University of Bangladesh to apply and... So that we do it directly join our community global optimum value 2 is obtained when 0 1! Start read this Blog is Just the List of problems for Dynamic Programming using quadrangle inequalities '' by Frances!, Dynamic Programming surprisingly, we think of algorithm, we can calculate directly and the … Dynamic Programming problems! Calculated using a persistent segment tree.However, to get it passed within the whereas... Is of O ( nlogn ) $to$ N $dp divide and conquer optimization$ K \$ non-intersecting non-empty subsegments that. Good enough could read Chinese, cdq ’ s divide-and-conquer is a good reference applications... Value 2 is present in exactly one subsegment 수 있다 has found applications in fields...	2021-01-19 21:46:09	{"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": 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": 0.6254611611366272, "perplexity": 1362.1205778970964}, "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/1610703519784.35/warc/CC-MAIN-20210119201033-20210119231033-00428.warc.gz"}
http://kldns.net/confidence-interval/standard-error-confidence-limits.html	## How To Fix Standard Error Confidence Limits (Solved) Home > Confidence Interval > Standard Error Confidence Limits # Standard Error Confidence Limits ## Contents A small version of such a table is shown in Table 1. If we draw a series of samples and calculate the mean of the observations in each, we have a series of means. Easton and John H. A 95% confidence interval for the unknown mean is ((101.82 - (1.960.49)), (101.82 + (1.960.49))) = (101.82 - 0.96, 101.82 + 0.96) = (100.86, 102.78). his comment is here This formula is only approximate, and works best if n is large and p between 0.1 and 0.9. The standard error estimated using the sample standard deviation is 2.56. This is usually the case even with finite populations, because most of the time, people are primarily interested in managing the processes that created the existing finite population; this is called One of the children had a urinary lead concentration of just over 4.0 mmol /24h. click for more info ## 95 Confidence Interval Formula When the sample size is large, say 100 or above, the t distribution is very similar to the standard normal distribution. The standard error of the mean of one sample is an estimate of the standard deviation that would be obtained from the means of a large number of samples drawn from ISBN 0-7167-1254-7 , p 53 ^ Barde, M. (2012). "What to use to express the variability of data: Standard deviation or standard error of mean?". Confidence intervals are not just for means Confidence intervals are most often computed for a mean. The first column, df, stands for degrees of freedom, and for confidence intervals on the mean, df is equal to N - 1, where N is the sample size. Confidence intervals provide the key to a useful device for arguing from a sample back to the population from which it came. Because the 5,534 women are the entire population, 23.44 years is the population mean, μ {\displaystyle \mu } , and 3.56 years is the population standard deviation, σ {\displaystyle \sigma } Standard Error Formula The ages in one such sample are 23, 27, 28, 29, 31, 31, 32, 33, 34, 38, 40, 40, 48, 53, 54, and 55. ISBN 0-521-81099-X ^ Kenney, J. The mean of all possible sample means is equal to the population mean. For some more definitions and examples, see the confidence interval index in Valerie J. Furthermore, it is a matter of common observation that a small sample is a much less certain guide to the population from which it was drawn than a large sample. So the standard error of a mean provides a statement of probability about the difference between the mean of the population and the mean of the sample. 90 Confidence Interval This is also the standard error of the percentage of female patients with appendicitis, since the formula remains the same if p is replaced by 100-p. If the population standard deviation is finite, the standard error of the mean of the sample will tend to zero with increasing sample size, because the estimate of the population mean As the r gets smaller the SEM gets larger. ## 95 Confidence Interval Calculator Therefore the confidence interval is computed as follows: Lower limit = 16.362 - (2.013)(1.090) = 14.17 Upper limit = 16.362 + (2.013)(1.090) = 18.56 Therefore, the interference effect (difference) for the http://handbook.cochrane.org/chapter_7/7_7_7_2_obtaining_standard_errors_from_confidence_intervals_and.htm The true standard error of the mean, using σ = 9.27, is σ x ¯ = σ n = 9.27 16 = 2.32 {\displaystyle \sigma _{\bar {x}}\ ={\frac {\sigma }{\sqrt 95 Confidence Interval Formula The SEM can be added and subtracted to a students score to estimate what the students true score would be. 95% Confidence Interval However, the concept is that if we were to take repeated random samples from the population, this is how we would expect the mean to vary, purely by chance. Retrieved 17 July 2014. this content For each sample, calculate a 95% confidence interval. Dividing the difference by the standard deviation gives 2.62/0.87 = 3.01. Another way of looking at this is to see that if you chose one child at random out of the 140, the chance that the child's urinary lead concentration will exceed How To Calculate Confidence Interval In Excel The SEM is an estimate of how much error there is in a test. A standard error may then be calculated as SE = intervention effect estimate / Z. One of the children had a urinary lead concentration of just over 4.0 mmol /24h. weblink His true score is 88 so the error score would be 6. Then the standard error of each of these percentages is obtained by (1) multiplying them together, (2) dividing the product by the number in the sample, and (3) taking the square Standard Error Vs Standard Deviation Blackwell Publishing. 81 (1): 75–81. Here the size of the sample will affect the size of the standard error but the amount of variation is determined by the value of the percentage or proportion in the ## Confidence interval for a proportion In a survey of 120 people operated on for appendicitis 37 were men. Student B has an observed score of 109. This approximate formula is for moderate to large sample sizes; the reference gives the exact formulas for any sample size, and can be applied to heavily autocorrelated time series like Wall For the purpose of hypothesis testing or estimating confidence intervals, the standard error is primarily of use when the sampling distribution is normally distributed, or approximately normally distributed. Standard Error Of The Mean For example, if p = 0.025, the value z* such that P(Z > z) = 0.025, or P(Z < z) = 0.975, is equal to 1.96. For example, a series of samples of the body temperature of healthy people would show very little variation from one to another, but the variation between samples of the systolic blood Swinscow TDV, and Campbell MJ. Some of these are set out in table 2. http://kldns.net/confidence-interval/standard-error-and-95-confidence-limits-example.html The first step is to obtain the Z value corresponding to the reported P value from a table of the standard normal distribution. Note that this does not mean that we would expect, with 95% probability, that the mean from another sample is in this interval. The level C of a confidence interval gives the probability that the interval produced by the method employed includes the true value of the parameter . Notice that s x ¯ = s n {\displaystyle {\text{s}}_{\bar {x}}\ ={\frac {s}{\sqrt {n}}}} is only an estimate of the true standard error, σ x ¯ = σ n However, the mean and standard deviation are descriptive statistics, whereas the standard error of the mean describes bounds on a random sampling process. Perspect Clin Res. 3 (3): 113–116. How many standard deviations does this represent? The standard deviation of the age for the 16 runners is 10.23, which is somewhat greater than the true population standard deviation σ = 9.27 years. In other words, the more people that are included in a sample, the greater chance that the sample will accurately represent the population, provided that a random process is used to We will finish with an analysis of the Stroop Data. The standard deviation of all possible sample means is the standard error, and is represented by the symbol σ x ¯ {\displaystyle \sigma _{\bar {x}}} . A better method would be to use a chi-squared test, which is to be discussed in a later module. This may sound unrealistic, and it is. The graphs below show the sampling distribution of the mean for samples of size 4, 9, and 25. This observation is greater than 3.89 and so falls in the 5% of observations beyond the 95% probability limits. They will show chance variations from one to another, and the variation may be slight or considerable. Video 1: A video summarising confidence intervals. (This video footage is taken from an external site. df 0.95 0.99 2 4.303 9.925 3 3.182 5.841 4 2.776 4.604 5 2.571 4.032 8 2.306 3.355 10 2.228 3.169 20 2.086 2.845 50 2.009 2.678 100 1.984 2.626 You True Scores / Estimating Errors / Confidence Interval / Top Estimating Errors Another way of estimating the amount of error in a test is to use other estimates of error. Table 2 shows that the probability is very close to 0.0027. A t table shows the critical value of t for 47 - 1 = 46 degrees of freedom is 2.013 (for a 95% confidence interval). Table 1: Mean diastolic blood pressures of printers and farmers Number Mean diastolic blood pressure (mmHg) Standard deviation (mmHg) Printers 72 88 4.5 Farmers 48 79 4.2 To calculate the standard This can be obtained from a table of the standard normal distribution or a computer (for example, by entering =abs(normsinv(0.008/2) into any cell in a Microsoft Excel spreadsheet).	2018-01-21 20:11:12	{"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": 0.8649876117706299, "perplexity": 455.4737511073876}, "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-2018-05/segments/1516084890874.84/warc/CC-MAIN-20180121195145-20180121215145-00355.warc.gz"}
https://www.physicsforums.com/threads/could-you-give-an-example-in-latex.313718/	# Could you give an example in Latex? 1. May 13, 2009 ### Loren Booda I would like an example in Latex of a product with indices n=1 to N for bN, all minus 1, finally equal to bZ. I have not been able to find the original Latex tutorials. 2. May 13, 2009 ### sylas In the meantime, do you mean $$b_Z = \prod_{n=1}^N (b_n - 1)$$ I doubt it's what you want, but it is hard to parse the English description. Cheers -- sylas 3. May 13, 2009 ### Loren Booda sylas, You did well to decipher my description, excepting that the "-1" would appear outside of the product. Thank you for cheering me up.	2018-02-22 19:52:49	{"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": 0.46508798003196716, "perplexity": 2477.661908050946}, "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-2018-09/segments/1518891814249.56/warc/CC-MAIN-20180222180516-20180222200516-00723.warc.gz"}
https://cloud.originlab.com/doc/en/X-Function/ref/imgResize	# 2.6.5.4 imgResize Resize Resize image ## Command Line Usage 1. imgResize lock:=1 unit:=pixel w:=50; 2. imgResize lock:=1 unit:=percent h:=120 interpolate:=resample; 3. imgResize lock:=0 w:=120 h:=130 interpolate:=normal img:=mat(1) oimg:=mat(2); ## Variables Display Name Variable Name I/O and Type Default Value Description Lock Proportions lock Input int 1 Specifies whether or not to keep the aspect ratio of the original image. Unit unit Input int percent This variable works with two other variables, w and h. It specifies the units for them. Option list • pixel:Pixel The width (w) and height (h) of the output image are measured in pixels. • percent :Percent The width (w) and height (h) of the output image are measured in percentage of the width and height of the input image. Width w Input/Output double Specifies the width of the output image. Height h Input/Output double Specifies the height of the output image. Input Matrix img Input Image <active> Specifies the source image to be manipulated. The default input is the active image. Output Image oimg Output Image <input> Specifies the output image. By default, it is the same as the input image. See the syntax here. Interpolation interpolate Input int normal Specifies the interpolation method which is used when the image is resized. Option list • normal:Normal Uses nearest neighbor interpolation to produce the resized image. • resample:Resample Uses bilinear interpolation and averaging to produce a higher-quality image. • bicubic :Bicubic Uses bicubic interpolation and averaging to produce a higher quality image. This method is slower than Resample. ## Description This X-Function can be used to resize the input image to any size. ## Examples In this example, we use the imgResize function to resize the input image. We change its width to 300 pixels and its height to 200 pixels: 1. Create a new matrix and import car.bmp under \Samples\Image Processing and Analysis folder into it. 2. From the Origin menu, select Matrix: Set Dimensions. We can see the dimension of the image is 500*375. Click Cancel to close the dialog. 3. Make the input image is active and select Image: Geometric Transform: Resize. This opens the dialog of the imgResize X-Function. 4. In the imgResize X-Function dialog, change the settings as the screenshot below and click OK to close the dialog. A new image is created. 5. When the new image is active, choose Matrix: Set Dimensions from the Origin menu to view the dimension of this image. The original image The output image ## Algorithm The resizing is implemented with L_SizeBitmap() function from LEADTOOLS Main API. Please refer to the LEADTOOLS Main API Help file, Version 14 and read the L_SizeBitmap topic. ## References LEADTOOLS Main API Help file, Version 14	2022-05-28 03:23:13	{"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": 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": 0.3249261677265167, "perplexity": 5682.562188691673}, "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-2022-21/segments/1652663012542.85/warc/CC-MAIN-20220528031224-20220528061224-00778.warc.gz"}
https://brilliant.org/problems/root-and-quadratic-together/	Quadratic and Roots go Hand in Hand Algebra Level 5 Let $f\left( x \right) =3{ x }^{ 2 }-7x+c$ where $c$ is variable cofficient with $x > \frac 7 6$. Determine the value of $c$ such that $f\left( x \right)$ touches $g\left( x \right) =\frac { \sqrt { 12x+49-12c } +7 }{ 6 }$ Give your answer as $\lfloor c \rfloor$. Source Given to me as challenge by me friend Lakshay kumar × Problem Loading... Note Loading... Set Loading...	2020-01-26 13:39:49	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 7, "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": 0.29432666301727295, "perplexity": 4207.817170864743}, "config": {"markdown_headings": false, "markdown_code": true, "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-2020-05/segments/1579251688806.91/warc/CC-MAIN-20200126104828-20200126134828-00152.warc.gz"}
http://people.ee.duke.edu/~drsmith/transformation-optics/cloaking.htm	# Transformation Optics \| Making Things Invisible ## Cloaking with Transformation Optics There are many possible routes to making objects seem invisible--after all, magicians have been doing this for centuries! The magician can make an object invisible to an audience by cleverly positioning mirrors around the object to deflect the light away from the object. The mirrors trick our eyes, making us believe that light from somewhere in the distance passed through empty space. The trick works because we intuitively understand that light travels in a straight line; when something causes light to bend or be rerouted, we can get confused and fooled by the light that finally reaches our eyes. The diagram below shows a simple example of a light source whose beam is rerouted around an object--in this case a rabbit, seeming as if it passed through empty space. While not the grand stage illusion a magician might concoct, this example shows how light can be managed to make an object seem as if it isn’t there. Cloaking with Transformation Optics - While the use of mirrors provides one scheme for achieving invisibility, there are a number of problems with this approach that can be easily pointed out. Most notably, the invisibility effect, as pictured, works with the observer situated at only one position. If the observer moves around, he or she will quickly discover they've been tricked, since they'll see the mirrors and maybe other gizmos where only the rabbit should be. Magicians go to great lengths to ensure the audience has a limited perspective of the illusion, keeping their apparatus out of view and hidden in the shadows. It is a very challenging task to truly make an object appear invisible, regardless of where the observer and the light source are located. Assuming that we can't change the optical properties of the object we are trying to conceal, we must instead try to imaging wrapping the object with some sort of cloak that will render both the hidden object and itself invisible, appearing as if light had passed through empty space. It's what the magician tried to do, but could only achieve with a bunch of heavy mirrors, and only if the observer were fixed in one location. It's very hard to imagine what sort of collection of mirrors or materials could allow light to detour around the concealed object, looking as though it had passed through space, no matter what direction it came from. Fortunately, it turns out we don't have to sort out all of those details to figure out how to make an invisibility cloak. At least conceptually, the design process turns out to be remarkably simple, and starts just with a little bit of imagination. First, we need a way to visualize empty space. We can do this by imagining a set of points fixed in space, which we connect by lines to form a grid. Space, of course, is continuous, so that there are an infinite number of points that can be specified in Cartesian coordinates by $(x, y, z)$; but we visualize space by plotting lines of $x$ along intervals of $y$, and lines of $y$ along intervals of $x$. Since light travels in a straight line we can easily depict the trajectory of a ray of light in space by just plotting a line. We've indicated a ray by the blue line on the grid, shown below, which represents the path that a ray of light will take in space. We often think of rays when we think of light, but in reality light is an electromagnetic wave, which varies throughout space. We can depict a wave by indicating its oscillations throughout space, as shown on the plot below (right). Here, the wave is a simple sinusoidal variation, with the wavelength being twice the distance between the white lines. This sort of depiction of a wave is more common at lower frequencies, where the earlier experiments first took place. Although it makes great sense to visualize empty space with a simple grid as above, we don't have to do it that way. We can create a coordinate transformation of any sort, and make the same sort of grid using the new transformed variables. Why would we want to do that? Let's not answer that right away, but just show the result first. We initially create a function that transforms us from our usual coordinates $(x, y)$ to a new set of coordinates $(x', y')$. We can write down any transformation we like that allows us to compute $x'(x, y)$ and $y'(x, y)$. If we want to create a cloak, it's important that the transformed coordinates become identical to the original coordinates at some point, otherwise we have to transform all of space. A common type of transformation is a radial transformation, that pushes all of the space within some circular region into a shell. The transformation can be visualized by plotting the same lines of constant $x$ or constant $y$, but in the new frame $(x', y')$. It's sometimes useful to write the transformation as a matrix equation relating the new coordinates to the old, like ${\bf{x'}} = {\bf{\Lambda x}}$. The result of this little mathematical exercise is the plot shown below. We haven't done anything real, we just are now looking at space in a different but very interesting way. First off, there are no grid lines in the core of the space. That means if the grid lines were to represent paths that light can take, no light can seemingly reach a portion of space. It's as if we created a hole in space. But, again, this is just an illusion of plotting space in a different set of coordinates! Now, if we actually could warp space physically, then this grid shows what would happen. Our original ray of light, shown by the straight blue line above, would take a curved trajectory in the transformed space. In fact, all rays of light entering the transformed region would be pushed away from the center. Anything placed within the center region would effectively be invisible, since no light could ever come into contact with it. The same sort of cloaking picture is true for a wave. If we plot the sinusoidal wave in the new coordinates, then we get the sort of picture shown above and to the right. Here, the flat wavefronts that enter the transform region are bent away from the core, and are then restored as they exit the other side. Comparing these visualizations of warped space to the mirror analogy at the top of the page, we can see that we have achieved invisibility, at least in principle. Rays of light are redirected around the object or region to be concealed, in this case by opening up a hole in space and pushing all the space from a circular region into a shell. Outside the shell, space is not affected. In addition, because we are just transforming space, we have created a shell that does not scatter light or wavesnothing reflects from the shell, and light that passes through returns to its trajectory as if it had passed through empty space. Unlike the magicians