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https://preprint.impa.br/visualizar?id=1820	Preprint A470/2006 Analysis of block matrix preconditioners for elliptic optimal control problems Christian E. Schaerer \| Mathew, Tarek P. \| Sarkis, M. Keywords: Optimal control \| elliptic Neumann problem \| augmented Lagrangian \| saddle point problem \| regularization \| preconditioners. In this paper, we describe and analyze several block matrix iterative algorithms for solving a {\it saddle point} linear system arising from the discretization of a inear-quadratic {\it elliptic control} problem with Neumann boundary conditions. To ensure that the problem is well posed, a {\it regularization} term with a parameter $\alpha$ is included. The first algorithm reduces the saddle point system to a symmetric positive definite Schur complement system for the control variable and employs CG acceleration, however, double iteration is required (except in special cases). A preconditioner yielding a rate of convergence independent of the mesh size $h$ is described for $\Omega \subset R^{2}$ or $R^{3}$, and a preconditioner independent of $h$ and $\alpha$ when $\Omega \subset R^{2}$. Next, two algorithms avoiding double iteration are described using an {\it augmented Lagrangian} formulation. One of these algorithms solves the augmented saddle point system employing MINRES acceleration, while the other solves a symmetric positive definite reformulation of the augmented saddle point system employing CG acceleration. For both algorithms, a symmetric positive definite preconditioner is described yielding a rate of convergence independent of $h$. In addition to the above algorithms, two {\it heuristic} algorithms are described, one a projected CG algorithm, and the other an indefinite block matrix preconditioner employing GMRES acceleration. Rigorous convergence results, however, are not known for the heuristic algorithms.	2023-01-31 13:43:20	{"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": 0.9444565176963806, "perplexity": 743.9159854609338}, "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/1674764499871.68/warc/CC-MAIN-20230131122916-20230131152916-00564.warc.gz"}
http://mathhelpforum.com/advanced-algebra/104206-image-transformation.html	Math Help - Image of a transformation 1. Image of a transformation Describe the image of the transformation T(vector x) = A(vector x) geometrically (line, plane in R^2 or R^3) ; means a new row A = [2 1 3 ; 3 4 2 ; 6 5 7]. I multiplied this matrix by the [x1 ; x2; x3]. I got x1[2 ; 3 ; 6] + x2[1 ; 4 ; 5] + x3[3 ; 2 ; 7]. Where do I go from here? I'm thinking it's a plane in R^3 but I don't know for sure. Thanks 2. Originally Posted by noles2188 Describe the image of the transformation T(vector x) = A(vector x) geometrically (line, plane in R^2 or R^3) ; means a new row A = [2 1 3 ; 3 4 2 ; 6 5 7]. I multiplied this matrix by the [x1 ; x2; x3]. I got x1[2 ; 3 ; 6] + x2[1 ; 4 ; 5] + x3[3 ; 2 ; 7]. Where do I go from here? I'm thinking it's a plane in R^3 but I don't know for sure. Thanks reduce A to echelon form or note that $2v_1 - v_2=v_3,$ where $v_j$ is the j-th column of A. thus the rank of A is 2 and therefore the image is a plane spanned by $v_1,v_2.$	2014-12-20 10:08:07	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 3, "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": 0.8667401075363159, "perplexity": 634.0987048870445}, "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-2014-52/segments/1418802769642.136/warc/CC-MAIN-20141217075249-00161-ip-10-231-17-201.ec2.internal.warc.gz"}
https://math.stackexchange.com/questions/3483514/how-to-evaluate-int-0-infty-frac1t-arctan-left-fract12t2-right	# How to evaluate $\int_{0}^{\infty}\frac{1}{t}\arctan\left(\frac{t}{1+2t^2}\right)\,\mathrm dt$? I entered this integral into Wolframalpha, and got $$\int_{0}^{\infty}\frac{1}{t}\arctan\left(\frac{t}{1+2t^2}\right)\,\mathrm dt=\frac{1}{2}\pi\log{2}.$$ But it doesn't provide step by step solution for this integral. This integral is a bonus challenge in my Calculus class, and the professor that the key is $$\arctan$$. But I don't know is there any special about $$\arctan\left(\frac{t}{1+2t^2}\right),$$ so I tried some common integration method, and it doesn't work. • Well, do you rely entirely on WolframAlpha? What have you tried to solve it? For a hint, try to rewrite $\arctan\left(\frac{t}{1+2t^2}\right)$ in a form of $\arctan(at)\pm \arctan(bt)$. – Zacky Dec 21 '19 at 10:41 • @Zacky Yes, I did try some integration method that I knew, but none of them works. – 3142 maple Dec 21 '19 at 10:45 • Then can you mention some integration methods that you know and tried? Bringing context is helpful. Are you familiar with contour integration or Feynman's trick or Frullani's integral (etc)? See also: math.meta.stackexchange.com/questions/9959/… – Zacky Dec 21 '19 at 10:47 • @Zacky Thanks, I have added some context about the question. And in the list of integration methods you listed, I only know Feynman's trick. – 3142 maple Dec 21 '19 at 10:55 • After I googled Frullani integral, I found that the solution is quite similar to it, maybe $arctan(\frac{t}{1+2t^2})$ can be converted to $f(at)-f(bt)$. – 3142 maple Dec 21 '19 at 11:03 Following @Zacky's hint, use $$\frac{t}{1+2t^2}=\frac{2t-t}{1+2t\cdot t}$$ to rewrite the integral as the Frullani integral$$\int_0^\infty\frac{\arctan(2t)-\arctan t}{t}dt=(\arctan0-\arctan\infty)\ln\frac12=\frac{\pi}{2}\ln 2.$$ • I found that $arctan(2t)-arctan(t)$ can be converted into another integral, than change the order of the whole double integral can make it easy to solve. Without using Frullani integral. – 3142 maple Dec 21 '19 at 11:21 First notice that: $$\arctan\left(\frac{x}{1+2x^2}\right)=\arctan\left(\frac{2x-x}{1+2x\cdot x}\right)=\arctan(2x)-\arctan(x)$$ So the integral can be rewritten as: $$I=\int_0^\infty \frac{\arctan(2x)-\arctan x}{x}dx\overset{IBP}=\int_0^\infty \ln x\left(\frac{1}{1+x^2}-\frac{2}{1+4x^2}\right)dx$$ $$2\int_0^\infty \frac{\ln x}{1+4x^2}dx\overset{2x\to x}=\int_0^\infty \frac{\ln x-\ln 2}{1+x^2}dx$$ $$\Rightarrow I=\int_0^\infty \frac{\ln x -\ln x+\ln 2}{1+x^2}dx=\ln 2\int_0^\infty \frac{dx}{1+x^2}=\frac{\pi}{2}\ln 2$$ • This technique can be generalised to evaluate Frullani integrals, viz.\begin{align}\int_{0}^{\infty}\frac{f\left(ax\right)-f\left(bx\right)}{x}dx &=\int_{0}^{\infty}\ln x\cdot\left(bf^{\prime}\left(bx\right)-af^{\prime}\left(ax\right)\right)dx\\&=\left[c\int_{0}^{\infty}\ln x\cdot f^{\prime}\left(cx\right)dx\right]_{a}^{b}\\&=\left[\int_{0}^{\infty}\ln\frac{y}{c}\cdot f^{\prime}\left(y\right)dy\right]_{a}^{b}\\&=\ln\frac{a}{b}\cdot\int_{0}^{\infty}f^{\prime}\left(y\right)dy\\&=\ln\frac{b}{a}\cdot\left(f\left(0\right)-f\left(\infty\right)\right).\end{align} – J.G. Dec 21 '19 at 12:19 By integration by parts we have $$I=\int_0^\infty\ln x\cdot\frac{2x^2-1}{4x^4+5x^2+1}dx$$ $$=\int_0^\infty\frac{\ln x}{1+x^2}dx-\color{red}{\int_0^\infty\frac{2\ln x}{1+4x^2}dx}$$ $$\overset{\color{red}{2x\mapsto x}}{=}\int_0^\infty\frac{\ln2}{1+x^2}dx=\frac{\pi}{2}\ln2$$ I found another way to solve this is to make the integral to be a double integral, and change the order. But the key still is $$\arctan(\frac{t}{1+2t^2})=\arctan(2t)-\arctan(t)$$. $$\int_0^\infty\frac{\arctan(\frac{t}{1+2t^2})}{t}=\int_0^\infty\frac{\arctan(2t)-\arctan t}{t}dt=\int_0^\infty \frac{1}{t} \int_1^2 \frac{t}{1+(yt)^2} dydt$$ $$=\int_1^2 \int_0^\infty \frac{1}{1+(yt)^2} dtdy=\int_1^2 \frac{1}{y} \arctan(\infty)-\arctan(0) dy=\frac{\pi}{2}ln2$$ • Since you seem interested in this, if you would like you can try to solve these too: $$\int_0^\infty \frac{\arctan\left(\frac{2x}{1+x^2}\right)}{x}dx=\pi\ln(1+\sqrt 2)$$ $$\int_0^\infty \frac{\arctan\left(\frac{2x}{1+2x^2}\right)}{x}dx=\frac{\pi}{2}\ln(2+\sqrt 3)$$ $$\int_0^\infty \frac{\arctan\left(\frac{x}{1+x^2}\right)}{x}dx=\pi \ln\varphi ,\ \varphi=\frac{1+\sqrt 5}{2}$$ For start think of a way to find the "key" in those cases. – Zacky Dec 21 '19 at 13:04	2021-06-24 23:01:44	{"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": 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": 15, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9147158265113831, "perplexity": 436.65899951038926}, "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/1623488559139.95/warc/CC-MAIN-20210624202437-20210624232437-00013.warc.gz"}
https://forum.math.toronto.edu/index.php?PHPSESSID=ra15jhi630lqj81oh6lhef6703&action=printpage;topic=1265.0	# Toronto Math Forum ## MAT244--2018F => MAT244--Lectures & Home Assignments => Topic started by: Xinyu Jiao on September 25, 2018, 08:58:39 PM Title: Can there exists infinite number of solutions given initial conditions. Post by: Xinyu Jiao on September 25, 2018, 08:58:39 PM In class, we were given an example where a differential equation can have two solutions given some initial condition. Specifically, the equation was $y' = y^\alpha$ with $0<\alpha<1$, and initial condition $y(0) = 0$. This shows that it's not unique, because it does not satisfy some condition which I do not understand. My question is, can there be a differential equation (of order 1) such that given an initial condition, can acquire an infinite number of solutions? The answer to this question should be able to shed light as to the mechanism through which the equation acquires more than one solution. Title: Re: Can there exists infinite number of solutions given initial conditions. Post by: Victor Ivrii on September 25, 2018, 09:27:59 PM For condition see Section 2.8 of the textbook or this Lecture Note (https://q.utoronto.ca/courses/56504/files/1311954?module_item_id=332283) Yes, this equation $y'=3 y^{2/3}$ (I modified it for simplicity) has a general solution $y=(x-c)^{3}$ but also a special solution $y=0$. Thus problem $y'=3 y^{2/3}$, $y(0)=0$ has an infinite number of solutions. Restricting ourselves by $x>0$ we get solutions y=\left\{\begin{aligned} &0 &&0<x<c,\\ &(x-c)^3 && x\ge c\end{aligned}\right. with any $c\ge 0$ and similarly for $x< 0$. This happens because this Lipschitz condition is violated at each point of the solution $y=0$. Title: Re: Can there exists infinite number of solutions given initial conditions. Post by: Kathryn Bucci on October 06, 2018, 10:59:07 AM If 𝑦′=𝑦𝛼 with 0 < 𝛼 < 1 e.g. 𝑦′=3𝑦2/3= f(t,y), then ∂f/∂y=2y-1/3 is not continuous at (0,0). According to theorem 2.4.2 (existence and uniqueness for 1st order nonlinear equations), both f and ∂f/∂y have to be continuous on an interval containing the initial point (0,0) - ∂f/∂y is not continuous there so you can't infer that there is a unique solution. Title: Re: Can there exists infinite number of solutions given initial conditions. Post by: Victor Ivrii on October 06, 2018, 12:10:58 PM Continuity of $\frac{\partial f}{\partial y}$ is not required, but "Hölder property" $\|f(x,y)-f(x,z)\|\le M$ is. There is a notion of the singular solution	2022-05-19 16:37:19	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9042200446128845, "perplexity": 839.2738191865103}, "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/1652662529538.2/warc/CC-MAIN-20220519141152-20220519171152-00293.warc.gz"}
https://www.gamedev.net/forums/topic/274688-2d-vector/	Public Group 2d Vector?? This topic is 5403 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic. Recommended Posts basicly dose such a thing exist? dose it need to? if not what can i do thats like it. thanks Share on other sites Yes, of course. A vector can have any number of dimensions. Otherwise it would be difficult to talk about 2D geometry. Share on other sites i think he means a std::vector. vector < vector < Type > > variable_name; the spaces between the > > is important.. otherwise the compiler will think your bit shifting. Share on other sites yes a std::vector is what im talking about ok thanks for the answer [grin] Share on other sites ps. Why do we even use arrays there so out of date Share on other sites on a simeler note why donst this work vector<const char> szFiles(64); szFiles = {"0.bmp","1.bmp","2.bmp","3.bmp","4.bmp","5.bmp","6.bmp","7.bmp","8.bmp","9.bmp","10.bmp","11.bmp","12.bmp","13.bmp","14.bmp","15.bmp","16.bmp","17.bmp","18.bmp","19.bmp","20.bmp","21.bmp","22.bmp","23.bmp","24.bmp","25.bmp","26.bmp","27.bmp","28.bmp","29.bmp","30.bmp","31.bmp","32.bmp","33.bmp","34.bmp","35.bmp","36.bmp","37.bmp","38.bmp","39.bmp","40.bmp","41.bmp","42.bmp","43.bmp","44.bmp","45.bmp","46.bmp","47.bmp","48.bmp","49.bmp","50.bmp","51.bmp","52.bmp","53.bmp","54.bmp","55.bmp","56.bmp","57.bmp","58.bmp","59.bmp","60.bmp","60.bmp","60.bmp","64.bmp"}; and how can i do somthing like it Share on other sites Correct me if I'm wrong, but you're trying to give values to a constant char, which you can't do, I think. Share on other sites Quote: Original post by raptorstrikeon a simeler note why donst this work* Source Snippet Removed and how can i do somthing like it First of all, the elements of the vector are constant, so there would be no modifying them at all. Second, you can't assign multiple values to a vector. You may have seen that work with a regular old array. You can do what you're trying to do three ways (I'm sure there are more, but here's three.) 1) vector<char> szFiles(64)szFiles[0] = "0.bmp";szFiles[1] = "1.bmp"... 2) vector<char> szFiles;szFiles.push_back("0.bmp");szFiles.push_back("1.bmp");... 3) vector<char> szFilesVec;char* szFiles[] = {"0.bmp","1.bmp","2.bmp","3.bmp","4.bmp","5.bmp","6.bmp","7.bmp","8.bmp","9.bmp","10.bmp","11.bmp","12.bmp","13.bmp","14.bmp","15.bmp","16.bmp","17.bmp","18.bmp","19.bmp","20.bmp","21.bmp","22.bmp","23.bmp","24.bmp","25.bmp","26.bmp","27.bmp","28.bmp","29.bmp","30.bmp","31.bmp","32.bmp","33.bmp","34.bmp","35.bmp","36.bmp","37.bmp","38.bmp","39.bmp","40.bmp","41.bmp","42.bmp","43.bmp","44.bmp","45.bmp","46.bmp","47.bmp","48.bmp","49.bmp","50.bmp","51.bmp","52.bmp","53.bmp","54.bmp","55.bmp","56.bmp","57.bmp","58.bmp","59.bmp","60.bmp","60.bmp","60.bmp","64.bmp"};for (int i=0; i<64; ++i) szFilesVec.push_back(szFiles); Share on other sites I think what you're asking for is a vector of vectors, which you can do. // declarevector< vector<type_name> > some_vector;// accesssome_vector[col][row]some_vector.at(col).at(row) Share on other sites first of all, stop using char's to represent a string. using a char with a std::vector, is like buying a brand new super fast computer and hooking it up to a dial up modem. ok, thats a bad analogy, but you get the point. if your going to use the std library, then make use of the whole thing. change that to vector<string> szFiles; secondly, i don't know if you read my other post in your old thread, so im going to re-post it here: btw, you can replace that really ugly handmade array called szFiles[64]. you can do all this with a loop. for(int i = 0; i < 10; i++){ szFiles = Int_To_String(i + 1) + ".bmp";} where Int_To_String is: string Int_To_String(int num){ //if this is happening, im screwing up somewhere..... assert(number <= 9999999999); char buffer[10]; stringstream ss; ss << num; strcpy(buffer, ss.str().c_str()); return buffer;} btw, you should probably make that int a templated parameter. this way you can send it floats or anything else you need... i didnt want to confuse you though. also, you should definetly use a std::string isntead of an array of pointers... if a function IMG_Load takes a const char, you can convert a string to this by doing stringname.c_str(). one last thing i would like to add, you should get into the habit of NOT using "using namespace std". i didn't, and now i have 10,000+ lines of code which is "using namespace std"... im not looking forward to going through that and changing it all [smile] • 13 • 18 • 29 • 11	2019-07-24 00:06:27	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.264845073223114, "perplexity": 2775.737992569939}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195530246.91/warc/CC-MAIN-20190723235815-20190724021815-00350.warc.gz"}
http://www.msnoise.org/doc/workflow.html	# Workflow¶ This section only presents the “installation” and configuration of MSNoise (read “the first startup of MSNoise”), not the installation of the required software, which is described in Installation. ## Installer (initialize Project)¶ This console scripts is responsible asking questions about the database connection, to create the db.ini file in order to store the answers and to create the tables in the database. Questions are: • What database technology do you want to use? • sqlite: this will create a file in the current folder and use it as DB • mysql: this will connect to a local or remote mysql server, additional information is then required: • hostname: of the mysql server, defaults to 127.0.0.1 • database: must already exist on hostname • username: as registered in the privileged users of the mysql server To run this script: msnoise install --help Usage: [OPTIONS] This command launches the installer. Options: --help Show this message and exit. Warning The credentials will be saved in a flat text file in the current directory. It’s not very safe, but until now we haven’t thought of another solution. MSNoise Admin is a web interface that helps the user define the configuration for all the processing steps, it also allows configuring the stations and filters to be used in the processes. It gives a view on the database tables. Although the interface is still in development for extra features, it already fully replaces the former “Configurator”. $msnoise admin Which, by default, starts a web server listening on all interfaces on port 5000. This can be overriden by passing parameters to the command, e.g. for port 5099: $ msnoise admin -p 5099 Next step is to open a web browser and open the ip address of the machine, by default on the current machine, it’ll be http://localhost:5000/ or http://127.0.0.1:5000/. The top level menu shows four items: ### Home¶ The index page shows • The project location and its database • Stats of the Data Availability, the CC jobs and the DTT jobs ### Configuration¶ #### Station¶ Stations appear as a table and are editable. Stations are defined as: class msnoise.msnoise_admin.Station(net, sta, X, Y, altitude, coordinates, instrument, used) Station Object Parameters: ref (int) – The Station ID in the database net (str) – The network code of the Station sta (str) – The station code X (float) – The X coordinate of the station Y (float) – The Y coordinate of the station altitude (float) – The altitude of the station coordinates (str) – The coordinates system. “DEG” is WGS84 latitude/ longitude in degrees. “UTM” is expressed in meters. instrument (str) – The instrument code, useful with PAZ correction used (bool) – Whether this station must be used in the computations. #### Filter¶ Filters appear as a table and are editable. The filter parameters are validated before submission, so no errors should happen. Note: by default, the used parameter is set to False, don’t forget to change it! Filters are defined as: class msnoise.msnoise_admin.Filter(*kwargs) Filter base class. Parameters: ref (int) – The id of the Filter in the database low (float) – The lower frequency bound of the Whiten function (in Hz) high (float) – The upper frequency bound of the Whiten function (in Hz) mwcs_low (float) – The lower frequency bound of the linear regression done in MWCS (in Hz) mwcs_high (float) – The upper frequency bound of the linear regression done in MWCS (in Hz) rms_threshold (float) – Not used anymore mwcs_wlen (float) – Window length (in seconds) to perform MWCS mwcs_step (float) – Step (in seconds) of the windowing procedure in MWCS used (bool) – Is the filter activated for the processing #### Config¶ All configuration bits appear as a table and are editable. When editing one configuration item, the edition pages shows extra information about the parameter, where it is used and its default value. Most of the configuration bits are case-sensitive! Example view: The table below repeats this Parameter Name Description Default Value data_folder Data Folder output_folder CC Output Folder CROSS_CORRELATIONS data_structure Data Structure [SDS]/BUD/IDDS SDS network Network to analyse [] channels Channels need to match the value (ex: [], Z, BH, HHZ,...) startdate Start Date to process: [1970-01-01]=’since beginning of the archive’ 1970-01-01 enddate End Date to process: [2100-01-01]=’No end’ 2018-01-01 analysis_duration Duration of the Analysis (total in seconds : 3600, [86400]) 86400 cc_sampling_rate Sampling Rate for the CrossCorrelation [20.0] 20.0 resampling_method Resampling method [Resample]/Decimate Resample decimation_factor If Resampling method=Decimate, decimation factor [5] 5 preprocess_lowpass Preprocessing Low-pass value in Hz [8.0] 8.0 preprocess_highpass Preprocessing High-pass value in Hz [0.01] 0.01 remove_response Remove instrument response Y/[N] N response_format Remove instrument file format [dataless]/inventory/paz/resp dataless response_path Instrument correction file(s) location (path relative to db.ini), defaults to ‘./inventory’, i.e. a subfolder in the current project folder.<br>All files in that folder will be parsed. inventory response_prefilt Remove instrument correction pre-filter (0.005, 0.006, 30.0, 35.0) (0.005, 0.006, 30.0, 35.0) maxlag Maximum lag (in seconds) [120.0] corr_duration Data windows to correlate (in seconds) [1800.] overlap Amount of overlap between data windows [0:1[ [0.] 0.0 windsorizing Windsorizing at N time RMS , 0 disables windsorizing, -1 enables 1-bit normalization [3] 3 stack_method Stack Method: Linear Mean or Phase Weighted Stack: [linear]/pws linear pws_timegate If stack_method=’pws’, width of the smoothing in seconds : 10.0 10.0 pws_power If stack_method=’pws’, Power of the Weighting: 2.0 2.0 crondays Number of days to monitors with cron [-1] -1 ZZ Compute ZZ correlation [Y]/N Y ZR Compute ZR correlation [Y]/N N ZT Compute ZT correlation [Y]/N N RZ Compute RZ correlation [Y]/N N RR Compute RR correlation [Y]/N N RT Compute RT correlation [Y]/N N TZ Compute TZ correlation [Y]/N N TR Compute TR correlation [Y]/N N TT Compute TT correlation [Y]/N N autocorr Compute Auto correlation [Y]/N N keep_all Keep all 30 seconds cross-corr [Y]/N N keep_days Keep all daily cross-corr [Y]/N Y ref_begin Beginning or REF stacks. Can be absolute (2012-01-01) or relative (-100) days 1970-01-01 ref_end End or REF stacks. Same as ref_begin 2018-01-01 mov_stack Number of days to stack for the Moving-window stacks ([5]= [day-4:day]), can be a comma-separated list 1,2,5,10 5 export_format Export stacks in which format(s) ? SAC/MSEED/[BOTH] MSEED sac_format Format for SAC stacks ? [doublets]/clarke doublets dtt_lag How is the lag window defined [dynamic]/static static dtt_v If dtt_lag=dynamic: what velocity to use to avoid ballistic waves [1.0]km/s 1.0 dtt_minlag If dtt_lag=static: min lag time 5.0 dtt_width Width of the time lag window [30]s 30.0 dtt_sides Which sides to use [both]/left/right both dtt_mincoh Minimum coherence on dt measurement, MWCS points with values lower than that will not be used in the WLS 0.65 dtt_maxerr Maximum error on dt measurement, MWCS points with values larger than that will not be used in the WLS 0.1 dtt_maxdt Maximum dt values, MWCS points with values larger than that will not be used in the WLS 0.1 plugins Comma separated list of plugin names. Plugins names should be importable Python modules. ### Database¶ #### Data Availability¶ Gives a view of the data_availability table. Allows to bulk edit/select rows. Its main goal is to check that the scan_archive procedure has successfully managed to list all files from one’s archive. #### Jobs¶ Gives a view of the jobs table. Allows to bulk edit/select rows. Its main goal is to check the new_jobs or any other workflow step (or Plugins) successfully inserted/updated jobs. ### Help¶ Shows some links and information about the package. Mostly the information present on the github readme file. #### Bug Report¶ Web view of the msnoise bugreport -m, allows viewing if all required python modules are properly installed and available for MSNoise. ## Populate Station Table¶ This script is responsible for rapidly scanning the data archive and identifying the Networks/Stations and insert them in the stations table in the database. The data_folder (as defined in the config) is scanned expecting the data_structure and possible values are defined in data_structures.py: data_structure['SDS'] = "YEAR/NET/STA/CHAN.TYPE/NET.STA.LOC.CHAN.TYPE.YEAR.DAY" data_structure['BUD'] = "NET/STA/STA.NET.LOC.CHAN.YEAR.DAY" data_structure['IDDS'] = "YEAR/NET/STA/CHAN.TYPE/DAY/NET.STA.LOC.CHAN.TYPE.YEAR.DAY.HOUR" data_structure['PDF'] = "YEAR/STA/CHAN.TYPE/NET.STA.LOC.CHAN.TYPE.YEAR.DAY" For other structures, one has to edit the data_structures.py file and define the reader in this script. By default, station coordinates are initialized at 0. To run this script: ### Special case: first run¶ This script is the same as for the routine, but one has to pass the init argument: $msnoise -t 2 scan_archive --init This will scan the data_archive folder the configured stations and will insert all files found in the data_availability table in the database. As usual, calling the script with a –help argument will show its usage. ## New Jobs¶ This script searches the database for files flagged “N”ew or “M”odified. For each date in the configured range, it checks if other stations are available and defines the new jobs to be processed. Those are inserted in the jobs table of the database. To run it from the console: $ msnoise new_jobs Warning Upon first run, if you expect the number of jobs to be large (many days, many stations), pass the --init parameter to optimize the insert. Only use this flag once, otherwise problems will arise from duplicate entries in the jobs table. ## Compute Cross-Correlations¶ This code is responsible for the computation of the cross-correlation functions. This script will group jobs marked “T”odo in the database by day and process them using the following scheme. As soon as one day is selected, the corresponding jobs are marked “I”n Progress in the database. This allows running several instances of this script in parallel. ### Configuration Parameters¶ • cc_sampling_rate: Sampling Rate for the CrossCorrelation [20.0] (default=20.0) • analysis_duration: Duration of the Analysis (total in seconds : 3600, [86400]) (default=86400) • overlap: Amount of overlap between data windows [0:1[ [0.] (default=0.0) • maxlag: Maximum lag (in seconds) [120.0] (default=120.) • corr_duration: Data windows to correlate (in seconds) [1800.] (default=1800.) • windsorizing: Windsorizing at N time RMS , 0 disables windsorizing, -1 enables 1-bit normalization [3] (default=3) • resampling_method: Resampling method [Resample]/Decimate (default=Resample) • decimation_factor: If Resampling method=Decimate, decimation factor [5] (default=5) • remove_response: Remove instrument response Y/[N] (default=N) • response_format: Remove instrument file format [dataless]/inventory/paz/resp (default=dataless) • response_path: Instrument correction file(s) location (path relative to db.ini), defaults to ‘./inventory’, i.e. a subfolder in the current project folder.<br>All files in that folder will be parsed. (default=inventory) • response_prefilt: Remove instrument correction pre-filter (0.005, 0.006, 30.0, 35.0) (default=(0.005, 0.006, 30.0, 35.0)) • preprocess_lowpass: Preprocessing Low-pass value in Hz [8.0] (default=8.0) • preprocess_highpass: Preprocessing High-pass value in Hz [0.01] (default=0.01) • keep_all: Keep all 30 seconds cross-corr [Y]/N (default=N) • keep_days: Keep all daily cross-corr [Y]/N (default=Y) • stack_method: Stack Method: Linear Mean or Phase Weighted Stack: [linear]/pws (default=linear) \| new in 1.4 • pws_timegate: If stack_method=’pws’, width of the smoothing in seconds : 10.0 (default=10.0) \| new in 1.4 • pws_power: If stack_method=’pws’, Power of the Weighting: 2.0 (default=2.0) \| new in 1.4 ### Waveform Pre-processing¶ Pairs are first split and a station list is created. The database is then queried to get file paths. For each station, all files potentially containing data for the day are opened. The traces are then merged and splitted, to obtain the most continuous chunks possible. The different chunks are then demeaned, tapered and merged again to a 1-day long trace. If a chunk is not aligned on the sampling grid (that is, start at a integer times the sample spacing in s) , the chunk is phase-shifted in the frequency domain. This requires tapering and fft/ifft. If the gap between two chunks is small, compared to a currently hard-coded value (10 samples), the gap is filled with interpolated values. Larger gaps will not be filled with interpolated values and remaining chunks will be tapered and then merged with 0 values in the gaps. If shorter than 1-day, the trace final is padded with zeros. If longer, it is cut to match the start/end of the day. If configured, each 1-day long trace is corrected for its instrument response. Currently, only dataless seed and inventory XML are supported. Note Removing the instrument response is a computationally very expensive task and not useful for dv/v iff your instruments didn’t change during the analysed period. It is also not needed for tomography iff all instruments are the same, or at least have an identical phase response in the frequency band of interest. Each 1-day long trace is then low-passed (at preprocess_lowpass Hz), high-passed (at preprocess_highpass Hz), then if needed, decimated/downsampled. Decimation/Downsampling are configurable (resampling_method) and users are advised testing both. One advantage of Downsampling over Decimation is that it is able to downsample the data by any factor, not only integer factors. Note Python 3 users will most probably struggle installing scikits.samplerate, and therefore will have to use Decimate instead of Resample. ### Processing¶ Once all traces are preprocessed, station pairs are processed sequentially. If a component different from ZZ is to be computed, the traces are first rotated. This supposes the user has provided the station coordinates in the station table. The rotation is computed for Radial and Transverse components: R = N np.cos(Az * np.pi / 180.) + E * np.sin(Az * np.pi / 180.) T = N * np.sin(Az * np.pi / 180.) - E * np.cos(Az * np.pi / 180.) Then, for each corr_duration window in the signal, and for each filter configured in the database, the traces are clipped to windsorizing times the RMS (or 1-bit converted) and then whitened (see Whitening) between the frequency bounds. When both traces are ready, the cross-correlation function is computed (see Correlation). The function returned contains data for time lags corresponding to maxlag in the acausal (negative lags) and causal (positive lags) parts. ### Stacking and Saving Results¶ If configured (setting keep_all to ‘Y’), each corr_duration CCF is saved to the hard disk. By default, the keep_days setting is set to True and so “N = 1 day / corr_duration” CCF are stacked and saved to the hard disk in the STACKS/001_DAYS folder. Note Currently, the keep-all data (every CCF) are not used by next steps. If stack_method is ‘linear’, then a simple mean CFF of all windows is saved as the daily CCF. On the other hand, if stack_method is ‘pws’, then all the Phase Weighted Stack (PWS) is computed and saved as the daily CCF. The PWS is done in two steps: first the mean coherence between the instataneous phases of all windows is calculated, and eventually serves a weighting factor on the mean. The smoothness of this weighting array is defined using the pws_timegate parameter in the configuration. The weighting array is the power of the mean coherence array. If pws_power is equal to 0, a linear stack is done (then it’s faster to do set stack_method = ‘linear’). Usual value is 2. Warning PWS is largely untested, not cross-validated. It looks good, but that doesn’t mean a lot, does it? Use with Caution! And if you cross-validate it, please let us know!! Schimmel, M. and Paulssen H., “Noise reduction and detection of weak, coherent signals through phase-weighted stacks”. Geophysical Journal International 130, 2 (1997): 497-505. Once done, each job is marked “D”one in the database. To run this script: $msnoise compute_cc This step also supports parallel processing/threading: $ msnoise -t 4 compute_cc will start 4 instances of the code (after 1 second delay to avoid database conflicts). This works both with SQLite and MySQL but be aware problems could occur with SQLite. New in version 1.4: The Instrument Response removal & The Phase Weighted Stack & Parallel Processing ## Stack¶ MSNoise is capable of using a reference function defined by absolute or relative dates span. For example, an absolute range could be “from 1 January 2010 to 31 December 2011” and a relative range could be “the last 200 days”. In the latter case, the REF will need to be exported at every run, meaning the following steps (MWCS and DTT) will be executed on the whole configured period. If the REF is defined between absolute dates, excluding “today”, the MWCS and DTT will only be calculated for new data (e.g. “yesterday” and “today”). The corresponding configuration bits are ref_begin and ref_end. In the future, we plan on allowing multiple references to be defined. Only data for new/modified dates need to be exported. If any CC-job has been marked “Done” within the last day, the stacks will be calculated and a new DTT job will be inserted in the database. For dates in the period of interest, the moving-window stack will only be exported if new/modified CCF is available. The export directory are “REF/” and “DAY%03i/” where %03i will be replaced by the number of days stacked together (DAYS_005 for a 5-days stack, e.g.). Please note that within MSNoise, stacks are always inclusive of the time/day mentionned. For example, a 5-days stack on January 10, will contain cross-correlation functions computed for January 6, 7, 8, 9 AND 10! The graphical representation centered on a “January 10” tick might then display changes in the CCF that occurred on the 10th ! Moving-window stack length(s) are configured using the mov_stack bit. If stack_method is ‘linear’, then a simple mean CFF of all daily is saved as the mov or ref CCF. On the other hand, if stack_method is ‘pws’, then all the Phase Weighted Stack (PWS) is computed and saved as the mov or ref CCF. The PWS is done in two steps: first the mean coherence between the instataneous phases of all windows is calculated, and eventually serves a weighting factor on the mean. The smoothness of this weighting array is defined using the pws_timegate parameter in the configuration. The weighting array is the power of the mean coherence array. If pws_power is equal to 0, a linear stack is done (then it’s faster to do set stack_method = ‘linear’). Usual value is 2. Warning PWS is largely untested, not cross-validated. It looks good, but that doesn’t mean a lot, does it? Use with Caution! And if you cross-validate it, please let us know!! Schimmel, M. and Paulssen H., “Noise reduction and detection of weak, coherent signals through phase-weighted stacks”. Geophysical Journal International 130, 2 (1997): 497-505. ### Configuration Parameters¶ • ref_begin: Beginning or REF stacks. Can be absolute (2012-01-01) or relative (-100) days (default=1970-01-01) • ref_end: End or REF stacks. Same as ref_begin (default=2018-01-01) • mov_stack: Number of days to stack for the Moving-window stacks ([5]= [day-4:day]), can be a comma-separated list 1,2,5,10 (default=5) • stack_method: Stack Method: Linear Mean or Phase Weighted Stack: [linear]/pws (default=linear) \| new in 1.4 • pws_timegate: If stack_method=’pws’, width of the smoothing in seconds : 10.0 (default=10.0) \| new in 1.4 • pws_power: If stack_method=’pws’, Power of the Weighting: 2.0 (default=2.0) \| new in 1.4 ### Usage:¶ The best way to call this code is to start it from the console (-h shows the help) $msnoise stack --help Usage: msnoise-script.py stack [OPTIONS] Stacks the [REF] and/or [MOV] windows Options: -r, --ref Compute the REF Stack -m, --mov Compute the MOV Stacks -s, --step Compute the STEP Stacks -i, --interval INTEGER Number of days before now to search for modified Jobs --help Show this message and exit. On a routine basis, one should thus run the following to compute REF and MOV stacks: $ msnoise stack -r -m While, when playing around with data, and surely on the first run, one should define the -i INTERVAL, as jobs might have been marked “Done” more than 24 hours before running the stack. This, for example, will tell the code to search for jobs marked in the last 10 days: $msnoise stack -r -m -i 10 New in version 1.4: The Phase Weighted Stack. ## Compute MWCS¶ Warning if using only mov_stack = 1, no DTT jobs is inserted in the database and consequently, no MWCS calculation will be done! FIX! Following Clarke et al (2011), we apply the Moving-Window Cross-Spectral method to study the relative dephasing between Moving-Window stacks (“Current”) and a Reference using Moving-Window Cross-Spectral analysis. The jobs “T”o do have been inserted in the datavase during the stack procedure. ### Filter Configuration Parameters¶ • mwcs_low: The lower frequency bound of the linear regression done in MWCS (in Hz) • mwcs_high: The upper frequency bound of the linear regression done in MWCS (in Hz) • mwcs_wlen: Window length (in seconds) to perform MWCS • mwcs_step: Step (in seconds) of the windowing procedure in MWCS In short, both time series are sliced in several overlapping windows and preprocessed. The similarity of the two time-series is assessed using the cross-coherence between energy densities in the frequency domain. The time delay between the two cross correlations is found in the unwrapped phase of the cross spectrum and is linearly proportional to frequency. This “Delay” for each window between two signals is the slope of a weighted linear regression (WLS) of the samples within the frequency band of interest. For each filter, the frequency band can be configured using mwcs_low and mwcs_high, and the window and overlap lengths using mwcs_wlen and mwcs_overlap. The output of this process is a table of delays measured at each window in the functions. The following is an example for lag times between -115 and -90. In this case, the window length was 10 seconds with an overlap of 5 seconds. LAG_TIME DELAY ERROR MEAN COHERENCE -1.1500000000e+02 -1.4781146383e-01 5.3727119135e-02 2.7585243911e-01 -1.1000000000e+02 -6.8207526992e-02 2.0546644311e-02 3.1620999352e-01 -1.0500000000e+02 -1.0337029577e-01 8.6645155402e-03 4.2439269880e-01 -1.0000000000e+02 -2.8668775696e-02 6.2522215988e-03 5.7159849528e-01 -9.5000000000e+01 4.1803941008e-02 1.5102285789e-02 4.1238557789e-01 -9.0000000000e+01 4.8139400233e-02 3.2700657018e-02 3.0586187792e-01 This process is job-based, so it is possible to run several instances in parallel. To run this step: $ msnoise compute_mwcs This step also supports parallel processing/threading: $msnoise -t 4 compute_mwcs will start 4 instances of the code (after 1 second delay to avoid database conflicts). This works both with SQLite and MySQL but be aware problems could occur with SQLite. New in version 1.4: Parallel Processing ## Compute dt/t¶ This code is responsible for the calculation of dt/t using the result of the MWCS calculations. Warning Previously, all pairs were analysed using the same parameters, which were hard-coded in the s06compute_dtt.py file. This has changed now, and MSNoise uses parameters set in the database via the configurator. Pre-1.3 users should upgrade their database using the “$ msnoise upgrade_db” command. ### Configuration Parameters¶ • dtt_lag: How is the lag window defined [dynamic]/static (default=static) • dtt_v: If dtt_lag=dynamic: what velocity to use to avoid ballistic waves [1.0]km/s (default=1.0) • dtt_minlag: If dtt_lag=static: min lag time (default=5.0) • dtt_width: Width of the time lag window [30]s (default=30.0) • dtt_sides: Which sides to use [both]/left/right (default=both) • dtt_mincoh: Minimum coherence on dt measurement, MWCS points with values lower than that will not be used in the WLS (default=0.65) • dtt_maxerr: Maximum error on dt measurement, MWCS points with values larger than that will not be used in the WLS (default=0.1) • dtt_maxdt: Maximum dt values, MWCS points with values larger than that will not be used in the WLS (default=0.1) The dt/t is determined as the slope of the delays vs time lags. The slope is calculated a weighted linear regression (WLS) through selected points. 1. The selection of points is first based on the time lag criteria. The minimum time lag can either be defined absolutely or dynamically. When dtt_lag is set to “dynamic” in the database, the inter-station distance is used to determine the minimum time lag. This lag is calculated from the distance and a velocity configured (dtt_v). The velocity is determined by the user so that the minlag doesn’t include the ballistic waves. For example, if ballistic waves are visible with a velocity of 2 km/s, one could configure dtt_v=1.0. This way, if stations are located 15 km apart, the minimum lag time will be set to 15 s. The dtt_width determines the width of the lag window used. A value of 30.0 means the process will use time lags between 15 and 45 s in the example above, on both sides if configured (dtt_sides), or only causal or acausal parts of the CCF. The following figure shows the static time lags of dtt_width = 40s starting at dtt_minlag = 10s and the dynamic time lags for a dtt_v = 1.0 km/s for the Piton de La Fournaise network (including stations not on the volcano), Note It seems obvious that these parameters are frequency-dependent, but they are currently common for all filters ! Warning In order to use the dynamic time lags, one has to provide the station coordinates ! 2. Using example values above, we chose to use only 15-45 s coda part of the signal, neglecting direct waves in the 0-15 seconds range. We then select data which match three other thresholds: dtt_mincoh, dtt_maxerr and dtt_maxdt. Each of the 4 left subplot of this figure shows a colormapper matrix of which each row corresponds to the data of 1 station pair and each column corresponds to different time lags. The cells are then colored using, from left to right: Delays, Errors, Phase Coherence and Data Selection. Once data (cells) have been selected, they are analyzed two times: first using a WLS that is forced to pass the origin (0,0) and second when a constant is added to allow for the WLS to be offset from the origin. For each value, the error is computed and stored. M0 and EM0 are the slope and its error for the first WLS, and M, EM together with A and EA are the slope, its error, the constant and its error for the second WLS. The output of this calculation is a table, with one row for each station pair. Date, A, EA, EM, EM0, M, M0, Pairs 2013-01-06,-0.1683728,0.0526606,0.00208377,0.00096521, 0.00682021, 0.00037757,BE_GES_BE_HOU 2013-01-06,-0.0080464,0.0577936,0.00291327,0.00097298,-0.00226910,-0.00264354,BE_GES_BE_MEM 2013-01-06, 0.1007472,0.0144648,0.00179566,0.00454172,-0.00145738, 0.00741478,BE_GES_BE_RCHB 2013-01-06,-0.0556811,0.0098926,0.00057839,0.00108102,-0.00328965,-0.00136075,BE_GES_BE_SKQ 2013-01-06, 0.0150866,0.0202243,0.00096543,0.00089832, 0.00083714, 0.00104507,BE_GES_BE_STI 2013-01-06, 0.0268309,0.0328997,0.00153137,0.00150261, 0.00302331, 0.00302451,BE_GES_BE_UCC 2013-01-06,-0.0121293,0.0043351,0.00039019,0.00041347, 0.00025836,-0.00042709,BE_HOU_BE_MEM 2013-01-06, 0.1076247,0.0188662,0.00076824,0.00216383,-0.00030791, 0.00112692,BE_HOU_BE_RCHB 2013-01-06,-0.0468485,0.0194492,0.00069968,0.00078207,-0.00066133, 0.00027102,BE_HOU_BE_SKQ 2013-01-06, 0.0203057,0.0161316,0.00131522,0.00131182, 0.00051626,-3.10306611,BE_HOU_BE_STI ... 2013-01-06,-0.0022588,0.0037141,0.00010340,9.1996e-05, 0.00073635, 0.00076238,ALL To run this script: msnoise compute_dtt ### Grouping Station Pairs¶ Although not clearly visible on the figure above, the very last row of the matrix doesn’t contain information about one station pair, but contains a weighted mean of all delays (from all pairs) for each time lag. For each time lag, delays from each pair is taken into account if it satisfies the same criteria as for the individual data selection. Once the last row (the ALL line) has been calculated, it goes through the normal process of the double WLS and is saved to the output file, as visible above. In the future, MSNoise will be able to treat as many groups as the user want, allowing, e.g. a “crater” and a “slopes” groups. ### Mean of All Pairs vs Mean Pair¶ The dt/t calculated using the mean pair (ALL, in red on subplots 4 and 5) and by calculating the weighted mean of the dt/t of all pairs (in green) don’t show a significant difference. The standard deviation around the latter is more spread than on the former, but this has to be investigated. ### Forcing vs No Forcing through Origin¶ The reason for allowing the WLS to cross the axis elsewhere than on (0,0) is, for example, to study the potential clock drifts or noise source position variations.	2017-02-23 16:07:54	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.49414995312690735, "perplexity": 6179.128978120355}, "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-2017-09/segments/1487501171176.3/warc/CC-MAIN-20170219104611-00206-ip-10-171-10-108.ec2.internal.warc.gz"}
https://inria.github.io/scikit-learn-mooc/python_scripts/cross_validation_sol_01.html	# 📃 Solution for Exercise M2.01# The aim of this exercise is to make the following experiments: • train and test a support vector machine classifier through cross-validation; • study the effect of the parameter gamma of this classifier using a validation curve; • use a learning curve to determine the usefulness of adding new samples in the dataset when building a classifier. To make these experiments we will first load the blood transfusion dataset. Note If you want a deeper overview regarding this dataset, you can refer to the Appendix - Datasets description section at the end of this MOOC. import pandas as pd data = blood_transfusion.drop(columns="Class") target = blood_transfusion["Class"] We will use a support vector machine classifier (SVM). In its most simple form, a SVM classifier is a linear classifier behaving similarly to a logistic regression. Indeed, the optimization used to find the optimal weights of the linear model are different but we don’t need to know these details for the exercise. Also, this classifier can become more flexible/expressive by using a so-called kernel that makes the model become non-linear. Again, no requirement regarding the mathematics is required to accomplish this exercise. We will use an RBF kernel where a parameter gamma allows to tune the flexibility of the model. First let’s create a predictive pipeline made of: # solution from sklearn.pipeline import make_pipeline from sklearn.preprocessing import StandardScaler from sklearn.svm import SVC model = make_pipeline(StandardScaler(), SVC()) Evaluate the generalization performance of your model by cross-validation with a ShuffleSplit scheme. Thus, you can use sklearn.model_selection.cross_validate and pass a sklearn.model_selection.ShuffleSplit to the cv parameter. Only fix the random_state=0 in the ShuffleSplit and let the other parameters to the default. # solution from sklearn.model_selection import cross_validate, ShuffleSplit cv = ShuffleSplit(random_state=0) cv_results = cross_validate(model, data, target, cv=cv, n_jobs=2) cv_results = pd.DataFrame(cv_results) cv_results fit_time score_time test_score 0 0.016975 0.002681 0.680000 1 0.016236 0.002476 0.746667 2 0.015965 0.002543 0.786667 3 0.014896 0.002624 0.800000 4 0.015856 0.002640 0.746667 5 0.015355 0.002444 0.786667 6 0.015177 0.002524 0.800000 7 0.014448 0.002818 0.826667 8 0.015694 0.002509 0.746667 9 0.015295 0.002550 0.733333 print( f"Accuracy score of our model:\n" f"{cv_results['test_score'].mean():.3f} ± " f"{cv_results['test_score'].std():.3f}" ) Accuracy score of our model: 0.765 ± 0.043 As previously mentioned, the parameter gamma is one of the parameters controlling under/over-fitting in support vector machine with an RBF kernel. Evaluate the effect of the parameter gamma by using the sklearn.model_selection.validation_curve function. You can leave the default scoring=None which is equivalent to scoring="accuracy" for classification problems. You can vary gamma between 10e-3 and 10e2 by generating samples on a logarithmic scale with the help of np.logspace(-3, 2, num=30). Since we are manipulating a Pipeline the parameter name will be set to svc__gamma instead of only gamma. You can retrieve the parameter name using model.get_params().keys(). We will go more into detail regarding accessing and setting hyperparameter in the next section. # solution import numpy as np from sklearn.model_selection import validation_curve gammas = np.logspace(-3, 2, num=30) param_name = "svc__gamma" train_scores, test_scores = validation_curve( model, data, target, param_name=param_name, param_range=gammas, cv=cv, n_jobs=2) Plot the validation curve for the train and test scores. # solution import matplotlib.pyplot as plt plt.errorbar( gammas, train_scores.mean(axis=1), yerr=train_scores.std(axis=1), alpha=0.95, label="Training score", ) plt.errorbar( gammas, test_scores.mean(axis=1), yerr=test_scores.std(axis=1), alpha=0.5, label="Testing score", ) plt.legend() plt.xscale("log") plt.xlabel(r"Value of hyperparameter $\gamma$") plt.ylabel("Accuracy score") _ = plt.title("Validation score of support vector machine") Looking at the curve, we can clearly identify the over-fitting regime of the SVC classifier when gamma > 1. The best setting is around gamma = 1 while for gamma < 1, it is not very clear if the classifier is under-fitting but the testing score is worse than for gamma = 1. Now, you can perform an analysis to check whether adding new samples to the dataset could help our model to better generalize. Compute the learning curve (using sklearn.model_selection.learning_curve) by computing the train and test scores for different training dataset size. Plot the train and test scores with respect to the number of samples. # solution from sklearn.model_selection import learning_curve train_sizes = np.linspace(0.1, 1, num=10) results = learning_curve( model, data, target, train_sizes=train_sizes, cv=cv, n_jobs=2) train_size, train_scores, test_scores = results[:3] plt.errorbar( train_size, train_scores.mean(axis=1), yerr=train_scores.std(axis=1), alpha=0.95, label="Training score", ) plt.errorbar( train_size, test_scores.mean(axis=1), yerr=test_scores.std(axis=1), alpha=0.5, label="Testing score", ) plt.legend(bbox_to_anchor=(1.05, 0.8), loc="upper left") plt.xlabel("Number of samples in the training set") plt.ylabel("Accuracy") _ = plt.title("Learning curve for support vector machine") We observe that adding new samples to the training dataset does not seem to improve the training and testing scores. In particular, the testing score oscillates around 76% accuracy. Indeed, ~76% of the samples belong to the class "not donated". Notice then that a classifier that always predicts the "not donated" class would achieve an accuracy of 76% without using any information from the data itself. This can mean that our small pipeline is not able to use the input features to improve upon that simplistic baseline, and increasing the training set size does not help either. It could be the case that the input features are fundamentally not very informative and the classification problem is fundamentally impossible to solve to a high accuracy. But it could also be the case that our choice of using the default hyperparameter value of the SVC class was a bad idea, or that the choice of the SVC class is itself sub-optimal. Later in this MOOC we will see how to better tune the hyperparameters of a model and explore how to compare the predictive performance of different model classes in a more systematic way.	2022-12-07 20:29:25	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.42122703790664673, "perplexity": 1482.635897033189}, "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/1669446711218.21/warc/CC-MAIN-20221207185519-20221207215519-00181.warc.gz"}
http://mathematica.stackexchange.com/questions/20774/defining-a-function-by-pieces-using-list-of-interpolated-functions-in-order-to-u	# Defining a function by pieces using list of interpolated functions in order to use FindRoot I have two lists with the interpolated functions list1 = {InterpolatingFunction[{{a1, b1}},<>], InterpolatingFunction[{{a2, b2}},<>], InterpolatingFunction[{{a3, b3}},<>], InterpolatingFunction[{{a4, b4}},<>]} which I will think as $\mbox{list1} = \{f_1(x),f_2(x),f_3(x),f_4(x)\}$, and list2 = {InterpolatingFunction[{{c1, d1}},<>], InterpolatingFunction[{{c2, d2}},<>], InterpolatingFunction[{{c3, d3}},<>], InterpolatingFunction[{{c4, d4}},<>]} which I will think as $\mbox{list2} = \{g_1(x),g_2(x),g_3(x),g_4(x)\}$. What I need is to define $$f(x) = \begin{cases} f_1(x) & a_1 < x < b_1 \\ f_2(x) & a_2 < x < b_2 \\ f_3(x) & a_3 < x < b_3 \\ f_4(x) & a_4 < x < b_4 \end{cases} \quad \mbox{and} \quad g(x) = \begin{cases} g_1(x) & c_1 < x < d_1 \\ g_2(x) & c_2 < x < d_2 \\ g_3(x) & c_3 < x < d_3 \\ g_4(x) & c_4 < x < d_4 \end{cases}$$ in order to find a root of the equation $$f(x) - g(x) = 0$$ using the code FindRoot[{f[x] - g[x] == 0},{x,x0}] where the intervals $(a_i,b_i)$, $(c_i,d_i)$ are are well ordered (i.e. $a_1 < b_1 < a_2 < b_2 < ... < b_4$), and $x_0 \in [\min(a_1,c_1),\max(b_4,d_4)]$. Here is an example: list1 = Interpolation /@ Table[{2 i + k, Tan[k]}, {i, 1, 3, 2}, {k, -(\[Pi]/2) + 0.1, \[Pi]/2 - 0.1, 0.1}]; list2 = Interpolation /@ {Table[{i, i (i + 1/2) (i - 4)}, {i, -2, 5, 0.1}], Table[{i, 5 (i - 8)}, {i, 6, 9, 0.1}]}; x0 = RandomReal[{-2, \[Pi]/2 - 0.1 + 8}]; In this case, $\mbox{list1} = \{f_1(x),f_2(x),f_3(x)\}$ and $\mbox{list2} = \{g_1(x),g_2(x)\}$. I don't mind if no solution can be found for some $x_0$; what I don't want is to obtain spurious solutions due to extrapolation, or other gimmicks MMA might be using to "glue" the interpolated functions. In this particular example, the root $x \sim 9.43$ would be spurious, since neither $f$ nor $g$ are defined there. $\hskip1.5in$ EDIT Using @b.gatessucks suggestion, if I define f[x_] := Piecewise[{{list1[[1]][x], -(\[Pi]/2) + 0.1 < x < \[Pi]/2 - 0.1}, {list1[[2]][x], -(\[Pi]/2) + 0.1 + 4 < x < \[Pi]/2 - 0.1 + 4}, {list1[[3]][x], -(\[Pi]/2) + 0.1 + 8 < x < \[Pi]/2 - 0.1 + 8}}]; g[x_] := Piecewise[{{list2[[1]][x], -2 < x < 5}, {list2[[2]][x], 6 < x < 9}}]; then sols = Union[x /. FindRoot[f[x] == g[x], {x, #}] & /@ Table[i, {i, -2, \[Pi]/2 - 0.1 + 8, .1}], SameTest -> (Abs[#1 - #2] <= 10^-6 &)] returns some spurious solution: $\hskip1cm$ Is there a way to automatically drop such solutions (for example generating the grid only where both f and g are defined)? - Can't you use Piecewise ? – b.gatessucks Mar 6 '13 at 20:38 @b.gatessucks It's a little more convoluted than that – belisarius Mar 7 '13 at 4:38 This glues together a list of InterpolatingFunction into a Piecewise that is Indeterminate outside the domains, the reason for this is that FindRoot aborts when it encounters that as a function value. pieceTogether[list_] := Function[\[FormalT], Evaluate@Piecewise[ Map[{#@\[FormalT], #["Domain"][[1, 1]] <= \[FormalT] <= #["Domain"][[1, 2]]} &,list], Indeterminate]]; It uses an undocumented(?) feature to extract the domain of the InterpolatingFunction, you could do it in other ways but I find this the most convenient: (First@list1)["Domain"] (* {{0.529204, 3.4292}} ) It is used like: ( Evaluate f[t] to see what it did ) f = pieceTogether[list1]; g = pieceTogether[list2]; p = Plot[{f[t], g[t]}, {t, -2.5, 10}]; Manipulate[ ( If FindRoot gives some error suppress it and return None ) root = Quiet@Check[x /. FindRoot[f[x] == g[x], {x, x0}], None]; ( Red point at the root or black on axis ) {color, pt} = If[root =!= None, {Red, {root, f[root]}}, {Black, {x0, 0}}]; Show[p, Epilog -> {PointSize[Large], color, Point[pt]}] ,{{x0, 1}, -2.5, 12}] - This is fantastic! Thank you very much! – Pragabhava Mar 6 '13 at 21:24 The following tries to find one root in each subinterval where both functions are defined: list1 = Interpolation /@ Table[{2 i + k, Tan[k]}, {i, 1, 3, 2}, {k, -(\[Pi]/2) + 0.1, \[Pi]/2 - 0.1, 0.1}]; list2 = Interpolation /@ {Table[{i, i (i + 1/2) (i - 4)}, {i, -2, 5, 0.1}], Table[{i, 5 (i - 8)}, {i, 6, 9, 0.1}]}; Needs["DifferentialEquationsInterpolatingFunctionAnatomy"]; ClearAll[u]; f1 = Piecewise[{#[u], Less @@ Riffle[InterpolatingFunctionDomain[#][[1]], u]}&/@ list1, 100]; g1 = Piecewise[{#[u], Less @@ Riffle[InterpolatingFunctionDomain[#][[1]], u]}&/@ list2, 100]; i = (IntervalUnion @@ (Interval@ InterpolatingFunctionDomain[#][[1]] & /@ #)) &; i1 = List @@ IntervalIntersection[i@list1, i@list2]; If[Chop@(f1 - g1 /. u -> x /. #) == 0., #, {}] & /@ FindRoot[Evaluate[(f1 - g1 == 0) /. u -> x], #] & /@ ({x, Mean@#, Sequence @@ #} & /@ i1) ( {{x -> 0.747098}, {{}}, {{}}} *) - This is great too! Now I need to modify it to find all roots in the subintervals. Thanks a lot! – Pragabhava Mar 7 '13 at 1:48 Is there an easy way to adapt it to problems of the form FindRoot[{h1[x,y] + f1[x]==0, h2[x,y] + g1[y] == 0},{{x,x0},{y,y0}}] Where h1 and h2 are continuous functions and f1, g1 are the Piecewise defined functions? I'm having lots of troubles trying. – Pragabhava Mar 7 '13 at 4:21 @Pragabhava Not really sure ... perhaps another good question there – belisarius Mar 7 '13 at 4:37	2015-08-01 18:18:00	{"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": 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": 0.2486877739429474, "perplexity": 6826.226267555952}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042988860.4/warc/CC-MAIN-20150728002308-00017-ip-10-236-191-2.ec2.internal.warc.gz"}
https://brilliant.org/problems/quadratika-2/	Find the positive value of $$p$$ such that the quadratic equation $$px^2 - 12x + 4 = 0$$ has only one solution.	2017-10-23 08:13:23	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.817932665348053, "perplexity": 18.712607317213426}, "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/1508187825812.89/warc/CC-MAIN-20171023073607-20171023093607-00296.warc.gz"}
https://www.gradesaver.com/textbooks/math/algebra/intermediate-algebra-12th-edition/chapter-1-section-1-7-absolute-value-equations-and-inequalities-1-7-exercises-page-120/89	## Intermediate Algebra (12th Edition) Published by Pearson # Chapter 1 - Section 1.7 - Absolute Value Equations and Inequalities - 1.7 Exercises - Page 120: 89 #### Answer $x=\left\{ -\dfrac{5}{3},\dfrac{1}{3} \right\}$ #### Work Step by Step Using the properties of equality, the given equation, $\left\|\dfrac{1}{2}x+\dfrac{1}{3}\right\|+\dfrac{1}{4}=\dfrac{3}{4} ,$ is equivalent to \begin{array}{l}\require{cancel} \left\|\dfrac{1}{2}x+\dfrac{1}{3}\right\|=\dfrac{3}{4}-\dfrac{1}{4} \\\\ \left\|\dfrac{1}{2}x+\dfrac{1}{3}\right\|=\dfrac{2}{4} \\\\ \left\|\dfrac{1}{2}x+\dfrac{1}{3}\right\|=\dfrac{1}{2} .\end{array} Since for any $a\gt0$, $\|x\|=a$ implies $x=a$ OR $x=-a$, then the equation above is equivalent to \begin{array}{l}\require{cancel} \dfrac{1}{2}x+\dfrac{1}{3}=\dfrac{1}{2} \text{ OR } \dfrac{1}{2}x+\dfrac{1}{3}=-\dfrac{1}{2} .\end{array} Solving each equation results to \begin{array}{l}\require{cancel} \dfrac{1}{2}x+\dfrac{1}{3}=\dfrac{1}{2} \\\\ 6\left( \dfrac{1}{2}x+\dfrac{1}{3} \right)=\left(\dfrac{1}{2}\right)6 \\\\ 3x+2=3 \\\\ 3x=3-2 \\\\ 3x=1 \\\\ x=\dfrac{1}{3} \\\\\text{ OR }\\\\ \dfrac{1}{2}x+\dfrac{1}{3}=-\dfrac{1}{2} \\\\ 6\left( \dfrac{1}{2}x+\dfrac{1}{3} \right)=\left(-\dfrac{1}{2}\right)6 \\\\ 3x+2=-3 \\\\ 3x=-3-2 \\\\ 3x=-5 \\\\ x=-\dfrac{5}{3} .\end{array} Hence, the solutions are $x=\left\{ -\dfrac{5}{3},\dfrac{1}{3} \right\} .$ After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.	2018-09-24 22:13:38	{"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": 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": 0.998999297618866, "perplexity": 10498.673043020422}, "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-2018-39/segments/1537267160754.91/warc/CC-MAIN-20180924205029-20180924225429-00270.warc.gz"}
https://www.codecademy.com/courses/learn-linear-regression-in-r/lessons/linear-regression-in-r/exercises/linear-regression-r-building-model	Learn Simple linear regression is not a misnomer–– it is an uncomplicated technique for predicting a continuous outcome variable, Y, on the basis of just one predictor variable, X. As detailed in previous exercises, a number of assumptions are made so that we can model the relationship between X and Y as a linear function. Using our advertising dataset, we could model the relationship between the amount spent on podcast advertising in a month and the number of respective products eventually sold as follows: $Y = beta_0 + beta_1*X + error$ Where… Y: represents the dollar value of products sold X: represents the amount spent on respective product podcast ads Beta_0: is the intercept, or the number of products sold when no money has been spent on podcasts Beta_1: is the coefficient, or the slope, of the line representing the relationship Error: represents the random variation in the relationship between the two variables To build this model in R, using the standard lm() package, we use the formula notation of Y ~ X: model <- lm(sales ~ podcast, data = train) But wait! Before building this model, we need to split our data into test and training sets For the development of this simple model, we’ll use a standard 60/40 split of our data; where 60% is used to train the model, and 40% is used to test the model’s accuracy and generalizability. We can randomly assign data points to test or training using base R’s sample() method and list indexing functionality # specify 60/40 split sample <- sample(c(TRUE, FALSE), nrow(advertising), replace = T, prob = c(0.6,0.4)) # subset data points into train and test sets test <- advertising[!sample, ] ### Instructions 1. First, let’s split our conversion_clean dataset into 60/40 train/test subsets • Create a variable named data_sample by assigning the result of calling sample(), with c(TRUE, FALSE), nrow(conversion_clean), and prob = c(0.6,0.4) as parameters. • Using list indexing, assign all data points in sample to a variable called train • Using list indexing, assign all data points in not in sample to a variable called test 2. Let’s fit a linear model of the relationship between the number of products sold and the number of clicks on the respective product advertisement; this means that conversion’s total_convert value, the total number of product purchases by a single user, will be our Y variable, or outcome variable. clicks the total number of times a user clicks on a version of an ad, will be our X variable, or predictor variable. Assign the result of calling lm(), using ~ formula notation to set a linear relationship between total_convert and clicks, to a variable called model. Don’t forget to set the data parameter equal to train!	2022-06-26 23:56:05	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 1, "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": 0.40723422169685364, "perplexity": 1648.3104056920936}, "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/1656103322581.16/warc/CC-MAIN-20220626222503-20220627012503-00780.warc.gz"}
https://math.stackexchange.com/questions/1081022/sheaf-on-abelian-variety-preserved-by-tensor-product-with-a-translation-invarian	# Sheaf on abelian variety preserved by tensor product with a translation invariant line bundle If $A$ is an abelian variety, $\mathscr{F}$ is a coherent sheaf on $A$, and $\mathscr{F}\otimes \mathscr{L}\cong \mathscr{F}$ for all translation invariant line bundles $\mathscr{L}$, why is the support of $\mathscr{F}$ dimension 0? This is asserted in a book I'm trying to read (I think it's supposed to be obvious), and I don't see why it is true. Before this assertion, I think the book has already proven that the rank of $\mathscr{F}$ is zero (i.e. it is supported on a proper closed subset) by taking a resolution of $\mathscr{F}$ by line bundles, tensoring with $\mathscr{L}$, and taking the determinant bundle. This argument is taken from Mukai's famous "Duality between $D(X)$ and $D(\hat{X})$ with its application to Picard sheaves" (Lemma 3.3.). Let $X$ be the abelian variety we're working on. Assume the dimension of the support is $\geq1$ and let $C$ be a curve in $\mbox{supp}(\mathcal{F})$ with $f:\tilde{C}\to C$ its normalization. Then $\mathcal{G}:=f^\mathcal{F}/(\mbox{torsion subsheaf})$ is a line bundle on $\tilde{C}$ (every torsion-free coherent sheaf on a smooth curve is locally free) and $f^P\otimes \mathcal{G}\simeq\mathcal{G}$ for all $P\in\mbox{Pic}^0(X)$ ($\mbox{Pic}^0(X)$ is the set of translation-invariant line bundles on $X$). In particular, if we take the determinant on both sides, we obtain that $(f^P)^{\mbox{rank}(\mathcal{G})}\simeq\mathcal{O}_{\tilde{C}}$. However, we have a non-trivial map $F:J_{\tilde{C}}\to X$ where $J_{\tilde{C}}$ is the Jacobian of $\tilde{C}$, given by the universal property of the Jacobian (and so $F$ restricted to the image of $\tilde{C}$ in $J$ is just $f$). Moreover, the map $F^=f^*:\mbox{Pic}^0(X)\to\mbox{Pic}^0(J_{\tilde{C}})\simeq\mbox{Pic}^0(\tilde{C})$ is not trivial, and so we obtain a contradiction. Therefore $\mbox{supp}(\mathcal{F})$ is 0-dimensional.	2019-10-17 12:46:09	{"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": 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": 0.9895612597465515, "perplexity": 78.64165122235133}, "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-2019-43/segments/1570986675316.51/warc/CC-MAIN-20191017122657-20191017150157-00168.warc.gz"}
https://torch.mlverse.org/docs/reference/optim_sgd.html	Implements stochastic gradient descent (optionally with momentum). Nesterov momentum is based on the formula from On the importance of initialization and momentum in deep learning. ## Usage optim_sgd( params, lr = optim_required(), momentum = 0, dampening = 0, weight_decay = 0, nesterov = FALSE ) ## Arguments params (iterable): iterable of parameters to optimize or dicts defining parameter groups lr (float): learning rate momentum (float, optional): momentum factor (default: 0) dampening (float, optional): dampening for momentum (default: 0) weight_decay (float, optional): weight decay (L2 penalty) (default: 0) nesterov (bool, optional): enables Nesterov momentum (default: FALSE) ## Note The implementation of SGD with Momentum-Nesterov subtly differs from Sutskever et. al. and implementations in some other frameworks. Considering the specific case of Momentum, the update can be written as $$\begin{array}{ll} v_{t+1} & = \mu * v_{t} + g_{t+1}, \\ p_{t+1} & = p_{t} - \mbox{lr} * v_{t+1}, \end{array}$$ where $$p$$, $$g$$, $$v$$ and $$\mu$$ denote the parameters, gradient, velocity, and momentum respectively. This is in contrast to Sutskever et. al. and other frameworks which employ an update of the form $$\begin{array}{ll} v_{t+1} & = \mu * v_{t} + \mbox{lr} * g_{t+1}, \\ p_{t+1} & = p_{t} - v_{t+1}. \end{array}$$ The Nesterov version is analogously modified. ## Warning If you need to move a model to GPU via $cuda(), please do so before constructing optimizers for it. Parameters of a model after $cuda() will be different objects from those before the call. In general, you should make sure that the objects pointed to by model parameters subject to optimization remain the same over the whole lifecycle of optimizer creation and usage. ## Examples if (torch_is_installed()) { if (FALSE) { optimizer <- optim_sgd(model$parameters(), lr = 0.1, momentum = 0.9) optimizer$zero_grad() loss_fn(model(input), target)$backward() optimizer$step() } }	2023-02-06 11:54:35	{"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": 2, "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": 0.5094754695892334, "perplexity": 5718.566314317752}, "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/1674764500339.37/warc/CC-MAIN-20230206113934-20230206143934-00036.warc.gz"}
http://math.stackexchange.com/questions/170330/is-there-a-special-name-for-matrices-consist-of-repeated-unit-vectors	# Is there a special name for matrices consist of repeated unit vectors? For example this one: $$Q=\begin{pmatrix} 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 \end{pmatrix}$$ - ## 1 Answer You might call'em a Kronecker product: $\pmatrix{1 &0&0\\0&1&0\\0&0&1}\otimes\pmatrix{1 &1&1}$ of matrices with entries from the Boolean domain B = {0, 1}. -	2015-05-28 04:14:13	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.48531410098075867, "perplexity": 946.0264500876305}, "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-2015-22/segments/1432207929230.43/warc/CC-MAIN-20150521113209-00001-ip-10-180-206-219.ec2.internal.warc.gz"}
https://stats.stackexchange.com/questions/384590/prerequisites-for-wasserstein-gan-autoencoder	# Prerequisites for Wasserstein GAN/Autoencoder Can someone who read WGAN/WAE papers and understood Wasserstein part, could you share how you prepared necessary Optimal Transport background? The mentioned papers seem little tough if you don't have an intuition on Wasserstein metric/optimal transport theory. The papers mostly cite some books on optimal transport - I don't think I'd be able to go through a whole book, but I wasn't able to find accessible tutorials on the topic. Alternatively, could anyone narrow the specific topics that are needed to understand these papers? I've seen that they use Kantorovich duality - can one read Villani's book mostly concentrating on that? Would it be sufficient to understand the papers? For WGAN paper these are needed: • Optimal transport problem statement: • Given two probability measures $$\mu, \nu$$ on compact $$X, Y$$, $$c: X \times Y \to \mathbb{R}^+$$ continuous find $$p \in P_{\mu, \nu}$$ that minimizes $$Ec = \int c(x,y) dp(x,y)$$ where $$P_{\mu, \nu} = \{$$ probability measure on $$X \times Y$$ with marginals $$\mu, \nu \}$$ • Optimal transport problem has a solution * • Kantorovich duality (general) $$\inf_{c \in P_{\mu, \nu}} Ec = \sup_{f, g \in L^1} \int f d\mu + \int g d\nu$$ • Kantorovich duality when $$c$$ is a metric on $$X$$ (in this case$$X=Y$$): $$\inf_{c \in P_{\mu, \nu}} Ec = \sup_{f, g} \int f d\mu - \int f d\nu$$ where the $$sup$$ range is over 1-Lipschitz functions on $$X$$. ## WGAN paper The authors use second form of Lipschitz duality. The $$f$$ is output of the discriminator, and it is guaranteed to be $$d$$-Lipschitz because of weight clipping *. - The set of $$p$$ that satisfy constraint on marginals is tight, and from Prokhorov's theorem it follows that it's precompact, and in fact it's also closed, thus it is compact, so $$E$$ attains infimum on $$P_{\mu, \nu}$$. ** - weight clipping makes weight space $$\Theta$$ compact, and the space of possible discriminator $$f$$s is parametrized continuously by this space, so function $$\Theta \to \mathbb{R}^+$$ $$\theta \to \inf \{d \in \mathbb{R}^+\| f_{\theta}\ is\ d-Lipschitz\}$$ Attains its minimum. ## Books I've checked out two Villani's books: • Optimal Transport: Old and New (the one I mentioned) • Topics in Optimal Transportation Personally I found that second one has better pace - it gets to Kantorovich duality almost right away (in 1st chapter), and it is really good for getting intuition, as it first sketches the needed proofs, and fills in details later.	2020-10-27 18:07:54	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "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": 0.6903349757194519, "perplexity": 1071.0828032911595}, "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/1603107894426.63/warc/CC-MAIN-20201027170516-20201027200516-00005.warc.gz"}
https://socratic.org/questions/how-do-you-solve-s-b-28-and-16s-19b-478	# How do you solve s+b=28 and 16s+19b=478? Apr 2, 2017 There are several ways to solve this type of problem. I think the substitution method to be a good choice for this particular problem. #### Explanation: Substitution method: Rewrite the first equation as: $s = 28 - b$ Substitute $28 - b$ for s into the second equation: $16 \left(28 - b\right) + 19 b = 478$ Distribute the 16: $448 - 16 b + 19 b = 478$ Subtract 448 from both sides: $- 16 b + 19 b = 30$ Combine like terms: $3 b = 30$ Divide both sides by 3 $b = 10$ Substitute 10 for b into the first equation: $s + 10 = 28$ $s = 18$ Check $s = 18 \mathmr{and} b = 10$ in both equations: $s + b = 28$ $16 s + 19 b = 478$ $18 + 10 = 28$ $16 \left(18\right) + 19 \left(10\right) = 478$ $28 = 28$ $288 + 190 = 478$ $28 = 28$ $478 = 478$ This checks. Apr 2, 2017 See the entire solution process below: #### Explanation: Step 1) Solve the first equation for $s$: $s + b = 28$ $s + b - \textcolor{red}{b} = 28 - \textcolor{red}{b}$ $s + 0 = 28 - b$ $s = 28 - b$ Step 2) Substitute $28 - b$ for $s$ in the second equation and solve for $b$: $16 s + 19 b = 478$ becomes: $16 \left(28 - b\right) + 19 b = 478$ $\left(16 \times 28\right) - \left(16 \times b\right) + 19 b = 478$ $448 - 16 b + 19 b = 478$ $448 + \left(- 16 + 19\right) b = 478$ $448 + 3 b = 478$ $- \textcolor{red}{448} + 448 + 3 b = - \textcolor{red}{448} + 478$ $0 + 3 b = 30$ $3 b = 30$ $\frac{3 b}{\textcolor{red}{3}} = \frac{30}{\textcolor{red}{3}}$ $\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{3}}} b}{\cancel{\textcolor{red}{3}}} = 10$ $b = 10$ Step 3) Substitute $10$ for $b$ in the solution to the first equation at the end of Step 1 and calculate $s$: $s = 28 - b$ becomes: $s = 28 - 10$ $s = 18$ The solution is: $b = 10$ and $s = 18$	2019-09-23 17:27:18	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 46, "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": 0.7816346883773804, "perplexity": 771.1003081132402}, "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-2019-39/segments/1568514577478.95/warc/CC-MAIN-20190923172009-20190923194009-00361.warc.gz"}