approach with mirrors, our potential invisibility cloak works no matter where the observer is and no matter where the light comes from. Transformation Media - Our proposed invisibility cloak is an intriguing device, but suffers from a major implementation problem: We cannot warp space. One might be inclined to dismiss the entire concept as unreasonable, except that there is an interesting way to exploit the transformations and the very elegant means of managing light that is implied. Maxwell's equations in media, the equations that govern light and other electromagnetic waves, can be written to have the same form in any set of coordinates. The trick is that the presence of a medium enters Maxwell's equations as a couple of parameters that indicate the electric and magnetic response of a medium. When we transform our coordinate system, we can apply the transformation to these parameters such that the rest of the equations look identical as they did before the transformation. So, what changed? When we apply the transformations to the material parameters, we end up creating the specification for a new material. That material accomplishes exactly our goal, causing light to behave as if we had actually warped space! It's actually a stunning result, and results in a remarkably simple recipe. One doesn't have to understand the details and all the symbols to appreciate just how simple the recipe actually is. If our coordinate transformation is expressed as ${\bf{x'}} = {\bf{\Lambda x}}$, then the material parameters required to actually implement the transformation are specified by ${\bf{\varepsilon '}} = \frac{{\bf{\Lambda}}{\bf{\varepsilon}}{\bf{\Lambda}}^T}{\left\| {\bf{\Lambda }} \right\|} {\bf{\mu '}} = \frac{{\bf{\Lambda}}{\bf{\mu}}{\bf{\Lambda}}^T}{\left\| {\bf{\Lambda }} \right\|}$ That's all there is to it. Just from those relatively simple equations, we can obtain the design of a material that will cause light to propagate as if we had warped space. Now it's fair to ask, what's the catch? The catch is that the required material generally must be anisotropic (having different properties along different directions), and must have both electric and magnetic response. Moreover, those properties must vary throughout space--or at least over the transformed region. So a transformation optics solution requires a very complicated medium, or what we might call a transformation medium, to achieve the desired management of light. Trying to achieve a general transformation optical medium with conventional materials would be a difficult, if not an impossible task. However, we are not bound to using conventional materials when we create a transformation medium. Modern artificial materials, now often called metamaterials, have provided us with enough new capabilities that we can actually often achieve the incredibly demanding transformation optical properties. Metamaterials, combined with the transformation optical design approach, have now provided us with a new route to invisibility! # Useful References Controlling electromagnetic fields J. B. Pendry, D. Schurig, D. R. Smith Science 312, 1780 (2006) Metamaterial electromagnetic cloak at microwave frequencies D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, D. R. Smith Science 314, 977 (2006)	2017-09-19 13:18:24	{"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": 2, "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": 0.5801746249198914, "perplexity": 425.628278900204}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "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-2017-39/segments/1505818685698.18/warc/CC-MAIN-20170919131102-20170919151102-00646.warc.gz"}
https://www.jobilize.com/course/section/generalizations-and-nomenclature-by-openstax?qcr=www.quizover.com	# Hypothesis testing Page 1 / 2 Suppose you measure a collection of scalars ${x}_{1},,{x}_{N}$ . You believe the data is distributed in one of two ways. Your first model, call it ${H}_{0}$ , postulates the data to be governed by the density ${f}_{0}(x)$ (some fixed density). Your second model, ${H}_{1}$ , postulates a different density ${f}_{1}(x)$ . These models, termed hypotheses , are denoted as follows: ${H}_{0}:({x}_{n}, {f}_{0}(x)),n=1N$ ${H}_{1}:({x}_{n}, {f}_{1}(x)),n=1N$ A hypothesis test is a rule that, given a measurement $x$ , makes a decision as to which hypothesis best "explains" the data. Suppose you are confident that your data is normally distributed with variance 1, but you are uncertain aboutthe sign of the mean. You might postulate ${H}_{0}:({x}_{n}, (-1, 1))$ ${H}_{1}:({x}_{n}, (1, 1))$ These densities are depicted in . Assuming each hypothesis is a priori equally likely, an intuitively appealing hypothesis test is to compute the sample mean $\langle x\rangle =\frac{1}{N}\sum_{n=1}^{N} {x}_{n}$ , and choose ${H}_{0}$ if $\langle x\rangle \le 0$ , and ${H}_{1}$ if $\langle x\rangle > 0$ . As we will see later, this test is in fact optimal under certain assumptions. ## Generalizations and nomenclature The concepts introduced above can be extended inseveral ways. In what follows we provide more rigorous definitions, describe different kinds of hypothesis testing, andintroduce terminology. ## Data In the most general setup, the observation is a collection ${x}_{1},,{x}_{N}$ of random vectors. A common assumption, which facilitates analysis, is that the data are independent and identicallydistributed (IID). The random vectors may be continuous, discrete, or in some cases mixed. It is generally assumedthat all of the data is available at once, although for some applications, such as Sequential Hypothesis Testing , the data is a never ending stream. ## Binary versus m-ary tests When there are two competing hypotheses, we refer to a binary hypothesis test. When the number of hypotheses is $M\ge 2$ , we refer to an M-ary hypothesis test. Clearly, binary is a special case of $M$ -ary, but binary tests are accorded a special status for certain reasons. These includetheir simplicity, their prevalence in applications, and theoretical results that do not carry over to the $M$ -ary case. ## Phase-shift keying Suppose we wish to transmit a binary string of length $r$ over a noisy communication channel. We assign each of the $M=2^{r}$ possible bit sequences to a signal ${s}^{k}$ , $k=\{1, , M\}$ where ${s}_{n}^{k}=\cos (2\pi {f}_{0}n+\frac{2\pi (k-1)}{M})$ This symboling scheme is known as phase-shift keying (PSK). After transmitting a signal across the noisy channel, the receiver faces an $M$ -ary hypothesis testing problem: ${H}_{0}:x={s}^{1}+w$ ${H}_{M}:x={s}^{M}+w$ where $(w, (0, ^{2}I))$ . In many binary hypothesis tests, one hypothesis represents the absence of a ceratinfeature. In such cases, the hypothesis is usually labelled ${H}_{0}$ and called the null hypothesis. The other hypothesis is labelled ${H}_{1}$ and called the alternative hypothesis. ## Waveform detection Consider the problem of detecting a known signal $s=\left(\begin{array}{c}{s}_{1}\\ \\ {s}_{N}\end{array}\right)$ in additive white Gaussian noise (AWGN). This scenario is common in sonar and radar systems. Denotingthe data as $x=\left(\begin{array}{c}{x}_{1}\\ \\ {x}_{N}\end{array}\right)$ , our hypothesis testing problem is ${H}_{0}:x=w$ ${H}_{1}:x=s+w$ where $(w, (0, ^{2}I))$ . ${H}_{0}$ is the null hypothesis, corresponding to the absence of a signal. are nano particles real yeah Joseph Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master? no can't Lohitha where we get a research paper on Nano chemistry....? nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review Ali what are the products of Nano chemistry? There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others.. learn Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level learn da no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts Bhagvanji hey Giriraj Preparation and Applications of Nanomaterial for Drug Delivery revolt da Application of nanotechnology in medicine has a lot of application modern world Kamaluddeen yes narayan what is variations in raman spectra for nanomaterials ya I also want to know the raman spectra Bhagvanji I only see partial conversation and what's the question here! what about nanotechnology for water purification please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment. Damian yes that's correct Professor I think Professor Nasa has use it in the 60's, copper as water purification in the moon travel. Alexandre nanocopper obvius Alexandre what is the stm is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.? Rafiq industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong Damian How we are making nano material? what is a peer What is meant by 'nano scale'? What is STMs full form? LITNING scanning tunneling microscope Sahil how nano science is used for hydrophobicity Santosh Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq Rafiq what is differents between GO and RGO? Mahi what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq Rafiq if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION Anam analytical skills graphene is prepared to kill any type viruses . Anam Any one who tell me about Preparation and application of Nanomaterial for drug Delivery Hafiz what is Nano technology ? write examples of Nano molecule? Bob The nanotechnology is as new science, to scale nanometric brayan nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale Damian Is there any normative that regulates the use of silver nanoparticles? what king of growth are you checking .? Renato how did you get the value of 2000N.What calculations are needed to arrive at it Privacy Information Security Software Version 1.1a Good Got questions? Join the online conversation and get instant answers!	2021-05-10 19:01:35	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 37, "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": 0.6893340945243835, "perplexity": 1435.4557328579385}, "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/1620243991759.1/warc/CC-MAIN-20210510174005-20210510204005-00452.warc.gz"}
https://stats.stackexchange.com/questions/567402/the-distribution-of-the-difference-between-two-correlated-non-central-t-distribu?noredirect=1	# The distribution of the difference between two correlated non-central t distribution Suppose a binormal population $$\{X, Y\}$$ with means $$\mathbf{\mu} = \{\mu_1,\mu_2\} \ne \{0,0\}$$ and covariance $$\Sigma= \sigma^2\begin{bmatrix}1 & \rho\\ \rho &1 \end{bmatrix}$$. Let $$S^2$$ be an estimator of $$\sigma^2$$ with $$f$$ degrees of freedom. It is known that the variates $$\{X/S, Y/S\}$$ follows a bivariate non-central $$t$$ distribution (e.g., Kshirsagar, 1961). Walgren (1980) derived the distribution of the product $$X/S \times Y/S$$. Is there a derivation for their difference, $$X/S - Y/S$$? Edit: In the present context, I use the pooled standard deviation, that is the mean of the separate standard deviations, $$S_X = \sum_{i=1}^n (X_i-\bar{X})^2 / (n-1)$$ $$S_Y = \sum_{i=1}^n (Y_i-\bar{Y})^2 / (n-1)$$ with $$S = \sqrt{ (S_X^2 + S_Y^2)/2 }$$. This estimate of $$\sigma$$ is independent of both $$X$$ and $$Y$$, and as shown here is a chi-square distribution with degrees of freedom $$2(n-1)/(1+\rho^2)$$. • Please state which estimator of $\sigma^2$ you have in mind. (At least three would be natural in this setting and there are many others one could use as well.) – whuber Mar 10, 2022 at 23:25 • I edited the question. Mar 11, 2022 at 6:51 • I don't have a "derivation" but I'm getting $\frac{\Gamma \left(\frac{v+1}{2}\right) \left(2 (1-\rho ) \sigma ^2 v\right)^{v/2} \left(2 (1-\rho ) \sigma ^2 v+(-\mu_1 +\mu_2+z)^2\right)^{-\frac{1}{2} (v+1)}}{\sqrt{\pi } \Gamma \left(\frac{v}{2}\right)}$ for the pdf using Mathematica with $Z=X/S-Y/S$. – JimB Mar 11, 2022 at 7:03 • Do you mean that $(n-1)S^2/\sigma^2$ has a chi-square distribution with degrees of freedom $2(n-1)/(1+\rho^2)$ ? – JimB Mar 11, 2022 at 19:40 • @JimB The result from Ben in the link provided indicates that $2(n-1)/(1+\rho^2) S^2 / \sigma^2 \sim (1+\rho^2)/(2(n-1)) \chi^2_v$ in which $v = 2(n-1)/(1+\rho^2)$. Mar 11, 2022 at 20:01 Here is an approach that shows how to obtain a numerical approximation to the probability density function of $$Z=X/S-Y/S$$. (I haven't been successful in finding an analytic solution.) Using the joint density of $$X/S$$ and $$Y/S$$ found in Kshirsagar 1961 (as given in the question): r = {{1, \[Rho]}, {\[Rho], 1}}; (* Correlation matrix ) \[Mu] = {\[Mu]x, \[Mu]y}; t = {x, y}; f = 2 (n - 1)/(1 + \[Rho]^2); ( Degrees of freedom for estimate of S^2 ) jointPDF = (Exp[-\[Mu] . Inverse[r] . \[Mu]/(2 \[Sigma]^2)]/(\[Pi] f Sqrt[ Det[r]] Gamma[f/2])) Sum[(2^(\[Alpha]/2) (t . Inverse[r] . \[Mu])^\[Alpha] Gamma[(f + 2 + \[Alpha])/2])/ (\[Sigma]^\[Alpha] f^(\[Alpha]/2) \[Alpha]! (1 + t . Inverse[r] . t/f)^((f + 2 + \[Alpha])/2)), {\[Alpha], 0, \[Infinity]}]; jointPDF = FullSimplify[jointPDF, Assumptions -> {-1 < \[Rho] < 1, \[Sigma] > 0, n > 1, n \[Element] Integers, \[Mu]x \[Element] Reals, \[Mu]y \[Element] Reals, x \[Element] Reals, y \[Element] Reals}] A more readable version of the code is below: The result is The pdf of the difference $$Z = X/S - Y/S$$ can be found numerically by replacing $$y$$ with $$x-z$$ and then integrating over $$x$$: (* Numerical estimate of pdf of X/S - Y/S for a few values of n \ (sample size for estimating \[Sigma])) pdfz100 = Table[{z, NIntegrate[ jointPDF /. {y -> x - z, n -> 100, \[Sigma] -> 2, \[Rho] -> 1/2, \[Mu]x -> 1, \[Mu]y -> 3}, {x, -\[Infinity], \[Infinity]}]}, {z, -6, 3, 1/10}]; pdfz4 = Table[{z, NIntegrate[ jointPDF /. {y -> x - z, n -> 4, \[Sigma] -> 2, \[Rho] -> 1/2, \[Mu]x -> 1, \[Mu]y -> 3}, {x, -\[Infinity], \[Infinity]}]}, {z, -6, 3, 1/10}]; pdfz2 = Table[{z, NIntegrate[ jointPDF /. {y -> x - z, n -> 2, \[Sigma] -> 2, \[Rho] -> 1/2, \[Mu]x -> 1, \[Mu]y -> 3}, {x, -\[Infinity], \[Infinity]}]}, {z, -6, 3, 1/10}]; ListPlot[{pdfz100, pdfz4, pdfz2}, Joined -> True, ImageSize -> Large, PlotLegends -> {"n = 100", "n = 4", "n = 2"}, PlotLabel -> Style["\[Sigma] = 2, \[Rho] = 1/2, \!$$\SubscriptBox[$\[Mu]$$, \ $$x$$]$ = 1, \!$$\SubscriptBox[$\[Mu]$$, $$y$$]$ = 3", Bold, 18]] The results follow: As a check one should perform some simulations. parms = {\[Sigma] -> 2, \[Rho] -> 1/2, \[Mu]x -> 1, \[Mu]y -> 3, n -> 2}; nsim = 100000; ( Number of simulations ) ( Data for x and y ) data = RandomVariate[ BinormalDistribution[{\[Mu]x, \[Mu]y}, {\[Sigma], \[Sigma]}, \[Rho]] /. parms, nsim]; ( Data to for estimating S ) xy = RandomVariate[ BinormalDistribution[{mu1, mu2}, {\[Sigma], \[Sigma]}, \[Rho]] /. parms, {nsim, n /. parms}]; s = Sqrt[(Variance[#[[All, 1]]]/2 + Variance[#[[All, 2]]]/2) & /@ xy]; ( Z = X/S - Y/S ) zz = data[[All, 1]]/s - data[[All, 2]]/s; ( Numerically estimate the pdf of z ) pdfz = Table[{z, NIntegrate[jointPDF /. y -> x - z /. parms, {x, -\[Infinity], \[Infinity]}]}, {z, Quantile[zz, 0.005], Quantile[zz, 0.995], (Quantile[zz, 0.995] - Quantile[zz, 0.005])/200}]; ( Plot the results *) Show[Histogram[zz, "FreedmanDiaconis", "PDF"], ListPlot[pdfz, Joined -> True, PlotRange -> All]] There seems to be a match.	2023-02-01 19:34:21	{"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": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 24, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2757472097873688, "perplexity": 4061.6604425082814}, "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/1674764499949.24/warc/CC-MAIN-20230201180036-20230201210036-00043.warc.gz"}
https://zbmath.org/?q=an%3A0886.73031	# zbMATH — the first resource for mathematics On the asymptotic behaviour of sensitive shells with small thickness. (English. Abridged French version) Zbl 0886.73031 Summary: Sensitivity is a type of instability that appears in the limit behavior of certain shells as the thickness $$\varepsilon$$ tends to zero. We consider the behavior for small $$\varepsilon >0$$ in two cases. In the first case (elliptic shell clamped on a part of the boundary and free on the remainder), a Fourier expansion shows that the components of order $$k$$ grow exponentially with $$k$$ up to a saturation value $$\sim \log \varepsilon^{-1}$$. In the second example (elliptic shell submitted to $$u_3=0$$ on the boundary, $$u_3$$ is normal component of the displacement), a boundary layer appears with thickness and amplitude of orders $$\varepsilon^{1/2}$$ and $$\varepsilon^{-1/2}$$, respectively. ##### MSC: 74K15 Membranes 35Q72 Other PDE from mechanics (MSC2000) Full Text:	2021-05-10 05:38:12	{"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": 0.5888944268226624, "perplexity": 724.9008925115147}, "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/1620243989030.87/warc/CC-MAIN-20210510033850-20210510063850-00574.warc.gz"}
https://www.physicsforums.com/threads/linear-algebra-unitary-matrix.408753/	Homework Help: Linear Algebra: Unitary matrix 1. Jun 8, 2010 Niles 1. The problem statement, all variables and given/known data Hi My teacher told us that if we have a unitary matrix U, then $$\sum\limits_p {\left\| {U_{np} } \right\|^2 } = 1$$ Is that really correct? I thought he should be summing over n, not p. 2. Jun 8, 2010 Dick Use that if U is unitary, then the hermitian conjugate of U is unitary also to show you can sum over either index. 3. Jun 8, 2010 Niles Hmm, all I know is that U-1=UH. I cannot see how that helps me. 4. Jun 8, 2010 Dick Define V=U^H. Then V also satisfies V^(-1)=V^H. So V is also unitary. The sum over the second index for U is the same as the sum over the first index for V. 5. Jun 8, 2010 Niles I see, very smart. Thanks. Have a nice day.	2018-09-26 13:01:52	{"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": 0.8712549805641174, "perplexity": 1224.9630566651756}, "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-2018-39/segments/1537267164925.98/warc/CC-MAIN-20180926121205-20180926141605-00015.warc.gz"}
https://physics.stackexchange.com/questions/416011/newtons-bucket-artificial-gravity-absolute-rotation-and-machs-principle	# Newton's Bucket, Artificial Gravity, Absolute Rotation, and Mach's Principle I have been trying to understand how we can talk about absolute rotation in general relativity. I get that it is an area of active debate with some adherents of Mach's Principle and others believing that there simply exists absolute rotation. I think the best way of confronting the issue is trying to work with the simplest situation I can think of, and it seems to me that Mach's Principle cannot survive this situation. So here is the thought experiment: You are a on a cylindrically symmetric spaceship without any other objects in the universe. You start with everything at rest: you feel no forces, motion is described by the Minkowski metric. Then you start a large flywheel in the center of the ship rotating quite quickly. To preserve angular momentum, the ship will rotate in the opposite direction. You are now rotating with the ship, so you feel "artificial gravity", a force which forces you to the outer rim of the ship (you would call it a centrifugal force classically). We can perform an easy experiment that seems to show that we are rotating and in which direction: simply throw one ball in each tangential direction, one will fall slower and one will fall faster than a dropped ball. But given a relativistic framework it seems in bad taste to appeal to an absolute spacetime which we are rotating relative to, so why can't we claim that we on the spaceship are at rest and the flywheel in the center is rotating very quickly? Is there a way we could write a stress-energy tensor which would accurately describe motion in the spaceship without claiming a distinct "non-rotating frame"? Machians seem to be able to avoid absolute rotation by claiming all rotation is relative to distant bodies, but without any other bodies in the universe, what is our reference? This leads some to conclude that Newton's Bucket would not have the surface of the water become concave by the "rotation" in a universe without other bodies, but in our universe we started with a stationary ship, in a frame where we could use the Minkowski metric. Transforming the metric into the new (relatively rotating) frame would predict geodesic motion that give the effects of "artificial gravity", so there must clearly be rotational effects at play in this example. But if there were to be an observer which only came into existence after the ship had already started rotating, she could not know that in the past both the ship and the wheel had been at relative rest and the Minkowski metric applied, so how could she have a reference for the rotation. The only way all this seems possible to explain to me is to claim an absolute rotation which is not in reference to any other bodies. How can Mach's Principle survive this? Is there a valid way to write a stress-energy tensor in a cooridinate system which "thinks of" the spaceship as at rest and the rotating flywheel and/or the mass energy of the ship as giving all the odd effects we would like to attribute to rotation? More simply: is there any way to think of the spaceship as not rotating? It is my inclination that absolute rotation cannot be right as it seems to put us right back to the days before Einstein, but the conclusions seem difficult to escape. • Possible duplicate of Is Mach's Principle Wrong? See also other questions in the Related column. – sammy gerbil Jul 10 '18 at 18:53 • That question is in a similar vein but my question is primarily focused on if we can escape the idea of absolute rotation in even the simplest possible case, using Mach's principle as the most common formulation of a relative rotation. The primary question in my post is: is there a way we can think about the spaceship is not rotating? – Keefer Rowan Jul 10 '18 at 18:58 • Have you looked at the other questions in the Related column, eg Why does rotation simulate gravity if motion is relative? – sammy gerbil Jul 10 '18 at 19:08 • As an observer, I don't care if this question is close to a duplicate. It's perfectly asked and frames the question in a different way than I've read it before. – BuckFilledPlatypus Jul 10 '18 at 19:21 • Yeah, I think I've look at all other pertinent questions. In the question you linked for example, everyone gives one of two answers either a. rotation is absolute or b. rotation is relative, but relative to a background universe of bodies. The first answer still applies if that's what you want to claim, but if you want to claim rotation is relative (which is to me the more appealing option) my though experiment introduces new problems by removing the background of stars/galaxy to give as reference. It seems the notion of relative rotation can't survive the simplest example. – Keefer Rowan Jul 10 '18 at 19:23 It is my inclination that absolute rotation cannot be right as it seems to put us right back to the days before Einstein, but the conclusions seem difficult to escape. No, this is just a philosophical bias, which is not borne out at all by the actual math. In the very early days, it was thought that velocity was absolute. Then Galilean relativity came along and said the opposite. If one didn't pay attention, one might think that Galilean relativity means nothing is absolute: that is, "absolute acceleration cannot be right because it seems to put us right back to the days before Galileo". But that simply isn't true. You can't just say that because one thing isn't absolute, a completely different thing isn't absolute either -- that is lazy philosophizing. The same thing holds for angular velocity. You might argue that angular velocity is also called a velocity, so it has to be relative like linear velocity. But that's a rather superficial resemblance. In my book angular velocity is not a velocity at all, but rather a particular kind of periodic acceleration. And we know acceleration is absolute. To put it another way: we go out and observe certain symmetries of the universe. Translation invariance tells us position isn't absolute, boost invariance tells us velocity isn't absolute, and rotational invariance tells us angular orientation isn't absolute. There is no such observed symmetry for angular velocity. Is there a way we could write a stress-energy tensor which would accurately describe motion in the spaceship without claiming a distinct "non-rotating frame"? [...] If there were to be an observer which only came into existence after the ship had already started rotating, she could not know that in the past both the ship and the wheel had been at relative rest and the Minkowski metric applied, so how could she have a reference for the rotation. In the formalism of general relativity, the structure of rotating and non-rotating frames is already put in from the outset, in the form of the Levi-Civita connection. This is prior to the notion of any observer or any particular matter content. This makes general relativity not obey Mach's principle, though Einstein himself didn't like this. Specifically, suppose we are in Minkowski spacetime, where the connection is flat. An inertial frame is one where the connection coefficients are all zero. This is preserved by Lorentz transformations, but not by going to a rotating frame. Since the connection coefficients may be measured locally, an observer can find which frames are inertial even if they have no angular reference whatsoever. (The stress-energy tensor is found in just the usual way, but its conservation law $D_\mu T^{\mu\nu} = 0$ depends directly on the connection. The same goes for the geodesic equation.)	2019-08-25 03:06:56	{"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": 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": 0.6677661538124084, "perplexity": 282.1373450061701}, "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/1566027322170.99/warc/CC-MAIN-20190825021120-20190825043120-00048.warc.gz"}
https://socratic.org/questions/green-light-has-a-wavelength-of-5200-a-how-do-you-calculate-the-energy-of-one-ph	# Green light has a wavelength of 5200 A. How do you calculate the energy of one photon of green light? Aug 15, 2017 $E = 3.8 \times {10}^{-} 19$ $\text{J}$ #### Explanation: We're asked to calculate the energy of one photon of a green light, given its wavelength of $5200$ $\text{Å}$. To do this, we can use the equation ul(E = (hc)/f where • $E$ is the energy of the photon (in joules) • $h$ is Planck's constant, equal to $6.626 \times {10}^{-} 34$ $\text{J"•"s}$ • $c$ is the speed of light in vacuum, $299792458$ $\text{m/s}$ • $f$ is the frequency of the photon (in meters) We need to convert from ångströms to meters, using the conversion factor $1$ $\text{m}$ $= {10}^{10}$ $\text{Å}$: 5200cancel("Å")((1color(white)(l)"m")/(10^10cancel("Å"))) = color(red)(ul(5.2xx10^-7color(white)(l)"m" The energy of the photon is thus $\textcolor{b l u e}{E} = \left(\left(6.626 \times {10}^{-} 34 \textcolor{w h i t e}{l} \text{J"•cancel("s"))(299792458cancel("m/s")))/(color(red)(5.2xx10^-7cancel("m"))) = color(blue)(ulbar(\|stackrel(" ")(" "3.8xx10^-19color(white)(l)"J"" }\right) \|\right)$ rounded to $2$ significant figures.	2020-07-06 18:23:57	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 20, "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": 0.9117056131362915, "perplexity": 535.0760420364493}, "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/1593655881763.20/warc/CC-MAIN-20200706160424-20200706190424-00533.warc.gz"}
https://nbviewer.jupyter.org/github/applied-bioinformatics/built-iab/blob/master/IAB-notebooks/5/2.ipynb	5.2 Glossary ¶ 5.2.1 Pairwise alignment (noun)¶ A hypothesis about which bases or amino acids in two biological sequences are derived from a common ancestral base or amino acid. By definition, the aligned sequences will be of equal length with gaps (usually denoted with -, or . for terminal gaps) indicating hypothesized insertion deletion events. A pairwise alignment may be represented as follows: ACC---GTAC CCCATCGTAG 5.2.2 kmer (noun)¶ A kmer is simply a word (or list of adjacent characters) in a sequence of length k. For example, the overlapping kmers in the sequence ACCGTGACCAGTTACCAGTTTGACCAA are as follows: In [1]: import skbio skbio.DNA('ACCGTGACCAGTTACCAGTTTGACCAA').kmer_frequencies(k=5, overlap=True) It is common for bioinformaticians to substitute the value of k for the letter k in the word kmer. For example, you might here someone say "we identified all seven-mers in our sequence", to mean they identified all kmers of length seven.	2021-06-23 21:26:13	{"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": 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": 0.6581588983535767, "perplexity": 3659.7313787378694}, "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-2021-25/segments/1623488540235.72/warc/CC-MAIN-20210623195636-20210623225636-00389.warc.gz"}
https://mathhelpboards.com/authors/klaas-van-aarsen-178/?s=97b3882b6307de5c7f3d92596696315a	• ## Klaas van Aarsen There is no available content written by Klaas van Aarsen • ### Recent Forum Posts #### Re: Maximum value a function satisfying a differential equation can achieve. When I throw it at Octave online, I get: So it seems it is none of the above, but the answer $3$ is close. Klaas van Aarsen Today, 17:57 #### Re: MHB's future - sell, merge, or archive I'm more active on FMH, and still not great there, but I think a merge, in my humble opinion, would be a good thing for the help community. The firemath Today, 14:51 #### Maximum value a function satisfying a differential equation can achieve. Let $f:\mathbb R\to \mathbb R$ be a twice-differentiable function such that $f(x)+f^{\prime\prime}(x)=-x\|\sin(x)\|f'(x)$ for $x\geq 0$. Assume that $f(0)=-3$ caffeinemachine Today, 14:20 #### Re: Transportation calculus Thank you a lot for this help Skeeter, It really helped me with my work. You are a wonderful person. ducduy Yesterday, 23:00 #### Re: 3.4.6 limit of a power function Prove It is not giving you the answer he is giving you a suggestion that you can use the limit he posted. See if there is any kind of substitution you topsquark Yesterday, 17:02	2020-04-07 23:57:33	{"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": 0.7750972509384155, "perplexity": 1986.4209598743946}, "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-2020-16/segments/1585371806302.78/warc/CC-MAIN-20200407214925-20200408005425-00033.warc.gz"}
http://mathoverflow.net/questions/70457/open-mapping-theorem-for-riemann-surfaces	# Open mapping theorem for Riemann surfaces What restriction must one impose on a Riemann surface M in order for all biholomorphic $f:M\to\mathbb{C}$ to be open mappings, aka mappings of $M$ onto open subsets $f(M)\subset\mathbb{C}$? - The open mapping theorem from complex analysis carries over to Riemann surfaces basically immediately. – Jack Huizenga Jul 16 '11 at 0:03 Clarification: If you don't require Riemann surfaces to be connected then the correct statement would be: A holomorphic map $f : X \rightarrow Y$ between Riemann surfaces is an open map provided that it is not constant on any connected component of $X$.	2016-05-03 20:52:15	{"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": 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": 0.9519506096839905, "perplexity": 291.0991523411134}, "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-2016-18/segments/1461860121776.48/warc/CC-MAIN-20160428161521-00137-ip-10-239-7-51.ec2.internal.warc.gz"}
https://grazeconomics.wordpress.com/2020/01/20/intro-to-econ-tenth-lecture-aside-equal-opportunities/	# Intro to Econ: Tenth Lecture Aside – Equal Opportunities In this post I want to use the model and insight of the previous post to talk about equal opportunities. With this I mean the idea that everyone has the same access to education. I will argue that it is not for fairness but for efficiency reasons why a social planner might prefer a world with equal opportunities. I should also add that this post is a bit fanciful and one could possibly disagree with the way the argument goes. Take it with a grain of salt. Recall the example from the previous post. We had a small world consisting of three jobs and three people. The people were called Yvonne, Jacqueline, and Babette, and the jobs that they could have (we need one person doing each job) were manager, electro technician, and recycling expert. The following table states the potential added value (recall post on GDP) in thousands of euros per year that each person could create by working in one of the possible jobs. $\begin{tabular}{c\|ccc} & manager & technician & recycling expert \\ \hline Yvonne & 120 & 160 & 40 \\ Jacqueline & 200 & 220 & 140 \\ Babette & 100 & 100 & 180 \\ \end{tabular}$ Recall that this little job market had only one possible stable allocation (or matching) of people to jobs, the unique total added value maximizing one, in which Yvonne is technician, Jacqueline is manager, and Babette recycling expert. The total added value in this case is 160+200+180=540 thousand of euros per year. We also noted that this model does not allow us to make strong conclusions about the salaries in this job market. There is a range of possible (unequal) splits of value added between employees and employers. It was clear, however, that Jacqueline, by virtue of her high levels of added value, would have to make considerably more than Yvonne in this highly stylized job market. In fact, I believe it was 60 thousand euros a year more. But otherwise salaries can be higher or lower on the whole. It is just as stable to pay Yvonne and Babette 50 thousand a year and Jacqueline 110 thousand a year as it is to pay everyone 30 thousand a year more. This is so in this model and I am not completely sure whether this would hold up in better models that we would build if our primary interest was salaries. But as it is, this model leaves room for unionized bargaining, which does happen in many countries in the world (Austria, for instance). So perhaps this feature is not completely silly either. With unionized bargaining I mean that the employees all get together and negotiate a sort of base wage together as one negotiator against the also possibly unionized collective of employers as one other negotiator. But I wanted to address a different problem in this post. What do we expect to happen if we do and if we don’t have equal opportunities? One could argue that education is a way for people’s potential added value numbers to be generated. A person without an education could do very few jobs, that’s the idea at least. To be an engineer, for instance, you probably need to study engineering at least. Let us suppose for the moment that Yvonne simply has no access to education. And without education the added value she can provide is very limited in all three positions. In fact assume it is limited to 20 thousand a year in each job. The new situation can then be depicted in the following table. $\begin{tabular}{c\|ccc} & manager & technician & recycling expert \\ \hline Yvonne & 20 & 20 & 20 \\ Jacqueline & 200 & 220 & 140 \\ Babette & 100 & 100 & 180 \\ \end{tabular}$ Given the insight from the previous post we know that the only stable allocation or matching is the total added value maximizing one. Here this means that Yvonne will be manager (a very bad one and also badly paid), Jacqueline technician, and Yvonne recycling expert. The total added value is 20+220+180=420 thousand a year. By assumption this is less than in the case when Yvonne is educated. But more interestingly, now Jacqueline has less competition (when educated Yvonne is also not such a bad manager) and now Jacqueline needs to be paid at least 180 thousand a year. Also Babette now needs to be paid at least 80 thousand a year. This means that if you ask Jacqueline and Babette in this small world if they would like Yvonne to be educated they might well say no. Because then Yvonne could compete with them for these jobs which might mean that they might be paid less. As I said before, this post should be taken with a grain of salt. The model completely ignores the realistic possibility that an educated (and this could also mean well-trained, it does not have to mean “high-brow” education) person could create altogether new job possibilities, from which (almost) everyone in the economy could benefit. As an example consider someone who is educated in specific ways that enables him or her to start something like Microsoft or Google. Such a person then becomes a new employer and, in our model here, would create a fourth column in the matrix and thus new scope for added value. But I could imagine that in some societies the elites, being the only ones with access to education, worry that if everyone gets an education, then their possibly not so clever offspring will have a hard time getting well-paid jobs. It is otherwise difficult to see how one could be against equal opportunities, not for fairness but for efficiency reasons. But the model is probably really too simple to provide the definite answer to this question.	2020-02-19 22:45:38	{"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": 2, "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": 0.4161764979362488, "perplexity": 900.434320370138}, "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/1581875144429.5/warc/CC-MAIN-20200219214816-20200220004816-00453.warc.gz"}
https://www.gradesaver.com/textbooks/math/calculus/thomas-calculus-13th-edition/chapter-10-infinite-sequences-and-series-section-10-4-comparison-tests-exercises-10-4-page-591/30	## Thomas' Calculus 13th Edition Let $u_n=\dfrac{(\ln n)^2}{(n)^{3/2}}$ and $v_n=\dfrac{1}{ (n)^{5/4}}$ Now, $\lim\limits_{n \to \infty}\dfrac{u_n}{v_n} =\lim\limits_{n \to \infty}\dfrac{\dfrac{(\ln n)^2}{(n)^{(3/2)}}}{1/(n)^{(5/4)}}$ $\lim\limits_{n \to \infty} \dfrac{8 (\ln n)}{(n)^{1/4}}=0$ Thus, the series converges by the limit comparison test.	2020-03-31 07:23:46	{"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": 0.9995219707489014, "perplexity": 161.39182999798595}, "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-2020-16/segments/1585370500331.13/warc/CC-MAIN-20200331053639-20200331083639-00140.warc.gz"}
https://www.jiskha.com/display.cgi?id=1288753825	# Integration posted by . 1. [integration] (3x+7sin(x))^2 dx i tried 3 different methods of integration but all of my answers are wrong. 2. [Integration]from 0 to 5 (3w-3)/(w+6)dw i divided the numerator by the denominator and then tried to solve the question but im getting the wrong answer. ## Similar Questions 1. ### Competetion help on Integration If there four options are given to a Defenite Integration then what shortcut trick we should choose to get the correct answer. Like in Indefinte Integration we find the differtation of the answer to reach the function given to integrate. … 2. ### calc evaluate the integral: y lny dy i know it's integration by parts but i get confused once you have to do it the second time Leibnitz rule (a.k.a. product rule): d(fg) = f dg + g df y lny dy = d[y^2/2 ln(y)] - y/2 dy ----> Integral … 3. ### integration sqrt2 times e^t times t^(1/2) dt bounded between 0 and 1. i tried integration by parts.. but it keeps repeating.. please help. thank you! 4. ### culteral Diversity In Milton Gorton's theory of assimilation, the crucial step is from a. Integration to acculteration b. Acculteration to integration c. Assimilation to plurism d. Anglo-conformity to the melting pot e. Integration to intermarriage 5. ### Calc ..basic integration I forgto how to integrate...its been to long. integration(Sqrt(1+2t^2+t^4)) 6. ### math(easy integration) find the integration of absolute x from -4 to 2? 7. ### calculus reverse the order of integration integration 1-e integration 0-lnx dy dx 8. ### calculus The following definite integral can be evaluated by subtracting F(B) - F(A), where F(B) and F(A) are found from substituting the limits of integration. \int_{0}^{4} \frac{1600 x +1200 }{(2 x^2 +3 x +1)^5}dx After substitution, the … 9. ### AP Calc B/C the integration of (e^x)/(1-E^2x)^3/2 with respect to x We are currently doing integration by tables, but I can't find the formula that I should use! 10. ### Organic Chemistry (i can't upload an image so i have to explain the spectrum) My spectrum has peaks at 0.85 ppm (integration = 1.49) 1.2 ppm (integration = 7.93) 1.5 ppm (integration = 1.11) 3.6 ppm (integration = 1) the only one that shows clear peaks … More Similar Questions	2017-11-24 05:35:53	{"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": 0.8548431396484375, "perplexity": 5084.073069025039}, "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-2017-47/segments/1510934807089.35/warc/CC-MAIN-20171124051000-20171124071000-00574.warc.gz"}
https://dsp.stackexchange.com/questions/82058/are-there-any-techniques-that-can-achieve-higher-transmission-rate-than-shannon	# Are there any techniques that can achieve higher transmission rate than Shannon capacity? The maximum bit rate that can be transmitted over a channel with bandwidth B is determined by Shannon C=B log(1+S/N) Are there any techniques that could break this limit? • It's sorta like asking if there is a technique that can break the Conservation of Energy principle. Mar 19 at 20:18	2022-08-11 21:46:15	{"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": 0.8029299974441528, "perplexity": 446.13064255521357}, "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/1659882571502.25/warc/CC-MAIN-20220811194507-20220811224507-00291.warc.gz"}