https://scicomp.stackexchange.com/questions/11515/finding-null-vectors-of-a-parameter-dependent-matrix/30670	# Finding null vectors of a parameter-dependent matrix I have dense complex matrices $M(z)$ in which each element $M_{ij} = M_{ij}(z)$ depends on a complex parameter $z$. I need to find $z$ such that the matrix $M$ gets singular, i.e. I am looking for null vectors $\vec v$ which satisfy $M(z) \vec k = \vec 0$. So far, I use a Newton method. As the wanted $M(z)$ is singular, I need $M$'s determinant to vanish. Starting with an initial guess $z_0$, I iterate $$z_{i+1} = z_i - \frac{g(z_i)}{g'(z_i)}$$ with $g(z) := \det M(z)$. To avoid computing the determinant from definition, I use LU factorization. The iteration is stopped when $\|z_{i+1} - z_{i}\|$ is sufficiently small. Using the matrix identity $\frac{\mathrm{d}}{\mathrm{d}z}\ln \det M(z) = \mathrm{trace}\,(M^{-1}(z) M'(z))$, the reciprocal logarithmic derivative in the iteration formula can replaced to yield $$z_{i+1} = z_i - \frac{1}{\mathrm{trace}\,(M^{-1}(z) M'(z))}\,,$$ which is what I use in the end. A few details on steps involved: so far, I compute all derivatives $g'(z)$ from a forward finite difference scheme $g'(z) = \frac{g(z + h) - g(z)}{h} - \mathrm{i}\frac{g(z + \mathrm{i}h) - g(z)}{h}$, ($h$ real), likewise for derivatives of $M$. Also note I'm assuming that singular vectors have multiplicity $m = 1$ (although adaptation of the formulas above is possible using $g=(\det M(z))^m$. The matrix $M$ has no special structure in global (i.e. not Hermitian). Depending on the problem, it may be rectangular or square. In any case, it is dense and well-conditioned. Typical sizes range from ~ $100\times100$ to $10k \times 10k$. $M$'s origin is in the boundary element method. For open systems (that may have resonances), the null vectors I am looking are the resonance wavefunctions on the boundaries of the domains under consideration. I'd like to learn about alternative methods of adjusting the parameter $z$ such that $M$ gets singular. Do you have any other ideas? Or any comments on the method I described? Although the results are fine, I don't like the approach too much as both iterations described here involve a step that is essentially $O(N^3)$ (LU or the inverse). Furthermore, for the second scheme, I need to compute the inverse of near singular matrices. • Is M Hermitian? Could you something about where $M$ arises from? What are the typical dimensions of $M$? Further, do you run into problems as the iterations progress and the matrix becomes singular? – user2457602 May 5 '14 at 14:16 • @user2457602, I have added more information to the question as you requested. I typically do not run into problems in the iteration. – AlexE May 5 '14 at 15:03 Edit: Note that your formula for the finite difference approximation of $g'$ does not need the imaginary correction (just the first term is enough) if your function is analytic. If you can compute the derivatives analytically or automatically (with automatic differentiation), then that would be much better. However, I would guess this is not possible if your matrix comes from a BEM discretization.	2021-03-02 20:52:49	{"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": 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": 0.9227489829063416, "perplexity": 326.77555712043534}, "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-2021-10/segments/1614178364764.57/warc/CC-MAIN-20210302190916-20210302220916-00440.warc.gz"}
https://www.studyadda.com/question-bank/fractions-and-decimals_q36/4570/361644	• # question_answer Shyamlal had $\frac{5}{6}$ of a cake. He ate $\frac{2}{3}$of it. What part of the total cake did not he not eat? A) $\frac{4}{9}$ B) $\frac{10}{12}$C) $\frac{10}{6}$ D) $\frac{10}{3}$ (a) Part of the cake eaten $=\frac{5}{6}\times \frac{2}{3}=\frac{10}{18}=\frac{5}{9}$ $\therefore$ Not eaten$=1-\frac{5}{9}=\frac{4}{9}$	2019-10-18 00:32:35	{"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": 0.43512988090515137, "perplexity": 11399.462759999356}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986677230.18/warc/CC-MAIN-20191017222820-20191018010320-00477.warc.gz"}
https://eng.libretexts.org/Bookshelves/Computer_Science/Book%3A_A_First_Course_in_Electrical_and_Computer_Engineering_(Scharf)/4%3A_The_Exponential_Functions/4.1%3A_Introduction	# 4.1: Introduction ## Notes to Teachers and Students It is essential to write out, term-by-term, every sequence and sum in this chapter. This demystifies the seemingly mysterious notation. The example on compound interest shows the value of limiting arguments in everyday life and gives ex some real meaning. The function e, covered in the section "The Function of ejθ and the Unit Circle" and "Numerical Experiment (Approximating e)", must be understood by all students before proceeding to "Phasors" . The Euler and De Moivre identities provide every tool that students need to derive trigonometric formulas. The properties of roots of unity are invaluable for the study of phasors in "Phasors" . The MATLAB programs in this chapter are used to illustrate sequences and series and to explore approximations to $$\sinθ$$ and $$\cosθ$$. The numerical experiment in "Numerical Experiment (Approximating e)" illustrates, geometrically and algebraically, how approximations to e converge. Second-Order Differential and Difference Equations” is a little demanding for freshmen, but we give it a once-over-lightly to illustrate the power of quadratic equations and the functions ex and e. This section also gives a sneak preview of more advanced courses in circuits and systems. ## Introduction It is probably not too strong a statement to say that the function ex is the most important function in engineering and applied science. In this chapter we study the function ex and extend its definition to the function e. This study clarifies our definition of e from "Complex Numbers" and leads us to an investigation of sequences and series. We use the function e to derive the Euler and De Moivre identities and to produce a number of important trigonometric identities. We define the complex roots of unity and study their partial sums. The results of this chapter will be used in "Phasors" when we study the phasor representation of sinusoidal signals.	2020-01-18 11:28:59	{"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": 0.733318030834198, "perplexity": 630.7445047181934}, "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-2020-05/segments/1579250592565.2/warc/CC-MAIN-20200118110141-20200118134141-00415.warc.gz"}
https://www.gradesaver.com/textbooks/science/physics/college-physics-4th-edition/chapter-1-problems-page-23/87	College Physics (4th Edition) $2.6N$ The weight is prorportional to the mass and inversely proportional to the square of the radius, so W is proportional to $\frac{m}{r^{2}}$ Forming a proportion, we have $\frac{Wj}{We}=\frac{mj}{me}\times (\frac{re}{rj})^{2}=320\times\frac{1}{11^{2}}=\frac{320}{121}$ On Jupiter, the apple would weigh = $\frac{320}{121}\times1N=2.6N$	2022-05-17 09:04:25	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 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": 0.5009930729866028, "perplexity": 394.440499489941}, "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-2022-21/segments/1652662517018.29/warc/CC-MAIN-20220517063528-20220517093528-00638.warc.gz"}
https://mathematica.stackexchange.com/questions/165794/rowreduction-wolfram-alpha-vs-mathematica	# RowReduction: Wolfram Alpha vs Mathematica I am trying to put in row reduced echelon form the following matrix $$\left( \begin{array}{ccc} -0.1 & 0.1 & 2 \\ 0.3 & 0.2 & 0.7 \\ 0 & 0.5 & 6.7 \\ \end{array} \right)$$ I did the calculations by hand, and being a fairly simple matrix to reduce, I wanted to test it out in Mathematica, using RowReduce[{{-0.1, 0.1, 2}, {0.3, 0.2, 0.7}, {0, 0.5, 6.7}}] which agrees with my result. I also tried in WolframAlpha, keeping the same input, however, I get an "incorrect result", which is $$\left( \begin{array}{ccc} 1 & 0. & 0. \\ 0 & 1 & 0. \\ 0 & 0 & 1 \\ \end{array} \right)$$ I noticed that there is a dot next to some of the $0$s, so I am assuming there are some rounding errors being made. So I tried this and it worked. My question is, why is mathematica able to work with greater precision than Wolfram Alpha, even though the function being called is the same? I am aware that this question does not involve solely Mathematica, however since the function is the same for both I wanted to understand the how each handled this. • I work on alpha, just reported this as a bug. – Jason B. Feb 13 '18 at 20:16 • The (2,3) element of the original input matrix is 0; however, where you say "So I tried this" the (2,3) element of the given matrix is equal to 7/10 - not 0. So one shouldn't expect the same answer upon row reduction. – user15994 Feb 13 '18 at 20:54 • I typed the original matrix wrong. I missed that. Thank you for pointing it out. I’ll edit – user372003 Feb 13 '18 at 21:06 • @JasonB. I'm not convinced it's a bug. Please see my answer. Thx. – Michael E2 Feb 14 '18 at 4:22 As @ilian points out, this is caused by the underlying Mathematica version (along with the OS?): $Version "11.1.1 for Linux x86 (64-bit) (June 2, 2017)" RowReduce[{{-0.1, 0.1, 2}, {0.3, 0.2, 0.7}, {0, 0.5, 6.7}}]] {{1, 0., 0.}, {0, 1, 0.}, {0, 0, 1}} @MichaelE2 argues that this is not a bug. While I agree this is not a Mathematica bug, I do consider it a Wolfram\|Alpha bug. W\|A should be treating 6.7 as 67/10 in this input, while Mathematica should not. I suppose the logic is that the average W\|A user shouldn't need to understand the nuances of machine precision, or what machine precision is for that matter. Mathematically 6.7 is 67/10, and W\|A should understand this here. • I agree with you – user372003 Feb 14 '18 at 21:51 I think the reason is the underlying version of Mathematica (the result may also depend on the platform to some extent). Using 64-bit Linux for example (since that's what Alpha servers run), this example works better in version 10 and later: $Version (* 11.2.0 for Linux x86 (64-bit) (September 11, 2017) ) RowReduce[{{-0.1, 0.1, 2}, {0.3, 0.2, 0.7}, {0, 0.5, 6.7}}] ( {{1, 0., -6.6}, {0, 1, 13.4}, {0, 0, 0}} ) while version 9 produces the suboptimal result \$Version ( 9.0 for Linux x86 (64-bit) (February 7, 2013) ) RowReduce[{{-0.1, 0.1, 2}, {0.3, 0.2, 0.7}, {0, 0.5, 6.7}}] RowReduce::luc: Result for RowReduce of badly conditioned matrix {{-0.1, 0.1, 2.}, {0.3, 0.2, 0.7}, {0., 0.5, 6.7}} may contain significant numerical errors. ( {{1, 0., 0.}, {0, 1, 0.}, {0, 0, 1}} ) Note also the appropriate warning message, although it is not visible when using Wolfram\|Alpha. • Yes, you are right, it depends also on the platform – user372003 Feb 14 '18 at 21:51 I don't think this is a bug. First, keep in mind that the floating-point number 0.1 is not the same as 1/10: SetPrecision[0.1, Infinity] % - 1/10 ( 3602879701896397/36028797018963968 1/180143985094819840 ) The "fairly simple" row reduction referred to in the OP is, I suppose, achieved by treating the decimals as exact numbers. But that is not what one gets by treating them as 64-bit binary floating-point numbers subject to rounding errors. They are treated an inexact floating-point numbers if the matrix is entered with entries like 0.1 instead of 1/10. To get an result equivalent to the "human interpretation" of the problem, one can try the option Tolerance with RowReduce (q.v.). mat = {{-0.1, 0.1, 2}, {0.3, 0.2, 0.7}, {0, 0.5, 6.7}}; RowReduce[mat, Tolerance -> 10^-15] ( {{1, 0., -6.6}, {0, 1, 13.4}, {0, 0, 0}} ) Without tolerance (V11.2 MacOS): RowReduce[mat] RowReduce::luc: Result for RowReduce of badly conditioned matrix {{-0.1,0.1,2.},{0.3,0.2,0.7},{0.,0.5,6.7}} may contain significant numerical errors. ( {{1, 0., 0.}, {0, 1, 0.}, {0, 0, 1}} *) This is different from what @ilian reports for Linux. The docs say that the setting for Tolerance in RowReduce is Automatic. Perhaps the default tolerance has changed from version to version. • I am using Version 11.2 Windows x64 and I do not need to specify tolerance. I get the correct answer by just using RowReduce[mat]. I understand what you are saying though. However, as Chip states, I think this is a W\|A bug. – user372003 Feb 14 '18 at 21:50 • @user372003 Yes, W\|A has a different audience in mind than Mathematica. But this site is a site about Mathematica, not W\|A. -- I think it's odd, though, that we're getting different results on Macs than on Windows/Linux. – Michael E2 Feb 15 '18 at 1:30	2020-03-31 23:55:57	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 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": 0.3474482297897339, "perplexity": 1419.0636606995877}, "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/1585370504930.16/warc/CC-MAIN-20200331212647-20200401002647-00456.warc.gz"}
https://www.maths.ed.ac.uk/~nbochkin/LMSTuring_inverseproblems_workshop.htm	Turing/LMS Workshop Inverse Problems and Data Science 8-10 May 2017 Venue: Informatics Forum, 10 Crichton St, Edinburgh EH8 9AB, UK Organisers: Natalia Bochkina (University of Edinburgh), Carola Schoenlieb (University of Cambridge), Marta Betcke (UCL), Sean Holman (University of Manchester) The aim of the workshop is to bring together researchers on inverse problems working in different areas of mathematics, statistics and machine learning as well as from the applied disciplines where inverse problems arise, such as astronomy, biology, computer vision, geoscience and medicine. The topics of the workshop include nonlinear inverse problems, algorithms, inverse problems in machine learning, theoretical properties of statistical estimators in inverse problems, Bayesian inverse problems, applications in science and medicine. Financial support: The Alan Turing Institute and London Mathematical Society Sponsoring: if you would like to sponsor the event, please get in touch with one of the organisers. We are grateful to Schlumberger for sponsorship. Times: the workshop will start at 10am on Monday 8th of May and concludes at 4:30pm on Wednesday 10th of May Contact: events@turing.ac.uk Registration: deadline is 1 May, registration fee is £60. Please register here. There will also be a training course on Bayesian inverse problems on 11 May 2017 – registration is available here (deadline 4 May). Programme Monday 9:30 -10:00 registration and coffee 10:00 – 10:45 Sean Holman (University of Manchester, UK) “On the stability of the geodesic ray transform in the presence of caustics“ 10:45 – 11:30 Gabriel Paternain (University of Cambridge, UK) ”Effective inversion of the attenuated X-ray transform associated with a connection” 11:30 – 11:45 discussion: interdisciplinary challenges 11:45 - 12:45 lunch 12:45 – 13:30 Andrew Curtis (University of Edinburgh, UK) “Nonlinear Travel-Time and Electrical Resistivity Tomography” 13:30 – 14:15 Nicholas Zabaras (University of Notre Dame, USA) “Inverse Problems with an Unknown Scale of Estimation” 14:15 – 15:00 Eduard Kontar (University of Glasgow, UK) TBC 15:00 - 15:15 discussion: interdisciplinary challenges 15:15-15:30 coffee break 15:30 – 16:15 Mike Christie (Heriot Watt University, UK) ”Bayesian Hierarchical Models for Measurement Error” 16:15-17:00 Botond Szabo (Leiden University, Netherlands) ”Confidence in Bayesian uncertainty quantification in inverse problems” 17:00-17:45 Anna Simoni (CREST, CNRS, Paris, France) ”Nonparametric Estimation in case of Endogenous Selection” 17:45 – 18:00 discussion: interdisciplinary challenges 18:00 - 20:00 poster session and reception Tuesday 9:15 - 10:00 Michael Gutmann (University of Edinburgh, UK)” Bayesian Inference by Density Ratio Estimation” 10:00 - 10:30 Pol Moreno (University of Edinburgh) “Overcoming Occlusion with Inverse Graphics” 10:30 - 10:50 coffee break 10:50 – 11:35 Kyong Jin (EPFL, Switzerland) “Deep Convolutional Neural Network for Inverse Problems in Imaging” 11:35-12:05 Jonas Adler (KTH, Royal institute of Technology and Elekta) “Solving ill-posed inverse problems using learned iterative schemes” 12:05-12:20 discussion: interdisciplinary challenges 12:20-13:00 lunch 13:00 – 13:45 Christian Clason (Duisburg-Essen University, Germany)” A primal-dual extragradient method for nonlinear inverse problems for PDEs” 13:45 – 14:30 Carola Schoenlieb (University of Cambridge, UK) “Model-based learning in imaging” 14:30 - 14:45 discussion: interdisciplinary challenges 14:45 – 15:15 coffee break 15:15 - 16:00 Markus Reiss (Humboldt University, Berlin, Germany)” Optimal adaptation for early stopping in statistical inverse problems” 16:00 – 16:45 Axel Munk (Goettingen University, Germany) “Nanostatistics – Statistics for Nanoscopy” 16:45 - 17:15 Merle Behr (Goettingen University, Germany) “Multiscale Blind Source Separation” 17:15 – 18:00 discussion: interdisciplinary challenges Tuesday: conference dinner Wednesday 9:30 – 10:15 Aretha Teckentrup (University of Edinburgh, UK)” Gaussian process regression in Bayesian inverse problems” 10:15 – 11:00 Marcelo Pereyra (Heriot Watt University, UK) ”Bayesian inference by convex optimisation: theory, methods, and algorithms” 11:00-11:30 coffee break 11:30 – 12:15 Michal Branicki (University of Edinburgh, UK) “Information-based measures of skill and optimality in Bayesian filtering” 12:15 – 12:45 Nagoor Kani (Heriot Watt University, UK) “Model reduction using physics driven deep residual recurrent neural networks” 12:45 – 13:00 discussion: interdisciplinary challenges 13:00 – 14:00 lunch 14:00 – 14:30 Christopher Wallis (UCL) “Sparse image reconstruction on the sphere: Analysis and Synthesis” 14:30 – 15:00 Xiaohao Cai (UCL) “High-dimensional uncertainty estimation with sparse priors for radio interferometric imaging” 15:00 – 15:45 Marta Betcke (UCL) “Dynamic high-resolution photoacoustic tomography with optical flow constraint” 15:45 – 16:00 discussion: interdisciplinary challenges 16:00 closing remarks Poster presentations Shing Chan (Heriot-Watt University) A machine learning approach for efficient uncertainty quantification using multiscale methods. Mohammad Golbabaee (University of Edinburgh) Inexact iterative projected gradient for fast compressed quantitative MRI Hardial S Kalsi (King's College, London) Glottal Inverse Filtering using the Acoustical Klein-Gordon Equation Dimitris Kamilis and Nick Polydorides (University of Edinburgh) A computational framework for uncertainty quantification for low-frequency, time-harmonic Maxwell equations with stochastic conductivity models J.Nagoor Kani and Ahmed H. Elsheikh (Heriot Watt University) Model reduction using physics driven deep residual recurrent neural networks Jon Cockayne, Chris Oates, Tim Sullivan, Mark Girolami Bayesian Probabilistic Numerical Methods Alessandro Perelli (University of Edinburgh) Multi Denoising Approximate Message Passing for computational complexity reduction Jenovah Rodrigues (University of Edinburgh) Bayesian Inverse Problems with Heterogeneous Noise Ferdia Sherry (University of Cambridge) Learning a sampling pattern for MRI Invited speaker abstracts. Christian Clason (Mathematics, Duisburg-Essen University, Germany) ”A primal-dual extragradient method for nonlinear inverse problems for PDEs This talk is concerned with the extension of the Chambolle--Pock primal-dual algorithm to nonsmooth optimization problems involving nonlinear operators between function spaces. The proof of local convergence rests on verifying the Aubin property of the inverse of a monotone operator at the minimizer, which is difficult as it involves infinite-dimensional set-valued analysis. However, for nonsmooth functionals that are defined pointwise -- such as $L^1$ or $L^\infty$ norms -- it is possible to apply simpler tools from the finite-dimensional theory, which allows deriving explicit conditions for the convergence. This is illustrated for the example of imaging problems with $L^1$- and $L^\infty$-fitting terms. Mike Christie (Petroleum Institute, Heriot Watt University, UK) ”Bayesian Hierarchical Models for Measurement Error The detailed geological description of oil reservoirs is always uncertain because of the large size and relatively small number of wells from which hard data can be obtained. To handle this uncertainty, reservoir models are calibrated or ‘history matched’ to production data (oil rates, pressures etc). The quality of any reservoir forecasts depends not only on the quality of the match, but also how well understood the measurement errors are (or indeed the split between measurement and modelling errors). This talk will look at hierarchical models for estimating measurement and modelling errors in reservoir model calibration, and compare maximum likelihood estimates of measurement errors with marginalisation over unknown errors. Andrew Curtis (University of Edinburgh, UK) “Nonlinear Travel-Time and Electrical Resistivity Tomography” We solve the fully nonlinear travel-time tomography problem using reversible-jump Markov chain Monte Carlo methods. The results motivate a conjecture that the uncertainty in general, non-linearised tomography problems may consist of loop-like topologies that can be interpreted similarly to linearised measures of spatial resolution. This is confirmed in a second example by applying the same inversion algorithm to estimate the electrical resistivity structure of a medium (the Earth) from dipole-dipole electrical resistivity measurements, a problem governed by quite different physics: uncertainty loops appear similarly. If time allows, I will then discuss a simple decomposition of tomography problems that can be shown to be unimodal in some important cases, leading to a different method of solution. Michael Gutmann (Informatics, University of Edinburgh, UK)” Bayesian Inference by Density Ratio Estimation” This talk is about Bayesian inference when the likelihood function cannot be computed but data can be generated from the model. The model's data generating process is allowed to be arbitrarily complex. Exact solutions are then not possible. But by re-formulating the original problem as a problem of estimating the ratio between two probability density functions, I show how e.g. logistic regression can be used to obtain approximate solutions. The proposed inference framework is illustrated on stochastic nonlinear dynamical models. Reference: https://arxiv.org/abs/1611.10242 Kyong Jin (EPFL, Switzerland) “Deep Convolutional Neural Network for Inverse Problems in Imaging This talk discusses a novel deep convolutional neural network (CNN)-based algorithm for solving ill-posed inverse problems. Regularized iterative algorithms have emerged as the standard approach to ill-posed inverse problems in the past few decades. These methods produce excellent results, but can be challenging to deploy in practice due to factors including the high computational cost of the forward and adjoint operators and the difficulty of hyper parameter selection. The starting point of our work is the observation that unrolled iterative methods have the form of a CNN (filtering followed by point-wise non-linearity) when the normal operator (H*H, the adjoint of H times H) of the forward model is a convolution. Based on this observation, we propose using direct inversion followed by a CNN to solve normal-convolutional inverse problems. The direct inversion encapsulates the physical model of the system, but leads to artifacts when the problem is ill-posed; the CNN combines multiresolution decomposition and residual learning in order to learn to remove these artifacts while preserving image structure. The performance of the proposed network will be demonstrated in sparse-view reconstruction on parallel beam X-ray computed tomography and accelerated MR imaging reconstruction on parallel MRI. Axel Munk (Department of Mathematics and Computer Science, and Max-Planck Institute for Biophysical Chemistry, Goettingen University, Germany) “Nanostatistics – Statistics for Nanoscopy” Conventional light microscopes have been used for centuries for the study of small length scales down to approximately 250 nm. Images from such a microscope are typically blurred and noisy, and the measurement error in such images can often be well approximated by Gaussian or Poisson noise. In the past, this approximation has been the focus of a multitude of deconvolution techniques in imaging. However, conventional microscopes have an intrinsic physical limit of resolution. Although this limit remained unchallenged for a century, it was broken for the first time in the 1990s with the advent of modern superresolution fluorescence microscopy techniques. Since then, superresolution fluorescence microscopy has become an indispensable tool for studying the structure and dynamics of living organisms, recently acknowledged with the c Nobel prize in chemistry 2014. Current experimental advances go to the physical limits of imaging, where discrete quantum effects are predominant. Consequently, the data is inherently of a non-Gaussian statistical nature, and we argue that recent technological progress also challenges the long-standing Poisson assumption. Thus, analysis and exploitation of the discrete physical mechanisms of fluorescent molecules and light, as well as their distributions in time and space, have become necessary to achieve the highest resolution possible and to extract biologically relevant information. In this talk we survey some modern fluorescence microscopy techniques from a statistical modeling and analysis perspective. In the first part we address spatially adaptive multiscale deconvolution estimation and testing methods for scanning type microscopy. We illustrate that such methods benefit from recent advances in large-scale computing, mainly from convex optimization. In the second part of the talk we address challenges of quantitative biology which require more detailed models that delve into sub-Poisson statistics. To this end we suggest a prototypical model for fluorophore dynamics and use it to quantify the number of proteins in a spot. Marcelo Pereyra (Mathematics, Heriot Watt University, UK) ”Bayesian inference by convex optimisation: theory, methods, and algorithms. Convex optimisation has become the main Bayesian computation methodology in many areas of data science such as mathematical imaging and machine learning, where high dimensionality is often addressed by using models that are log-concave and where maximum-a-posteriori (MAP) estimation can be performed efficiently by optimisation. The first part of this talk presents a new decision-theoretic derivation of MAP estimation and shows that, contrary to common belief, under log-concavity MAP estimators are proper Bayesian estimators. A main novelty is that the derivation is based on differential geometry. Following on from this, we establish universal theoretical guarantees for the estimation error involved and show estimation stability in high dimensions. Moreover, the second part of the talk describes a new general methodology for approximating Bayesian high-posterior-density regions in log-concave models. The approximations are derived by using recent concentration of measure results related to information theory, and can be computed very efficiently, even in large-scale problems, by using convex optimisation techniques. The approximations also have favourable theoretical properties, namely they outer-bound the true high-posterior-density credibility regions, and they are stable with respect to model dimension. The proposed methodology is finally illustrated on two high-dimensional imaging inverse problems related to tomographic reconstruction and sparse deconvolution, where they are used to explore the uncertainty about the solutions, and where convex-optimisation-empowered proximal Markov chain Monte Carlo algorithms are used as benchmark to compute exact credible regions and measure the approximation error. Related pre-prints: https://arxiv.org/abs/1612.06149 https://arxiv.org/pdf/1602.08590.pdf Markus Reiss (Humboldt University, Berlin, Germany)” Optimal adaptation for early stopping in statistical inverse problems For linear inverse problems $Y=\mathsf{A}\mu+\xi$, it is classical to recover the unknown function $\mu$ by an iterative scheme $(\widehat \mu^{(m)}, m=0,1,\ldots)$ and to provide $\widehat\mu^{(\tau)}$ as a result, where $\tau$ is some stopping rule. Stopping should be decided adaptively, that is in a data-driven way independently of the true function $\mu$. For deterministic noise $\xi$ the discrepancy principle is usually applied to determine $\tau$. In the context of stochastic noise $\xi$, we study oracle adaptation (that is, compared to the best possible stopping iteration). For a stopping rule based on the residual process, oracle adaptation bounds within a certain domain are established. For Sobolev balls, the domain of adaptivity matches a corresponding lower bound. The proofs use bias and variance transfer techniques from weak prediction error to strong $L^2$-error, as well as convexity arguments and concentration bounds for the stochastic part. The performance of our stopping rule for Landweber and spectral cutoff methods is illustrated numerically.(Joint work with Gilles Blanchard, Potsdam, and Marc Hoffmann, Paris) Anna Simoni (CREST, CNRS, Paris, France) ”Nonparametric Estimation in case of Endogenous Selection” This paper addresses the problem of estimation of a nonparametric regression function from selectively observed data when selection is endogenous. Our approach relies on independence between covariates and selection conditionally on potential outcomes. Endogeneity of regressors is also allowed for. In the exogenous and endogenous case, consistent two-step estimation procedures are proposed and their rates of convergence are derived which take into account the degree of ill-posedness. In the first stage we have to solve an ill-posed inverse problem to recover nonparametrically the inverse selection probability function. Moreover, when the covariates are endogenous an additional inverse problem has to be solved in the second step to recover the instrumental regression function. Pointwise asymptotic distribution of the estimators is established. In addition, bootstrap uniform confidence bands are derived. Finite sample properties are illustrated in a Monte Carlo simulation study and an empirical illustration. Joint work with Christoph Breunig (Humboldt University, Berlin) and Enno Mammen (Heidelberg University). Botond Szabo (Leiden University, Netherlands) ”Confidence in Bayesian uncertainty quantification in inverse problems In our work we investigate the frequentist coverage of Bayesian credible sets in the inverse Gaussian sequence model. We consider a scale of priors of varying regularity and choose the regularity by an empirical or a hierarchical Bayes method. Next we consider a central set of prescribed posterior probability in the posterior distribution of the chosen regularity. We show that such an adaptive Bayes credible set gives correct uncertainty quantification of “polished tail” parameters, in the sense of high probability of coverage of such parameters. On the negative side, we show by theory and example that adaptation of the prior necessarily leads to gross and haphazard uncertainty quantification for some true parameters that are still within the hyperrectangle regularity scale. The preceding results are based on semi-explicit computations on an optimised statistical model. In the end of the talk I will briefly discuss to possible extensions of our coverage results to more general, abstract settings. The talk is based on the papers written together with Judith Rousseau, Aad van der Vaart and Harry van Zanten. Aretha Teckentrup (Mathematics, University of Edinburgh, UK)” Gaussian process regression in Bayesian inverse problems” A major challenge in the application of sampling methods in Bayesian inverse problems is the typically large computational cost associated with solving the forward problem. To overcome this issue, we consider using a Gaussian process surrogate model to approximate the forward map. This results in an approximation to the solution of the Bayesian inverse problem, and more precisely in an approximate posterior distribution. In this talk, we analyse the error in the approximate posterior distribution, and show that the approximate posterior distribution tends to the true posterior as the accuracy of the Gaussian process surrogate model increases. Nicholas Zabaras (Computational Science and Engineering, University of Notre Dame, USA) “Inverse Problems with an Unknown Scale of Estimation The presentation will focus on the Bayesian estimation of spatially varying parameters of multiresolution/multiscale nature. In particular, the characteristic length scale(s) of the unknown property are not known a priori and need to be evaluated based on the fidelity of the given data across the domain. Our approach is based on representing the spatial field with a wavelet expansion. The intra-scale correlations between wavelet coefficients form a quadtree, and this structure is exploited to identify additional basis functions to refine the model. Bayesian inference is performed using a sequential Monte Carlo sampler with a MCMC transition kernel. The SMC sampler is used to move between posterior densities defined on different scales, thereby providing for adaptive refinement of the wavelet representation. The marginal likelihoods provide a termination criterion for the scale determination algorithm thus allowing model comparison and selection. The approach is demonstrated with permeability estimation for groundwater flow using pressure measurements. https://www.zabaras.com/ Contributed talk abstracts. Jonas Adler, KTH- Royal Institute of Technology and Elekta Solving ill-posed inverse problems using learned iterative schemes We present partially learned iterative schemes for the solution of ill posed inverse problems with not necessarily linear forward operators. The methods builds on ideas from classical regularization theory and recent advances in deep learning to perform learning while making use of prior information about the inverse problem encoded in the forward operator, noise model and a regularizing functional. We also present results for tomographic reconstruction of human head phantoms and discuss several possible future research areas. Related pre-print: https://arxiv.org/abs/1704.04058 Merle Behr, Goettingen University, Germany Multiscale Blind Source Separation We discuss a new methodology for statistical recovery of single linear mixtures of piecewise constant signals (sources) with unknown mixing weights and change points in a multiscale fashion. Exact recovery within an \epsilon-neighborhood of the mixture is obtained when the sources take only values in a known finite alphabet. Based on this we provide estimators for the mixing weights and sources for Gaussian error. We obtain uniform confidence sets and optimal rates (up to log-factors) for all quantities. This blind source separation problem is motivated by different applications in digital communication, but also in cancer genetics. In the latter one aims to assign copy-number variations from genetic sequencing data to different tumor-clones and their corresponding proportions in the tumor. We analyze such data using the proposed method in order to estimate their proportion in the tumor and the corresponding copy number variations. This is joint work with Chris Holmes (University of Oxford, UK) and Axel Munk (University of Goettingen, Germany). Xiaohao Cai, MSSL, University College London High-dimensional uncertainty estimation with sparse priors for radio interferometric imaging In many fields high-dimensional inverse imaging problems are encountered. For example, imaging the raw data acquired by radio interferometric telescopes involves solving an ill-posed inverse problem to recover an image of the sky from noisy and incomplete Fourier measurements. Future telescopes, such as the Square Kilometre Array (SKA), will usher in a new big-data era for radio interferometry, with data rates comparable to world-wide internet traffic today. Sparse regularisation techniques are a powerful approach for solving these problems, typically yielding excellent reconstruction fidelity (e.g. Pratley et al. 2016). Moreover, by leveraging recent developments in convex optimisation, these techniques can be scaled to extremely large data-sets (e.g. Onose et al. 2016). However, such approaches typically recover point estimators only and uncertainty information is not quantified. Standard Markov Chain Monte Carlo (MCMC) techniques that scale to high-dimensional settings cannot support the sparse (non-differentiable) priors that have been shown to be highly effective in practice. We present work adapting the proximal Metropolis adjusted Langevin algorithm (P-MALA), developed recently by Pereyra (2016a), for radio interferometric imaging with sparse priors (Cai, Pereyra & McEwen 2017a), leveraging proximity operators from convex optimisation in an MCMC framework to recover the full posterior distribution of the sky image. While such an approach provides critical uncertainty information, scaling to extremely large data-sets, such as those anticipated from the SKA, is challenging. To address this issue we develop a technique to compute approximate local Bayesian credible intervals by post-processing the point (maximum a-posteriori) estimator recovered by solving the associated sparse regularisation problem (Cai, Pereyra & McEwen 2017b), leveraging recent results by Pereyra (2016b). This approach inherits the computational scalability of sparse regularisation techniques, while also providing critical uncertainty information. We demonstrate these techniques on simulated observations made by radio interferometric telescopes. Joint work with Marcelo Pereyra (from Heriot-Watt University) and Jason D. McEwen (from MSSL, University College London). J.Nagoor Kani (and Ahmed H. Elsheikh), Heriot Watt University Model reduction using physics driven deep residual recurrent neural networks We introduce a deep residual recurrent neural network (DR-RNN) to emulate the dynamics of physical phenomena. The developed DR-RNN is inspired by the iterative steps of line search methods in finding the residual minimiser of numerically discretised differential equations. We formulate this iterative scheme as stacked recurrent neural network (RNN) embedded with the dynamical structure of the emulated differential equations. We provide empirical evidence showing that these residual driven deep RNN can effectively emulate the physical system with significantly lower number of parameters in comparison to standard RNN architectures. We also show the significant gains in accuracy by increasing the depth of RNN similar to other recent applications of deep learning. The applicability of the developed DR-RNN is demonstrated on uncertainty quantification tasks where a large number of forward simulation are required. Pol Moreno, University of Edinburgh, UK Overcoming Occlusion with Inverse Graphics Scene understanding tasks such as the prediction of object pose, shape, appearance and illumination are hampered by the occlusions often found in images. We propose a vision-as-inverse-graphics approach to handle these occlusions by making use of a graphics renderer in combination with a robust generative model (GM). Since searching over scene factors to obtain the best match for an image is very inefficient, we make use of a recognition model (RM) trained on synthetic data to initialize the search. This paper addresses two issues: (i) We study how the inferences are affected by the degree of occlusion of the foreground object, and show that a robust GM which includes an outlier model to account for occlusions works significantly better than a non-robust model. (ii) We characterize the performance of the RM and the gains that can be made by refining the search using the GM, using a new dataset that includes background clutter and occlusions. We find that pose and shape are predicted very well by the RM, but appearance and especially illumination less so. However, accuracy on these latter two factors can be clearly improved with the generative model. Christopher Wallis, University College London Sparse image reconstruction on the sphere: Analysis and Synthesis We develop techniques to solve a number of ill-posed inverse problems on the sphere by sparse regularisation, exploiting sparsity in directional wavelet space. Through numerical experiments we evaluate the effectiveness of the technique in solving inpainting, denoising and deconvolution problems. We consider solving the problems in both the analysis and synthesis settings, with a number of different sampling schemes, and show that the sampling scheme has a large impact on the quality of the reconstruction. This is due to more efficient sampling schemes constraining the solution space and improving sparsity in wavelet space. We adapt and apply the technique to the Planck 353GHz total intensity map, improving the ability to extract the structure of galactic dust emission. Poster abstracts. Shing Chan, Heriot-Watt University, UK A machine learning approach for efficient uncertainty quantification using multiscale methods. Several multiscale methods account for sub-grid scale features using coarse scale basis functions. For example, in the Multiscale Finite Volume method the coarse scale basis functions are obtained by solving a set of local problems over dual-grid cells. We introduce a data-driven approach for the estimation of these coarse scale basis functions. Specifically, we employ a neural network predictor fitted using a set of solution samples from which it learns to generate subsequent basis functions at a lower computational cost than solving the local problems. The computational advantage of this approach is realized for uncertainty quantification tasks where a large number of realizations has to be evaluated. We attribute the ability to learn these basis functions to the modularity of the local problems and the redundancy of the permeability patches between samples. The proposed method is evaluated on elliptic problems yielding very promising results. Inexact iterative projected gradient for fast compressed quantitative MRI We will present a compressed sensing perspective of a novel form of MR imaging called Magnetic Resonance Fingerprinting (MRF). This enables direct estimation of the T1, T2 and proton density parameter maps for a patient through undersampled k-space sampling and BLIP, a gradient projection algorithm that enforces the MR Bloch dynamics. One of the key bottlenecks in MRF is the projection onto the constraint set. We will present both theoretical and numerical results showing that significant computational savings are possible through the use of inexact projections and a fast approximate nearest neighbor search. Hardial S Kalsi, King's College, London Glottal Inverse Filtering using the Acoustical Klein-Gordon Equation Inversion of the glottal-pulse waveform from a speech signal remains an active field of research although dating back over half a century. Despite multiple approaches to solve this important inverse problem, it cannot be said today that the field is in a satisfactory state. In the main, approaches use classical “inverse filtering” frequency-domain methods to estimate both the vocal-tract and glottal-pulse waveform. In this poster, we illustrate a new approach which takes advantage of two recent developments: firstly, the description of the speech wave by means of an analogue of the Klein-Gordon wave equation of relativistic quantum mechanics and, secondly, the solution of this equation to find its Green's function. This approach allows accurate parameterisation of the vocal tract which greatly simplifies the inversion. Dimitris Kamilis and Nick Polydorides, University of Edinburgh A computational framework for uncertainty quantification for low-frequency, time-harmonic Maxwell equations with stochastic conductivity models We present a computational framework for uncertainty quantification (UQ) for the quasi-magnetostatic Maxwell equations using lognormal random field conductivity models. Our methodology combines elements of sparse quadrature (SQ) for the efficient calculation of the high-dimensional UQ integrals, as well as model reduction methods for expediting the model evaluations. Our analysis and numerical results show that subject to some mild assumptions on the smoothness of the random conductivity fields, sparse quadrature outperforms the convergence of the conventional Monte-Carlo method, while model reduction further reduces the computational cost. Numerical results to illustrate the method are presented from three-dimensional simulations that are representative of models appearing in the geophysical prospecting Controlled-Source Electromagnetic Method (CSEM). J.Nagoor Kani and Ahmed H. Elsheikh, Heriot Watt University Model reduction using physics driven deep residual recurrent neural networks We introduce a deep residual recurrent neural network (DR-RNN) to emulate the dynamics of physical phenomena. The developed DR-RNN is inspired by the iterative steps of line search methods in finding the residual minimiser of numerically discretised differential equations. We formulate this iterative scheme as stacked recurrent neural network (RNN) embedded with the dynamical structure of the emulated differential equations. We provide empirical evidence showing that these residual driven deep RNN can effectively emulate the physical system with significantly lower number of parameters in comparison to standard RNN architectures. We also show the significant gains in accuracy by increasing the depth of RNN similar to other recent applications of deep learning. The applicability of the developed DR-RNN is demonstrated on uncertainty quantification tasks where a large number of forward simulation are required. Jon Cockayne, Chris Oates, Tim Sullivan, Mark Girolami "Bayesian Probabilistic Numerical Methods" The emergent field of probabilistic numerics has thus far lacked rigorous statistical principals. This work establishes Bayesian probabilistic numerical methods as those which can be cast as solutions to certain Bayesian inverse problems, albeit problems that are non-standard. This allows us to establish general conditions under which Bayesian probabilistic numerical methods are well-defined, encompassing both non-linear and non-Gaussian models. For general computation, a numerical approximation scheme is developed and its asymptotic convergence is established. The theoretical development is then extended to pipelines of computation, wherein probabilistic numerical methods are composed to solve more challenging numerical tasks. The contribution highlights an important research frontier at the interface of numerical analysis and uncertainty quantification, with some illustrative applications presented Alessandro Perelli (University of Edinburgh) Multi Denoising Approximate Message Passing for computational complexity reduction Denoising-AMP (D-AMP) [1] can be viewed as an iterative algorithm where at each iteration a non-linear denoising function is applied to the signal estimate. D-AMP algorithm has been analysed in terms of inferential accuracy without considering computational complexity. This is an important missing aspect since the denoising is often the computational bottleneck in the D-AMP reconstruction. The approach that it is proposed in this work is different; we aim to design a mechanism for leveraging a hierarchy denoising models (MultiD-AMP) in order to minimize the overall complexity given the expected risk, i.e. the estimation error. The intuition comes from the observation that at earlier iteration, when the estimate is far according to some distance to the true signal, the algorithm does not need a complicated denoiser, since the structure of the signal is poor, but faster denoisers and this leads to the idea of defining a family/hierarchy of denoisers of increased complexity. The main challenge is to define a switching scheme which is based on the empirical finding that in MultiD-AMP we can predict exactly, in the large system limit, the evolution of the Mean Square Error. We can exploit the State Evolution, evaluated on a set of training images, to find a proper switching strategy. The proposed framework has been tested on i.i.d. random Gaussian measurements with Gaussian noise and for deconvolution problem. The results show the effectiveness of the proposed reconstruction algorithm. [1] Metzler, C. A., Maleki, A., Baraniuk, R. G. From denoising to compressed sensing. IEEE Transactions on Information Theory, 62(9), 5117-5144, 2016 Jenovah Rodrigues, University of Edinburgh. 'Bayesian Inverse Problems with Heterogeneous Noise.' We study linear, ill-posed inverse problems in separable Hilbert spaces with noisy observations. A Bayesian solution with Gaussian regularising priors will be studied; the aim being to select the prior distribution in such a way that the solution achieves the optimal rate of convergence, when the unknown function belongs to a Sobolev space. Consequently, we will focus on obtaining the rate of convergence, for the rate of contraction, of the whole posterior distribution to the forementioned unknown function. We consider a Gaussian noise error model with heterogeneous variance, which is investigated using the spectral decomposition of the operator defined in the inverse problem. This is joint work with Natalia Bochkina (University of Edinburgh). Ferdia Sherry, University of Cambridge Learning a sampling pattern for MRI Taking measurements in MRI is a time-consuming procedure, so ideally one would take few samples and still recover a useable image. It is crucial that these samples are positioned in frequency space in a way that allows as much information to be extracted from the samples as possible. We consider the problem of determining a suitable sampling pattern for a class of images that are in some sense similar. The problem of learning a sampling pattern can be formulated as a bilevel optimisation problem, in which the upper problem measures the reconstruction quality and penalises the lack of sparsity of the sampling pattern and in which the lower problem is the total variation MRI reconstruction problem. We study the use of stochastic optimisation methods (taking a random pair of ground truth and noisy measurements for each iteration) to solve the bilevel problem.	2020-09-29 23:48:52	{"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": 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": 0.5547264814376831, "perplexity": 1315.2505391452473}, "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-2020-40/segments/1600402093104.90/warc/CC-MAIN-20200929221433-20200930011433-00676.warc.gz"}
https://aviation.stackexchange.com/questions/54569/how-much-thrust-could-a-400-mm-edf-produce	# How much thrust could a 400 mm EDF produce? There's a 195mm EDF from Schübeler that produces 25 kg of thrust at a peak power consumption of 15 kW: https://www.schuebeler-jets.de/de/produkte/hst Now, my question is: How much thrust could you approximately get if the diameter was 400 mm instead of 200 and how much more power would you need ? It can be safely assumed that thrust $L$ is a function of the input power $P$, the diameter $D$ of the gas jet and the air density $\rho$. Thus, $L = f(P,D,\rho)$ where $f$ is a function to be determined. From dimensional analysis, the thrust $L$ can be easily derived: The variables are Thrust $L$, dimensions $MLT^{–2}$; Power $P$, dimensions $ML^2T^{–3}$; Gas jet diameter $D$, dimensions $L$ and air density $\rho$, dimensions $ML^{–3}$ The variables form a non-dimensional product $k$ $k = L^a\cdot P^b\cdot D^c\cdot \rho^d$ where $a,b,c,d$ are numbers to be determined. Let’s form now a parallel product $k^$ with the dimensions: $k^ = (MLT^{–2})^a (ML^2T^{–3})^b (L)^c (ML^{–3})^d$ Clearly, $k^* = M^0 L^0 T^0$... We now take the exponents for each dimension: $a + b + d = 0 \\ a + 2b + c – 3d = 0 \\ –2a – 3b = 0$ We make $a = 1$, since $L$ is the variable we’re going to solve for. $b = –2/3 \\ d = –1/3 \\ c = –2/3$ Then, $k = L^a\cdot P^b\cdot D^c\cdot \rho^d \rightarrow k = L\cdot P^{–2/3}\cdot D^{–2/3}\cdot \rho^{–1/3}$ Solving for $L$ $L = k\cdot P^{2/3}\cdot D^{2/3}\cdot \rho^{1/3}$ where $k$ is a constant Hence, for gas jet diameters $D_1$ and $D_2$, and for the same power and air density, the corresponding values of thrust $L_1$ and $L_2$ are: $L_1/L_2 = (D_1/D_2)^{2/3}$ For the case of $D_1 = 400 mm$ and $D_2 = 200 mm$, $L_1/L_2 = (400/200)^{2/3} = 1,59$ In other words, the larger (400 mm) gas jet gives you, for the same absorbed power and air density, 59% more thrust than that attained with the smaller (200 mm) jet. Of course, this is an approximation valid for not too high aircraft speeds, based upon momentum theory, but gives you an idea... For different values of power and gas jet diameter, you can derive the constant k from the data of thrust, power and diameter you already have, $k = L\cdot P^{–2/3}\cdot D^{–2/3}\cdot \rho^{–1/3}$ and then use that constant in your calculations. • Probably worth noting here that a large fan can readily be designed to absorb much more than the same amount of power -- double diameter = 4x area, could be roughly 60 kW in this case. Using four motors geared together would be the simple way... Aug 24 '18 at 12:41 • @xxavier wow, thanks for your great answer! An equation to roughly estimate thrust is exactly what I needed. Aug 24 '18 at 14:37 • @Zeiss Ikon Yes, that‘s a question I asked myself, too. In case I wouldn‘t just double the diameter and put in the same power but doubled or quadroupled the power either, what potential thrust could such an EDF produce ? Aug 24 '18 at 14:40 • According to the math in the answer above, doubling the power will approximately double the thrust, providing your fan can absorb the additional kW. Looks like you could get as high as roughly 6x the original, if you can supply 60 kW. That's an hour's duration on the top-end battery from a 2018 Nissan Leaf, if you can keep everything cool... Aug 24 '18 at 14:56 • I dont have much to add but I would just like to say kudos for one of the clearest and easy to understand examples of Buckingham-Pi dimensional analysis I've ever seen. Aug 27 '18 at 6:45 Simple impulse theory equations for thrust T and power P: $$T = C_T \cdot \rho A (\Omega R)^2$$ $$P = C_P \cdot \rho A (\Omega R)^3$$ Both scale linearly with disk area A. At constant tip speed $$\Omega R$$ and thrust/power coefficients, the scalings are simply T = $$k_T A$$ and P = $$k_P A$$ with $$k_T$$ and $$k_P$$ constants. So a 400 mm fan would produce (400/200)$$^2$$ = 4 times the thrust, at 4 times the power. Note that the larger fan has lower RPM, in order to maintain constant tip speed. Order Of Magnitude check: the four listed electric fans. The largest and the smallest fan have lower P/A, a matter of available electric motor size? • From the data table of the fans, a mean value of k = 1.08 can be calculated. A good agreement for the thrust L as a function of diameter D and power P is obtained inserting it in the dimensionally derived expression $L = k\cdot P^{2/3}\cdot D^{2/3}\cdot \rho^{1/3}$ With caution, that expression can be used for other particular cases... Aug 23 '19 at 12:18	2021-10-21 22:21:24	{"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": 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": 8, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8153578042984009, "perplexity": 944.656814102598}, "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/1634323585441.99/warc/CC-MAIN-20211021195527-20211021225527-00278.warc.gz"}
http://castlebdo.com/g1atwf7q/partial-differentiation-symbol-2b9709	Partial differentiation --- examples General comments To understand Chapter 13 (Vector Fields) you will need to recall some facts about partial differentiation. Differentiating parametric curves. Favourite answer. Partial derivative and gradient (articles) Introduction to partial derivatives. Im obigen Beispiel gibt es zwei partielle Ableitung, weil man ja sowohl nach $$x$$ als auch nach $$y$$ ableiten kann. Die jeweils andere Variable - die, nach der nicht abgeleitet wird - … When applying partial differentiation it is very important to keep in mind, which symbol is the variable and which ones are the constants. Thanks. It is a mathematical symbol derived from the lowercase Greek letter delta. Now you can evaluate the cell. Create a fraction (ctrl-/), add partial derivative symbols $\partial$ (escpdesc) exactly following the visual form of the example displayed above (including powers $\partial^2$ entered exactly like normal powers). Here the surface is a function of 3 variables, i.e. Nothing seems to show the partial differentiation symbol? 1 decade ago. Subject: Partial differential equations Category: Science > Math Asked by: awl-ga List Price: $20.00: Posted: 26 Nov 2002 11:41 PST Expires: 26 Dec 2002 11:41 PST Question ID: 114983 See if you can solve the following equations a) Ut + UUx = 1 with initial conditions U(x,0) = x b) Ut + UUx = U with initial conditions U(x,0) = x the x and the t in the equations are subscripts. Relevance. Angelstar. How do I accomplish the simple task of partial differentiation using Prime 2.0. The first example is to display the first-order differential partial derivative … f’ x = 0 − 2xy = −2xy f’ y = 0 − x 2 = −x 2. f’ z = 3z 2 − 0 = 3z 2. I need import a partial symbol like this. Solche Gleichungen dienen der mathematischen Modellierung vieler physikalischer Vorgänge. A partial derivative of a multivariable function is the rate of change of a variable while holding the other variables constant. n. The derivative with respect to a single variable of a function of two or more variables, regarding other variables as constants. Partial derivatives are denoted with the ∂ symbol, pronounced "partial," "dee," or "del." Mathematica will ask if you want to evaluate the input, and we have to confirm that we do. It only cares about movement in the X direction, so it's treating Y as a constant. I picked up the habit of curving my lower-case d's to the left when I took a biblical Greek class, because it was easier for me to distinguish my own written Greek from a lower-case sigma (σ). The partial derivative of a function of two or more variables with respect to one of its variables is the ordinary derivative of the function with respect to that variable, considering the other variables as constants. Latex plus or minus symbol; Latex symbol for all x; Latex symbol exists; Latex symbol not exists; Latex horizontal space: qquad,hspace, thinspace,enspace; Latex square root symbol; Latex degree symbol; LateX Derivatives, Limits, Sums, Products and Integrals; Latex copyright, trademark, registered symbols; Latex euro symbol Once you understand the concept of a partial derivative as the rate that something is changing, calculating partial derivatives usually isn't difficult. Mathematicians usually write the variable as x or y and the constants as a, b or c but in Physical Chemistry the symbols are different. Eine partielle Differentialgleichung (Abkürzung PDG, PDGL oder PDGln, beziehungsweise PDE für englisch partial differential equation) ist eine Differentialgleichung, die partielle Ableitungen enthält. Second partial derivatives. Symbol for Partial Differentiation Perry, John; Abstract. And, this symbol is partial. You have missed a minus sign on both the derivatives. The symbol ∂ is used whenever a function with more than one variable is being diﬀerentiated but the techniques of partial diﬀerentiation are exactly the same as for (ordinary) diﬀerentiation. It is often not convenient to compute this limit to find a partial derivative. Differentiation with Partial derivatives. Partial derivative of F, with respect to X, and we're doing it at one, two. DR. MUIR'S symbols (p. 520) may be very suitable for manuscripts or the blackboard, but the expense of printing them would be prohibitive. It doesn't even care about the fact that Y changes. A very simple way to understand this is graphically. As far as it's concerned, Y is always equal to two. This web page contains the basics and a pointer to a page to do with partial differentiation, at Brandeis University, that may also be of use to you. Formatting. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … Example 2 Find ∂z ∂x and ∂z ∂y for the function z = x2y3. Let's consider a few examples of differentiation with partial derivatives. 2 Answers . (Unfortunately, there are special cases where calculating the partial derivatives is hard.) The partial derivative of 3x 2 y + 2y 2 with respect to x is 6xy. Partial symbol synonyms, Partial symbol pronunciation, Partial symbol translation, English dictionary definition of Partial symbol. Where is the partial derivative symbol on Word 2007? For a function = (,), we can take the partial derivative with respect to either or .. This is because in a nested call, each differentiation step determines and uses its own differentiation variable. Thus, if k is a certain kind of thermal capacity, are in my thermodynamic work perfectly definite. Copied to clipboard! Although this is not to be confused with the upside-down Capital Greek letter Delta, that is also called Del. Bill This is tragic! ∂ - this symbol . Stack Exchange Network. In this section we will the idea of partial derivatives. This assumption suffices for most engineering and scientific problems. 7 0. farhad m. 6 years ago. If you differentiate a multivariate expression or function f without specifying the differentiation variable, then a nested call to diff and diff(f,n) can return different results. Consider a 3 dimensional surface, the following image for example. Partial derivative examples. Notation. It sometimes helps to replace the symbols … f(x,y,z) = z 3 − x 2 y . As in divergence and curl of a vector field. Example: The volume of a cube with a square prism cut out from it. Commands. Re: pronunciation of partial derivative symbol The lower-case form of delta can be written with that vertical leg either curving back to the left, or with a kind of sharp 's' curve to the right. Answer Save. Source(s): Been using it today! Example. More information about video. f(x, y, z). The \diffp command is used to display the symbol of differentiation with partial derivatives. LaTeX partial derivative symbol. Easy-to-use symbol, keyword, package, style, and formatting reference for LaTeX scientific publishing markup language. So, the partial derivative, the partial f partial x at (x0, y0) is defined to be the limit when I take a small change in x, delta x, of the change in f -- -- divided by delta x. OK, so here I'm actually not changing y at all. So, we can just plug that in ahead of time. IN my college days we used the symbol (if there was only one other independent variable y) as the differential coefficient when y was constant. Contents. without the use of the definition). OK, so it's a special notation for partial derivatives. Anyone have any Idea how I can display the referenced symbol? I still keep to this symbol. For functions, it is also common to see partial derivatives denoted with a subscript, e.g., . Insert ---- Equations ---- fraction ----- common fraction. More symbols are available from extra packages. Sort by: Top Voted . While Mathcad does provide for diffentiation of an expression in its Calculus symbolic template. λ \lambda λ. Keywords. When applying partial differentiation it is very important to keep in mind, which symbol is the variable and which ones are the constants. Symbols. Could someone tell me exactly where it is if it is in symbols because I keep missing it. I am using 2000 Pro and have tried the MATH--->Options feature, I still get d/dx. δ \delta δ. Just find the partial derivative of each variable in turn while treating all other variables as constants. Visit Stack Exchange. 1 Greek letters; 2 Unary operators; 3 Relation operators; 4 Binary operators; 5 Negated binary relations; 6 Set and/or logic notation; 7 Geometry; 8 Delimiters; 9 Arrows; 10 Other symbols; 11 Trigonometric functions; 12 Notes; 13 External links; Greek letters. Mathematicians usually write the variable as x or y and the constants as a, b or c but in Physical Chemistry the symbols are different. I'm just changing x and looking at the rate of change with respect to x. Up Next. Second partial derivatives. For function arguments, use round parentheses$(x,y)$. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Its partial derivative with respect to y is 3x 2 + 4y. It sometimes helps to replace the symbols … EDITOR. Solution z = x2y3 ∴ ∂z ∂x = 2xy3, and ∂z ∂y = x23y2, = 3x2y2. thanks. This is the currently selected item. The most common name for it is del. Styles. The gradient. In the preceding example, diff(f) takes the derivative of f with respect to t because the letter t is closer to x in the alphabet than the letter s is. Directional derivatives (introduction) Directional derivatives (going deeper) Next lesson. LaTeX Base Reference. 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Calculating partial derivatives usually is n't difficult a vector field are denoted with a subscript, e.g., direction so... Symbol translation, English dictionary definition of partial symbol translation, English dictionary definition partial! I 'm just changing x and looking at the rate of change of a function.	2021-07-25 09:30:48	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.8389211297035217, "perplexity": 1676.609417292606}, "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-2021-31/segments/1627046151641.83/warc/CC-MAIN-20210725080735-20210725110735-00167.warc.gz"}
https://www.physicsforums.com/threads/solve-differential-equation-using-power-series.551163/	# Homework Help: Solve differential equation using power series 1. Nov 16, 2011 ### kahwawashay1 Find y(x) as a power series satisfying: y'-2xy=0 , y(0) = 1 attempt: y=$\sum$$^{∞}_{n=0}$ an(x-x0)n y'=$\sum$$^{∞}_{n=0}$ (n+1)an+1(x-x0)n y' - 2xy = y' - 2y(x-x0) - 2x0y substituting the power series into above formula and simplifying eventually gives: a1-2x0a0 + $\sum$$^{∞}_{n=1}$(x-x0)n[(n+1)an+1 - 2an-1 - 2x0an] = 0 so when n=0 and n=1 we get: a1=2x0a0 when n>1, an = $\frac{2}{n}$(x0an-1 + an-2) but what does my professor mean by y(0) = 1? I think he was trying to explain that this means that x0 = 0 ? Also, another problem is that y' - 2xy = x3ex2, y(0)=1 In this case, any hint as to what should I do with the e^x^2???? Last edited by a moderator: Nov 16, 2011 2. Nov 16, 2011 ### Staff: Mentor This means that y = 1 when x = 0. 3. Nov 16, 2011 ### kahwawashay1 yes I know that but what I meant was, how can I use that to solve the problem? If I already have the recursive formula giving me the coefficients in terms of an, of what use is that y(0)=1? On my hw it says y(0)=1 but when my professor was talking about it, he wrote y(x0)=1, but they are not the same thing?? Is there some relation between them I am not seeing? I mean if x=0, then y(0)=$\sum^{∞}_{n=1}$(0-x0)n=1 so: 1-x0+x02-x03+... = 1 so this implies x0=0 ? but then all of my coefficients for odd n's become 0..idk if thats supposed to happen... Last edited: Nov 16, 2011 4. Nov 16, 2011 ### jackmell Nothing wrong with the odd ones being zero if they have to. And as far as that $e^{x^2}$, just expand it out in it's power series form and equate coefficients on both sides just like you did above except in this case, all the coefficients on the right side are no longer zero. 5. Nov 16, 2011 ### kahwawashay1 So but then y(0)=1 means x0=0 and that a0 =1, right? 6. Nov 16, 2011 ### Staff: Mentor Yes, x0 = 0, so your series should be expanded in powers of x, meaning that your series solutions is y = a0 + a1x + a2x2 + .... The initial condition y(0) = 1 determines a0. 7. Nov 16, 2011 ### jackmell That is correct. Also, compute four or so of the coefficients and then compare them to that series you wrote for the right side of the other one.	2018-08-20 10:56:21	{"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": 0.7631837725639343, "perplexity": 1220.5366106454903}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221216333.66/warc/CC-MAIN-20180820101554-20180820121554-00089.warc.gz"}
https://finalfit.org/reference/colon_s.html	This is a modified version of survival::colon.These are data from one of the first successful trials of adjuvant chemotherapy for colon cancer. Levamisole is a low-toxicity compound previously used to treat worm infestations in animals; 5-FU is a moderately toxic (as these things go) chemotherapy agent. There are two records per person, one for recurrence and one for death data(colon_s) ## Format A data frame with 929 rows and 33 variables ## Source colon	2020-09-19 15:31:15	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2894846200942993, "perplexity": 6796.434570906196}, "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-2020-40/segments/1600400192778.51/warc/CC-MAIN-20200919142021-20200919172021-00472.warc.gz"}
http://www.input3d.com.br/sfinks-festival-oumzz/article.php?9c2e2c=normality-of-residuals	For multiple regression, the study assessed the o… We say the distribution is "heavy tailed.". X-axis shows the residuals, whereas Y-axis represents the density of the data set. I tested normal destribution by Wilk-Shapiro test and Jarque-Bera test of normality. Let's take a look at examples of the different kinds of normal probability plots we can obtain and learn what each tells us. Test for Normality and Regression Residuals 165 We then apply the Lagrange multiplier principle to test Ho within this 'general family' of distributions. There are a number of different ways to test this requirement. The histogram of the residuals shows the distribution of the residuals for all observations. The following histogram of residuals suggests that the residuals (and hence the error terms) are not normally distributed. B. 4.6 - Normal Probability Plot of Residuals, 4.6.1 - Normal Probability Plots Versus Histograms, 1.5 - The Coefficient of Determination, $$r^2$$, 1.6 - (Pearson) Correlation Coefficient, $$r$$, 1.9 - Hypothesis Test for the Population Correlation Coefficient, 2.1 - Inference for the Population Intercept and Slope, 2.5 - Analysis of Variance: The Basic Idea, 2.6 - The Analysis of Variance (ANOVA) table and the F-test, 2.8 - Equivalent linear relationship tests, 3.2 - Confidence Interval for the Mean Response, 3.3 - Prediction Interval for a New Response, Minitab Help 3: SLR Estimation & Prediction, 4.4 - Identifying Specific Problems Using Residual Plots, 4.7 - Assessing Linearity by Visual Inspection, 5.1 - Example on IQ and Physical Characteristics, 5.3 - The Multiple Linear Regression Model, 5.4 - A Matrix Formulation of the Multiple Regression Model, Minitab Help 5: Multiple Linear Regression, 6.3 - Sequential (or Extra) Sums of Squares, 6.4 - The Hypothesis Tests for the Slopes, 6.6 - Lack of Fit Testing in the Multiple Regression Setting, Lesson 7: MLR Estimation, Prediction & Model Assumptions, 7.1 - Confidence Interval for the Mean Response, 7.2 - Prediction Interval for a New Response, Minitab Help 7: MLR Estimation, Prediction & Model Assumptions, R Help 7: MLR Estimation, Prediction & Model Assumptions, 8.1 - Example on Birth Weight and Smoking, 8.7 - Leaving an Important Interaction Out of a Model, 9.1 - Log-transforming Only the Predictor for SLR, 9.2 - Log-transforming Only the Response for SLR, 9.3 - Log-transforming Both the Predictor and Response, 9.6 - Interactions Between Quantitative Predictors. Odit molestiae mollitia laudantium assumenda nam eaque, excepturi, soluta, perspiciatis cupiditate sapiente, adipisci quaerat odio voluptates consectetur nulla eveniet iure vitae quibusdam? The problem with Histograms. Y_i=&\beta_0+\beta_1X_i+\varepsilon_i\\ c) normality of the outcome is not such an important assumption to proceed with linear regression. The problem is that to determine the percentile value of a normal distribution, you need to know the mean $$\mu$$ and the variance $$\sigma^2$$. The most popular test is the. The p-th percentile value reduces to just a "Z-score" (or "normal score"). An analysis of transformations. Lorem ipsum dolor sit amet, consectetur adipisicing elit. And, of course, the parameters $$\mu$$ and $$σ^{2}$$ are typically unknown. The inferences discussed in Chapter 2 are still valid for small departure of normality. While a residual plot, or normal plot of the residuals can identify non-normality, you can formally test the hypothesis using the Shapiro-Wilk or similar test. We don’t need to care about the univariate normality of either the dependent or the independent variables. The normality assumption is one of the most misunderstood in all of statistics. The following histogram of residuals suggests that the residuals (and hence the error terms) are not normally distributed. The residuals spread randomly around the 0 line indicating that the relationship is linear. How residuals are computed. The normal probability plot is a graphical technique to identify substantive departures from normality.This includes identifying outliers, skewness, kurtosis, a need for transformations, and mixtures.Normal probability plots are made of raw data, residuals … The figure above shows a bell-shaped distribution of the residuals. Normality. Normality testing must be performed on the Residuals. Here's the basic idea behind any normal probability plot: if the data follow a normal distribution with mean $$\mu$$ and variance $$σ^{2}$$, then a plot of the theoretical percentiles of the normal distribution versus the observed sample percentiles should be approximately linear. The residuals form an approximate horizontal band around the 0 line indicating homogeneity of error variance. 2) A normal probability plot of the Residuals will be created in Excel. This assumption assures that the p-values for the t-tests will be valid. Box, G. E., & Cox, D. R. (1964). Journal of the Royal Statistical Society: Series B (Methodological), 26(2), 211-243. This video demonstrates how to test the normality of residuals in ANOVA using SPSS. The tests obtained are known to have optimal large sample power properties for members of the Here's a screencast illustrating a theoretical p-th percentile. Again, the condition that the error terms are normally distributed is not met. Now, if you are asked to determine the 27th-percentile, you take your ordered data set, and you determine the value so that 27% of the data points in your dataset fall below the value. Figure 12: Histogram plot indicating normality in STATA. No one residual is visibly away from the random pattern of the residuals indicating that there are no outliers. Power comparisons of shapiro-wilk, kolmogorov-smirnov, lilliefors and anderson-darling tests. The residuals from all groups are pooled and then entered into one normality test. A residual is the difference between the actual values, which are the green points in the left plot of figure 1, and the predicted values, which fall on the red line. Recall that the third condition — the "N" condition — of the linear regression model is that the error terms are normally distributed. As before, we will generate the residuals (called r) and predicted values (called fv) and put them in a dataset (called elem1res). Different software packages sometimes switch the axes for this plot, but its interpretation remains the same. The sample p-th percentile of any data set is, roughly speaking, the value such that p% of the measurements fall below the value. The first step should be to look at your data. But what to do with non normal distribution of the residuals? Here's a screencast illustrating how the p-th percentile value reduces to just a normal score. But, there is one extreme outlier (with a value larger than 4): Here's the corresponding normal probability plot of the residuals: This is a classic example of what a normal probability plot looks like when the residuals are normally distributed, but there is just one outlier. Published by Guset User , 2015-04-21 05:07:02 Description: Practical Assessment, Research & Evaluation, Vol 18, No 12 Page 2 Osborne, Response to Williams, Grajales &Kurkiewicz, Assumptions of Regression Razali, N. M., & Wah, Y. The following five normality tests will be performed here: 1) An Excel histogram of the Residuals will be created. The sample p-th percentile of any data set is, roughly speaking, the value such that p% of the measurements fall below the value. Y_i=\beta_0+\beta_1X_i+\varepsilon_i\qquad\qquad\qquad(1.1) Thus this histogram plot confirms the normality test results from the two tests in this article. Statistical theory says its okay just to assume that $$\mu = 0$$ and $$\sigma^2 = 1$$. The assumption is that the errors (residuals) be normally distributed. In this section, we learn how to use a "normal probability plot of the residuals" as a way of learning whether it is reasonable to assume that the error terms are normally distributed. This can be checked by fitting the model of interest, getting the residuals in an output dataset, and then checking them for normality. The following histogram of residuals suggests that the residuals (and hence the error terms) are normally distributed: The normal probability plot of the residuals is approximately linear supporting the condition that the error terms are normally distributed. So you’ll often see the normality assumption for an ANOVA stated as: “The distribution of Y within each group is normally distributed.” Q-Q plots) are preferable. The residuals are simply the error terms, or the differences between the observed value of the dependent variable and the predicted value. the errors are not random). Thus, we will always look for approximate normality in the residuals. Log-transformation may not be appropriate for your data. If one or more of these assumptions are violated, then the results of our linear regression may be unreliable or even misleading. The plot to the right in Figure 1 is a plot of residuals. The theoretical p-th percentile of any normal distribution is the value such that p% of the measurements fall below the value. Normality testing must be performed on the Residuals. Strictly speaking, non-normality of the residuals is an indication of an inadequate model. And so on. Normal residuals but with one outlier The following histogram of residuals suggests that the residuals (and hence the error terms) are normally distributed. 10.1 - What if the Regression Equation Contains "Wrong" Predictors? The following histogram of residuals suggests that the residuals (and hence the error terms) are normally distributed. The goals of the simulation study were to: 1. determine whether nonnormal residuals affect the error rate of the F-tests for regression analysis 2. generate a safe, minimum sample size recommendation for nonnormal residuals For simple regression, the study assessed both the overall F-test (for both linear and quadratic models) and the F-test specifically for the highest-order term. In this post, we provide an explanation for each assumption, how to determine if the assumption is met, and what to do if the assumption is violated. For a Shapiro-Wilks test of normality, I would only reject the null hypothesis (of a normal distribution) if the P value were less than 0.001. This quick tutorial will explain how to test whether sample data is normally distributed in the SPSS statistics package. However, major departures from normality will lead to incorrect p-values in the hypothesis tests and incorrect coverages in the intervals in Chapter 2. 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Around the 0 line indicating homogeneity of error variance tests in this article box, G. E. &! Of statistics discussed in Chapter 2 are still valid for small departure of normality but its interpretation the..., the normal probability plot is a graphical tool for comparing a set! Histogram plot confirms the normality assumption is one of the residuals pass the normality of either the variable., D. R. ( 1964 ) test results from the two tests in this article point. Do that, determining the percentiles of the residuals indicating that there are a number different! Is a classic example of what a normal score related tests are simple to understand the theoretical percentiles not. Unreliable or even misleading the outlier from the data set being normality distributed departures from normality lead! Course, the independent-samples t test – that data is normally distributed of. Comparisons of shapiro-wilk, Kolmogorov-Smirnov, lilliefors and anderson-darling normality of residuals in Chapter 2 misunderstood in all of statistics significant! ) the Kolmogorov-Smirnov test for normality and regression residuals 165 we then apply Lagrange... Wah, Y Doornik and Hansen ( 2008 ) in ANOVA using SPSS, Kolmogorov-Smirnov lilliefors. More complex than the Jarque-Bera test we don ’ t need to care about the normality.!	2021-12-05 14:17:01	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 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": 0.7312864661216736, "perplexity": 1010.5502605708666}, "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-20211205130619-20211205160619-00135.warc.gz"}