https://www.onlinemathsschool.us/2018/11/limit-of-function.html	Limit of a Function The word "limit" has often been heard in our daily lives. For example, someone says, "My patience limit is almost gone" or "The credit card you use is almost close to the limit." Words like boundaries, almost, and limits have meaning to or close to something, very close, but unable to reach or not can be exactly the same. The use of these words has a relationship with the word limit which we will study below. Understanding the Concept of Limit Functions at a Point Limit is the basic concept for differential calculus and integral calculus material. In mathematical language, limits explain the value of a function if it is approached from a certain point, Why should it be approached from a certain point and not exactly at that particular point? This is because not all functions have values at all points. For example, the function $f (x) = \frac{x^2-1}{x-1}$ does not have a value (meaning) at x=1 because f(1) has an uncertain value of 0/0. If we take x values from greater than 1 (from the right) and smaller than 1 (from the left) close to 1 then the value of f(x) tends to approach the value 2. This value 2 is called the limit value of the function $f (x) = \frac {x^2-1}{x-1}$ when x approaches 1 from the left and right. Then the limit value of the function is called the limit function $f (x) = \frac{x^2-1}{x-1}$. Mathematically written: $\lim_{x \rightarrow 1} f (x) = \frac{x^2-1}{x-1} = 2$. A function f(x) is said to have a limit value at point x=a if the limit function f(x) for x approaches a from the left is equal to the limit function f(x) for x close to a from the right. Suppose the limit function f(x) for x approaches a from the left is $L_1$ and the limit function f (x) for x approaches a from the right is $L_2$. If $L_1 \neq L_2$, the limit function f(x) for x approaches a no. Conversely, if $L_1 = L_2 = L$, the limit of the function f(x) for x approaches a is L. Mathematically, the definition of the limit of the function is given in the following definition. Definition: A function y = f (x) is defined for x around a (then a is called the limit point), then $\lim_{x \rightarrow a} f (x) = L$ if and only if $\lim_{x \rightarrow a^-} f (x) = \lim_{x \rightarrow a^+} f (x) = L$. From the limit function definition above, the limit function search can be done using a numerical approach such as compiling a table of function values by taking the function domain from the left and right of a limit point; and the function graph approach is to see the graphic image of the function both from the left and the right of the limit point to know intuition the limit value of the function exists or does not exist. Able to apply the main limit theaters The following are limit Theorems which are useful in determining the limits of a function. Because the function you want to specify the limit can be in the form of numbers, fillings, times, and for functions that have known limits. Because if we only use a numerical approach or this graph is very inefficient and effective. 1. If f(x)=k then $\lim_{x \rightarrow a} f (x) = k$ (for each constant k and a real number). 2. If k is a constant $\lim_{x \rightarrow a} k.f(x) = k \lim_{x \rightarrow a} f(x)$ 3. $\lim_{x \rightarrow a} [f (x) + g (x)] = \lim_{x \rightarrow a} f (x) + \lim_{x \rightarrow a} g (x)$ 4. $\lim_{x \rightarrow a} [f (x) - g (x)] = \lim_ {x \rightarrow a} f (x) - \lim_ {x \rightarrow a} g (x)$ 5. $\lim_{x \rightarrow a} [f (x) \times g (x)] = \lim_{x \rightarrow a} f (x) \times \lim_{x \rightarrow a} g (x)$ 6. $\lim_{x \rightarrow a} \frac{f (x)}{g (x)} = \frac {\lim_ {x \rightarrow a} f (x)}{\lim_{x \rightarrow a} g (x)}$ 7. $\lim_{x \rightarrow a} [f (x)]^n = [\lim_{x \rightarrow a} f (x)]^n$ 8. $\lim_{x \rightarrow a} {^n \sqrt{f (x)}} = {^n \sqrt{\lim_{x \rightarrow a} f (x)}}$ Determine the Limit of the Algebraic Function Form $\lim_{x \rightarrow a} f (x)$ Determining $\lim_{x \rightarrow a} f (x)$ can be done by direct substitution method which is looking for the function value f(x) at x = a. Because, if f(x) has a meaningful value at x = a (defined at x=a) then $\lim_{x \rightarrow a} f (x) = f (a)$. Form $\lim_{x \rightarrow \infty} \frac{f (x)}{g (x)}$ To specify $\lim_{x \rightarrow \infty} \frac{f (x)}{g (x)}$, we must first understand why $\lim_{x \rightarrow \infty} \frac{1}{x} = 0$. We understand it intuitively that if 1 is divided by many numbers into infinity, the result tends to go to 0. Note the graphical function image $f (x) = \frac{1}{x}$ following, when $x \rightarrow \infty$ then $f (x) \rightarrow 0$ Because $\lim_{x \rightarrow \infty} \frac{1}{x} = 0$ then for every n positive number and a real number, $\lim_{x \rightarrow \infty} \frac{a}{x^n} = 0$. With this knowledge, determining the limit of the algebraic function $\lim_{x \rightarrow \infty} \frac{f (x)}{g (x)}$ (certain form) is done by dividing the numerator f(x) and part denominator g(x) with $x^n$, where n is the highest power of f(x) or g(x). Problems example: $\lim_{x \rightarrow \infty} \frac{2x^2 + 3x-1}{x + 2}$ Settlement: \begin{align} \lim_{x \rightarrow \infty} \frac{2x^2 + 3x-1}{x + 2} & = \lim_{x \rightarrow \infty} \frac{\frac{2x^2 + 3x-1}{x^2}}{\frac{x + 2}{x ^ 2}} \\ & = \lim_{x \rightarrow \infty} \frac{2+ \frac{3}{ x} - \frac{1}{x^2}}{\frac{1}{x} + \frac{2}{x ^ 2}} \\ & = \infty \end{align} $\lim_{x \rightarrow \infty} \frac{(1-2x)^2}{\sqrt{8x^4-1}}$ Settlement: \begin{align} \lim_{x \rightarrow \infty} \frac{(1-2x)^2}{\sqrt{8x^4-1}} & = \lim_{x \rightarrow \infty} \frac{\frac{(1-2x)^2}{x^2}}{\frac{\sqrt{8x^4-1}}{x^2}} \\ & = \lim_{x \rightarrow \infty } \frac{(\frac{1-2x}{x})^2}{\frac{\sqrt{8x ^ 4-1}}{\sqrt{x^4}}} \\ & = \lim_{ x \rightarrow \infty} \frac{(\frac{1}{x} - 2)^2}{\sqrt{8- \frac{1}{x^4}}} \\ & = \frac{4}{\sqrt{8}} \\ & = \frac{4}{2 \sqrt{2}} \\ & = \frac{2}{\sqrt{2}} \\ & = \sqrt{2} \end{align} Form $\lim_{x \rightarrow \infty} [\sqrt{f (x)} - \sqrt{g (x)}]$ The limit of the function in the form of $\lim_{x \rightarrow \infty} [\sqrt{f (x)} - \sqrt{g (x)}]$ can be solved by multiplying by the opposite factor, $\frac{\sqrt{f (x)} + \sqrt{g (x)}}{\sqrt{f (x)} + \sqrt {g (x)}}$ so that it becomes the form $\lim_{x \rightarrow \infty} \frac{j (x)}{k (x)}$. Problems example: $\lim_{x \rightarrow \infty} [\sqrt{2x-1} - \sqrt{3x + 5}]$ Settlement: \begin{align} & \lim_{x \rightarrow \infty} [\sqrt{2x-1} - \sqrt{3x + 5}] \\ & = \lim_{x \rightarrow \infty} [\sqrt{2x-1} - \sqrt{3x + 5}] \times \frac{\sqrt{2x-1} + \sqrt{3x + 5}}{\sqrt{2x-1} + \sqrt{3x + 5 }} \\ & = \lim_{x \rightarrow \infty} \frac{{2x-1} - (3x + 5)}{\sqrt{2x-1} + \sqrt{3x + 5}} \\ & = \lim_{x \rightarrow \infty} \frac{-x-6}{\sqrt{2x-1} + \sqrt{3x + 5}} = - \infty \end{align} Form $\lim_{x \rightarrow \infty} [\sqrt{ax^2 + bx + c} - \sqrt{px^2 + qx + r} ]$ This form often appears in high school / equivalent national exam questions and can be completed quickly using the following conditions. 1. If a = p then $\lim_{x \rightarrow \infty} [\sqrt{ax^2 + bx + c} - \sqrt{px^2 + qx + r}] = \frac{bq}{2 \sqrt{a}}$ 2. If a > p then $\lim_{x \rightarrow \infty} [\sqrt{ax^2 + bx + c} - \sqrt{px^2 + qx + r}] = \infty$ 3. If a < p is $\lim_{x \rightarrow \infty} [\sqrt{ax^2 + bx + c} - \sqrt{px^2 + qx + r}] = - \infty$ Problems example: Calculate $\lim_{x \rightarrow \infty} [\sqrt{3x^2-4x + 8} - \sqrt{3x^2-2x + 7}]$ Settlement: Because a = p then \begin{align} \lim_{x \rightarrow \infty} [\sqrt{3x^2-4x + 8} - \sqrt{3x^2-2x + 7}] & = \frac{bq}{2 \sqrt{a}} \\ & = \frac{-4 - (- 2)}{2 \sqrt{3}} \\ & = \frac{-2}{2 \sqrt{3}} \\ & = - \frac{1}{\sqrt{3}} \\ & = - \frac{\sqrt{3}}{3} \end{align} Determining the Limit of Trigonometric Functions In some cases, the limit resolution of trigonometric functions is almost the same as the completion of the limit of algebraic functions, for example by direct substitution methods. If the direct substitution method produces an indeterminate value then it is done by factoring method so that it is not indeterminate anymore. The trigonometric formulas you have learned and the main limit theorem are useful in completing the limits of trigonometric functions. Problems example: 1. $\lim_{x \rightarrow \frac{\pi}{4}} sin \ x = sin \ \frac{\pi}{4} = \frac{1}{2} \sqrt{2}$ 2. \begin{align} \lim_{x \rightarrow 0} (cos^2 \ x - sin^2 \ x) & = (\lim_{x \rightarrow 0} cos \ x)^2 - (\lim_{x \rightarrow 0} sin \ x)^2 \\ & = (1)^2- (0)^2 \\ & = 1 \end{align} Because $sin \ 2x = 2sin \ x \ cos \ x$ then \begin{align} \lim_{x \rightarrow 0} \frac{sin \ x}{sin \ 2x} & = \lim_{x \rightarrow 0} \frac{sin \ x}{2sin \ x \ cos \ x} \\ & = \lim_{x \rightarrow 0} \frac{1}{2 \ cos \ x} \\ & = \frac{1}{2 \ cos \ (0)} \\ & = \frac{1}{2 \times 1} \\ & = \frac{1}{2} \end{align} The limits of the trigonometric function can also be solved using formulas. The trigonometric function formulas in question are: $\lim_{x \rightarrow 0} \frac{sin \ x}{x} = \lim_{x \rightarrow 0} \frac{x}{sin \ x} = 1$ $\lim_{x \rightarrow 0} \frac{tan \ x}{x} = \lim_{x \rightarrow 0} \frac{x}{tan \ x} = 1$ The limit formula for the basic trigonometric function above can be expanded. Suppose u is a function of x and if $x \rightarrow 0$ then $u \rightarrow 0$, so that the formulas can be written to be: 1. $\lim_{u \rightarrow 0} \frac{sin \ u}{u} = \lim_{u \rightarrow 0} \frac{u}{sin \ u} = 1$ 2. $\lim_{u \rightarrow 0} \frac{tan \ u}{u} = \lim_{u \rightarrow 0} \frac{u}{tan \ u} = 1$ Problems example: $\lim_{u \rightarrow 0} \frac{1 - cos \ x}{x^2}$ Settlement: If we do the direct substitution method, it turns out the result is in the form of indeterminate 0/0. This problem can be solved by using trigonometric similarities $cos \ 2x = 1 - 2 \ sin^2 \ x$ to change $1 - cos \ x$ so we can use the formula $\lim_{u \rightarrow 0} \frac{sin \ u}{u} = \lim_{u \rightarrow 0} \frac{u}{sin \ u} = 1$. Because $cos \ 2x = 1 - 2 \ sin^2 \ x$ then $cos \ x = 1 - 2 \ sin^2 \ (\frac{1}{2} x) \Leftrightarrow 1-cos \ x = 2 \ sin^2 \ (\frac{1}{2} x)$, and by specifying $u = \frac{1}{2} x$ then \begin{align} \lim_{x \rightarrow 0} \frac{1 - cos \ x}{x^2} & = \lim_{x \rightarrow 0} \frac{2 \ sin^2 \ \frac{ 1}{2} x}{x^2} \\ & = \lim_{x \rightarrow 0} \frac{2 \ sin^2 \frac{1}{2} x}{x^2} \times \frac{\frac{1}{4}}{\frac{1}{4}} \\ & = \lim_{x \rightarrow 0} \frac{1}{2} \ \frac{sin^2 \ \frac{1}{2} x}{(\frac{1}{2} x)^2} \\ & = \lim_{u \rightarrow 0} \frac{1}{2} \ \frac{sin^2 \ u} {u^2} \\ & = \frac{1}{2} \lim_{u \rightarrow 0}(\frac{sin \ u}{u})^2 \\ & = \frac{1}{2} (\lim_{u \rightarrow 0} \frac{sin \ u}{u})^2 \\ & = \frac{1}{2}. (1)^2 \\ & = \frac{1}{2} \end{align}	2018-12-15 19:55:39	{"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": 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": 7, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9999963045120239, "perplexity": 409.9149377760702}, "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-2018-51/segments/1544376826968.71/warc/CC-MAIN-20181215174802-20181215200802-00234.warc.gz"}
https://support.sas.com/documentation/cdl/en/statug/66103/HTML/default/statug_qreg_details06.htm	# The QUANTREG Procedure ### Confidence Interval The QUANTREG procedure provides three methods to compute confidence intervals for the regression quantile parameter : sparsity, rank, and resampling. The sparsity method is the most direct and the fastest, but it involves estimation of the sparsity function, which is not robust for data that are not independently and identically distributed. To deal with this problem, the QUANTREG procedure computes a Huber sandwich estimate by using a local estimate of the sparsity function. The rank method, which computes confidence intervals by inverting the rank score test, does not suffer from this problem, but it uses the simplex algorithm and is computationally expensive with large data sets. The resampling method, which uses the bootstrap approach, addresses these problems, but at a computation cost. Based on these properties, the QUANTREG uses a combination of the resampling and rank methods as the default. For data sets with more than either 5,000 observations or 20 variables, the QUANTREG procedure uses the MCMB resampling method; otherwise it uses the rank method. You can request a particular method by using the CI= option in the PROC QUANTREG statement. #### Sparsity Consider the linear model and assume that , , are iid with a distribution F and a density , where in a neighborhood of . Under some mild conditions where and See Koenker and Bassett (1982b). This asymptotic distribution for the regression quantile can be used to construct confidence intervals. However, the reciprocal of the density function which is called the sparsity function, must first be estimated. Since can be estimated by the difference quotient of the empirical quantile function—that is, where is an estimate of and is a bandwidth that tends to zero as . The QUANTREG procedure provides two bandwidth methods. The Bofinger bandwidth is an optimizer of mean squared error for standard density estimation, and the Hall-Sheather bandwidth is based on Edgeworth expansions for studentized quantiles, where is the second derivative of and satisfies for the construction of confidence intervals. The quantity is not sensitive to f and can be estimated by assuming f is Gaussian. can be estimated by the empirical quantile function of the residuals from the quantile regression fit, or the empirical quantile function of regression proposed by Bassett and Koenker (1982), The QUANTREG procedure interpolates the first empirical quantile function and gets the piecewise linear version is set to a constant if falls outside . This estimator of the sparsity function is sensitive to the iid assumption. Alternately, Koenker and Machado (1999) considered the non-iid case. By assuming local linearity of the conditional quantile function in x, they proposed a local estimator of the density function by using the difference quotient. A Huber sandwich estimate of the covariance and standard error is computed and used to construct the confidence intervals. One difficulty with this method is the selection of the bandwidth when using the difference quotient. With a small sample size, either the Bofinger or the Hall-Sheather bandwidth tends to be too large to assure local linearity of the conditional quantile function. The QUANTREG procedure uses a heuristic bandwidth selection in these cases. By default, the QUANTREG procedure computes non-iid confidence intervals. You can request iid confidence intervals with the IID option in the PROC statement. #### Inversion of Rank Tests The classical theory of rank tests can be extended to test the hypothesis : in the linear regression model . Here . See Gutenbrunner and Jureckova (1992) for more details. By inverting this test, confidence intervals can be computed for the regression quantiles that correspond to . The rank score function can be obtained by solving the dual problem For a fixed quantile , integrating with respect to the -quantile score function yields the -quantile scores Under the null hypothesis : for large n, where . Let Then from the constraint in the full model. In order to obtain confidence intervals for , a critical value can be specified for . The dual vector is a piecewise constant in , and can be altered without compromising the optimality of as long as the signs of the residuals in the primal quantile regression problem do not change. When gets to such a boundary, the solution does change, but can be restored by taking one simplex pivot. The process can continue in this way until exceeds the specified critical value. Since is piecewise constant, interpolation can be used to obtain the desired level of confidence interval; see Koenker and d’Orey (1994). #### Resampling The bootstrap can be implemented to compute confidence intervals for regression quantile estimates. As in other regression applications, both the residual bootstrap and the xy-pair bootstrap can be used. The former assumes iid random errors and resamples from the residuals, while the later resamples xy pairs and accommodates some forms of heteroscedasticity. Koenker (1994) considered a more interesting resampling mechanism, resampling directly from the full regression quantile process, which he called the Heqf bootstrap. In contrast with these bootstrap methods, Parzen, Wei, and Ying (1994) observed that which is the estimating equation for the regression quantile, is a pivotal quantity for the quantile regression parameter . In other words, the distribution of can be generated exactly by a random vector , which is a weighted sum of independent, re-centered Bernoulli variables. They further showed that for large n, the distribution of can be approximated by the conditional distribution of , where solves an augmented quantile regression problem with n + 1 observations with and sufficiently large for a given realization of u. By exploiting the asymptotically pivot role of the quantile regression gradient condition, this approach also achieves some robustness to certain heteroscedasticity. Although the bootstrap method by Parzen, Wei, and Ying (1994) is much simpler, it is too time-consuming for relatively large data sets, especially for high-dimensional data sets. The QUANTREG procedure implements a new, general resampling method developed by He and Hu (2002), which is referred to as the Markov chain marginal bootstrap (MCMB). For quantile regression, the MCMB method has the advantage that it solves p one-dimensional equations instead of p-dimensional equations, as do the previous bootstrap methods. This greatly improves the feasibility of the resampling method in computing confidence intervals for regression quantiles.	2021-09-17 12:08:57	{"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": 0.8662430644035339, "perplexity": 818.3538092851832}, "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-2021-39/segments/1631780055645.75/warc/CC-MAIN-20210917120628-20210917150628-00675.warc.gz"}
https://socratic.org/questions/what-is-the-square-root-of-50-the-square-root-of-8-1#564832	What is the square root of 50 + the square root of 8? Mar 5, 2018 See explanation. Explanation: $\sqrt{50} + \sqrt{8} = \sqrt{2 \cdot 25} + \sqrt{2 \cdot 4} = 5 \sqrt{2} + 2 \sqrt{2} = 7 \sqrt{2}$ Mar 5, 2018 $7 \sqrt{2}$ Explanation: $\textcolor{m a \ge n t a}{\sqrt{50} + \sqrt{8}}$ $\textcolor{p u r p \le}{\implies \sqrt{2 \times 5 \times 5} + \sqrt{2 \times 2 \times 2}}$ $\textcolor{\mathmr{and} a n \ge}{\implies 5 \sqrt{2} + 2 \sqrt{2}}$ $\textcolor{g r e e n}{\implies 7 \sqrt{2}}$ Hope it helps :)	2021-10-28 18:30:58	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 6, "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": 0.6661396622657776, "perplexity": 1406.8784403437357}, "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/1634323588398.42/warc/CC-MAIN-20211028162638-20211028192638-00474.warc.gz"}
https://tz.vtlgbtcaucus.org/10508-what-bacteria-results-in-a-gram-ve-cocci-and-catalas.html	# What bacteria results in a Gram +ve cocci and catalase +ve? What test comes next? We are searching data for your request: Forums and discussions: Manuals and reference books: Data from registers: Wait the end of the search in all databases. Upon completion, a link will appear to access the found materials. # Results so far. We are trying to determine unknown microorganisms in intro to microbiology course. I first did gram stain and they were all cocci morphology, purple color and clumped together (actually I'm not so sure if clumped is right word, but they definitely were not in chains and I didn't see any isolated coccus, so if that's qualifies as clumped then they were clumped. So I have a gram+ here. Next I did catalase test and got bubbles on slide; I put thick chunk of the unknown on slide and dropped 2-3 drops of $${H}_{2}{O}_2$$; instant bubbling. # Next test. So now I will test for coagulase but I'm uncertain what a positive or negative result will tell me about he organism and whether this is the terminal test (for ID-ing organism I mean) My notes say that positive for coagulase indicates S. aureus. But does that mean that it is unnecessary to do the Mannitol test at that point then? Or is Mannitol used as a confirmation test? Further, if unknown tests negative for coagulase, my notes say that if its also novobiocin sensitive (pos) then it is S. epidermidis. But my same question then is, is it necessary to still do the mannitol test? At this point you have staph, you can use this flow chart (from here) to figure out what staph exactly. If the test comes back as coagulase positive be very careful as Staph aureus is pathogenic. ## DNase Test: Principle, Procedure, Results DNA hydrolysis test or Deoxyribonuclease (DNase) test is used to determine the ability of an organism to hydrolyze DNA and utilize it as a source of carbon and energy for growth. An agar medium DNase agar, a differential medium is used to test the ability of an organism to produce deoxyribonuclease or DNase. This medium is pale green in color because of the DNA-methyl green (indicator) complex (Note: Methyl green is a cation that binds to the negatively-charged DNA). It also contains nutrients for the bacteria. Figure -1: DNA Hydrolysis test A. Positive Staphylococcus aureus B. Positive Serratia marcescens C. Negative: Staphylococcus epidermidis If the organism that grows in the medium produces Deoxyribonuclease, it breaks down DNA into smaller fragments. When the DNA is broken down, it no longer binds to the methyl green, and green color fades and the colony is surrounded by a colorless zone (See fig-1). #### Requirements: 1. Media: DNase Agar or DNase agar with Methyl green indicator. 2. Reagent: Hydrochloric acid (1mol/L) only when DNase agar without an indicator is used 3. Others: Inoculating loop, Bunsen burner ### Procedure of DNase (DNA hydrolysis test) 1. Dry the surface of agar plates before use. Each plate may be divided into sections by drawing lines on the bottom of the plate. 