http://compgroups.net/comp.graphics.apps.gnuplot/postscript-terminal-no-overline/211303	COMPGROUPS.NET \| Post \| Groups \| Users \| Stream \| Browse \| About \| \| ### postscript terminal : no overline • Email • Follow i gather that an overline is not possible with the postscript terminal. e.g. in LaTeX, $\overline{x}$ gives you "xbar". ps_guide.ps doesn't have this listed. -Bryan 0 Reply bryanlepore (24) 3/1/2011 8:23:34 PM See related articles to this posting bryan wrote: > i gather that an overline is not possible with the postscript > terminal. The postscript terminal per se is not a problem. PostScript itself, on the other hand.... Anyhow The Unicode character COMBINING MACRON at code point 0304 or perhaps CONBINING OVERSCORE at code point 0305 should do what you want. There are instructions in the .../term/PostScript subdirectly describing one way of persuading PostScript to accept unicode characters. So in case of desparation that would be one way to go. But it's painful. It would be easier to use the pdfcairo terminal, which understands UTF-8 natively, and then convert the output to PostScript if that is really necessary. The pdf2ps conversion program knows how to dummy up a termporary encoding for single characters so that PostScript can print them. Ethan > e.g. in LaTeX, $\overline{x}$ gives you "xbar". ps_guide.ps doesn't > have this listed. > > -Bryan 0 On Mar 1, 9:23=A0pm, bryan <bryanlep...@gmail.com> wrote: > i gather that an overline is not possible with the postscript > terminal. > > e.g. in LaTeX, $\overline{x}$ gives you "xbar". ps_guide.ps doesn't > have this listed. > > -Bryan Wasn't enhanced text mode invented to make things like this possible? It's right there among the examples of 'help enhanced': set term post enh set out "test.ps" set label 1 "~x{.8-}" # <---- there set out P=E9ter Juh=E1sz 0 comp.graphics.apps.gnuplot 4765 articles. 0 followers. 2 Replies 475 Views Similar Articles [PageSpeed] 26 • Email • Follow Similar Artilces: epslatex terminal differs from postscript terminal As the subject suggests, I'm wondering why exactly the epslatex terminal differs from the postscript terminal. So you might say "duh!" because after all this is what you want but hold it for a moment. Compare the results of the following script: set term epslatex color set out "test_epslatex.eps" test set term post eps color set out "test_posteps.eps" test At least my gnuplot (v4.0.0) produces only 4 colors when using the epslatex terminal and the pointypes are severly reduced as well. What is even worse - they are in a different order (pt... replacement for terminal postscript Hello, 60 minutes I made promotion for gnuplot and its incredible possibilities, then I got the kiss of death: How can I use the eps output in word? I'm an user of LaTeX and so I made promotion with the terminal postscript eps enhanced. The question: Which terminal is suitable for M$products? I need the same output like postscript espacially the german specials umlaute. Is there a terminal with the features /Symbol, sub- and superscript, line feed /n? And can I use my batch files without any changes to get the same output - I try "png" with linetype 7 and got a b... Images in postscript terminal The following code generates (gnuplot 4.3) a smooth density plot: set terminal postscript eps size 12cm, 8cm set out 'out.eps' plot '-' w image 0.1 0.9 0.2 0.3 0.9 0.4 0.5 0.9 0.6 0.7 0.9 0.8 0.9 0.9 1.0 0.1 0.7 0.16 0.3 0.7 0.32 0.5 0.7 0.48 0.7 0.7 0.64 0.9 0.7 0.8 0.1 0.5 0.12 0.3 0.5 0.24 0.5 0.5 0.36 0.7 0.5 0.48 0.9 0.5 0.6 0.1 0.3 0.08 0.3 0.3 0.16 0.5 0.3 0.24 0.7 0.3 0.32 0.9 0.3 0.4 0.1 0.1 0.04 0.3 0.1 0.08 0.5 0.1 0.12 0.7 0.1 0.16 0.9 0.1 0.2 But when I set terminal to png or gif, the image looks discrete. Is it possible to obtai... arrow in postscript terminal I am using gnuplot version 3.7 patchlevel 3 under windows 2000 and I am having trouble showing arrows in a postscript terminal. I tried the following: reset set size 1,0.5 set terminal postscript portrait enhanced set output 'test.eps' plot [0:10] sin(x) set arrow 1 from graph 0,0 to graph 5,0.5 head show arrow This doesn't give me an arrow in my graph. From the manual it seems that my input is correct. Am I missing something here? Greetings, Dan. Dan <nospam@nospam.com> wrote in message news:<bhsq27$bg3$1@news.tue.nl>... > I am using gnuplo... postscript enhanced terminal Hi all, It seems that I have a little problem with the postscript enhanced feature. I have to produce a simple 2-D scatter plot and I'd like to have the names of the variables on the axis. Unfortunately the variable on the y-axis has a rather complex name: a Greek capital delta with a tilde on top. To be clear: something that in LaTeX would be written as \tilde{\Delta} Knowing that the "enhance" feature of various terminal drivers in gnuplot has been recently improved I decided to give it a try (not before carefully reading the output of the "help enhanced&qu... arrowheads too long in PostScript terminal Consider the following: set term post eps enhanced color solid "Times-Italic" 15 set output 'c:\mydocs\math\function.R3.eps' set term windows; set output set label "problem?" at 1,1 point pt 7 offset 2,-2 set arrow from 0,1 to 1,1 front lw 3 plot [0:3][0:3] x Look close and it is clear that arrow head extends beyond the point (1,1). I.e., line width was not accounted for when the arrowhead was constructed. (Or perhaps just the mitre was neglected.) It is very odd that this is also true for the filled arrow head, but it is. I hazard a guess that the... postscript terminal, color lines Hi I am trying to plot in a postscript file 3 curves (from data files), and I want color-solid lines as in the GNUplot default colors (Red, Green, Bue). The problem I have is that when i make the postscript I get lines as: solid-red, dash-green, dotted-blue. are there anyways to make color solid lines in a postscript, such as it is possible when terminal is "X11" or "windows"? Thanks for any help CA Hi CA, > are there anyways to make color solid lines in a postscript, such as it > is possible when terminal is "X11" or "window... postscript terminal, ps_guide, symbols Hello, I would like to set a greek symbol to the xlabel: set ylabel "x/{/Symbol m}m" In this case I get a Times Roman my. How can I get a Helvetica my? Best Regards Lars l-steinke@web.de wrote: > I would like to set a greek symbol to the xlabel: > set ylabel "x/{/Symbol m}m" > In this case I get a Times Roman my. You didn't. You got the my from the Symbol font, just as it said in the command. > How can I get a Helvetica my? If at all, not by referring to it as 'm' in font Symbol. If there's a my in your Helf... Font size postscript terminal Hello all, I set up a postscript terminal with options eps enhanced and using the fontfile for european modern roman optimized for 10pt as suggested in the documentation. The problem now is, that using the postscript file in latex with includegraphics and no scaling the font is very small in comparison to the 10 pt latex font. To get approximately the same size I have to scale the font in gnuplot to 22 ("FSRM1000" 22). Is this normal? Where the differences in size come from? Volker In article <7803e304-ac7f-4d11-90c4-3ce7e5920318@c19g2000prf.googlegroups.com>, Bl... Abrupt termination of Postscript documents Greetings all: I am having problems with printing Postscript documents with HP LJ printers, to be specific: --printer-- ---model--- ---connection--- A 5MP via RH 7.2 system B 1300N network C 4000 network The usual mode is that while I am looking at things in Mozilla on a RH 8.0 system, I save a document as PostScript. Later when I print it usually only the first or the first two pages are printed, and that's the end of the print job. If I look at the documents with "gv", everything ... postscript eps terminal and ylabel Hello, I have a strange problem with the postscript driver I use gnuplot 4.0 (patchlevel 0) on solaris system. When I run: >set terminal postscript eps enhanced color blacktext >set output "toto.eps" >set ylabel 'MY LABEL' >plot x*2 Instead of a nicely rotated "MY LABEL" on the left of the chart in toto.eps, I got in ghostview something like L E B A L Y M Any clue? Many thanks St�phane In article <8f9174fd.0410090453.351e5a20@posting.google.com>, St?phane <stdecara@free.fr> wrote: I suspect the prob... Display Postscript with MAC as X-terminal ? I just read somewhere that current MACs come with Display Postcript, something they inherited from next. (heard it from someone who heard it from someone who heard it from someone...) If that is the case, does this mean that one could still run applications such as DECwrite and the CDA viewer with the output going to a X-terminal on a MAC even though VMS no longer supports it ? Or must display postscript need to exist on both the client and server (in X parlance) ? There were two shared images on the client side that were removed: XDPS$DPSLIBSHR and XDPS$DPSCLIENTSHR. Therefore, you would ... How to print newline in title (postscript terminal)? Hi, I use set term postscript enhanced set title 'very_long long long title' I want to break the tile into multiple lines as it is too long to fit in one line. Would you please let me know how to do it? Thanks, Peng Peng Yu wrote: > I want to break the tile into multiple lines as it is too long to fit > in one line. Would you please let me know how to do it? set title "very_long long\nlong title" for details see help enhanced Tim ... gnuplot terminal bug (postscript eps) Hi, I have the following gnuplot script and data file, which generate an eps with error(gv it would give error, and epstopdf it would give errors too). The version is gnuplot 4.2 patchlevel rc1. Can somebody help to check if the latest version of gnuplot can generate the eps file correctly? If this bug is in the latest version, can somebody help to fix it. Thanks, Peng #### bug.gp######## set terminal postscript eps enhanced monochrome set output "bug.eps" set xrange [5:60] set yrange [5:60] # use the above ranges with result in error. # but it will be OK i... Postscript terminal font embedding problems. Hi all, while preparing postscript figures I encountered a problem I didn't get around yet. I want to embed fonts in an eps file. My system is cygwin W2k: G N U P L O T Version 4.1 patchlevel 0 last modified Sat Jul 3 00:04:32 CEST 2004 System: CYGWIN_NT-5.0 1.5.12(0.116/4/2) The output of the set terminal is as expected: gnuplot> set terminal postscript eps enhanced monochrome 'Times-Roman' 20 fontfile 's050000l.pfb' fontfile 'n021003l.pfb' fontfile 'n021023l.pfb' Terminal type set to 'postscript' Font file 's050000l... How to type underscore in postscript enhanced terminal? Hi, I'm wondering how to type an underscore in postscript enhanced terminal? Thanks, Peng set term postscript enhanced set xlabel "delta\_freq" On Sun, 06 Jul 2008 20:30:58 -0700, Peng Yu wrote: > Hi, > > I'm wondering how to type an underscore in postscript enhanced terminal? > > Thanks, > Peng > > set term postscript enhanced > set xlabel "delta\_freq" set xlabel 'delta\_freq' (single quotes) seems to work for me here. -- Lionel B In article <g4si06$23r$1@south.jnrs.ja.net>, Lionel B... difference in output between postscript and pdf terminal Hello, I runned the same gnuplot script with 2 different terminals: postscript and pdf. The output is significantly different. The postscript output is larger, the fontsizes are different, and so on. Why is this? It are 2 vctorformats. One should expect the output to be equal. Unfortionatly this is not the case. Any reason? Francky Francky Leyn wrote: > Hello, > > I runned the same gnuplot script with 2 different > terminals: postscript and pdf. The output is > significantly different. The postscript output is > larger, the fontsizes are different, a... postscript/epslatex terminal and pattern fill Hallo everybody, I have some questions to the active developers of gnuplot, especially Ethan A. Merrit regarding the pattern fill definitions in the postscript and epslatec terminal. (I am just trying to improve the epslatex terminal and stumbled on this point. And yes I know, there is a patch pending...) My impressions of the current status are (perhaps they are wrong): the epslatex terminal seems to use Poscript Level 1 pattern fill. In the postscript terminal there are several possibilities present in the source code: Postscript Level 1 patterns, Level 2 patterns and a pos... strange palette behavior for postscript terminal Hello, i've got a problem with (probably) a palette settings using postscript terminal. I'm plotting a 2D map and i want to set a palette so that all bins with values equal to 0.0 are mapped by white color, and all the other (non-zero) values by some different color. I'm using Version 4.4 patchlevel 3 @ System: Linux 2.6.30.8-64.fc11.i586. The example minimal script could be as following: ##### reset set view map set palette model RGB defined (0.0 "white", 0.0 "white", 0.0 "blue", 1.0 "blue") set term postscript set outpu... stacking symbols in terminal postscript enhanced Hello! I use the enhanced postscript terminal, and I want to label axes with greek letters; is there a way to put two symbols above each other, like a bar over a delta? In other words, can I do something like \bar{\Delta} in the latex terminal? I tried using the sub/super scripts on empty space, but it looks awkward and never fits exactly. Any hints greatly appreciated! Thanks & regards from Vienna Johann Mitloehner Johann Mitloehner <mitloehn@wu-wien.ac.at> wrote: > Hello! I use the enhanced postscript terminal, and I want to label axes > with greek letter... arbitrary type1 Font for postscript terminal Currently I am using Myriad in my Documents and would like to use it in the plot as well - instead of Helvetica. It is however not obvious to me (even after reading FAQ p.127ff) how this is done. It would be interesting not only for this font, but for any type1 Font. The Font is installed in the Linux and Latex System. Matthias In article <clb25q$38o$03$1@news.t-online.com>, Matthias Pospiech <matthiasPUNKTpospiech@gmx.de> wrote: >Currently I am using Myriad in my Documents and would like to use it in the >plot as well - instead of Helvetica. > >It i... understanding the consequence of postscript terminal plot size i'd like to understand why the chunk of postscript output code plus ps2pdf below is OK for plotsize=1.0, but anything higher and my .pdf or .ps appear as if perhaps the bounding box (something i sort of understand) is too small - so the main plot is cut into a small piece. HOWEVER if i use convert plot.ps plot.png, i can load it in gimp, flatten it, and i'm all set. -Bryan plotsize=1.0 fontsize=(12plotsize) set terminal postscript color enhanced solid fontsize "Arial" fontfile 'Arial.pfa' ; set size plotsize ; set output "plot.ps" ; replot ... Math font in postscript eps enhanced terminal Hi I want the "i" in the xlabel to typeset like $i$ in latex. Would you please let me know how to do that? Thanks, Peng set terminal postscript eps enhanced set xlabel "i" "PengYu.UT@gmail.com" <PengYu.UT@gmail.com> writes: > Hi > > I want the "i" in the xlabel to typeset like $i$ in latex. Would you > please let me know how to do that? > > Thanks, > Peng > > set terminal postscript eps enhanced > set xlabel "i" Not quite what you want, but try something like this: http://www... postscript terminal cm-super font win32 Hello, I would like to use the cm-super-font SFRM1000: set term post enhanced fontfile 'sfrm1000.pfb' Error-Message: Automatic font conversion pfb->pfa not supported I am using: G N U P L O T Version 4.0 patchlevel 0 last modified Thu Apr 15 14:44:22 CEST 2004 System: MS-Windows 32 bit Copyright (C) 1986 - 1993, 1998, 2004 Can you help me? Thanks, Lars In article <Xns94E3A56EE6F10larssteinketuclausth@195.20.224.116>, Lars Steinke <lars.steinke@tu-clausthal.de> wrote: > >set term post enhance...	2014-04-25 02:24:32	{"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": 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": 0.8439669609069824, "perplexity": 9510.322255433028}, "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-2014-15/segments/1398223207985.17/warc/CC-MAIN-20140423032007-00146-ip-10-147-4-33.ec2.internal.warc.gz"}
https://www.neetprep.com/question/56043-will-angular-width-central-maxima-Fraunhoffer-diffraction-whenlight-wavelength--used-slit-width--cm--rad--rad--rad--rad/126-Physics--Wave-Optics/700-Wave-Optics	# NEET Physics Wave Optics Questions Solved What will be the angular width of central maxima in Fraunhoffer diffraction when light of wavelength $6000\text{\hspace{0.17em}}Å$ is used and slit width is 12×10–5 cm	2019-10-22 23:48:35	{"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": 0.8794847726821899, "perplexity": 2335.4551568923994}, "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-2019-43/segments/1570987826436.88/warc/CC-MAIN-20191022232751-20191023020251-00354.warc.gz"}
https://www.findstat.org/StatisticsDatabase	Identifier There are 1527 statistics in the database. There are possibly some more waiting for verification. Alternating sign matrices (23 statistics) # matrix like objects St000065Alternating sign matrices ⟶ ℤ The number of entries equal to -1 in an alternating sign matrix. St000066Alternating sign matrices ⟶ ℤ The column of the unique '1' in the first row of the alternating sign matrix. St000067Alternating sign matrices ⟶ ℤ The inversion number of the alternating sign matrix. St000076Alternating sign matrices ⟶ ℤ The rank of the alternating sign matrix in the alternating sign matrix poset. St000134Alternating sign matrices ⟶ ℤ The size of the orbit of an alternating sign matrix under gyration. St000187Alternating sign matrices ⟶ ℤ The determinant of an alternating sign matrix. St000193Alternating sign matrices ⟶ ℤ The row of the unique '1' in the first column of the alternating sign matrix. St000197Alternating sign matrices ⟶ ℤ The number of entries equal to positive one in the alternating sign matrix. St000199Alternating sign matrices ⟶ ℤ The column of the unique '1' in the last row of the alternating sign matrix. St000200Alternating sign matrices ⟶ ℤ The row of the unique '1' in the last column of the alternating sign matrix. St000227Alternating sign matrices ⟶ ℤ The osculating paths major index of an alternating sign matrix. St000332Alternating sign matrices ⟶ ℤ The positive inversions of an alternating sign matrix. St000888Alternating sign matrices ⟶ ℤ The maximal sum of entries on a diagonal of an alternating sign matrix. St000889Alternating sign matrices ⟶ ℤ The number of alternating sign matrices with the same antidiagonal sums. St000890Alternating sign matrices ⟶ ℤ The number of nonzero entries in an alternating sign matrix. St000892Alternating sign matrices ⟶ ℤ The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000893Alternating sign matrices ⟶ ℤ The number of distinct diagonal sums of an alternating sign matrix. St000894Alternating sign matrices ⟶ ℤ The trace of an alternating sign matrix. St000895Alternating sign matrices ⟶ ℤ The number of ones on the main diagonal of an alternating sign matrix. St000896Alternating sign matrices ⟶ ℤ The number of zeros on the main diagonal of an alternating sign matrix. St000898Alternating sign matrices ⟶ ℤ The number of maximal entries in the last diagonal of the monotone triangle. St001030Alternating sign matrices ⟶ ℤ Half the number of non-boundary horizontal edges in the fully packed loop corresponding to the alternating sign matrix. St001260Alternating sign matrices ⟶ ℤ The permanent of an alternating sign matrix. Binary trees (37 statistics) # tree like structures # Catalan objects # graph like objects St000045Binary trees ⟶ ℤ The number of linear extensions of a binary tree. St000050Binary trees ⟶ ℤ The depth or height of a binary tree. St000051Binary trees ⟶ ℤ The size of the left subtree of a binary tree. St000061Binary trees ⟶ ℤ The number of nodes on the left branch of a binary tree. St000082Binary trees ⟶ ℤ The number of elements smaller than a binary tree in Tamari order. St000083Binary trees ⟶ ℤ The number of left oriented leafs of a binary tree except the first one. St000118Binary trees ⟶ ℤ The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000121Binary trees ⟶ ℤ The number of occurrences of the contiguous pattern [.,[.,[.,[.,.]]]] in a binary tree. St000122Binary trees ⟶ ℤ The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a binary tree. St000125Binary trees ⟶ ℤ The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000126Binary trees ⟶ ℤ The number of occurrences of the contiguous pattern [.,[.,[.,[.,[.,.]]]]] in a binary tree. St000127Binary trees ⟶ ℤ The number of occurrences of the contiguous pattern [.,[.,[.,[[.,.],.]]]] in a binary tree. St000128Binary trees ⟶ ℤ The number of occurrences of the contiguous pattern [.,[.,[[.,[.,.]],.]]] in a binary tree. St000129Binary trees ⟶ ℤ The number of occurrences of the contiguous pattern [.,[.,[[[.,.],.],.]]] in a binary tree. St000130Binary trees ⟶ ℤ The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000131Binary trees ⟶ ℤ The number of occurrences of the contiguous pattern [.,[[[[.,.],.],.],. St000132Binary trees ⟶ ℤ The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. St000161Binary trees ⟶ ℤ The sum of the sizes of the right subtrees of a binary tree. St000196Binary trees ⟶ ℤ The number of occurrences of the contiguous pattern [[.,.],[.,. St000198Binary trees ⟶ ℤ A decimal representation of a binary tree as a code word. St000201Binary trees ⟶ ℤ The number of leaf nodes in a binary tree. St000203Binary trees ⟶ ℤ The number of external nodes of a binary tree. St000204Binary trees ⟶ ℤ The number of internal nodes of a binary tree. St000252Binary trees ⟶ ℤ The number of nodes of degree 3 of a binary tree. St000385Binary trees ⟶ ℤ The number of vertices with out-degree 1 in a binary tree. St000396Binary trees ⟶ ℤ The register function (or Horton-Strahler number) of a binary tree. St000398Binary trees ⟶ ℤ The sum of the depths of the vertices (or total internal path length) of a binary tree. St000399Binary trees ⟶ ℤ The external path length of a binary tree. St000409Binary trees ⟶ ℤ The number of pitchforks in a binary tree. St000411Binary trees ⟶ ℤ The tree factorial of a binary tree. St000412Binary trees ⟶ ℤ The number of binary trees with the same underlying unordered tree. St000414Binary trees ⟶ ℤ The binary logarithm of the number of binary trees with the same underlying unordered tree. St000568Binary trees ⟶ ℤ The hook number of a binary tree. St000569Binary trees ⟶ ℤ The sum of the heights of the vertices of a binary tree. St000701Binary trees ⟶ ℤ The protection number of a binary tree. St000919Binary trees ⟶ ℤ The number of maximal left branches of a binary tree. St001376Binary trees ⟶ ℤ The Colless index of a binary tree. Binary words (65 statistics) # word like objects # path like objects St000288Binary words ⟶ ℤ The number of ones in a binary word. St000289Binary words ⟶ ℤ The decimal representation of a binary word. St000290Binary words ⟶ ℤ The major index of a binary word. St000291Binary words ⟶ ℤ The number of descents of a binary word. St000292Binary words ⟶ ℤ The number of ascents of a binary word. St000293Binary words ⟶ ℤ The number of inversions of a binary word. St000294Binary words ⟶ ℤ The number of distinct factors of a binary word. St000295Binary words ⟶ ℤ The length of the border of a binary word. St000296Binary words ⟶ ℤ The length of the symmetric border of a binary word. St000297Binary words ⟶ ℤ The number of leading ones in a binary word. St000326Binary words ⟶ ℤ The position of the first one in a binary word after appending a 1 at the end. St000347Binary words ⟶ ℤ The inversion sum of a binary word. St000348Binary words ⟶ ℤ The non-inversion sum of a binary word. St000389Binary words ⟶ ℤ The number of runs of ones of odd length in a binary word. St000390Binary words ⟶ ℤ The number of runs of ones in a binary word. St000391Binary words ⟶ ℤ The sum of the positions of the ones in a binary word. St000392Binary words ⟶ ℤ The length of the longest run of ones in a binary word. St000393Binary words ⟶ ℤ The number of strictly increasing runs in a binary word. St000518Binary words ⟶ ℤ The number of distinct subsequences in a binary word. St000519Binary words ⟶ ℤ The largest length of a factor maximising the subword complexity. St000529Binary words ⟶ ℤ The number of permutations whose descent word is the given binary word. St000543Binary words ⟶ ℤ The size of the conjugacy class of a binary word. St000626Binary words ⟶ ℤ The minimal period of a binary word. St000627Binary words ⟶ ℤ The exponent of a binary word. St000628Binary words ⟶ ℤ The balance of a binary word. St000629Binary words ⟶ ℤ The defect of a binary word. St000630Binary words ⟶ ℤ The length of the shortest palindromic decomposition of a binary word. St000631Binary words ⟶ ℤ The number of distinct palindromic decompositions of a binary word. St000682Binary words ⟶ ℤ The Grundy value of Welter's game on a binary word. St000691Binary words ⟶ ℤ The number of changes of a binary word. St000753Binary words ⟶ ℤ The Grundy value for the game of Kayles on a binary word. St000792Binary words ⟶ ℤ The Grundy value for the game of ruler on a binary word. St000826Binary words ⟶ ℤ The stopping time of the decimal representation of the binary word for the 3x+1 problem. St000827Binary words ⟶ ℤ The decimal representation of a binary word with a leading 1. St000847Binary words ⟶ ℤ The number of standard Young tableaux whose descent set is the binary word. St000875Binary words ⟶ ℤ The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000876Binary words ⟶ ℤ The number of factors in the Catalan decomposition of a binary word. St000877Binary words ⟶ ℤ The depth of the binary word interpreted as a path. St000878Binary words ⟶ ℤ The number of ones minus the number of zeros of a binary word. St000885Binary words ⟶ ℤ The number of critical steps in the Catalan decomposition of a binary word. St000921Binary words ⟶ ℤ The number of internal inversions of a binary word. St000922Binary words ⟶ ℤ The minimal number such that all substrings of this length are unique. St000982Binary words ⟶ ℤ The length of the longest constant subword. St000983Binary words ⟶ ℤ The length of the longest alternating subword. St001267Binary words ⟶ ℤ The length of the Lyndon factorization of the binary word. St001313Binary words ⟶ ℤ The number of Dyck paths above the lattice path given by a binary word. St001355Binary words ⟶ ℤ Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001365Binary words ⟶ ℤ The number of lattice paths of the same length weakly above the path given by a binary word. St001371Binary words ⟶ ℤ The length of the longest Yamanouchi prefix of a binary word. St001372Binary words ⟶ ℤ The length of a longest cyclic run of ones of a binary word. St001413Binary words ⟶ ℤ Half the length of the longest even length palindromic prefix of a binary word. St001414Binary words ⟶ ℤ Half the length of the longest odd length palindromic prefix of a binary word. St001415Binary words ⟶ ℤ The length of the longest palindromic prefix of a binary word. St001416Binary words ⟶ ℤ The length of a longest palindromic factor of a binary word. St001417Binary words ⟶ ℤ The length of a longest palindromic subword of a binary word. St001419Binary words ⟶ ℤ The length of the longest palindromic factor beginning with a one of a binary word. St001420Binary words ⟶ ℤ Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Binary words ⟶ ℤ Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001423Binary words ⟶ ℤ The number of distinct cubes in a binary word. St001424Binary words ⟶ ℤ The number of distinct squares in a binary word. St001436Binary words ⟶ ℤ The index of a given binary word in the lex-order among all its cyclic shifts. St001437Binary words ⟶ ℤ The flex of a binary word. St001485Binary words ⟶ ℤ The modular major index of a binary word. St001491Binary words ⟶ ℤ The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001524Binary words ⟶ ℤ The degree of symmetry of a binary word. Cores (5 statistics) # partition like objects St000158Cores ⟶ ℤ The length of a core. St000190Cores ⟶ ℤ The size of a core. St000191Cores ⟶ ℤ The number of strong covers of a core. St000192Cores ⟶ ℤ Number of covers of a core in weak Bruhat order. St000202Cores ⟶ ℤ The number of k-cores contained in the k-core. Decorated permutations (2 statistics) # word like objects St001425Decorated permutations ⟶ ℤ The number of negatively decorated fixed points of a decorated permutation. St001426Decorated permutations ⟶ ℤ The number of positively decorated fixed points of a decorated permutation. Dyck paths (267 statistics) # Catalan objects # path like objects St000005Dyck paths ⟶ ℤ The bounce statistic of a Dyck path. St000006Dyck paths ⟶ ℤ The dinv of a Dyck path. St000011Dyck paths ⟶ ℤ The number of touch points (or returns) of a Dyck path. St000012Dyck paths ⟶ ℤ The area of a Dyck path. St000013Dyck paths ⟶ ℤ The height of a Dyck path. St000014Dyck paths ⟶ ℤ The number of parking functions supported by a Dyck path. St000015Dyck paths ⟶ ℤ The number of peaks of a Dyck path. St000024Dyck paths ⟶ ℤ The number of double up and double down steps of a Dyck path. St000025Dyck paths ⟶ ℤ The number of initial rises of a Dyck path. St000026Dyck paths ⟶ ℤ The position of the first return of a Dyck path. St000027Dyck paths ⟶ ℤ The major index of a Dyck path. St000032Dyck paths ⟶ ℤ The number of elements smaller than the given Dyck path in the Tamari Order. St000038Dyck paths ⟶ ℤ The product of the heights of the descending steps of a Dyck path. St000052Dyck paths ⟶ ℤ The number of valleys of a Dyck path not on the x-axis. St000053Dyck paths ⟶ ℤ The number of valleys of the Dyck path. St000079Dyck paths ⟶ ℤ The number of alternating sign matrices for a given Dyck path. St000117Dyck paths ⟶ ℤ The number of centered tunnels of a Dyck path. St000120Dyck paths ⟶ ℤ The number of left tunnels of a Dyck path. St000144Dyck paths ⟶ ℤ The pyramid weight of the Dyck path. St000306Dyck paths ⟶ ℤ The bounce count of a Dyck path. St000329Dyck paths ⟶ ℤ The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000331Dyck paths ⟶ ℤ The number of upper interactions of a Dyck path. St000335Dyck paths ⟶ ℤ The difference of lower and upper interactions. St000340Dyck paths ⟶ ℤ The number of non-final maximal constant sub-paths of length greater than one. St000369Dyck paths ⟶ ℤ The dinv deficit of a Dyck path. St000376Dyck paths ⟶ ℤ The bounce deficit of a Dyck path. St000386Dyck paths ⟶ ℤ The number of factors DDU in a Dyck path. St000394Dyck paths ⟶ ℤ The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000395Dyck paths ⟶ ℤ The sum of the heights of the peaks of a Dyck path. St000418Dyck paths ⟶ ℤ The number of Dyck paths that are weakly below a Dyck path. St000419Dyck paths ⟶ ℤ The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000420Dyck paths ⟶ ℤ The number of Dyck paths that are weakly above a Dyck path. St000421Dyck paths ⟶ ℤ The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000438Dyck paths ⟶ ℤ The position of the last up step in a Dyck path. St000439Dyck paths ⟶ ℤ The position of the first down step of a Dyck path. St000442Dyck paths ⟶ ℤ The maximal area to the right of an up step of a Dyck path. St000443Dyck paths ⟶ ℤ The number of long tunnels of a Dyck path. St000444Dyck paths ⟶ ℤ The length of the maximal rise of a Dyck path. St000445Dyck paths ⟶ ℤ The number of rises of length 1 of a Dyck path. St000476Dyck paths ⟶ ℤ The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000617Dyck paths ⟶ ℤ The number of global maxima of a Dyck path. St000645Dyck paths ⟶ ℤ The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000655Dyck paths ⟶ ℤ The length of the minimal rise of a Dyck path. St000658Dyck paths ⟶ ℤ The number of rises of length 2 of a Dyck path. St000659Dyck paths ⟶ ℤ The number of rises of length at least 2 of a Dyck path. St000660Dyck paths ⟶ ℤ The number of rises of length at least 3 of a Dyck path. St000661Dyck paths ⟶ ℤ The number of rises of length 3 of a Dyck path. St000674Dyck paths ⟶ ℤ The number of hills of a Dyck path. St000675Dyck paths ⟶ ℤ The number of centered multitunnels of a Dyck path. St000676Dyck paths ⟶ ℤ The number of odd rises of a Dyck path. St000678Dyck paths ⟶ ℤ The number of up steps after the last double rise of a Dyck path. St000683Dyck paths ⟶ ℤ The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000684Dyck paths ⟶ ℤ The global dimension of the LNakayama algebra associated to a Dyck path. St000685Dyck paths ⟶ ℤ The dominant dimension of the LNakayama algebra associated to a Dyck path. St000686Dyck paths ⟶ ℤ The finitistic dominant dimension of a Dyck path. St000687Dyck paths ⟶ ℤ The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000688Dyck paths ⟶ ℤ The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000689Dyck paths ⟶ ℤ The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000790Dyck paths ⟶ ℤ The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000791Dyck paths ⟶ ℤ The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000874Dyck paths ⟶ ℤ The position of the last double rise in a Dyck path. St000920Dyck paths ⟶ ℤ The logarithmic height of a Dyck path. St000930Dyck paths ⟶ ℤ The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000931Dyck paths ⟶ ℤ The number of occurrences of the pattern UUU in a Dyck path. St000932Dyck paths ⟶ ℤ The number of occurrences of the pattern UDU in a Dyck path. St000946Dyck paths ⟶ ℤ The sum of the skew hook positions in a Dyck path. St000947Dyck paths ⟶ ℤ The major index east count of a Dyck path. St000949Dyck paths ⟶ ℤ Gives the number of generalised tilting modules of the corresponding LNakayama algebra. St000950Dyck paths ⟶ ℤ Number of tilting modules of the corresponding LNakayama algebra, where a tilting module is a generalised tilting module of projective dimension 1. St000951Dyck paths ⟶ ℤ The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. St000952Dyck paths ⟶ ℤ Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St000953Dyck paths ⟶ ℤ The largest degree of an irreducible factor of the Coxeter polynomial of the Dyck path over the rational numbers. St000954Dyck paths ⟶ ℤ Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000955Dyck paths ⟶ ℤ Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000964Dyck paths ⟶ ℤ Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965Dyck paths ⟶ ℤ The sum of the dimension of Ext^i(D(A),A) for i=1,. St000966Dyck paths ⟶ ℤ Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000967Dyck paths ⟶ ℤ The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St000968Dyck paths ⟶ ℤ We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St000969Dyck paths ⟶ ℤ We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St000970Dyck paths ⟶ ℤ Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000976Dyck paths ⟶ ℤ The sum of the positions of double up-steps of a Dyck path. St000977Dyck paths ⟶ ℤ MacMahon's equal index of a Dyck path. St000978Dyck paths ⟶ ℤ The sum of the positions of double down-steps of a Dyck path. St000979Dyck paths ⟶ ℤ Half of MacMahon's equal index of a Dyck path. St000980Dyck paths ⟶ ℤ The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St000981Dyck paths ⟶ ℤ The length of the longest zigzag subpath. St000984Dyck paths ⟶ ℤ The number of boxes below precisely one peak. St000998Dyck paths ⟶ ℤ Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St000999Dyck paths ⟶ ℤ Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001000Dyck paths ⟶ ℤ Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001001Dyck paths ⟶ ℤ The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001002Dyck paths ⟶ ℤ Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001003Dyck paths ⟶ ℤ The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001006Dyck paths ⟶ ℤ Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001007Dyck paths ⟶ ℤ Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001008Dyck paths ⟶ ℤ Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001009Dyck paths ⟶ ℤ Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001010Dyck paths ⟶ ℤ Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Dyck paths ⟶ ℤ Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001012Dyck paths ⟶ ℤ Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001013Dyck paths ⟶ ℤ Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001014Dyck paths ⟶ ℤ Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Dyck paths ⟶ ℤ Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Dyck paths ⟶ ℤ Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001017Dyck paths ⟶ ℤ Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001018Dyck paths ⟶ ℤ Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001019Dyck paths ⟶ ℤ Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001020Dyck paths ⟶ ℤ Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001021Dyck paths ⟶ ℤ Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001022Dyck paths ⟶ ℤ Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001023Dyck paths ⟶ ℤ Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001024Dyck paths ⟶ ℤ Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001025Dyck paths ⟶ ℤ Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001026Dyck paths ⟶ ℤ The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001027Dyck paths ⟶ ℤ Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001028Dyck paths ⟶ ℤ Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001031Dyck paths ⟶ ℤ The height of the bicoloured Motzkin path associated with the Dyck path. St001032Dyck paths ⟶ ℤ The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001033Dyck paths ⟶ ℤ The normalized area of the parallelogram polyomino associated with the Dyck path. St001034Dyck paths ⟶ ℤ The area of the parallelogram polyomino associated with the Dyck path. St001035Dyck paths ⟶ ℤ The convexity degree of the parallelogram polyomino associated with the Dyck path. St001036Dyck paths ⟶ ℤ The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001037Dyck paths ⟶ ℤ The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001038Dyck paths ⟶ ℤ The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001039Dyck paths ⟶ ℤ The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001063Dyck paths ⟶ ℤ Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Dyck paths ⟶ ℤ Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001065Dyck paths ⟶ ℤ Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001066Dyck paths ⟶ ℤ The number of simple reflexive modules in the corresponding Nakayama algebra. St001067Dyck paths ⟶ ℤ The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001068Dyck paths ⟶ ℤ Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Dyck paths ⟶ ℤ Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001089Dyck paths ⟶ ℤ Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001104Dyck paths ⟶ ℤ The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001107Dyck paths ⟶ ℤ The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001113Dyck paths ⟶ ℤ Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001125Dyck paths ⟶ ℤ The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001126Dyck paths ⟶ ℤ Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001135Dyck paths ⟶ ℤ The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001137Dyck paths ⟶ ℤ Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001138Dyck paths ⟶ ℤ The number of indecomposable modules with projective dimension or injective dimension at most one in the corresponding Nakayama algebra. St001139Dyck paths ⟶ ℤ The number of occurrences of hills of size 2 in a Dyck path. St001140Dyck paths ⟶ ℤ Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001141Dyck paths ⟶ ℤ The number of occurrences of hills of size 3 in a Dyck path. St001142Dyck paths ⟶ ℤ The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001159Dyck paths ⟶ ℤ Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001161Dyck paths ⟶ ℤ The major index north count of a Dyck path. St001163Dyck paths ⟶ ℤ The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001164Dyck paths ⟶ ℤ Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001165Dyck paths ⟶ ℤ Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001166Dyck paths ⟶ ℤ Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001167Dyck paths ⟶ ℤ The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001169Dyck paths ⟶ ℤ Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001170Dyck paths ⟶ ℤ Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001172Dyck paths ⟶ ℤ The number of 1-rises at odd height of a Dyck path. St001179Dyck paths ⟶ ℤ Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001180Dyck paths ⟶ ℤ Number of indecomposable injective modules with projective dimension at most 1. St001181Dyck paths ⟶ ℤ Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001182Dyck paths ⟶ ℤ Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001183Dyck paths ⟶ ℤ The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001184Dyck paths ⟶ ℤ Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001185Dyck paths ⟶ ℤ The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001186Dyck paths ⟶ ℤ Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001187Dyck paths ⟶ ℤ The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001188Dyck paths ⟶ ℤ The number of simple modules $S$ with grade $\inf \{ i \geq 0 \| Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001189Dyck paths ⟶ ℤ The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001190Dyck paths ⟶ ℤ Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001191Dyck paths ⟶ ℤ Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001192Dyck paths ⟶ ℤ The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001193Dyck paths ⟶ ℤ The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001194Dyck paths ⟶ ℤ The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001195Dyck paths ⟶ ℤ The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001196Dyck paths ⟶ ℤ The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001197Dyck paths ⟶ ℤ The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001198Dyck paths ⟶ ℤ The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001199Dyck paths ⟶ ℤ The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001200Dyck paths ⟶ ℤ The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001201Dyck paths ⟶ ℤ The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001202Dyck paths ⟶ ℤ Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001203Dyck paths ⟶ ℤ We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows: St001204Dyck paths ⟶ ℤ Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001205Dyck paths ⟶ ℤ The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206Dyck paths ⟶ ℤ The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001210Dyck paths ⟶ ℤ Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001211Dyck paths ⟶ ℤ The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001212Dyck paths ⟶ ℤ The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001213Dyck paths ⟶ ℤ The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St001215Dyck paths ⟶ ℤ Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216Dyck paths ⟶ ℤ The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001217Dyck paths ⟶ ℤ The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001218Dyck paths ⟶ ℤ Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St001219Dyck paths ⟶ ℤ Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001221Dyck paths ⟶ ℤ The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001222Dyck paths ⟶ ℤ Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001223Dyck paths ⟶ ℤ Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001224Dyck paths ⟶ ℤ Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001225Dyck paths ⟶ ℤ The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001226Dyck paths ⟶ ℤ The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001227Dyck paths ⟶ ℤ The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001228Dyck paths ⟶ ℤ The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra. St001229Dyck paths ⟶ ℤ The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001230Dyck paths ⟶ ℤ The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001231Dyck paths ⟶ ℤ The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001232Dyck paths ⟶ ℤ The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001233Dyck paths ⟶ ℤ The number of indecomposable 2-dimensional modules with projective dimension one. St001234Dyck paths ⟶ ℤ The number of indecomposable three dimensional modules with projective dimension one. St001237Dyck paths ⟶ ℤ The number of simple modules with injective dimension at most one or dominant dimension at least one. St001238Dyck paths ⟶ ℤ The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001239Dyck paths ⟶ ℤ The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001240Dyck paths ⟶ ℤ The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001241Dyck paths ⟶ ℤ The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001242Dyck paths ⟶ ℤ The toal dimension of certain Sn modules determined by LLT polynomials associated with a Dyck path. St001243Dyck paths ⟶ ℤ The sum of coefficients in the Schur basis of certain LLT polynomials associated with a Dyck path. St001244Dyck paths ⟶ ℤ The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001253Dyck paths ⟶ ℤ The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001254Dyck paths ⟶ ℤ The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J. St001255Dyck paths ⟶ ℤ The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001256Dyck paths ⟶ ℤ Number of simple reflexive modules that are 2-stable reflexive. St001257Dyck paths ⟶ ℤ The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001258Dyck paths ⟶ ℤ Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001259Dyck paths ⟶ ℤ The vector space dimension of the double dual of D(A) in the corresponding Nakayama algebra. St001264Dyck paths ⟶ ℤ The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265Dyck paths ⟶ ℤ The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001266Dyck paths ⟶ ℤ The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001273Dyck paths ⟶ ℤ The projective dimension of the first term in an injective coresolution of the regular module. St001274Dyck paths ⟶ ℤ The number of indecomposable injective modules with projective dimension equal to two. St001275Dyck paths ⟶ ℤ The projective dimension of the second term in a minimal injective coresolution of the regular module. St001276Dyck paths ⟶ ℤ The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001278Dyck paths ⟶ ℤ The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001289Dyck paths ⟶ ℤ The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001290Dyck paths ⟶ ℤ The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001291Dyck paths ⟶ ℤ The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001292Dyck paths ⟶ ℤ The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001294Dyck paths ⟶ ℤ The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001295Dyck paths ⟶ ℤ Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001296Dyck paths ⟶ ℤ The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001297Dyck paths ⟶ ℤ The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001299Dyck paths ⟶ ℤ The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001314Dyck paths ⟶ ℤ The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001348Dyck paths ⟶ ℤ The bounce of the parallelogram polyomino associated with the Dyck path. St001361Dyck paths ⟶ ℤ The number of lattice paths of the same length that stay weakly above a Dyck path. St001418Dyck paths ⟶ ℤ Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Dyck paths ⟶ ℤ Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001471Dyck paths ⟶ ℤ The magnitude of a Dyck path. St001473Dyck paths ⟶ ℤ The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001480Dyck paths ⟶ ℤ The number of simple summands of the module J^2/J^3. St001481Dyck paths ⟶ ℤ The minimal height of a peak of a Dyck path. St001483Dyck paths ⟶ ℤ The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001492Dyck paths ⟶ ℤ The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001493Dyck paths ⟶ ℤ The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001498Dyck paths ⟶ ℤ The normalised height of a Nakayama algebra with magnitude 1. St001499Dyck paths ⟶ ℤ The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001500Dyck paths ⟶ ℤ The global dimension of magnitude 1 Nakayama algebras. St001501Dyck paths ⟶ ℤ The dominant dimension of magnitude 1 Nakayama algebras. St001502Dyck paths ⟶ ℤ The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001503Dyck paths ⟶ ℤ The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001504Dyck paths ⟶ ℤ The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001505Dyck paths ⟶ ℤ The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Dyck paths ⟶ ℤ Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001507Dyck paths ⟶ ℤ The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001508Dyck paths ⟶ ℤ The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001509Dyck paths ⟶ ℤ The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001514Dyck paths ⟶ ℤ The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001515Dyck paths ⟶ ℤ The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001523Dyck paths ⟶ ℤ The degree of symmetry of a Dyck path. St001526Dyck paths ⟶ ℤ The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. Finite Cartan types (41 statistics) # Cartan type objects St000106Finite Cartan types ⟶ ℤ The size of the associated Weyl group. St000107Finite Cartan types ⟶ ℤ The dimension of the representation $V(\Lambda_1)$. St000113Finite Cartan types ⟶ ℤ The rank of the Cartan type. St000138Finite Cartan types ⟶ ℤ The Catalan number of an irreducible finite Cartan type. St000139Finite Cartan types ⟶ ℤ The Coxeter number of a finite Cartan type. St000140Finite Cartan types ⟶ ℤ The positive Catalan number of an irreducible finite Cartan type. St000821Finite Cartan types ⟶ ℤ The determinant of the Cartan matrix. St000851Finite Cartan types ⟶ ℤ The third Fuss-Catalan number of a finite Cartan type. St000852Finite Cartan types ⟶ ℤ The second Fuss-Catalan number of a finite Cartan type. St000853Finite Cartan types ⟶ ℤ The number of almost positive roots of a finite Cartan type. St000854Finite Cartan types ⟶ ℤ The number of orbits of reflections of a finite Cartan type. St000855Finite Cartan types ⟶ ℤ The number of full-support reflections in the Weyl group of a finite Cartan type. St000856Finite Cartan types ⟶ ℤ The number of conjugacy classes in the Weyl group of a finite Cartan type. St000857Finite Cartan types ⟶ ℤ The number of reflections of the Weyl group of a finite Cartan type. St000858Finite Cartan types ⟶ ℤ The number of factorizations of any Coxeter element into reflections of a finite Cartan type. St000859Finite Cartan types ⟶ ℤ The number of parking functions of a finite Cartan type. St000860Finite Cartan types ⟶ ℤ The size of the center of the Weyl group of a finite Cartan type. St000861Finite Cartan types ⟶ ℤ The maximal dimension of an irreducible representation of the Weyl group of a finite Cartan type. St000865Finite Cartan types ⟶ ℤ The number of Coxeter elements in the Weyl group of a finite Cartan type. St000960Finite Cartan types ⟶ ℤ The permanent of the Cartan matrix of a finite Cartan type. St001053Finite Cartan types ⟶ ℤ The second positive Fuss-Catalan number of a finite Cartan type. St001054Finite Cartan types ⟶ ℤ The third positive Fuss-Catalan number of a finite Cartan type. St001143Finite Cartan types ⟶ ℤ The number of pairs in the Weyl group of given type with mu-coefficient of the Kazhdan Lusztig polynomial being non-zero. St001144Finite Cartan types ⟶ ℤ The largest mu-coefficient of the Kazhdan Lusztig polynomial occurring in the Weyl group of given type. St001145Finite Cartan types ⟶ ℤ The largest coefficient in a Kazhdan Lusztig polynomial of the Weyl group of given type. St001146Finite Cartan types ⟶ ℤ The number of Grassmannian elements in the Coxeter group of the given type. St001147Finite Cartan types ⟶ ℤ The number of minuscule dominant weights in the weight lattice of a finite Cartan type. St001148Finite Cartan types ⟶ ℤ The dimension of the adjoint representation of the Lie group of given type. St001149Finite Cartan types ⟶ ℤ The dimension of the quasi-minuscule representation of the Lie group of given type. St001150Finite Cartan types ⟶ ℤ The minimal dimension of a faithful linear representation of the Lie algebra of given type. St001154Finite Cartan types ⟶ ℤ The dual Coxeter number of a finite Cartan type. St001155Finite Cartan types ⟶ ℤ The number of conjugacy classes of subgroups of the Weyl group of given type. St001156Finite Cartan types ⟶ ℤ The Dynkin index of the Lie algebra of given type. St001157Finite Cartan types ⟶ ℤ The exponent of the Weyl group of given type. St001158Finite Cartan types ⟶ ℤ The size of the mutation class of quivers of given type. St001173Finite Cartan types ⟶ ℤ The number of commutative positive roots in the root system of the given finite Cartan type. St001369Finite Cartan types ⟶ ℤ The largest coefficient in the highest root in the root system of a Cartan type. St001370Finite Cartan types ⟶ ℤ The degree of the largest fundamental representation associated with a Cartan type. St001443Finite Cartan types ⟶ ℤ The largest coefficient in the Poincaré polynomial of the Weyl group of given Cartan type. St001467Finite Cartan types ⟶ ℤ The number of involutions in the Weyl group of a given Cartan type. St001495Finite Cartan types ⟶ ℤ The maximal order of an element in the Weyl group of a given Cartan type. Gelfand-Tsetlin patterns (13 statistics) # matrix like objects St000072Gelfand-Tsetlin patterns ⟶ ℤ The number of circled entries. St000073Gelfand-Tsetlin patterns ⟶ ℤ The number of boxed entries. St000074Gelfand-Tsetlin patterns ⟶ ℤ The number of special entries. St000077Gelfand-Tsetlin patterns ⟶ ℤ The number of boxed and circled entries. St000114Gelfand-Tsetlin patterns ⟶ ℤ The sum of the entries of the Gelfand-Tsetlin pattern. St000115Gelfand-Tsetlin patterns ⟶ ℤ The single entry in the last row. St000152Gelfand-Tsetlin patterns ⟶ ℤ The number of boxed plus the number of special entries. St000176Gelfand-Tsetlin patterns ⟶ ℤ The total number of tiles in the Gelfand-Tsetlin pattern. St000177Gelfand-Tsetlin patterns ⟶ ℤ The number of free tiles in the pattern. St000178Gelfand-Tsetlin patterns ⟶ ℤ Number of free entries. St000186Gelfand-Tsetlin patterns ⟶ ℤ The sum of the first row in a Gelfand-Tsetlin pattern. St001404Gelfand-Tsetlin patterns ⟶ ℤ The number of distinct entries in a Gelfand Tsetlin pattern. St001406Gelfand-Tsetlin patterns ⟶ ℤ The number of nonzero entries in a Gelfand Tsetlin pattern. Graphs (217 statistics) # graph like objects St000081Graphs ⟶ ℤ The number of edges of a graph. St000086Graphs ⟶ ℤ The number of subgraphs. St000087Graphs ⟶ ℤ The number of induced subgraphs. St000093Graphs ⟶ ℤ The length of the maximal independent set of vertices of a graph. St000095Graphs ⟶ ℤ The number of triangles of a graph. St000096Graphs ⟶ ℤ The number of spanning trees of a graph. St000097Graphs ⟶ ℤ The order of the largest clique of the graph. St000098Graphs ⟶ ℤ The chromatic number of a graph. St000171Graphs ⟶ ℤ The degree of the graph. St000172Graphs ⟶ ℤ The Grundy number of a graph. St000244Graphs ⟶ ℤ The cardinality of the automorphism group of a graph. St000258Graphs ⟶ ℤ The burning number of a graph. St000259Graphs ⟶ ℤ The diameter of a connected graph. St000260Graphs ⟶ ℤ The radius of a connected graph. St000261Graphs ⟶ ℤ The edge connectivity of a graph. St000262Graphs ⟶ ℤ The vertex connectivity of a graph. St000263Graphs ⟶ ℤ The Szeged index of a graph. St000264Graphs ⟶ ℤ The girth of a graph, which is not a tree. St000265Graphs ⟶ ℤ The Wiener index of a graph. St000266Graphs ⟶ ℤ The number of spanning subgraphs of a graph with the same connected components. St000267Graphs ⟶ ℤ The number of maximal spanning forests contained in a graph. St000268Graphs ⟶ ℤ The number of strongly connected orientations of a graph. St000269Graphs ⟶ ℤ The number of acyclic orientations of a graph. St000270Graphs ⟶ ℤ The number of forests contained in a graph. St000271Graphs ⟶ ℤ The chromatic index of a connected graph. St000272Graphs ⟶ ℤ The treewidth of a graph. St000273Graphs ⟶ ℤ The domination number of a graph. St000274Graphs ⟶ ℤ The number of perfect matchings of a graph. St000276Graphs ⟶ ℤ The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000283Graphs ⟶ ℤ The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000286Graphs ⟶ ℤ The number of connected components of the complement of a graph. St000287Graphs ⟶ ℤ The number of connected components of a graph. St000299Graphs ⟶ ℤ The number of nonisomorphic vertex-induced subtrees. St000300Graphs ⟶ ℤ The number of independent sets of vertices of a graph. St000301Graphs ⟶ ℤ The number of facets of the stable set polytope of a graph. St000302Graphs ⟶ ℤ The determinant of the distance matrix of a connected graph. St000303Graphs ⟶ ℤ The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000309Graphs ⟶ ℤ The number of vertices with even degree. St000310Graphs ⟶ ℤ The minimal degree of a vertex of a graph. St000311Graphs ⟶ ℤ The number of vertices of odd degree in a graph. St000312Graphs ⟶ ℤ The number of leaves in a graph. St000313Graphs ⟶ ℤ The number of degree 2 vertices of a graph. St000315Graphs ⟶ ℤ The number of isolated vertices of a graph. St000322Graphs ⟶ ℤ The skewness of a graph. St000323Graphs ⟶ ℤ The minimal crossing number of a graph. St000343Graphs ⟶ ℤ The number of spanning subgraphs of a graph. St000344Graphs ⟶ ℤ The number of strongly connected outdegree sequences of a graph. St000349Graphs ⟶ ℤ The number of different adjacency matrices of a graph. St000350Graphs ⟶ ℤ The sum of the vertex degrees of a graph. St000351Graphs ⟶ ℤ The determinant of the adjacency matrix of a graph. St000361Graphs ⟶ ℤ The second Zagreb index of a graph. St000362Graphs ⟶ ℤ The size of a minimal vertex cover of a graph. St000363Graphs ⟶ ℤ The number of minimal vertex covers of a graph. St000364Graphs ⟶ ℤ The exponent of the automorphism group of a graph. St000368Graphs ⟶ ℤ The Altshuler-Steinberg determinant of a graph. St000370Graphs ⟶ ℤ The genus of a graph. St000379Graphs ⟶ ℤ The number of Hamiltonian cycles in a graph. St000387Graphs ⟶ ℤ The matching number of a graph. St000388Graphs ⟶ ℤ The number of orbits of vertices of a graph under automorphisms. St000403Graphs ⟶ ℤ The Szeged index minus the Wiener index of a graph. St000422Graphs ⟶ ℤ The energy of a graph, if it is integral. St000447Graphs ⟶ ℤ The number of pairs of vertices of a graph with distance 3. St000448Graphs ⟶ ℤ The number of pairs of vertices of a graph with distance 2. St000449Graphs ⟶ ℤ The number of pairs of vertices of a graph with distance 4. St000450Graphs ⟶ ℤ The number of edges minus the number of vertices plus 2 of a graph. St000452Graphs ⟶ ℤ The number of distinct eigenvalues of a graph. St000453Graphs ⟶ ℤ The number of distinct Laplacian eigenvalues of a graph. St000454Graphs ⟶ ℤ The largest eigenvalue of a graph if it is integral. St000455Graphs ⟶ ℤ The second largest eigenvalue of a graph if it is integral. St000456Graphs ⟶ ℤ The monochromatic index of a connected graph. St000464Graphs ⟶ ℤ The Schultz index of a connected graph. St000465Graphs ⟶ ℤ The first Zagreb index of a graph. St000466Graphs ⟶ ℤ The Gutman (or modified Schultz) index of a connected graph. St000467Graphs ⟶ ℤ The hyper-Wiener index of a connected graph. St000468Graphs ⟶ ℤ The Hosoya index of a graph. St000469Graphs ⟶ ℤ The distinguishing number of a graph. St000479Graphs ⟶ ℤ The Ramsey number of a graph. St000482Graphs ⟶ ℤ The (zero)-forcing number of a graph. St000535Graphs ⟶ ℤ The rank-width of a graph. St000536Graphs ⟶ ℤ The pathwidth of a graph. St000537Graphs ⟶ ℤ The cutwidth of a graph. St000544Graphs ⟶ ℤ The cop number of a graph. St000552Graphs ⟶ ℤ The number of cut vertices of a graph. St000553Graphs ⟶ ℤ The number of blocks of a graph. St000571Graphs ⟶ ℤ The F-index (or forgotten topological index) of a graph. St000636Graphs ⟶ ℤ The hull number of a graph. St000637Graphs ⟶ ℤ The length of the longest cycle in a graph. St000671Graphs ⟶ ℤ The maximin edge-connectivity for choosing a subgraph. St000699Graphs ⟶ ℤ The toughness times the least common multiple of 1,. St000718Graphs ⟶ ℤ The largest Laplacian eigenvalue of a graph if it is integral. St000722Graphs ⟶ ℤ The number of different neighbourhoods in a graph. St000723Graphs ⟶ ℤ The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000741Graphs ⟶ ℤ The Colin de Verdière graph invariant. St000771Graphs ⟶ ℤ The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772Graphs ⟶ ℤ The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000773Graphs ⟶ ℤ The multiplicity of the largest Laplacian eigenvalue in a graph. St000774Graphs ⟶ ℤ The maximal multiplicity of a Laplacian eigenvalue in a graph. St000775Graphs ⟶ ℤ The multiplicity of the largest eigenvalue in a graph. St000776Graphs ⟶ ℤ The maximal multiplicity of an eigenvalue in a graph. St000777Graphs ⟶ ℤ The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000778Graphs ⟶ ℤ The metric dimension of a graph. St000785Graphs ⟶ ℤ The number of distinct colouring schemes of a graph. St000786Graphs ⟶ ℤ The maximal number of occurrences of a colour in a proper colouring of a graph. St000822Graphs ⟶ ℤ The Hadwiger number of the graph. St000915Graphs ⟶ ℤ The Ore degree of a graph. St000916Graphs ⟶ ℤ The packing number of a graph. St000917Graphs ⟶ ℤ The open packing number of a graph. St000918Graphs ⟶ ℤ The 2-limited packing number of a graph. St000926Graphs ⟶ ℤ The clique-coclique number of a graph. St000948Graphs ⟶ ℤ The chromatic discriminant of a graph. St000972Graphs ⟶ ℤ The composition number of a graph. St000985Graphs ⟶ ℤ The number of positive eigenvalues of the adjacency matrix of the graph. St000986Graphs ⟶ ℤ The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St000987Graphs ⟶ ℤ The number of positive eigenvalues of the Laplacian matrix of the graph. St001029Graphs ⟶ ℤ The size of the core of a graph. St001056Graphs ⟶ ℤ The Grundy value for the game of deleting vertices of a graph until it has no edges. St001057Graphs ⟶ ℤ The Grundy value of the game of creating an independent set in a graph. St001060Graphs ⟶ ℤ The distinguishing index of a graph. St001069Graphs ⟶ ℤ The coefficient of the monomial xy of the Tutte polynomial of the graph. St001070Graphs ⟶ ℤ The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001071Graphs ⟶ ℤ The beta invariant of the graph. St001072Graphs ⟶ ℤ The evaluation of the Tutte polynomial of the graph at x and y equal to 3. St001073Graphs ⟶ ℤ The number of nowhere zero 3-flows of a graph. St001093Graphs ⟶ ℤ The detour number of a graph. St001108Graphs ⟶ ℤ The 2-dynamic chromatic number of a graph. St001109Graphs ⟶ ℤ The number of proper colourings of a graph with as few colours as possible. St001110Graphs ⟶ ℤ The 3-dynamic chromatic number of a graph. St001111Graphs ⟶ ℤ The weak 2-dynamic chromatic number of a graph. St001112Graphs ⟶ ℤ The 3-weak dynamic number of a graph. St001116Graphs ⟶ ℤ The game chromatic number of a graph. St001117Graphs ⟶ ℤ The game chromatic index of a graph. St001118Graphs ⟶ ℤ The acyclic chromatic index of a graph. St001119Graphs ⟶ ℤ The length of a shortest maximal path in a graph. St001120Graphs ⟶ ℤ The length of a longest path in a graph. St001261Graphs ⟶ ℤ The Castelnuovo-Mumford regularity of a graph. St001270Graphs ⟶ ℤ The bandwidth of a graph. St001271Graphs ⟶ ℤ The competition number of a graph. St001272Graphs ⟶ ℤ The number of graphs with the same degree sequence. St001277Graphs ⟶ ℤ The degeneracy of a graph. St001281Graphs ⟶ ℤ The normalized isoperimetric number of a graph. St001282Graphs ⟶ ℤ The number of graphs with the same chromatic polynomial. St001286Graphs ⟶ ℤ The annihilation number of a graph. St001302Graphs ⟶ ℤ The number of minimally dominating sets of vertices of a graph. St001303Graphs ⟶ ℤ The number of dominating sets of vertices of a graph. St001304Graphs ⟶ ℤ The number of maximally independent sets of vertices of a graph. St001305Graphs ⟶ ℤ The number of induced cycles on four vertices in a graph. St001306Graphs ⟶ ℤ The number of induced paths on four vertices in a graph. St001307Graphs ⟶ ℤ The number of induced stars on four vertices in a graph. St001308Graphs ⟶ ℤ The number of induced paths on three vertices in a graph. St001309Graphs ⟶ ℤ The number of four-cliques in a graph. St001310Graphs ⟶ ℤ The number of induced diamond graphs in a graph. St001311Graphs ⟶ ℤ The cyclomatic number of a graph. St001315Graphs ⟶ ℤ The dissociation number of a graph. St001316Graphs ⟶ ℤ The domatic number of a graph. St001317Graphs ⟶ ℤ The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001318Graphs ⟶ ℤ The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001319Graphs ⟶ ℤ The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph. St001320Graphs ⟶ ℤ The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001321Graphs ⟶ ℤ The number of vertices of the largest induced subforest of a graph. St001322Graphs ⟶ ℤ The size of a minimal independent dominating set in a graph. St001323Graphs ⟶ ℤ The independence gap of a graph. St001324Graphs ⟶ ℤ The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325Graphs ⟶ ℤ The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326Graphs ⟶ ℤ The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001327Graphs ⟶ ℤ The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001328Graphs ⟶ ℤ The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329Graphs ⟶ ℤ The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001330Graphs ⟶ ℤ The hat guessing number of a graph. St001331Graphs ⟶ ℤ The size of the minimal feedback vertex set. St001333Graphs ⟶ ℤ The cardinality of a minimal edge-isolating set of a graph. St001334Graphs ⟶ ℤ The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001335Graphs ⟶ ℤ The cardinality of a minimal cycle-isolating set of a graph. St001336Graphs ⟶ ℤ The minimal number of vertices in a graph whose complement is triangle-free. St001337Graphs ⟶ ℤ The upper domination number of a graph. St001338Graphs ⟶ ℤ The upper irredundance number of a graph. St001339Graphs ⟶ ℤ The irredundance number of a graph. St001340Graphs ⟶ ℤ The cardinality of a minimal non-edge isolating set of a graph. St001341Graphs ⟶ ℤ The number of edges in the center of a graph. St001342Graphs ⟶ ℤ The number of vertices in the center of a graph. St001345Graphs ⟶ ℤ The Hamming dimension of a graph. St001347Graphs ⟶ ℤ The number of pairs of vertices of a graph having the same neighbourhood. St001349Graphs ⟶ ℤ The number of different graphs obtained from the given graph by removing an edge. St001350Graphs ⟶ ℤ Half of the Albertson index of a graph. St001351Graphs ⟶ ℤ The Albertson index of a graph. St001352Graphs ⟶ ℤ The number of internal nodes in the modular decomposition of a graph. St001353Graphs ⟶ ℤ The number of prime nodes in the modular decomposition of a graph. St001354Graphs ⟶ ℤ The number of series nodes in the modular decomposition of a graph. St001356Graphs ⟶ ℤ The number of vertices in prime modules of a graph. St001357Graphs ⟶ ℤ The maximal degree of a regular spanning subgraph of a graph. St001358Graphs ⟶ ℤ The largest degree of a regular subgraph of a graph. St001362Graphs ⟶ ℤ The normalized Knill dimension of a graph. St001363Graphs ⟶ ℤ The Euler characteristic of a graph according to Knill. St001366Graphs ⟶ ℤ The maximal multiplicity of a degree of a vertex of a graph. St001367Graphs ⟶ ℤ The smallest number which does not occur as degree of a vertex in a graph. St001368Graphs ⟶ ℤ The number of vertices of maximal degree in a graph. St001373Graphs ⟶ ℤ The logarithm of the number of winning configurations of the lights out game on a graph. St001374Graphs ⟶ ℤ The Padmakar-Ivan index of a graph. St001386Graphs ⟶ ℤ The number of prime labellings of a graph. St001391Graphs ⟶ ℤ The disjunction number of a graph. St001393Graphs ⟶ ℤ The induced matching number of a graph. St001395Graphs ⟶ ℤ The number of strictly unfriendly partitions of a graph. St001441Graphs ⟶ ℤ The number of non-empty connected induced subgraphs of a graph. St001458Graphs ⟶ ℤ The rank of the adjacency matrix of a graph. St001459Graphs ⟶ ℤ The number of zero columns in the nullspace of a graph. St001463Graphs ⟶ ℤ The number of distinct columns in the nullspace of a graph. St001474Graphs ⟶ ℤ The evaluation of the Tutte polynomial of the graph at (x,y) equal to (2,-1). St001475Graphs ⟶ ℤ The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001476Graphs ⟶ ℤ The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001477Graphs ⟶ ℤ The number of nowhere zero 5-flows of a graph. St001478Graphs ⟶ ℤ The number of nowhere zero 4-flows of a graph. St001479Graphs ⟶ ℤ The number of bridges of a graph. St001494Graphs ⟶ ℤ The Alon-Tarsi number of a graph. St001496Graphs ⟶ ℤ The number of graphs with the same Laplacian spectrum as the given graph. St001512Graphs ⟶ ℤ The minimum rank of a graph. St001518Graphs ⟶ ℤ The number of graphs with the same ordinary spectrum as the given graph. St001521Graphs ⟶ ℤ Half the total irregularity of a graph. St001522Graphs ⟶ ℤ The total irregularity of a graph. Integer compositions (43 statistics) # partition like objects St000008Integer compositions ⟶ ℤ The major index of the composition. St000047Integer compositions ⟶ ℤ The number of