2. Inoculate the test agar medium: There are two types of inoculation that can be done. • Touch a colony of the organism under test with a loop and inoculate it onto a small area of the DNase test agar plate, in the middle of one of the marked sections to form a thick plaque of growth 5-10 mm in diameter after incubation. • Incubate the plate at 37°C for 18-24hr. • Use a heavy inoculum and draw a line 3-4 cm long from the rim to the center of the DNase test agar plate • Incubate the plate at 37°C for 18-24hr. 1. When using DNase agar without the indicator, • Flood the plate with 1N Hydrochloric Acid. • Leave the plate to stand for a few minutes to allow the reagent to absorb into the plate. Decant excess hydrochloric acid and then examine the plate within 5 minutes against a dark background. Fig:2: DNase Test: M. catarrhalis (+ve) and N.gonorrhoeae (-ve). When DNase is produced by organisms, an acidic end product is formed and the pH indicator changes from red (alkaline) to yellow (acid). ## Starch Hydrolysis Test: Principle, Procedure, Results Starch hydrolysis test is used to determine if the organism is capable of breaking down starch into maltose through the activity of the extra-cellular α-amylase enzyme. Starch, the most important source of carbohydrate for humans, is a polysaccharide mixture of two polymers, amylose, and amylopectin, the latter being predominant. Amylose is a linear polysaccharide of several thousand α-D-glucose linked by 1,4-α-glycosidic bonds. Amylopectin is a branched-chain polysaccharide composed of glucose units linked primarily by α-1,4-glycosidic bonds but with occasional α-1,6-glycosidic bonds, which are responsible for the branching. ### Principle Starch molecules are too large to enter the bacterial cell, so only bacteria that secrete exoenzymes (α -amylase and oligo-1,6-glucosidase) are able to hydrolyze starch into subunits (dextrin, maltose, or glucose). These molecules are readily transported into the bacterial cell to be used in metabolism. In starch hydrolysis test (also known as amylase test), we use starch agar, which is a differential nutritive medium. The test organisms are inoculated onto a starch plate and incubated at 30°C until growth is seen (i.e. up to 48 hours). The Petri plate is then flooded with an iodine solution. If there is no enzyme present, and therefore no hydrolysis, the amylose and iodine react together to form a blue color. Depending on the concentration of the iodine used, iodine turns blue, purple, or black in the presence of starch. When bacteria capable of producing α-amylase and oligo-1,6-glucosidase are grown on starch agar, they secrete enzymes into the surrounding areas and hydrolyze the starch. As no amylose is present in the medium surrounding the bacterial colony, clearing around the bacterial growth is seen (there is no color development). ## Diseases caused by S.pyogenes Mnemonic: Diseases caused by Streptococcus pyogenes: NIPPLES: Necrotising fasciitis and myositis Impetigo Pharyngitis Pneumonia Lymphangitis Erysipelas and cellulitis Scarlet fever/ Streptococcal TSS 1. Necrotizing fasciitis (NF) 2. Streptococcal toxic shock syndrome (STSS) 3. Cellulitis 4. Bacteremia 5. Pneumonia 6. Puerperal sepsis ### Key Tests that are used to identify S. pyogenes: The sample for the isolation/identification of S. pyogenes is either pharyngeal exudates, pus, blood, tissue, or body fluids depending on the sites and nature of infection. ## Acinetobacter: Disease, Properties, Resistance Acinetobacter is a group of bacteria commonly found in soil, water, and dry environments. Acinetobacter poses very little risk to immune-competent people and the infections are mainly confined in healthcare settings housing very ill patients. People with a weakened immune system are susceptible to infections with Acinetobacter. They acquire Acinetobacter infections by person-to-person contact or contact with contaminated surfaces. Immunocompromised patients i.e. people who have weakened immune systems, chronic lung disease, or diabetes are susceptible to this infection. Very ill patients on a ventilator, those with a prolonged hospital stay, persons having invasive devices like urinary catheters are at greater risk of Acinetobacter infections. Outbreaks of Acinetobacter infections typically occur in intensive care units (ICU). Acinetobacter can live on the skin and may survive in the environment/inanimate surfaces for several days. Acinetobacter is associated with skin colonization of hospital personnel and may also “colonize” or live in a patient without causing infection or symptoms, especially in tracheostomy sites or open wounds. While there are many species of Acinetobacter and all can cause human disease, Acinetobacter baumannii for about 80% of reported infections. Acinetobacter causes a variety of diseases, ranging from pneumonia to serious blood or wound infections, and the symptoms vary depending on the disease. It is an important cause of ventilator-associated pneumonia and catheter-related bacteremia. Biochemical Properties: : Gram-negative cocci or coccobacilli 1. Oxygen requirement: Strictly aerobic 2. Growth requirements: Non-fastidious: Non fermentative: Negative Positive (+ve): Negative (-ve): Positive (+ve) some species may not give a positive citrate utilization test.: Negative (-ve): Negative except. A.haemolyticus 3. Chloramphenicol: Resistant: +ve Drug Resistance and Antibiotics in use Acinetobacter is often resistant to many commonly prescribed antibiotics. Multiple Drug Resistance (MDR) patterns observed in Acinetobacter baumannii (MDR-AB) currently pose significant challenges for the management and treatment of infections. CDC has categorized Multidrug-resistant Acinetobacter as a serious threats to public health. There are few antimicrobial agents that are commonly used for the treatment of infections with Acinetobacter baumannii . ## Introduction Actinomyces spp and Propionibacterium propionicus (previously Arachnia propionica) are members of a large group of pleomorphic Gram-positive bacteria, many of which fhave some tendency toward mycelial growth. Both are members of the oral flora of humans or animals. Actinomyces species, in particular, are major components of dental plaque. A israelii, A gerencseriae (previously A israelii serotype II), and P propionicus cause actinomycosis in humans and animals. Other species of Actinomyces can be involved in mixed anaerobic and other infections, where they may not always play an obviously pathogenic role. In addition, some coryneform bacteria (diphtheroids) isolated from clinical samples, which had been placed into the Centers for Disease Control Coryneform groups 1, 2 and E, have been identified as new species of Actinomyces. ## Contents The genus Corynebacterium was created by Lehmann and Neumann in 1896 as a taxonomic group to contain the bacterial rods responsible for causing diphtheria. The genus was defined based on morphological characteristics. Based on studies of 16S-rRNA, they have been grouped into the subdivision of gram-positive eubacteria with high G:C content, with close phylogenetic relationship to Arthrobacter, Mycobacterium, Nocardia, and Streptomyces. [7] The term comes from Greek κορύνη, korýnē 'club, mace, staff, knobby plant bud or shoot' [8] and βακτήριον, baktḗrion 'little rod'. [9] The term "diphtheroids" is used to represent corynebacteria that are non-pathogenic for example, C. diphtheriae would be excluded. [ citation needed ] The term diphtheroid comes from Greek διφθέρα, diphthérā 'prepared hide, leather'. [10] [11] Comparative analysis of corynebacterial genomes has led to the identification of several conserved signature indels which are unique to the genus. Two examples of these conserved signature indels are a two-amino-acid insertion in a conserved region of the enzyme phosphoribose diphosphate:decaprenyl-phosphate phosphoribosyltransferase and a three-amino-acid insertion in acetate kinase, both of which are found only in Corynebacterium species. Both of these indels serve as molecular markers for species of the genus Corynebacterium. Additionally, 16 conserved signature proteins, which are uniquely found in Corynebacterium species, have been identified. Three of the conserved signature proteins have homologs found in the genus Dietzia, which is believed to be the closest related genus to Corynebacterium. In phylogenetic trees based on concatenated protein sequences or 16S rRNA, the genus Corynebacterium forms a distinct clade, within which is a distinct subclade, cluster I. The cluster is made up of the species C. diphtheriae, C. pseudotuberculosis, C. ulcerans, C. aurimucosum, C. glutamicum, and C. efficiens. This cluster is distinguished by several conserved signature indels, such as a two-amino-acid insertion in LepA and a seven- or eight-amino-acid insertions in RpoC. Also, 21 conserved signature proteins are found only in members of cluster I. Another cluster has been proposed, consisting of C. jeikeium and C. urealyticum, which is supported by the presence of 19 distinct conserved signature proteins which are unique to these two species. [12] Corynebateria have a high G+C content ranging from 46-74 mol%. [13] The principal features of the genus Corynebacterium were described by Collins and Cummins in 1986. [14] They are gram-positive, catalase-positive, non-spore-forming, non-motile, rod-shaped bacteria that are straight or slightly curved. [15] Metachromatic granules are usually present representing stored phosphate regions. Their size falls between 2 and 6 μms in length and 0.5 μm in diameter. The bacteria group together in a characteristic way, which has been described as the form of a "V", "palisades", or "Chinese characters". They may also appear elliptical. They are aerobic or facultatively anaerobic, chemoorganotrophs. They are pleomorphic through their lifecycles, they occur in various lengths, and they frequently have thickenings at either end, depending on the surrounding conditions. [16] ### Cell wall Edit The cell wall is distinctive, with a predominance of mesodiaminopimelic acid in the murein wall [4] [15] and many repetitions of arabinogalactan, as well as corynemycolic acid (a mycolic acid with 22 to 26 carbon atoms), bound by disaccharide bonds called L-Rhap-(1 → 4)--D-GlcNAc-phosphate. These form a complex commonly seen in Corynebacterium species: the mycolyl-AG–peptidoglican (mAGP). [17] ### Culture Edit Corynebacteria grow slowly, even on enriched media. In terms of nutritional requirements, all need biotin to grow. Some strains also need thiamine and PABA. [14] Some of the Corynebacterium species with sequenced genomes have between 2.5 and 3.0 million base pairs. The bacteria grow in Loeffler's medium, blood agar, and trypticase soy agar (TSA). They form small, grayish colonies with a granular appearance, mostly translucent, but with opaque centers, convex, with continuous borders. [15] The color tends to be yellowish-white in Loeffler's medium. In TSA, they can form grey colonies with black centers and dentated borders that look similar to flowers (C. gravis), or continuous borders (C. mitis), or a mix between the two forms (C. intermedium). Corynebacterium species occur commonly in nature in the soil, water, plants, and food products. [4] [15] The nondiphtheiroid Corynebacterium species can even be found in the mucosa and normal skin flora of humans and animals. [4] [15] Unusual habitats, such as the preen gland of birds have been recently reported for Corynebacterium uropygiale. [18] Some species are known for their pathogenic effects in humans and other animals. Perhaps the most notable one is C. diphtheriae, which acquires the capacity to produce diphtheria toxin only after interacting with a bacteriophage. [19] [20] Other pathogenic species in humans include: C. amycolatum, C. striatum, C. jeikeium, C. urealyticum, and C. xerosis [21] [22] [23] [24] [25] all of these are important as pathogens in immunosuppressed patients. Pathogenic species in other animals include C. bovis and C. renale. [26] This genus has been found to be part of the human salivary microbiome. [27] The most notable human infection is diphtheria, caused by C. diphtheriae. It is an acute and contagious infection characterized by pseudomembranes of dead epithelial cells, white blood cells, red blood cells, and fibrin that form around the tonsils and back of the throat. [28] In developed countries, it is an uncommon illness that tends to occur in unvaccinated individuals, especially school-aged children, elderly, neutropenic or immunocompromised patients, and those with prosthetic devices such as prosthetic heart valves, shunts, or catheters. It is more common in developing countries [29] It can occasionally infect wounds, the vulva, the conjunctiva, and the middle ear. It can be spread within a hospital. [30] The virulent and toxigenic strains are lysogenic, and produce an exotoxin formed by two polypeptide chains, which is itself produced when a bacterium is transformed by a gene from the β prophage. [19] [20] Several species cause disease in animals, most notably C. pseudotuberculosis, which causes the disease caseous lymphadenitis, and some are also pathogenic in humans. Some attack healthy hosts, while others tend to attack the immunocompromised. Effects of infection include granulomatous lymphadenopathy, pneumonitis, pharyngitis, skin infections, and endocarditis. Corynebacterial endocarditis is seen most frequently in patients with intravascular devices. [31] Several species of Corynebacterium can cause trichomycosis axillaris. [32] C. striatum may cause axillary odor. [33] C. minutissimum causes erythrasma. Nonpathogenic species of Corynebacterium are used for very important industrial applications, such as the production of amino acids, [34] [35] nucleotides, and other nutritional factors (Martín, 1989) bioconversion of steroids [36] degradation of hydrocarbons [37] cheese aging [38] and production of enzymes. [39] Some species produce metabolites similar to antibiotics: bacteriocins of the corynecin-linocin type, [30] [40] [41] antitumor agents, [42] etc. One of the most studied species is C. glutamicum, whose name refers to its capacity to produce glutamic acid in aerobic conditions. [43] This is used in the food industry as monosodium glutamate in the production of soy sauce and yogurt. [ citation needed ] Species of Corynebacterium have been used in the mass production of various amino acids including glutamic acid, a food additive that is made at a rate of 1.5 million tons/ year. The metabolic pathways of Corynebacterium have been further manipulated to produce lysine and threonine. [ citation needed ] L-Lysine production is specific to C. glutamicum in which core metabolic enzymes are manipulated through genetic engineering to drive metabolic flux towards the production of NADPH from the pentose phosphate pathway, and L-4-aspartyl phosphate, the commitment step to the synthesis of L-lysine, lysC, dapA, dapC, and dapF. These enzymes are up-regulated in industry through genetic engineering to ensure adequate amounts of lysine precursors are produced to increase metabolic flux. Unwanted side reactions such as threonine and asparagine production can occur if a buildup of intermediates occurs, so scientists have developed mutant strains of C. glutamicum through PCR engineering and chemical knockouts to ensure production of side-reaction enzymes are limited. Many genetic manipulations conducted in industry are by traditional cross-over methods or inhibition of transcriptional activators. [44] Expression of functionally active human epidermal growth factor has been brought about in C. glutamicum, [45] thus demonstrating a potential for industrial-scale production of human proteins. Expressed proteins can be targeted for secretion through either the general secretory pathway or the twin-arginine translocation pathway. [46] Unlike gram-negative bacteria, the gram-positive Corynebacterium species lack lipopolysaccharides that function as antigenic endotoxins in humans. [ citation needed ] Most species of corynebacteria are not lipophilic. [ citation needed ] ### Nonlipophilic Edit The nonlipophilic bacteria may be classified as fermentative and nonfermentative: • Patients presenting with an infection need accurate diagnosis of the infective organism they are infected with to begin targeted treatment. • To achieve this, a number of investigations can be carried out which can be used to identify the exact organism, including what drugs can be used to treat the organism. • A gram stain is the main investigation used to identify bacterial infections - it involves looking at the colour and shape of a stained bacterial sample from the patient down a microscope. • A serology is a common investigation used to detect antigens/antibodies within the patient's blood. It can be used for diagnosing all micro-organism types. • A PCR test can be used to detect DNA/RNA within the fluids of the human body and can be used to diagnose any infection - but it takes a long time and is expensive. It can be used for diagnosing all micro-organism types. • A blood culture can be taken to grow any bacteria that may be present within the usually sterile blood of the patient. Whatever grows in the culture can then be tested for antibacterial susceptibility. However, this process takes a long time and is not suitable for hyper-acute situations. • In emergency/acute situations such as sepsis or meningitis (although blood/CSF samples will be taken) treatment is usually given empirically – antibiotics against the most likely causative organism without definite diagnosis using the above techniques When a patient presents with an infection, it can often be difficult to tell the exact organism they are infected with from clinical signs and history alone. Many symptoms are common among many micro-organisms and thus investigations need to be carried out to confirm a diagnosis and begin targeted treatment. This article covers a handful of diagnostic investigations that can be requested by a doctor to help with specific diagnosis of an infection. Some of these tests can be used for diagnosing other diseases that are not infectious in cause, but this is not within the scope of this article. Details of the methods of carrying out these investigations are not required past this level for medical students, but accurate interpretation of their results is vital. Gram staining Used for: Bacterial infections Sample collected via: Sputum sample, stool sample, blood sample, urine sample A gram stain involves looking at the shape and staining of a bacterial species under a microscope after carrying out a gram stain (using crystal violet and safranin). Gram staining differentiates bacteria by the chemical and physical properties of their cell walls through detecting peptidoglycan, which is present in the cell wall of gram-positive bacteria. To understand gram staining, an understanding of the different shapes and components of the cell walls of bacteria is required. The gram stain is almost always the first step in the preliminary identification of a bacterial organism. While gram staining is a valuable diagnostic tool in both clinical and research settings, not all bacteria can be definitively classified by this technique. How is it done? There are 4 basic steps to performing a gram-stain: 1. A primary stain of Crystal Violet is applied to a heat-fixed smear of a bacterial culture. This is stain is absorbed by the peptidoglycan in the walls of gram-positive bacteria. Therefore, gram positive cells will appear violet/purple in colour. 2. Iodide is added to keep the stain inside the cells and stop it from being washed out. 3. Rapid decolorization with ethanol or acetone. This essentially ‘washes out’ any remaining stain from the slide or from any cells which haven’t taken up the stain. 4. The gram-negative cells which haven’t taken up the initial stain will now be transparent due to the decolourization and can not be visualised. The sample is now counter-stained with safranin to visualise these cells which were not stained in the first step. Safranin is a pink colour therefore, gram-negative cells are pink in colour. Bacterial shapes Bacterial shapes can be either: • Coccus shaped (pl. cocci) – these bacteria look circular • Bacillus shaped (pl. bacilli) – these bacteria look like long tubes and are sometimes referred to as “rod” bacteria • Spirillus/Spiral – specific to a handful of gram-negative bacteria, this is a rare finding on a gram stain Image: The 3 shapes of bacteria. Note that cocci can arrange into singular cocci, diplococci (pairs of cocci) and larger groups such as strips and bunches Creative commons source by CKRobinson [CC BY-SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0)] The arrangement of the bacteria within space is also important. For example: • Streptococcus species are commonly arranged in strips of cocci (Remember: STREPS are STRIPS) • Staphylococcus species are usually arranged in bunches (similar to grapes) This information can be very useful in distinguishing between different species that may appear similar on a gram stain. Figure: Gram Stain of a Streptococcus species. Notice the long chains of cocci ("strips") and purple colouring as all streptococci are gram positive Figure: Gram Stain of a Staphylococcus species. Notice the bunches of cocci. Creative commons source by Y Tambe [CC BY-SA 3.0 (http://creativecommons.org/licenses/by-sa/3.0/)] Bacterial cell wall components Bacterial cell wall components often define how they infect the host, as well as their virulence and potential to cause disease. The cell wall components are also key in how an organism gram stains. There are two main categories of bacteria: Bacterial cell wall compositions vary based upon the category of bacteria: • Gram-positive cell walls contain a plasma membrane and a LARGE peptidoglycan cell layer • Gram-negative cell walls contain a plasma membrane a SMALL amount of peptidoglycan and then are often further encapsulated in a lipopolysaccharide cell wall (LPS) – this is a virulence factor that often leads to gram negative infections being more severe. Diagram: A Gram-positive bacterial cell wall with its main components SimpleMed original by Marcus Judge Diagram: A Gram-negative bacterial cell wall with its main components. Note the LPS outer capsule that causes serious disease and is a virulence factor SimpleMed original by Marcus Judge Interpreting a gram stain To reiterate interpretation is carried out by analysing the colour and shape of the bacteria on the microscope slide. Peptidoglycan absorbs the crystal violet stain used in gram stains. As such: • A gram-positive bacterium is PURPLE/VIOLET on a gram stain as it has more peptidoglycan. • A gram-negative bacterium is PINK on a gram stain. As previously discussed, the shape of a bacterium can be either cocci, bacillus or spiral. Figure: Gram Positive vs Gram Negative gram stain Converting this information into a diagnosis Upon knowing the colour and shape of a bacterium the current differentials can be supported or rejected. This is why knowledge of the shape and gram of common species of bacterium is important for a physician Here are some useful examples: Gram positive cocci – (Purple circles) • Staphylococcus species e.g. Staphylococcus aureus (causes cellulitis and skin abscesses) • Streptococcus species e.g. Streptococcus pneumoniae (causes URT infections and pneumonia, is usually a diplococcus i.e. in pairs) Often treated with: Penicillins and Carbapenems Gram positive bacilli – (Purple rods) • Clostridium difficile (also referred to as C.diff)(a hospital acquired infection that causes diarrhoea and other digestive complications, notable for its’ antibiotic resistance and spore forming ability making it hard to eradicate) • Bacillus anthracis (commonly known as anthrax, this is a notifiable disease i.e. it must be reported to public health England if suspected) • Listeria monocytogenes (causes listeriosis which can be an extremely severe life threatening condition) Often treated with: Penicillins, Carbapenems and Erythromycin (macrolides) Gram negative cocci – (pink circles) • Neisseria meningitidis (causes meningococcal meningitis) • Neisseria gonorrhoeae (causes gonorrhoea) Often treated with: Cephalosporins