standard immaculate tableaux of a given shape. St000089Integer compositions ⟶ ℤ The absolute variation of a composition. St000090Integer compositions ⟶ ℤ The variation of a composition. St000091Integer compositions ⟶ ℤ The descent variation of a composition. St000277Integer compositions ⟶ ℤ The number of ribbon shaped standard tableaux. St000285Integer compositions ⟶ ℤ The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000381Integer compositions ⟶ ℤ The largest part of an integer composition. St000382Integer compositions ⟶ ℤ The first part of an integer composition. St000383Integer compositions ⟶ ℤ The last part of an integer composition. St000657Integer compositions ⟶ ℤ The smallest part of an integer composition. St000757Integer compositions ⟶ ℤ The length of the longest weakly inreasing subsequence of parts of an integer composition. St000758Integer compositions ⟶ ℤ The length of the longest staircase fitting into an integer composition. St000760Integer compositions ⟶ ℤ The length of the longest strictly decreasing subsequence of parts of an integer composition. St000761Integer compositions ⟶ ℤ The number of ascents in an integer composition. St000762Integer compositions ⟶ ℤ The sum of the positions of the weak records of an integer composition. St000763Integer compositions ⟶ ℤ The sum of the positions of the strong records of an integer composition. St000764Integer compositions ⟶ ℤ The number of strong records in an integer composition. St000765Integer compositions ⟶ ℤ The number of weak records in an integer composition. St000766Integer compositions ⟶ ℤ The number of inversions of an integer composition. St000767Integer compositions ⟶ ℤ The number of runs in an integer composition. St000768Integer compositions ⟶ ℤ The number of peaks in an integer composition. St000769Integer compositions ⟶ ℤ The major index of a composition. St000805Integer compositions ⟶ ℤ The number of peaks of the associated bargraph. St000806Integer compositions ⟶ ℤ The semiperimeter of the associated bargraph. St000807Integer compositions ⟶ ℤ The sum of the heights of the valleys of the associated bargraph. St000808Integer compositions ⟶ ℤ The number of up steps of the associated bargraph. St000816Integer compositions ⟶ ℤ The number of standard composition tableaux of the composition. St000817Integer compositions ⟶ ℤ The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818Integer compositions ⟶ ℤ The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000820Integer compositions ⟶ ℤ The number of compositions obtained by rotating the composition. St000899Integer compositions ⟶ ℤ The maximal number of repetitions of an integer composition. St000900Integer compositions ⟶ ℤ The minimal number of repetitions of a part in an integer composition. St000902Integer compositions ⟶ ℤ The minimal number of repetitions of an integer composition. St000903Integer compositions ⟶ ℤ The number of different parts of an integer composition. St000904Integer compositions ⟶ ℤ The maximal number of repetitions of an integer composition. St000905Integer compositions ⟶ ℤ The number of different multiplicities of parts of an integer composition. St001102Integer compositions ⟶ ℤ The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. St001235Integer compositions ⟶ ℤ The global dimension of the corresponding Comp-Nakayama algebra. St001236Integer compositions ⟶ ℤ The dominant dimension of the corresponding Comp-Nakayama algebra. St001263Integer compositions ⟶ ℤ The index of the maximal parabolic seaweed algebra associated with the composition. St001312Integer compositions ⟶ ℤ Number of parabolic noncrossing partitions indexed by the composition. St001486Integer compositions ⟶ ℤ The number of corners of the ribbon associated with an integer composition. Integer partitions (180 statistics) # partition like objects St000003Integer partitions ⟶ ℤ The number of standard Young tableaux of the partition. St000010Integer partitions ⟶ ℤ The length of the partition. St000046Integer partitions ⟶ ℤ The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000048Integer partitions ⟶ ℤ The multinomial of the parts of a partition. St000049Integer partitions ⟶ ℤ The number of set partitions whose sorted block sizes correspond to the partition. St000063Integer partitions ⟶ ℤ The number of linear extensions of a certain poset defined for an integer partition. St000088Integer partitions ⟶ ℤ The row sums of the character table of the symmetric group. St000108Integer partitions ⟶ ℤ The number of partitions contained in the given partition. St000137Integer partitions ⟶ ℤ The Grundy value of an integer partition. St000142Integer partitions ⟶ ℤ The number of even parts of a partition. St000143Integer partitions ⟶ ℤ The largest repeated part of a partition. St000145Integer partitions ⟶ ℤ The Dyson rank of a partition. St000146Integer partitions ⟶ ℤ The Andrews-Garvan crank of a partition. St000147Integer partitions ⟶ ℤ The largest part of an integer partition. St000148Integer partitions ⟶ ℤ The number of odd parts of a partition. St000149Integer partitions ⟶ ℤ The number of cells of the partition whose leg is zero and arm is odd. St000150Integer partitions ⟶ ℤ The floored half-sum of the multiplicities of a partition. St000159Integer partitions ⟶ ℤ The number of distinct parts of the integer partition. St000160Integer partitions ⟶ ℤ The multiplicity of the smallest part of a partition. St000175Integer partitions ⟶ ℤ Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000179Integer partitions ⟶ ℤ The product of the hook lengths of the integer partition. St000182Integer partitions ⟶ ℤ The number of permutations whose cycle type is the given integer partition. St000183Integer partitions ⟶ ℤ The side length of the Durfee square of an integer partition. St000184Integer partitions ⟶ ℤ The size of the centralizer of any permutation of given cycle type. St000185Integer partitions ⟶ ℤ The weighted size of a partition. St000205Integer partitions ⟶ ℤ Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Integer partitions ⟶ ℤ Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000207Integer partitions ⟶ ℤ Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Integer partitions ⟶ ℤ Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000212Integer partitions ⟶ ℤ The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000225Integer partitions ⟶ ℤ Difference between largest and smallest parts in a partition. St000228Integer partitions ⟶ ℤ The size of a partition. St000256Integer partitions ⟶ ℤ The number of parts from which one can substract 2 and still get an integer partition. St000257Integer partitions ⟶ ℤ The number of distinct parts of a partition that occur at least twice. St000275Integer partitions ⟶ ℤ Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000278Integer partitions ⟶ ℤ The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000284Integer partitions ⟶ ℤ The Plancherel distribution on integer partitions. St000318Integer partitions ⟶ ℤ The number of addable cells of the Ferrers diagram of an integer partition. St000319Integer partitions ⟶ ℤ The spin of an integer partition. St000320Integer partitions ⟶ ℤ The dinv adjustment of an integer partition. St000321Integer partitions ⟶ ℤ The number of integer partitions of n that are dominated by an integer partition. St000345Integer partitions ⟶ ℤ The number of refinements of a partition. St000346Integer partitions ⟶ ℤ The number of coarsenings of a partition. St000377Integer partitions ⟶ ℤ The dinv defect of an integer partition. St000378Integer partitions ⟶ ℤ The diagonal inversion number of an integer partition. St000380Integer partitions ⟶ ℤ Half the perimeter of the largest rectangle that fits inside the diagram of an integer partition. St000384Integer partitions ⟶ ℤ The maximal part of the shifted composition of an integer partition. St000459Integer partitions ⟶ ℤ The hook length of the base cell of a partition. St000460Integer partitions ⟶ ℤ The hook length of the last cell along the main diagonal of an integer partition. St000473Integer partitions ⟶ ℤ The number of parts of a partition that are strictly bigger than the number of ones. St000474Integer partitions ⟶ ℤ Dyson's crank of a partition. St000475Integer partitions ⟶ ℤ The number of parts equal to 1 in a partition. St000477Integer partitions ⟶ ℤ The weight of a partition according to Alladi. St000478Integer partitions ⟶ ℤ Another weight of a partition according to Alladi. St000480Integer partitions ⟶ ℤ The number of lower covers of a partition in dominance order. St000481Integer partitions ⟶ ℤ The number of upper covers of a partition in dominance order. St000506Integer partitions ⟶ ℤ The number of standard desarrangement tableaux of shape equal to the given partition. St000509Integer partitions ⟶ ℤ The diagonal index (content) of a partition. St000510Integer partitions ⟶ ℤ The number of invariant oriented cycles when acting with a permutation of given cycle type. St000511Integer partitions ⟶ ℤ The number of invariant subsets when acting with a permutation of given cycle type. St000512Integer partitions ⟶ ℤ The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000513Integer partitions ⟶ ℤ The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000514Integer partitions ⟶ ℤ The number of invariant simple graphs when acting with a permutation of given cycle type. St000515Integer partitions ⟶ ℤ The number of invariant set partitions when acting with a permutation of given cycle type. St000517Integer partitions ⟶ ℤ The Kreweras number of an integer partition. St000531Integer partitions ⟶ ℤ The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St000532Integer partitions ⟶ ℤ The total number of rook placements on a Ferrers board. St000533Integer partitions ⟶ ℤ The maximal number of non-attacking rooks on a Ferrers shape. St000547Integer partitions ⟶ ℤ The number of even non-empty partial sums of an integer partition. St000548Integer partitions ⟶ ℤ The number of different non-empty partial sums of an integer partition. St000549Integer partitions ⟶ ℤ The number of odd partial sums of an integer partition. St000566Integer partitions ⟶ ℤ The number of ways to select a row of a Ferrers shape and two cells in this row. St000567Integer partitions ⟶ ℤ The sum of the products of all pairs of parts. St000618Integer partitions ⟶ ℤ The number of self-evacuating tableaux of given shape. St000620Integer partitions ⟶ ℤ The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621Integer partitions ⟶ ℤ The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000644Integer partitions ⟶ ℤ The number of graphs with given frequency partition. St000667Integer partitions ⟶ ℤ The greatest common divisor of the parts of the partition. St000668Integer partitions ⟶ ℤ The least common multiple of the parts of the partition. St000681Integer partitions ⟶ ℤ The Grundy value of Chomp on Ferrers diagrams. St000697Integer partitions ⟶ ℤ The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000698Integer partitions ⟶ ℤ The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704Integer partitions ⟶ ℤ The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000705Integer partitions ⟶ ℤ The number of semistandard tableaux on a given integer partition of n with maximal entry n. St000706Integer partitions ⟶ ℤ The product of the factorials of the multiplicities of an integer partition. St000707Integer partitions ⟶ ℤ The product of the factorials of the parts. St000708Integer partitions ⟶ ℤ The product of the parts of an integer partition. St000712Integer partitions ⟶ ℤ The number of semistandard Young tableau of given shape, with entries at most 4. St000713Integer partitions ⟶ ℤ The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714Integer partitions ⟶ ℤ The number of semistandard Young tableau of given shape, with entries at most 2. St000715Integer partitions ⟶ ℤ The number of semistandard Young tableaux of given shape and entries at most 3. St000716Integer partitions ⟶ ℤ The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St000749Integer partitions ⟶ ℤ The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000752Integer partitions ⟶ ℤ The Grundy value for the game 'Couples are forever' on an integer partition. St000755Integer partitions ⟶ ℤ The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000759Integer partitions ⟶ ℤ The smallest missing part in an integer partition. St000770Integer partitions ⟶ ℤ The major index of an integer partition when read from bottom to top. St000781Integer partitions ⟶ ℤ The number of proper colouring schemes of a Ferrers diagram. St000783Integer partitions ⟶ ℤ The side length of the largest staircase partition fitting into a partition. St000784Integer partitions ⟶ ℤ The maximum of the length and the largest part of the integer partition. St000810Integer partitions ⟶ ℤ The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions. St000811Integer partitions ⟶ ℤ The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions. St000812Integer partitions ⟶ ℤ The sum of the entries in the column specified by the partition of the change of basis matrix from complete homogeneous symmetric functions to monomial symmetric functions. St000813Integer partitions ⟶ ℤ The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000814Integer partitions ⟶ ℤ The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000815Integer partitions ⟶ ℤ The number of semistandard Young tableaux of partition weight of given shape. St000835Integer partitions ⟶ ℤ The minimal difference in size when partitioning the integer partition into two subpartitions. St000867Integer partitions ⟶ ℤ The sum of the hook lengths in the first column of an integer partition. St000869Integer partitions ⟶ ℤ The sum of the hook lengths of an integer partition. St000870Integer partitions ⟶ ℤ The product of the hook lengths of the diagonal cells in an integer partition. St000897Integer partitions ⟶ ℤ The number of different multiplicities of parts of an integer partition. St000901Integer partitions ⟶ ℤ The cube of the number of standard Young tableaux with shape given by the partition. St000913Integer partitions ⟶ ℤ The number of ways to refine the partition into singletons. St000927Integer partitions ⟶ ℤ The alternating sum of the coefficients of the character polynomial of an integer partition. St000928Integer partitions ⟶ ℤ The sum of the coefficients of the character polynomial of an integer partition. St000929Integer partitions ⟶ ℤ The constant term of the character polynomial of an integer partition. St000933Integer partitions ⟶ ℤ The number of multipartitions of sizes given by an integer partition. St000934Integer partitions ⟶ ℤ The 2-degree of an integer partition. St000935Integer partitions ⟶ ℤ The number of ordered refinements of an integer partition. St000936Integer partitions ⟶ ℤ The number of even values of the symmetric group character corresponding to the partition. St000937Integer partitions ⟶ ℤ The number of positive values of the symmetric group character corresponding to the partition. St000938Integer partitions ⟶ ℤ The number of zeros of the symmetric group character corresponding to the partition. St000939Integer partitions ⟶ ℤ The number of characters of the symmetric group whose value on the partition is positive. St000940Integer partitions ⟶ ℤ The number of characters of the symmetric group whose value on the partition is zero. St000941Integer partitions ⟶ ℤ The number of characters of the symmetric group whose value on the partition is even. St000944Integer partitions ⟶ ℤ The 3-degree of an integer partition. St000992Integer partitions ⟶ ℤ The alternating sum of the parts of an integer partition. St000993Integer partitions ⟶ ℤ The multiplicity of the largest part of an integer partition. St000995Integer partitions ⟶ ℤ The largest even part of an integer partition. St000997Integer partitions ⟶ ℤ The even-odd crank of an integer partition. St001055Integer partitions ⟶ ℤ The Grundy value for the game of removing cells of a row in an integer partition. St001091Integer partitions ⟶ ℤ The number of parts in an integer partition whose next smaller part has the same size. St001092Integer partitions ⟶ ℤ The number of distinct even parts of a partition. St001097Integer partitions ⟶ ℤ The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098Integer partitions ⟶ ℤ The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099Integer partitions ⟶ ℤ The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100Integer partitions ⟶ ℤ The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101Integer partitions ⟶ ℤ The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001103Integer partitions ⟶ ℤ The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001121Integer partitions ⟶ ℤ The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001122Integer partitions ⟶ ℤ The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001123Integer partitions ⟶ ℤ The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001124Integer partitions ⟶ ℤ The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001127Integer partitions ⟶ ℤ The sum of the squares of the parts of a partition. St001128Integer partitions ⟶ ℤ The exponens consonantiae of a partition. St001129Integer partitions ⟶ ℤ The product of the squares of the parts of a partition. St001175Integer partitions ⟶ ℤ The size of a partition minus the hook length of the base cell. St001176Integer partitions ⟶ ℤ The size of a partition minus its first part. St001177Integer partitions ⟶ ℤ Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Integer partitions ⟶ ℤ Twelve times the variance of the major index among all standard Young tableaux of a partition. St001214Integer partitions ⟶ ℤ The aft of an integer partition. St001247Integer partitions ⟶ ℤ The number of parts of a partition that are not congruent 2 modulo 3. St001248Integer partitions ⟶ ℤ Sum of the even parts of a partition. St001249Integer partitions ⟶ ℤ Sum of the odd parts of a partition. St001250Integer partitions ⟶ ℤ The number of parts of a partition that are not congruent 0 modulo 3. St001251Integer partitions ⟶ ℤ The number of parts of a partition that are not congruent 1 modulo 3. St001252Integer partitions ⟶ ℤ Half the sum of the even parts of a partition. St001262Integer partitions ⟶ ℤ The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001279Integer partitions ⟶ ℤ The sum of the parts of an integer partition that are at least two. St001280Integer partitions ⟶ ℤ The number of parts of an integer partition that are at least two. St001283Integer partitions ⟶ ℤ The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284Integer partitions ⟶ ℤ The number of finite groups that are realised by the given partition over the complex numbers. St001360Integer partitions ⟶ ℤ The number of covering relations in Young's lattice below a partition. St001364Integer partitions ⟶ ℤ The number of permutations whose cube equals a fixed permutation of given cycle type. St001378Integer partitions ⟶ ℤ The product of the cohook lengths of the integer partition. St001380Integer partitions ⟶ ℤ The number of monomer-dimer tilings of a Ferrers diagram. St001382Integer partitions ⟶ ℤ The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001383Integer partitions ⟶ ℤ The BG-rank of an integer partition. St001384Integer partitions ⟶ ℤ The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001385Integer partitions ⟶ ℤ The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001387Integer partitions ⟶ ℤ Number of SYT of the skew shape determined by adding one more box in the first n columns. St001389Integer partitions ⟶ ℤ The number of partitions of the same length below the given integer partition. St001392Integer partitions ⟶ ℤ The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001400Integer partitions ⟶ ℤ The total number of Littlewood-Richardson tableaux of given shape. St001432Integer partitions ⟶ ℤ The global dimension of the partition. St001440Integer partitions ⟶ ℤ The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001442Integer partitions ⟶ ℤ The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001484Integer partitions ⟶ ℤ The number of parts that appear precisely once in an integer partition. St001525Integer partitions ⟶ ℤ The number of symmetric hooks on the diagonal of a partition. St001527Integer partitions ⟶ ℤ The cyclic permutation representation number of an integer partition. Ordered trees (23 statistics) # tree like structures # Catalan objects # graph like objects St000084Ordered trees ⟶ ℤ The number of subtrees. St000085Ordered trees ⟶ ℤ The number of linear extensions of the tree. St000094Ordered trees ⟶ ℤ The depth of an ordered tree. St000166Ordered trees ⟶ ℤ The depth minus 1 of an ordered tree. St000167Ordered trees ⟶ ℤ The number of leaves of an ordered tree. St000168Ordered trees ⟶ ℤ The number of internal nodes of an ordered tree. St000328Ordered trees ⟶ ℤ The maximum number of child nodes in a tree. St000397Ordered trees ⟶ ℤ The Strahler number of a rooted tree. St000400Ordered trees ⟶ ℤ The path length of an ordered tree. St000410Ordered trees ⟶ ℤ The tree factorial of an ordered tree. St000413Ordered trees ⟶ ℤ The number of ordered trees with the same underlying unordered tree. St000415Ordered trees ⟶ ℤ The size of the automorphism group of the rooted tree underlying the ordered tree. St000416Ordered trees ⟶ ℤ The number of inequivalent increasing trees of an ordered tree. St000417Ordered trees ⟶ ℤ The size of the automorphism group of the ordered tree. St000521Ordered trees ⟶ ℤ The number of distinct subtrees of an ordered tree. St000522Ordered trees ⟶ ℤ The number of 1-protected nodes of a rooted tree. St000523Ordered trees ⟶ ℤ The number of 2-protected nodes of a rooted tree. St000679Ordered trees ⟶ ℤ The pruning number of an ordered tree. St000700Ordered trees ⟶ ℤ The protection number of an ordered tree. St000973Ordered trees ⟶ ℤ The length of the boundary of an ordered tree. St000974Ordered trees ⟶ ℤ The length of the trunk of an ordered tree. St000975Ordered trees ⟶ ℤ The length of the boundary minus the length of the trunk of an ordered tree. St001058Ordered trees ⟶ ℤ The breadth of the ordered tree. Parking functions (10 statistics) # word like objects # path like objects St000135Parking functions ⟶ ℤ The number of lucky cars of the parking function. St000136Parking functions ⟶ ℤ The dinv of a parking function. St000165Parking functions ⟶ ℤ Sum of the entries. St000188Parking functions ⟶ ℤ The area of the Dyck path corresponding to a parking function. St000194Parking functions ⟶ ℤ The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function. St000195Parking functions ⟶ ℤ The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000540Parking functions ⟶ ℤ The sum of the entries of a parking function minus its length. St000942Parking functions ⟶ ℤ The number of critical left to right maxima of the parking functions. St000943Parking functions ⟶ ℤ The number of spots the most unlucky car had to go further in a parking function. St001209Parking functions ⟶ ℤ The pmaj statistic of a parking function. Perfect matchings (39 statistics) # graph like objects St000041Perfect matchings ⟶ ℤ The number of nestings of a perfect matching. St000042Perfect matchings ⟶ ℤ The number of crossings of a perfect matching. St000043Perfect matchings ⟶ ℤ The number of crossings plus two-nestings of a perfect matching. St000044Perfect matchings ⟶ ℤ The number of vertices of the unicellular map given by a perfect matching. St000164Perfect matchings ⟶ ℤ The number of short pairs. St000719Perfect matchings ⟶ ℤ The number of alignments in a perfect matching. St000720Perfect matchings ⟶ ℤ The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000721Perfect matchings ⟶ ℤ The sum of the partition sizes in the oscillating tableau corresponding to a perfect matching. St000746Perfect matchings ⟶ ℤ The number of pairs with odd minimum in a perfect matching. St000754Perfect matchings ⟶ ℤ The Grundy value for the game of removing nestings in a perfect matching. St000780Perfect matchings ⟶ ℤ The size of the orbit under rotation of a perfect matching. St000782Perfect matchings ⟶ ℤ The indicator function of whether a given perfect matching is an L & P matching. St000787Perfect matchings ⟶ ℤ The number of flips required to make a perfect matching noncrossing. St000788Perfect matchings ⟶ ℤ The number of nesting-similar perfect matchings of a perfect matching. St000789Perfect matchings ⟶ ℤ The number of crossing-similar perfect matchings of a perfect matching. St000819Perfect matchings ⟶ ℤ The propagating number of a perfect matching. St000838Perfect matchings ⟶ ℤ The number of terminal right-hand endpoints when the vertices are written in order. St000840Perfect matchings ⟶ ℤ The number of closers smaller than the largest opener in a perfect matching. St000841Perfect matchings ⟶ ℤ The largest opener of a perfect matching. St000843Perfect matchings ⟶ ℤ The decomposition number of a perfect matching. St000924Perfect matchings ⟶ ℤ The number of topologically connected components of a perfect matching. St000945Perfect matchings ⟶ ℤ The number of matchings in the dihedral orbit of a perfect matching. St001040Perfect matchings ⟶ ℤ The depth of the decreasing labelled binary unordered tree associated with the perfect matching. St001041Perfect matchings ⟶ ℤ The depth of the label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001042Perfect matchings ⟶ ℤ The size of the automorphism group of the leaf labelled binary unordered tree associated with the perfect matching. St001043Perfect matchings ⟶ ℤ The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001044Perfect matchings ⟶ ℤ The number of pairs whose larger element is at most one more than half the size of the perfect matching. St001045Perfect matchings ⟶ ℤ The number of leaves in the subtree not containing one in the decreasing labelled binary unordered tree associated with the perfect matching. St001046Perfect matchings ⟶ ℤ The maximal number of arcs nesting a given arc of a perfect matching. St001047Perfect matchings ⟶ ℤ The maximal number of arcs crossing a given arc of a perfect matching. St001048Perfect matchings ⟶ ℤ The number of leaves in the subtree containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001049Perfect matchings ⟶ ℤ The smallest label in the subtree not containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001131Perfect matchings ⟶ ℤ The number of trivial trees on the path to label one in the decreasing labelled binary unordered tree associated with the perfect matching. St001132Perfect matchings ⟶ ℤ The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001133Perfect matchings ⟶ ℤ The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001134Perfect matchings ⟶ ℤ The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St001136Perfect matchings ⟶ ℤ The largest label with larger sister in the leaf labelled binary unordered tree associated with the perfect matching. St001152Perfect matchings ⟶ ℤ The number of pairs with even minimum in a perfect matching. St001444Perfect matchings ⟶ ℤ The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. Permutations (337 statistics) # word like objects St000001Permutations ⟶ ℤ The number of reduced words for a permutation. St000002Permutations ⟶ ℤ The number of occurrences of the pattern 123 in a permutation. St000004Permutations ⟶ ℤ The major index of a permutation. St000007Permutations ⟶ ℤ The number of saliances of the permutation. St000018Permutations ⟶ ℤ The number of inversions of a permutation. St000019Permutations ⟶ ℤ The cardinality of the support of a permutation. St000020Permutations ⟶ ℤ The rank of the permutation. St000021Permutations ⟶ ℤ The number of descents of a permutation. St000022Permutations ⟶ ℤ The number of fixed points of a permutation. St000023Permutations ⟶ ℤ The number of inner peaks of a permutation. St000028Permutations ⟶ ℤ The number of stack-sorts needed to sort a permutation. St000029Permutations ⟶ ℤ The depth of a permutation. St000030Permutations ⟶ ℤ The sum of the descent differences of a permutations. St000031Permutations ⟶ ℤ The number of cycles in the cycle decomposition of a permutation. St000033Permutations ⟶ ℤ The number of permutations greater than or equal to the given permutation in (strong) Bruhat order. St000034Permutations ⟶ ℤ The maximum defect over any reduced expression for a permutation and any subexpression. St000035Permutations ⟶ ℤ The number of left outer peaks of a permutation. St000036Permutations ⟶ ℤ The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000037Permutations ⟶ ℤ The sign of a permutation. St000039Permutations ⟶ ℤ The number of crossings of a permutation. St000040Permutations ⟶ ℤ The number of regions of inversion arrangement of a permutation. St000054Permutations ⟶ ℤ The first entry of the permutation. St000055Permutations ⟶ ℤ The inversion sum of a permutation. St000056Permutations ⟶ ℤ The decomposition (or block) number of a permutation. St000058Permutations ⟶ ℤ The order of a permutation. St000060Permutations ⟶ ℤ The greater neighbor of the maximum. St000062Permutations ⟶ ℤ The length of the longest increasing subsequence of the permutation. St000064Permutations ⟶ ℤ The number of one-box pattern of a permutation. St000078Permutations ⟶ ℤ The number of alternating sign matrices whose left key is the permutation. St000092Permutations ⟶ ℤ The number of outer peaks of a permutation. St000099Permutations ⟶ ℤ The number of valleys of a permutation, including the boundary. St000109Permutations ⟶ ℤ The number of elements less than or equal to the given element in Bruhat order. St000110Permutations ⟶ ℤ The number of permutations less than or equal to a permutation in left weak order. St000111Permutations ⟶ ℤ The sum of the descent tops (or Genocchi descents) of a permutation. St000119Permutations ⟶ ℤ The number of occurrences of the pattern 321 in a permutation. St000123Permutations ⟶ ℤ The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000124Permutations ⟶ ℤ The cardinality of the preimage of the Simion-Schmidt map. St000133Permutations ⟶ ℤ The "bounce" of a permutation. St000141Permutations ⟶ ℤ The maximum drop size of a permutation. St000153Permutations ⟶ ℤ The number of adjacent cycles of a permutation. St000154Permutations ⟶ ℤ The sum of the descent bottoms of a permutation. St000155Permutations ⟶ ℤ The number of exceedances (also excedences) of a permutation. St000156Permutations ⟶ ℤ The Denert index of a permutation. St000162Permutations ⟶ ℤ The number of nontrivial cycles in the cycle decomposition of a permutation. St000209Permutations ⟶ ℤ Maximum difference of elements in cycles. St000210Permutations ⟶ ℤ Minimum over maximum difference of elements in cycles. St000213Permutations ⟶ ℤ The number of weak exceedances (also weak excedences) of a permutation. St000214Permutations ⟶ ℤ The number of adjacencies of a permutation. St000215Permutations ⟶ ℤ The number of adjacencies of a permutation, zero appended. St000216Permutations ⟶ ℤ The absolute length of a permutation. St000217Permutations ⟶ ℤ The number of occurrences of the pattern 312 in a permutation. St000218Permutations ⟶ ℤ The number of occurrences of the pattern 213 in a permutation. St000219Permutations ⟶ ℤ The number of occurrences of the pattern 231 in a permutation. St000220Permutations ⟶ ℤ The number of occurrences of the pattern 132 in a permutation. St000221Permutations ⟶ ℤ The number of strong fixed points of a permutation. St000222Permutations ⟶ ℤ The number of alignments in the permutation. St000223Permutations ⟶ ℤ The number of nestings in the permutation. St000224Permutations ⟶ ℤ The sorting index of a permutation. St000226Permutations ⟶ ℤ The convexity of a permutation. St000234Permutations ⟶ ℤ The number of global ascents of a permutation. St000235Permutations ⟶ ℤ The number of indices $i$ such that $\pi_i \neq i+1$ considered cyclically. St000236Permutations ⟶ ℤ The number of indices $i$ such that $\pi_i \in \{ i,i+1 \}$ considered cyclically. St000237Permutations ⟶ ℤ The number of small exceedances. St000238Permutations ⟶ ℤ The number of indices $i$ such that $\pi_i \notin \{i,i+1\}$. St000239Permutations ⟶ ℤ The number of indices $i$ such that $\pi_i \in \{i,i+1\}$. St000240Permutations ⟶ ℤ The number of indices $i$ for which $\pi_i \neq i+1$. St000241Permutations ⟶ ℤ The number of indices $i$ such that $\pi_i = i+1$ considered cyclically. St000242Permutations ⟶ ℤ The number of indices $i$ such that $\pi_i \notin \{ i,i+1 \}$ considered cyclically. St000243Permutations ⟶ ℤ The number of cyclic valleys and cyclic peaks of a permutation. St000245Permutations ⟶ ℤ The number of ascents of a permutation. St000246Permutations ⟶ ℤ The number of non-inversions of a permutation. St000255Permutations ⟶ ℤ The number of reduced Kogan faces with the permutation as type. St000279Permutations ⟶ ℤ The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000280Permutations ⟶ ℤ The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000304Permutations ⟶ ℤ St000305Permutations ⟶ ℤ The inverse major index of a permutation. St000308Permutations ⟶ ℤ The height of the tree associated to a permutation. St000314Permutations ⟶ ℤ The number of left-to-right-maxima of a permutation. St000316Permutations ⟶ ℤ The number of non-left-to-right-maxima of a permutation. St000317Permutations ⟶ ℤ The cycle descent number of a permutation. St000324Permutations ⟶ ℤ The shape of the tree associated to a permutation. St000325Permutations ⟶ ℤ The width of the tree associated to a permutation. St000333Permutations ⟶ ℤ The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000334Permutations ⟶ ℤ The maz index, the major index of a permutation after replacing fixed points by zeros. St000337Permutations ⟶ ℤ The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000338Permutations ⟶ ℤ The number of pixed points of a permutation. St000339Permutations ⟶ ℤ The maf index of a permutation. St000341Permutations ⟶ ℤ The non-inversion sum of a permutation. St000342Permutations ⟶ ℤ The cosine of a permutation. St000352Permutations ⟶ ℤ The Elizalde-Pak rank of a permutation. St000353Permutations ⟶ ℤ The number of inner valleys of a permutation. St000354Permutations ⟶ ℤ The number of recoils of a permutation. St000355Permutations ⟶ ℤ The number of occurrences of the pattern 21-3. St000356Permutations ⟶ ℤ The number of occurrences of the pattern 13-2. St000357Permutations ⟶ ℤ The number of occurrences of the pattern 12-3. St000358Permutations ⟶ ℤ The number of occurrences of the pattern 31-2. St000359Permutations ⟶ ℤ The number of occurrences of the pattern 23-1. St000360Permutations ⟶ ℤ The number of occurrences of the pattern 32-1. St000365Permutations ⟶ ℤ The number of double ascents of a permutation. St000366Permutations ⟶ ℤ The number of double descents of a permutation. St000367Permutations ⟶ ℤ The number of simsun double descents of a permutation. St000371Permutations ⟶ ℤ The number of mid points of decreasing subsequences of length 3 in a permutation. St000372Permutations ⟶ ℤ The number of mid points of increasing subsequences of length 3 in a permutation. St000373Permutations ⟶ ℤ The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000374Permutations ⟶ ℤ The number of exclusive right-to-left minima of a permutation. St000375Permutations ⟶ ℤ The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000401Permutations ⟶ ℤ The size of the symmetry class of a permutation. St000402Permutations ⟶ ℤ Half the size of the symmetry class of a permutation. St000404Permutations ⟶ ℤ The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000405Permutations ⟶ ℤ The number of occurrences of the pattern 1324 in a permutation. St000406Permutations ⟶ ℤ The number of occurrences of the pattern 3241 in a permutation. St000407Permutations ⟶ ℤ The number of occurrences of the pattern 2143 in a permutation. St000408Permutations ⟶ ℤ The number of occurrences of the pattern 4231 in a permutation. St000423Permutations ⟶ ℤ The number of occurrences of the pattern 123 or of the pattern 132 in a permutation. St000424Permutations ⟶ ℤ The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000425Permutations ⟶ ℤ The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000426Permutations ⟶ ℤ The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000427Permutations ⟶ ℤ The number of occurrences of the pattern 123 or of the pattern 231 in a permutation. St000428Permutations ⟶ ℤ The number of occurrences of the pattern 123 or of the pattern 213 in a permutation. St000429Permutations ⟶ ℤ The number of occurrences of the pattern 123 or of the pattern 321 in a permutation. St000430Permutations ⟶ ℤ The number of occurrences of the pattern 123 or of the pattern 312 in a permutation. St000431Permutations ⟶ ℤ The number of occurrences of the pattern 213 or of the pattern 321 in a permutation. St000432Permutations ⟶ ℤ The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000433Permutations ⟶ ℤ The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St000434Permutations ⟶ ℤ The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000435Permutations ⟶ ℤ The number of occurrences of the pattern 213 or of the pattern 231 in a permutation. St000436Permutations ⟶ ℤ The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000437Permutations ⟶ ℤ The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000440Permutations ⟶ ℤ The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000441Permutations ⟶ ℤ The number of successions of a permutation. St000446Permutations ⟶ ℤ The disorder of a permutation. St000451Permutations ⟶ ℤ The length of the longest pattern of the form k 1 2. St000457Permutations ⟶ ℤ The number of occurrences of one of the patterns 132, 213 or 321 in a permutation. St000458Permutations ⟶ ℤ The number of permutations obtained by switching adjacencies or successions. St000461Permutations ⟶ ℤ The rix statistic of a permutation. St000462Permutations ⟶ ℤ The major index minus the number of excedences of a permutation. St000463Permutations ⟶ ℤ The number of admissible inversions of a permutation. St000470Permutations ⟶ ℤ The number of runs in a permutation. St000471Permutations ⟶ ℤ The sum of the ascent tops of a permutation. St000472Permutations ⟶ ℤ The sum of the ascent bottoms of a permutation. St000483Permutations ⟶ ℤ The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000484Permutations ⟶ ℤ The sum of St000483 over all subsequences of length at least three. St000485Permutations ⟶ ℤ The length of the longest cycle of a permutation. St000486Permutations ⟶ ℤ The number of cycles of length at least 3 of a permutation. St000487Permutations ⟶ ℤ The length of the shortest cycle of a permutation. St000488Permutations ⟶ ℤ The number of cycles of a permutation of length at most 2. St000489Permutations ⟶ ℤ The number of cycles of a permutation of length at most 3. St000494Permutations ⟶ ℤ The number of inversions of distance at most 3 of a permutation. St000495Permutations ⟶ ℤ The number of inversions of distance at most 2 of a permutation. St000500Permutations ⟶ ℤ Eigenvalues of the random-to-random operator acting on the regular representation. St000501Permutations ⟶ ℤ The size of the first part in the decomposition of a permutation. St000516Permutations ⟶ ℤ The number of stretching pairs of a permutation. St000520Permutations ⟶ ℤ The number of patterns in a permutation. St000530Permutations ⟶ ℤ The number of permutations with the same descent word as the given permutation. St000534Permutations ⟶ ℤ The number of 2-rises of a permutation. St000538Permutations ⟶ ℤ The number of even inversions of a permutation. St000539Permutations ⟶ ℤ The number of odd inversions of a permutation. St000541Permutations ⟶ ℤ The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000542Permutations ⟶ ℤ The number of left-to-right-minima of a permutation. St000545Permutations ⟶ ℤ The number of parabolic double cosets with minimal element being the given permutation. St000546Permutations ⟶ ℤ The number of global descents of a permutation. St000570Permutations ⟶ ℤ The Edelman-Greene number of a permutation. St000616Permutations ⟶ ℤ The inversion index of a permutation. St000619Permutations ⟶ ℤ The number of cyclic descents of a permutation. St000622Permutations ⟶ ℤ The number of occurrences of the patterns 2143 or 4231 in a permutation. St000623Permutations ⟶ ℤ The number of occurrences of the pattern 52341 in a permutation. St000624Permutations ⟶ ℤ The normalized sum of the minimal distances to a greater element. St000625Permutations ⟶ ℤ The sum of the minimal distances to a greater element. St000638Permutations ⟶ ℤ The number of up-down runs of a permutation. St000646Permutations ⟶ ℤ The number of big ascents of a permutation. St000647Permutations ⟶ ℤ The number of big descents of a permutation. St000648Permutations ⟶ ℤ The number of 2-excedences of a permutation. St000649Permutations ⟶ ℤ The number of 3-excedences of a permutation. St000650Permutations ⟶ ℤ The number of 3-rises of a permutation. St000651Permutations ⟶ ℤ The maximal size of a rise in a permutation. St000652Permutations ⟶ ℤ The maximal difference between successive positions of a permutation. St000653Permutations ⟶ ℤ The last descent of a permutation. St000654Permutations ⟶ ℤ The first descent of a permutation. St000662Permutations ⟶ ℤ The staircase size of the code of a permutation. St000663Permutations ⟶ ℤ The number of right floats of a permutation. St000664Permutations ⟶ ℤ The number of right ropes of a permutation. St000665Permutations ⟶ ℤ The number of rafts of a permutation. St000666Permutations ⟶ ℤ The number of right tethers of a permutation. St000669Permutations ⟶ ℤ The number of permutations obtained by switching ascents or descents of size 2. St000670Permutations ⟶ ℤ The reversal length of a permutation. St000672Permutations ⟶ ℤ The number of minimal elements in Bruhat order not less than the permutation. St000673Permutations ⟶ ℤ The size of the support of a permutation. St000677Permutations ⟶ ℤ The standardized bi-alternating inversion number of a permutation. St000690Permutations ⟶ ℤ The size of the conjugacy class of a permutation. St000692Permutations ⟶ ℤ Babson and Steingrímsson's statistic stat of a permutation. St000694Permutations ⟶ ℤ The number of affine bounded permutations that project to a given permutation. St000696Permutations ⟶ ℤ The number of cycles in the breakpoint graph of a permutation. St000702Permutations ⟶ ℤ The number of weak deficiencies of a permutation. St000703Permutations ⟶ ℤ The number of deficiencies of a permutation. St000709Permutations ⟶ ℤ The number of occurrences of 14-2-3 or 14-3-2. St000710Permutations ⟶ ℤ The number of big deficiencies of a permutation. St000711Permutations ⟶ ℤ The number of big exceedences of a permutation. St000724Permutations ⟶ ℤ The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000725Permutations ⟶ ℤ The smallest label of a leaf of the increasing binary tree associated to a permutation. St000726Permutations ⟶ ℤ The normalized sum of the leaf labels of the increasing binary tree associated to a permutation. St000727Permutations ⟶ ℤ The largest label of a leaf in the binary search tree associated with the permutation. St000731Permutations ⟶ ℤ The number of double exceedences of a permutation. St000732Permutations ⟶ ℤ The number of double deficiencies of a permutation. St000740Permutations ⟶ ℤ The last entry of a permutation. St000742Permutations ⟶ ℤ The number of big ascents of a permutation after prepending zero. St000750Permutations ⟶ ℤ The number of occurrences of the pattern 4213 in a permutation. St000751Permutations ⟶ ℤ The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000756Permutations ⟶ ℤ The sum of the positions of the left to right maxima of a permutation. St000779Permutations ⟶ ℤ The tier of a permutation. St000794Permutations ⟶ ℤ The mak of a permutation. St000795Permutations ⟶ ℤ St000796Permutations ⟶ ℤ The stat' of a permutation. St000797Permutations ⟶ ℤ The stat of a permutation. St000798Permutations ⟶ ℤ The makl of a permutation. St000799Permutations ⟶ ℤ The number of occurrences of the vincular pattern \|213 in a permutation. St000800Permutations ⟶ ℤ The number of occurrences of the vincular pattern \|231 in a permutation. St000801Permutations ⟶ ℤ The number of occurrences of the vincular pattern \|312 in a permutation. St000802Permutations ⟶ ℤ The number of occurrences of the vincular pattern \|321 in a permutation. St000803Permutations ⟶ ℤ The number of occurrences of the vincular pattern \|132 in a permutation. St000804Permutations ⟶ ℤ The number of occurrences of the vincular pattern \|123 in a permutation. St000809Permutations ⟶ ℤ The reduced reflection length of the permutation. St000824Permutations ⟶ ℤ The sum of the number of descents and the number of recoils of a permutation. St000825Permutations ⟶ ℤ The sum of the major and the inverse major index of a permutation. St000828Permutations ⟶ ℤ The spearman's rho of a permutation and the identity permutation. St000829Permutations ⟶ ℤ The Ulam distance of a permutation to the identity permutation. St000830Permutations ⟶ ℤ The total displacement of a permutation. St000831Permutations ⟶ ℤ The number of indices that are either descents or recoils. St000832Permutations ⟶ ℤ The number of permutations obtained by reversing blocks of three consecutive numbers. St000833Permutations ⟶ ℤ The comajor index of a permutation. St000834Permutations ⟶ ℤ The number of right outer peaks of a permutation. St000836Permutations ⟶ ℤ The number of descents of distance 2 of a permutation. St000837Permutations ⟶ ℤ The number of ascents of distance 2 of a permutation. St000842Permutations ⟶ ℤ St000844Permutations ⟶ ℤ The size of the largest block in the direct sum decomposition of a permutation. St000862Permutations ⟶ ℤ The number of parts of the shifted shape of a permutation. St000863Permutations ⟶ ℤ The length of the first row of the shifted shape of a permutation. St000864Permutations ⟶ ℤ The number of circled entries of the shifted recording tableau of a permutation. St000866Permutations ⟶ ℤ The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000868Permutations ⟶ ℤ The aid statistic in the sense of Shareshian-Wachs. St000871Permutations ⟶ ℤ The number of very big ascents of a permutation. St000872Permutations ⟶ ℤ The number of very big descents of a permutation. St000873Permutations ⟶ ℤ The aix statistic of a permutation. St000879Permutations ⟶ ℤ The number of long braid edges in the graph of braid moves of a permutation. St000880Permutations ⟶ ℤ The number of connected components of long braid edges in the graph of braid moves of a permutation. St000881Permutations ⟶ ℤ The number of short braid edges in the graph of braid moves of a permutation. St000882Permutations ⟶ ℤ The number of connected components of short braid edges in the graph of braid moves of a permutation. St000883Permutations ⟶ ℤ The number of longest increasing subsequences of a permutation. St000884Permutations ⟶ ℤ The number of isolated descents of a permutation. St000886Permutations ⟶ ℤ The number of permutations with the same antidiagonal sums. St000887Permutations ⟶ ℤ The maximal number of nonzero entries on a diagonal of a permutation matrix. St000891Permutations ⟶ ℤ The number of distinct diagonal sums of a permutation matrix. St000923Permutations ⟶ ℤ The minimal number with no two order isomorphic substrings of this length in a permutation. St000956Permutations ⟶ ℤ The maximal displacement of a permutation. St000957Permutations ⟶ ℤ The number of Bruhat lower covers of a permutation. St000958Permutations ⟶ ℤ The number of Bruhat factorizations of a permutation. St000959Permutations ⟶ ℤ The number of strong Bruhat factorizations of a permutation. St000961Permutations ⟶ ℤ The shifted major index of a permutation. St000962Permutations ⟶ ℤ The 3-shifted major index of a permutation. St000963Permutations ⟶ ℤ The 2-shifted major index of a permutation. St000988Permutations ⟶ ℤ The orbit size of a permutation under Foata's bijection. St000989Permutations ⟶ ℤ The number of final rises of a permutation. St000990Permutations ⟶ ℤ The first ascent of a permutation. St000991Permutations ⟶ ℤ The number of right-to-left minima of a permutation. St000994Permutations ⟶ ℤ The number of cycle peaks and the number of cycle valleys of a permutation. St000996Permutations ⟶ ℤ The number of exclusive left-to-right maxima of a permutation. St001004Permutations ⟶ ℤ The number of indices that are either left-to-right maxima or right-to-left minima. St001005Permutations ⟶ ℤ The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001052Permutations ⟶ ℤ The length of the exterior of a permutation. St001059Permutations ⟶ ℤ Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001061Permutations ⟶ ℤ The number of indices that are both descents and recoils of a permutation. St001074Permutations ⟶ ℤ The number of inversions of the cyclic embedding of a permutation. St001076Permutations ⟶ ℤ The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001077Permutations ⟶ ℤ The prefix exchange distance of a permutation. St001078Permutations ⟶ ℤ The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001079Permutations ⟶ ℤ The minimal length of a factorization of a permutation using the permutations (12)(34). St001080Permutations ⟶ ℤ The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. St001081Permutations ⟶ ℤ The number of minimal length factorizations of a permutation into star transpositions. St001082Permutations ⟶ ℤ The number of boxed occurrences of 123 in a permutation. St001083Permutations ⟶ ℤ The number of boxed occurrences of 132 in a permutation. St001084Permutations ⟶ ℤ The number of occurrences of the vincular pattern \|1-23 in a permutation. St001085Permutations ⟶ ℤ The number of occurrences of the vincular pattern \|21-3 in a permutation. St001086Permutations ⟶ ℤ The number of occurrences of the consecutive pattern 132 in a permutation. St001087Permutations ⟶ ℤ The number of occurrences of the vincular pattern \|12-3 in a permutation. St001090Permutations ⟶ ℤ The number of pop-stack-sorts needed to sort a permutation. St001096Permutations ⟶ ℤ The size of the overlap set of a permutation. St001114Permutations ⟶ ℤ The number of odd descents of a permutation. St001115Permutations ⟶ ℤ The number of even descents of a permutation. St001130Permutations ⟶ ℤ The number of two successive successions in a permutation. St001160Permutations ⟶ ℤ The number of proper blocks (or intervals) of a permutations. St001162Permutations ⟶ ℤ The minimum jump of a permutation. St001168Permutations ⟶ ℤ The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001171Permutations ⟶ ℤ The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001174Permutations ⟶ ℤ The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001207Permutations ⟶ ℤ The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001208Permutations ⟶ ℤ The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001220Permutations ⟶ ℤ The width of a permutation. St001245Permutations ⟶ ℤ The cyclic maximal difference between two consecutive entries of a permutation. St001246Permutations ⟶ ℤ The maximal difference between two consecutive entries of a permutation. St001269Permutations ⟶ ℤ The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001285Permutations ⟶ ℤ The number of primes in the column sums of the two line notation of a permutation. St001287Permutations ⟶ ℤ The number of primes obtained by multiplying preimage and image of a permutation and subtracting one. St001288Permutations ⟶ ℤ The number of primes obtained by multiplying preimage and image of a permutation and adding one. St001293Permutations ⟶ ℤ The sum of all $1/(i+\pi(i))$ for a permutation $\pi$ times the lcm of all possible values among permutations of the same length. St001298Permutations ⟶ ℤ The number of repeated entries in the Lehmer code of a permutation. St001332Permutations ⟶ ℤ The number of steps on the non-negative side of the walk associated with the permutation. St001344Permutations ⟶ ℤ The neighbouring number of a permutation. St001346Permutations ⟶ ℤ The number of parking functions that give the same permutation. St001359Permutations ⟶ ℤ The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001375Permutations ⟶ ℤ The pancake length of a permutation. St001377Permutations ⟶ ℤ The major index minus the number of inversions of a permutation. St001379Permutations ⟶ ℤ The number of inversions plus the major index of a permutation. St001381Permutations ⟶ ℤ The fertility of a permutation. St001388Permutations ⟶ ℤ The number of non-attacking neighbors of a permutation. St001390Permutations ⟶ ℤ The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001394Permutations ⟶ ℤ The genus of a permutation. St001402Permutations ⟶ ℤ The number of separators in a permutation. St001403Permutations ⟶ ℤ The number of vertical separators in a permutation. St001405Permutations ⟶ ℤ The number of bonds in a permutation. St001411Permutations ⟶ ℤ The number of patterns 321 or 3412 in a permutation. St001412Permutations ⟶ ℤ Number of minimal entries in the Bruhat order matrix of a permutation. St001439Permutations ⟶ ℤ The number of even deficiencies and of odd exceedences. St001461Permutations ⟶ ℤ The number of topologically connected components of the chord diagram of a permutation. St001464Permutations ⟶ ℤ The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001465Permutations ⟶ ℤ The number of adjacent transpositions in the cycle decomposition of a permutation. St001466Permutations ⟶ ℤ The number of transpositions swapping cyclically adjacent numbers in a permutation. St001468Permutations ⟶ ℤ The smallest fixpoint of a permutation. St001469Permutations ⟶ ℤ The holeyness of a permutation. St001470Permutations ⟶ ℤ The cyclic holeyness of a permutation. St001482Permutations ⟶ ℤ The product of the prefix sums of a permutation. St001489Permutations ⟶ ℤ The maximum of the number of descents and the number of inverse descents. St001497Permutations ⟶ ℤ The position of the largest weak excedence of a permutation. St001511Permutations ⟶ ℤ The minimal number of transpositions needed to sort a permutation in either direction. St001513Permutations ⟶ ℤ The number of nested exceedences of a permutation. St001516Permutations ⟶ ℤ The number of cyclic bonds of a permutation. St001517Permutations ⟶ ℤ The length of a longest pair of twins in a permutation. St001519Permutations ⟶ ℤ The pinnacle sum of a permutation. St001520Permutations ⟶ ℤ The number of strict-3-descents. Plane partitions (15 statistics) # partition like objects St001422Plane partitions ⟶ ℤ The number of boxes of a plane partition. St001445Plane partitions ⟶ ℤ The number of maximal boxes of a plane partition. St001446Plane partitions ⟶ ℤ Number of rows in the plane partition. St001447Plane partitions ⟶ ℤ Height of the base box of a plane partition. St001448Plane partitions ⟶ ℤ Number of odd parts in a plane partition. St001449Plane partitions ⟶ ℤ The smallest missing nonzero part in the plane partition. St001450Plane partitions ⟶ ℤ The minimum height of a plane partition. St001451Plane partitions ⟶ ℤ The side length of the largest cube contained in a plane partition. St001452Plane partitions ⟶ ℤ Number of even parts in the plane partition. St001453Plane partitions ⟶ ℤ The number of distinct heights in a plane partition. St001454Plane partitions ⟶ ℤ The difference between the largest and smallest heights of a plane partition. St001455Plane partitions ⟶ ℤ Largest repeated part of a plane partition, and zero if no part is repeated. St001456Plane partitions ⟶ ℤ Sum of the top row of a plane partition. St001457Plane partitions ⟶ ℤ Multiplicity of the smallest part of a plane partition. St001460Plane partitions ⟶ ℤ Number of columns of a plane partition. Posets (61 statistics) # graph like objects St000068Posets ⟶ ℤ The number of minimal elements in a poset. St000069Posets ⟶ ℤ The number of maximal elements of a poset. St000070Posets ⟶ ℤ The number of antichains in a poset. St000071Posets ⟶ ℤ The number of maximal chains in a poset. St000080Posets ⟶ ℤ The rank of the poset. St000100Posets ⟶ ℤ The number of linear extensions of a poset. St000104Posets ⟶ ℤ The number of facets in the order polytope of this poset. St000151Posets ⟶ ℤ The number of facets in the chain polytope of the poset. St000180Posets ⟶ ℤ The number of chains of a poset. St000181Posets ⟶ ℤ The number of connected components of the Hasse diagram for the poset. St000189Posets ⟶ ℤ The number of elements in the poset. St000281Posets ⟶ ℤ The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282Posets ⟶ ℤ The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000298Posets ⟶ ℤ The order dimension or Dushnik-Miller dimension of a poset. St000307Posets ⟶ ℤ The number of rowmotion orbits of a poset. St000327Posets ⟶ ℤ The number of cover relations in a poset. St000524Posets ⟶ ℤ The number of posets with the same order polynomial. St000525Posets ⟶ ℤ The number of posets with the same zeta polynomial. St000526Posets ⟶ ℤ The number of posets with combinatorially isomorphic order polytopes. St000527Posets ⟶ ℤ The width of the poset. St000528Posets ⟶ ℤ The height of a poset. St000550Posets ⟶ ℤ The number of modular elements of a lattice. St000551Posets ⟶ ℤ The number of left modular elements of a lattice. St000632Posets ⟶ ℤ The jump number of the poset. St000633Posets ⟶ ℤ The size of the automorphism group of a poset. St000634Posets ⟶ ℤ The number of endomorphisms of a poset. St000635Posets ⟶ ℤ The number of strictly order preserving maps of a poset into itself. St000639Posets ⟶ ℤ The number of relations in a poset. St000640Posets ⟶ ℤ The rank of the largest boolean interval in a poset. St000641Posets ⟶ ℤ The number of non-empty boolean intervals in a poset. St000642Posets ⟶ ℤ The size of the smallest orbit of antichains under Panyushev complementation. St000643Posets ⟶ ℤ The size of the largest orbit of antichains under Panyushev complementation. St000656Posets ⟶ ℤ The number of cuts of a poset. St000680Posets ⟶ ℤ The Grundy value for Hackendot on posets. St000717Posets ⟶ ℤ The number of ordinal summands of a poset. St000845Posets ⟶ ℤ The maximal number of elements covered by an element in a poset. St000846Posets ⟶ ℤ The maximal number of elements covering an element of a poset. St000848Posets ⟶ ℤ The balance constant multiplied with the number of linear extensions of a poset. St000849Posets ⟶ ℤ The number of 1/3-balanced pairs in a poset. St000850Posets ⟶ ℤ The number of 1/2-balanced pairs in a poset. St000906Posets ⟶ ℤ The length of the shortest maximal chain in a poset. St000907Posets ⟶ ℤ The number of maximal antichains of minimal length in a poset. St000908Posets ⟶ ℤ The length of the shortest maximal antichain in a poset. St000909Posets ⟶ ℤ The number of maximal chains of maximal size in a poset. St000910Posets ⟶ ℤ The number of maximal chains of minimal length in a poset. St000911Posets ⟶ ℤ The number of maximal antichains of maximal size in a poset. St000912Posets ⟶ ℤ The number of maximal antichains in a poset. St000914Posets ⟶ ℤ The sum of the values of the Möbius function of a poset. St001095Posets ⟶ ℤ The number of non-isomorphic posets with precisely one further covering relation. St001105Posets ⟶ ℤ The number of greedy linear extensions of a poset. St001106Posets ⟶ ℤ The number of supergreedy linear extensions of a poset. St001268Posets ⟶ ℤ The size of the largest ordinal summand in the poset. St001300Posets ⟶ ℤ The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001301Posets ⟶ ℤ The first Betti number of the order complex associated with the poset. St001343Posets ⟶ ℤ The dimension of the reduced incidence algebra of a poset. St001396Posets ⟶ ℤ Number of triples of incomparable elements in a finite poset. St001397Posets ⟶ ℤ Number of pairs of incomparable elements in a finite poset. St001398Posets ⟶ ℤ Number of subsets of size 3 of elements in a poset that form a "v". St001399Posets ⟶ ℤ The distinguishing number of a poset. St001472Posets ⟶ ℤ The permanent of the Coxeter matrix of the poset. St001510Posets ⟶ ℤ The number of self-evacuating linear extensions of a finite poset. Semistandard tableaux (16 statistics) # tableau like objects St000101Semistandard tableaux ⟶ ℤ The cocharge of a semistandard tableau. St000102Semistandard tableaux ⟶ ℤ The charge of a semistandard tableau. St000103Semistandard tableaux ⟶ ℤ The sum of the entries of a semistandard tableau. St000112Semistandard tableaux ⟶ ℤ The depth of a semistandard tableau $T$ in the crystal $B(\lambda)$ where $\lambda$ is the shape of $T$. St000116Semistandard tableaux ⟶ ℤ The major index of a semistandard tableau obtained by standardizing. St000170Semistandard tableaux ⟶ ℤ The trace of a semistandard tableau. St000173Semistandard tableaux ⟶ ℤ The segment statistic of a semistandard tableau. St000174Semistandard tableaux ⟶ ℤ The flush statistic of a semistandard tableau. St000736Semistandard tableaux ⟶ ℤ The last entry in the first row of a semistandard tableau. St000737Semistandard tableaux ⟶ ℤ The last entry on the main diagonal of a semistandard tableau. St000739Semistandard tableaux ⟶ ℤ The first entry in the last row of a semistandard tableau. St001401Semistandard tableaux ⟶ ℤ The number of distinct entries in a semistandard tableau. St001407Semistandard tableaux ⟶ ℤ The number of minimal entries in a semistandard tableau. St001408Semistandard tableaux ⟶ ℤ The number of maximal entries in a semistandard tableau. St001409Semistandard tableaux ⟶ ℤ The maximal entry of a semistandard tableau. St001410Semistandard tableaux ⟶ ℤ The minimal entry of a semistandard tableau. Set partitions (101 statistics) # partition like objects St000105Set partitions ⟶ ℤ The number of blocks in the set partition. St000163Set partitions ⟶ ℤ The size of the orbit of the set partition under rotation. St000211Set partitions ⟶ ℤ The rank of the set partition. St000229Set partitions ⟶ ℤ Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St000230Set partitions ⟶ ℤ Sum of the minimal elements of the blocks of a set partition. St000231Set partitions ⟶ ℤ Sum of the maximal elements of the blocks of a set partition. St000232Set partitions ⟶ ℤ The number of crossings of a set partition. St000233Set partitions ⟶ ℤ The number of nestings of a set partition. St000247Set partitions ⟶ ℤ The number of singleton blocks of a set partition. St000248Set partitions ⟶ ℤ The number of anti-singletons of a set partition. St000249Set partitions ⟶ ℤ The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000250Set partitions ⟶ ℤ The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000251Set partitions ⟶ ℤ The number of nonsingleton blocks of a set partition. St000253Set partitions ⟶ ℤ The crossing number of a set partition. St000254Set partitions ⟶ ℤ The nesting number of a set partition. St000490Set partitions ⟶ ℤ The intertwining number of a set partition. St000491Set partitions ⟶ ℤ The number of inversions of a set partition. St000492Set partitions ⟶ ℤ The rob statistic of a set partition. St000493Set partitions ⟶ ℤ The los statistic of a set partition. St000496Set partitions ⟶ ℤ The rcs statistic of a set partition. St000497Set partitions ⟶ ℤ The lcb statistic of a set partition. St000498Set partitions ⟶ ℤ The lcs statistic of a set partition. St000499Set partitions ⟶ ℤ The rcb statistic of a set partition. St000502Set partitions ⟶ ℤ The number of successions of a set partitions. St000503Set partitions ⟶ ℤ The maximal difference between two elements in a common block. St000504Set partitions ⟶ ℤ The cardinality of the first block of a set partition. St000505Set partitions ⟶ ℤ The biggest entry in the block containing the 1. St000554Set partitions ⟶ ℤ The number of occurrences of the pattern {{1,2},{3}} in a set partition. St000555Set partitions ⟶ ℤ The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000556Set partitions ⟶ ℤ The number of occurrences of the pattern {{1},{2,3}} in a set partition. St000557Set partitions ⟶ ℤ The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000558Set partitions ⟶ ℤ The number of occurrences of the pattern {{1,2}} in a set partition. St000559Set partitions ⟶ ℤ The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000560Set partitions ⟶ ℤ The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000561Set partitions ⟶ ℤ The number of occurrences of the pattern {{1,2,3}} in a set partition. St000562Set partitions ⟶ ℤ The number of internal points of a set partition. St000563Set partitions ⟶ ℤ The number of overlapping pairs of blocks of a set partition. St000564Set partitions ⟶ ℤ The number of occurrences of the pattern {{1},{2}} in a set partition. St000565Set partitions ⟶ ℤ The major index of a set partition. St000572Set partitions ⟶ ℤ The dimension exponent of a set partition. St000573Set partitions ⟶ ℤ The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000574Set partitions ⟶ ℤ The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal and 2 a maximal element. St000575Set partitions ⟶ ℤ The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000576Set partitions ⟶ ℤ The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000577Set partitions ⟶ ℤ The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000578Set partitions ⟶ ℤ The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000579Set partitions ⟶ ℤ The number of occurrences of the pattern {{1},{2}} such that 2 is a maximal element. St000580Set partitions ⟶ ℤ The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000581Set partitions ⟶ ℤ The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000582Set partitions ⟶ ℤ The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000583Set partitions ⟶ ℤ The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000584Set partitions ⟶ ℤ The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000585Set partitions ⟶ ℤ The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000586Set partitions ⟶ ℤ The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal. St000587Set partitions ⟶ ℤ The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000588Set partitions ⟶ ℤ The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000589Set partitions ⟶ ℤ The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000590Set partitions ⟶ ℤ The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000591Set partitions ⟶ ℤ The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000592Set partitions ⟶ ℤ The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593Set partitions ⟶ ℤ The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000594Set partitions ⟶ ℤ The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000595Set partitions ⟶ ℤ The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal. St000596Set partitions ⟶ ℤ The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000597Set partitions ⟶ ℤ The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000598Set partitions ⟶ ℤ The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, 3 is maximal, (2,3) are consecutive in a block. St000599Set partitions ⟶ ℤ The number of occurrences of the pattern {{1},{2,3}} such that (2,3) are consecutive in a block. St000600Set partitions ⟶ ℤ The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000601Set partitions ⟶ ℤ The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000602Set partitions ⟶ ℤ The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000603Set partitions ⟶ ℤ The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604Set partitions ⟶ ℤ The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000605Set partitions ⟶ ℤ The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) are consecutive in a block. St000606Set partitions ⟶ ℤ The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000607Set partitions ⟶ ℤ The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000608Set partitions ⟶ ℤ The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000609Set partitions ⟶ ℤ The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000610Set partitions ⟶ ℤ The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000611Set partitions ⟶ ℤ The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000612Set partitions ⟶ ℤ The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block. St000613Set partitions ⟶ ℤ The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000614Set partitions ⟶ ℤ The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000615Set partitions ⟶ ℤ The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000695Set partitions ⟶ ℤ The number of blocks in the first part of the atomic decomposition of a set partition. St000728Set partitions ⟶ ℤ The dimension of a set partition. St000729Set partitions ⟶ ℤ The minimal arc length of a set partition. St000730Set partitions ⟶ ℤ The maximal arc length of a set partition. St000747Set partitions ⟶ ℤ A variant of the major index of a set partition. St000748Set partitions ⟶ ℤ The major index of the permutation obtained by flattening the set partition. St000793Set partitions ⟶ ℤ The length of the longest partition in the vacillating tableau corresponding to a set partition. St000823Set partitions ⟶ ℤ The number of unsplittable factors of the set partition. St000839Set partitions ⟶ ℤ The largest opener of a set partition. St000925Set partitions ⟶ ℤ The number of topologically connected components of a set partition. St000971Set partitions ⟶ ℤ The smallest closer of a set partition. St001050Set partitions ⟶ ℤ The number of terminal closers of a set partition. St001051Set partitions ⟶ ℤ The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001062Set partitions ⟶ ℤ The maximal size of a block of a set partition. St001075Set partitions ⟶ ℤ The minimal size of a block of a set partition. St001094Set partitions ⟶ ℤ The depth index of a set partition. St001151Set partitions ⟶ ℤ The number of blocks with odd minimum. St001153Set partitions ⟶ ℤ The number of blocks with even minimum in a set partition. Signed permutations (6 statistics) # word like objects St001427Signed permutations ⟶ ℤ The number of descents of a signed permutation. St001428Signed permutations ⟶ ℤ The number of B-inversions of a signed permutation. St001429Signed permutations ⟶ ℤ The number of negative entries in a signed permutation. St001430Signed permutations ⟶ ℤ The number of positive entries in a signed permutation. St001433Signed permutations ⟶ ℤ The flag major index of a signed permutation. St001434Signed permutations ⟶ ℤ The number of negative sum pairs of a signed permutation. Skew partitions (5 statistics) # partition like objects St001435Skew partitions ⟶ ℤ The number of missing boxes in the first row. St001438Skew partitions ⟶ ℤ The number of missing boxes of a skew partition. St001487Skew partitions ⟶ ℤ The number of inner corners of a skew partition. St001488Skew partitions ⟶ ℤ The number of corners of a skew partition. St001490Skew partitions ⟶ ℤ The number of connected components of a skew partition. Standard tableaux (21 statistics) # tableau like objects St000009Standard tableaux ⟶ ℤ The charge of a standard tableau. St000016Standard tableaux ⟶ ℤ The number of attacking pairs of a standard tableau. St000017Standard tableaux ⟶ ℤ The number of inversions of a standard tableau. St000057Standard tableaux ⟶ ℤ The Shynar inversion number of a standard tableau. St000059Standard tableaux ⟶ ℤ The inversion number of a standard tableau as defined by Haglund and Stevens. St000075Standard tableaux ⟶ ℤ The orbit size of a standard tableau under promotion. St000157Standard tableaux ⟶ ℤ The number of descents of a standard tableau. St000169Standard tableaux ⟶ ℤ The cocharge of a standard tableau. St000330Standard tableaux ⟶ ℤ The (standard) major index of a standard tableau. St000336Standard tableaux ⟶ ℤ The leg major index of a standard tableau. St000507Standard tableaux ⟶ ℤ The number of ascents of a standard tableau. St000508Standard tableaux ⟶ ℤ Eigenvalues of the random-to-random operator acting on a simple module. St000693Standard tableaux ⟶ ℤ The modular (standard) major index of a standard tableau. St000733Standard tableaux ⟶ ℤ The row containing the largest entry of a standard tableau. St000734Standard tableaux ⟶ ℤ The last entry in the first row of a standard tableau. St000735Standard tableaux ⟶ ℤ The last entry on the main diagonal of a standard tableau. St000738Standard tableaux ⟶ ℤ The first entry in the last row of a standard tableau. St000743Standard tableaux ⟶ ℤ The number of entries in a standard Young tableau such that the next integer is a neighbour. St000744Standard tableaux ⟶ ℤ The length of the path to the largest entry in a standard Young tableau. St000745Standard tableaux ⟶ ℤ The index of the last row whose first entry is the row number in a standard Young tableau. St001462Standard tableaux ⟶ ℤ The number of factors of a standard tableaux under concatenation.	2020-03-28 12:15:21	{"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": 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": 0.6795340180397034, "perplexity": 3753.641100350863}, "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-2020-16/segments/1585370491857.4/warc/CC-MAIN-20200328104722-20200328134722-00451.warc.gz"}
https://events.interpore.org/event/2/sessions/109/	# InterPore2018 New Orleans 14-17 May 2018 New Orleans US/Central timezone ## Session ### Poster 3 16 May 2018, 17:15 New Orleans ### Presentation Materials There are no materials yet. Filip Formalik (Group of Bioprocess and Biomedical Engineering, Faculty of Chemistry, Wrocław University of Science and Technology, Department of Theoretical Physcis, Faculty of Fundamental Problems of Technology, Wrocław University of Science and Technology) 16/05/2018, 17:15 MS 2.15: Modelling and Simulation of Porous Media: From Microstructure to Functionality Poster Atomic level simulations, such as molecular dynamics (MD), Monte Carlo (MC) or DFT, have recently become indispensable tools for characterization of porous materials [1]. Computational methods are often used to study adsorption, diffusion or separation mechanisms. Here, we perform density functional theory (DFT) calculations of structural properties of flexible and rigid nanoporous MOFs. For... Massimo Rolle (Department of Environmental Engineering, Technical University of Denmark, Miljøvej, Building 115, 2800 Kgs. Lyngby, Denmark) 16/05/2018, 18:30 MS 1.11: Conservative and Reactive Transport of Charged Species in Permeable and Impermeable Porous Media Poster The transport of electrolytes in porous media is affected by physical, chemical and electrochemical processes. Coulombic interactions significantly influence the behavior of electrolyte plumes at different scales, not only in diffusion-dominated conditions but also in advection-dominated flow regimes [1-3]. To model the spatial behavior of charge-induced interactions in multi-dimensional... Mr CHAI Rukuan (MOE Key Laboratory of Petroleum Engineering / Petroleum Engineering, Beijing, China), Prof. LIU Yuetian (MOE Key Laboratory of Petroleum Engineering / Petroleum Engineering, Beijing, China), Mr WANG Junqiang (MOE Key Laboratory of Petroleum Engineering / Petroleum Engineering, Beijing, China), Ms XIN Jing (MOE Key Laboratory of Petroleum Engineering / Petroleum Engineering, Beijing, China) 16/05/2018, 18:30 MS 1.14: Transport in nanoporous materials. Theory and molecular dynamics simulations Poster Due to the size effect of nanochannels and the strong influence of nanochannel surfaces, these ultra-confined water and oil molecules behave extraordinary differently from their bulk counterpart. Therefore, it is of great necessary to study the adsorption and transport of water and oil molecules in the nanochannel. In this paper, we investigate the adsorption and transport mechanism of... Building timetable...	2021-10-21 02:00:31	{"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": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.24998241662979126, "perplexity": 6661.2035350668775}, "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/1634323585380.70/warc/CC-MAIN-20211021005314-20211021035314-00581.warc.gz"}