and fluoroquinolones Gram negative bacilli – (pink rods) Often treated with: Cephalosporins and fluoroquinolones This is an Advert - we use these to keep SimpleMed free! If you see something you like, please click on it - it supports the site :) Other investigative techniques: Definition: Serology refers to the study of serum (blood plasma without clotting factors) but in this context is used specifically to refer to diagnostic techniques within this field of study. Used for: All types of infections and other non-infective conditions e.g. rheumatological conditions. Used to detect: Bacteraemia (bacteria present in the bloodstream)/Syphilis/Hepatitis/HIV Sample collected via: Blood (usually), other bodily fluids Serological tests rely on the formation of antibodies in response to an infection (against a given microorganism). Presence of these antibodies in the blood implies that the body has begun to mount an immune response (which may be ineffective) against the organism. This diagnostic test is not suitable in immunocompromised individuals as they may lack the ability to form antibodies. There are several serology techniques that can be used depending on the antibodies being studied. These include: ELISA, agglutination, precipitation, complement-fixation, fluorescent antibodies and chemiluminescence to name but a few examples (detailed knowledge of these tests is not required for this topic). These techniques often involve the use of complementary antibodies to detect disease marker antibodies taken from the patient’s serum. The complementary antibodies may have a fluorescent marker or cause an observable enzymatic reaction in response to signal a positive test. Full Blood Count Definition: A very common test carried out on the patient’s blood to detect the levels and characteristics of certain benchmark cells within the blood. Anomalies in the number of red blood cells, their size and the number of immune cells/inflammatory markers etc. can be indicative of a disease. This investigation is not inherently diagnostic but can often show if a patient is infected and what cells are being affected within the blood. Used for: All types of infections Used to detect: • Sepsis (raised inflammatory markers and either extremely high or low neutrophil count depending on stage of progression) • AIDS (T-cell count below 200 cells/μL) • Epstein Barr Virus (EBV) (high lymphocytes early on in infection, low B-cell count late) • Any other acute viral or bacterial infections (raised inflammatory markers) Sample collected via: Blood A full blood count is almost always requested when a patient comes in acutely ill. It provides information such as: • White cell count • Neutrophil count: Raised levels may indicate bacterial infection. They may also be raised in acute viral infections. • Lymphocyte count: Higher with some viral infections such as Epstein-Barr virus (although B-cell count will drop later on in this disease’s progression when it begins to infect B-cells). Counts may be decreased by HIV infection, as this causes the destruction of T-cells. • Raised Monocytes: May be raised in bacterial infection, tuberculosis and malaria. • Eosinophils: Increased in parasitic infections. See our dedicated article in the haematology section Definition: Polymerase chain reaction (PCR) is a technique that is used to amplify trace amounts of DNA (and in some instances, RNA) located in or on almost any liquid or surface where DNA strands may be deposited. Thus, a sample of fluid or blood taken from an infected patient can be passed through a PCR machine to amplify the specific DNA/RNA strands of certain viruses/bacteria that the technician is looking for. The amplified genetic information can then be used to match the sample with a microorganism on record. Used for: Every infection and organism – the gold standard of infection investigations Used to detect: Everything Sample collected via: Blood, faecal sample, urine sample, throat swab, sputum sample A PCR test is diagnostically the most accurate form of investigation commonly carried out, as it can detect the exact DNA of a certain microorganism within a sample and can be automated (reducing human error). It can even detect specific mutations that could cause resistance to certain medications. This allows further personalisation of treatment strategy to increase treatment efficacy. PCR tests are powerful tools however, they take considerable time to carry out and are far more expensive to run than a gram stain or basic serology. They can however, be used in viral infections to detect the specific infective organism. This is an Advert - we use these to keep SimpleMed free! If you see something you like, please click on it - it supports the site :) Blood culture Definition: As the name implies this technique is simply culturing a blood sample (the growing any bacteria found within the blood on a plate). It is used to detect infections have haematogenous spread (through the blood stream). This is possible as the bloodstream is usually a sterile environment therefore any positive culture indicates pathology. The technique can be used to grow large enough amounts of the bacteria from within a patient’s bloodstream to carry out tests such as antibiotic susceptibility (to identify the best antibiotic to treat that strain of the bacteria). Used for: Suspected Bacterial infections, particularly those with likely antibiotic resistance. Used to detect: Bacteria Sample collected via: Blood The bacteria are given an ideal growth environment on agar jelly. This allows rapid growth and detection of any bacteria within the blood. MIC (minimum inhibitory concentration) and susceptibility testing can then be carried out on the bacteria. This test is often paired with a gram-stain test, allowing for an excellent and full diagnosis of the infection present within the patient. It is worth remembering that all these techniques do take time, therefore in acute emergency situations such as Sepsis empirical treatment must be given. Therefore, it is also useful to understand the most likely causative bacteria for infection in specific patient groups. ## Streptococcus pneumoniae symptoms? Streptococcus pneumonia paves the way to a lot of pneumococcal diseases in people that have very low immune systems, children and elderly people. These diseases are contagious and can spread from a person to another. Along with that, it could also be life-threatening. Therefore, it is recommended to watch out for the symptoms of pneumococcal disease. The infections of the pneumococcal disease mostly happen around the sinuses, bloodstream, lungs, middle ear and meninges which is the lining of the spinal cord and brain which ultimately results in meningitis. Hence, to mention a few of the streptococcus pneumoniae symptoms are: • Cough • Chills and fever • Difficulty in breathing • Rapid breathing • Pain in the chest • Stiff neck • Disorientation or confusion • Sensitivity to light • Increased heart rate • A sensation of cold and/or shivering and shaking • Discomfort and pain • Sweaty skin • Short breath • Sleepiness • Ear pain • The swollen or red eardrum • Bloodstained sputum • Nausea and vomiting • Drowsiness ## Abstract Group B streptococcal infection (Streptococcus agalactiae) is one of the leading causes of life-threatening disease in the early neonatal period, resulting in sepsis, pneumonia, and meningitis. During invasive infections, an excessive release of pro-inflammatory cytokine, such as interleukin-6 (IL-6), thus IL-6 gene is significant, as a diagnostic marker of systemic infection of the newborns. The present study aimed to describe the epidemiology diagnostic of GBS disease in neonatal by phenotypic and genotypic methods. Nine hundred and ninety-six samples were taken at Maternity and Children Hospital, Jeddah, Saudi Arabia for a period of one year (2011–2012). Results indicated that out of 217 infected samples, twenty (9.23.0%) were positive for group B Streptococci bacteria. This study also shows that female infants are more susceptible than males. The level of IL-6 was higher in mothers above 30 years. Twenty positive Streptococci group B isolates showed bands with the cylE gene primers in the border between 228 bp, 267 bp and 50 bp. Molecular detection by Real time polymerase chain reaction was also done to detect the target (Sip gene) encoding the Sip surface immunogenic protein. Specific primers and TaqMan probe were chosen for this purpose. A Real-time PCR method targeting the sip gene of GBS in neonates after delivery has been evaluated.	2022-05-24 12:54:13	{"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": 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": 1, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.43655094504356384, "perplexity": 8273.0745322502}, "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-2022-21/segments/1652662572800.59/warc/CC-MAIN-20220524110236-20220524140236-00487.warc.gz"}
http://epplus.codeplex.com/discussions/283139	# Can't open xlsx file djarvis8 Dec 15, 2011 at 8:02 PM I am trying to do the most simplest thing- open an existing test.xlsx file but always fail on accessing package.Workbook. ```FileInfo existingFile = new FileInfo("TestSpreadsheets\\test1.xlsx"); using (ExcelPackage package = new ExcelPackage(existingFile)) { ExcelWorkbook workbook = package.Workbook; // throws exception }``` "Index was out of range. Must be non-negative and less than the size of the collection.". The file does exist, I can see that under the package info, along with the size and various other things. ddelella Dec 19, 2011 at 6:10 PM You use the FileInfo object to check if the path is valid before attempting to load the ExcelPackage. In my code I do not use the \\ escape character in my paths and it works just fine. The only strange thing to keep in mind is that when referencing Worksheets(x) the index is not 0 based, it begins at 1.	2017-08-21 12:12:54	{"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": 0.8762974143028259, "perplexity": 976.3935931283036}, "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/1502886108268.39/warc/CC-MAIN-20170821114342-20170821134342-00118.warc.gz"}
https://pure.mpg.de/pubman/faces/ViewItemOverviewPage.jsp?itemId=item_1833358	English # Item ITEM ACTIONSEXPORT Released Report #### Exact ground states of Ising spin classes: new experimental results with a branch and cut algorithm ##### MPS-Authors /persons/resource/persons45092 Mutzel, Petra Algorithms and Complexity, MPI for Informatics, Max Planck Society; ##### External Resource No external resources are shared ##### Fulltext (public) MPI-I-95-1-004.pdf (Any fulltext), 130KB ##### Supplementary Material (public) There is no public supplementary material available ##### Citation Diehl, M., De Simone, C., Jünger, M., Mutzel, P., Reinelt, G., & Rinaldi, G.(1995). Exact ground states of Ising spin classes: new experimental results with a branch and cut algorithm (MPI-I-1995-1-004). Saarbrücken: Max-Planck-Institut für Informatik. Cite as: http://hdl.handle.net/11858/00-001M-0000-0014-A765-7 ##### Abstract In this paper we study 2-dimensional Ising spin glasses on a grid with nearest neighbor and periodic boundary interactions, based on a Gaussian bond distribution, and an exterior magnetic field. We show how using a technique called branch and cut, the exact ground states of grids of sizes up to $100\times 100$ can be determined in a moderate amount of computation time, and we report on extensive computational tests. With our method we produce results based on more than $20\,000$ experiments on the properties of spin glasses whose errors depend only on the assumptions on the model and not on the computational process. This feature is a clear advantage of the method over other more popular ways to compute the ground state, like Monte Carlo simulation including simulated annealing, evolutionary, and genetic algorithms, that provide only approximate ground states with a degree of accuracy that cannot be determined a priori. Our ground state energy estimation at zero field is~$-1.317$.	2021-08-05 21:17:43	{"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": 0.5943408012390137, "perplexity": 1695.8588756286294}, "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/1627046157039.99/warc/CC-MAIN-20210805193327-20210805223327-00425.warc.gz"}
https://socratic.org/questions/what-is-fe-2-so-4-3-called	What is Fe_2(SO_4)_3 called? $\text{Iron (III) sulfate}$ or $\text{ferric sulfate}$ Either name is still commonly used. Of course, it is formulated on the basis of electrical neutrality. 2 equiv $F {e}^{3 +}$, the ferric ion, and 3 equiv $S {O}_{4}^{2 -}$ give a neutral charge.	2020-09-27 20:18:27	{"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": 0.9080162048339844, "perplexity": 2313.329347474707}, "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-2020-40/segments/1600401578485.67/warc/CC-MAIN-20200927183616-20200927213616-00133.warc.gz"}
https://ftp.aimsciences.org/article/doi/10.3934/amc.2012.6.259	Article Contents Article Contents # List decoding of matrix-product codes from nested codes: An application to quasi-cyclic codes • A list decoding algorithm for matrix-product codes is provided when $C_1, ..., C_s$ are nested linear codes and $A$ is a non-singular by columns matrix. We estimate the probability of getting more than one codeword as output when the constituent codes are Reed-Solomon codes. We extend this list decoding algorithm for matrix-product codes with polynomial units, which are quasi-cyclic codes. Furthermore, it allows us to consider unique decoding for matrix-product codes with polynomial units. Mathematics Subject Classification: Primary: 94B05; Secondary: 94B35. Citation: • [1] P. Beelen and K. Brander, Key equations for list decoding of Reed-Solomon codes and how to solve them, J. Symbolic Comput., 45 (2010), 773-786.doi: 10.1016/j.jsc.2010.03.010. [2] T. Blackmore and G. H. Norton, Matrix-product codes over $\mathbb F_q$, Appl. Algebra Engrg. Comm. Comput., 12 (2001), 477-500.doi: 10.1007/PL00004226. [3] I. I. Dumer, Concatenated codes and their multilevel generalizations, in "Handbook of Coding Theory,'' North-Holland, Amsterdam, (1998), 1911-1988. [4] P. Elias, List decoding for noisy channels, Rep. No. 335, Research Laboratory of Electronics, MIT, Cambridge, MA, 1957. [5] V. Guruswami and A. Rudra, Better binary list decodable codes via multilevel concatenation, IEEE Trans. Inform. Theory, 55 (2009), 19-26.doi: 10.1109/TIT.2008.2008124. [6] V. Guruswami and M. Sudan, Improved decoding of Reed-Solomon and algebraic-geometry codes, IEEE Trans. Inform. Theory, 45 (1999), 1757-1767.doi: 10.1109/18.782097. [7] F. Hernando, K. Lally and D. Ruano, Construction and decoding of matrix-product codes from nested codes, Appl. Algebra Engrg. Comm. Comput., 20 (2009), 497-507.doi: 10.1007/s00200-009-0113-5. [8] F. Hernando and D. Ruano, New linear codes from matrix-product codes with polynomial units, Adv. Math. Commun., 4 (2010), 363-367.doi: 10.3934/amc.2010.4.363. [9] T. Kasami, A Gilbert-Varshamov bound for quasi-cyclic codes of rate 1/2, IEEE Trans. Inform. Theory, IT-20 (1974), 679.doi: 10.1109/TIT.1974.1055262. [10] K. Lally, Quasicyclic codes - some practical issues, in "Proceedings of 2002 IEEE International Symposium on Information Theory,'' 2002. [11] K. Lally and P. Fitzpatrick, Algebraic structure of quasicyclic codes, Discrete Appl. Math., 111 (2001), 157-175.doi: 10.1016/S0166-218X(00)00350-4. [12] K. Lee and M. E. O'Sullivan, List decoding of Reed-Solomon codes from a Gröbner basis perspective, J. Symbolic Comput., 43 (2008), 645-658.doi: 10.1016/j.jsc.2008.01.002. [13] R. R. Nielsen and T. Høholdt, Decoding Reed-Solomon codes beyond half the minimum distance, in "Coding Theory, Cryptography and Related Areas (Guanajuato, 1998),'' Springer, Berlin, (2000), 221-236. [14] F. Özbudak and H. Stichtenoth, Note on Niederreiter-Xing's propagation rule for linear codes, Appl. Algebra Engrg. Comm. Comput., 13 (2002), 53-56.doi: 10.1007/s002000100091. [15] W. C. Schmid and R. Schürer, "Mint,'' Dept. of Mathematics, University of Salzburg, http://mint.sbg.ac.at/about.php [16] J. M. Wozencraft, List decoding, in "Quarterly Progress Report,'' MIT, Cambridge, MA, (1958), 90-95.	2023-03-26 08:23:36	{"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": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.29702287912368774, "perplexity": 3058.7667990054915}, "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-2023-14/segments/1679296945440.67/warc/CC-MAIN-20230326075911-20230326105911-00362.warc.gz"}
https://es.mathworks.com/help/signal/ug/distortion-measurements.html	# Distortion Measurements Generate 2048 samples of a sinusoid of frequency 2.5 kHz sampled at 50 kHz. Add white Gaussian noise such that the signal-to-noise ratio (SNR) is 80 dB. Fs = 5e4; f0 = 2.5e3; N = 2048; t = (0:N-1)/Fs; SNR = 80; x = cos(2pif0t); x = x+randn(size(x))std(x)/db2mag(SNR); Pass the result through a weakly nonlinear amplifier represented by a polynomial. The amplifier introduces spurious tones at the frequencies of the harmonics. amp = [1e-5 5e-6 -1e-3 6e-5 1 25e-3]; x = polyval(amp,x); Plot the signal spectrum and annotate the SNR, verifying that it has the expected value. The snr function computes the power ratio of the fundamental to the noise floor and ignores the DC component and the harmonics. snr(x,Fs); Plot the signal spectrum and annotate the total harmonic distortion (THD). The thd function computes the power ratio of the harmonics to the fundamental and ignores the DC component and the noise floor. thd(x,Fs); Plot the signal spectrum and annotate the signal to noise and distortion ratio (SINAD). The sinad function computes the power ratio of the fundamental to the harmonics and the noise floor. It ignores only the DC component. Verify that the SNR, THD, and SINAD obey the equation $1{0}^{-SNR/10}+1{0}^{THD/10}=1{0}^{-SINAD/10}.$ lhs = 10^(-snr(x,Fs)/10)+10^(thd(x,Fs)/10) lhs = 7.2203e-08	2022-12-03 17:16:07	{"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": 0.7232308387756348, "perplexity": 1944.9988057718049}, "config": {"markdown_headings": true, "markdown_code": false, "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/1669446710933.89/warc/CC-MAIN-20221203143925-20221203173925-00324.warc.gz"}
https://physics.stackexchange.com/questions/211762/what-is-the-reason-significance-of-using-sum-limits-n-1-inftyn-rightarro	# What is the reason/significance of using $\sum\limits_{n=1}^{\infty}n\rightarrow-\frac{1}{12}$? What is the reason/significance of using a trick equation in the Volume I - String Theory - Joseph Polchinsky? I have no doubts at all that the author knows extremely well the subject and that this is not an error. So my question is only related to the reason/significance of the usage of this equation. $$\begin{equation} \sum\limits_{n=1}^{\infty}n\rightarrow-\frac{1}{12}\tag{1.3.32}\end{equation}$$ • This particular sum is also discussed here, here and here, and on Math.SE here. See also this Phys.SE post. Also related Phys.SE post here. – Qmechanic Oct 10 '15 at 16:02 Here are two common ways of obtaining that result. 1) Analytic continuation of the zeta function. For $\operatorname{Re} s>1$, the Riemann zeta function is defined by $$\zeta(s)=\sum_{n=1}^\infty n^{-s}$$ This function satisfies the Riemann functional equation $$\zeta(s)=2^s\pi^{s-1}\sin\left(\frac{\pi s}{2}\right)\Gamma(1-s)\zeta(1-s)$$ which is valid on the whole complex plane. (Here $\Gamma(z)$ is the gamma function.) Now, your sum is evidently $\zeta(-1)$, which can't be calculated directly using the sum definition, but using this analytic continuation one finds $\zeta(-1)=-1/12$. This is not really saying that that sum converges to $-1/12$, but rather that the analytic continuation of a similar sum is $-1/12$. 2) Physically motivated. Physically, the $n$ here means that we have modes of the quantum field with wave number $n$. The energy of each mode is given by $\omega_n =n\pi/d$, where $d$ is the separation of the plates. These physical plates won't be able to contain waves with arbitrary energy, those will leak out and no longer contribute to the energy between the plates. So, we introduce a factor $\mathrm{e}^{-a\omega_n/\pi}$ with $a$ chosen so that modes with $\omega_n\gg\pi/a$ do not contribute to the sum. So now we have (we call the sum $\zeta(-1)$ for convenience) $$\zeta(-1)=\sum_{n=1}^\infty n\to\sum_{n=1}^\infty n\mathrm{e}^{-an/d}=-d\frac{\partial}{\partial a}\sum_{n=1}^\infty \mathrm{e}^{-an/d}=-d\frac{\partial}{\partial a}\frac{1}{1-\mathrm{e}^{-a/d}}=\frac{d\mathrm{e}^{a/d}}{(e^{a/d}-1)^2}$$ Now we take $a$ to be small and expand $$\zeta(-1)\to\frac{d^2}{a^2}-\frac{1}{12}+\mathcal{O}(a^2)$$ Up to an overall factor the force between the plates is given by something like $\mathrm{d}\zeta(-1)/\mathrm{d}d$. The Taylor series diverges (it has to, we started with a divergent series), but we do not observe this in nature. (There is a force between the plates, but it surely isn't infinite.) Now, it turns out that when you do the same calculation for for the other plate, the infinite terms cancel exactly. And in the limit of $a\to0$, all the terms with $a$ vanish. So, physically, it was as if we had taken $\zeta(-1)=-1/12$ all along. • Good, but there are many more ways to calculate this sum... – Luboš Motl Oct 10 '15 at 15:44 • @LubošMotl Edited the first sentence, these are the two methods I have seen in textbooks. What are some others? – Ryan Unger Oct 10 '15 at 15:46 • @0celo7 This answer by Luboš is relevant. – Danu Oct 10 '15 at 17:10 The significance is that it actually works! Actually $\sum\limits_{n=1}^{\infty}n= \infty$ but this usually does not really help in evaluate physical problems (in this case string theorie). So one idea is to find a representative value or property of such a sum in order to proceed with the analysis. In this particular case the zeta function worked. As far as I know nobody "knows" why it worked here, and why one gets meaningful results when assuming the sum equals $-\frac{1}{12}$.	2021-06-15 07:43:12	{"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": 0.8991481065750122, "perplexity": 205.06321383606465}, "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-2021-25/segments/1623487617599.15/warc/CC-MAIN-20210615053457-20210615083457-00431.warc.gz"}