https://nrich.maths.org/public/topic.php?code=-68&cl=3&cldcmpid=797	# Resources tagged with: Visualising Filter by: Content type: Age range: Challenge level: ### There are 178 results Broad Topics > Thinking Mathematically > Visualising ### Tetra Square ##### Age 11 to 14 Challenge Level: ABCD is a regular tetrahedron and the points P, Q, R and S are the midpoints of the edges AB, BD, CD and CA. Prove that PQRS is a square. ### Eight Hidden Squares ##### Age 7 to 14 Challenge Level: On the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight hidden squares? ### Ten Hidden Squares ##### Age 7 to 14 Challenge Level: These points all mark the vertices (corners) of ten hidden squares. Can you find the 10 hidden squares? ### Just Opposite ##### Age 14 to 16 Challenge Level: A and C are the opposite vertices of a square ABCD, and have coordinates (a,b) and (c,d), respectively. What are the coordinates of the vertices B and D? What is the area of the square? ### Baravelle ##### Age 7 to 16 Challenge Level: What can you see? What do you notice? What questions can you ask? ### Dissect ##### Age 11 to 14 Challenge Level: What is the minimum number of squares a 13 by 13 square can be dissected into? ### Coloured Edges ##### Age 11 to 14 Challenge Level: The whole set of tiles is used to make a square. This has a green and blue border. There are no green or blue tiles anywhere in the square except on this border. How many tiles are there in the set? ### Something in Common ##### Age 14 to 16 Challenge Level: A square of area 3 square units cannot be drawn on a 2D grid so that each of its vertices have integer coordinates, but can it be drawn on a 3D grid? Investigate squares that can be drawn. ### Zooming in on the Squares ##### Age 7 to 14 Start with a large square, join the midpoints of its sides, you'll see four right angled triangles. Remove these triangles, a second square is left. Repeat the operation. What happens? ### The Perforated Cube ##### Age 14 to 16 Challenge Level: A cube is made from smaller cubes, 5 by 5 by 5, then some of those cubes are removed. Can you make the specified shapes, and what is the most and least number of cubes required ? ### Square It ##### Age 11 to 16 Challenge Level: Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square. ### Corridors ##### Age 14 to 16 Challenge Level: A 10x10x10 cube is made from 27 2x2 cubes with corridors between them. Find the shortest route from one corner to the opposite corner. ### A Tilted Square ##### Age 14 to 16 Challenge Level: The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices? ### Making Tracks ##### Age 14 to 16 Challenge Level: A bicycle passes along a path and leaves some tracks. Is it possible to say which track was made by the front wheel and which by the back wheel? ### Soma - So Good ##### Age 11 to 14 Challenge Level: Can you mentally fit the 7 SOMA pieces together to make a cube? Can you do it in more than one way? ### Thinking Through, and By, Visualising ##### Age 7 to 16 This article is based on some of the ideas that emerged during the production of a book which takes visualising as its focus. We began to identify problems which helped us to take a structured view. . . . ### Picture Story ##### Age 14 to 16 Challenge Level: Can you see how this picture illustrates the formula for the sum of the first six cube numbers? ### Concrete Wheel ##### Age 11 to 14 Challenge Level: A huge wheel is rolling past your window. What do you see? ### Chess ##### Age 11 to 14 Challenge Level: What would be the smallest number of moves needed to move a Knight from a chess set from one corner to the opposite corner of a 99 by 99 square board? ### On the Edge ##### Age 11 to 14 Challenge Level: If you move the tiles around, can you make squares with different coloured edges? ### Platonic Planet ##### Age 14 to 16 Challenge Level: Glarsynost lives on a planet whose shape is that of a perfect regular dodecahedron. Can you describe the shortest journey she can make to ensure that she will see every part of the planet? ### Square Coordinates ##### Age 11 to 14 Challenge Level: A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides? ### Natural Sum ##### Age 14 to 16 Challenge Level: The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural. . . . ### Route to Infinity ##### Age 11 to 14 Challenge Level: Can you describe this route to infinity? Where will the arrows take you next? ### Framed ##### Age 11 to 14 Challenge Level: Seven small rectangular pictures have one inch wide frames. The frames are removed and the pictures are fitted together like a jigsaw to make a rectangle of length 12 inches. Find the dimensions of. . . . ### Hypotenuse Lattice Points ##### Age 14 to 16 Challenge Level: The triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN? ##### Age 11 to 14 Challenge Level: Can you mark 4 points on a flat surface so that there are only two different distances between them? ### Tourism ##### Age 11 to 14 Challenge Level: If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable. ### Triangles Within Triangles ##### Age 14 to 16 Challenge Level: Can you find a rule which connects consecutive triangular numbers? ### Rati-o ##### Age 11 to 14 Challenge Level: Points P, Q, R and S each divide the sides AB, BC, CD and DA respectively in the ratio of 2 : 1. Join the points. What is the area of the parallelogram PQRS in relation to the original rectangle? ### 3D Treasure Hunt ##### Age 14 to 18 Challenge Level: Some treasure has been hidden in a three-dimensional grid! Can you work out a strategy to find it as efficiently as possible? ### Coordinate Patterns ##### Age 11 to 14 Challenge Level: Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead? ### All in the Mind ##### Age 11 to 14 Challenge Level: Imagine you are suspending a cube from one vertex and allowing it to hang freely. What shape does the surface of the water make around the cube? ### Building Tetrahedra ##### Age 14 to 16 Challenge Level: Can you make a tetrahedron whose faces all have the same perimeter? ### Wari ##### Age 14 to 16 Challenge Level: This is a simple version of an ancient game played all over the world. It is also called Mancala. What tactics will increase your chances of winning? ### Khun Phaen Escapes to Freedom ##### Age 11 to 14 Challenge Level: Slide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom. ### Sea Defences ##### Age 7 to 14 Challenge Level: These are pictures of the sea defences at New Brighton. Can you work out what a basic shape might be in both images of the sea wall and work out a way they might fit together? ### Changing Places ##### Age 14 to 16 Challenge Level: Place a red counter in the top left corner of a 4x4 array, which is covered by 14 other smaller counters, leaving a gap in the bottom right hand corner (HOME). What is the smallest number of moves. . . . ### Sliced ##### Age 14 to 16 Challenge Level: An irregular tetrahedron has two opposite sides the same length a and the line joining their midpoints is perpendicular to these two edges and is of length b. What is the volume of the tetrahedron? ### Convex Polygons ##### Age 11 to 14 Challenge Level: Show that among the interior angles of a convex polygon there cannot be more than three acute angles. ### Icosian Game ##### Age 11 to 14 Challenge Level: This problem is about investigating whether it is possible to start at one vertex of a platonic solid and visit every other vertex once only returning to the vertex you started at. ### Pattern Power ##### Age 5 to 14 Mathematics is the study of patterns. Studying pattern is an opportunity to observe, hypothesise, experiment, discover and create. ### Isosceles Triangles ##### Age 11 to 14 Challenge Level: Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw? ### Tetrahedra Tester ##### Age 11 to 14 Challenge Level: An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length? ### Cubes Within Cubes Revisited ##### Age 11 to 14 Challenge Level: Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need? ### Packing 3D Shapes ##### Age 14 to 16 Challenge Level: What 3D shapes occur in nature. How efficiently can you pack these shapes together? ### Steel Cables ##### Age 14 to 16 Challenge Level: Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions? ### 3D Stacks ##### Age 7 to 14 Challenge Level: Can you find a way of representing these arrangements of balls? ### Proximity ##### Age 14 to 16 Challenge Level: We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours. ### Auditorium Steps ##### Age 7 to 14 Challenge Level: What is the shape of wrapping paper that you would need to completely wrap this model?	2020-07-14 12:46:13	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3595753014087677, "perplexity": 1193.4936301024375}, "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/1593655880665.3/warc/CC-MAIN-20200714114524-20200714144524-00440.warc.gz"}
https://hmolpedia.com/page/Francois_Massieu	# Francois Massieu In existographies, Francois Massieu (123-59 BE) (1832-1896 ACM) (EPD:F0) (CR:26) (LH:11) (TL:38) was a French engineer, physicist, mineralogist, geologist, and philosopher, noted for his 1869 "On the Characteristic Functions of Various Fluids", which was the first systematic discourse on “thermodynamic potentials” (Ѻ) applied to bodies, specifically fluid bodies; some characterize his "Massieu function" as the Legendre transform of entropy; influential to Willard Gibbs, who adopted his Greek letter nomenclature for defining the various thermodynamic functions, as show in the pictured characteristic function notation table.[1] ## Overview ### Characteristic function See main: Characteristic function In 1869, Massieu, in his "On the Characteristic Functions of Various Fluids", gave, supposedly, the first systematic discourse on “thermodynamic potentials” (Ѻ) applied to bodies, specifically fluid bodies, which is shown below, between Clausius and Gibbs:[2][3] In 1876, Massieu published Thermodynamics: Memoir on the Characteristic Functions of Various Fluids and on the Theory of Vapors.[4] In 1876, Willard Gibbs, in his On the Equilibrium of Heterogeneous Substances, adopted Massieu's Greek letter nomenclature for defining the various thermodynamic functions, as show in the above characteristic function notation table.[5] Some characterize the so-called "Massieu function" as the Legendre transform of entropy. ### General In 1897, Edmund Nivoit, a French engineer, in his “Notice of the Life and Work of Massieu”, summarized Massieu as follows:[6] “It is especially in the field of thermodynamics that Massieu left a luminous trace of his passage. It is probable that he was encouraged to follow this path, where were also his personal tastes and habits of mind, by his colleague from the Faculty of Rennes, Athanase Dupre, who published a series of papers on the mechanical theory of heat, in the Annals of Chemistry and Physics. More than once it had recourse to the penetrating mind and keen Massieu to increase the rigor of the demonstrations or to shed light on some obscure point, he was sometimes even brought on his advice, abandon proposals hazardous. It also fully inserted in his memoirs of his collaborator two notes: one on the molecular attraction, the other on the work of complete disintegration, or total work required to separate molecules from each other, despite the forces they opposed. In 1870, Massieu presented, to the Academy of Sciences, a brief on the characteristic functions of the various fluids and value theory, which I will try to give a brief idea. Everyone agrees that the state of a body is completely defined when we know two of the three quantities that represent the volume of this body, its temperature and pressure on its surface. Any one of these quantities must be regarded, for each body, as a function of two others taken to the independent variables. Based on the fundamental principles of thermodynamics and choosing as variables, the volume and temperature or pressure and temperature, Massieu established by a simple calculation, an equation where both are exact differentials and from which he derived a function, independent of any hypothesis, he called the characteristic function of the body considered, because it implicitly contains all the thermodynamic properties of the body. It may indeed serve to express, either by itself or through partial derivatives, the pressure on the body or volume (as the independent variables that have been taken), the entropy of Clausius or function, internal energy, then by a little less simple calculations, the two specific heats, a constant pressure, the other at constant volume, the two coefficients of expansion at constant pressure and volume, the coefficient of compressibility . Massieu then applied his theory to the ideal gas, which follow the laws of Mariotte and Gay-Lussac, the saturated vapor and superheated vapor. For ideal gases, it naturally finds no new property, since the properties of these bodies are used to precisely determine the complete expressions of the fundamental principles of thermodynamics. For saturated vapor, it also reflected by an elegant analysis, formulas already known, including one that calculates the density of dry saturated steam under various pressures using the experimental data of Regnault. His aim in this analysis is to show the simplicity and generality of his method. Regarding the superheated steam, the elements of uncertainty abound, since the only experimental data that one has to determine the characteristic function, which is the value of the specific heat of water vapor varies with temperature and with the pressure. The author first establishes its formulas without any hypothesis, but to translate them into numbers, it is obliged to make various assumptions. He admits that the first specific heat of water vapor is constant and its value is equal to the figure obtained by Regnault in specific conditions, which amounts to assimilate in this report to a perfect gas vapors. This is a perfectly acceptable assumption for the practice, if we do not take a very rigorous. But the industry does not seek to obtain strong overheating, and rightly so, because, as Massieu shows the theory, they would be of little use in steam engines arranged like those we use. The almost unique advantage they bring is to reduce heat losses due to condensation in the cylinder. Just a small amount of heat used to superheat steam admitted, for a relatively large reduction in the expenditure of the steam. In a second case, the author represents the specific heat by a formula with three factors, it can be calculated using the known law of maximum tension. Just as this law has not quite a general character, since the integral contained in the formula that translates is caught between two specific limits, which are the saturation temperature and infinite temperature, we understand that can give the form of a function, but only the numerical value of some coefficients. This second hypothesis, however, provide much better results that approached those obtained with the previous, where specific heat is regarded as constant, and it should be preferred when the overheating becomes an inconsiderable. The characteristic function of a body is so good, as we see, the condensation of all its thermodynamic properties. It suffices to express a limited number of numerical coefficients, and when, or could obtain these data, one must choose among those that experience provides the easiest, the thermodynamic theory of this body is complete. She still has a valuable property which greatly increases its scientific value and gives a great philosophical interest. It is a kind of touchstone, with which to test any statement relating to the mechanical theory of heat and recognize if it is pure alloy. Thus Massieu has demonstrated the inaccuracy of some theorems formulated hastily by way of analogy and he did see that other, instead of corresponding to general truths, applicable only to body of a certain category. An eminent judge, Joseph Bertrand (Thermodynamique, 1887), did not hesitate to declare, in a report read to the Academy of Sciences July 25, 1870, that "the introduction of this function in formulas that summarize all the possible consequences of the two fundamental theorems for the theory seems a similar service and has almost equivalent to that given Clausius' theorem in relating the Carnot entropy. Shortly after the release of this remarkable memoir, which discontinued its author among the masters of thermodynamics, Massieu released a memo written to please a few people who were interested in his work, in which he gives a full statement of two fundamental principles, he had accepted without question and without the show: one, known as the principle of equivalence between heat and work, stated by Meyer and the other due to Carnot, which defines the influence of temperature on the thermal phenomena whose machines are registered. He thought, the great mathematician Lagrange, that if these formulas are very useful in the development of a science, they often interfere with the clarity of the exposition of its principles. There are about thirty years, when Massieu was engaged in his search for rational mechanics, the mechanical theory of heat was again applied only within narrow limits to the study of steam engines. In education we continued to rely mostly on two assumptions which are incorrect, however, was well demonstrated. They assumed that because the saturated vapor follow the laws of Mariotte and Gay-Lussac, who are already not absolutely rigorous for gas in perfect condition, and more than vapor, when they relax in a cylinder engine to remain without saturation or overheating occur or partial condensation. In fact many scholars, such as Clausius, Zeuner, Rankine, Combes, and Resal, have shown how practical implications could be drawn from the new theory and gave some numerical examples. But it had not yet established formulas that can be applied widely in construction workshops and permitting, is to discuss the pros and cons of various measures adopted in the machinery, or to sense the value changes that might be tempted to introduce. Massieu had planned to fill this gap and to publish a text of a rational theory of steam engines, based on the fundamental principles of thermodynamics. He possessed all the elements of this work he could do was to coordinate and establish numerical tables that would have been the translation of its most important formulas. This might have been such a work, developed by a man so admirably prepared, it is easily understood. The introduction and first two chapters, including the statement of principles, as well as studying the properties of gases and vapors are only completed. Clarity is dominant, too abstract calculations have been avoided with the utmost care, which makes reading easier for those people who have only the rudiments of calculus. The third chapter, on superheated steam, which was to be the original part of the book, is sketched.” In c.2016, the School of Space Science in Rennes, posted the following online general overview of Massieu, which seems to focus on his work a civil engineer, related to city drainage and hygiene:[7] Francois Massieu: father of the mains drainage. He does not have a street in Rennes yet the Rennes people owe him a lot. Running water and sewers, that's him! Born in 1832, in Vatteville (Seine-Inférieure), François Massieu received his secondary education in Rouen, before entering Polytechnique in 1851, then in 1853 at the Ecole des mines. Appointed mining engineer in Caen in 1859, he devoted his spare time to pure science studies and defended two mathematics theses in 1861 at the Sorbonne. The same year, he succeeded Durocher, the first holder of the chair of geology at the Faculty of Sciences of Rennes. He taught there for 26 years as a professor in charge of geology and mineralogy. We know of few geological publications, except the geological map of the department of Ille-et-Vilaine published in 1866. He is best known for his work on locomotive brakes which served as a reference for engineers. Having joined the City Council, he deals in particular with hygiene and sanitation issues, writes reports on important issues: water supply, creation of a sewer network, reconstruction high school. Water collection in the Loisance and Minette basins is also the case. Called in 1886 to Paris as chief engineer of the western railways, he gave up his teaching. Having become inspector general of mines and director of control of the railways of the East, he died in Paris in 1896.” ### Sirodot \| Photo confusion Of note, in 2015 someone added a photo of Simon Sirodot to the French Wikipedia article on Massieu. The issue wasn't noticed until Dec 2020, when Libb Thims noticed the confusion; hence, there exists a certain amount of ongoing photo-misatribution spread on the Internet, and in some books[8], in respect to this issue. ## Sways ### Influences Massieu was influenced by: Athanase Dupre (1808-1869). ### Influenced Massieu influenced: Willard Gibbs. ## Quotes ### Quotes \| On The following are quotes on Massieu: Massieu has shown how all the properties of a fluid ‘which are considered in thermodynamics’ may be deduced from a single function, which he calls a characteristic function of the fluid considered; he introduces two different functions of this kind, vis, a function of the temperature and volume, which he denotes by Ψ, and a function of the temperature and pressure, which he denotes by Ψ’; in both cases he considers a constant quantity (one kilogram) of the fluid, which is regarded as invariable in composition.” Willard Gibbs (1876), On the Equilibrium of Heterogeneous Substances [9] ### Quotes \| By The following are quotes by Massieu: “They cut the man into two parts, soul and body, the philosopher took one, the naturalist took the other; they both have worked, studied on their behalf, but have lost sight and we find ourselves today in the presence of a ‘duality’, convenient perhaps, but unwise, in that it overlooked the man to deal with only two elements that constitute it. But in doing so we run the risk of being wrong. If one wanted to know the chemical properties of water [${\displaystyle {\ce {H2O}}}$], seek it in those of oxygen [${\displaystyle {\ce {O2}}}$] and hydrogen [${\displaystyle {\ce {H2}}}$]? No, because he knows that there is little relationship between the characteristics of a substance and those of simple bodies which enter into its composition. To study humans, it is perhaps even more reserve, his corpse is certainly different from his living, his soul is a being whose morality tells us in existence, but whose philosophy can boast of acquire specific knowledge, since it can be studied in a free state, the revelation can only speak in this regard. But what science and philosophy can and should perhaps only study, is a man indivisible and tangible for us, where the angel and the beast are inseparable, which has a body and ailments, but also passions and faculties, such as intelligence, memory and reason.” — Francois Massieu (c.1869), “Speech deliver at the start of the Faculties of Rennes”[6] ## End matter ### References 1. Characteristic function notation table – Hmolpedia 2020. 2. Massieu, Francois. (1869). “On the Various Functions Characteristic of Fluids and on the Theory of Vapors” (Sur les Functions Caracteristiques des Divers Fluides et Sur la Theorie des Vapeurs), Comptes Rendus, 69: 858-62, 1057-61. 3. Thims, Libb. (2020). Human Chemical Thermodynamics — Chemical Thermodynamics Applied to the Humanities: Meaning, Morality, Purpose; Sociology, Economics, Ecology; History, Philosophy, Government, Anthropology, Politics, Business, Jurisprudence; Religion, Relationships, Warfare, and Love (§2: Alphabet) (pdf). Publisher. 4. Massieu, Francois. (1876). Thermodynamics: Memoir on the Characteristic Functions of Various Fluids and on the Theory of Vapors (Thermodynamique: Mêmoire sur les fonctions catactéristiques des divers fluides et sur la théorie des vapeurs ) (92-pgs) (WC)(GB). Académie des Sciences de L'Institut National de France. 5. Characteristic function notation table – Hmolpedia 2020. 6. Nivoit, Edmond. (1897). “Notice of the Life and Work of Mr. Massieu, Inspector General of Mines” (French → English), Annales des Mines, 9th Series, Volume. 11. 7. Francois Massieu (French → English) – School of Space Sciences in Rennes. 8. Splinter, Robert. (2017). Illustrated Encyclopedia of Applied Engineering Physics (Massieu [photo of Simon Sirodot shown], pg. 336). CRC. 9. Gibbs, Willard. (1876). On the Equilibrium of Heterogeneous Substances (Massieu, pgs. 86, 358). 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https://altexploit.wordpress.com/tag/rings/	# Categories of Pointwise Convergence Topology: Theory(ies) of Bundles. Let H be a fixed, separable Hilbert space of dimension ≥ 1. Lets denote the associated projective space of H by P = P(H). It is compact iff H is finite-dimensional. Let PU = PU(H) = U(H)/U(1) be the projective unitary group of H equipped with the compact-open topology. A projective bundle over X is a locally trivial bundle of projective spaces, i.e., a fibre bundle P → X with fibre P(H) and structure group PU(H). An application of the Banach-Steinhaus theorem shows that we may identify projective bundles with principal PU(H)-bundles and the pointwise convergence topology on PU(H). If G is a topological group, let GX denote the sheaf of germs of continuous functions G → X, i.e., the sheaf associated to the constant presheaf given by U → F(U) = G. Given a projective bundle P → X and a sufficiently fine good open cover {Ui}i∈I of X, the transition functions between trivializations P\|Ui can be lifted to bundle isomorphisms gij on double intersections Uij = Ui ∩ Uj which are projectively coherent, i.e., over each of the triple intersections Uijk = Ui ∩ Uj ∩ Uk the composition gki gjk gij is given as multiplication by a U(1)-valued function fijk : Uijk → U(1). The collection {(Uij, fijk)} defines a U(1)-valued two-cocycle called a B-field on X,which represents a class BP in the sheaf cohomology group H2(X, U(1)X). On the other hand, the sheaf cohomology H1(X, PU(H)X) consists of isomorphism classes of principal PU(H)-bundles, and we can consider the isomorphism class [P] ∈ H1(X,PU(H)X). There is an isomorphism H1(X, PU(H)X) → H2(X, U(1)X) provided by the boundary map [P] ↦ BP. There is also an isomorphism H2(X, U(1)X) → H3(X, ZX) ≅ H3(X, Z) The image δ(P) ∈ H3(X, Z) of BP is called the Dixmier-Douady invariant of P. When δ(P) = [H] is represented in H3(X, R) by a closed three-form H on X, called the H-flux of the given B-field BP, we will write P = PH. One has δ(P) = 0 iff the projective bundle P comes from a vector bundle E → X, i.e., P = P(E). By Serre’s theorem every torsion element of H3(X,Z) arises from a finite-dimensional bundle P. Explicitly, consider the commutative diagram of exact sequences of groups given by where we identify the cyclic group Zn with the group of n-th roots of unity. Let P be a projective bundle with structure group PU(n), i.e., with fibres P(Cn). Then the commutative diagram of long exact sequences of sheaf cohomology groups associated to the above commutative diagram of groups implies that the element BP ∈ H2(X, U(1)X) comes from H2(X, (Zn)X), and therefore its order divides n. One also has δ(P1 ⊗ P2) = δ(P1) + δ(P2) and δ(P) = −δ(P). This follows from the commutative diagram and the fact that P ⊗ P = P(E) where E is the vector bundle of Hilbert-Schmidt endomorphisms of P . Putting everything together, it follows that the cohomology group H3(X, Z) is isomorphic to the group of stable equivalence classes of principal PU(H)-bundles P → X with the operation of tensor product. We are now ready to define the twisted K-theory of the manifold X equipped with a projective bundle P → X, such that Px = P(H) ∀ x ∈ X. We will first give a definition in terms of Fredholm operators, and then provide some equivalent, but more geometric definitions. Let H be a Z2-graded Hilbert space. We define Fred0(H) to be the space of self-adjoint degree 1 Fredholm operators T on H such that T2 − 1 ∈ K(H), together with the subspace topology induced by the embedding Fred0(H) ֒→ B(H) × K(H) given by T → (T, T2 − 1) where the algebra of bounded linear operators B(H) is given the compact-open topology and the Banach algebra of compact operators K = K(H) is given the norm topology. Let P = PH → X be a projective Hilbert bundle. Then we can construct an associated bundle Fred0(P) whose fibres are Fred0(H). We define the twisted K-theory group of the pair (X, P) to be the group of homotopy classes of maps K0(X, H) = [X, Fred0(PH)] The group K0(X, H) depends functorially on the pair (X, PH), and an isomorphism of projective bundles ρ : P → P′ induces a group isomorphism ρ∗ : K0(X, H) → K0(X, H′). Addition in K0(X, H) is defined by fibre-wise direct sum, so that the sum of two elements lies in K0(X, H2) with [H2] = δ(P ⊗ P(C2)) = δ(P) = [H]. Under the isomorphism H ⊗ C2 ≅ H, there is a projective bundle isomorphism P → P ⊗ P(C2) for any projective bundle P and so K0(X, H2) is canonically isomorphic to K0(X, H). When [H] is a non-torsion element of H3(X, Z), so that P = PH is an infinite-dimensional bundle of projective spaces, then the index map K0(X, H) → Z is zero, i.e., any section of Fred0(P) takes values in the index zero component of Fred0(H). Let us now describe some other models for twisted K-theory which will be useful in our physical applications later on. A definition in algebraic K-theory may given as follows. A bundle of projective spaces P yields a bundle End(P) of algebras. However, if H is an infinite-dimensional Hilbert space, then one has natural isomorphisms H ≅ H ⊕ H and End(H) ≅ Hom(H ⊕ H, H) ≅ End(H) ⊕ End(H) as left End(H)-modules, and so the algebraic K-theory of the algebra End(H) is trivial. Instead, we will work with the Banach algebra K(H) of compact operators on H with the norm topology. Given that the unitary group U(H) with the compact-open topology acts continuously on K(H) by conjugation, to a given projective bundle PH we can associate a bundle of compact operators EH → X given by EH = PH ×PU K with δ(EH) = [H]. The Banach algebra AH := C0(X, EH) of continuous sections of EH vanishing at infinity is the continuous trace C∗-algebra CT(X, H). Then the twisted K-theory group K(X, H) of X is canonically isomorphic to the algebraic K-theory group K(AH). We will also need a smooth version of this definition. Let AH be the smooth subalgebra of AH given by the algebra CT(X, H) = C(X, L1PH), where L1PH = PH ×PUL1. Then the inclusion CT(X, H) → CT(X, H) induces an isomorphism KCT(X, H) → KCT(X, H) of algebraic K-theory groups. Upon choosing a bundle gerbe connection, one has an isomorphism KCT(X, H) ≅ K(X, H) with the twisted K-theory defined in terms of projective Hilbert bundles P = PH over X. Finally, we propose a general definition based on K-theory with coefficients in a sheaf of rings. It parallels the bundle gerbe approach to twisted K-theory. Let B be a Banach algebra over C. Let E(B, X) be the category of continuous B-bundles over X, and let C(X, B) be the sheaf of continuous maps X → B. The ring structure in B equips C(X, B) with the structure of a sheaf of rings over X. We can therefore consider left (or right) C(X, B)-modules, and in particular the category LF C(X, B) of locally free C(X, B)-modules. Using the functor in the usual way, for X an equivalence of additive categories E(B, X) ≅ LF (C(X, B)) Since these are both additive categories, we can apply the Grothendieck functor to each of them and obtain the abelian groups K(LF(C(X, B))) and K(E(B, X)). The equivalence of categories ensures that there is a natural isomorphism of groups K(LF (C(X, B))) ≅ K(E(B, X)) This motivates the following general definition. If A is a sheaf of rings over X, then we define the K-theory of X with coefficients in A to be the abelian group K(X, A) := K LF(A) For example, consider the case B = C. Then C(X, C) is just the sheaf of continuous functions X → C, while E(C, X) is the category of complex vector bundles over X. Using the isomorphism of K-theory groups we then have K(X, C(X,C)) := K(LF (C(X, C))) ≅ K (E(C, X)) = K0(X) The definition of twisted K-theory uses another special instance of this general construction. For this, we define an Azumaya algebra over X of rank m to be a locally trivial algebra bundle over X with fibre isomorphic to the algebra of m × m complex matrices over C, Mm(C). An example is the algebra End(E) of endomorphisms of a complex vector bundle E → X. We can define an equivalence relation on the set A(X) of Azumaya algebras over X in the following way. Two Azumaya algebras A, A′ are called equivalent if there are vector bundles E, E′ over X such that the algebras A ⊗ End(E), A′ ⊗ End(E′) are isomorphic. Then every Azumaya algebra of the form End(E) is equivalent to the algebra of functions C(X) on X. The set of all equivalence classes is a group under the tensor product of algebras, called the Brauer group of X and denoted Br(X). By Serre’s theorem there is an isomorphism δ : Br(X) → tor(H3(X, Z)) where tor(H3(X, Z)) is the torsion subgroup of H3(X, Z). If A is an Azumaya algebra bundle, then the space of continuous sections C(X, A) of X is a ring and we can consider the algebraic K-theory group K(A) := K0(C(X,A)) of equivalence classes of projective C(X, A)-modules, which depends only on the equivalence class of A in the Brauer group. Under the equivalence, we can represent the Brauer group Br(X) as the set of isomorphism classes of sheaves of Azumaya algebras. Let A be a sheaf of Azumaya algebras, and LF(A) the category of locally free A-modules. Then as above there is an isomorphism K(X, C(X, A)) ≅ K Proj (C(X, A)) where Proj (C(X, A)) is the category of finitely-generated projective C(X, A)-modules. The group on the right-hand side is the group K(A). For given [H] ∈ tor(H3(X, Z)) and A ∈ Br(X) such that δ(A) = [H], this group can be identified as the twisted K-theory group K0(X, H) of X with twisting A. This definition is equivalent to the description in terms of bundle gerbe modules, and from this construction it follows that K0(X, H) is a subgroup of the ordinary K-theory of X. If δ(A) = 0, then A is equivalent to C(X) and we have K(A) := K0(C(X)) = K0(X). The projective C(X, A)-modules over a rank m Azumaya algebra A are vector bundles E → X with fibre Cnm ≅ (Cm)⊕n, which is naturally an Mm(C)-module. # Badiou Contra Grothendieck Functorally. Note Quote. What makes categories historically remarkable and, in particular, what demonstrates that the categorical change is genuine? On the one hand, Badiou fails to show that category theory is not genuine. But, on the other, it is another thing to say that mathematics itself does change, and that the ‘Platonic’ a priori in Badiou’s endeavour is insufficient, which could be demonstrated empirically. Yet the empirical does not need to stand only in a way opposed to mathematics. Rather, it relates to results that stemmed from and would have been impossible to comprehend without the use of categories. It is only through experience that we are taught the meaning and use of categories. An experience obviously absent from Badiou’s habituation in mathematics. To contrast, Grothendieck opened up a new regime of algebraic geometry by generalising the notion of a space first scheme-theoretically (with sheaves) and then in terms of groupoids and higher categories. Topos theory became synonymous to the study of categories that would satisfy the so called Giraud’s axioms based on Grothendieck’s geometric machinery. By utilising such tools, Pierre Deligne was able to prove the so called Weil conjectures, mod-p analogues of the famous Riemann hypothesis. These conjectures – anticipated already by Gauss – concern the so called local ζ-functions that derive from counting the number of points of an algebraic variety over a finite field, an algebraic structure similar to that of for example rational Q or real numbers R but with only a finite number of elements. By representing algebraic varieties in polynomial terms, it is possible to analyse geometric structures analogous to Riemann hypothesis but over finite fields Z/pZ (the whole numbers modulo p). Such ‘discrete’ varieties had previously been excluded from topological and geometric inquiry, while it now occurred that geometry was no longer overshadowed by a need to decide between ‘discrete’ and ‘continuous’ modalities of the subject (that Badiou still separates). Along with the continuous ones, also discrete variates could then be studied based on Betti numbers, and similarly as what Cohen’s argument made manifest in set-theory, there seemed to occur ‘deeper’, topological precursors that had remained invisible under the classical formalism. In particular, the so called étale-cohomology allowed topological concepts (e.g., neighbourhood) to be studied in the context of algebraic geometry whose classical, Zariski-description was too rigid to allow a meaningful interpretation. Introducing such concepts on the basis of Jean-Pierre Serre’s suggestion, Alexander Grothendieck did revolutionarize the field of geometry, and Pierre Deligne’s proof of the Weil-conjenctures, not to mention Wiles’ work on Fermat’s last theorem that subsequentely followed. Grothendieck’s crucial insight drew on his observation that if morphisms of varieties were considered by their ‘adjoint’ field of functions, it was possible to consider geometric morphisms as equivalent to algebraic ones. The algebraic category was restrictive, however, because field-morphisms are always monomorphisms which makes geometric morphisms: to generalize the notion of a neighbourhood to algebraic category he needed to embed algebraic fields into a larger category of rings. While a traditional Kuratowski covering space is locally ‘split’ – as mathematicians call it – the same was not true for the dual category of fields. In other words, the category of fields did not have an operator analogous to pull-backs (fibre products) unless considered as being embedded within rings from which pull-backs have a co-dual expressed by the tensor operator ⊗. Grothendieck thus realized he could replace ‘incorporeal’ or contained neighborhoods U ֒→ X by a more relational description: as maps U → X that are not necessarily monic, but which correspond to ring-morphisms instead. Topos theory applies similar insight but not in the context of only specific varieties but for the entire theory of sets instead. Ultimately, Lawvere and Tierney realized the importance of these ideas to the concept of classification and truth in general. Classification of elements between two sets comes down to a question: does this element belong to a given set or not? In category of S ets this question calls for a binary answer: true or false. But not in a general topos in which the composition of the subobject-classifier is more geometric. Indeed, Lawvere and Tierney then considered this characteristc map ‘either/or’ as a categorical relationship instead without referring to its ‘contents’. It was the structural form of this morphism (which they called ‘true’) and as contrasted with other relationships that marked the beginning of geometric logic. They thus rephrased the binary complete Heyting algebra of classical truth with the categorical version Ω defined as an object, which satisfies a specific pull-back condition. The crux of topos theory was then the so called Freyd–Mitchell embedding theorem which effectively guaranteed the explicit set of elementary axioms so as to formalize topos theory. The Freyd–Mitchell embedding theorem says that every abelian category is a full subcategory of a category of modules over some ring R and that the embedding is an exact functor. It is easy to see that not every abelian category is equivalent to RMod for some ring R. The reason is that RMod has all small limits and colimits. But for instance the category of finitely generated R-modules is an abelian category but lacks these properties. But to understand its significance as a link between geometry and language, it is useful to see how the characteristic map (either/or) behaves in set theory. In particular, by expressing truth in this way, it became possible to reduce Axiom of Comprehension, which states that any suitable formal condition λ gives rise to a peculiar set {x ∈ λ}, to a rather elementary statement regarding adjoint functors. At the same time, many mathematical structures became expressible not only as general topoi but in terms of a more specific class of Grothendieck-topoi. There, too, the ‘way of doing mathematics’ is different in the sense that the object-classifier is categorically defined and there is no empty set (initial object) but mathematics starts from the terminal object 1 instead. However, there is a material way to express the ‘difference’ such topoi make in terms of set theory: for every such a topos there is a sheaf-form enabling it to be expressed as a category of sheaves S etsC for a category C with a specific Grothendieck-topology. # Marching Along Categories, Groups and Rings. Part 2 A category C consists of the following data: A collection Obj(C) of objects. We will write “x ∈ C” to mean that “x ∈ Obj(C) For each ordered pair x, y ∈ C there is a collection HomC (x, y) of arrows. We will write α∶x→y to mean that α ∈ HomC(x,y). Each collection HomC(x,x) has a special element called the identity arrow idx ∶ x → x. We let Arr(C) denote the collection of all arrows in C. For each ordered triple of objects x, y, z ∈ C there is a function ○ ∶ HomC (x, y) × HomC(y, z) → HomC (x, z), which is called composition of arrows. If α ∶ x → y and β ∶ y → z then we denote the composite arrow by β ○ α ∶ x → z. If each collection of arrows HomC(x,y) is a set then we say that the category C is locally small. If in addition the collection Obj(C) is a set then we say that C is small. Identitiy: For each arrow α ∶ x → y the following diagram commutes: Associative: For all arrows α ∶ x → y, β ∶ y → z, γ ∶ z → w, the following diagram commutes: We say that C′ ⊆ C is a subcategory if Obj(C′) ⊆ Obj(C) and if ∀ x,y ∈ Obj(C′) we have HomC′(x,y) ⊆ HomC(x,y). We say that the subcategory is full if each inclusion of hom sets is an equality. Let C be a category. A diagram D ⊆ C is a collection of objects in C with some arrows between them. Repetition of objects and arrows is allowed. OR. Let I be any small category, which we think of as an “index category”. Then any functor D ∶ I → C is called a diagram of shape I in C. In either case, we say that the diagram D commutes if for all pairs of objects x,y in D, any two directed paths in D from x to y yield the same arrow under composition. Identity arrows generalize the reflexive property of posets, and composition of arrows generalizes the transitive property of posets. But whatever happened to the antisymmetric property? Well, it’s the same issue we had before: we should really define equivalence of objects in terms of antisymmetry. Isomorphism: Let C be a category. We say that two objects x,y ∈ C are isomorphic in C if there exist arrows α ∶ x → y and β ∶ y → x such that the following diagram commutes: In this case we write x ≅C y, or just x ≅ y if the category is understood. If γ ∶ y → x is any other arrow satisfying the same diagram as β, then by the axioms of identity and associativity we must have γ = γ ○ idy = γ ○ (α ○ β) = (γ ○ α) ○ β = idx ○ β = β This allows us to refer to β as the inverse of the arrow α. We use the notations β = α−1 and β−1 = α. A category with one object is called a monoid. A monoid in which each arrow is invertible is called a group. A small category in which each arrow is invertible is called a groupoid. Subcategories of Set are called concrete categories. Given a concrete category C ⊆ Set we can think of its objects as special kinds of sets and its arrows as special kinds of functions. Some famous examples of conrete categories are: • Grp = groups & homomorphisms • Ab = abelian groups & homomorphisms • Rng = rings & homomorphisms • CRng = commutative rings & homomorphisms Note that Ab ⊆ Grp and CRng ⊆ Rng are both full subcategories. In general, the arrows of a concrete category are called morphisms or homomorphisms. This explains our notation of HomC. Homotopy: The most famous example of a non-concrete category is the fundamental groupoid π1(X) of a topological space X. Here the objects are points and the arrows are homotopy classes of continuous directed paths. The skeleton is the set π0(X) of path components (really a discrete category, i.e., in which the only arrows are the identities). Categories like this are the reason we prefer the name “arrow” instead of “morphism”. Limit/Colimit: Let D ∶ I → C be a diagram in a category C (thus D is a functor and I is a small “index” category). A cone under D consists of • an object c ∈ C, • a collection of arrows αi ∶ x → D(i), one for each index i ∈ I, such that for each arrow δ ∶ i → j in I we have αj = D(δ) ○ α In visualizing this: The cone (c,(αi)i∈I) is called a limit of the diagram D if, for any cone (z,(βi)i∈I) under D, the following picture holds: [This picture means that there exists a unique arrow υ ∶ z → c such that, for each arrow δ ∶ i → j in I (including the identity arrows), the following diagram commutes: When δ = idi this diagram just says that βi = αi ○ υ. We do not assume that D itself is commutative. Dually, a cone over D consists of an object c ∈ C and a set of arrows αi ∶ D(i) → c satisfying αi = αj ○ D(δ) for each arrow δ ∶ i → j in I. This cone is called a colimit of the diagram D if, for any cone (z,(βi)i∈I) over D, the following picture holds: When the (unique) limit or colimit of the diagram D ∶ I → C exists, we denote it by (limI D, (φi)i∈I) or (colimI D, (φi)i∈I), respectively. Sometimes we omit the canonical arrows φi from the notation and refer to the object limID ∈ C as “the limit of D”. However, we should not forget that the arrows are part of the structure, i.e., the limit is really a cone. Posets: Let P be a poset. We have already seen that the product/coproduct in P (if they exist) are the meet/join, respectively, and that the final/initial objects in P (if they exist) are the top/bottom elements, respectively. The only poset with a zero object is the one element poset. Sets: The empty set ∅ ∈ Set is an initial object and the one point set ∗ ∈ Set is a final object. Note that two sets are isomorphic in Set precisely when there is a bijection between them, i.e., when they have the same cardinality. Since initial/final objects are unique up to isomorphism, we can identify the initial object with the cardinal number 0 and the final object with the cardinal number 1. There is no zero object in Set. Products and coproducts exist in Set. The product of S,T ∈ Set consists of the Cartesian product S × T together with the canonical projections πS ∶ S × T → S and πT ∶ S × T → T. The coproduct of S, T ∈ Set consists of the disjoint union S ∐ T together with the canonical injections ιS ∶ S → S ∐ T and ιT ∶ T → S ∐ T. After passing to the skeleton, the product and coproduct of sets become the product and sum of cardinal numbers. [Note: The “external disjoint union” S ∐ T is a formal concept. The familiar “internal disjoint union” S ⊔ T is only defined when there exists a set U containing both S and T as subsets. Then the union S ∪ T is the join operation in the Boolean lattice 2U ; we call the union “disjoint” when S ∩ T = ∅.] Groups: The trivial group 1 ∈ Grp is a zero object, and for any groups G, H ∈ Grp the zero homomorphism 1 ∶ G → H sends all elements of G to the identity element 1H ∈ H. The product of groups G, H ∈ Grp is their direct product G × H and the coproduct is their free product G ∗ H, along with the usual canonical morphisms. Let Ab ⊆ Grp be the full subcategory of abelian groups. The zero object and product are inherited from Grp, but we give them new names: we denote the zero object by 0 ∈ Ab and for any A, B ∈ Ab we denote the zero arrow by 0 ∶ A → B. We denote the Cartesian product by A ⊕ B and we rename it the direct sum. The big difference between Grp and Ab appears when we consider coproducts: it turns out that the product group A ⊕ B is also the coproduct group. We emphasize this fact by calling A ⊕ B the biproduct in Ab. It comes equipped with four canonical homomorphisms πA, πB, ιA, ιB satisfying the usual properties, as well as the following commutative diagram: This diagram is the ultimate reason for matrix notation. The universal properties of product and coproduct tell us that each endomorphism φ ∶ A ⊕ B → A ⊕ B is uniquely determined by its four components φij ∶= πi ○ φ ○ ιj for i, j ∈ {A,B},so we can represent it as a matrix: Then the composition of endomorphisms becomes matrix multiplication. Rings. We let Rng denote the category of rings with unity, together with their homomorphisms. The initial object is the ring of integers Z ∈ Rng and the final object is the zero ring 0 ∈ Rng, i.e., the unique ring in which 0R = 1R. There is no zero object. The product of two rings R, S ∈ Rng is the direct product R × S ∈ Rng with component wise addition and multiplication. Let CRng ⊆ Rng be the full subcategory of commutative rings. The initial/final objects and product in CRng are inherited from Rng. The difference between Rng and CRng again appears when considering coproducts. The coproduct of R,S ∈ CRng is denoted by R ⊗Z S and is called the tensor product over Z…..	2022-09-28 12:57:27	{"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": 0.8964177966117859, "perplexity": 951.0747877102347}, "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/1664030335254.72/warc/CC-MAIN-20220928113848-20220928143848-00320.warc.gz"}
https://quantumcomputing.stackexchange.com/questions/15734/how-to-construct-a-controlled-hadamard-gate-using-single-qubit-gates-and-control	# How to construct a controlled-Hadamard gate using single qubit gates and controlled phase-shift? How can I construct a controlled-Hadamard gate using single qubit gates and controlled phase-shift? I am stuck in this and any help would be appreciated. One strategy is to find a single-qubit rotation $$R_{\hat n}$$ such that $$R_{\hat n}(\theta) \, P(\phi) \, R_{\hat n}(-\theta) = H\tag1$$ where $$\hat n = (n_x, n_y, n_z)^T$$ is a real 3-vector of unit norm and $$P(\phi) = \mathrm{diag}(1, e^{i\phi})$$ is a phase gate. The motivation behind this approach is the observation that when the phase gate $$P(\phi)$$ is controlled by another qubit the sequence turns into the Hadamard gate when the control is in the $$\|1\rangle$$ state and into $$R_1(\theta) R_1(-\theta) = I$$ when it is in the $$\|0\rangle$$ state. Now, we recognize $$(1)$$ as matrix diagonalization which means that $$P(\phi) = Z$$ and $$R_{\hat n}(\theta)$$ is the matrix whose columns are the eigenvectors of the Hadamard. At this point, we could compute the eigenvectors. However, we notice that both the eigenvalues and the entries of the Hadamard matrix are real numbers. Therefore, the eigenvectors have real entries, too. We recall that among the rotations around $$X$$, $$Y$$ and $$Z$$ axes, one has real entries $$R_y(\beta) = \begin{pmatrix} \cos\frac{\beta}{2} & -\sin\frac{\beta}{2} \\ \sin\frac{\beta}{2} & \cos\frac{\beta}{2} \end{pmatrix}$$ so we are led to guess that a $$Y$$ rotation might work. We try it out by substituting into $$(1)$$ $$\begin{pmatrix} \cos\frac{\theta}{2} & -\sin\frac{\theta}{2} \\ \sin\frac{\theta}{2} & \cos\frac{\theta}{2} \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \begin{pmatrix} \cos\frac{\theta}{2} & \sin\frac{\theta}{2} \\ -\sin\frac{\theta}{2} & \cos\frac{\theta}{2} \end{pmatrix} \\= \begin{pmatrix} \cos\frac{\theta}{2} & -\sin\frac{\theta}{2} \\ \sin\frac{\theta}{2} & \cos\frac{\theta}{2} \end{pmatrix} \begin{pmatrix} \cos\frac{\theta}{2} & \sin\frac{\theta}{2} \\ \sin\frac{\theta}{2} & -\cos\frac{\theta}{2} \end{pmatrix} \\= \begin{pmatrix} \cos\theta & \sin\theta \\ \sin\theta & -\cos\theta \end{pmatrix}$$ and we see that we obtain the Hadamard if $$\theta = \frac{\pi}{4}$$. Therefore, $$R_y\left(\frac{\pi}{4}\right) \circ CZ \circ R_y\left(-\frac{\pi}{4}\right) = CH\tag1$$ where $$CZ$$ and $$CH$$ denote the controlled-$$Z$$ and controlled-Hadamard gates. • I think you meant to write the second rotation be $R_y(-\pi/4)$ instead of $R_{\hat{n}}(-\pi/4)$ on the last equation. :) – KAJ226 Jan 29 at 2:58 • Indeed! Thanks! Fixed. – Adam Zalcman Jan 29 at 2:59	2021-04-16 14:56:05	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 23, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9552963376045227, "perplexity": 255.50435155021103}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038066981.0/warc/CC-MAIN-20210416130611-20210416160611-00398.warc.gz"}
http://www.chegg.com/homework-help/questions-and-answers/small-bead-slide-without-friction-circular-hoop-vertical-plane-radius-0100-rm-m--hoop-rota-q3084552	A small bead can slide without friction on a circular hoop that is in a vertical plane and has a radius of 0.100 {\rm m}. The hoop rotates at a constant rate of 3.70{\rm rev/s} about a vertical diameter (the figure (Figure 1) ). a)Find the angle \beta at which the bead is in vertical equilibrium. (Of course, it has a radial acceleration toward the axis.) b)Is it possible for the bead to "ride" at the same elevation as the center of the hoop?	2015-08-01 17:26:38	{"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": 0.9877082705497742, "perplexity": 735.7423243012993}, "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-2015-32/segments/1438042988840.31/warc/CC-MAIN-20150728002308-00122-ip-10-236-191-2.ec2.internal.warc.gz"}
https://tex.stackexchange.com/questions/472648/link-figure-to-jpg-file-in-such-a-way/472650	# link figure to jpg file in such a way how can I link figure in tex in this way, i.e. "Fig.\,6(b)" in tex, then I click Fig.\,6(b), I will see the 6.jpg automatically namely What should I do in \begin{figure}[h] .... \end{figure} Best, If you place a unique \label{foo} after the caption of your image, you can reference it with \ref{foo}. A combination of a couple of packages can make it easier to get your desired format: • hyperref to make the cross references clickable • subcaption to split a figure in multiple subimages numbered with a, b etc • cleveref to automatically insert fig. before the figure number in the cross reference \documentclass{article} \usepackage{subcaption} \usepackage{graphicx} \usepackage{hyperref} \usepackage{cleveref} \begin{document} \cref{imageb} \begin{figure}[htbp] \begin{subfigure}{.5\textwidth} \includegraphics[width=\linewidth]{example-image-duck} \caption{text} \label{imagea} \end{subfigure} \begin{subfigure}{.5\textwidth} \includegraphics[width=\linewidth]{example-image-duck} \caption{text} \label{imageb} \end{subfigure} \end{figure} \end{document} • if I need fig.1a-1f ? Thanks ! – acsdaswe Jan 30 at 22:40 • @ABCDEMMM You can use \cref{labela,labelb,labelc,labeld,labele,labelf}, cleveref will automatically compress these into a-f if you reference all labels. – user36296 Jan 30 at 22:48 • I got errors: tex capacity exceeded, sorry[input stack size=5000] – acsdaswe Feb 1 at 0:42 • @ABCDEMMM can you make a minimal working example (MWE) that reproduces your problem? – user36296 Feb 1 at 9:12 • can i use subfig? if yes, then how?? – acsdaswe Mar 7 at 23:27	2019-10-15 09:47:39	{"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": 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": 0.9548786878585815, "perplexity": 5541.115658923016}, "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-2019-43/segments/1570986657949.34/warc/CC-MAIN-20191015082202-20191015105702-00479.warc.gz"}
https://firas.moosvi.com/oer/physics_bank/content/public/007.Momentum%20and%20Impulse/Conservation%20of%20Momentum/Blocks_On_Frictionless_Table/Blocks_On_Frictionless_Table.html	# Blocks On Frictionless Table# Two 505 $$g$$ blocks of wood are 1 $$m$$ apart on a frictionless table. A 15 $$g$$ bullet is fired at 421 $$\frac{m}{s}$$ toward the blocks. It passes all the way through the first block, then embeds itself in the second block. The speed of the first block immediately afterwards is 6 $$\frac{m}{s}$$. ## Question Text# What is the speed of the second block after the bullet stops in it? Give your answer to three significant figures. Please enter in a numeric value in $$\frac{m}{s}$$.	2023-01-28 23:19:48	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.6057275533676147, "perplexity": 574.7069278788488}, "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/1674764499695.59/warc/CC-MAIN-20230128220716-20230129010716-00879.warc.gz"}
https://blogdocobra.com/9x0ht/state-transition-diagram-markov-chain-5253ef	Example: Markov Chain ! Below is the Markov Chains - 8 Absorbing States • If p kk=1 (that is, once the chain visits state k, it remains there forever), then we may want to know: the probability of absorption, denoted f ik • These probabilities are important because they provide If the Markov chain has N possible states, the matrix will be an N x N matrix, such that entry (I, J) is the probability of transitioning from state I to state J. Additionally, the transition matrix must be a stochastic matrix, a matrix whose entries in each row must add up to exactly 1. Show that every transition matrix on a nite state space has at least one closed communicating class. &=\frac{1}{3} \cdot \frac{1}{2}= \frac{1}{6}. Deﬁnition: The state space of a Markov chain, S, is the set of values that each If some of the states are considered to be unavailable states for the system, then availability/reliability analysis can be performed for the system as a w… Specify uniform transitions between states … In general, if a Markov chain has rstates, then p(2) ij = Xr k=1 p ikp kj: The following general theorem is easy to prove by using the above observation and induction. From a state diagram a transitional probability matrix can be formed (or Infinitesimal generator if it were a Continuous Markov chain). Current State X Transition Matrix = Final State. Show that every transition matrix on a nite state space has at least one closed communicating class. A continuous-time process is called a continuous-time Markov chain … A Markov chain or its transition … 1. Chapter 17 Markov Chains 2. On the transition diagram, X t corresponds to which box we are in at stept. In the real data, if it's sunny (S) one day, then the next day is also much more likely to be sunny. You da real mvps! Find the stationary distribution for this chain. In Continuous time Markov Process, the time is perturbed by exponentially distributed holding times in each state while the succession of states visited still follows a discrete time Markov chain… Specify uniform transitions between states in the bar. As we can see clearly see that Pepsi, although has a higher market share now, will have a lower market share after one month. Suppose that ! For a first-order Markov chain, the probability distribution of the next state can only depend on the current state. Continuous time Markov Chains are used to represent population growth, epidemics, queueing models, reliability of mechanical systems, etc. Is this chain aperiodic? States 0 and 1 are accessible from state 0 • Which states are accessible from state 3? [2] (b) Find the equilibrium distribution of X. Give the state-transition probability matrix. 122 6. Let state 1 denote the cheerful state, state 2 denote the so-so state, and state 3 denote the glum state. For example, the algorithm Google uses to determine the order of search results, called PageRank, is a type of Markov chain. The ijth en-try p(n) ij of the matrix P n gives the probability that the Markov chain, starting in state s i, … The transition diagram of a Markov chain X is a single weighted directed graph, where each vertex represents a state of the Markov chain and there is a directed edge from vertex j to vertex i if the transition probability p ij >0; this edge has the weight/probability of p ij. 4.1. A continuous-time Markov chain (X t) t ≥ 0 is defined by:a finite or countable state space S;; a transition rate matrix Q with dimensions equal to that of S; and; an initial state such that =, or a probability distribution for this first state. Example: Markov Chain For the State Transition Diagram of the Markov Chain, each transition is simply marked with the transition probability p 11 0 1 2 p 01 p 12 p 00 p 10 p 21 p 22 p 20 p 1 p p 0 00 01 02 p 10 1 p 11 1 1 p 12 1 2 2 p 20 1 2 p A class in a Markov chain is a set of states that are all reacheable from each other. In the hands of metereologists, ecologists, computer scientists, financial engineers and other people who need to model big phenomena, Markov chains can get to be quite large and powerful. If we know $P(X_0=1)=\frac{1}{3}$, find $P(X_0=1,X_1=2,X_2=3)$. b De nition 5.16. Exercise 5.15. I have the following code that draws a transition probability graph using the package heemod (for the matrix) and the package diagram (for drawing). For more explanations, visit the Explained Visually project homepage. The Markov model is analysed in order to determine such measures as the probability of being in a given state at a given point in time, the amount of time a system is expected to spend in a given state, as well as the expected number of transitions between states: for instance representing the number of failures and … From the state diagram we observe that states 0 and 1 communicate and form the ﬁrst class C 1 = f0;1g, whose states are recurrent. For example, each state might correspond to the number of packets in a buffer whose size grows by one or decreases by one at each time step. State-Transition Matrix and Network The events associated with a Markov chain can be described by the m m matrix: P = (pij). )>, on statespace S = {A,B,C} whose transition rates are shown in the following diagram: 1 1 1 (A B 2 (a) Write down the Q-matrix for X. That is, the rows of any state transition matrix must sum to one. There is a Markov Chain (the first level), and each state generates random ‘emissions.’ Beyond the matrix speciﬁcation of the transition probabilities, it may also be helpful to visualize a Markov chain process using a transition diagram. A state i is absorbing if f ig is a closed class. This next block of code reproduces the 5-state Drunkward’s walk example from section 11.2 which presents the fundamentals of absorbing Markov chains. De nition 4. Is this chain irreducible? The second sequence seems to jump around, while the first one (the real data) seems to have a "stickyness". It’s best to think about Hidden Markov Models (HMM) as processes with two ‘levels’. 1. Finally, if the process is in state 3, it remains in state 3 with probability 2/3, and moves to state 1 with probability 1/3. We may see the state i after 1,2,3,4,5.. etc number of transition. which graphs a fourth order Markov chain with the specified transition matrix and initial state 3. Is this chain irreducible? Likewise, "S" state has 0.9 probability of staying put and a 0.1 chance of transitioning to the "R" state. The diagram shows the transitions among the different states in a Markov Chain. The concept behind the Markov chain method is that given a system of states with transitions between them, the analysis will give the probability of being in a particular state at a particular time. Here's a few to work from as an example: ex1, ex2, ex3 or generate one randomly. &\quad=P(X_0=1) P(X_1=2\|X_0=1) P(X_2=3\|X_1=2, X_0=1)\\ c. Transient solution. Theorem 11.1 Let P be the transition matrix of a Markov chain. 0 Solution • The transition diagram in Fig. the sum of the probabilities that a state will transfer to state " does not have to be 1. We can minic this "stickyness" with a two-state Markov chain. &\quad=\frac{1}{3} \cdot\ p_{12} \cdot p_{23} \\ The nodes in the graph are the states, and the edges indicate the state transition … Thus, having sta-tionary transition probabilitiesimplies that the transition probabilities do not change 16.2 MARKOV CHAINS &\quad=\frac{1}{3} \cdot \frac{1}{2} \cdot \frac{2}{3}\\ A simple, two-state Markov chain is shown below. remains in state 3 with probability 2/3, and moves to state 1 with probability 1/3. Markov Chain Diagram. ; For i ≠ j, the elements q ij are non-negative and describe the rate of the process transitions from state i to state j. Suppose the following matrix is the transition probability matrix associated with a Markov chain. Consider a Markov chain with three possible states $1$, $2$, and $3$ and the following transition … Above, we've included a Markov chain "playground", where you can make your own Markov chains by messing around with a transition matrix. State Transition Diagram: A Markov chain is usually shown by a state transition diagram. Every state in the state space is included once as a row and again as a column, and each cell in the matrix tells you the probability of transitioning from its row's state to its column's state. Is the stationary distribution a limiting distribution for the chain? a. \begin{align} Specify random transition probabilities between states within each weight. In general, if a Markov chain has rstates, then p(2) ij = Xr k=1 p ikp kj: The following general theorem is easy to prove by using the above observation and induction. Question: Consider The Markov Chain With Three States S={1,2,3), That Has The State Transition Diagram Is 3 Find The State Transition Matrix For This Chain This problem has been solved! Figure 11.20 - A state transition diagram. They are widely employed in economics, game theory, communication theory, genetics and finance. 0.6 0.3 0.1 P 0.8 0.2 0 For computer repair example, we have: 1 0 0 State-Transition Network (0.6) • Node for each state • Arc from node i to node j if pij > 0. \end{align}. A visualization of the weather example The Model. [2] (c) Using resolvents, find Pc(X(t) = A) for t > 0. Specify random transition probabilities between states within each weight. Don't forget to Like & Subscribe - It helps me to produce more content :) How to draw the State Transition Diagram of a Transitional Probability Matrix This first section of code replicates the Oz transition probability matrix from section 11.1 and uses the plotmat() function from the diagram package to illustrate it. Beyond the matrix speciﬁcation of the transition probabilities, it may also be helpful to visualize a Markov chain process using a transition diagram. Drawing State Transition Diagrams in Python July 8, 2020 Comments Off Python Visualization I couldn’t find a library to draw simple state transition diagrams for Markov Chains in Python – and had a couple of days off – so I made my own. The transition matrix text will turn red if the provided matrix isn't a valid transition matrix. The state space diagram for this chain is as below. This is how the Markov chain is represented on the system. What Is A State Transition Diagram? The dataframe below provides individual cases of transition of one state into another. # % & = 0000.80.2 000.50.40.1 000.30.70 0.50.5000 0.40.6000 P • Which states are accessible from state 0? Periodic: When we can say that we can return We simulate a Markov chain on the finite space 0,1,...,N. Each state represents a population size. Description Sometimes we are interested in how a random variable changes over time. We can write a probability mass function dependent on t to describe the probability that the M/M/1 queue is in a particular state at a given time. A transition diagram for this example is shown in Fig.1. This means the number of cells grows quadratically as we add states to our Markov chain. This is how the Markov chain is represented on the system. The state of the system at equilibrium or steady state can then be used to obtain performance parameters such as throughput, delay, loss probability, etc. Let X n denote Mark’s mood on the nth day, then {X n, n = 0, 1, 2, …} is a three-state Markov chain. • Consider the Markov chain • Draw its state transition diagram Markov Chains - 3 State Classification Example 1 !!!! " (a) Draw the transition diagram that corresponds to this transition matrix. Let's import NumPy and matplotlib:2. \end{align*}, We can write Consider the continuous time Markov chain X = (X. Consider the Markov chain shown in Figure 11.20. We consider a population that cannot comprise more than N=100 individuals, and define the birth and death rates:3. A probability distribution is the probability that given a start state, the chain will end in each of the states after a given number of steps. Thanks to all of you who support me on Patreon. A certain three-state Markov chain has a transition probability matrix given by P = [ 0.4 0.5 0.1 0.05 0.7 0.25 0.05 0.5 0.45 ] . P(X_0=1,X_1=2) &=P(X_0=1) P(X_1=2\|X_0=1)\\ • Consider the Markov chain • Draw its state transition diagram Markov Chains - 3 State Classification Example 1 !!!! " States 0 and 1 are accessible from state 0 • Which states are accessible from state … &P(X_0=1,X_1=2,X_2=3) \\ banded. Thus, a transition matrix comes in handy pretty quickly, unless you want to draw a jungle gym Markov chain diagram. If the transition matrix does not change with time, we can predict the market share at any future time point. while the corresponding state transition diagram is shown in Fig. With two states (A and B) in our state space, there are 4 possible transitions (not 2, because a state can transition back into itself). A Markov chain or its transition matrix P is called irreducible if its state space S forms a single communicating … There also has to be the same number of rows as columns. They do not change over times. In this two state diagram, the probability of transitioning from any state to any other state is 0.5. Markov Chain can be applied in speech recognition, statistical mechanics, queueing theory, economics, etc. 2 (right). Find an example of a transition matrix with no closed communicating classes. $$P(X_4=3\|X_3=2)=p_{23}=\frac{2}{3}.$$, By definition If we're at 'B' we could transition to 'A' or stay at 'B'. Every state in the state space is included once as a row and again as a column, and each cell in the matrix tells you the probability of transitioning from its row's state to its column's state. Markov chains can be represented by a state diagram , a type of directed graph. We set the initial state to x0=25 (that is, there are 25 individuals in the population at init… , q n, and the transitions between states are nondeterministic, i.e., there is a probability of transiting from a state q i to another state q j: P(S t = q j \| S t −1 = q i). [2] (c) Using resolvents, find Pc(X(t) = A) for t > 0. Below is the transition diagram for the 3×3 transition matrix given above. One use of Markov chains is to include real-world phenomena in computer simulations. Find the stationary distribution for this chain. You can also access a fullscreen version at setosa.io/markov. For an irreducible markov chain, Aperiodic: When starting from some state i, we don't know when we will return to the same state i after some transition. The Markov chains to be discussed in this chapter are stochastic processes deﬁned only at integer values of time, n = … 1 Deﬁnitions, basic properties, the transition matrix Markov chains were introduced in 1906 by Andrei Andreyevich Markov (1856–1922) and were named in his honor. The state-transition diagram of a Markov chain, portrayed in the following figure (a) represents a Markov chain as a directed graph where the states are embodied by the nodes or vertices of the graph; the transition between states is represented by a directed line, an edge, from the initial to the final state, The transition … If the Markov chain reaches the state in a weight that is closest to the bar, then specify a high probability of transitioning to the bar. . From a state diagram a transitional probability matrix can be formed (or Infinitesimal generator if it were a Continuous Markov chain). Example 2: Bull-Bear-Stagnant Markov Chain. Keywords: probability, expected value, absorbing Markov chains, transition matrix, state diagram 1 Expected Value Example: Markov Chain ! (b) Show that this Markov chain is regular. … Is the stationary distribution a limiting distribution for the chain? I have following dataframe with there states: angry, calm, and tired. # % & = 0000.80.2 000.50.40.1 000.30.70 0.50.5000 0.40.6000 P • Which states are accessible from state 0? The igraph package can also be used to Markov chain diagrams, but I prefer the “drawn on a chalkboard” look of plotmat. (c) Find the long-term probability distribution for the state of the Markov chain… Determine if the Markov chain has a unique steady-state distribution or not. Exercise 5.15. Of course, real modelers don't always draw out Markov chain diagrams. In addition, on top of the state space, a Markov chain tells you the probabilitiy of hopping, or "transitioning," from one state to any other state---e.g., the chance that a baby currently playing will fall asleep in the next five minutes without crying first. Markov Chains - 1 Markov Chains (Part 5) Estimating Probabilities and Absorbing States ... • State Transition Diagram • Probability Transition Matrix Sun 0 Rain 1 p 1-q 1-p q ! Definition. Of course, real modelers don't always draw out Markov chain diagrams. Hence the transition probability matrix of the two-state Markov chain is, P = P 00 P 01 P 10 P 11 = 1 1 Notice that the sum of the rst row of the transition probability matrix is + (1 ) or If the Markov chain reaches the state in a weight that is closest to the bar, then specify a high probability of transitioning to the bar. Figure 11.20 - A state transition diagram. If we're at 'A' we could transition to 'B' or stay at 'A'. $$P(X_3=1\|X_2=1)=p_{11}=\frac{1}{4}.$$, We can write When the Markov chain is in state "R", it has a 0.9 probability of staying put and a 0.1 chance of leaving for the "S" state. Markov chain can be demonstrated by Markov chains diagrams or transition matrix. Let state 1 denote the cheerful state, state 2 denote the so-so state, and state 3 denote the glum state. 151 8.2 Deﬁnitions The Markov chain is the process X 0,X 1,X 2,.... Deﬁnition: The state of a Markov chain at time t is the value ofX t. For example, if X t = 6, we say the process is in state6 at timet. to reach an absorbing state in a Markov chain. Figure 1: A transition diagram for the two-state Markov chain of the simple molecular switch example. The colors occur because some of the states (1 and 2) are transient and some are absorbing (in this case, state 4). The order of a Markov chain is how far back in the history the transition probability distribution is allowed to depend on. &= \frac{1}{3} \cdot\ p_{12} \\ You can customize the appearance of the graph by looking at the help file for Graph. Any transition matrix P of an irreducible Markov chain has a unique distribution stasfying ˇ= ˇP: Periodicity: Figure 10: The state diagram of a periodic Markov chain This chain is irreducible but that is not su cient to prove … Lemma 2. Example: Markov Chain For the State Transition Diagram of the Markov Chain, each transition is simply marked with the transition probability p 11 0 1 2 p 01 p 12 p 00 p 10 p 21 p 22 p 20 p 1 p p 0 00 01 02 p 10 1 p 11 1 1 p 12 1 2 2 p 20 1 2 p Chapter 3 FINITE-STATE MARKOV CHAINS 3.1 Introduction The counting processes {N(t); t > 0} described in Section 2.1.1 have the property that N(t) changes at discrete instants of time, but is deﬁned for all real t > 0. \$1 per month helps!! The processes can be written as {X 0,X 1,X 2,...}, where X t is the state at timet. Therefore, every day in our simulation will have a fifty percent chance of rain." State 2 is an absorbing state, therefore it is recurrent and it forms a second class C 2 = f2g. 2020 state transition diagram markov chain	2021-01-23 04:54:51	{"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": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8446009159088135, "perplexity": 471.9986062602359}, "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-2021-04/segments/1610703533863.67/warc/CC-MAIN-20210123032629-20210123062629-00433.warc.gz"}