https://mathoverflow.net/questions/141378/is-this-lemma-in-elementary-linear-algebra-new	# Is this lemma in elementary linear algebra new? Is anyone familiar with the following, or anything close to it? Lemma. Suppose $A$, $B$ are nonzero finite-dimensional vector spaces over an infinite field $k$, and $V$ a subspace of $A\otimes_k B$ such that (1) For every nonzero $a\in A$ there exists nonzero $b\in B$ such that $a\otimes b\in V$, and likewise, (2) For every nonzero $b\in B$ there exists nonzero $a\in A$ such that $a\otimes b\in V$. Then (3) $\dim_k(V) \geq \dim_k(A) + \dim_k(B) - 1$. Remarks: The idea of (1) and (2) is that the spaces $A$ and $B$ are minimal for "supporting" $V$; that is, if we replace $A$ or $B$ by any proper homomorphic image, and we map $A\otimes B$ in the obvious way into the new tensor product, then that map will not be one-one on $V$. The result is equivalent to saying that if one is given a finite-dimensional subspace $V$ of a tensor product $A\otimes B$ of arbitrary vector spaces, then one can replace $A$, $B$ by images whose dimensions sum to $\leq \dim(V) + 1$ without hurting $V$. In the lemma as stated, if we take for $A$ a dual space $C^*$, and interpret $A\otimes B$ as $\mathrm{Hom}(C,B)$, then the hypothesis again means that $C$ and $B$ are minimal as spaces "supporting" $V$, now as a subspace of $\mathrm{Hom}(C,B)$; namely, that restricting to any proper subspace of $C$, or mapping onto any proper homomorphic image of $B$, will reduce the dimension of $V$. In the statement of the lemma, where I assumed $k$ infinite, I really only need its cardinality to be at least the larger of $\dim_k A$ and $\dim_k B$. The proof is harder than I would have thought; my write-up is 3.3K. I will be happy to show it if the result is new. • Welcome to Mathoverflow! – user6976 Sep 5 '13 at 21:09 • I don't have time to think about this right now, but it seems strikingly familiar to the following theorem of Hopf. If $f:A \otimes B\to C$ is a linear map which is injective on each factor separately, then $\dim f(A\otimes B) \geq \dim A + \dim B - 1.$ However, this theorem is true over $\mathbb{C}$ but false over $\mathbb{R}$ (the proof is given by algebraic topology), so maybe it is only a superficial observation. – Jack Huizenga Sep 6 '13 at 0:50 • This feels like a statement from projective geometry. $\mathbb{P}(V) \subseteq \mathbb{P}(A \otimes B)$ somehow "intersects enough" $\mathbb{P}(A) \times \mathbb{P}(B)$ so that $\dim(\mathbb{P}(V)) \geq \dim(\mathbb{P}(A) \times \mathbb{P}(B))$. – Martin Brandenburg Sep 6 '13 at 1:23 • Probably also related: Flanders' theorem (§8.3 in Prasolov's Linear Algebra book www2.math.su.se/~mleites/Prasolov/prasLinAlg/pr-linAlg-main.dvi ). – darij grinberg Sep 6 '13 at 2:30 • Somehow it feels that the following "dual" result is very closely related, and could for some fields also yield your inequality (by splitting the tensor product into "separable" and "joint" spaces, whose dimensions add up to $d_Ad_B$): On the maximal dimension of a completely entangled subspace..." by K. Parathasarathy; ias.ac.in/mathsci/vol114/nov2004/Pm2342.pdf --- in particular, your subspaces have the "separable" state property, while the cited paper considers "full entangled" subspaces. – Suvrit Sep 6 '13 at 15:04 This is a nice lemma: I know a good deal of similar results but this one is unknown to me. I believe it is suitable, as an answer, to give a proof that works with no restriction on the cardinality of the underlying field $F$. I will frame the answer in terms of matrix spaces. Thus, we have a linear subspace $V \subset M_{n,p}(F)$ such that, for every non-zero vector $X \in F^n$, the space $V$ contains a rank $1$ matrix with column space $F X$ and, for every non-zero vector $Y \in F^p$, the space $V$ contains a rank $1$ matrix with row space $F Y^t$. Note that those assumptions are unchanged in multiplying $V$ by invertible matrices (be it on the left or on the right). The proof works by induction on $p$. The case where $p=1$ or $n=1$ is obvious. Assume now that $p>1$ and $n>1$. The discussion is split into two cases, where the standard basis of $F^p$ is denoted by $(e_1,\dots,e_p)$. Case 1: $V e_p=F^n$. Then, one writes every matrix $M$ of $V$ as $M=\begin{bmatrix} A(M) & C(M) \end{bmatrix}$ where $A(M) \in M_{n,p-1}(F)$ and $C(M) \in F^n$. With our assumptions, we find rank $1$ matrices $M_1,\dots,M_{p-1}$ in $V$ with respective row spaces $F e_1^t,\dots,F e_{p-1}^t$. Then, $M_1,\dots,M_{p-1}$ are linearly independent and all belong to the kernel of $V \ni M \mapsto C(M)$. Using the rank theorem, one deduces that $\dim V \geq (p-1)+\dim C(V)=(p-1)+n$. Case 2 : $V e_p \subsetneq F^n$. Multiplying $V$ on the left by a well-chosen invertible matrix, we lose no generality in assuming that $V e_p \subset F^{n-1} \times \{0\}$. In other words, every matrix $M$ of $V$ may be written as $$M=\begin{bmatrix} A(M) & C(M) \\ R(M) & 0 \end{bmatrix}$$ where $A(M)$ is an $(n-1) \times (p-1)$ matrix, $R(M)$ is a row matrix and $C(M)$ is a column matrix. Then, we note that $A(V)$ satisfies the same set of assumptions as $V$: indeed, if we take a non-zero row $L \in M_{1,p-1}(F)$, then we know that $V$ contains a rank $1$ matrix $M_1$ whose row space is spanned by $\begin{bmatrix} L & 1 \end{bmatrix}$. Obviously the last row of $M_1$ is zero whence $A(M_1)$ is non-zero and its row space is included in $F L$. One works likewise to obtain the remaining part of the condition. Thus, by induction one finds $$\dim A(V) \geq (n-1)+(p-1)-1.$$ Finally, we know that $V$ must contain a non-zero matrix $M_2$ with $A(M_2)=0$ and $C(M_2)=0$, and that it must contain a non-zero matrix $M_3$ with $A(M_3)=0$ and $R(M_3)=0$. Obviously, $M_2$ and $M_3$ are linearly independent vectors in the kernel of $V \ni M \mapsto A(M)$. Using the rank theorem, one concludes that $$\dim V \geq 2+\dim A(V) \geq 2+(n-1)+(p-1)-1=n+p-1.$$ • This is a nice proof. Although it seems to be algebraic, I think it is geometric in disguise. Perhaps someone can formulate this proof coordinate-free, and/or in terms of the the intersection of $\mathbb{P}(V) \subseteq \mathbb{P}(A \otimes B)$ with $\mathbb{P}(A) \times \mathbb{P}(B) \hookrightarrow \mathbb{P}(A \otimes B)$? – Martin Brandenburg Sep 6 '13 at 19:20 As suggested by Martin, there is a geometric interpretation of this lemma. Though the proof is probably not shorter than the one proposed by Clément. Nevertheless, this is the kind of very classical reasonning one encouters in the study of secant varieties. Let us put $$a = dim A$$ and $$b = dim B$$. If $$a \otimes b \in A \otimes B$$, I denote its image in $$\mathbb{P}(A \otimes B)$$ by $$[a \otimes b]$$. I denote by $$X_{A,B} = \{(a,b), \textrm{such that} [a \otimes b] \in \mathbb{P}(V) \}$$. This is clearly equal to the scheme $$(\mathbb{P}(A) \times \mathbb{P}(B)) \cap \mathbb{P}(V)$$ (I'll consider only schemes with reduced structure here). Let us consider the natural projections $$p_A : X_{A,B} \longrightarrow \mathbb{P}(A)$$ and $$p_B : X_{A,B} \longrightarrow B$$. The hypothesis given by the OP show that $$p_A$$ and $$p_B$$ are surjective. Denote by $$\gamma_A$$ the dimension of the generic fiber of $$p_A$$, by $$\gamma_B$$ the dimension of the generic fiber of $$p_B$$, by $$X_A$$ a maximal dimensional irreducible component of the scheme $$p_A^{-1}(p_A(X_{A,B}))$$ and by $$X_B$$ a maximal dimensional irreducible component of the scheme $$p_B^{-1}(p_B(X_{A,B}))$$. The theorem of the dimension gives $$dim \ X_A = a-1 + \gamma_A$$ and $$dim \ X_B =b-1 + \gamma_B$$. The secant variety $$S(X_A,X_B)$$ (that is the closure of variety of lines joining a point of $$X_A$$ and a point of $$X_B$$) is included in $$\mathbb{P}(V)$$ and the goal will be to bound below its dimension to get a bound for $$dim \ \mathbb{P}(V)$$. The dimension of $$S(X_A,X_B)$$ is equal to $$\dim \ X_A + \dim \ X_B +1 - \delta$$, where $$\delta$$ is the secant defect of $$S(X_A,X_B)$$. Concretely, if $$M$$ is a generic point of $$S(X_A,X_B)$$, then $$\delta$$ is the dimension of the scheme: $$\{[a_1 \otimes b_1] \in X_A, \ \textrm{s.t. \exists [a_2 \otimes b_2] \in X_B and (x,y) \in \mathbb{k}^2 with M = x.a_1\otimes b_1 + y.a_2 \otimes b_2} \}.$$ It is well known that the secant defect of $$S(\mathbb{P}(A) \times \mathbb{P}(B),\mathbb{P}(A) \times \mathbb{P}(B))$$ is $$2$$. Indeed, the parameter family to decompose a rank $$2$$ matrix as a sum of two rank $$1$$ matrices is $$\mathbb{P}^1 \times \mathbb{P}^1$$. (short explanation : as one only needs to construct one of these rank $$1$$ matrices : choose the image (choice of a $$\mathbb{k}^1$$ in the image of the rank $$2$$ matrix, which is isomorphic to $$\mathbb{k}^2$$) and choose a hyperplane containing the kernel of the rank $$2$$ matrix). Assume that $$S(X_A,X_B)$$ consists only of rank $$1$$ matrices. Since $$X_A$$ surjects onto $$A$$ and $$X_B$$ surjects onto $$B$$, we easily deduce that $$X_A = \mathbb{P}(A) \times b_0$$ and $$X_B = a_0 \times \mathbb{P}(B)$$ for some $$a_0$$ and $$b_0$$ fixed. The dimension of $$S(X_A,X_B)$$ is then obviously seen to be $$a-1+b-1-0 = a+b-2$$ and then we have $$dim \ \mathbb{P}(V) \geq a+b-2$$. Assume that the $$S(X_A,X_B)$$ contains a matrix of rank $$2$$. Then, the generic $$M \in S(X_A,X_B)$$ has rank $$2$$. Since $$X_A \subset \mathbb{P}(A) \times \mathbb{P}(B)$$, $$X_B \subset \mathbb{P}(A) \times \mathbb{P}(B)$$ and the secant defect of $$S(\mathbb{P}(A) \times \mathbb{P}(B),\mathbb{P}(A) \times \mathbb{P}(B))$$ is $$2$$, we deduce that $$\delta \leq 2$$. As a consequence, $$dim \ S(X_A,X_B) \geq a-1 + \gamma_A + b-1 + \gamma_B +1 - \delta \geq a+b-3.$$ If $$\delta \leq 1$$, then we get in fact: $$dim \ S(X_A,X_B) \geq a-1 + \gamma_A + b-1 + \gamma_B-1 \geq a+b-2,$$ and this implies that $$dim \ \mathbb{P}(V) \geq a+b-2$$, which is what we wanted. If $$\delta = 2$$, then the dimension of $$\{[a_1 \otimes b_1] \in X_A, \ \textrm{s.t. \exists [a_2 \otimes b_2] \in X_B and (x,y) \in \mathbb{k}^2 with M = x.a_1\otimes b_1 + y.a_2 \otimes b_2} \}$$ is $$2$$. In view of the explicit decomposition of a rank $$2$$ matrix as the sum of two rank $$1$$ matrices, this implies that for every $$a$$ in $$\mathbb{P}(A)$$, there is at least a $$\mathbb{P}^1$$ of $$b \in \mathbb{P}(B)$$ such that $$[a \otimes b] \in X_A$$. We deduce that $$\gamma_A \geq 1$$ and finally: $$\dim S(X_A,X_B) \geq a-1+1 +b-1 + 1 -2 = a+b-2,$$ which again implies $$dim \ \mathbb{P}(V) \geq a+b-2$$.	2020-10-27 12:55:08	{"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": 83, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9665061831474304, "perplexity": 111.22583966337483}, "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/1603107894175.55/warc/CC-MAIN-20201027111346-20201027141346-00478.warc.gz"}
https://math.stackexchange.com/questions/1808002/computing-a-three-dimensional-lebesgue-measure-of-a-bounded-set	# Computing a three-dimensional Lebesgue measure of a bounded set How can I compute the three-dimensional Lebesgue-measure of the set $A$ which is bounded by the areas $x+y+z =6$, $x=0$, $z=0$ and $x+2y=4$? A hint on how I solve problems like this in general would be much appreciated. I need to know how to compute the measure/volume by exclusively using integrals. • Sorry of being (too?) late, but I've posted another answer that I've been composing anyway. – Han de Bruijn Jun 5 '16 at 18:07 Your set $A$ is a three-sided pyramid standing on the $(x,y)$-plane, and having its top at $(0,0,6)$. The base $B$ of this pyramid is a triangle bounded by the lines $x=0$, $x+y=6$, and $x+2y=4$. Its vertices are $(0,2)$, $(0,6)$, and $(8,-2)$. The set $B$ can be described as follows: $$B:=\left\{(x,y)\>\biggm\|\>0\leq x\leq 8, \ 2-{x\over2}\leq y\leq 6-x\right\}\ .$$ The walls $x=0$ and $y=2-{x\over2}$ of $A$ are vertical, whereas the third wall is given by $z=6-x-y=:h(x,y)$. It follows that $${\rm vol}(A)=\int_B h(x,y)\>{\rm d}(x,y)=\int_0^8\int_{2-x/2}^{6-x}(6-x-y)\>dy\>dx=\ldots={64\over3}\ .$$ Isn't that measure the same as the volume of the tetrahedral pyramid, bounded by those areas? The lines $x+y+0=6$ and $x+4y=0$ intersect at $(8,-2,0)$, which is thus one of the vertices of the pyramid. The other vertices are $(0,0,0)$ and $(0,6,0)$ and $(0,0,6)$. The required volume can be calculated in several ways. One way is the area of the triangle at the bottom times one-third of the pyramid's height: $8 \times 6/2 \times 6/3 = \large 48$ . • Oh wow thanks for the effort, your drawings helped me a lot. I made a mistake though, it should be $x+2y=4$ instead of $x+4y=0$, but that I should be able to fix myself now. Now if I want to compute the volume by using integrals, how do I determine the upper and lower limits of the integrals? – Tesla Jun 3 '16 at 17:10 • And yes, of course it is the same as the volume. But our upcoming exam will be about computing volumes/measures by using integrals (Tonelli etc).. – Tesla Jun 3 '16 at 18:37 AfterMath. By "exclusively using integrals" has been done by Christian Blatter, so here comes the "like this in general" part, as formulated in the question too. Hope it is appreciated by the OP. ## Linear Tetrahedron Let's consider the simplest non-trivial finite element shape in 3-D, which is a linear tetrahedron. Function behaviour is approximated inside such a tetrahedron by a linear interpolation between the function values at the vertices, also called nodal points. Let $T$ be such a function, and $x,y,z$ coordinates, then: $$T = A.x + B.y + C.z + D$$ Where the constants A, B, C, D are yet to be determined. Substitute $x=x_k$ , $y=y_k$ , $z=z_k$ with $k=0,1,2,3$. Start with: $$T_0 = A.x_0 + B.y_0 + C.z_0 + D$$ Clearly, the first of these equations can already be used to eliminate the constant $D$, once and forever: $$T - T_0 = A.(x - x_0) + B.(y - y_0) + C.(z - z_0)$$ Then the constants $A$ , $B$ , $C$ are determined by: $$\begin{array}{ll} T_1 - T_0 = A.(x_1 - x_0) + B.(y_1 - y_0) + C.(z_1 - z_0) \\ T_2 - T_0 = A.(x_2 - x_0) + B.(y_2 - y_0) + C.(z_2 - z_0) \\ T_3 - T_0 = A.(x_3 - x_0) + B.(y_3 - y_0) + C.(z_3 - z_0) \end{array}$$ Three equations with three unknowns. A solution can be found: $$\left[ \begin{array}{c} A \\ B \\ C \end{array} \right] = \left[ \begin{array}{ccc} x_1-x_0 & y_1-y_0 & z_1-z_0 \\ x_2-x_0 & y_2-y_0 & z_2-z_0 \\ x_3-x_0 & y_3-y_0 & z_3-z_0 \end{array} \right]^{-1} \left[ \begin{array}{c} T_1-T_0 \\ T_2-T_0 \\ T_3-T_0 \end{array} \right]$$ It is concluded that $A,B,C$ and hence $(T-T_0)$ must be a linear expression in the $(T_k-T_0)$: $$T - T_0 = \xi.(T_1 - T_0) + \eta.(T_2 - T_0) + \zeta.(T_3 - T_0)$$ $$= \left[ \begin{array}{ccc} \xi & \eta & \zeta \end{array} \right] \left[ \begin{array}{c} T_1-T_0 \\ T_2-T_0 \\ T_3-T_0 \end{array} \right]$$ See above: $$= \left[ \begin{array}{ccc} \xi & \eta & \zeta \end{array} \right] \left[ \begin{array}{ccc} x_1-x_0 & y_1-y_0 & z_1-z_0 \\ x_2-x_0 & y_2-y_0 & z_2-z_0 \\ x_3-x_0 & y_3-y_0 & z_3-z_0 \end{array} \right] \left[ \begin{array}{c} A \\ B \\ C \end{array} \right]$$ See above: $$= T - T_0 = \left[ \begin{array}{ccc} x-x_0 & y-y_0 & z-z_0 \end{array} \right] \left[ \begin{array}{c} A \\ B \\ C \end{array} \right]$$ Hence: $$\begin{array}{ll} x - x_0 = \xi .(x_1 - x_0) + \eta.(x_2 - x_0) + \zeta.(x_3 - x_0) \\ y - y_0 = \xi .(y_1 - y_0) + \eta.(y_2 - y_0) + \zeta.(y_3 - y_0) \\ z - z_0 = \xi .(z_1 - z_0) + \eta.(z_2 - z_0) + \zeta.(z_3 - z_0) \end{array}$$ But also: $$T - T_0 = \xi.(T_1 - T_0) + \eta.(T_2 - T_0) + \zeta.(T_3 - T_0)$$ Therefore the same expression holds for the function $T$ as well as for the coordinates $x,y,z$. This is called an isoparametric transformation. It is remarked without proof that the local coordinates $\xi,\eta,\zeta$ within a tetrahedron can be interpreted as sub-volumes, spanned by the vectors $\vec{r}_k-\vec{r}_0$ and $\vec{r}-\vec{r}_0$ where $\vec{r}=(x,y,z)$ and $k=1,2,3$. The general tetrahedron is thus mapped upon a special tetrahedron. This special tetrahedron is commonly called the parent tetrahedron. It rests in $(\xi,\eta,\zeta)$ space and it has (unit) vertices $(0,0,0)$ , $(1,0,0)$ , $(0,1,0)$ , $(0,0,1)$ . The inside ($\to$ volume) of both tetrahedrons is defined by: $$\xi > 0 \quad ; \quad \eta > 0 \quad ; \quad \zeta > 0 \quad ; \quad \xi + \eta + \zeta < 1$$ Now for the volume of the general tetrahedron. It follows from the above that: $$\begin{array}{ll} dx = (x_1 - x_0)\,d\xi + (x_2 - x_0)\,d\eta + (x_3 - x_0)\,d\zeta \\ dy = (y_1 - y_0)\,d\xi + (y_2 - y_0)\,d\eta + (y_3 - y_0)\,d\zeta \\ dz = (z_1 - z_0)\,d\xi + (z_2 - z_0)\,d\eta + (z_3 - z_0)\,d\zeta \end{array}$$ Consequently: $$dx\,dy\,dz = \begin{vmatrix} (x_1 - x_0) & (x_2 - x_0) & (x_3 - x_0) \\ (y_1 - y_0) & (y_2 - y_0) & (y_3 - y_0) \\ (z_1 - z_0) & (z_2 - z_0) & (z_3 - z_0) \end{vmatrix} d\xi\,d\eta\,d\zeta$$ Upon integration, we only have to calculate for the special case: the parent tetrahedron. Christian Blatter has shown how to do such a thing: $$\iiint d\xi\,d\eta\,d\zeta = \frac{1}{6}$$ The general case follows from this: $$\iiint dx\,dy\,dz = \frac{1}{6} \begin{vmatrix} (x_1 - x_0) & (x_2 - x_0) & (x_3 - x_0) \\ (y_1 - y_0) & (y_2 - y_0) & (y_3 - y_0) \\ (z_1 - z_0) & (z_2 - z_0) & (z_3 - z_0) \end{vmatrix}$$ Let's work it out for the example at hand, where the vertices $(x_k,y_k,z_k)$ for $\;k=0,1,2,3$ , upon (renewed) calculation, are given by: $(0,2,0)$, $(8,-2,0)$ , $(0,6,0)$ , $(0,2,4)$. So the volume is: $$\frac{1}{6} \begin{vmatrix} (x_1 - x_0) & (x_2 - x_0) & (x_3 - x_0) \\ (y_1 - y_0) & (y_2 - y_0) & (y_3 - y_0) \\ (z_1 - z_0) & (z_2 - z_0) & (z_3 - z_0) \end{vmatrix} = \frac{1}{6} \begin{vmatrix} 8 & 0 & 0 \\ -4 & 4 & 0 \\ 0 & 0 & 4 \end{vmatrix} = \frac{64}{3}$$ Further Notes. The theory of the linear tetrahedron is the 3-D generalization of an analogous theory for the 2-D linear triangle: As far as the latter reference is concerned, there exists a "closed" triangle equation in 2-D. And there exists a "closed" tetraedron equation in 3-D as well: $$T(x,y,z) = \min( \xi , \eta , \zeta, 1 - \xi - \eta - \zeta )$$ Our "inside/outside" function $T$ is zero at the boundaries of the tetrahedron, positive inside and negative outside. Quite the same is the case with more familiar closed 3-D equations, like the one of a sphere: $\;S(x,y,z) = R^2 - (x-a)^2 - (y-b)^2 - (z-c)^2$ . The maximum of the function $T$ is reached for $\xi = \eta = \zeta = 1 - \xi - \eta - \zeta = 1/4$ , hence at the midpoint (barycenter) of the tetrahedron. • Oh wow, I will definitely have a detailed look on this solution when I'm back home. I hope you put that much effort into it because you enjoy yourself thinking about problems in different ways because I would never ask for so much work.. – Tesla Jun 5 '16 at 18:09 • @Sigma: Don't worry. Most of the material has been there already in LaTeX (PDF) format. Only the last part concerning the volumes is really "new". And of course I'm doing mathematics for fun, a great deal :-) – Han de Bruijn Jun 5 '16 at 19:57	2019-06-19 11:39:29	{"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": 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": 0.9267792105674744, "perplexity": 273.75688512296523}, "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-2019-26/segments/1560627998959.46/warc/CC-MAIN-20190619103826-20190619125826-00103.warc.gz"}
http://sunglee.us/mathphysarchive/?p=2590	# An Algebra Problem on Twitter Problem: Given $x,y\geq 0$ satisfying $$\label{eq:ellipse}x+y+\sqrt{2x^2+2xy+3y^2}=4$$ prove $x^2y<4$. (Hat tip: Sam Walters) Solution. First rewrite \eqref{eq:ellipse} as $$\label{eq:ellipse2}\sqrt{2x^2+2xy+3y^2}=4-x-y$$ Squaring \eqref{eq:ellipse} we obtain an equation of ellipse $$\label{eq:ellipse3}(x+4)^2+2(y+2)^2=40$$ (Figire 1) Figure 1 Graphically we see that the inequality holds as shown in Figure 2. Figure 2. Ellipse (x+4)^2+2(y+2)^2=40, x=0..-4+sqrt(40) (red) and y=4/x^2 (blue) Suppose $x>0$ (for $x=0$ the inequality $x^2y<4$ is trivial). Since $x>0,y>0$ then, $$x^2y<4\Longleftrightarrow (y+2)^2<\left(\frac{4}{x^2}+2\right)^2$$ Solve \eqref{eq:ellipse3} for $(y+2)^2$. $$\label{eq:ellipse4}(y+2)^2=20-\frac{(x+4)^2}{2}$$ Now subtract $\left(\frac{4}{x^2}+2\right)^2$ from the RHS of \eqref{eq:ellipse4}. $$20-\frac{(x+4)^2}{2}-\left(\frac{4}{x^2}+2\right)^2=\frac{-x^6-8x^5+16x^4-32x^2-32}{2x^4}<0$$ since $-x^6-8x^5+16x^4-32x^2-32<0$ for $0<x<-4+\sqrt{40}$ as shown in Figure 3. Figure 3. The graph of f(x)=-x^6-8x^5+16x^4-32x^2-32, x=0..-4+sqrt(40) Update: Republic of Math graphically came up with a sharper inequality $x^2y<1$. The graphics can be seen here. As you can see in the graphics, there is still room for even (slightly) more sharp inequality. In fact $x^2y<0.9$ as you can see in Figure 4 below. Figure 4. Ellipse (x+4)^2+2(y+2)^2=40, x=0..-4+sqrt(40) (red) and y=0.9/x^2 (blue) Update: While I could not analytically find the smallest value of $a>0$ such that $x^2y<a$, I found graphically that $a$ can be as small as $0.789$. Figure 5 and Figure 6 are the graphs of $f(x)=-x^6-8x^5+16x^4-8ax^2-2a^2$ for $0\leq x\leq -4+\sqrt{40}$ for $a=0.789$ and for $a=0.788$, respectively. For $a=0.789$, $f(x)<0$ on $[0,-4+\sqrt{40}]$. Figure 5. The graph of f(x)=-x^6-8x^5+16x^4-8ax^2-2a^2 (for a=0.789), x=0..-4+sqrt(40) But with $a=0.788$ $f(x)$ is no longer negative for all $x$ in $[0,-4+\sqrt{40}]$. Figure 5. The graph of f(x)=-x^6-8x^5+16x^4-8ax^2-2a^2 (for a=0.788), x=0..-4+sqrt(40) This site uses Akismet to reduce spam. Learn how your comment data is processed.	2018-10-21 21:00:51	{"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": 4, "x-ck12": 0, "texerror": 0, "math_score": 0.9066523909568787, "perplexity": 780.9795816626837}, "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-2018-43/segments/1539583514355.90/warc/CC-MAIN-20181021203102-20181021224602-00124.warc.gz"}