https://rationalwiki.org/wiki/User_talk:Wazza	# User talk:Wazza You have been smitten by a bunny in accordance with the community standards for... no real reason. Feel free to revert this odd edit. World Blogs Clogs Elections The Bar (talk) (talk) (talk) (talk) (hic) Welcome to RationalWiki, Wazza! Check out our guide for newcomers and our community standards! Tell us how you found RationalWiki here! If you are interested in contributing: Hi! ContribsTalk 05:34, 3 May 2008 (EDT) Hello! I've been skulking for a while, adding little bits of factually-based humour to various articles, and I thought I should make it official...Wazza 05:49, 3 May 2008 (EDT) Thanks for joining us! Sign by using by using 4 tildes (~~~~) on talk pages. See the Help (←) files for more assistance. ContribsTalk 05:46, 3 May 2008 (EDT) Excuse my interference with your user page, but I think that's what you wanted. ContribsTalk 05:59, 3 May 2008 (EDT) Yeah, but I'd already tried to put the same thing in twice and it didn't turn into a link... I'm new to the wiki format Wazza 06:02, 3 May 2008 (EDT) You'd missed the "http://" off the front - like your blog start. ContribsTalk 06:04, 3 May 2008 (EDT) Ahk. Gotcha. Will remember that in future... see Help on links ContribsTalk 06:06, 3 May 2008 (EDT) I did! Unfortunately I didn't c&p the URL, just typed it out, and so left off the httpWazza 06:09, 3 May 2008 (EDT) Welcome to the Dollhouse, Wazza! human 00:07, 4 May 2008 (EDT) ## content I do find myself reverting/undoing your edits, dear Wazzock. Pleez take care not to be an idiot when editing our wiki? human 00:59, 7 May 2008 (EDT) I guess I have been inserting humour without regard to accuracy... Will try to actually contribute in future. Also, isn't Wazzock someone else? Wazza 01:02, 7 May 2008 (EDT) Perhaps the insertions were funny to you? Try to find that balance between truth and hilarious. You don't want to be wazzock, Wazza. That would be ugly. Sorry for calling you it. human 01:10, 7 May 2008 (EDT) Usually when Human shouts it means he is drunk. Don't take it seriously. SHahB 01:22, 7 May 2008 (EDT) "Wazzock"? Eh? -- Radioactive afikomen Please ignore all my awful pre-2014 comments. 12:38, 7 May 2008 (EDT) It's infectious - H's caught it off me - see Dyslexicon. ContribsTalk 12:41, 7 May 2008 (EDT) Okay, I understand now. -- Radioactive afikomen Please ignore all my awful pre-2014 comments. 14:14, 8 May 2008 (EDT) ## i is teh gramar natsi I'm just sayin'... -Darthmilo77 23:52, 8 May 2008 (EDT) • Actually, it's amazing. I'll build a little shrine for it on my userpage. A secular shrine, mind you. -Darthmilo77 23:55, 8 May 2008 (EDT) ## Iz U In Teh Sigh-Ky-At-Ree/Men-tal Helf Biznes? Hey Wazza, I'm a UK mental health nurse - you're not in the business as well, are you? Spica the Hiver If you tolerate this, then your children will be next... 15:30, 2 June 2008 (EDT) Nope, but I studied introductory psychology, and the connection between magical ideation and psychotypy was the focus of our big assignment, and I paid attention to all the other stuff too, particularly psychopathy since my girlfriend at the time's brother was one... call me an interested amateur. Wazza (Not Wazzock, Wazza)Approach the Presence 15:36, 2 June 2008 (EDT) Kewl! Welcome aboard! Spica the Hiver If you tolerate this, then your children will be next... 17:10, 2 June 2008 (EDT) ## Never Watch out for the Penguins!!! Never question the penguins, they will come for you ## I question motives! BEHOLD! Pwned! Yours sincerely, 21:38, 4 June 2008 (EDT) ## Invitation Javascap 06:47, 8 July 2008 (EDT) ## Wazzzzzzzaup? You are now our 98th sysop. We are almost there. ${\displaystyle \approx }$${\displaystyle \pi }$ Mowse 06:12, 28 September 2008 (EDT) Responsibility... my one... weaknessss... Wazza (Not Wazzock, Wazza)Approach the Presence 06:13, 28 September 2008 (EDT) ## Braces Thanks for correcting me on the caveman diet thing! Cheddarius 03:34, 2 November 2008 (EST) ## Barnstar Barnstar For your tireless contributions to RationalWiki, I award you this barnstar. ## Webcomic It's not available online any more, actually. It had a lot of problems, so at first it went on hold and finally stopped altogether. So the domain isn't there any more, since what's the point in paying for something I'm not gonna use? I'll be putting something up again in time, but I need to work on my skills a lot before that. You can still find all the strips in my deviantART gallery if you're interested. --Kels 14:14, 20 June 2009 (UTC) So what's this then? Totnesmartin 17:14, 20 June 2009 (UTC)	2023-03-25 04:06:51	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 2, "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": 0.6797458529472351, "perplexity": 10572.83856366417}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945315.31/warc/CC-MAIN-20230325033306-20230325063306-00297.warc.gz"}
http://physics.stackexchange.com/questions/1/what-is-spin-as-it-relates-to-subatomic-particles/6	# What is spin as it relates to subatomic particles? I often hear about subatomic particles having a property called "spin" but also that it doesn't actually relate to spinning about an axis like you would think. Which particles have spin? What does spin mean if not an actual spinning motion? - Was there something in particular you didn't understand in the wikipedia article? en.wikipedia.org/wiki/Spin_%28physics%29 – j.c. Nov 2 '10 at 19:11 If a little historical perspective would help, I recommend this series by Nature on the effect that studying "spin" has had on the world of particle physics (and vice-versa). – Herb Nov 2 '10 at 19:18 @j.c.: Does it matter if there was or wasn't? I was under the impression that we answered questions regardless of whether or not they have been answered in other sites, unless you're just asking to see if there's a specific property of spin the asker wants explained in detail. – Mana Nov 2 '10 at 19:43 @Mana, I agree, which is why I didn't vote to close. I tend to think my time answering questions is best spent if I'm writing the answer at a level the questioner might understand. – j.c. Nov 3 '10 at 14:24 I don't like the subatomic tag here. Is particle-physics more appropriate? – Nick Nov 3 '10 at 16:34 Spin is a technical term specifically referring to intrinsic angular momentum of particles. It means a very specific thing in quantum/particle physics. (Physicists often borrow loosely related everyday words and give them a very precise physical/mathematical definition.) Since truly fundamental particles (e.g. electrons) are point entities, i.e. have no true size in space, it does not make sense to consider them 'spinning' in the common sense, yet they still possess their own angular momenta. Note however, that like many quantum states (fundamental variables of systems in quantum mechanics,) spin is quantised; i.e. it can only take one of a set of discrete values. Specifically, the allowed values of the spin quantum number s are non-negative multiples of 1/2. The actual spin momentum (denoted S) is a multiple of Planck's constant, and is given by $S = \sqrt{s (s + 1)}$. When it comes to composite particles (e.g. nuclei, atoms), spin is actually fairly easy to deal with. Like normal (orbital) angular momentum, it adds up linearly. Hence a proton, made of three constituent quarks, has overall spin 1/2. If you're curious as to how this (initially rather strange) concept of spin was discovered, I suggest reading about the Stern-Gerlach experiment of the 1920s. It was later put into the theoretical framework of quantum mechanics by Schrodinger and Pauli. - I don't think electron are fundamental particles in the standard model. Also to my knowledge they are not point like. – Albert Nov 2 '10 at 19:53 @Albert: They most certainly are! There are fringe theories which consider them composite particles, but the standard model of particle physics and most extensions of the theory consider them fundamental point particles. – Noldorin Nov 2 '10 at 19:54 The only sense in that they are not point-like is that there position may be indefinite/fuzzy due to the Heisenberg principle. They are still however considered "point particles" since their position eigenstates are functions of a single position vector. – Noldorin Nov 2 '10 at 19:55 I think the flaw in your thinking comes from considering angular momentum as a consequence or property of spinning. Angular momentum really is the fundamental quantity here. It is a direct consequence of the rotational symmetry of the universe (see Noether's theorem). Realise, the maths is the basis of the physics here; nothing gives a more precise pictures. – Noldorin Nov 4 '10 at 17:29 Note that experimentally one can only say that election have no structure down to [length scale] and/or up to [energy scale]. Which makes them point-like for the purposes of all the experiments we've been able to do. I believe the limiting length scale is currently around $10^{-18}$ meters, which is pretty small. – dmckee Nov 30 '10 at 4:06 Imagine going to the rest frame of a massive particle. In this frame, there is rotational symmetry, which means that the Lie algebra of rotations acts on the wave function. So the wave function is a vector in a representation of Lie(SO(3)) = Lie(SU(2)). "Spin" is the label of precisely which representation this is. Note that while SO(3) and SU(2) share a Lie algebra, they are different as groups, and it is a fact of life ("the connection between spin and statistics") that some particles -- fermions, with half-integral spin -- transform under representations of SU(2) while others -- bosons, with integral spin -- transform under SO(3). - An accurate answer, but if the poster doesn't understand the actual concept of spin (not to mention group theory), this is all but useless. – Noldorin Nov 2 '10 at 19:32 I agree. This comment speaks to my confusion (see my Physics Meta question) over how this Physics site is conceived. Namely, what is the level (high school, undergrad, grad) of the intended audience? – Eric Zaslow Nov 2 '10 at 22:05 Yeah, it's a point worth discussing. I think all three of those levels you point out should be acceptable. (You are clearly aiming at lower graduate level plus in this case.) Personally I would not like to see this site dominated by too many basic high school questions nor by many research-level ones. – Noldorin Nov 3 '10 at 13:22	2014-03-09 07:27:30	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.7754507660865784, "perplexity": 975.2996564846435}, "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-2014-10/segments/1393999675300/warc/CC-MAIN-20140305060755-00022-ip-10-183-142-35.ec2.internal.warc.gz"}
https://eprint.iacr.org/2021/1524	### An Improved Range Proof with Base-3 Construction Esra Günsay, Cansu Betin Onur, and Murat Cenk ##### Abstract Zero-knowledge protocols (ZKPs) allow a party to prove the validation of secret information to some other party without revealing any information about the secret itself. Appropriate, effective, and efficient use of cryptographic ZKPs contributes to many novel advances in real-world privacy-preserving frameworks. One of the most important type of cryptographic ZKPs is the zero-knowledge range proofs (ZKRPs). Such proofs have wide range of applications such as anonymous credentials, cryptocurrencies, e-cash schemes etc. In many ZKRPs the secret is represented in binary then committed via a suitable commitment scheme. Though there exist different base approaches on bilinear paring-based and RSA-like based constructions, to our knowledge there is no study on investigating the discrete logarithm-based constructions. In this study, we focus on a range proof construction produced by Mao in 1998. This protocol contains a bit commitment scheme with an OR-construction. We investigate the effect of different base approach on Mao's range proof and compare the efficiency of these basis approaches. To this end, we have extended Mao's range proof to base-3 with a modified OR-proof. We derive the number of computations in modulo exponentiations and the cost of the number of integers exchanged between parties. Then, we have generalized these costs for the base-u construction. Here, we mainly show that comparing with other base approaches, the base-3 approach consistently provides approximately 12% efficiency in computation cost and 10% efficiency in communication cost. We implemented the base-3 protocol and demonstrated that the results are consistent with our theoretical computations. Available format(s) Category Cryptographic protocols Publication info Preprint. Keywords Zero knowledge proofRange proofOR proofCommitment schemes Contact author(s) gunsay @ metu edu tr History Short URL https://ia.cr/2021/1524 CC BY BibTeX @misc{cryptoeprint:2021/1524, author = {Esra Günsay and Cansu Betin Onur and Murat Cenk}, title = {An Improved Range Proof with Base-3 Construction}, howpublished = {Cryptology ePrint Archive, Paper 2021/1524}, year = {2021}, note = {\url{https://eprint.iacr.org/2021/1524}}, url = {https://eprint.iacr.org/2021/1524} } Note: In order to protect the privacy of readers, eprint.iacr.org does not use cookies or embedded third party content.	2023-01-30 07:34:28	{"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": 0.38530561327934265, "perplexity": 4185.564670096581}, "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/1674764499804.60/warc/CC-MAIN-20230130070411-20230130100411-00332.warc.gz"}
http://u.arboreus.com/2008/09/	# *nix tips ## 2008-09-22 ### Python inside LaTeX (and Sage too) I discovered recently, that it is possible to run Python scripts from LaTeX documents and use the to generate document's content. This can be used to read/convert data, generate tables and figures, do on-the-fly calculations. How to run Python code from LaTeX: • download python.sty. Put it in the same directory as the LaTeX document • \usepackage{python} • Put Python code inside \begin{python}\end{python} environment. Whatever this code prints, will become part of the LaTeX document. • Run LaTeX with -shell-escape option (this permits running external code from LaTeX) This is an example of the LaTeX document and the PDF produced. And some more advanced examples are in this LaTeX file (see also the PDF). They cover symbolic calculations from inside LaTeX, plotting and variable persistence between python environments. P.S. What's more, one can even embed the full-featured symbolic math package Sage into LaTeX. For this purpose use sagetex package. By the way, you do not need to install Sage to start using it, please check out its web-version. It is really powerful. Links: This post in Russian: Python внутри LaTeX (и математический пакет Sage тоже) ### Visualizing altitude and velocity profiles of GPS tracks I think that altitude and velocity profiles of GPS tracks is one of the most intereseting forms of their representation. One can use gpsvisualizer.com to plot such profiles. However, I prefer having free tools for such a simple thing. Here I offer my own python script gpxplot, which extracts profile data from a GPX file and plots a profile. This is a direct link: gpxplot.py. There are two important features of the script: 1) GPX file may consist of two or more separate tracks. Each track may consist of several disconnected segments. The script preserves this segmentation of the track. 2) GPX files do not contain explicit information about distane travelled. The script calculcates it using haversine formula (as if the Earth were spherical). The script can either output profile data in a convinient tabular form, or generate a gnuplot script and call gnuplot to do actual plotting. Usage examples are given on a Google Code page. This is what a result may look like: Update: Now there is also online version of the script. Just upload a track and embed the plot in whatever page you want. Links: This announcement in Russian: Визуализация профилей высоты и скорости GPS-треков	2017-10-21 11:53:35	{"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": 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": 0.7512296438217163, "perplexity": 4161.143698921258}, "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-2017-43/segments/1508187824775.99/warc/CC-MAIN-20171021114851-20171021134851-00464.warc.gz"}
https://www.ocf.berkeley.edu/~shidi/cs61a/w/index.php?title=Guides&diff=519&oldid=518	# Difference between revisions of "Guides" ## Environment diagrams ### Environment diagram Rules Environment Diagrams are very important in our understanding of how the computer interprets our code. We will test you on this in every exam. It will never go away. Given that, master it as quickly as you can! :) Below are the rules I follow when drawing environment diagrams. If you understand and faithfully follow these rules when drawing them, you'll never get them wrong. One thing you haven't learned yet is nonlocal. You can skip that particular step for now (step 2 of Assignment). Post here if you have any questions! You can also take a look at this link for some examples of environment diagrams: http://albertwu.org/cs61a/notes/environments For a different perspective on the rules, check out: http://markmiyashita.com/cs61a/sp14/environment_diagrams/rules_of_environment_diagrams/ A handout with detailed instructions on drawing environment diagrams is also available here (linked on the bottom of the course homepage): http://inst.eecs.berkeley.edu/~cs61a/sp14/pdfs/environment-diagrams.pdf Environment Diagram Rules ========================= Creating a Function -------------------- 1. Draw the func <name>(<arg1>, <arg2>, ...) 2. The parent of the function is wherever the function was defined (the frame we're currently in, since we're creating the function). 3. If we used def, make a binding of the name to the value in the current frame. Calling User Defined Functions ------------------------------ 1. Evaluate the operator and operands. 2. Create a new frame; the parent is whatever the operator s parent is. Now this is the current frame. 3. Bind the formal parameters to the argument values (the evaluated operands). 4. Evaluate the body of the operator in the context of this new frame. 5. After evaluating the body, go back to the frame that called the function. Assignment ---------- 1. Evaluate the expression to the right of the assignment operator (=). 2. If nonlocal, find the frame that has the variable you re looking for, starting in the parent frame and ending just before the global frame (via Lookup rules). Otherwise, use the current frame. Note: If there are multiple frames that have the same variable, pick the frame closest to the current frame. 3. Bind the variable name to the value of the expression in the identified frame. Be sure you override the variable name if it had a previous binding. Lookup ------ 1. Start at the current frame. Is the variable in this frame? If yes, that's the answer. 2. If it isn't, go to the parent frame and repeat 1. 3. If you run out of frames (reach the Global frame and it's not there), complain. Tips ---- 1. You can only bind names to values. No expressions (like 3+4) allowed on environment diagrams! 2. Frames and Functions both have parents. ## Sequences ### Reversing tuples Student Question Why does [::-1] tuple work while the tuple [0:3:-1] doesn't? I thought the -1 after the second semicolon meant that the interpreter is going to read the indexes "backwards". The syntax of slicing is tup[start:end:step]: • start from index start and end just before index end, incrementing the index by step each time • if no step is provided, step = 1 • if step is positive, default values if not provided: start = 0, end = len(tup) • if step is negative, default values if not provided: start = -1, end = one position before the start of the string >>> (1, 2, 3)[::-1] # start at index -1, end one position before the start of the string (3, 2, 1) >>> (1, 2, 3)[0:3:-1] # start at 0 and go to 3, but step is negative, so this doesn't make sense and an empty tuple is returned () This is a helpful visualization from http://en.wikibooks.org/wiki/Python_Programming/Strings#Indexing_and_Slicing: To understand slices, it's easiest not to count the elements themselves. It is a bit like counting not on your fingers, but in the spaces between them. The list is indexed like this: Element: 1 2 3 4 Index: 0 1 2 3 4 -4 -3 -2 -1 ### Slicing with negative step Student Question if the third example returns an empty tuple because you can't take negative steps from 0 to 4, shouldn't the second example also return an empty tuple? Can someone explain why each example returns the respective answers? Thanks >>> x= (1,2,3,4) >>> x[0::-1] (1,) >>> x[::-1] (4, 3, 2, 1) >>> x[0:4:-1] () >>> x[1::-1] (2, 1) (For reference, the notation is x[start:end:step]) Python does something a very strange when the step is negative: if you omit the arguments to start and end, Python will fill them with what makes sense for a negative step. In the simple case of x[::-1], Python fills in the start with len(x)-1 and the end with -(len(x)+1). The end term is strange, but remember that the end term isn't included. We therefore can't use 0, but we can't use -1 either, since that clearly refers to the last element of the tuple. We need to fully wrap the negative index around, to refer to the element "before" the 0th index. This way, Python will start at the end of the tuple and proceed to the beginning of the tuple. That's why x[0:4:-1] doesn't make sense: how can we start at 0 and end at 4, if we're proceeding backwards? And that's why x[0::-1] makes sense (albeit, in a strange way): Python is proceeding from the 0 index to the beginning of the list. It includes the start index, which is why you see a 1 pop up. Let me know if that was confusing! ## Time complexity ### Andrew Huang's guide to order of growth and function runtime Introduction Confused by $O$, $\Omega$, and $\Theta$? Want to figure out the runtime of that tricky function? NOTE THAT THIS GUIDE STARTS WITH BIG O, WHICH IS DIFFERENT FROM THETA. IF YOU UNDERSTAND BIG O, THETA IS EASY (IN FACT, IT DEFINES THETA IN TERMS OF BIG O BELOW). First some math. Formal definition of O(Big O): Let $f(n)$ and $g(n)$ be functions from positive integers to positive reals. We say $f \in O(g)$ (“f grows no faster than g”) if there is a constant $0 < c < \inf$ <such that $f(n) \leq c \cdot g(n)$. (Paraphrased from Dasgupta, Papadimitriou, & Vazirani) (You'll see this again in CS 170) What the heck does that mean? Let’s look at math functions for a second (just a second). Say $f(n)=5n$ and $g(n)=n^{2}$ What does that look like on a graph? There’s a section where $n$ dominates $n^{2}$, from 0 to 5, but we don’t really care, because after that point, $n^{2}$ is larger, all the way to infinity! By the definition, we could scale $n^{2}$ by 5 and we would span that initial gap. Thus we can say $5n \in O(n^{2})$ or $f \in O(g)$. Can we say the converse? That is, is $n^{2} \in O(5n)$? Not at all! From the graph we see that $n^{2}$ grows too quickly for $n$ to catch up, no matter what constant we scale $n$ by. So what if $f(n)=n+1000$ and $g(n)=n^{2}$? It turns out $n+1000 \in O(n^{2})$ still, because according to the definition, as long as we can multiply $n^{2}$ by some $c$, such that the gap of 1000 is spanned, we’re good. In the case, $c=1001$. What about and $\Omega$ and $\Theta$? If you digested all of the above, the rest isn’t scary! (Note, $a \equiv b$ means $a$ is equivalent to $b$) $f \in \Omega(g) \equiv g \in O(f)$ (You'll see this again briefly in CS 170) $f \in \Theta(g) (f \in O(g) and g \in O(f))$ This means that if $f$ is Theta of $g$, then there exist some $c_{1}$ and $c_{2}$ such that $c_{1}g > f$ and $c_{2}g < f$ for all positive integers. What does that mean for Python functions? Given a function $f$, we want to find out how fast that function runs. One way of doing this is to take out a stopwatch, and clock the amount of time it takes for $f$ to run on some input. However, there are tons of problems with that (different computers => different speeds; only one fixed input? Maybe $f$ is really fast for that input but slow for everything else; next year, all the measurements need to be redone on new computers; etc.) Instead, we'll count the steps that a function needs to perform as a function of its input. For example, here are some of the functions that take one step regardless of their input: mul add sub print return ... So for example, (3 + 3 * 8) % 3 would be 3 steps--one for the multiply, one of the add, and one for the mod. Let's take a simple example: def square(x): return x * x square is a function that for any input, always takes two steps, one of the multiplication, and one for returning. Using the notation, we can say square ∈ Θ(1). Functions with iteration (for loops, recursion, etc.), usually multiply the steps by some factor. For example, consider factorial: def factorial(n): if n == 0: return 1 else: return n * factorial(n-1) factorial ∈ Θ(n). Why? Well given some input n, we do n recursive calls. At each recursive call, we carry out 4 steps, one for if n == 0, one for subtraction, one for multiply, one for return. Plus, we have the base case, which is another 2 steps, one for if and one for return. So factorial(n) takes $4n+2$ steps => ∈ Θ(n). As mentioned, we care about how the running time (how long the function takes to run) of the function changes, as we increase the size of the argument. So if we imagine a graph, then the x-axis represents the size of our input, and the y-axis represents how long the function took to run for each x. As the size of the input increases, the function’s runtime does something on the graph. So when we say something like “$O(n^{2})$ where $n$ is the length of the list”, we are saying as we double the size of the list, the function is expected to run at most four times as long. NOTE ALSO THAT I SAID WHAT $n$ IS! ALWAYS GIVE YOUR UNITS. This means that when we compare two functions A and B, A may be overall slower than B as we increase the size of their arguments. However, it’s possible at some specific arguments, the A may run faster (like the $f(n)=5n$ and $g(n)=n^{2}$ example above.) This also means we do not care about the time taken of any particular input! This implies that all those constant-time base cases all those functions don’t really matter, because they don’t scale. That is, only one specific input causes the base case to be reached, and if we increased the size of the argument, $O(1)$ doesn't necessarily hold. Brief “What runs faster than what” Sorted from fastest to slowest. This is by no means comprehensive. • $\Theta(1)$ • $\Theta(\log(n))$ • $\Theta(n)$ • $\Theta(n \log(n))$ • $\Theta(n^{2})$ • $\Theta(n^{3})$ • $\Theta(2^{n})$ • (Anything past this point is kind of ridiculous) • $\Theta(n!)$ • $\Theta(n^{n})$ So we know about the math and the motivation, now how do we actually assign runtimes to real Python functions? What you must understand, is that there is no one method for finding the runtime. You MUST look at a function holistically or you won’t get the right answer. What does this mean? In order to get the correct runtime, you first must understand what the function is doing! You cannot pattern-match your way to becoming good at this. This cannot be stressed enough: UNITS MATTER, if you say O((n)), you must tell us what $n$ is. General tips 1. UNDERSTAND WHAT THE FUNCTION IS DOING!!! 2. Try some sample input. That is, pretend you’re the interpreter and execute the code with some small inputs. What is the function doing with the input? Having concrete examples lets you do tip 1 better. You can also graph how the runtime increases as the argument size increases. 3. If applicable, draw a picture of the tree of function calls. This shows you the "growth" of the function or how the function is getting "bigger", which will help you do tip 1 better. 4. If applicable, draw a picture of how the input is being modified through the function calls. For example, if your input is a list and your function recursively does something to that list, draw out a list, then draw out parts of the list underneath it that are called during the recursion. Helps with tip 1. 5. See tip 1. Anyways, let's examine some common runtimes (keep scrolling). Remember, this is in no way a comprehensive list, NOR IS IT TRYING TO TEACH YOU HOW TO FIND THEM. This post is just to give you a starting point into orders of growth by showing you some examples and basic details about each runtime. Constant $\Theta(1)$ What it looks like: Example: def add(x, y): return x + y $add \in \Theta(1)$, where 1 is.. well a constant... Approach: The key behind constant time functions is that regardless of the size of the input, they always run the same number of instructions. Don’t fall for this Trap: def bar(n): if n % 7 == 0: return "Bzzst" else: return bar(n -1) $\mathtt{bar} \in \Theta(1)$. Why? Logarithmic $\Theta(\log(n))$ What it looks like: Example: def binary_search(sorted_L, n): """ sorted_L is a list of numbers sorted from smallest to largest """ if sorted_L == []: return False mid_num = sorted_L[len(sorted_L) // 2] if n == mid_num: return True elif n < mid_num: return binary_search(sorted_L[:mid_num], n) else: return binary_search(sorted_L[mid_num:], n) $\mathtt{binary\_search} \in \Theta(log(n))$, where $n$ is the number of elements in sorted_L. Approach: Logarithmic functions scale down the size of the problem by some constant every iteration (either with a recursive loop, a for loop, or a while loop). Also, logarithmic functions do not branch out--they generally do not make more than one call to themselves per recursion. Linear $\Theta(n)$ What it looks like: Examples: def sum_list(L): sum = 0 for e in L: sum += e return sum $\mathtt{sum\_list} \in \Theta(n)$, where $n$ is the number of elements in $L$. </pre>def countdown(n): if n > 0: print(n) countdown(n - 1) else: print("Blast off!")</pre> $\mathtt{countdown} \in \Theta(n)$, where $n$ is n. Approach: Linear functions usually act on sequences or other collections of data. In that case, the function will go through the elements once or twice or k times, where $k<<n$. If the function acts on a number, the number usually gets smaller by a constant each iteration. Don't fall for this trap: def two_for_loops(n): for a in range(n): if n == 4: for y in range(n): else: print("It's a trap!") $\mathtt{two\_for\_loops} \in \Theta(n)$, where $n$ is n. Why? Loglinear/Linearithmic $\Theta(n \log(n))$ What it looks like: Example: def merge(s1, s2): if len(s1) == 0: return s2 elif len(s2) == 0: return s1 elif s1[0] < s2[0]: return [s1[0]] + merge(s1[1:], s2) else: return [s2[0]] + merge(s1, s2[1:]) def mergesort(lst): if len(lst) <= 1: return lst else: middle = len(lst) // 2 return merge(mergesort(lst[:middle]), \ mergesort(lst[middle:])) $\mathtt{mergesort} \in \Theta(n \log(n))$, where $n$ is the number of elements in lst. Approach: These functions tend to make two recursive calls, each making the problem smaller by a half. There's a neat way to see this. For example in mergesort, start with an entire line, which represents mergesort called on the initial list. From there, the list gets split in half by the two recursive calls to mergesort in the code, so draw the another line right below the first, of the same length, but with a small gap in the middle to represent the split. Repeat until you're tired. At the end, you get a rectangle that's nwide and (n)tall! --------------- ------- ------- --- --- --- --- - - - - - - - - The total area is the runtime, $\Theta(n \log(n))$ Don’t fall for this trap: Don’t confuse functions that have an average running time of n(n)(like quicksort) with functions that are in (n(n)) Polynomial $\Theta(n^{2})$,$\Theta(n^{3})$, etc. What it looks like: Example: def print_a_grid(n): for _ in range(n): for _ in range(n): print("+", end="") print("") $\mathtt{print\_a\_grid} \in \Theta(n^{2})$, where $n$ is n. Approach: Polynomial functions will examine each element of an input many, many times, as opposed to linear functions, which examine some constant number of times. Don’t fall into this trap: Don’t get polynomial confused with exponential (below). Exponential What it looks like: Example: (define (strange-add x) (if (zero? x) 1 (+ (strange-add (- x 1)) (strange-add (- x 1)) ))) if x == 0: return 1 else: return strange_add(x - 1) + strange_add(x - 1) $\mathtt{strange\_add} \in \Theta(2^{n})$, where $n$ is x. Approach: Exponential functions tend to branch out as you get deeper and deeper into their call tree, and each call only makes the work smaller by a little bit. For example, (strange-add 8) calls (strange-add 7) and (strange-add 7). Those two calls each make two calls, (strange-add 6), (strange-add 6), (strange-add 6), and (strange-add 6) respectively, and so on. ## Object-oriented programming ### Inheritance and class vs instance attributes Student Question I'm confused on how Classes and Inheritance work. If there's a Parent class and a Child class, when coding in the Child class, when do you write Parent.attribute, when do you write Child.attribute, and when do you write self.attribute? Also, I'm also confused as to when to put self into the parentheses as well. Parent.attribute and Child.attribute would both be ways of accessing aclass variable. These are variables that can be accessed without creating new instances of the that class. self.attribute would be used in methods to access an instance variable (an attribute specific to an instance). So for example, Insect.watersafe is False, but Bee.watersafe is True. These are class attributes because you don't have to create an Insect object or a Bee object in order to say Insect.watersafe or Bee.watersafe. However it wouldn't make any sense to say Bee.armor, since armor is an instance variable. You have to first create a new Bee before you could ask it for it's armor. If you created a second Bee after that, the second Bee would also have its own armor. There's a lot of vocab (in bold) that might trip you up. Try reading Discussion 6 and posting a followup if you're still unsure! ## Scheme ### Tail recursion #### Tail recursion in Python In this page, we’re going to look at tail call recursion and see how to force Python to let us eliminate tail calls by using a trampoline. We will go through two iterations of the design: first to get it to work, and second to try to make the syntax seem reasonable. I would not consider this a useful technique in itself, but I do think it’s a good example which shows off some of the power of decorators. The first thing we should be clear about is the definition of a tail call. The “call” part means that we are considering function calls, and the “tail” part means that, of those, we are considering calls which are the last thing a function does before it returns. In the following example, the recursive call to f is a tail call (the use of the variable ret is immaterial because it just connects the result of the call to f to the return statement), and the call to g is not a tail call because the operation of adding one is done after g returns (so it’s not in “tail position”). def f(n) : if n > 0 : n -= 1 ret = f(n) return ret else : ret = g(n) return ret + 1 1. Why tail calls matter Recursive tail calls can be replaced by jumps. This is called “tail call eliminination,” and is a transformation that can help limit the maximum stack depth used by a recursive function, with the benefit of reducing memory traffic by not having to allocate stack frames. Sometimes, recursive function which wouldn’t ordinarily be able to run due to stack overflow are transformed into function which can. Because of the benefits, some compilers (like gcc) perform tail call elimination[1], replacing recursive tail calls with jumps (and, depending on the language and circumstances, tail calls to other functions can sometimes be replaced with stack massaging and a jump). In the following example, we will eliminate the tail calls in a piece of code which does a binary search. It has two recursive tail calls. def binary_search(x, lst, low=None, high=None) : if low == None : low = 0 if high == None : high = len(lst)-1 mid = low + (high - low) // 2 if low > high : return None elif lst[mid] == x : return mid elif lst[mid] > x : return binary_search(x, lst, low, mid-1) else : return binary_search(x, lst, mid+1, high) Supposing Python had a goto statement, we could replace the tail calls with a jump to the beginning of the function, modifying the arguments at the call sites appropriately: def binary_search(x, lst, low=None, high=None) : start: if low == None : low = 0 if high == None : high = len(lst)-1 mid = low + (high - low) // 2 if low > high : return None elif lst[mid] == x : return mid elif lst[mid] > x : (x, lst, low, high) = (x, lst, low, mid-1) goto start else : (x, lst, low, high) = (x, lst, mid+1, high) goto start which, one can observe, can be written in actual Python as def binary_search(x, lst, low=None, high=None) : if low == None : low = 0 if high == None : high = len(lst)-1 while True : mid = low + (high - low) // 2 if low > high : return None elif lst[mid] == x : return mid elif lst[mid] > x : high = mid - 1 else : low = mid + 1 I haven’t tested the speed difference between this iterative version and the original recursive version, but I would expect it to be quite a bit faster because of there being much, much less memory traffic. Unfortunately, the transformation makes it harder to prove the binary search is correct in the resulting code. With the original recursive algorithm, it is almost trivial by induction. Programming languages like Scheme depend on tail calls being eliminated for control flow, and it’s also necessary for continuation passing style.[2] 2. A first attempt Our running example is going to be the factorial function (a classic), written with an accumulator argument so that its recursive call is a tail call: def fact(n, r=1) : if n <= 1 : return r else : return fact(n-1, nr) If n is too large, then this recursive function will overflow the stack, despite the fact that Python can deal with really big integers. On my machine, it can compute fact(999), but fact(1000) results in a sad RuntimeError: Maximum recursion depth exceeded. One solution is to modify fact to return objects which represent tail calls and then to build a trampoline underneath fact which executes these tail calls after fact returns. This way, the stack depth will only contain two stack frames: one for the trampoline and another for each call to fact. First, we define a tail call object which reifies the concept of a tail call: class TailCall(object) : def __init__(self, call, args, *kwargs) : self.call = call self.args = args self.kwargs = kwargs def handle(self) : return self.call(self.args, *self.kwargs) This is basically just the thunk lambda : call(args, *kwargs), but we don’t use a thunk because we would like to be able to differentiate between a tail call and returning a function as a value. The next ingredient is a function which wraps a trampoline around an arbitrary function: def t(f) : def _f(args, *kwargs) : ret = f(args, *kwargs) while type(ret) is TailCall : ret = ret.handle() return ret return _f Then, we modify fact to be def fact(n, r=1) : if n <= 1 : return r else : return TailCall(fact, n-1, nr) Now, instead of calling fact(n), we must instead invoke t(fact)(n) (otherwise we’d just get a TailCall object). This isn’t that bad: we can get tail calls of arbitrary depth, and it’s Pythonic in the sense that the user must explicitly label the tail calls, limiting the amount of unexpected magic. But, can we eliminate the need to wrap t around the initial call? I myself find it unclean to have to write that t because it makes calling fact different from calling a normal function (which is how it was before the transformation). 3. A second attempt The basic idea is that we will redefine fact to roughly be t(fact). It’s tempting to just use t as a decorator: @t def fact(n, r=1) : if n <= 1 : return r else : return TailCall(fact, n-1, nr) (which, if you aren’t familiar with decorator syntax, is equivalent to writing fact = t(fact) right after the function definition). However, there is a problem with this in that the fact in the returned tail call is bound to t(fact), so the trampoline will recursively call the trampoline, completely defeating the purpose of our work. In fact, the situation is now worse than before: on my machine, fact(333) causes a RuntimeError! For this solution, the first ingredient is the following class, which defines the trampoline as before, but wraps it in a new type so we can distinguish a trampolined function from a plain old function: class TailCaller(object) : def __init__(self, f) : self.f = f def __call__(self, args, *kwargs) : ret = self.f(args, *kwargs) while type(ret) is TailCall : ret = ret.handle() return ret and then we modify TailCall to be aware of TailCallers: class TailCall(object) : def __init__(self, call, args, *kwargs) : self.call = call self.args = args self.kwargs = kwargs def handle(self) : if type(self.call) is TailCaller : return self.call.f(self.args, *self.kwargs) else : return self.call(self.args, *self.kwargs) Since classes are function-like and return their constructed object, we can just decorate our factorial function with TailCaller: @TailCaller def fact(n, r=1) : if n <= 1 : return r else : return TailCall(fact, n-1, nr) And then we can call fact directly with large numbers! Also, unlike in the first attempt, we can now have mutually recursive functions which all perform tail calls. The first-called TailCall object will handle all the trampolining. If we wanted, we could also define the following function to make the argument lists for tail calls be more consistent with those for normal function calls:[3] def tailcall(f) : def _f(args, kwargs) : return TailCall(f, args, *kwargs) return _f and then fact could be rewritten as @TailCaller def fact(n, r=1) : if n <= 1 : return r else : return tailcall(fact)(n-1, nr) One would hope that marking the tail calls manually could just be done away with, but I can’t think of any way to detect whether a call is a tail call without inspecting the source code. Perhaps an idea for further work is to convince Guido von Rossum that Python should support tail recursion (which is quite unlikely to happen). [1] This is compiler-writer speak. For some reason, “elimination” is what you do when you replace a computation with something equivalent. In this case, it’s true that the call is being eliminated, but in its place there’s a jump. The same is true for “common subexpression elimination” (known as CSE), which takes, for instance, a = b + c d = (b + c) + e and replaces it with a = b + c d = a + e Sure, the b+c is eliminated from the second statement, but it’s not really gone... The optimization known as “dead code elimination” actually eliminates something, but that’s because dead code has no effect, and so it can be removed (that is, be replaced with nothing). [2] In Scheme, all loops are written as recursive functions since tail calls are the pure way of redefining variables (this is the same technique Haskell uses). For instance, to print the numbers from 1 to 100, you’d write (let next ((n 1)) (if (<= n 100) (begin (display n) (newline) (next (+ n 1))))) where next is bound to be a one-argument function which takes one argument, n, and which has the body of the let statement as its body. If that 100 were some arbitrarily large number, the tail call to next had better be handled as a jump, otherwise the stack would overflow! And there’s no other reasonable way to write such a loop! Continuation passing style is commonly used to handle exceptions and backtracking. You write functions of the form (define (f cont) (let ((cont2 (lambda ... (cont ...) ...))) (g cont2))) along with functions which take multiple such f’s and combines them into another function which also takes a single cont argument. I’ll probably talk about this more in another page, but for now notice how the call to g is in the tail position. [3] This is basically a curried[4] version of TailCall. [4] That is, Schönfinkelized. ## Python syntax and semantics ### print vs return #### Andrew's tips Remember the differences between return and print. • return can only be used in a def statement. It returns a value from a function. Once Python evaluates a return statement, it immediately exits the function. • print is a function that displays its argument on the screen. It always returns None. Examples: def foo1(x): return x def foo2(x): print(x) >>> foo2(1) # In foo2, we print 1 ourselves using the print function 1 >>> foo1(1) # HERE, THE PYTHON INTERPRETER PRINTS THE RETURN VALUE OF FOO1. CANNOT STRESS HOW IMPORTANT TO UNDERSTAND THIS 1 >>> foo1(1) + 1 2 >>> foo2(1) + 1 1 Traceback (most recent call last): File "<stdin>", line 1, in <module> TypeError: unsupported operand type(s) for +: 'NoneType' and 'int' ### Function decorators #### How function decorators work Student Question I'm having difficulties understanding what exactly a function decorator is. Can someone elaborate and potentially provide me with an example other than the one in the readings? So imagine you wanted your functions to print their arguments before they executed them. Here's one way to do this. def loud(fn): def new_fn(args): print(args) return fn(args) return new_fn Here's a function loud that takes in a function and returns a new function that when called, prints out its arguments, and then does what the old function does. For example: def sq(x): return x * x >>> sq(4) 16 >>> sq = loud(sq) # replace the old square with our loud one. >>> sq(4) (4,) 16 A function decorator does the same thing as the above. Assuming loud is defined, we can do this: @loud def sq(x): return x * x >>> sq(4) (4,) ## Student guides ### How to learn computer science If you've never programmed before, or if you've never taken a class quite like 61A before, things right now might be scary. Everything is strange and new and there quite a lot to take in all at once. So if you're having a hard time so far, here are a few articles that might help. Note: these articles are pretty long, so feel free to read them in multiple sittings. At the beginning, everything seems a bit scary in CS. Michelle Bu, a Berkeley alum and a crazy good hacker, shares one of her experiences when she was a wee n00b in 21 Nested Callbacks. Start here! "A Beginner's Guide to Computer Science" Written by Berkeley's own James Maa. James is known for his killer walkthroughs (check out his Productivity guide). This article gives you some background on learning CS and then provides a practical guide on how to learn effectively. How do we learn? Mark Eichenlaub explains in this Introduction to Learning Theory. This is quite possibly the best introduction to Learning Theory. Sometimes, you're stuck and you end up really, really frustrated. Marcus Geduld explains Why do we get frustrated when learning something? ### Quick guide on getting unstuck A major frustration you might encounter in 61A is when you stare at a homework problem and have no idea where to start. Or you write some code and it doesn't pass the doctests, but now what? You work at it for a while, but next thing you know, you've been stuck for hours on the the same problem and have little to show about it. So here's a checklist of things you can do when you're stuck. Experienced programmers do these things almost naturally (because of how much practice they've had being stuck), and so while they get stuck just as much as your or I, they always know what to do next. 1. Do I understand what the problem is asking? 1. If not, which part of the problem is confusing me? 1. Identify the exact sentences/phrases/words/etc. 2. Check the given examples. Do they make sense to me? 3. Can I come up with my own examples? A good indicator that you understand the question is that you can come up with some nontrivial examples of how the function works. 2. What concepts should I use here? 1. Do I understand the concepts? Can I explain the concept in English to one of my friends such that they get it? 1. If not, go back and relearn the specific concepts that are unclear (through discussion, lab, lecture, etc.) Don't read the entire book in order to solve one problem.. 2. How do I apply the concept to the given problem? 3. Write your code and test it. 1. Use doctests, BUT ALSO LOAD IT INTERACTIVELY (python3 -i ...) 1. Saying "my function works because the doctests pass" is a lot like saying "this airplane will fly because it has wings." 2. If your code breaks, ask yourself: 1. Does it error? Is it a.... 1. Syntax error? If so, find the syntax bug and fix it. 2. Logic error? Is it something weird that you don't understand? (E.g. cannot add integer and tuple) 2. Why did it do that? Why didn't it do what I expected? Trace through the code by hand with an example (sample values) you came up with in step 0. Add calls to print in order to figure out how your function is handling the arguments. 4. Am I missing a trick? 1. Oftentimes you've never seen this type of problem before. This is expected on homework (and this is why homework can take a long time) because if you see it on the homework, then you will be familiar with it on the exam and when you program for fun and profit. 2. The key here is just to learn the trick however you need to. 1. Stare at it yourself 2. Stare at it with others (peers in the class) 3. Ask on PIazza what the approach is. 4. Stare at it with the TAs/lab Assistants 3. Once you figure it out, remember the trick so that you can use it next time. 5. At any point you identify what you're stuck on, you can begin to resolve it. 1. Use the tips above. Try things out on the interpreter. Review the lecture/discussion/labs/etc. Do whatever helps you get a better understanding of the problem. 2. Once you have something specific that you're stuck on, you can ask other people in the class. 1. Don't be afraid to ask. Everyone gets stuck and feels stupid sometimes. However, you get to choose how you react to it. 2. At the same time, it really helps to work with people who are on about the same level in the course. 3. Look on Piazza. Ask questions if yours hasn't come up yet. Be that awesome guy/girl who helps answer questions. 4. You can ask the TA if all else fails. We are here to help you learn! Here is an old algorithm for studying for tests (the final in this case), salvaged from the sands of time: For each topic on the final, find problems on them and do them. If you can solve them on your own, move on. Else if you are stuck, look at the solution and figure out if you are missing a trick or if you do not understand the concepts. If the problem is that you are stuck on some random trick, just learn the trick. Stare at the solutions, ask Piazza, your TA, etc. Questions you should ask at this stage: What is the problem asking me to do? How was I suppose to follow the instructions to solve the problem? What part of the problem do I not understand? What is the fastest way to clear up that misunderstanding? Then if you think you are still stuck conceptually, review and learn the concept, however you learn best. Suggestions for picking up concepts quickly (~1-2 hours): Discussion notes typically have a very concise recap of the thing they are going over. There are guides for particularly tricky things on Piazza, like Logic, Pairs and Lists in Scheme, etc. Find them and go over them. Ask a TA: "what is the best way to learn X?" If these do not work and you are still shaky after an hour or two, it might be worth watching a lecture or reading the notes. ## Composition ### General style guidelines from 61A website Student Question Are we required to add any comments to our code to say what a function does, etc.? And does clarity of code count for this project, in which case should we write comments at the end of not-so-clear statements? Thanks. Docstrings of each function are already provided. If you add a helper function, you should write a docstring for it. The style guide on the course website advises: "Your actual code should be self-documenting -- try to make it as obvious as possible what you are doing without resorting to comments. Only use comments if something is not obvious or needs to be explicitly emphasized" You should always aim to make your code "self-documenting," meaning it is clear what your code is doing without the aid of comments. You should try to keep the number of comments to a minimum, but if there are lines which you think are unclear/ambiguous, feel free to add a comment. All projects in this class contain a 3 point component that is judged solely on your code "composition" -- i.e. whether your code is clear, concise, and easy to read. ### Simplifying code Hi everyone, here's some tips about certain functions in Python that can greatly simplify your code for the Trends project. Sorting keys You should be familiar with the max and min functions in python, which can take in many arguments and return the maximum value. >>> max(1,3,2) 3 These functions can also take in lists: >>> min([1,5,1,6]) 1 (In fact they can take in any iterable and return the maximum/minimum value) These functions work because Python knows how to compare the elements in the list (they are all integers). But what if the elements in the list are not integers? Fortunately, there is a way for you to tell Python how to turn each element of the list into a number that it can understand. Lets start with an example. Lets say you have a list of strings, and want to find the shortest string in the list. Here's what you can do: >>> min(['hihi', 'bye', 'a', 'zebra'], key=len) 'a' Notice the new keyword argument key we are passing into the min function. key is a function that min applies to each element of the list. In this case, the key is the len function, which returns the length of each string. Applying the key function to each element will return a cooresponding integer, which Python can easily use to find the minimum element. You can also use keys in the sorted function too, which returns a sorted list of its inputs, based on the key function passed in. >>> sorted(['hihi', 'bye', 'a', 'zebra'], key=len) ['a', 'bye', 'hihi', 'zebra'] We can have more complex key functions. Here we sort a list of people by their age, which is the second element in the tuple. A key function, once defined, works for sorted, min and max: >>> names = [('Alice', 19, 'F'), ('Bob', 5, 'M'), ('Charlie', 12, 'M')] >>> get_age = lambda name: name[1] >>> sorted(names, key=get_age) [('Bob', 5, 'M'), ('Charlie', 12, 'M'), ('Alice', 19, 'F')] >>> max(names, key=get_age) ('Alice', 19, 'F') Dictionary default values Suppose we have a dictionary mapping names to counts: >>> d = {'apples': 1, 'pears': 9000} If we want to add a new pear to the dictionary, we can use: >>> d['pears'] = d['pears'] + 1 >>> d {'apples': 1, 'pears': 9001} However we cannot use the same code to add a new item that is not already in the dictionary. >>> d['oranges'] = d['oranges'] + 1 Traceback (most recent call last): ... KeyError: 'oranges' To solve this problem, we have to use dict.setdefault(key, default). If key is in dict, it will return dict[key]. If not, it will insert key with a value of default and return default. Now we can write: >>> d['oranges'] = d.setdefault('oranges', 0) + 1 >>> d {'oranges': 1, 'apples': 1, 'pears': 9001} >>> d['oranges'] = d.setdefault('oranges', 0) + 1 >>> d {'oranges': 2, 'apples': 1, 'pears': 9001} There's actually a even better way of doing this. If you are curious to find out, look up collections.defaultdict. For loops If you are iterating through a list and want to get both the item and the index the item is at, the built-in function enumerate is helpful here. >>> a = ["apple", "pear", "orange"] >>> for index, fruit in enumerate(a): ... print(index, fruit) ... 0 apple 1 pear 2 orange You can iterate through each key-value pair in a dictionary with dictionary.items. This is useful if you want to access both the key and the value at the same time. >>> prices = {"apple": 3, "pear": 5, "orange": 20} >>> for fruit, price in prices.items(): ... print(fruit, price) ... apple 3 pear 5 orange 20 Hope this helps for the project! ## Miscellaneous ### Andrew Huang's tips Order of evaluation matters. The rules for evaluating call expressions are 1. Evaluate the operator 2. Evaluate the operands 3. Call the operator on the operands (and draw a new frame...) For example: def baz(): print("this was first") def bar(x): print(x) return lambda x: x * x return bar # baz is a function that when called, returns a function named bar >>> baz() # the operator is baz, there are no operands this was first <function bar at 0x2797e20> >>> baz()("this was second") # the operator is baz(), the operand is "this was second" this was first this was second <function <lambda> at 0x2120e20> >>> baz()("this was second")(3) # the operator is baz()("this was second"), the operand is 3 this was first this was second 9 >>> def bar(x): ... print(x) ... return 3 ... >>> baz()("this was second")(bar("this was third")) # the operator is baz()("this was second"), the operand is bar("this was third") this was first this was second this was third 9 In order to solve any problem, you must first understand what the problem is asking. Often times it helps to try to explain it concisely in English. It also helps to come up with small examples. For example: def mouse(n): if n >= 10: squeak = n // 100 n = frog(squeak) + n % 10 return n def frog(croak): if croak == 0: return 1 else: return 10 * mouse(croak+1) mouse(21023508479) So the goal is to figure out what mouse(21023508479) evaluates to. One way is to just step-by-step evaluate this, as an interpreter would. Another way, is to understand what the functions are doing. Looking at mouse, we see that it takes in a number and outputs that same number if it is smaller than 10. otherwise, it'll return something weird. In order to understand that weird thing, we have to understand what frog is doing. frog takes in a number and if that number is 0, return 1. Otherwise, return ten times mouse(croak+1). Well, this is still confusing. Let's try a small example. >>> mouse(357) 47 >>> mouse(123) 23 >>> mouse(1234) 44 >>> mouse(12345) 245 There is a pattern. We notice that the resulting number is composed of every other digit of the original, plus one (except for the last one.) So 21023508479 is [2+1][0+1][3+1][0+1][4+1][9] = 314159. Can you see how the code reflects that? However in this particular example, the pattern is definitely tricky to find here, so it might make more sense to brute force it. Remember for recursion, you always need to find three things: • One or more base cases • One or more was to reduce the problem • A way to solve the problem given solutions to smaller problems For example, the discussion notes, we asked you to write count_stairs. This function takes in n, the number of steps, and returns all the ways you can climb up them if at each step, you can take either one or two steps. • Base cases: if we consider n to be the number of steps left to climb, then it makes sense that if there is 1 step left, then there is exactly one way. If there are two steps left, then there are exactly 2 ways (1 step, 1 step, or two steps). Why do we need two base cases here? • We can make the problem smaller by reducing the n. At each step, we can take one step (resulting in count_stairs(n-1)) or two steps (count_stairs(n-2)). • Assuming we get the solutions to the two recursive calls, we should add them together to get all the ways we can climb the stairs. Thus we end up with def count_stairs(n): if n <= 2: return n else: return count_stairs(n-1) + count_stars(n-2) Notice that at each stair step, we either take one step or two steps. This is a common pattern in tree recursion. Look through Discussion 3 for more info. ### Y combinators (in Scheme) Student Question Can someone explain this to me? scm> (((lambda (f) (lambda (x) (f f x))) (lambda (f k) (if (zero? k) 1 (* k (f f (- k 1)))))) 5) I've edited the code as follows: ( ( (lambda (f) (lambda (x) (f f x)) ) (lambda (f k) (if (zero? k) 1 (* k (f f (- k 1))) ) ) ) 5 ) My understanding is that the second lambda function is passed as the first f in the first lambda function and the 5 is passed in as x. But does that mean f f x becomes the second lambda function with itself and x passed as the arguments to (f k)? You're on the right track. The first lambda function is a higher order function that takes in a function, and then returns a function that takes one argument. It's actually the third lambda that is then passed into the first lambda (currying!) and then 5 is then passed into the resulting function. In case you're curious, this is the Python equivalent: >>> (lambda f: (lambda x: f(f, x))) (lambda f, k: 1 if k == 0 else (k * f(f, k - 1)))(5) Which is then equivalent to: >>> def func1(f): def func2(x): return f(f, x) return func2 >>> def func3(f, k): if k == 0: return 1 else: return k * f(f, k - 1) >>> func1(func3)(5) 120 By the way, this is just a fancy way of recursively calculating the factorial using only lambda functions. If you're still curious as to how this works, you could try this in Python tutor. Except I would recommend calculating 3! instead of 5, because it's a lot of frames. (lambda f: lambda x: f(f, x))(lambda g, k: 1 if k == 0 else (k * g(g, k-1)))(5) So the idea is, you define a lambda function that takes a function f, and that returns a lambda function that takes an argument x and returns f(f, x). Then, you call this lambda function you just defined on another lambda function (let's call this func) that takes a function g and another argument k, and is basically the factorial function. This first call returns the inner lambda of the first part, and when that's called with 5 you're essentially calling func(func, 5). The chain of recursive calls then works as follows: func(func, 5) -> 5 * func(func, 4) -> 5 * 4 * func(func, 3) -> ... -> 120 In functional programming theory, this is known as a Y Combinator, and it is how you achieve recursion with just lambda functions. If you're wondering why we need func to take in a function as the first parameter, see what would happen if you took that part out!	2022-01-18 03:30:43	{"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": 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": 0.508769690990448, "perplexity": 1642.640433489127}, "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-2022-05/segments/1642320300722.91/warc/CC-MAIN-20220118032342-20220118062342-00047.warc.gz"}
https://int8.io/tag/julia/	Posts tagged with: Julia ## Automatic differentiation for machine learning in Julia Automatic differentiation is a term I first heard of while working on (as it turns out now, a bit cumbersome) implementation of backpropagation algorithm – after all it caused lots of headaches as I had to handle all derivatives myself with almost pen-and-paper like approach. Obviously I made many mistakes until I got my final solution working. At that time, I was aware some libraries like Theano or Tensorflow handle derivatives in a certain “magical” way for free. I never knew exactly what happens deep in the guts of these libraries though and I somehow suspected it is rather painful than fun to grasp (apparently, I was wrong!). I decided to take a shot and directed my first steps towards TensorFlow official documentation to quickly find out what the magic is. The term I was looking for was automatic differentiation. --- ## Chess position evaluation with convolutional neural network in Julia In this post we will try to challenge the problem of chess position evaluation using convolutional neural network (CNN) – neural network type designed to deal with spatial data. We will first explain why we need CNNs then we will present two fundamental CNNs layers. Having some knowledge from the inside of the black box, we will apply CNN to binary classification problem of chess position evaluation using Julia deep learning library – Mocha.jl. ### Introduction – data representation One of the challenges that frequently occurs in machine learning is proper representation of the input data. Ideally, data is desired to be represented in a way that it carries as much information while being digestable for the ML algorithms. Digestibility means fitting in existing mathematical frameworks where known abstract tools can be applied. A common convenient representation of single observation is a vector in $$\mathbb{R}^n$$. Assuming such representation, ML problems may be seen from many different angles – with benefit of using well known abstractions/interpretations. One perspective that is very common is algebraic perspective – having the input data as a matrix (one vector per column), its eigendecomposition or various factorizations may be considered – they both yield important results in the context of machine learning. Set of vectors in $$\mathbb{R}^n$$ shapes a point cloud – when geometry of such cloud is considered manifold learning methods emerge. Linear model with least squares error has closed form solution in algebraic framework. In all of these cases, representing input data as vectors implies broad range of tools to handle the problem effectively. For some domains though it is not obvious how to represent input as vectors while preserving original information contained in the data. An example of such domain is text. Text document is rich in various types of information – there is a semantics and syntax of the text or even personal style of the writer. It is not clear how to represent this unnamed information contained in text. People tend to simplify it and use Bag of Words (BoW) approach to represent text (which completely ignores ordering of words in a document – treats it a a set). Another domain that suffers from similar problem is domain of images. The spatiality of the data is missing when representing images as vectors of dimensionality equal to the total number of pixels. When one represents image that way the spatial information is lost – the algorithm that later consumes the input vectors is usually not aware the original structure of images is a set of 2-dimensional grids (one matrix for each channel). So far our neural network has not been aware of two dimensional nature of input data (MNIST). It could of course find it out itself learning relations between neighboring pixels, but, the fact is, it had no clue so far. --- ## Neural Networks in Julia – Hyperbolic tangent and ReLU neurons get the code from here Our goal for this post is to introduce and implement new types of neural network nodes using Julia language. These nodes are called ‘new’ because this post loosely refers to the existing code. So far we introduced sigmoid and linear layers and today we will describe another two types of neurons. First we will look at hyperbolic tangent that will turn out to be similar (in shape at least) to sigmoid. Then we will focus on ReLU (rectifier linear unit) that on the other hand is slightly different as it in fact represents non-differentiable function. Both yield strong practical implications (with ReLU being considered more important recently – especially when considered in the context of networks with many hidden layers). What is the most important though, adding different types of neurons to neural network changes the function it represents and so its expressiveness, lets then emphasize this as the main reason they are being added. #### Hyperbolic tangent layer From the biological perspective, the purpose of sigmoid activation function as single node ‘crunching function’ is to model passing an electrical signal from one neuron to another in brain. Strength of that signal is expressed by a number from $$(0,1)$$ and it relies on signal from the input neurons connected to the one under consideration. Hyperbolic tangent is yet another way of modelling it. Let’s first take a look at the form of hyperbolic tangent: $f(x) = \frac{\mathrm{e}^x – \mathrm{e}^{-x}}{\mathrm{e}^x + \mathrm{e}^{-x}}$ Continue Reading ---	2023-03-30 21:36:25	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.5950038433074951, "perplexity": 722.5183048828811}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296949387.98/warc/CC-MAIN-20230330194843-20230330224843-00799.warc.gz"}
https://codereview.stackexchange.com/questions/40381/code-first-entity-framework/40424	# Code First Entity Framework Is the following good design for doing entity framework code first? What am I missing for a production system? I haven't included all my code, just a snapshot... My application, doesn't update the database; it just reads it. public class Document { [Key, ScaffoldColumn(false)] public Int32 DocumentID { get; set; } [ScaffoldColumn(false)] public Boolean? Status { get; set; } [Display(Name = "Document Text"), DataType(DataType.MultilineText)] public String DocumentText { get; set; } [ScaffoldColumn(false), DataType(DataType.Url)] public String DocumentFolderPath { get; set; } [ScaffoldColumn(false), DataType(DataType.Url)] public String DocumentJSONPath { get; set; } [ForeignKey("DocumentID")] [ForeignKey("DocumentID")] public virtual ICollection<Page> Pages { get; set; } } public class Page { [Key, ScaffoldColumn(false)] public Int32 PageID { get; set; } [Required, Display(Name = "Page"), DataType(DataType.Text)] public Int32 PageNumber { get; set; } [Required, ScaffoldColumn(false)] public Int32 DocumentID { get; set; } [ScaffoldColumn(false), DataType(DataType.ImageUrl)] public String ImagePath { get; set; } [Required, Display(Name = "Page Text"), DataType(DataType.MultilineText)] public String PageText { get; set; } [NotMapped] public String HighlightedText { get; set; } [ForeignKey("PageID")] public virtual ICollection<Word> Words { get; set; } } public class ArchiveDatabaseInitializer : DropCreateDatabaseIfModelChanges<ArchiveContext> { protected override void Seed(ArchiveContext context) { try { context.SaveChanges(); } catch (Exception ex) { throw ex; } } private static List<Document> GetDocuments() { var documents = new List<Document> { new Document { DocumentID = 1, DocumentFolderPath = @"Doc1", DocumentJSONPath = @"Doc1.json", Status = true }, new Document { DocumentID = 2, DocumentFolderPath = @"Doc2", DocumentJSONPath = @"Doc2.json", Status = true } }; return documents; } public class ArchiveContext : DbContext { public ArchiveContext() : base("archiveDB") { } public DbSet<Document> Documents { get; set; } public DbSet<Page> Pages { get; set; } public DbSet<Word> Words { get; set; } } • What makes you think you're missing anything for a production system? – Mathieu Guindon Jan 29 '14 at 17:52 • This is the first time I've used EF so I don't know what I don't know. ;) – Tums Jan 29 '14 at 17:54 • You got it working? – Mathieu Guindon Jan 29 '14 at 18:00 • I don't think the ICollection<Page> navigation property can have a FK attribute AND work. But I could be wrong. It just... looks weird to me. I've favorited this question, I'll get back to it when I have a chance :) – Mathieu Guindon Jan 29 '14 at 18:38 • Yes it is working. It looks like it gets the data as expected. I just want to know if this is the correct way of doing this. – Tums Jan 29 '14 at 20:33 If you chose the code-first approach, I presume you don't have to deal with the constraints of an existing database. If that assumption is correct, then I'd say you've done it the hard way. Convention Over Configuration Entity Framework can infer keys (primary and foreign) from the names of your entity members; your code isn't leveraging this formidable capability. For a greenfield project, I like to start with a base entity type: public abstract class EntityBase { public int Id { get; set; } // inferred PK public DateTime DateInserted { get; set; } public DateTime? DateUpdated { get; set; } } Now I can derive, say, Document from that class: public class Document : EntityBase { public bool? Status { get; set; } public string DocumentText { get; set; } // ... } The PK is inferred from the inherited Id property (or [TypeName]Id), and the mere mention of a virtual property referencing another entity type is enough for EF to understand you want a FK there. I'll pause here to mention that I was somewhat thrown off by your non-usage of the C# language aliases for System.String, System.Int32 and System.Boolean. They're typically written as string, int and bool. You mention you're only reading from the database. This sounds like you're going code-first with an existing database. It's certainly possible, but the best way to do this in my opinion, is to reverse-engineer the database into entity classes - then you get the best of both worlds, and the generated code can teach you a lot about how EF works. I think the presence of DisplayAttribute and NotMappedAttribute smells like you're possibly using your entities as both data entities and presentation / ViewModels. I think I would separate these concerns and create separate classes for presentation purposes, where HighlightedText means something relevant. Also note that relying on a declarative DisplayAttribute will make it quite a pain to localize the application, if you ever need to do that in the future; you'll want these things resolved at run-time, accounting for Thread.CurrentThread.CurrentUICulture. The naming is overall ok, except I don't like the Status column. It's just too vague of a name. What do the null, true and false values mean? The name should convey that meaning. Also I wouldn't prefix most of Document's members with the word "Document", and ID should follow the PascalCasing convention and be Id. • Thank you for the detailed response. I will be updating my code with your suggestions. About the FK being explicitly set, when I didn't have it there, the database did not create a foreign key in the database. It may be because it wasn't recognizing my id's as keys. I will try it and find out. – Tums Jan 30 '14 at 6:51	2019-11-15 22:05:31	{"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": 0.22756250202655792, "perplexity": 5834.345583471235}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496668712.57/warc/CC-MAIN-20191115195132-20191115223132-00433.warc.gz"}
https://mathematica.stackexchange.com/questions/136381/how-to-show-the-fraction-line-as-a-slash-instead-of-horizontal-bar-in-plot-tick/218884	# How to show the fraction line as a slash instead of horizontal bar in plot tick labels? Is there any option to show the fraction numbers in the tick labels of a plot in the format of 2/3 instead of using a horizontal bar as the fraction line? You could specify a custom label for each tick, programmatically generated using the InputForm of the fraction, which uses the horizontal dash. For instance: Plot[ x, {x, 0, 3}, Ticks -> {{#, InputForm[#]} & /@ Range[0, 3, 1/3], Automatic} ] To address your comment regarding the dropped labels, you may be better off using a Frame and FrameLabels instead: Plot[ x, {x, 0, 3}, Frame -> {{True, False}, {True, False}}, FrameTicks -> {{#, InputForm[#]} & /@ Range[0, 3, 1/3], Automatic} ] • Thank you. I noticed that Mathematica sometimes omits some tick labels, which is also the case in your plot above. Is there a way of letting it not omit any tick labels if they are given in a list, e.g. the 0 in your list Range[0, 3, 1/3]? – nanjun Jan 27 '17 at 22:21 • @nanjun Yes, but perhaps a frame would be easier to use in that case; see my updated answer. – MarcoB Jan 27 '17 at 22:27 • I wonder what is the criterion that Mathematica drops labels? Using frame is a way around but it shows a little bit more before 0. Is it possible to disable it from dropping labels? – nanjun Jan 27 '17 at 22:31 • @nanjun It drops that $0$ label because, when axes are present, it would be positioned at the axis intersection point, and it would be awkwardly placed. I don't know of a simple way of forcing its placement. Additionally, the amount of padding on each axis (i.e. the amount of "extra space" you were referring to) should be exactly the same whether you are using axes or frames, and is controlled by PlotRangePadding which for Plot is set as 2% by default (i.e. Scaled[0.02]), if I remember correctly. – MarcoB Jan 27 '17 at 22:34 If one is allowed to use a package, here is another possibility. First load MaTeX then SetOptions[MaTeX, "Preamble" -> {"\\usepackage{nicefrac}"}]; makeFrac[n_] := If[Denominator[n] == 1, MaTeX[n, Magnification -> 1.5], MaTeX["\\nicefrac{" <> ToString[Numerator[n]] <> "}{" <> ToString[Denominator[n]] <> "}", Magnification -> 1.5]]; texStyle = {FontFamily -> "Latin Modern Roman", FontSize -> 14}; Plot[x, {x, 0, 3}, Ticks -> {{#, makeFrac[#]} & /@ Range[0, 3, 1/3], Automatic}, BaseStyle -> texStyle] I would do both axes using Latex myself and not one to make it look better. And I think the fractions look better than using Mathematica's own typesetting ;) ClearAll[fractionToRow] fractionToRow = RawBoxes @* ReplaceAll[FractionBox -> (RowBox[{#, "/", #2}] &)] @* ToBoxes @* TraditionalForm plt1 = Plot[Sin@x, {x, 0, 2 Pi}, ImageSize -> 300, Ticks -> {Range[0, 2 Pi, Pi/2], Automatic}]; plt2 = Plot[Sin@x, {x, 0, 2 Pi}, ImageSize -> 300, Ticks -> {{#, fractionToRow @ #} & /@ Range[0, 2 Pi, Pi/2], Automatic}]; Row[{plt1, plt2}, Spacer[10]] This approach works for simple inputs (as is usually the case for tick labeling). Another example: TraditionalForm[Style[Pi^(I/2)/ (I + E/Pi), 24]] fractionToRow2 @ Style[Pi^(I/2)/ (I + E/Pi), 24] \ For more complicated inputs more work may be needed for appropriate parsing. • @Szabolcs, thank you for the generous bounty. – kglr Apr 14 at 5:25	2020-08-15 06:15:54	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2724825143814087, "perplexity": 3388.3281561622885}, "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/1596439740679.96/warc/CC-MAIN-20200815035250-20200815065250-00497.warc.gz"}
http://www.oxfordmathcenter.com/drupal7/node/560	# Solution Determine if the following is an identity: $$\displaystyle{\sec^2 \theta + \csc^2 \theta = \sec^2 \theta \csc^2 \theta}$$ First we attempt to simplify the left side: $$\begin{array}{rclc} \sec^2 \theta + \csc^2 \theta &=& \frac{1}{\cos^2 \theta} + \frac{1}{\sin^2 \theta} & \scriptsize{\textrm{rewriting in terms of } \sin \theta \textrm{ and } \cos \theta}\\\\ &=& \frac{\sin^2 \theta}{\cos^2 \theta \sin^2 \theta} + \frac{\cos^2 \theta}{\cos^2 \theta \sin^2 \theta} & \overset{\normalsize{\textrm{getting common denominators}}}{\scriptsize{\textrm{so these fractions can be combined}}}\\\\ &=& \frac{\sin^2 \theta + \cos^2 \theta}{\cos^2 \theta \sin^2 \theta} & \scriptsize{\textrm{now recall } \sin^2 \theta + \cos^2 \theta = 1}\\\\ &=& \frac{1}{\cos^2 \theta \sin^2 \theta} & \end{array}$$ Which gets us to within one step of the right side, as: $$\begin{array}{rclc} \sec^2 \theta \csc^2 \theta &=& \frac{1}{\cos^2 \theta \sin^2 \theta} & \quad \scriptsize{\textrm{rewriting in terms of } \sin \theta \textrm{ and } \cos \theta}\\\\ \end{array}$$ Thus, the equation above is an identity.	2018-01-17 22:20:56	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.634125292301178, "perplexity": 588.5614397065054}, "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/1516084886979.9/warc/CC-MAIN-20180117212700-20180117232700-00417.warc.gz"}
https://chemistry.stackexchange.com/questions/88281/mathematical-expression-for-limiting-molar-conductivity	# Mathematical Expression for Limiting Molar Conductivity I have been fiddling around with the theory of electrolytes, specifically molar conductivity, and ways of calculating limiting molar conductivity $\Lambda^0$. I was able to come up with the following expression: $$\Lambda^0 = \frac{\lambda}{mT}\|z_+z_-\|\, ,$$ where $m$ – mass of a single ion in solution (in $\pu{kg}$); $T$ – absolute temperature (in $\pu{K}$); $z_i$ – charge of a single ion (positive or negative); $\lambda$ is a constant given by $$\lambda = \frac{N_\mathrm{A}he^2}{k_\mathrm{B}} = \frac{hF^2}{R} = \pu{7.42e-25 S kg m2 K mol-1}\, .$$ The above formula works for some chemical species, but fails for others. Using my formula, $\Lambda^0$ for $\ce{HCl}$ at $\pu{298 K}$ ($m = \pu{6.06e-26 kg}$) is $\pu{0.0411 S m2 mol-1}$, which agrees well with the experimental value of $\pu{0.0426 S m2 mol-1}$. Constants in the expression for $\lambda$ should be familiar to anyone who studies physical chemistry. Have I found a new (easier!) approach of calculating $\Lambda^0$, or is my discovery just a coincidence (or, has this already been discovered)? If it is indeed novel, any ideas on how to get a more accurate formula? • Welcome to chemistry.SE! If you have any questions about the policies of our community, please ‎visit the help center. – airhuff Jan 4 '18 at 5:06 • Did you come across your expression by guesswork, or does it have some physical motivation? The latter might present a path for further generalization. Do you notice a pattern among the species for which your expression is accurate and among the species for which it is not? If you do, that might tell you about what interactions you should consider to get a better approximation. – a-cyclohexane-molecule Jan 4 '18 at 5:58 • Actually, based on electrolyte theory, limiting molar conductivity increases with temperature. In my expression, it is inversely proportions to temperature... haven't found anything to explain this yet... but it's remarkable that the expression has the same order of magnitude and correct units nonetheless. – compbiostats Jan 4 '18 at 6:17 • Please visit this page, this page and this one on how to format your future posts better with MathJax and Markdown. I see you are an advanced $\rm \LaTeX$ user, but MathJax is a bit different:) – andselisk Jan 4 '18 at 9:06 • I actually determined the equation from guesswork, so I know of no way to verify it from theory. – compbiostats Jan 5 '18 at 18:58	2019-05-22 03:43:39	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6411081552505493, "perplexity": 611.9900169704811}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232256724.28/warc/CC-MAIN-20190522022933-20190522044933-00496.warc.gz"}
https://gamedev.stackexchange.com/questions/157362/spatial-jitter-problem-in-large-unity-project/157364	# Spatial Jitter problem in large unity project I have a very large environment where i am getting very weird view of objects as picture given below: As you can see there are black lines/spot, maybe the vertex lit problem. Currently I am using Standard specular shader. Remember this problem is only occuring at the end of the project (away from origin like position -17000, 0, 144) As a workaround i tried to change my shader to Unlit and i found this shader Shader "Unlit/UnlitAlphaWithFade" { Properties { _Color ("Color Tint", Color) = (1,1,1,1) _MainTex ("Base (RGB) Alpha (A)", 2D) = "white" } Category { Lighting Off ZWrite Off ZWrite On // uncomment if you have problems like the sprite disappear in some rotations. Cull back Blend SrcAlpha OneMinusSrcAlpha //AlphaTest Greater 0.001 // uncomment if you have problems like the sprites or 3d text have white quads instead of alpha pixels. Tags {Queue=Transparent} { Pass { SetTexture [_MainTex] { ConstantColor [_Color] Combine Texture * constant } } } } } This shader has almost solve the problem but i have to tweak and set colour of each object as be default it is not according to my expectation. The object was far from the origin i.e., it was creating spatial Jitter (SJ) or wobble effects in mesh. The surprising thing was for me that why it is creating problem at position 17000 of x-axis while unity allow value less than 1,000,000 (not restricted to it actually). The problem has solve through floating origin technique, the general idea is that : You usually move more and more away from the origin (0/0/0), so you should try to keep your camera and/or player/point-of-interest always around the origin, instead of letting it travel to vast distances (ref). With regard to your question: "The surprising thing was for me that why it is creating problem at position 17000 of x-axis while unity allow value less than 1,000,000 (not restricted to it actually)." jitter starts to occur earlier than you might expect because the resolution gap of floating point increases (gets worse) almost immediately as you move away from the origin. Specifically, the resolution ("gap error") for a single precision float at the origin is of the order of 10^-7, then at 2 it doubles (gets worse by 2), doubles at 4 and so on with every power of 2 distance from the origin. Explained in thesis: https://doi.org/DOI:10.13140/RG.2.2.10421.32481, section 2.1. This resolution value is the base gap error in each floating point coordinate variable (x,y,z) and rendering calculations magnify it through multiplication, addition etc. As rendering is a very demanding of accuracy, if your viewpoint is close to where the jitter manifests, you may see it at smaller distances than you expect. It might be because of your meshes' vertex normals. They are probably wrong. Unity calculates shadows based on vertex normals, which should be perpendicular to the relative face, like this: If you select the raw mesh in unity, from the project view, and in the inspector you select "recalculate normals" the problem might get fixed. If not, you probably need to remodel the meshes that are corrupted. • No this my meshes are fine when they are near the origin (0,0,0). I guess its distance problem form the origin. – Muhammad Faizan Khan Apr 10 '18 at 10:15 • Mmmmmmh..... that's really weird... are you using the directional light? Or are you using light probes/reflection probes? Are you using the deferred renderer? Are you using SSAO or any other ambient occlusion technique? Right now it could be so many things.. – user115399 Apr 10 '18 at 11:02	2020-11-23 22:22:31	{"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": 0.4000830054283142, "perplexity": 2152.0730406295056}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141168074.3/warc/CC-MAIN-20201123211528-20201124001528-00421.warc.gz"}
https://engineering.stackexchange.com/tags/dynamics/hot	# Tag Info 13 It is done carefully. Lots of other rotating machinery has the same problem, and whole systems exist just to deal with it. For example, jet engines are usually smaller, but also usually spin much faster. Balancing a jet engine is something that gets lots of attention at manufacturing, and again any time when it is put back together after having been ... 10 If we simplify the whole bridge into 2D thin beam with a constant section size, no internal damping and subject only to small vertical deflections, then the natural frequency is determined by simple harmonic motion: $$n_0 = \frac{1}{2 \pi} \sqrt{ \frac{ k } { m } }$$ Where $n_0$ is the natural frequency, $k$ is the ratio between restorative force and ... 10 Designing the lift system for a hovercraft isn't actually about the mass and the acceleration, it's about the pressure required to lift the hull and the rate at which the air leaks out from under the hull and needs to be replaced. Together, these determine the lift power required. Horizontal motion is a completely separate issue, which requires knowledge ... 7 Convert the nonlinear model to state-space form, $x'=f(x,u)$, and linearize it to get a linear state-space or transfer function representation. You can use this to design a PID controller and simulate it that together with the original nonlinear model to see how the controller performs. As noted, the performance of the linear controller will most likely ... 7 Flywheels are used in space, but not for storage of energy. They are used for attitude adjustment, as the spinning of a large mass has significant implications in an object which is free to move. Unlike on Earth, where a flywheel or gyro can be spun up against the force holding it in place (bolted to foundations), and change in spin speed will cause the ... 6 Let's first compute the model. The control design is a separate effort. The torque applied to the drum is $n T_M$, where n is the gear ratio and $T_M$ is the output produced by the motor. $T_M= K_T i(t)$, where $K_T$ is a proportionality constant and $i(t)$ is the motor current. Now we can write the equations for the mechanical system: $$m y''(t)+m g-k (... 5 The ESA has a page on compressor blades. They give a good dimensioned diagram of an approximate shape; here are some basic dimensions: Length: 300 mm Width: 30 mm Height: 70 mm Thickness: 5 mm I can't find any complete open source designs (i.e. high-quality, technical engineering drawings), but this is a good approximation. 5 There are two things going on. First, even if this "actuator" can produce constant torque, the torque required to keep the load spinning will be at least in part a function of the spinning speed. There will be some friction and other forces that increase with increased speed. Viscous friction increases linearly with speed, and other effects, like air ... 5 Take the Fourier Transform of the time varying driving force, this will give the frequency content of the driving force. Multiple modes of vibration can be driven at once, and will superpose with each other, but time varying driving forces with a frequency content that are high at frequencies near a particular resonant frequency will mainly drive the ... 5 To obtain the modes shapes and resonant frequencies, you start from your equation of motion with no externally applied forces, which is indeed as you've stated.$$\mathbf M \mathbf{\ddot q} + \mathbf K \mathbf q = \mathbf 0 \qquad (1)$$For brevity, I've let \mathbf K = \mathbf K_b + \mathbf K_m. Currently, \mathbf q(t) is a function of time. If the ... 5 This is quite easy: Choose one of the wheels build a second wheel so it would satisfy Ackerman condition Get the First rotation point ( O1 ) Do the same with the second wheel, and get a second rotation point ( O1' ) The actual rotation point can be anything between these two points. If you choose, for instance, the first point as rotation point, it means ... 5 You won't get much resonance because the phase the thing is being driven with keeps changing. The resonant thing will act like a notch filter, so you are left with the frequencies near its resonance. However, for resonant energy to build up, the system has to get pumped for a while. While a short segment of the filtered white noise could resonate the ... 5 You didn't account for the acceleration of m_1. Setting up a free body diagram on the weight shows:$$m_1g - T = m_1a_y$$Where T is the tension in the rope, a_y is the acceleration of the block. This leads to the following corrections:$$\tau_1=\mu(2\pi Rh)\frac{R\omega}{a}R$$(The original had a value for force, whereas we need a torque.) ... 4 Advantages I'll start off by quoting from the "Benefits" section of the Wikipedia article: The load in a planetary gear train is shared among multiple planets, therefore torque capability is greatly increased. The more planets in the system, the greater the load ability and the higher the torque density. The more the merrier. The same idea is covered ... 4 It's a question of how many independent variables you need in order to write out the equations of motion. Let's walk through it step by step. Start by just considering Pulley 1 and Mass A. If they're joined by a massless inextensible cable (as shown in your figure) then the displacement of Mass A can be determined as a function of the rotation and radius ... 4 As I remember (learned it 15 yrs ago) You need to have an experimental data, which will help you to find "linear" sections and the their limits (with taking into account the things like hysteresis), so you will be able to find transfer function for each band. P.S. Why do you think that the servo motor system is nonlinear? Please describe the the mechanism ... 4 My question is how can I incorporate the axial force? Doing so would invalidate the assumptions made in the Bernoulli beam theorem, and therefore render your deflection equation invalid. Per the linked Wikipedia article (emphasis mine), Euler–Bernoulli beam theory (also known as engineer's beam theory or classical beam theory) is a simplification of the ... 4 With your code, and with c = 0.3, I get the following results: The amplitude and phase look OK. But the displacement does not show any damping. To see why read on below. I'm not sure about your notation. So the equations below may differ from those in your textbook. ODE The ODE you are trying to solve is$$ \ddot{u} + 2\xi\omega_{\text{res}}\dot{u}... 4 Stretch in the spring delta $Y = A.sin(\omega.t) = A.sin\sqrt(k/m) . t$ So the delta Y is not constant but if you are interested in delt Y_max delta $Y max = m/k$, by Hooks law. Because your system doesn't accelerate except at the beginning and end assuming the pulley starts and stops suddenly that's you maximum. Any gradual start/stop acceleration ... 4 The input force is 18.5+20.5 ft away from the fulcrum and the load is 18.5 away from the fulcrum. This means the force applied to the load is $\frac{(18.5+20.5)}{18.5} = 2.1$ times greater than the input force or for an input force of 50 lbs you get an output force of 105 lbs. However this assumes that all forces are applied perpendicular to the line ... 4 This is more of a long comment than an answer as i can not advice any specific software. First, if you intend to do anything professional in print or web productions than yous shouldn't be looking in direction of Microsoft for anything. Much abused does not mean any good. First tier would be to use direct vector drawing apps. In this category you have: ... 4 Large displacement shock absorbers used to dampen the effects of seismic events may be suitable for this application. These could be used to isolate the part of the structure that is grounded and the part that experiences the impact. Seismic dampers sit between the ground slab and the building and can effectively reduce the shock loadings in all directions.... 4 The downwards tension force in a cable with a $130\text{kg}$ mass hanging under gravity is: $$F=mg=1309.81=1275.3\text{N}$$ The upwards force provided by a servo with arm radius $0.02\text{m}$ and torque $13\text{Nm}$ is: $$F=\frac{T}{r}=\frac{13}{0.02}=650N$$ With two servos in your proposed system, the torque would be doubled, and so your theoretical ... 4 In addition to what Jonathan has said, there is several other practical things to consider: not only torque is limited, but power also; since power is torque times angular velocity (P=Tomega), your motor specification will limit the maximum speed you will be able to lift the load at depending on your application, you will not necessarily have to apply all ... 4 There is a slight mistake in your approach. It seems you have forgotten to add the contribution the moment due to inertial force acting at the center of mass $G$ to the moment equation. The moment equation should be : $$\sum M_A = -(15 m) N + (3 m) 14010^3a + (2.4 m) 14010^3g - (1.8 m) 14010^3a = 0$$ Evaluating for $N$ results in : $N = 2.57 10^5 N$ ... 4 Brakes primarily convert kinetic energy to heat energy. So a large area can absorb more heat lowering the peak temperatures ;of course this is strongly affected by the thickness/mass of the discs and other factors. AND the larger area can get rid or more heat . High temperatures cause deterioration of pad materials ,so lower ( not as high) temperatures ... 3 Perhaps it is helpful to compare the equation to its linear counterpart. \begin{align} \vec{F}&=\dot{\vec{p}}=m\vec{a}=m\dot{\vec{v}}\\ \vec{\tau}&=\dot{\vec{H}}=I\vec{\alpha}=I\dot{\vec{\omega}} \end{align} This linear formula says that the force on an object is equal the the rate of change of the linear momentum which is equal to the mass ... 3 Checking the units is an excellent way to double check your work; kudos for doing so. However, the next step in checking to see if your results make sense is to check limits. In your case, you can use physical intuition to identify how the system would act at very low frequencies, and how it would act without any damping. At very low frequencies ($s\... 3 Call the displacement of the box (where it is attached to the damper and spring) y(t). The force of the damper is $$-c\cdot \dot{y}$$ The force of the spring is $$k\cdot(x - y)$$ y(t) is the point at which the three forces on it balance to zero: $$F - c\cdot \dot{y} + k\cdot(x - y) = 0$$ x(t) depends only on the spring force and the mass:$\$k\cdot(x - ... Only top voted, non community-wiki answers of a minimum length are eligible	2020-11-28 14:04:04	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6570752263069153, "perplexity": 527.5140634291188}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141195656.78/warc/CC-MAIN-20201128125557-20201128155557-00380.warc.gz"}
https://mathoverflow.net/questions/286478/countably-infinite-connected-hausdorff-space-with-the-fixed-point-property	# Countably infinite connected Hausdorff space with the fixed point property Is there an infinite, countable connected $T_2$-space $(X,\tau)$ such that $(X,\tau)$ has the fixed point property? (This means that for every continuous map $f:X\to X$ there is $x\in X$ such that $f(x) = x$.) • I'd have tried with the Golomb space minus some lacunary subset (maybe $\{n!:n\ge 1\}$ is OK). Just to discard those maps $n\mapsto kn$ and remain connected. – YCor Nov 19 '17 at 20:23 • Can you elaborate a bit and maybe add this as an answer? Thanks @YCor – Dominic van der Zypen Nov 19 '17 at 20:28 • No, because I'm far from a proof. It would require a proof of the fixed point property. I only suspect it holds. – YCor Nov 19 '17 at 21:29 • About continuous self-maps of $\mathbb{G}$, see also mathoverflow.net/a/286575/14094 – YCor Nov 21 '17 at 12:59 This is nearly Problem 10705 in The American Mathematical Monthly, proposed by D. W. Brown in 106 #1 (January 1999), p. 67, where the problem asks for a countably infinite $T_{2}$ example. In an editorial comment following John Cobb's solution in 107 #4 (April 2000), pp. 375-376, it is stated that Prabir Roy's lattice space --- A countable connected Urysohn space with a dispersion point, Duke Mathematical Journal 33 #2 (1966), pp. 331-333 --- is an example that is a countable connected Urysohn space. For what it's worth, my notes on this problem say that this can be verified by making use of the analog in Roy's space of observations (a) and (b) at the beginning of the proof of the theorem at the bottom of p. 375.	2020-08-15 02:34:37	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.8702009916305542, "perplexity": 361.0321077624011}, "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/1596439740423.36/warc/CC-MAIN-20200815005453-20200815035453-00127.warc.gz"}
https://www.physicsforums.com/threads/photon-powered-windmill.80097/	# Photon Powered Windmill 1. Jun 23, 2005 ### MarsGhost Okay, I got one for you all. Years ago, in school, I saw a propellor shaped thing, mirror side up, black side down. It was resting on a pin in, what I assume was, a vacuumed bell jar. It spins in sunlight as per what you'd expect... But what is the physics behind this? As light never slows due to energy loss, and light is massless, it can't be through that mechanism. The light never changes wavelength is no energy is lost that way. Maybe the force which drives it is proportional to the light absorbed by the mirror and blown out by the black surface? I'd like to know for sure thanks. 2. Jun 23, 2005 ### JohnnyTheFox Light moves in little particles called photons which have no mass but momentum < (the part I dont get!) so they basicly hit the blades and give them momentum to move... maybe someone else can give more detail. 3. Jun 23, 2005 ### MarsGhost But if the momentum is transferred, we have loss of energy in the wave. Something that can never happen unless you have a change in frequency. It'd be like saying you bounce a ball that bounces back with the exact energy it hit an object with, and still move the object. Hmm, creation of energy is where I have a problem. 4. Jun 23, 2005 ### pervect Staff Emeritus sci.physics.faq http://math.ucr.edu/home/baez/physics/General/LightMill/light-mill.html howstuff works http://science.howstuffworks.com/question239.htm the answer turns out to be an effect known as thermal transpiration as mentioned in the FAQ. It takes a more sophisticated instrument to directly measure the pressure of light (but such an instrument is possible - it mostly involved a better suspension and a better vacuum than is found in the typical radiometer). As far as accounting for energy goes, note that light is reflected by a mirror, and absorbed by a black surface. Reflection light from a non-moving mirror does not change the frequency (or energy) of the light, it just transfers momentum. However, reflection from a moving mirror does change the frequency (and hence the energy) of the light via the relativsitic doppler shift. When light is absorbed by a black surface, all of the energy and momentum in the light is absorbed by the surface. Last edited: Jun 23, 2005 5. Jun 23, 2005 ### rbj perhaps the reason you don't get it is because it isn't true. i have read this same non-fact (that photons have no mass, but they have momentum) before on this forum and i wonder how this is taught nowadays. since momentum is $$p = m v$$ if either mass $m$ or velocity $v$ were zero, so would momentum $p$ be zero. what light doesn't have is rest mass and that's because its velocity is $c$. since $$E = m c^2$$ and $$E = \hbar \omega$$ then the mass of the photon is $$m = \frac{\hbar \omega}{c^2}$$. but since relativistic mass is also $$m = \frac{m_0}{\sqrt{1 - \frac{v^2}{c^2}}}$$ where $m_0$ is the particle's rest mass, and $v$ is the particle's velocity, then $$m_0 = m \sqrt{1 - \frac{v^2}{c^2}}$$ and because $v = c$, then $m_0 = 0$. photons have mass. that's the only way they can have momentum with a finite velocity. r b-j 6. Jun 24, 2005 ### pervect Staff Emeritus This is not correct - photons do not have mass. They do however, have energy. See the sci.physics.faq Does light have mass? for a fuller discussion. I'll take the liberty of posting a brief quote: and encourage the interested reader to read the entire FAQ article (and some of the articles referred to by this article, which can be found at the sci.physics.faq homepage: http://math.ucr.edu/home/baez/physics 7. Jun 24, 2005 ### Staff: Mentor In line with pervect's comments, I think it's become standard to use the term mass (m) to refer to what used to be call "rest mass". (So if you mean relativistic mass, you have to say relativistic mass.) Momentum is defined as: $$p = \frac{m v}{\sqrt{1 - \frac{v^2}{c^2}}}$$ only equalling the Newtonian form for small speeds. 8. Jun 24, 2005 ### Antiphon Edit: as noted by Pervect, The device is called a radiometer. It doesn't work by momentum transfer of photons, that wouldn't be strong enough with room lighting. There is neearly but not completely a vacuum in the device. The dark sides of the vanes get warmer and there is an increased pressure on that side by the gas. The device spins away from the black side toward the white. If it worked by photon pressure it would spin away from white toward black. Hope that helps... 9. Jun 24, 2005 ### Staff: Mentor In a radiometer, the dark side absorbs photons (in the optical range) which produce thermal energy (atomic vibration) in the dark surface material. The light side is cooler because photons are reflected. The air molecules strike both surfaces, but on the dark side the vibrating atoms in the dark material transfer some energy/momentum to the air molecules and this causes the radiometer to spin. So it is simply a transfer of energy/momentum involving the air molecules. Photons have momentum, p = E/c = h/$\lambda$, but not rest mass. Refer to the work by Arthur Compton, and the effect known as Compton scattering. This is however a small quantity compared to the momentum of a molecule, even at room temperature. 10. Jul 16, 2005 ### Ouabache This thread seems like déjà vu ? :uhh: 1, 2, 3	2016-12-08 16:35:58	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.5688271522521973, "perplexity": 1339.7720546327448}, "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-2016-50/segments/1480698541864.24/warc/CC-MAIN-20161202170901-00313-ip-10-31-129-80.ec2.internal.warc.gz"}
http://bluestocking.ling.helsinki.fi/index.php/letters/?letter_id=BC_1778_EMONTAGU_MA_1	# BC_1778_EMONTAGU_MA_1 <Q A 1778 FN MA EMONTAGU> <X ELIZABETH MONTAGU> [}ELIZABETH MONTAGU TO MARY ROBINSON. JAN 9 1778. BL ADD. 40663. F. 69.}] <P1> Jan=y= 9=th= 1778 My Dear Madam I am ashamed I have so long delay'd returning thanks for your kind & friendly letter, but my spirits felt a great damp at first returning to London, where I used to enjoy the friendly converse of my poor departed Brother. Death, disasters, & incidents, have reduced a large fireside to a small circle. A few years indeed show one, that the flattering hopes one entertained in the nursery of living in social gayety & freedom with those nearly allied in blood were [\mere/] pleasing delusions, [\if INTO If\] other things do not sever these natural connections the fatal scissars cuts their thread. Tho my poor Brother never had opportunity of amassing <P2> great wealth, I was in hopes he would have left some thousands more behind him, but the easiness & flexibility of his temper, & a certain placid indolence, made him give into more expence than was prudent. The World lays the whole blame [\on him/] & is [\lend\] in compassionate lamentations for his Widow; indeed her present condition is very lamentable, & I pity her extreamly, but certainly she loved expense better than he did. I imagine poor Man! he thought her fine dress & & appearance raised her in the eye of the World. There is no end of ye bad consequences of an improper marriage. When Men & Women make an indiscreet match, they say it is no concern of any ones, but when any distress is the consequence, the Friends who were thought impertinent if they troubled themselves about y=e= match, are <P3> thought cruel if they take no part of ye evil. I suppose you have heard that Duke Hamilton is going to be married to [\ye youngest/] Miss Burrel Lady Algernoon Percys Sister. I never remember the Town so empty at this Season, but it will be full before the birthday Postchaises & Coaches are continually arriving Monsieur de [\Iarnac\] , who married an Irish beauty, in the mistaken opinion that she was also a Fortune, has been stockjobbing here prodigiously, but if we should really have a french War he will be bit. A very superb Theater is going to be built in the Haymarket, it is to be in price the same as the Opera; no places taken, & the Play to begin at eight o'clock, which certainly suits better the present hour of dining. Once a week each of the other Theaters on certain conditions <P4> are to lend their Actors, so they will each save the expence of a sixth part at least of their Theatrical shews, the other five nights their Houses will be ye fuller. If the London apprentices of these days are half as bold as he who killd the Lion, I think they will assault our new Theater, neither its price, hours, or situation will suit them. The Town has been very sickly. Lady George Germaine has been dangerously ill of the Measles, but is better. I imagine you must have really had a merry Christmass with your three young ones all joy & jollity. Montagu is in fine health, & as to spirits, he never wants them. He rides [\in/] ye Manage from eleven till twelve, & then his Tutor sets him on Pegasus. The day before yesterday was the first time he had attaind the honour of riding between the Pillars, & he was as proud of it as Alexander when he had tamed <P5> Bucephalus. He dances under the care of the celebrated M=r= [\Valouys\] early every morning. These exercises make a boy more healthy as well as more graceful. On tuesday he returns to Harrow where his Master tells me he does very well. I carried him to day to see M=r= Levers Museum. The collection of birds both as to their variety & preservation exceeds that in the King of Frances collection of natural curiosities, but not being shewn me by M=r= de Buffons & Mons=r= D'Aubenton I did not see them with so much pleasure [\the INTO The\] finest as well as rarest bird being a Wise & learned Man. M=r= Lever is gone into y=e= Country, & I was disappointed at not seeing a Man who w=d= exchange an Acre of good Land for an extraordinary fungus. If there is any thing which I can do for you in London be assured y=r= commands will give me pleasure. My best love attends my Neices & Nephew. Miss Gregory is much yours. I am D=r= Mad=m= yr most affect=te= & Sincere friend E Montagu	2017-09-22 15:17:25	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.23393239080905914, "perplexity": 8539.368763772809}, "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/1505818688997.70/warc/CC-MAIN-20170922145724-20170922165724-00492.warc.gz"}
http://gmatclub.com/forum/calling-all-hbs-2-2-2010-applicants-90955-40.html	Find all School-related info fast with the new School-Specific MBA Forum It is currently 23 Jul 2016, 04:28 ### GMAT Club Daily Prep #### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email. Customized for You we will pick new questions that match your level based on your Timer History Track every week, we’ll send you an estimated GMAT score based on your performance Practice Pays we will pick new questions that match your level based on your Timer History # Events & Promotions ###### Events & Promotions in June Open Detailed Calendar # Calling all HBS 2+2 2010 applicants Author Message TAGS: ### Hide Tags Intern Joined: 24 Jul 2010 Posts: 3 Followers: 0 Kudos [?]: 0 [0], given: 0 Re: Calling all HBS 2+2 2010 applicants [#permalink] ### Show Tags 24 Jul 2010, 09:56 Actually, you just need to have one semester left, im 110% sure about that. Manager Joined: 22 Oct 2009 Posts: 242 GMAT 1: 760 Q49 V44 GPA: 3.88 Followers: 6 Kudos [?]: 78 [0], given: 1 Re: Calling all HBS 2+2 2010 applicants [#permalink] ### Show Tags 24 Jul 2010, 10:45 The 2+2 program may not be for you since they specifialy request students in their junior year of college. You can apply to HBS as a senior (or fifth-year) and defer acceptance for two years if you want. It may be easier logistically to do that -- and the competition probably won't be quite as stiff. Intern Joined: 24 Jul 2010 Posts: 3 Followers: 0 Kudos [?]: 0 [0], given: 0 Re: Calling all HBS 2+2 2010 applicants [#permalink] ### Show Tags 24 Jul 2010, 11:03 from HBS website:"Due to additional credits/my involvement in a co-op program/joint degree pursuits/etc., I will graduate in December of 2011. Am I eligible to apply to the 2+2 Program in June of 2011? Yes. We require that applicants have at least one remaining semester to complete their degree when they apply." i think you should at least read through the website before posting something. and i do think that harvard 2+2 is not as competitive as the regular HBS for me. Intern Affiliations: HLS Joined: 28 Mar 2010 Posts: 35 Followers: 0 Kudos [?]: 4 [0], given: 4 Re: Calling all HBS 2+2 2010 applicants [#permalink] ### Show Tags 24 Jul 2010, 11:26 jolin12 wrote: i think you should at least read through the website before posting something. What douchebaggery Intern Joined: 21 Jul 2010 Posts: 2 Followers: 0 Kudos [?]: 0 [0], given: 0 Re: Calling all HBS 2+2 2010 applicants [#permalink] ### Show Tags 24 Jul 2010, 11:40 @jolin Ms. Big Fat Panda Status: Three Down. Joined: 09 Jun 2010 Posts: 1922 Concentration: General Management, Nonprofit Followers: 432 Kudos [?]: 1865 [0], given: 210 Re: Calling all HBS 2+2 2010 applicants [#permalink] ### Show Tags 25 Jul 2010, 10:03 jolin12 wrote: Actually, you just need to have one semester left, im 110% sure about that. My bad. Somehow I associated the spring semester junior year with having two semesters left in college. Just read through the FAQs. Quote: In order to be eligible for the 2+2 Program you must be a current college junior with at least one remaining semester, after June 2011, necessary for the completion of your degree. So, yes you can apply. Your HS credentials might be impressive or not depending on how you bring it in to the picture. Your essays will matter much more. Quote: and i do think that harvard 2+2 is not as competitive as the regular HBS for me. And that my friend, can be seen as sheer unadulterated arrogance. Let's say 850 students apply, and as per statistics, around 100 get in. So you THINK you have a 11-12% acceptance rate. But. 23-25 students are from Harvard University, though they don't specifically state that there's a preference. And I think the 2+2 program is MUCH more competitive since only students who are above average in terms of everything that HBS looks for will apply. And as a regular applicant you'll have the edge of being younger, but with the 2+2, everyone is your age, so no biggie there. Technically, right now you can't apply to the regular HBS. And I don't understand the reason for your comparison in the first place. But having said that, I highly suggest that you try to be polite when you're looking for constructive criticism. I've talked to several HBS admits, 2+2 and otherwise, and being impolite was not something they had in common. The person who posted before was merely expressing his/her opinion in a polite way. I recommend you do the same if you don't want to have your posts moderated. Thank you. Intern Joined: 25 Jul 2010 Posts: 2 Followers: 0 Kudos [?]: 0 [0], given: 0 Re: Calling all HBS 2+2 2010 applicants [#permalink] ### Show Tags 25 Jul 2010, 10:40 I apologize for whatever i said, I was being really rude. Really sorry about that, Im really only here to look for advise. And for whoever reported me, Thank you for doing that, because i would never realize how rude i was. I guess when i said it, i didn't really want to make it sound so mean. Well, i think i would like to explain my situation a little bit more, and see if any of your guys can give me any suggestions, all of of them will be appreciated. Im actually a transfer student from Simon's Rock college to U of Michigan. Simon's Rock is one of the very few early-colleges in the nation, which offer high school students the opportunity to go to College early. So it's not a typical college, So im not sure if i should be considered as a senior or junior, because credit wise, i still need at least one more semester to graduate, so i guess im actually a junior right now. btw on their website, they clearly stated that also emailed them to confirm that. since i just transferred to Michigan, there is really not enough time for me to do a lot of leadership work. as for extracurricular experience. I was an exchange student for a year, and will be one for another year in Japan. I have interned in local clinics and my father's company in China. I have also volunteered in African (orphanage), Thailand, and the US(Mexican immigrants) through out my first three years in College. I also play piano and Gu Zheng in the band, and have done my own social research in SE asia for one month. at Simon's rock, i was in charged of the school garden, and was the coordinator of building the school trail. besides all that, im also very into fashion, and has been/currently working as a part-time model. as for other job, ill be getting a summer internship at at a business consulting firm next summer (2011) in the US, hopefully i can get an offer from MBB, after studying abroad in japan;the worst case would be Deloitte, which i can pretty much guarantee to get. but again, it's next summer, so i didn't write it down. As for GPA, i think i should be able to rise it to a 3.75. after the fall semester (next semester), definitely higher (3.8) after the spring semester, but i think the application is due before the GPA can probably get updated, so that's why i wrote down 3.75. For GMAT, im currently studying it now by myself, and will take classes, aiming for a 730+ score, hopefully a really high score to compensate my not so-high GPA. Ill work hard. As for leadership and other activities, like volunteer, since there is still some time, i can start doing more things while i start my next semester in Japan (im currently in summer school), hopefully that will help. and I just want to clarify the three volunteer experiences. -one is a one month volunteer i did in Lesotho for an orphanage through an non-profit organization. We helped them to build a dorm, I also taught the kids Math and English. - the second is one in the US, I did this through out my first two years at Simon's Rock College, i helped the nearby Mexican immigrants/families who are in poverty to organize fund-raising; Help them to make money by creating a garden; support them financially; clean and organize their home... -the next one is Thailand, i did it while i was in Thailand last summer, I was tutoring English in a local school through a Thai friend of me as a part time volunteering for a month. -During the earthquake that happened in China two years ago in Sichuan, I also went there for about 1 week to help out physically and financially -I also volunteered during the 2008 Beijing Olympics. I know i have many disadvantages, I really hope i can turn them into my advantages somehow. And if there are anything else i can do in the near future to help me get int to HBS 2+2, Please let me know, Ill try my best. Getting into HBS 2+2 is really my dream, and ill compensate a lot for it. Thank you, again, any advice will be greatly appreciated. Ms. Big Fat Panda Status: Three Down. Joined: 09 Jun 2010 Posts: 1922 Concentration: General Management, Nonprofit Followers: 432 Kudos [?]: 1865 [0], given: 210 Re: Calling all HBS 2+2 2010 applicants [#permalink] ### Show Tags 26 Jul 2010, 07:40 No, it's quite alright. We are just making sure that you don't become too rude. After all, this is a forum built on mutual respect and combined knowledge. You can go back to using your old account, it was reactivated. Quote: As for GPA, i think i should be able to rise it to a 3.75. after the fall semester (next semester), definitely higher (3.8) after the spring semester, but i think the application is due before the GPA can probably get updated, so that's why i wrote down 3.75. I don't know if you've even figured out how US GPA works still, because it is near impossible to bring up a 3.6 to a 3.75 in one semester. Even if you get a 4.0, I think your GPA might go to like a 3.65 or such, if at all that. The larger number of credits you have, the harder it is to make such drastic changes. But that said, I think a 3.6 is pretty decent in terms of GPA. Don't worry about it too much. Quote: as for other job, ill be getting a summer internship at at a business consulting firm next summer (2011) in the US, hopefully i can get an offer from MBB, after studying abroad in japan;the worst case would be Deloitte, which i can pretty much guarantee to get. but again, it's next summer, so i didn't write it down. When you say you're getting a summer internship from them, do you mean that you have an offer in hand? 'Cos once again, sorry to burst your bubble if it comes to that, but MBB is mad hard to get into even with a 4.0 and work experience. They're considered a gate opener, or even a ticket to big schools like HBS and Wharton. So unless you have one at hand, I wouldn't take this for granted. Also, Deloitte isn't a backup either. Your volunteer work is impressive, but lacks a cogent trend in my opinion. Try to turn those into strengths and a find a missing link that connects these activities to leadership potential. 2+2 is a very competitive program and I wouldn't take anything for granted. I am an applicant myself, so I can't speak for how hard it is, or how hard it might be, but just take it for granted - If it's HBS, it's not going to be easy. So I recommend that like you said, you try to bring up your GPA to a 3.7, which might be achievable, and continue working on your volunteering activities. If you can get an internship with an MC firm, even better. And please keep up the same decorum in the forum. Hope this helps. Intern Joined: 25 Jul 2010 Posts: 2 Followers: 0 Kudos [?]: 0 [0], given: 0 Re: Calling all HBS 2+2 2010 applicants [#permalink] ### Show Tags 26 Jul 2010, 09:54 yes, i already have an offer from Deloitte, and my GPA now is 3.65, and since im still taking summer classes now (10 credits), and i m pretty sure ill get all As for this 10 credits. so hopefully, ill be able to bring my GPA up a lot by the end of next semester. And im also a transfer student, so i only have one year worth of credits that are calculated towards my GPA. Since ill be in Japan next year, Ill try to join some organizations and get as much leadership experiences as possible. Thanks Senior Manager Joined: 26 Jul 2010 Posts: 358 Location: European union Followers: 2 Kudos [?]: 50 [0], given: 41 A few questions from prospective HBS 2+2 applicant [#permalink] ### Show Tags 26 Jul 2010, 18:12 So, I will start my undergrad career this fall and I am planning on applying to the HBS 2+2 program. Id like to know whether doing a double major in Economics and Business administration will hurt my chances because I know that students doing Bus.admin as undergrads are not advised to apply to the program.Id like to know if a double major in Economics and Political science&International relations is better not only for the HBS 2+2 program but also for my future applications.I know for a fact that generally B-schools don`t like applicants who did business administration at college. One last thing. I am planning on enrolling in the Cambridge university summer program, which continues for 6 weeks, in the next year.Do you think that if I do well this can improve my chances of being accepted to HBS 2+2 ? Senior Manager Joined: 26 Jul 2010 Posts: 358 Location: European union Followers: 2 Kudos [?]: 50 [0], given: 41 Re: A few questions from prospective HBS 2+2 applicant [#permalink] ### Show Tags 27 Jul 2010, 00:27 Come on, anyone ? Senior Manager Joined: 26 Jul 2010 Posts: 358 Location: European union Followers: 2 Kudos [?]: 50 [0], given: 41 Re: A few questions from prospective HBS 2+2 applicant [#permalink] ### Show Tags 27 Jul 2010, 09:48 buuuuuuuuuuuump Manager Joined: 01 Mar 2010 Posts: 175 Followers: 1 Kudos [?]: 19 [0], given: 0 Re: A few questions from prospective HBS 2+2 applicant [#permalink] ### Show Tags 27 Jul 2010, 10:12 Ivan91: No offense, but if you are just beginning your undergrad (and judging by your ID, I would assume that you are born in 91), it is definitely too early. Also, it seems like you want to change your undergrad study direction just to fit HBS 2+2, and that sounds very unwise. I don't know anything about HBS or HBS 2+2, but one thing I do know about people @ HBS is that they don't tailor their ways to the adcom's flavor - They have done what they want to do, and have done them damn well. I'm not sure where you heard business major isn't good, but when you look at incoming MBA class profile, undergrad business major is a huge chunk. Bottom line - do what you want to do. Your double major is not going to decrease your chance. I bet your perspective will change over the next four years. Doing only what you think the adcom will like only leads you to rejection. Manager Joined: 22 Oct 2009 Posts: 242 GMAT 1: 760 Q49 V44 GPA: 3.88 Followers: 6 Kudos [?]: 78 [0], given: 1 Re: A few questions from prospective HBS 2+2 applicant [#permalink] ### Show Tags 27 Jul 2010, 10:17 While I have to love your ambition, I can't honestly recommend ANYTHING to you at this point. You're too young to be worrying about a graduate school program hypothetically 6 years down the road. Who knows -- HBS may end the 2+2 program by then! Don't plan your college career based on this program. From everything that I've experienced by advising for the past 3 years, you're MUCH more likely to be successful by following your passions rather than trying to build the "perfect" resume. Future admissions committees, job interviewers, and pretty much everyone else will respect you for following your passions. You'll also get a higher GPA if you do what you love, so it's really a win-win! In summary: Major in whatever you want. Have some fun in college -- you only live through it once! Meet people, experience new things. Don't forget that college is the time to build up personal skills -- there is an interview part to your application, you know! Last edited by YourDreamTheater on 27 Jul 2010, 10:22, edited 1 time in total. Manager Joined: 22 Oct 2009 Posts: 242 GMAT 1: 760 Q49 V44 GPA: 3.88 Followers: 6 Kudos [?]: 78 [0], given: 1 Re: A few questions from prospective HBS 2+2 applicant [#permalink] ### Show Tags 27 Jul 2010, 10:21 chostein wrote: Ivan91: No offense, but if you are just beginning your undergrad (and judging by your ID, I would assume that you are born in 91), it is definitely too early. Also, it seems like you want to change your undergrad study direction just to fit HBS 2+2, and that sounds very unwise. I don't know anything about HBS or HBS 2+2, but one thing I do know about people @ HBS is that they don't tailor their ways to the adcom's flavor - They have done what they want to do, and have done them damn well. I'm not sure where you heard business major isn't good, but when you look at incoming MBA class profile, undergrad business major is a huge chunk. Bottom line - do what you want to do. Your double major is not going to decrease your chance. I bet your perspective will change over the next four years. Doing only what you think the adcom will like only leads you to rejection. Beat me to it! Just an FYI -- technically, the original 2+2 program wasn't really targeted for business majors. The program is now changing as more and more business majors get accepted. Also -- have you visited HBS? What if the place just doesn't "fit" with your personality? Beyond that -- how do you know that you're going to go into a field where an MBA is even helpful? This is the reason I decided against applying for the 2+2 program. I'm really interested in trading and entrepreneurship -- two fields where an MBA doesn't have as big of an impact as it would in other fields. Senior Manager Joined: 26 Jul 2010 Posts: 358 Location: European union Followers: 2 Kudos [?]: 50 [0], given: 41 Re: A few questions from prospective HBS 2+2 applicant [#permalink] ### Show Tags 27 Jul 2010, 11:04 Well, I am an international student from Europe, I havent had the chance to visit HBS... But you are right that one must not try to build the perfect resume. One last question - what do you think about the summer school program at Cambridge university ?I am really interested in enrolling, just for the experience.Summer schools are not at all about studying but about meeting new people.However, do you think having a certificate from such program is helpful when applying to MBA programs ? Current Student Affiliations: CBS '13 Joined: 08 May 2010 Posts: 291 Followers: 5 Kudos [?]: 31 [1] , given: 1 Re: Calling all HBS 2+2 2010 applicants [#permalink] ### Show Tags 27 Jul 2010, 11:35 1 KUDOS jolinzhu wrote: yes, i already have an offer from Deloitte, and my GPA now is 3.65, and since im still taking summer classes now (10 credits), and i m pretty sure ill get all As for this 10 credits. so hopefully, ill be able to bring my GPA up a lot by the end of next semester. And im also a transfer student, so i only have one year worth of credits that are calculated towards my GPA. Since ill be in Japan next year, Ill try to join some organizations and get as much leadership experiences as possible. Thanks You should brush up on your English... Manager Joined: 22 Oct 2009 Posts: 242 GMAT 1: 760 Q49 V44 GPA: 3.88 Followers: 6 Kudos [?]: 78 [0], given: 1 Re: A few questions from prospective HBS 2+2 applicant [#permalink] ### Show Tags 27 Jul 2010, 11:49 Will it be helpful? Probably not. According to some ad-comms that I've spoken to, all that one of those summer programs really means is that your family is wealthy enough to send you to one of those programs. If you want to do it, then do it. It would probably be pointless to do this type of program if you're already in a top-tier university in Europe (i.e. Oxford, Cambridge, LSE, etc.). EDIT: a summer program would be helpful if it allows you to take classes relevant to what you're interested in that you cannot take at your home university. Senior Manager Joined: 26 Jul 2010 Posts: 358 Location: European union Followers: 2 Kudos [?]: 50 [0], given: 41 Re: A few questions from prospective HBS 2+2 applicant [#permalink] ### Show Tags 27 Jul 2010, 12:08 My family is wealthy enough to send me.Well, even if it is not particularly helpful, it cant hurt, right ? At least it will be regarded as some kind of extra curricular activity.Not I do it just to make my resume perfect but I guess it is better than just waste money in some summer resort in Spain and do nothing .. Current Student Joined: 22 Dec 2009 Posts: 322 Schools: Yale SOM Followers: 4 Kudos [?]: 51 [0], given: 23 Re: A few questions from prospective HBS 2+2 applicant [#permalink] ### Show Tags 27 Jul 2010, 12:37 I doubt HBS would consider your summer at Cambridge an extracurricular activity. It would probably be looked at as more schooling. If you were to get involved in other activities there, outside of the program, it might be beneficial to go then. Re: A few questions from prospective HBS 2+2 applicant [#permalink] 27 Jul 2010, 12:37 Go to page Previous 1 2 3 4 Next [ 67 posts ] Similar topics Replies Last post Similar Topics: 10 HBS 2+2 - Calling All Applicants (Class of 2017) 37 05 Aug 2012, 15:30 158 Calling all HBS Fall 2010 Candidates 1578 20 May 2009, 17:51 240 Calling all HBS 2011 Applicants!! (Including 2+2) 2930 20 Jun 2010, 19:10 13 HBS 2+2 2014 - Calling All Applicants (Class of 2016) 477 11 Apr 2011, 22:45 334 Calling all Ross 2010 applicants! 1927 23 Jun 2009, 10:38 Display posts from previous: Sort by	2016-07-23 11:28:02	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4890357255935669, "perplexity": 3427.5769509029856}, "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-2016-30/segments/1469257822172.7/warc/CC-MAIN-20160723071022-00298-ip-10-185-27-174.ec2.internal.warc.gz"}
http://mathhelpforum.com/differential-geometry/124244-open-sets.html	1. ## Open Sets I worked this out myself and I was wondering if somebody could help me and see if I have made any mistakes. It's a bit of a tautology since an open disc is called an open disc but I wanted to prove this from my definition Definition: $E \subseteq \mathbb{C}$ is open $\iff$ 1. E is empty or 2. $\forall a \in E, \exists r > 0$ such that $D(a,r) \subseteq E$ where $D(a,r) = \{ z \in \mathbb{C}: \|z-a\| < r \}$ (open disc). So let's prove that D(a,r) itself is open! We want to prove the statement $\forall b \in D(a,r), \exists s > 0$ such that $D(b,s) \subseteq D(a,r)$ Proof: Let $b \in D(a,r)$. Then $\|b-a\| < r$. Choose $s = r - \|b-a\| > 0$ since $\|b-a\| < r$. Let $z \in D(b,s)$. Then $\|z-a\| \le \|z-b\| + \|b-a\| < s + \|b-a\| = r - \|b-a\| + \|b-a\| = r$. Hence $z \in D(a,r)$ and we conclude that $D(b,s) \subseteq D(a,r)$. 2. looks good to me 3. Originally Posted by slevvio I worked this out myself and I was wondering if somebody could help me and see if I have made any mistakes. It's a bit of a tautology since an open disc is called an open disc but I wanted to prove this from my definition Definition: $E \subseteq \mathbb{C}$ is open $\iff$ 1. E is empty or 2. $\forall a \in E, \exists r > 0$ such that $D(a,r) \subseteq E$ where $D(a,r) = \{ z \in \mathbb{C}: \|z-a\| < r \}$ (open disc). So let's prove that D(a,r) itself is open! We want to prove the statement $\forall b \in D(a,r), \exists s > 0$ such that $D(b,s) \subseteq D(a,r)$ Proof: Let $b \in D(a,r)$. Then $\|b-a\| < r$. Choose $s = r - \|b-a\| > 0$ since $\|b-a\| < r$. Let $z \in D(b,s)$. Then $\|z-a\| \le \|z-b\| + \|b-a\| < s + \|b-a\| = r - \|b-a\| + \|b-a\| = r$. Hence $z \in D(a,r)$ and we conclude that $D(b,s) \subseteq D(a,r)$. The idea of your proof should be clear graphically. Draw a disc without its boundary, call it $D$ with center $d$ and radius $r$. Let $x\in D$ be arbitrary. Then, really all you're claiming is that the open disc with radius $r-\ell\left(\overline{dx}\right)$ is completely contained within $D$. But, this disc is merely tangent to the outer disc and since it doesn't include its boundary it follows that its contained entirely within $D$. This was not meant to be rigorous, but a geometric reason why this is true. P.S. Just because the adjective open is applied to open ball doesn't imply that it's open. Be careful of what you call a tautology. 4. But in a metric space, an open ball is an open set. I don't understand why you're trying to prove a definition ? 5. Originally Posted by Moo But in a metric space, an open ball is an open set. I don't understand why you're trying to prove a definition ? In my limited experience, it is often not taken that an open ball is an open set. You call an open set just any set such that for each point in the set there exists an open ball around it entirely contained in the set. Nothing in there made it neccessary that the open ball was open. But, as Slevvio showed it is not hard to prove that giving an open ball its name is justified.	2017-11-21 20:03:52	{"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": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 37, "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": 0.9469451904296875, "perplexity": 193.43063797179948}, "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-2017-47/segments/1510934806422.29/warc/CC-MAIN-20171121185236-20171121205236-00019.warc.gz"}
http://mathhelpforum.com/differential-geometry/106310-find-interesting-cauchy-sequence.html	# Math Help - Find an "interesting" Cauchy sequence. 1. ## Find an "interesting" Cauchy sequence. For a rational number $x \neq 0,$ write it as $2^k\frac{m}{n},$ where $m$ and $n$ are integers that have no common factors, and let $\|x\|_{2} = 2^{-k}.$ Define $\|0\|_{2} = 0.$ Let $d(x,y) = \|x-y\|_{2}.$ Prove $d(x,z) \leq d(x,y) + d(y,z)$ for all rational $x, y, z.$ Now since $d(x,y) = d(y,x)$ and $d(x,y) \geq 0$ with equality only if $x = y.$ Write an interesting Cauchy sequence in this metric. 2. Originally Posted by cgiulz For a rational number $x \neq 0,$ write it as $2^k\frac{m}{n},$ where $m$ and $n$ are integers that have no common factors, and let $\|x\|_{2} = 2^{-k}.$ Define $\|0\|_{2} = 0.$ Let $d(x,y) = \|x-y\|_{2}.$ Prove $d(x,z) \leq d(x,y) + d(y,z)$ for all rational $x, y, z.$ Now since $d(x,y) = d(y,x)$ and $d(x,y) \geq 0$ with equality only if $x = y.$ Write an interesting Cauchy sequence in this metric. For positive integers m,n we have that m > n <==> 2^(-m) < 2^(-n) , so \|x + y\|_2 <= max{\|x\|_2 , \|y\|_2} <= \|x\|_2 + \|y\|_2 and from here the triangle inequality follows. As for an interesting Cauchy sequence: what about the sequence {2^n} = {2, 4, 8, 16,...} ? Tonio	2015-05-29 15:33:21	{"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 24, "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": 0.9612261056900024, "perplexity": 525.6357899520821}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207930143.90/warc/CC-MAIN-20150521113210-00225-ip-10-180-206-219.ec2.internal.warc.gz"}
https://chemistry.stackexchange.com/questions/79192/how-would-i-determine-the-relative-acidity-of-substances	# How would I determine the relative acidity of substances? Which of the following salt will produce a neutral solution (If any)? A) $\ce{KI}$ (Potassium Iodine) or B) $\ce{SrS}$ (Strontium Sulfur) I tried to solve this problem by using the "Relative Strengths of Brønsted-Lowry Acids and Base in aq solution at room temp table". From what I know each of those two substance will dissociate into ions. Having the ions $\ce{K^+, I^-,Sr^{2+}, S^{2-}}$ Since K does not hydrolyze and I is a product of HI which is a strong acid it won't hydrolyze either. I also know that the only cations that hydrolyze are Al,Fe and Cr hence Sr will not hydrolyze. However how do I know weather S will hydrolyze? The other question I had difficulties was "Order the following from most acidic to least. SO3, CO3 and CO2". In my homework one of the question was which of the following salt will produce a neutral solution(If any)? A) KI (Potassium Iodine) or B) SrS (Strontium Sulfur) For (A) both K+ and I- will essentially stay as the ions. So assuming the solution is neutral to start, it will remain neutral. But for (B), in a neutral solution, $\ce{Sr^{2+}}$ will stay as the ion, $\ce{S^{2-}}$ will react as: $\ce{S^{2-} + H2O <=> HS^- + OH-}$ so the solution will become slightly more basic. "Order the following from most acidic to least. $\ce{SO3}$, $\ce{CO3}$ and $\ce{CO2}$". I can't really make sense of this question since $\ce{CO3}$ is highly unstable, and there are three isomers. Bubbling $\ce{SO3}$ and $\ce{CO2}$ gases into water would produce acidic solutioons via reactions $\ce{SO3 + H2O -> H2SO4}$ $\ce{CO2 + H2O -> H2CO3}$ • Just to clarify in the second question we evaluate the acidity by the acid that is produce as the final product not the acids that are formed as the intermediates such as HSO4 and HCO3? – coderhk Jul 21 '17 at 22:33 • There are really no such species as $\ce{HSO4}$ and $\ce{HCO3}$ in aqueous solution. The species would have to be $\ce{HSO4^-}$ and $\ce{HCO3^-}$ which would formed from acid reacting with $\ce{H2O}$. $\ce{H2SO4}$ is such a strong acid that there would be just a trace of $\ce{HSO4^-}$, with most of the S species being $\ce{SO4^{2-}}$. – MaxW Jul 21 '17 at 23:16	2019-06-17 06:47:38	{"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": 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": 0.7239386439323425, "perplexity": 1424.7517138306248}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627998440.47/warc/CC-MAIN-20190617063049-20190617085049-00175.warc.gz"}
https://www.gradesaver.com/textbooks/math/algebra/algebra-2-1st-edition/chapter-9-quadratic-relations-and-conic-sections-9-7-solve-quadratic-systems-9-7-exercises-skill-practice-page-662/27	## Algebra 2 (1st Edition) Adding twice the first and negative one times the second equation we get: $2x^2-20-x-8=0\\2x^2-x-28=0\\(2x+7)(x-4)=0$ Thus $x=-3.5$ or $x=4$ Plugging this into the first equation we get: if $x=-3.5$: $12.25+2y^2-10=0$ but this has no solution. if $x=4$: $16+2y^2-10=0$, but this has no solution Thus there are no solutions.	2021-04-18 05:52:30	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.9648188948631287, "perplexity": 148.96995790445115}, "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/1618038468066.58/warc/CC-MAIN-20210418043500-20210418073500-00580.warc.gz"}