http://amakelov.github.io/tags/categories/index.html	The splitting lemma in abelian categories is a basic tool for decomposing objects into biproducts. It is at the heart of some powerful structure theorems, such as the ones for finitely generated abelian groups and more generally finitely generated modules over a PID. In this short post we state the basic decomposition result that any morphism in an abelian category decomposes canonically into an epimorphism followed by a monomorphism, and derive some very useful consequences. In this short post we define abelian categories (which takes some work), which were introduced as abstractions of some core properties of categories like abelian groups and modules over a ring; so for example most of homological algebra can be carried over any abelian category, which is neat.	2017-07-20 22:33:42	{"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": 0.8018840551376343, "perplexity": 291.80615499571974}, "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-2017-30/segments/1500549423512.93/warc/CC-MAIN-20170720222017-20170721002017-00699.warc.gz"}
https://physics.stackexchange.com/questions/472483/how-does-ice-melt-when-immersed-in-water/472586	# How does ice melt when immersed in water? When an ice cube is immersed in water at a room temperature, how is the thermal energy from the water transferred to the ice cube? • Infrared radiation from the water transfers thermal energy to the ice cube, which increases the ice cube particles KE store, breaking the intermolecular bonds of the ice cube, melting it. • The Brownian motion of the water particles causes them to collide with the ice cube, transferring KE to the ice cubes particles, increasing temperature, breaking intermolecular bonds and melting it. • – Steeven Apr 13 '19 at 13:14 • Does this rule out the first answer I gave? because infra red is a wave and doesn’t transfer matter whereas natural convection does “Heat convection occurs when bulk flow of a fluid (gas or liquid) carries heat along with the flow of matter in the fluid.”~wiki – Ubaid Hassan Apr 13 '19 at 13:34 • No no, it doesn't rule it out. Both are present at the same time, but radiation (following Stefan-Boltzmann's law) is very small at lower temperatures and becomes negligible in comparison to convection in a liquid. – Steeven Apr 13 '19 at 13:35 • When you say radiation, are you referring to infrared radiation? ~so to summarise, the heat transfer of water to ice is the combination of natural convection and thermal(including infrared) radiation? If so-does the transfer of KE by water particles colliding with the ice cube come into the picture at all? – Ubaid Hassan Apr 13 '19 at 13:44 • Water and ice are opaque (black) for thermal infrared. They are also at the same temperature, so radiating equally. – user137289 Apr 13 '19 at 21:24 ## Energy transfer methods In general, there exist three heat transfer mechanisms: • Thermal radiation transfers heat across a distance. More accurately, it is the transfer of wavelengths on the spectrum of light that when absorbed by the body is converted into heat). It follows Stefan-Boltzmann's law: $$\dot q_\text{rad}=\varepsilon\sigma_sA(T_1^4-T_2^4)$$ ($$\dot q$$ is energy per second transferred from body 1 to body 2, $$T$$ temperature, $$\varepsilon$$ emissivity, $$\sigma$$ the Stefan-Boltzmann constant, $$A$$ the radiating surface area.) • Thermal conduction transfers heat through a solid. It is defined for a continuum, a solid material, but can be thought of as heat passed on between neighbour particles. It follows Fourier's law: $$\dot q_\text{cond}=A\kappa\frac{\Delta T}{\Delta x}$$ ($$A$$ is area through which the heat flows, $$\kappa$$ thermal conductivity, $$\Delta T$$ temperature difference between two points, $$\Delta x$$ distance between those two points over which the heat is tranferred.) When you mention Brownian motion, it is relevant here with conduction: The random motion of particles, electrons etc. cause them to "bump into" and interact with neighbour particles. If one particles is more energetic, at a collision between particles they will share some of the kinetic energy. This is how thermal energy is conductively transferred. • Thermal convection transfers heat to/from a body by flowing close to it and deliver/absorb thermal energy to/from the surface. In some sense, it can be thought of as conduction between a fluid particle and a surface particle, where the fluid particle right after is replaced with a new, fresh one. Delivery/absorption of thermal energy from a single fluid particle is negligible as it carries a very little amount of energy, but with constant replacement of particles with newer ones, the energy transferred accumulates and becomes significant. This fluid-in-motion-induced heating/cooling effect is termed convection. It follows the relationship: $$\dot q_\text{conv}=Ah(T_\text{fluid}-T_\text{body})$$ $$A$$ is area exposed to the fluid. $$h$$ is the heat transfer coefficient and it highly depends on the scenario (the fluid, the flow, the surface interaction etc). $$h$$ is often experimentally determined beforehand. There are two types of thermal convection: • Natural convection caused purely by natural factors such as differences in temperature or density (the cooling water near the ice surface becomes denser and sinks, and is thus replaced by other warmer fluid molecules. In general, natural convection is the mechanism behind hot air rising and cold air falling and similar phenomena.) • Forced convection, which is fluid flow caused by non-natural mechanisms such as by a pump. In your case we have natural convection: The water particles near the ice surface deliver heat to the ice and in turn cool down. These now "colder" water particles are denser or "heavier" and will sink. New, warmer particles will take their place, ready to deliver more energy to the ice surface and repeat the process. ## Which is more dominant? The above three energy transfer factors are all the possibilities there are to transport energy. They are generally considered on equal terms as three distinct mechanisms with each their own energy transfer models. But, as you can see, convection is basically a "flow-version" of conduction if we consider it microscopically. • For thin fluids (with low viscosity), the convective effect of effective heating/cooling due to fluid motion is dominant. • For very thick fluids (with very high viscosity), so thick that you might mistake them for solids, heat can flow from particle to particle in a conductive manner, and conduction is dominant. • For some-what thick fluids, we may see a mix of these factors. The higher the heat capacity (corresponding to lower $$\kappa$$) of the fluid, the weaker is the conductive mechanism. In your case with water that has a rather low $$\kappa$$, we should be able to assume only a predominantly convective mechanism and no/negligible conduction over longer distances in the water. Thermal radiation could still be a factor as well, but at fairly low temperatures, radiation is low (note the power of 4 in the model) and possibly negligible. We end up with only convection (natural in your case) having a large influence in your case - in fluids, this is often the only effect that is relevant to consider, unless when sinking a glowing-hot metal into a very volatile liquid. This analysis can be verified by looking up numbers, as some comments ask for, of water and ice for the different models as well by comparing with the viscosity. I will not do this in this answer, but it should be fairly easy to find online; other answers are giving some of such numbers to justify the conclusion. • +100 if I could, but I have one final question. If thermal conduction is the transfer of heat “particle to particle” as you put it, then I assume you meant the transfer of KE between the particles. If so, the particles(and the bodies)must be touching so there should not be any “distance” between the “two points”. But this doesn’t fit the formula you gave where X =distance – Ubaid Hassan Apr 13 '19 at 14:42 • @UbaidHassan Yes, at the atomic level, heat and temperature is nothing but kinetic "vibrational" energy. Thermal conduction is not really defined for particle-to-particle. Fourier's law is found emperically under the assumption of a continuous material, and thus under the assumption that there is enough material for particle-particle interactions to be indistinguishable and only for their overall collective effect to play a role. Of this reason you will never hear conduction described for atomic particles; which is why it doesn't make much sense in your scenario either. – Steeven Apr 13 '19 at 14:54 • Can you give an estimate about the ratio of infrared? It's certainly minor, but how minor? Single-digit percent? – Peter - Reinstate Monica Apr 13 '19 at 17:24 • @PeterA.Schneider It should be much less than single digit percents. Consider the heat we get from the sun, at 6000K. If you have an object that's around 300K, that's 20 lower temperatures. Radiation is a 4th power effect, so that means the effects will be 20^4 less. That's 160,000 times less than the effect of the sun. The areas wont line up, obviously, so you'd have to do some conversions, but we're talking 5 orders of magnitude weaker than the sun. How long does it take the sun to melt an ice cube? – Cort Ammon Apr 14 '19 at 3:23 • @CortAmmon Thanks, this was the kind of estimate I had in mind -- I missed the 4th power in the Boltzmann equation. – Peter - Reinstate Monica Apr 14 '19 at 9:35 I am in complete disagreement with previous answers which consider convection as the main mechanism for heat transfer from liquid water to the ice cube. Convection is an important and dominant mechanism to maintain the liquid layers close to the ice surface at higher temperature. Thus, its main role is to ensure that at the surface between liquid and solid a constant difference of temperature is maintained. However, as a mechanism to carry energy from the liquid into the solid, convection simply does not exist! Unless one would think of fluid streams penetrating into the solid, which is not the case. Therefore we are left with conduction or radiation as possible ways to tranfer thermal energy from liquid water to the ice. A simple order-of-magnitude estimate, based on the formulae of the Stefan-Boltzmann's law and Fourier's law, taking into account the SI values of about $$10^{-7}$$ for $$\sigma_s$$, of about $$2$$ for $$\kappa$$ of ice, the values of the two temperatures and a value of $$\Delta x$$ of the order of a few interatomic distances, shows that the radiation contribution is negligible. An additional remark could be added on the microscopic description of the melting process. It is a well established observation that pre-melting, i.e. the melting of a solid starting from the surface layers, instead of than from the bulk, is a phenomenon present even in the case of ice. This observation would exclude the possibility that the melting process in the present case could start in the bulk of the ice. • So convection according to you is only the maintenance of a constant temperature of the liquid layers around the ice cube (in this case)? Then what actually is the mechanism of heat transfer to the ice cube from water? – Ubaid Hassan Apr 13 '19 at 21:44 • I wrote it above. Thermal conduction prevails by orders of magnitude on radiation. That is the only relevant mechanism to transfer thermal energy across the liquid solid border. Convection cennot play a direct role by definition. It plays an indirect role, as I tryed to explain. – GiorgioP Apr 13 '19 at 22:17 • I have added an explicit statement at the beginning of the third paragraph. In the original post it was implicit since, after exclusion of convection, I was considering the relative role played by conduction and radiation. – GiorgioP Apr 13 '19 at 22:29 • To provide a counterpoint, convection exists in the same way the sound barrier exists. While, at the microscopic level, convection is merely conduction, the macroscopic fluid flow in convection makes it so much more effective at transfering heat that we have to use entirely different equations to model it. Likewise, gas molecules simply move according to the equations of motion at any speed. However, there's a key point where the momentum of the gas particles becomes substantially more anisotropic (starts to have a direction), and when that happens we see shock waves anda "sound barrier." – Cort Ammon Apr 14 '19 at 16:48 • While you are right that convection and the sound barrier do not exist in the most strict of technical senses, I just wanted to make sure somebody doesn't get the wrong idea from the words. – Cort Ammon Apr 14 '19 at 16:49 Thermal energy transfer is in the form of heat from the water to the ice cube by natural convection. If the cube and water together form an isolated system (no heat transfer between them and their surroundings) the heat transfer will continue until all the ice is melted, or until the water temperature equals 0 C at which point any ice remaining will be in two phase thermal equilibrium with the water. Hope this helps • How do you know it's natural convection? – pentane Apr 13 '19 at 21:25 • @pentane There are two kinds of convection: forced and natural. Forced usually involves some kind of forced movement of the fluid over a surface. Say, by way of a fan, the wind, a pump for water, etc. Natural involves movement due to buoyancy, warm fluid rising over cool. – Bob D Apr 13 '19 at 21:53 • no I know what it is but how do you know an ice cube in a glass of water is convection. where's the flow? – pentane Apr 13 '19 at 21:57 # Heat Transfer Modes The three forms of heat transfer between a system and the surroundings are as follows: ## Conduction This is the transport of heat by particles exchanging their internal energy. It occurs by one of three modes -- molecular collisions (gases), collisions/vibrations (local in liquids and lattice in solids), and free electron transport (in conductors and semiconductors). Conduction requires (or sets up) a temperature gradient in the material that is transporting the heat. ## Convection This is the transport of heat content by the bulk motion of a fluid over an object. It occurs in one of two modes -- free or forced. In free convection, the fluid moves because it is subject to a buoyancy force. In forced convection, we push the fluid. Convection requires a temperature difference. Convection can be modeled using principles of conduction across a film between the fluid and the object. This is the transport of energy from an object as electromagnetic radiation. Radiation only requires that objects have a temperature. # The Melting Process To melt, atoms in a solid must gain enough energy to leave their bonds in the solid. Fusion is endothermic. The energy arrives as heat from the surroundings. It arrives by the motion of the hotter liquid water molecules hitting the colder solid. The energy difference between moving liquid molecules and static (vibrating) solid molecules is a temperature difference in internal energy coordinates. That temperature difference needs only be infinitesimal to support the flow of heat from hot to cold. Liquid water does not support free electrons (of course not!) nor does it support lattice vibrations (that is what is happening in the ice). So, the one mode of transport of heat is conduction by molecular collisions from liquid water to solid ice. The energy as heat can arrive by convection flow. When the system is in a gravitational field, and when the liquid immediately around the ice might become colder than the bulk water, the colder water will be denser. It will start to flow downward by natural convection. Thus, natural convection can be a factor in the heat flow. When the ice is floating on the water (typical), the colder water below the ice will fall down in the warmer water below it. As an inverse case, when you could put the ice cube at the bottom of a container and have hot water above it, you will shut down the natural convection mode. Think also about a cold penny that sits inserted into an insulated floor with hot air above it. The penny will have no natural convection modes because the cold air that might form around it is already denser than the hot air above it. This same thought is behind the formation of cold and hot fronts with thunderstorms in weather patterns. You did not say whether the tank was stirred. So we can ignore forced convection. The ice is radiating from it. The hotter water is radiating to the ice. The net radiation flux is to the ice from the water. ## Estimates of Magnitudes The temperatures of the solid ice and liquid water control the net radiation flux. When the liquid is only infinitesimally above the ice in temperature, the net radiation flux is ... small. Add to this that both ice and water have emissivities well below unity and their emissivities are comparable. At the end, you can pretty much say radiation is ... to be neglected. Natural convection, when it occurs, swamps conduction heat transfer (well, not literally of course). Presuming the ice is at the top allows for this. Saying the ice is surrounding by water and mixed with it will lower its contribution. At the end, we have conduction. Those "hotter" liquid water molecules are colliding constantly with those "colder" solid ice molecules (hot and cold as measures of internal energy). The transfer of heat is occurring constantly. A reference graph showing the variations in conductivity is found at this link. ## Remaining Clarification In pure materials (water), fusion occurs at a constant temperature. Never, ever can you discuss fusion as a process where the solid becomes hotter. The solid ice in this case stays pinned at one temperature as it completely melts. Inversely, you might find that when you mistakenly think the ice gets hotter during melting, you will immediately have to shut down any and all net heat transfer from the surroundings (liquid) to the system (ice). It is the second law of thermodynamics at play. • Liquids have a physics much closer to that of solids than gases (it is enough to compare the difference of densities to acknowledge it). Describing transport of energy in a liquid in term of collision is as good or as bad as using the same explanation for conduction in solids. – GiorgioP Apr 14 '19 at 17:13 • No doubt about important differences at level of the numbers. My point was about modeling the atomic dynamics of a liquid as collisions. The term collision is physically justified whenever an important change of momentum is concentrated in a short time interval. This is not the case for liquids. Atomic dynamic in liquids is much more complicate than phonon dynamics but collective modes (the equivalet of phonons in a harmonic solid) are routinely used to describe it. – GiorgioP Apr 15 '19 at 5:02 • A reasonable one-particle description of the atomic dynamics in dense liquids is a kind of superposition between diffusion and the so called cage motion which is the analogous of atomic vibration in a solid. The key point motivating my comment is that neither diffusion nor cage vibrations can be reasonably modeled as simple collisions. – GiorgioP Apr 15 '19 at 5:06 • @GiorgioP Much appreciated. I've modified my description to account for your insights in a manner that keeps it simple without I distorting the truth I hope. – Jeffrey J Weimer Apr 15 '19 at 12:40	2021-08-01 22:44:12	{"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": 22, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5592579245567322, "perplexity": 622.2718831763361}, "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/1627046154277.15/warc/CC-MAIN-20210801221329-20210802011329-00669.warc.gz"}
https://plainmath.net/90383/is-m-unique-given-a-m-2-where-a-and-m-ar	Is M unique given A=M^2 where A and M are real matrices? Is M unique given $A={M}^{2}$ where A and M are real matrices? My guessing is they are unique as I tried to diagonalize A to $PD{P}^{-1}$ and no matter how I order the eigenvalues in D, it still gives the same $M=P{D}^{\frac{1}{2}}{P}^{-1}$. But I am not sure this is true in general, since the diagonalization is too specific. You can still ask an expert for help • Questions are typically answered in as fast as 30 minutes Solve your problem for the price of one coffee • Math expert for every subject • Pay only if we can solve it No, if M is any $n×n$ matrix satisfying $A={M}^{2}$ (we then say that M is a square root of A) then −M is also a square root of A. For this kind of problem it helps to first think about the case where A and M are scalars then think about whether or not the same argument holds for matrices. Even if A is diagonalizable (which doesn't always hold), your argument using diagonalization doesn't really work here since there are many choices for ${D}^{1/2}$ (Note that ${D}^{1/2}$ is by definition any matrix satisfying ${D}^{1/2}{D}^{1/2}=D$). For instance, in the special case where A has n distinct nonzero eigenvalues, there are ${2}^{n}$ choices for this matrix. Explicitly, if $D=\mathrm{diag}\left({\lambda }_{1},\dots ,{\lambda }_{n}\right)$ then all square roots are of the form $\mathrm{diag}\left({\mu }_{1},\dots ,{\mu }_{n}\right)$ where μi is any number (real or complex) satisfying ${\mu }_{i}^{2}={\lambda }_{i}$ (there are exactly two of them for each i).	2022-09-30 07:29:29	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 67, "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": 0.9574983716011047, "perplexity": 223.4718472818391}, "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-2022-40/segments/1664030335444.58/warc/CC-MAIN-20220930051717-20220930081717-00511.warc.gz"}