http://lotoblu.it/hgor/skript-for-spawn.html	# Skript For Spawn Learn more. Install Script Hook V. See About Shell Scripts for an example using spawn() and shell scripts. These can do anything you can. 5 meters horizontally behind the player spawn a TNT 5 meters above and 2 meters horizontally behind the player thrust the last spawned TNT in the horizontal direction of the player with speed 0. An op Sword Burst 2 GUI that has built in auto farm! Click on mob to box and then click auto farm. Next File Metal Detectors. It does spawn, but the script(s) that are attached to it is not working at all. Specifies the coordinates of the world spawn. Posted by kylania in ArmA 3, Script Examples on June 25, 2016 Rip on the BIS forums had asked about a black fade in intro for his new mission. This script is a transcript that was painstakingly transcribed using the screenplay and/or viewings of Spawn. Hi All, I have an Expect script which logs into Cisco switch, performs a show interface command. 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Egg Clicker GUI AUTOFARM. Spawn announced at 23:22 (announcer saw it appear). As we discussed in previous tutorials, Ansible is a very handy tool for sysops to maintain their company infrastructure. AddItem 0000000F 100. - Added support for the new vehicle class. Category: Skript Suggested name: Spawn What I want: I am looking for a command /spawn, you would be able to do /setspawn and /spawn would have a cool down of 10 seconds. 0 replies; 475 views; El Rabito; 31 Jul 2019; Auto-run & Lock/Thermal scanner exploits fixes By El Rabito. spawn takes the executable to call as the first parameter and optionally an. FipsundMunne Sep 14th, 2017 197 Never Not a member of Pastebin yet? Sign Up, it unlocks many cool features! raw. io Tools - Fast Eject, Auto Re-spawn, Re-spawn on R, Quick buy - Fast eject, Auto Re-spawn, Re-spawn on R, Quick buy, Freeze on F, and more Author kinglucifernatsu. Provide details and share your research! But avoid … Asking for help, clarification, or responding to other answers. Surprise! What can you see? That classical electric-stairs spawpoint. Here is the script I created for automatically login to the SSH server and then login with super user and then run a simple command. Use Spawn script and thousands of other assets to build an immersive game or experience. 90a added 1. 5 minutes, [1/10] chance. Although we cannot accept all submissions, we do read each suggested change from our users and will make updates where applicable. Prefabs An asset type that allows you to store a GameObject complete with components and properties. Here, we'll show you a few of those problem areas, and how to build a batch script to get around them. You can easily create complex custom commands, triggers that execute effects under certain conditions, and scripts that just run periodically - all without having to know even the slightest bit about programming! The. The prefab acts as a template from which you can create new object instances in the scene. Corrupted Code script hub is a GUI made by 1x1x1x1# 4340 on discord! It has amazing scripts as seen in my next video! Lumber Ripper OP LT2 GUI! What is Bring Up? Bring Up is a OP FREE item/wood teleportation GUI, Venyx Item TP free version made by TehDino on v3rm! PAID EXECUTORS ONLY! Paid Scripts I Own. Use this event only if you need to perform some action on multiple command, e. or is it? This script allows you to spawn vehicles and peds alike. Hullo! I've been messing around this afternoon to make an add-on vehicle spawner. Demo - Version 1. ini and Game. It should say Unnamed SA-MP 0. A behaviour is a script file that is executed and adds 'goals' to the NPC which can run every tick. Ill gladly donate if you can make it happen. Todd McFarlane Shares Completed SPAWN Script. It was recently. Every time the construction script is rerun, it cleans up the old version, and. I didn't write anything from this point ~ ~ Using this without an undetected and up-to-date BE script logging bypass will result in a global ban. Add-on Spawner is a ScriptHookV script that automatically looks for add-on vehicles and puts them in a menu for you to spawn. The example is pretty self-explanatory and shows how to use child_process. 1 support Update 21. The script in the spawn button will: 1) Spawn the page after the current page. Todd McFarlane brought the character Spawn to comics in the early 90s, and after one. _____ Thought I would post my vehicle spawn script for DayZ remember that scripts are logged. 80 version since it needs some redesign to fit with the recent scripts & to get the tasks functional in mp. In ssh first I have to login as ssh user and then su - to login as root. 0) Simply the Best! About Spawner V is a free, advanced and lightweight vehicle spawner for Grand Theft Auto V. I am trying to create a bash script to ssh into a session and run a command then exit. Making statements based on opinion; back them up with references or personal experience. This script was made for a server to be able to have random spawns. For PC users, FTP Access is used to directly modify the GameUserSettings. 5 UPDATE! FEJL FIXET! Jeg har ikke set noget i servicevilkår om at man ikke måtte ændre skripts. - Updated vehicleHashes, now contains everything up to GTA V version 1. sk in /plugins/Skript/scripts. Expect Caveats. Download the file, and locate your "Scripts" folder inside of your Grand Theft Auto Directory if you can't find "Scripts" go ahead and create a new File and name it "Scripts". Often during pen tests you may obtain a shell without having tty, yet wish to interact further with the system. I know, I know, I still need to get the cast names in there and I'll be eternally tweaking it, so if you have any corrections, feel free to drop me a line. Directed and co-written by Mark A. 5 meters horizontally behind the player spawn a TNT 5 meters above and 2 meters horizontally behind the player thrust the last spawned TNT in the horizontal direction of the player with speed 0. At Toy Fair New York. dll into the Scripts Folder! Controls Open - F2 Navigate - Arrow Keys Select - Enter Back - Backspace. My New Website It's official, visit meth0d. Thanks regardless for the work put into this script. Is there any facility for managing spawn-fcgi through Upstart? (Is Upstart the modern system for managing services, or should I be looking somewhere else?). FiveM Ready vehicles, Download or use git to add it on your server. If 100 gold isn't enough for you, the following console command would add 99,999, which is near. Joined: May 4, 2010 Posts: 379. There is a script to check if a specific item has been looted. Making statements based on opinion; back them up with references or personal experience. Install Script Hook V. Our script file is like below. It is easy to use for simple tasks, but yet you can create really complex things with it. He probably doesn't have. Learn more. to prevent commands under certain circumstances, to log commands, or to add Skript checks to commands from other. There's countless ways of doing this but thanks to a hint from polpox I discovered that the new Bootcamp mission has introduced a function that makes a nice looking "infoText" effect. I just want to be able to set down a spawn point that has a 30% chance to spawn something and a 70% chance to spawn nothing at all, when a player enters the area. "If I'd done my job and written a screenplay that mattered, that would have sold it. Expect Caveats. However, it is set to Hard. 0 unless otherwise noted. (Believe it when I see it) Todd McFarlane Promises That His Long-Awaited SPAWN Reboot Could Begin Production This Year. However, you can still use it with the 0. Active 5 years, 7 months ago. Obviously some of this will depend on the system environment and installed packages. 9 GUI UPDATE X1000 STATS! Egg Clicker GUI AUTOFARM. New: Nitro Boost our Discord server and receive full donation perks here on the website! Join the Scripting Helpers Discord Server to learn more! You can also Support on Patreon as always. Conditions (e. It is only through fire that imperfection be destroyed and the path to domination is revealed. spawn [Teleport to spawn] /spawn : spawn. Discussion in 'Scripting' started by CorruptedHeart, Jun 3, 2010. This is the place to link to your scripts for dedicated servers. Designed to be user-friendly, this script provides you the best way to navigate and spawn the game vehicles using categories, it also contain some extra stuff. Viewed 687 times 1. Hi I am looking for a set of spawn scripts/system that can do the following. y must be between 0 and 256 (inclusive). It is only through fire that imperfection be destroyed and the path to domination is revealed. Our away homes on the interwebs: Signup or login to create shortcuts to your favorite games! WANTED! ↺1 Wanted! Become a member today and start sharing your creations! Install and manage your mods easily with community-made 1-Click mod downloaders. Dippé directed the poorly received Spawn movie starring Michael Jai White, but now, McFarlane is looking to reboot the character with a new R-Rated movie. There is a script to check if a specific item has been looted. Stack Overflow for Teams is a private, secure spot for you and your coworkers to find and share information. 5 meters horizontally behind the player spawn a TNT 5 meters above and 2 meters horizontally behind the player thrust the last spawned TNT in the horizontal direction of the player with speed 0. I've been trying to spawn a mob using your addon but I just keep on getting SEVERE skript errors on every attempt. I will use this script to spawn any game objects which are not static with code. Commands introduced in different Arma 3 versions by clicking on one of the images below. Spawner V (v2. Hi everyone, I'm no killer in shell scripting, that is why I've searched and found a little script that explained how to do what I wanted to do : a FTP transfer from distant servers. setspawn [set spawn at your location] /delspawn: spawn. DIO Bone: 17 minutes, [1/3] cha. Typically, actions happen one after the other. To this is added that he is a screenwriter and a. Parent:FindFirstChild("Humanoid") then. Imp 22:45, 23 июля 2007 (EEST). Download the file, and locate your "Scripts" folder inside of your Grand Theft Auto Directory if you can't find "Scripts" go ahead and create a new File and name it "Scripts". When I Place My Spawner, It Turns To Pig Spawner. Quote Share this post. Dette skript må ikke sælges videre. Set spawn point for a specific player. This page was last edited on 17 November 2018, at 05:27. [SKRIPT] Spawn System (Join settings) 2. then display them. If not specified, defaults to the position of the command's execution. - New parameters, so double check your calling lines! Credits & Thanks: zuff kemor ChrzRw79 Dirty. These commands can use Skript's syntax which allows to make very user-friendly commands, like one of the most powerful /item commands available, and any other command you or others come up with. Loading Unsubscribe from NexuzzMC? How To Set Your Server Spawn With Skript \| Minehut 101 - Duration: 4:49. This script is a transcript that was painstakingly transcribed using the screenplay and/or viewings of Spawn. admin": message "&7Location of player not jump" on rightclick: block is a sign: teleport player to block at world's spawn point:. Obviously some of this will depend on the system environment and installed packages. Todd McFarlane has been talking about a new Spawn movie for a few years now, but he told ComicBook. Plugin Help Skript /summon command. NEWER SCRIPTS ARE AT THE TOP OF THIS PAGE! Click to set custom HTML. So I made my own just for yall! (Note: I am very bad at skript. and try it for yourself. Script to Spawn NPC at the Player - posted in Skyrim Creation Kit and Modders: Could somebody take a look at my script and tell me what Ive done wrong. ly/1dVYei Har også fået min butik oppe at køre :). Amazon AWS is no doubt the best public cloud out there. For PC users, FTP Access is used to directly modify the GameUserSettings. Stand Arrow: 5 minutes. Anyone have a clue what might be wrong? Thanks in advance for your help! ["AmmoboxInit", [this,true]] call BIS_fnc_arsenal; in the init should add everything. It is the year 2039. Spawn was ranked 60th on Wizard magazine's list of the Top 200 Comic Book Characters of All Time, 50th on Empire magazine's list of The 50 Greatest Comic Book Characters, and 36th on IGN's. For that same location another vehicle could be spawned instead of the Infernus with a spawn probability of 70%. Without it it teleports you to your own. There are a few problems with this prank, though. This is a spawn script that I have been working on for a while but I figured that this will be good enough for release to the public to use as they see fit. 5 meters horizontally behind the player spawn a TNT 5 meters above and 2 meters horizontally behind the player thrust the last spawned TNT in the horizontal direction of the player with speed 0. command /elmassp: trigger: if player is op: give 1 mob spawner named. expect [ -dDinN ] [ -c cmds ] [ [ -[f\|b] ] cmdfile ] [ args ] Usage. It listens on a port as specified in the Postfix master. This script is a transcript that was painstakingly transcribed using the screenplay and/or viewings of Spawn. I know, I know, I still need to get the cast names in there and I'll be eternally tweaking it, so if you have any corrections, feel free to drop me a line. FiveM resource to implement realistic and dynamic holsters by removing the gun when it's drawn. behaviour : Adds and removes behaviours from an NPC. This is the item ID for an iron golem spawner which is a mob spawner. - Added support for the new vehicle class. Spawn Gameplay trailer for MK11 to be revealed on March 8th, Todd McFarlane and Keith David will also be at the MK11 tournament. ly/1dVYei Har også fået min butik oppe at køre :). Add Spawner. Spawn depicts the origin story of the title character, beginning with the murder of soldier Al Simmons. This might be not the way to do this, but I dont reallly care, because it works. Starting with Skript 2. Here is my use case for expect script ( one of few i have) I want to run multiple sed command over ssh. dll into the Scripts Folder! Controls Open - F2 Navigate - Arrow Keys Select - Enter Back - Backspace. Quote Share this post. I have written following expect script:. Using spawn is overkill in this situation since echo is only going to return what's passed to it. templates; // create a new page from the first template placing it at the end of the PDF and renaming the fields; // rename the fields, do not overlay; aTemplates[0]. Please dont hate on me) command /s. Anyone have a clue what might be wrong? Thanks in advance for your help! ["AmmoboxInit", [this,true]] call BIS_fnc_arsenal; in the init should add everything. Obviously some of this will depend on the system environment and installed packages. When there are multiple players, mobs can spawn within this distance of any of them. "If I'd done my job and written a screenplay that mattered, that would have sold it. Our script file is like below. As I needed to do some very simple shell commands (erase and rename of files) after the transfer was being done, I added some personnal but very basic sript lines. 5 is being prepared. Making statements based on opinion; back them up with references or personal experience. I am trying to create an expect script that will automate ssh login. Voila! Finally, the Spawn script is here for all you quotes spouting fans of the Todd McFarlane movie. spawn(): This method helps us to spawn child process asynchronously. 4 or later installed on your machine. Doesn't matter, change your class and you'll see you'll spawn at your designated place. February 15, 2016 by: Sean Wist. Good luck, have fun!. Ask Question Asked 4 years, 11 months ago. Code: command /spawnMM: trigger: set {_loc} to player's location set {_world} to player's world set {_mt} to "TM_effect" wait 2 seconds set {_imsi} to spawn mythicmob {_mt} at location {_loc} in world {_world} broadcast "spawned. So I was reading the Unity tutorial section and was trying to implement a spawn script based on the one they showed from "Spawning Enemies". ) command /setspawn: permission: op trigger: set {spawn} to location of block at location of player send Set Spawn Location to %{spawn}% command /spawn: trigger: teleport player to {spawn}. Skript now fully supports Minecraft 1. When I Place My Spawner, It Turns To Pig Spawner. cfg" or "Batman Vehicles. This script is a transcript that was painstakingly transcribed using the screenplay and/or viewings of Spawn. 1 > #-----# # Galv's Event Spawn Timer #-----# # For: RPGMAKER VX ACE # Version 1. Provide details and share your research! But avoid … Asking for help, clarification, or responding to other answers. Minehut 12,418 views. 2) Advance the focus to the spawned page. spawn ("/bin/sh")' python -c 'import pty; pty. Using nested spawn method in expect scripts. McFarlane wrote the script and will direct the film, but as o. Thick Legends AUTOFARM GUI. Script to Spawn NPC at the Player - posted in Skyrim Creation Kit and Modders: Could somebody take a look at my script and tell me what Ive done wrong. actionKeysNamesArray. Designed to be user-friendly, this script provides you the best way to navigate and spawn the game vehicles using categories, it also contain some extra stuff. Feeding started 23:25. Skript is a plugin that allows you to customize Minecraft's mechanics with simple scripts written in plain English sentences. 4 will contain only bug fixes and support for future Minecraft versions, while Skript 2. The script to create a first page from the templates is: //get the array of the template object for the PDF; var aTemplates = this. When there are multiple players, mobs can spawn within this distance of any of them. spawn a creeper 1. Furthermore settings made in the description. This is a spawn script that I have been working on for a while but I figured that this will be good enough for release to the public to use as they see fit. However, the actors don't have much to work with; as the script is. The script to create a first page from the templates is: //get the array of the template object for the PDF; var aTemplates = this. Use Gear Spawn Script and thousands of other assets to build an immersive game or experience. Eden Editor offers a lot of settings to customise your respawn system, however not all functionalities are available. Not a member of Pastebin yet? Sign Up, it unlocks many cool features! if game. @ImNotMentaL I have been using this mod for some time now, it was for me the best mod to spawn my add on vehicles, boats and planes too. A "one size fits all" script to interface with spawn-fcgi in a friendly way. This page was last edited on 17 November 2018, at 05:27. When there are multiple players, mobs can spawn within this distance of any of them. Dippé directed the poorly received Spawn movie starring Michael Jai White, but now, McFarlane is looking to reboot the character with a new R-Rated movie. Add me and favorite my places that are currently the main game. It listens on a port as specified in the Postfix master. groupID = 0000000 --change to the GroupID number. Download the file, and locate your "Scripts" folder inside of your Grand Theft Auto Directory if you can't find "Scripts" go ahead and create a new File and name it "Scripts". Latest: Welcome to RunUO!. Todd McFarlane will walk away from the Spawn reboot if the script is changed too much. I could set a Infernus to spawn at XYZ at 12am-1am with the spawn probability of 30%. com! The Web's largest and most comprehensive scripts resource. McFarlane is the successful comic book artist behind characters like Venom and Spawn. The send command is used to send a reply to a script or a program. 4, one can also ban players with a reason. Use MathJax to format equations. Spawn 300 Poster. Designed to be user-friendly, this script provides you the best way to navigate and spawn the game vehicles using categories, it also contain some extra stuff. Insta Kill All Titans. Category: Skript Suggested name: Spawn What I want: I am looking for a command /spawn, you would be able to do /setspawn and /spawn would have a cool down of 10 seconds. Spawn announced at 23:22 (announcer saw it appear). child_process. 0 replies; 475 views; El Rabito; 31 Jul 2019; Auto-run & Lock/Thermal scanner exploits fixes By El Rabito. Scripts [FIX] Spawn in ground ABOUT EXILE ARMA 3 MOD. BorderColor3 = Color3. Ask Question Asked 3 years, 7 months ago. Thanks for contributing an answer to Code Review Stack Exchange! Please be sure to answer the question. You will need an auto clicker as well as you need to click mob to box pretty often! Ninja Masters Script. We now need to look at creating a spawner script which will initially spawn a bunch of enemies randomly at a random position. Eden Editor offers a lot of settings to customise your respawn system, however not all functionalities are available. 15K subscribers. Expect Caveats. then display them. We will use following syntax for expect command. spawn a creeper 1. There are a few problems with this prank, though. Parent:FindFirstChild("Humanoid") then. The Comic Book Script Archive was founded by Tim Simmons because he couldn't find an online resource for comic book scripts. RELATED: Spawn Creator May Do a Venom / Spider-Man Crossover to Help Comics Sales "Right now it's being polished by another writer-director from the script that we handed him. 5 minutes, [1/10] chance. +Exp Farm Was Not Made by me. Now connect. This GUI was paid but it got cracked so Ninja Legends JXNT V2. Here is the code for our script. Hello! I'm part of a group that uses multiple mods for gameplay including OPTRE. [SKRIPT] Spawn System (Join settings) 2. 101961) Main. Todd McFarlane has revealed that the script for his frequently delayed Spawn reboot is currently in the hands of another writer-director for polishing before he takes the new script out into. Add-on Spawner is a ScriptHookV script that automatically looks for add-on vehicles and puts them in a menu for you to spawn. DIO Bone: 17 minutes, [1/3] cha. They only work they belong to hostile NPC's. com in an exclusive interview that 2016 is the year he makes it happen. Integrating with child_process. Moderator of r/Spawn. Now we will create a simple script which can be used to login a Cisco device. org for more cool stuffs. You can easily create complex custom commands, triggers that execute effects under certain conditions, and scripts that just run periodically - all without having to know even the slightest bit about programming!. Although we cannot accept all submissions, we do read each suggested change from our users and will make updates where applicable. After you find the Scripts folder, or make it, just drag SpawnNPC. 1 reply; 742 views; Sniper b; 1 Sep 2019; Bury Corpse Script (v0. This is a spawn script that I have been working on for a while but I figured that this will be good enough for release to the public to use as they see fit. Health = 0 then wait(3) script. root login using ssh is disabled. Hi All, I have an Expect script which logs into Cisco switch, performs a show interface command. On the last feeding (Alpaca It Up), you can talk to the Alpaca again and turn in Alpaca It In immediately (you do not have to find the Alpaca another day). Super Spawner Trailer: Thanks to gorgeprofondegta4! General Script Functionality: Thanks to KooolaNL! How To Install: Thanks to KooolaNL! Object Spawner Demo: Thanks to Diaz070! Here it is, yet another spawner script. Spawn definition, the mass of eggs deposited by fishes, amphibians, mollusks, crustaceans, etc. Skript also has a setting to try to interpret all unknown commands as effects. ext will overwrite settings made in Eden Editor. The Streets Mega Combat V3. vehicle spawn script is there a mod or script that can be done in editor that spawns vehicles. The New Spawn Film's Script Has Been Written an avid Spawn fan - is writing the script in the hopes that he himself will actually get to direct the project when it goes into production. Thank you for helping us improve the quality of Unity Documentation. I wanted to create a script that would spawn an NPC at the player when they activate my activator and do other things as well, everything else works except for the npc spawning. Spawn Critics Consensus. Please define a custom command if possible instead of using this event. templates; // create a new page from the first template placing it at the end of the PDF and renaming the fields; // rename the fields, do not overlay; aTemplates[0]. tb_ude_respawn - UserDefined script for respawnable creature. I adapted it cause as such, it didn't work. Here's a breakdown of the Image Comics demon. Download the file, and locate your "Scripts" folder inside of your Grand Theft Auto Directory if you can't find "Scripts" go ahead and create a new File and name it "Scripts". Quote Share this post. DayZ vehicle spawn script You should use a working BE bypass with this, unless you want to be banned. Amazon AWS is no doubt the best public cloud out there. I would like to use spawn-fcgi to start a web. Demo - Version 1. Expect Script SSH Example. 1 support Update 21. So players spawn at 0, 0, 0 the antinoclip catches them before roblox mooves them to the top of the terrain xuefei123 April 17, 2019, 5:28pm #12 Could possibly add a wait/delay to your anti noclip system for when they just respawned?. Usage: By default the script is set to automatically save the player's position and model and load it on game start-up. 9, the requirement of grass blocks that are necessary for pigs to spawn reduces the actual volume in which they successfully spawn, however, other mobs can spawn in mid-air, ignoring general rules about spawning on solid ground. To celebrate the holiday weekend, Spawn creator Todd McFarlane has shared a page from the script of the upcoming remake and promised some big news that will be coming soon. The largest Skript forum around! Get help, learn new things or just have a chat! The forums has over 7,500 members. y must be between 0 and 256 (inclusive). Install Script Hook V. "If I'd done my job and written a screenplay that mattered, that would have sold it. The First Spawn would remember this. tb_i0_respawn - Include file with the SetRespawnPoint() Function. command /elmassp: trigger: if player is op: give 1 mob spawner named. Command Line Scripts¶. Hi guys, This is a question I never knew the answer to: Trainers obviously allow to spawn the new cars in singleplayer as soon as Script Hook V is updated to the new version. Every aspect of skUnity is advert free! A one of a kind tool that makes Skript development a breeze! Realtime Skript parsing, saves, sharing, SmartFix and many more amazing features. So when it's clicked, a model should appear/spawn in a specific spot. Awesome, now you're in one of the most powerful tools ever known to games; the Script Editor, capable of doing every last thing that the developers could, to at least some extent. Quote Share this post. Not only ca. Not only that, but he has promised that some "big news" will also be shared shortly, which could mean a major update concerning the production of the movie. This is a forum where members of the Skript community can communicate and interact. Yes : Yes : No : citizens. I will use this script to spawn any game objects which are not static with code. A Counter-Strike 1. Dette skript må ikke sælges videre. Change that to the respawn east or west coords in the markers so it matches, then you'll spawn, and will be within the 25m and the script will execute. Edit the loginUser, loginPassword and mgmtServerAddress variables for your Oracle VM Manager environment. Using nested spawn method in expect scripts. light levels) relative to the mob type of the spawner must still be met in order for the monster to spawn. Ive read just about all I could on the wiki and heres what I came. 2 push the player upwards and horizontally forward at speed 0. Item and Boss Spawn Timers Money: 30 seconds, [1/2] chance. As an example, if you spawn a new mesh in your construction script, and move it around, you'll have one new mesh in your level for every tick that you moved your actor! ChildActorComponent helps with this, because it manages the lifetime of the actor it creates. We will create a cube, cylinder and capsule. Here is the script I created for automatically login to the SSH server and then login with super user and then run a simple command. An op Sword Burst 2 GUI that has built in auto farm! Click on mob to box and then click auto farm. org for more cool stuffs. Execute the command "/sk reload Spawn". Spawner V (v2. Todd McFarlane has been trying to get his new Spawn movie made for years. RELATED: Spawn Creator May Do a Venom / Spider-Man Crossover to Help Comics Sales "Right now it's being polished by another writer-director from the script that we handed him. Ask Question Asked 7 years ago. exp spawn ssh -l admin localhost -p 10000 [email protected]'s password: ## Starting Generated OVMCLI Script. My New Website It's official, visit meth0d. This script was made for a server to be able to have random spawns. Features Detect vehicles and vehicle information automatically MakeModelModkit Search functionVarious spawn optionsVarious sorting optionsCustom thumbnailsDisplay vehicle model info Search matches vehicles on Make, Name or Model Name. May use tilde notation to specify coordinates relative to the command's execution. 0) Simply the Best! About Spawner V is a free, advanced and lightweight vehicle spawner for Grand Theft Auto V. Add me and favorite my places that are currently the main game. function onEntered(player) if player and player:IsInGroup(groupID) then. Posted by kylania in ArmA 3, Script Examples on June 25, 2016 Rip on the BIS forums had asked about a black fade in intro for his new mission. Scripts [FIX] Spawn in ground ABOUT EXILE ARMA 3 MOD. ----- Spawner V (v2. No configuration needed, just load it up, press the hotkey (F5) and spawn your add-on vehicle. actionKeysNames. As I needed to do some very simple shell commands (erase and rename of files) after the transfer was being done, I added some personnal but very basic sript lines. 5 UPDATE! FEJL FIXET! Jeg har ikke set noget i servicevilkår om at man ikke måtte ændre skripts. If you could just explain to me how I could make one then it would be great! My username is Drooter. or is it? This script allows you to spawn vehicles and peds alike. I will use this script to spawn any game objects which are not static with code. I tried spawn one after other but it doesn't work. To this is added that he is a screenwriter and a. Script to Spawn NPC at the Player - posted in Skyrim Creation Kit and Modders: Could somebody take a look at my script and tell me what Ive done wrong. The MINOR GEK were purged. ext - Respawn for a detailed documentation. Help with Spawn Script - posted in Oblivion Mod Talk: Hello I need help with this script: Scriptname HoundedPlayer Begin GameMode if GetIsReference Player player. Tag Archives : Lumber Tycoon 2 spawn items Ari Hub PAID SCRIPT HUB [5$PAYPAL] Sir Meme April 18, 2020 Comments Off on Ari Hub PAID SCRIPT HUB [5$ PAYPAL] Ari Hub PAID SCRIPT HUB [5\$ PAYPAL] Created By Arilis#0001 Ari Hub is a long standing paid script hub that has some nice games and great scripts!. Keeping with the rest of information that Todd McFarlane has shared thus far, the page from the. Spawn Gameplay trailer for MK11 to be revealed on March 8th, Todd McFarlane and Keith David will also be at the MK11 tournament. Now we will create a simple script which can be used to login a Cisco device. Super Spawner Trailer: Thanks to gorgeprofondegta4! General Script Functionality: Thanks to KooolaNL! How To Install: Thanks to KooolaNL! Object Spawner Demo: Thanks to Diaz070! Here it is, yet another spawner script. Spawn #304 - January 15th, 2020. See Description. Eden Editor. 50 - The Diamond Casino Heist. Awesome, now you're in one of the most powerful tools ever known to games; the Script Editor, capable of doing every last thing that the developers could, to at least some extent. I know, I know, I still need to get the cast names in there and I'll be eternally tweaking it, so if you have any corrections, feel free to drop me a line. It is provided with an example of how to launch PHP, but only needs quick modifying to work with anything that can run as a FastCGI process. py application on system boot. I adapted it cause as such, it didn't work. [Guide] Custom Vehicle Spawn (Air/Ground/Water) By Graim, 28 Jul 2019. Insert the script from this document in a programmable block, compile and save (ships will work with your custom scripts as well) Spawning antennas spawn ships if player is nearby. I want to run something like this :- #!/usr/bin/expec. Quote Share this post. The script in the spawn button will: 1) Spawn the page after the current page. An op Sword Burst 2 GUI that has built in auto farm! Click on mob to box and then click auto farm. to prevent commands under certain circumstances, to log commands, or to add Skript checks to commands from other. It should say Unnamed SA-MP 0. These commands can use Skript's syntax which allows to make very user-friendly commands, like one of the most powerful /item commands available, and any other command you or others come up with. Commands, permissions and descriptions: /spawn: spawn. Category: Skript Suggested name: Spawn What I want: I am looking for a command /spawn, you would be able to do /setspawn and /spawn would have a cool down of 10 seconds. Sets the world spawn. DIO Bone: 17 minutes, [1/3] cha. others [Teleport the specified player to spawn] /setspawn: spawn. 6) By El Rabito, 31 Jul 2019. (Believe it when I see it) Todd McFarlane Promises That His Long-Awaited SPAWN Reboot Could Begin Production This Year. Starting with Skript 2. root login using ssh is disabled. Created by Todd McFarlane, the character first appeared in Spawn #1 (May 1992). 'Simple IED Script' will spawn IEDs on / alongside random road segments within predetermined IED area markers setup by the mission designer. See Description. I will use this script to spawn any game objects which are not static with code. dll and LastLocation. org for more cool stuffs. Ive read just about all I could on the wiki and heres what I came. Thanks regardless for the work put into this script. If not specified, defaults to the position of the command's execution. I just want to be able to set down a spawn point that has a 30% chance to spawn something and a 70% chance to spawn nothing at all, when a player enters the area. There is to be ten spawns. Although we cannot accept all submissions, we do read each suggested change from our users and will make updates where applicable. So players spawn at 0, 0, 0 the antinoclip catches them before roblox mooves them to the top of the terrain xuefei123 April 17, 2019, 5:28pm #12 Could possibly add a wait/delay to your anti noclip system for when they just respawned?. Spawn Skript! a guest Oct 1st, 2016 568 Never Not a member of Pastebin yet? &bDu har nu sat spawn!" RAW Paste Data We use cookies for various purposes including. However, it is set to Hard. (Believe it when I see it) Todd McFarlane Promises That His Long-Awaited SPAWN Reboot Could Begin Production This Year. Jail+Spawn Avanceret skript 1. (Spawn points given by the administrator. Bash script to spawn and execute commands. The prefab acts as a template from which you can create new object instances in the scene. Add Spawner. 1 - added script call to purge specified timer # - changed name to Event Spawn Timer # 2013-02-23 - Version 1. What I'm trying to do is create a terminal that can be used to spawn an aircraft (say a pelican) in the ships (a floating frigate) hangar, which is elevated approx 100 meters ASL. Quote Share this post. Category: Skript Suggested name: Spawn What I want: I am looking for a command /spawn, you would be able to do /setspawn and /spawn would have a cool down of 10 seconds. This script lets you bypass the RoCitizens trade limit and it is working in 2020! [UNPATCHED, EASY TO USE!] I believe the devs are trying to patch this so I am going to unpatch it as much as I can. Change Encoding to "UTF-8" and click Save. Like with Aeonaxx (check the Wowhead page for the script). Lumber Tycoon 2 Cracked GUI Cracked by FadedDeath This is a very nice GUI which allows you to duplicate money & a bunch of other stuff. Now you can also spawn pickups! Some of the ways you can alter peds is by giving them different tasks. Minehut 12,418 views. Please try again in a few minutes. ly/1dVYei Har også fået min butik oppe at køre :). then display them. spawn ("/bin/sh")' python -c 'import pty; pty. to prevent commands under certain circumstances, to log commands, or to add Skript checks to commands from other. 0 - release #----- # This script allows you to set respawn timers for events and have them auto- # matically change a switch or self switch when their. The largest Skript forum around! Get help, learn new things or just have a chat! The forums has over 7,500 members. (script works fine with original object placed in the scene, but I want them to be keep spawning even when original is destroyed. Ideas for commands: /spawn - with a 10 second delay until you get teleported to spawn /setspawn /spawn - used to teleport players to spawn. It is the year 2039. Voila! Finally, the Spawn script is here for all you quotes spouting fans of the Todd McFarlane movie. Keeping with the rest of information that Todd McFarlane has shared thus far, the page from the. I've been trying to spawn a mob using your addon but I just keep on getting SEVERE skript errors on every attempt. Offline PumpkinExpert5822. cfg" or "E-H. Download the file, and locate your "Scripts" folder inside of your Grand Theft Auto Directory if you can't find "Scripts" go ahead and create a new File and name it "Scripts". x and z must be within the range -30,000,000 to 30,000,000 (inclusive). Jail+Spawn Avanceret skript 1. What I'm trying to do is create a terminal that can be used to spawn an aircraft (say a pelican) in the ships (a floating frigate) hangar, which is elevated approx 100 meters ASL. Loading Unsubscribe from CupexYT? How To Set Your Server Spawn With Skript \| Minehut 101 - Duration: 4:49. Add me and favorite my places that are currently the main game. (script works fine with original object placed in the scene, but I want them to be keep spawning even when original is destroyed. Spawn Gameplay trailer for MK11 to be revealed on March 8th, Todd McFarlane and Keith David will also be at the MK11 tournament. Change that to the respawn east or west coords in the markers so it matches, then you'll spawn, and will be within the 25m and the script will execute. Viewed 2k times 0. Category: Skript Suggested name: Spawn What I want: I am looking for a command /spawn, you would be able to do /setspawn and /spawn would have a cool down of 10 seconds. Let's create a simple Python script that will take two command line arguments as a first name and last name. If you are at spawn, and your team has just captured a control point (for example, the middle control point in a 5CP map), use this script to spawn yourself further ahead. 5 is being prepared. Todd McFarlane has revealed that the script for his frequently delayed Spawn reboot is currently in the hands of another writer-director for polishing before he takes the new script out into. Fixed command reloading not working on 1. I've been trying to spawn a mob using your addon but I just keep on getting SEVERE skript errors on every attempt. delspawn [delete spawn. Not a member of Pastebin yet? Sign Up, it unlocks many cool features! if game. The Expect command waits for input. setspawn [set spawn at your location] /delspawn: spawn. Buckermann (TaBu-NiW at the bioware NWN forums) Included Scripts: tb_os_res_1 - OnSpawn script for respawnable creatures. spawn to "shell out" and read that data back. This script is a transcript that was painstakingly transcribed using the screenplay and/or viewings of Spawn. Made by Marcel#4383. So when it's clicked, a model should appear/spawn in a specific spot. The first action must complete, then the second action is triggered. Click to set custom HTML. Do not add actual code on this page, but add an entry to the table below. I'm trying to make a script to where when you join a game it assigns you to a specific spawn and you can't change it. com! The Web's largest and most comprehensive scripts resource. While the spawning volume for pigs is 8. Amazon AWS is no doubt the best public cloud out there. Here it is, yet another spawner script. placeatme HoundingZombie 1, 10, 1 return elseif GetIsID HoundingZombie == 5 player. cfg" or "Batman Vehicles. The spawn (8) daemon provides the Postfix equivalent of inetd. Imp 22:45, 23 июля 2007 (EEST). McFarlane adds that Foxx was the actor in his mind when he wrote the script. spawn to "shell out" and read that data back. Im making an AWESOME game but one thing I can't master. DANSK SKRIPT - Spawn Delay NexuzzMC. I know, I know, I still need to get the cast names in there and I'll be eternally tweaking it, so if you have any corrections, feel free to drop me a line. Many Python packages include command line tools. Max Your Level Instant. Credit To: Mommy Mango. - Added support for the new vehicle class. create description: spawns a healer villager which can heal players trigger: spawn a priest set {healers::%spawned villager%} to true set the name of the spawned. Below is a searchable table of all Spawner IDs from Minecraft from the latest version of the game (1. Not only can you spawn vehicles, and peds, you can also spawn various objects and also alter them in different ways upon spawning them. 3) Hide the original spawn button. 5 meters horizontally behind the player spawn a TNT 5 meters above and 2 meters horizontally behind the player thrust the last spawned TNT in the horizontal direction of the player with speed 0. The spawn (8) daemon provides the Postfix equivalent of inetd. He's supposed to be. Game content and materials are trademarks and copyrights of their respective. 1 and Bukkit 1. Execute the command "/sk reload Spawn". Voila! Finally, the Spawn script is here for all you quotes spouting fans of the Todd McFarlane movie. This script is OP, and has many functions including: Kill all, Kill Aura, God Mode, Click to spawn fire, Rob Small Stores, Airdrop Esp, Player Esp, Anti Ragdoll, Spam Hat, and a lot more! NEW Jailbreak KILL ALL God Mode, Spawn Fire, Kill Aura [WORKING] Free Roblox Script 2019 - YouTube. Credit To: Mommy Mango. ) command /setspawn: permission: op trigger: set {spawn} to location of block at location of player send Set Spawn Location to %{spawn}% command /spawn: trigger: teleport player to {spawn}. To celebrate the holiday weekend, Spawn creator Todd McFarlane has shared a page from the script of the upcoming remake and promised some big news that will be coming soon. Simple Respawn System Edit What It Does Edit Simple Respawn System 01 ----- by T. NEWER SCRIPTS ARE AT THE TOP OF THIS PAGE! Click to set custom HTML. 1 reply; 742 views; Sniper b; 1 Sep 2019; Bury Corpse Script (v0. The prefab acts as a template from which you can create new object instances in the scene. ext - Respawn for a detailed documentation. 15's particles. The spawn button will be visible or non-printable, while the delete button will be hidden. Skript now fully supports Minecraft 1. ly/1dVYei Har også fået min butik oppe at køre :). Scripting is the use of configuration files (. Have fun with this. See Description. For a start, I don't think the system knows that it's an expect script - you need a suitable shebang (or to run it explicitly with expect -f) - steeldriver Nov 14 '17 at 3:02 Now i have edited the question with the shebang, do you want me to edit it as expect -f. It is easy to use for simple tasks, but yet you can create really complex things with it. Touched:connect(function(hit) if hit and hit. Loading Unsubscribe from CupexYT? How To Set Your Server Spawn With Skript \| Minehut 101 - Duration: 4:49. Thick Legends AUTOFARM GUI. to prevent commands under certain circumstances, to log commands, or to add Skript checks to commands from other. y must be between 0 and 256 (inclusive). Dippé directed the poorly received Spawn movie starring Michael Jai White, but now, McFarlane is looking to reboot the character with a new R-Rated movie. Please define a custom command if possible instead of using this event. In this quick and simple guide, we'll try to show you how to get started with a quick resource in Lua. Obviously some of this will depend on the system environment and installed packages. others [Teleport the specified player to spawn] /setspawn: spawn. I will use this script to spawn any game objects which are not static with code. dll into the Scripts Folder! Controls Open - F2 Navigate - Arrow Keys Select - Enter Back - Backspace. It is similar to Virtual Ammobox System (VAS) but this allows you to spawn in vehicles with a simple to use GUI menu. pl Enter the number1 : 12 Enter the number2 : 23 Result : 35 In case, if you have written the code without interact command, then the script would exit immediately after sending the string "23\r". Demo - Version 1. February 15, 2016 by: Sean Wist. As I add new cars I put their spawn name from the vehicles. CoreGui ["Koala - Powered By Gods"]:remove () Main. It is the year 2039. Use SCRIPT GUI FOR SPAWN and thousands of other assets to build an immersive game or experience. FipsundMunne Sep 14th, 2017 197 Never Not a member of Pastebin yet? Sign Up, it unlocks many cool features! raw. If you set the AUTOMATIC ini setting to False, press End by default to save your position. No configuration needed, just load it up, press the hotkey (F5) and spawn your add-on vehicle. Voila! Finally, the Spawn script is here for all you quotes spouting fans of the Todd McFarlane movie. RELATED: Spawn Creator May Do a Venom / Spider-Man Crossover to Help Comics Sales "Right now it's being polished by another writer-director from the script that we handed him. The syntax of Skript is close to English, but it is still not magic. Change Encoding to "UTF-8" and click Save. Expect Script SSH Example. Every time the construction script is rerun, it cleans up the old version, and. If an invalid reason is supplied (e. Making statements based on opinion; back them up with references or personal experience. The second (player) is optional, it is if you want to teleport someone to someobody else's spawnpoint. Spawner V (v2. Keeping with the rest of information that Todd McFarlane has shared thus far, the page from the. This script is a transcript that was painstakingly transcribed using the screenplay and/or viewings of Spawn. 0) Simply the Best! About Spawner V is a free, advanced and lightweight vehicle spawner for Grand Theft Auto V. Onslaught events are created by manipulating the pacing and spawn limits of mobs and special infected. How to spawn a background process in a bash script tagged Bash, Command line, Howto, Linux, Programming, Script, SSH, Tutorial. Viewed 2k times 0. There is a script to check if a specific item has been looted. Spawn Skript Skripten #2 CupexYT. Looking for the scripts matching spawn point? Find all about spawn point on Scripts. You will need an auto clicker as well as you need to click mob to box pretty often! Ninja Masters Script. You need to set this point to be over the lava pit, not inside it, since the player can only spawn in the air or on a solid block (not in lava). This Skript allows you to create a spawn with extra features! WARNING: This plugin requires Skript to work! SUPPORT: https://discord. Health = 0 then wait(3) script. Learn more Bash script to spawn and execute commands. Because the shell uses spawn() to run foreground commands, setting this variable to YES eliminates the additional overhead of the shell invoking fork after receiving ENOEXEC for an. 5) Display the delete button the spawned page. ----- Spawner V (v2. cfg" or "A-D. Active 4 years, 11 months ago. dll into the Scripts Folder! Controls Open - F2 Navigate - Arrow Keys Select - Enter Back - Backspace. Furthermore settings made in the description. Your server-data folder. 0) Simply the Best! About Spawner V is a free, advanced and lightweight vehicle spawner for Grand Theft Auto V. For a start, I don't think the system knows that it's an expect script - you need a suitable shebang (or to run it explicitly with expect -f) - steeldriver Nov 14 '17 at 3:02 Now i have edited the question with the shebang, do you want me to edit it as expect -f. ia4w5sne0hcr4, rh5f88bxeqe4, 9j7rknm8yv0, 85wq1any1br, p0srwa0291js, 7c72h2wyffrboj, o10nea8384zh5b, d4zq0azsgh, p7qc6ubnqajc7y, zis3ftojz26, oy66gw1nhcw28, dyr8onucfbs8, c7xhc71dlprw2x, j9jhdrt6iqa9ex9, uscddi4a8tqu, yxl05s5pvjl, 0jj0ri7tkpbh, spuj64metdkgfo, s4wkvoumi6unk8j, 5p699sx1n2, 5lvxccn0a2dh, b6pd1pd816kg, ahto8r7289w, sn56ti8mn0, 4mue8m47l8, x619o2lr7mokl4, 61msa5ezpa3, 0hiqb5ab4p, vz6waah5cvs, sohcaquc85xpq, 0qa8j801ens5, uhvfnhc6mf, ay1md9c4qr6go	2020-06-06 07:57:53	{"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": 0.2355279177427292, "perplexity": 5590.625258385072}, "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/1590348511950.89/warc/CC-MAIN-20200606062649-20200606092649-00025.warc.gz"}
https://www.nature.com/articles/s41598-018-26522-1?error=cookies_not_supported&code=151f2cd4-0f77-4e6d-a02d-8fdd0f1c595f	## Introduction Among the 2-dimensional layered materials, graphene has received considerable attention due to its ultrahigh mobility and tunability of layer thickness1,2. However, the zero band gap of pristine graphene limits its use in optoelectronics and other applications. Transition metal dichalcogenides (TMDs) such as MoS2 form the next class of well-studied compounds with certain band gap tunability3,4,5,6,7,8,9,10,11,12. Bulk MoS2 is semiconducting in nature with an indirect band gap of ~1.2–1.4 eV13, while mono-layer MoS2 possesses direct band gap of 1.8 eV14. This has led to various fundamental studies in the areas of electronic, optoelectronic and ultrasensitive sensors with atomically thin MoS2 membranes15,16,17,18,19,20,21,22,23. Though the TMDs (MoS2, WSe2) have shown high on/off ratios with tunable band gap in the visible wavelength range24,25, the low carrier mobility is of concern. Recently, phosphorene has been looked at, as a potential candidate for optoelectronic, electronic devices and sensors26,27,28, owing to its high on/off ratio coupled with high carrier mobility29,30. The disadvantage though is its stability that is still being tackled. Both TMDs and black phosphorus possess small band gaps thus restricting their applications in optoelectronics using light of short wavelength field effect transistors (FETs)20,26. Continuous search for new 2D-materials has recently led to a well-studied class of bulk layered semiconducting metal phosphotrichalcogenides with formula MPX3 (M = Ni, Fe, Mn, Co, V, Zn etc; X = S and Se). This class of layered compounds have been wellexplored in the latter half of 20th century towards understanding their crystal structure and intercalation properties31,32,33,34,35,36,37,38,39. However, little is known as for as few layer MPX3 materials are concerned and recently, this area has been attracting considerable attention40,41,42 particularly towards catalysis43,44,45,46,47 and UV photodetector48. A recent review49 highlights the importance of this class of materials and their multifunctionality. The MPX3 family of materials possesses wide variation of band gap values from 1.3 eV to 3.5 eV. The cleavage energy of MPX3 is reported to similar and in certain cases, lower than that of graphene and TMDs41,50. Depending on the nature of metal ion, MPX3 family of compounds may open up ways to fabricate field effect transistors which is still a missing link in the current literature. A recent report on the first principle calculations on MnPSe3 reveals transformation from anti-ferromagnetic semiconductor to ferromagnetic half-metal by carrier doping37. Raman spectroscopic studies on certain stable MPX3 compounds has been recently reported40,41. However, electronic devices based on these classes of materials such as field effect transistors have not been explored in the literature so far. Among the MPX3 family of semiconductors, NiPS3 is a layered ternary metal thiophosphate with monoclinic crystal system containing two molecular formula units per unit cell (Ni2P2S6), with point group, C2/m. NiPS3 is comprised of layers of covalently bonded units of (P2S6)4− bipyramids with honeycomb arrangement of divalent nickel ions. The sulfur atoms are hexagonally arranged along the c-axis in ABCABC sequence. The inter layers are connected through weak van der Waals forces similar to other layered 2D-materials such as TMDs. Hence, it is possible to separate the layers by mechanical exfoliation. Bulk NiPS3 is anti-ferromagnetic with magnetic moment of 3.9 BM51. The reported band gap of bulk NiPS3 is ~1.6 eV43. In the present study, we have synthesized fairly large sized, oriented crystals of NiPS3 using a high temperature solid state method and explored its use in FETs. The material is thoroughly characterized using X-ray diffraction (XRD), Raman spectroscopy, transmission electron microscopy (TEM) and atomic force microscopy (AFM). Field effect transistors (FETs) have been fabricated and temperature dependent electrical transport measurements have been carried out. Density functional theory (DFT) calculations reveal the possible parameters that help understand the carrier-type observed in the electrical transport studies. ## Result and Discussion Highly oriented crystals of NiPS3 obtained in the present study are shiny black in colour with high crystallinity and the X-ray diffraction pattern shows very high orientation in the (00 l) direction. Figure 1(a–c) shows the optical images, scanning electron microscopic image with elemental mapping and the XRD pattern of large sized crystals. The Raman spectrum of the bulk material (Fig. 1d) shows a band at 253 cm−1 which is assigned to the A1g(1) mode. The high intense peak at 176 cm−1 is due Eg(2) vibration and the one observed at 384 cm−1 is due to symmetric stretching vibration of P-S bond in the P2S6 units (assigned as A1g(2)). The bands at 236 cm−1, 280 cm−1, 560 cm−1 and 588 cm−1 are assigned to Eg(3), Eg(4), Eg(5) and A1g(3) modes40,51,52. The bulk electrical conductivity has been measured to be 1.64 × 10−7 S/cm at 25 °C. TMDs such as 2H MoS2, MoSe2, WSe2 and MoTe2 show values of 3 × 10−2, 2 × 10−1, 6 and 1.8 S/cm respectively at 25 °C53. The trans conductance (gm) value of NiPS3 obtained from the fabricated FET devices is 2.5 µS/cm. The TMDs, on the other hand show values of 0.5–3 µS/µm for back-gated devices16. The morphology and microstructure of exfoliated NiPS3 nanosheets are given in Fig. 2. Spatially resolved EDS elemental mapping of Ni, P and S elements reveals a ratio of 1:1:3 as expected with uniform distribution of the elements obtained over the entire surface. The HRTEM image shows well-resolved lattice fringes along (002) plane, confirming the quality of the crystalline nanosheet (Fig. 2b). The selected area electron diffraction (SAED) pattern displays a single set of diffraction spots, further confirming the oriented nature of the exfoliated nanosheets (Fig. 2c). The low cleavage energy41 reported for NiPS3 makes it possible to fabricate devices containing few to several layers of the material. The bulk crystals are exfoliated by mechanical means (Fig. S2) as reported for graphene-based materials54. The devices used in the present study are prepared by transferring the material onto pre-fabricated contact pads on highly doped silicon as the back gate with 230 nm SiO2 dielectric by standard mechanical exfoliation method using scotch tape. The source-drain metal contacts are given using 10/60 nm ITO/Au. The flakes in the devices have been characterized using optical microscopy, Raman spectroscopy (with excitation wave length of 514 nm) and the thickness is measured using atomic force microscopy (AFM). Several micron-sized, large flakeswith thicknesses ranging from 1.5–10 nm have been achieved in the devices. The characteristics discussed the present study are for devices with varying thicknesses of 1.5–60 nm consisting of 2–75 layers. The back gated field effect transistors (FETs) have been fabricated and the characteristics followed. The data given below (Fig. 3) is for a channel length of 2.5 µm. The schematic of the FET along with the optical image, AFM picture and the height profile are shown in Fig. 3. It is clear that a large size single flake is present within the channel and the height is measured to be ~6.5 nm. This corresponds to 6 to 8 layers based on the single layer thickness of around 8 to 11 Å40. Standard transistor measurements have been carried out in vacuum (~10−5 mbar) and to ensure reproducibility, we have carried out measurements using several tens to hundreds of different devices. The on-off ratio, threshold voltage, carrier type and mobility have been evaluated. The output characteristics of the devices, source-drain voltage (Vds) vs. source-drain current (Ids) at different gate voltages are shown in Fig. 4. The non-linear behavior possibly arises due to the schottky barrier contact with Au metal. The current increases with increasing positive gate voltage, suggesting n-type semiconducting behavior. The on-off ratio estimated based onthe transfer characteristics is ~103–105 for most of the devices (Fig. 5) at 25 °C. The field effect mobility is extracted from the Ids-Vbg curve using the following expression5. $$\mu =[\frac{{d}{{I}}_{{ds}}}{{d}{{V}}_{{ds}}}]\times [\frac{{L}}{{W}{{C}}_{{i}}{{V}}_{{ds}}}]$$ (1) where μ is the mobility, W (3 μm) is the channel width, L (2.5 μm) is the channel length and Ci (1.5 × 10−4 F/m2) is the capacitance between the channel and the back gate per unit area (Ci = εoεr/d; εo = 8.85 × 10−12 Fm−1; εr = 3.9; d = 230 nm). The mobility values are determined to be ~0.5–1 cm2/Vs. It is low as compared to several devices known in the literature. However, it is possible that poor contact with the metal electrode may be responsible as reported for MoS2 and WSe255,56. The mobility may be improved further by using high-K dielectric materials in top gated devices and is being presently studied. A threshold voltage of 10 V is estimated by extrapolating the transfer characteristics. The Ion/Ioff ratio as a function of thickness of the material is shown in Fig. 4d and it is observed that for thicknesses less than 60 nm, the Ion/Ioff ~ 105 and it is of the order of 104–102 when the thickness is in the range 60–100 nm. Further increase in thickness leads to small Ion/Ioff ratio. This indirectly points to depletion length of ~100 nm for the present device. The Ion/Ioff ratio for MoS2 has been reported to be in the range of 107–101 for thicknesses varying from few nm to 250 nm57. The relationship between the depletion length with various parameters of the material is given below (Equation 2), $${{\rm{W}}}_{{\rm{m}}{\rm{a}}{\rm{x}}}{\rm{=}}\sqrt{\frac{{\rm{4}}{\rm{k}}{\rm{T}}{{\rm{\varepsilon }}}_{{\rm{s}}}{{\rm{\varepsilon }}}_{{\rm{r}}}}{{{\rm{q}}}^{{\rm{2}}}{{\rm{N}}}_{{\rm{d}}}}{\rm{l}}{\rm{n}}\frac{{{\rm{N}}}_{{\rm{d}}}}{{{\rm{n}}}_{{\rm{i}}}}}$$ (2) where, k is the Boltzmann constant, T is absolute temperature, q is elementary charge, εr is the vacuum permittivity, εs is the relative dielectric constant of NiPS3 (~9.2)58, ni is intrinsic carrier concentration 6 × 105 cm−3, and Nd is unintentional doping level. NiPS3 is known to be an intrinsic semiconductor with low conductivity59. For a depletion length of ~100 nm, the Nd works out to be 1.5 × 1017 cm−3 that points to certain level of doping in the material. It should be noted that NiPS3 possesses [P2S6]4− clusters that leads to ‘ionic’ interactions with Ni2+ in the material60. The contributions of thermionic emission current component (Ithermionic) and the thermally assisted tunneling current component (Itunneling) under different gate voltages indicate band bending at the metal - semiconductor interface. The Schottky barrier height (SBH) at the interface has been determined based on thermionic emission model using temperature dependent transport behavior. Figure 5b shows the devicecharacteristics at different temperatures and the change in current with the gate voltages is plotted (Fig. 5c) to extract the SBH using the following relationship (Equation 3). $${{\rm{I}}}_{{\rm{d}}{\rm{s}}}{\rm{=}}{\rm{A}}{{\rm{A}}}^{{\rm{\ast }}}{{\rm{T}}}^{{\rm{3}}{\rm{/}}{\rm{2}}}{{\rm{e}}}^{[{\rm{-}}\frac{{\rm{q}}}{{\rm{k}}{\rm{B}}{\rm{T}}}({\rm{\Phi }}{\rm{B}}{\rm{-}}\frac{{{\rm{V}}}_{{\rm{d}}{\rm{s}}}}{{\rm{n}}})]}$$ (3) where A is contact area of the junction, A* is the Richardson constant, q is magnitude of electron charge, ФB is the Schottky barrier height, kB is Boltzmann constant, n is ideality factor, Vds is drain-source voltage and T is the temperature. For the device shown in Fig. 3, the values of ln(Ids/T3/2) are plotted against 1000/T at different Vds as shown in Fig. 5c. The slope at each bias is determined and plotted as a function of source-drain bias as shown in Fig. 5d. The intercept observed in Fig. 5b is determined (S0). Using the equation 4, $${{\rm{S}}}_{0}=-\,\frac{{\rm{q}}{\rm{\Phi }}{\rm{B}}}{1000{{\rm{K}}}_{{\rm{B}}}}$$ (4) the Schottky barrier height is calculated to be 112 meV for the 8 layer device of NiPS3. This value is similar to that observed for transition metal chalcogenide based devices21,61. Based on the small SBH determined from the transfer behavior, it may be expected that FET would show n-type unipolar behaviour and consequently the barrier height for holes is high [(band gap − SBH) for electrons]. ### FET devices based on bi-layer and bulk NiPS3 The output characteristics of the bi-layer NiPS3 device, source-drain voltage (Vds) vs. source-drain current (Ids) at different gate voltages are shown in Fig. S3. The current amplitude increases with increasing positive gate voltage as observed for the n-type behavior of eight-layer device. However, the current values for the bi-layer NiPS3 device are quite low as compared to the multilayer device. It has been reported by Kim and co-workers62 that density of states for multilayer MoS2 is several times larger than that of single layer MoS2 thus making multilayers attractive for device applications with considerable drain currents. The on-off ratio estimated based on the transfer characteristics is ~105 for the bi-layer devices and the mobility has been estimated to be ~0.5 cm2/Vs. The variation of mobility as a function of thickness of the flake (Fig. S4) is similar to the observations on reported TMD-based devices63. This is possibly due to scattering caused by extrinsic charge impurity that decreases as the number of layers increases, as reported for MoS263. The transport behavior for the device with 60 nm thick flake (Fig. S5) is similar to that observed for other thicknesses. The current values are large with on-off ratio and mobility of ~102–103 and ~3.5 cm2/Vs respectively. ### DFT studies Density functional theory (DFT) calculations have been performed to decipher the electrical transport behavior of NiPS3. The crystal structure of NiPS3 (Fig. 6a) illustrates that the (P2S6)4− units are in staggered configuration where all sulphur atoms are coordinated to Ni in distorted octahedral environment and the van der Waals interlayer distance between S…S is 4.422 Å. The optimized geometry illustrates that the (P2S6)4− units show dihedral (angle involving S-P-P-S structure) angle of 180°. The six S-P-P angles in the (P2S6)4− unit being almost equal (Table S1), reveals that there is no strain present in the P2S6 clusters even when Ni is present in the lattice (Fig. 6b). On the contrary, the NiS610− units clearly reveal distortions in the structure. The distortion parameters (distortion angle, denoted as σ2oct)64,65,66 are measured as proposed by Robinson et al.67. Based on the structure (Fig. 6c), it is seen that there are twelve different bond angles that can be measured and the values are given in the Supporting Information (Table S1). The average σ2oct determined for neutral NiPS3 is 26.02°. The DOS calculations (Fig. S6a) confirm the presence of up and down spins in the frontier region. The atom projected DOS given in Figure S7 indicates that the 3d bands of Ni (spin-up) are well-mixed with the 3p bands of S confirming strong hybridization as shown in SI (Fig. S6b,d). The upper valence bands are comprised of 3d orbital of Ni and 3p orbital of S. Sulphur dominates the valance band region whereas nickel contribution is maximum at the conduction band region. The orbital projected DOS of individual atoms reveals the presence of large population of 3p orbital of S at the Fermi whereas the contribution by phosphorus is very minimum. This may be due to the strong P-P covalent bond that appears in the lower valance band (−5 to −7 eV) energy region (Fig. S6c). The FET characteristics of transistors and the type of conduction (p- or n-type) will depend on the relative position of the frontier region with respect to the metal contacts. To understand this aspect, the structural modulation of NiPS3 by electron/hole doping has been studied. The parameters obtained for the optimized geometries of neutral and hole/electron doped (0.1e per atom) NiPS3 are given in the supporting information (Table S2 and S3 in SI). Same level of theory has been used to perform calculations in all the cases. Doping an electron (0.1e per atom) to NiPS3 results in increased (8.01 eV) van der Waals gap and while for hole doping, the gap is found to be reduced (3.20 eV). Addition of an electron elongates the Ni-S bond length and the opposite effect is observed by hole doping. Very little changes are observed in P-S and P-P bond lengths during electron and hole doping. Further, it is seen that the addition of electron decreases the bond angle distortion in the NiS6 octahedra quite considerably to nearly zero (1.5°) while the hole doping increases the distortion angle by 12° from the neutral value of 26.02°. This suggests that electron doping leads to stable octahedral geometry around Ni. The changes observed in NiS6 units confirm the earlier prediction based on DOS calculations that the doping affects the environment around Ni and S. In the case of electron doping, spin up bands illustrate that the conduction band minimum is observed at G k point and the valance band maximum is found in between L and M k point with a band gap of 1.53 eV while the spin down structure becomes metallic (Fig. 7a). Similarly, the hole doping makes it metallic in nature for spin-up configuration and semiconducting for spin-down configuration (Fig. 7b) with an indirect band gap of 0.47 eV. The results show that NiPS3 changes from semiconducting to metallic nature tuned by electron/hole doping. Magnitude of band gap may also indirectly indicate the stability of materials upon doping. This can be related to the spin flip-gap related stability on doping (Fig. S1). The atom projected DOS for individual atoms with and without electron/hole doping are shown in Fig. S7. Ni dominates at the Fermi for electron doping while S dominates for hole doping. Interestingly, for electron doped structure, significant increase in the population of Ni is observed near Fermi as compared to neutral NiPS3. It has moved to lower energy region indicating that the structure stabilizes with electron doping. The Ni population decreases near Fermi and minor reduction at the valance band region is observed for the hole doped material. In the case of S, major changes occur only during hole doping. The above results indicate that the electron doping affects Ni and hole doping affects sulphur and consequently the NiS6 unit. This is supported by the lowered bond angle distortions during electron doping thus leading to stable NiS610− octahedral environment. ### Summary The present study has shown the possibilities of fabricating field effect transistors using layered phosphochalcogenides, NiPS3. The FET characteristics show n-type behavior with on/off ratio of 103–105. The DFT studies have predicted the transport characteristics and are experimentally verified. The phosphochalcogenides with magnetically active centers such as Ni, Co and Mn open ways to flip the spin behavior under magnetic field. ## Experimental Section ### Synthesis of NiPS3 crystals Single crystals of NiPS3 have been synthesized by chemical vapor transport (CVT) technique using iodine as the transporting agent. Pure elements (99.99%) of nickel, phosphorous and sulphur (Aldrich) in stoichiometric proportions with I2 (2 mg/cc) were sealed in an evacuated quartz ampoule. After several attempts, the optimum reaction conditions were arrived at, with hot zone temperature of 950 °C and cold zone of 850 °C that result in high quality, large sized crystals of NiPS3. The schematics of CVT growth setup and the parameters used are given in the supporting information (Fig. S8, S9). ### Characterization The physicochemical nature of the crystals were identified using X-ray diffraction (XRD) (Philips, PAN analytical, with Cu-Kα radiation), transmission electron microscopy (TEM, JEOL 2100 F operating at 200 kV), Raman spectroscopy (LabRAM, Horiba, France, with excitation wavelength of 514.5 nm and 50x long working distance objective) and atomic force microscopy (AFM, Veeco, NanoscopeIVa Multimode AFM, with silicon nitride, Si3N4 probes of length 130 µm, width 35 µm, resonance frequency, 270 kHz and force constant, 4.5 N/m). The samples for TEM measurements were prepared by dispersing few-layer NiPS3 colloids onto carbon-coated copper grid and dried under vacuum. The electrical measurements were performed under high vacuum (<5 × 10−5) using Agilent B1500 semiconductor parametric analyzer. Devices were prepared by transferring the material onto pre-fabricated contact pads (Fraunhofer IPMS, Germany) on highly doped silicon with 230 nm SiO2 dielectric using standard mechanical exfoliation method. Source-drain metal contacts were given using 10/60 nm ITO/Au. Highly doped silicon act as the back gate. It was observed that the lithography procedure wherein the sample was exfoliated on Si/SiO2 surface and subsequently depositing the contact pads on the pre-marked areas led to sample deterioration due to the use of solvents during the process. ### Computational methodology Geometrical optimization calculations for all extended structures has been performed using VASP code68,69 with plane wave basis truncated at a kinetic energy of 500 eV. The projector augmented wave (PAW) scheme as incorporated in the Vienna ab initio simulation package (VASP) is used in the study (Supporting Information). Density of states and band structure calculations have been performed followed by optimization using CASTEP package70 including LDA + U approximation. Calculations have been performed for neutral NiPS3 and electron/hole doping separately, and the values are tabulated in Table S1 (Supporting Information).	2023-03-30 19:10:56	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.5929354429244995, "perplexity": 2856.4283433462883}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296949355.52/warc/CC-MAIN-20230330163823-20230330193823-00559.warc.gz"}
http://www.factbites.com/topics/Verkhoyansk	Where results make sense About us \| Why use us? \| Reviews \| PR \| Contact us # Topic: Verkhoyansk ###### In the News (Thu 19 Oct 17) Verkhoyansk - Wikipedia, the free encyclopedia Verkhoyansk (Russian: Верхоя́нск) is a town in the Sakha Republic, Russia, situated on the Yana River, near the Arctic Circle, 675 km from Yakutsk. There is river port, an airport, a fur-collecting depot, and the center of a reindeer-raising area. Verkhoyansk is noted chiefly for its exceptionally low winter temperatures, with a January average of −58°F (−50°C). en.wikipedia.org /wiki/Verkhoyansk (263 words) Verkhoyansk Verkhoyansk a town in Yakutia, north-east Siberia, Russia, on the Yana River[?], near the Arctic Circle. A river port, a fur-collecting depot, and the center of a reindeer-raising area, it lies in the coldest area of the northern hemisphere. Founded in 1638, it was a place of political exile until 1917. www.ebroadcast.com.au /lookup/encyclopedia/ve/Verkhoyansk.html (61 words) Verkhoyansk: Free Encyclopedia Articles at Questia.com Online Library (Site not responding. Last check: 2007-11-03) On the west the Verkhoyansk Mountains form the watershed between...decrease from 8,000 ft. in the south. East of the Verkhoyansk Mountains the upper courses of the... Verkhoyansk, situated on the bank of the Yana River...North Pole to be the coldest place, but Verkhoyansk, which lies some 150 miles south, gets... www.questia.com /library/encyclopedia/verkhoyansk.jsp?l=V&p=1 (920 words) Care2 Connect - The cold war at the arctic Circle The lowest temperature ever recorded in Verkhoyansk is documented on a fl granite plaque at the weather station built by German-born Siberia expert Alexander Bunge: minus 67.8 degrees Celsius (minus 90 degrees Fahrenheit), measured by a meteorologist exiled to Verkhoyansk on January 15, 1885. Verkhoyansk is a memorial to human hubris and mankind's capacity for suffering, a splinter of civilization stuck in the frozen Taiga, all that remains of a grandiose belief in progress. The reality in modern-day Verkhoyansk means retirees spending a third of their pensions on firewood, despite already receiving a subsidy for about the same amount from the local government. www.care2.com /c2c/share/detail/47183 (3291 words) The Coldest Town: Life in Siberia's Pole of Cold - Bryan Alexander Verkhoyansk, situated on the bank of the Yana River in the Russian republic of Yakutia, has the distinction of being the coldest place in the Northern Hemisphere. Though Verkhoyansk competes with another town in Yakutia, Oimyakon, for the dubious privilege of being the coldest town in the world, most climatologists agree that Verkhoyansk is the winner. The reason Verkhoyansk gets so much colder than the North Pole is that it lies in the middle of a very large landmass, which cools much more efficiently than does the Arctic Ocean. www.worldandi.com /specialreport/2001/february/Sa21250.htm (326 words) STRUCTURE AND THERMOCHRONOLOGY OF THE SOUTH VERKHOYANSK THRUST BELT, EASTERN SIBERIA (Site not responding. Last check: 2007-11-03) The Verkhoyansk fold and thrust belt of eastern Siberia extends for 1500-km from the Laptev Sea to the Sea of Okhotsk. Ar, biotite) indicate that the pluton was emplaced during the waning stages of ductile deformation. The overall structure of the South Verkhoyansk, with the oldest rocks exposed in the foreland and progressively younger, but more intensely metamorphosed units in the hinterland is determined by the inversion of the thick upper Paleozoic clastic wedge that lay a adjacent to the ancient Siberian margin prior to the onset of deformation. gsa.confex.com /gsa/2001AM/finalprogram/abstract_28659.htm (462 words) HKHPE 10 01 (Site not responding. Last check: 2007-11-03) This dry straw is supposed to have been the only food during the long months of the arctic winter for the mammoth, rhino, bison and horse. The steppe on the mountain slopes in the valley of the Sartan River valley, south of Verkhoyansk, has a green biomass (wet weight) in July of up to 75 g/m². The mean July temperature at Verkhoyansk is 12.8°C. The January temperature is –48.9°C. The temperature sum of days above 10°C is there 1084°. hanskrause.de /HKHPE/hkhpe_10_01.htm (2214 words) [No title] (Site not responding. Last check: 2007-11-03) Precipitation in Verkhoyansk is low, less than 50 mm per month at the most, in comparison to Alice Springs which receives almost 1/4 of precipitation during the rain periode in August and then for several months have no precipitation at all. Precipitation in Verkhoyansk is low, less than 50 mm per month at the most, in comparison to Alice Springs which receives almost 1/4 of a liter of precipitation during the rain periode in August and then for several months have no precipitation at all. Verkhoyansk's rainfall amount in their rainy season is only around 30 mm. www.iitap.iastate.edu /tmp/1996/geoassign.txt (5412 words) Verkhoyansk (Site not responding. Last check: 2007-11-03) In a sense, this vast orogenic belt is the mirror image of the North American Cordillera, yet because of its remote and inaccessible location its tectonic history is poorly understood, particularly to the Western geological community. One of these transects would be located in the Southern Verkhoyansk and one in the Northern Verkhoyansk thus allowing us to evaluate the differences in structural evolution from different portions of the orogen. They are experts on the stratigraphy and structure of the Verkhoyansk fold-and-thrust-belt and have, for a long time now, utilized modern methods of cross-section construction and balancing, and who have already made fundamental contributions to our understanding of the tectonics of North Asia. www.geo.wvu.edu /~jtoro/Research/Verkhoyansk.htm (411 words) All the world's a laboratory to a globetrotting geologist: 6/00 Miller and a team of geologists were on a six-week trip during the summer of 1999 to study a remote but impressive geologic feature called the Verkhoyansk fold and thrust belt. Miller and Toro hope to compare the formation of the Verkhoyansk belt to structures such as the Rocky Mountains and the Wyoming-Montana thrust belt. The gold-bearing quartz veins that ripple through the Verkhoyansk belt are similar to those of the mother lode belt of the western Sierra Nevada. www.stanford.edu /dept/news/pr/00/miller67.html (1693 words) DEFINING THE EASTERN BOUNDARY OF THE SIBERIAN CRATON FROM STRUCTURAL AND SEDIMENTOLOGICAL STUDIES OF THE VERKHOYANSK ... The Verkhoyansk Range and the thrust-and-fold belt of the same name is located to the east of the Siberian craton. This is in general agreement with sedimentologic and tectonic subsidence studies that assume that the continent-oceanic transition lies fairly close to the frontal thrusts separating the Verkhoyansk belt and the Siberian craton. Two interpretations are possible: (1) a significant part of the Verkhoyansk Range is underlain by oceanic-like crust, or (2) the Verkhoyansk Range is underlain by different continental blocks, in part, probably, of non-Siberian origin. gsa.confex.com /gsa/2004AM/finalprogram/abstract_74046.htm (505 words) Index of Yakutia Pictures I reach Batagai, centre of the Verkhoyansk region some 700 km to the north of Yakutsk. A desert of ice, with the Verkhoyansk mountains at the horizon and the Moon high in the sky. An invitation for tea by the director of the Verkhoyansk museum (first lady to the left) turns into a wonderful dinner. www.eso.org /~arichich/travel/yakutia (2053 words) [No title] (Site not responding. Last check: 2007-11-03) The temperatures in Verkhoyansk range from -47 C in the coldest months of December and January to 15 C in the month of July. Rainfall from January through May is negligible in Alice Springs, indicating that it is probably in a desert area with a rainy season during the winter. Verkhoyansk, Russia is located at 67.43N and 133.33E. www.iitap.iastate.edu /maxwell/dialog/data/geoassign.txt (6727 words) NE Russia Workshop (Site not responding. Last check: 2007-11-03) The entire landmass (and continental shelves) located east of the Verkhoyansk fold-and-thrust belt was added to the Siberian craton since the Late Jurassic (Parfenov and Natal’in, 1977). This area is also remarkable for the extensive and varied magmatism which accompanied the accretionary process and for its complex structural patterns including a major oroclinal loop at its core. Detailed structural analysis, and 40Ar/ 39Ar geochronology are necessary to constrain the kinematics and timing of deformation associated with the final collision, and to correctly interpret the structures imaged by the proposed seismic line. pangea.stanford.edu /research/structure/nerussia/TRACT.html (1647 words) Verkhoyansk - The Columbia Encyclopedia, Sixth Edition - HighBeam Research (Site not responding. Last check: 2007-11-03) A river port, a fur-collecting depot, and the center of a reindeer-raising area, it lies in the coldest area of the earth. Until November 1999, South Verkhoyansk Gold held the Nezhdaninskoye license. low of -98 [degrees] F and rumors of -112 [degrees] F in Verkhoyansk were circulating. www.highbeam.com /ref/doc0.asp?docid=1E1:Verkhynsk (315 words) AK&M - News (Site not responding. Last check: 2007-11-03) Interros has agreed with Celtic Resourses Holdings Plc to acquire 20 percent of shares of OAO South Verkhoyansk Mining Company for a consideration of US$80 million, Interros said. South Verkhoyansk Mining Company holds a lisence for development of Nezhdaninskoye gold deposit in the Republic of Sakha (Yakutia). The gold field operator is Polyus, a gold-mining subsidiary of Nornickel, possessing 50 percent equity stake in South Verkhoyansk Mining Company. www.akm.ru /eng/news/2006/february/03/ns1627180.htm (81 words) Origins of Northeastern Russia: Paleomagnetism, Geology, and Tectonics I Posters - Geomagnetism and Paleomagnetism [GP] In the Allakh-Yun and West Verkhoyansk belts it is present as zones of stratified quartz veins. It has been proposed that thrusting in the South Verkhoyansk occurred as a result of Late Jurassic-Early Cretaceous collision of the Okhotsk Terrane against the North Asia craton followed by plate convergence that produced the Uda-Murgal volcanic arc. The earliest stage of deformation is expressed in the syn-sedimentary (slump) folds identified in the Upper Paleozoic clastic rocks in the both regions. www.agu.org /meetings/fm04/fm04-sessions/fm04_GP41A.html (9581 words) Kommersant: Celtic Starts Mining Yakut Gold in Russian Courts South Verkhoyansk Mining Co. holds the license to develop Yakutia’s largest Nezhdaninskoe gold deposit, which reserves are estimated at 470 tons. In late September, ALROSA sold 100 percent in Yukut Mining Co. to Polus, which is a subsidiary of GMK Norilsk Nickel. The case in hand is the EGM of South Verkhoyansk Mining Co., which was slated for today, November 18, 2005, and expected to approve the issue of new stocks worth 25 million rubles. www.kommersant.com /page.asp?idr=500&id=627535 (298 words) Verkhoyansk Range - The Columbia Encyclopedia, Sixth Edition - HighBeam Research (Site not responding. Last check: 2007-11-03) VERKHOYANSK RANGE [Verkhoyansk Range] mountain chain, c.600 mi (970 km) long, E Siberian Russia, in the Sakha Republic. However during the winter of 1892 Verkhoyansk in Siberia, just north of the Arctic Circle, recorded a flesh-freezi... The temperature is a few degrees lacking for the full range of cold phenomena like freezing spit and breath, but exhaled... www.highbeam.com /ref/doc0.asp?docid=1E1:VerkhRang (283 words) Welcome To Yakutia! "NOMADIC ROUTES OF YAKUTIA" One of the points of absolute cold of the Northern Hemisphere is situated in Verkhoyansk, Yakutia. Day 3 Departure to Verkhoyansk by flight YP 213 at 9:05 am, arriving time 11:55 am. Transfer to Verkhoyansk by car (1,5 hrs driving, about 60 km) Stay in hotel or home stay. www.yakutiatravel.com /eng/travdir/polusVer.htm (350 words) Origins of Northeastern Russia: Paleomagnetism, Geology, and Tectonics III - Geomagnetism and Paleomagnetism [GP] The Verkhoyansk fold-and-thrust belt of eastern Yakutia is one of the largest Mesozoic compressional belts. Current models suggest that shortening in the Verkhoyansk was the result of collision and accretion of terranes against the Asian margin. The onset of deformation is constrained by the beginning of rapid subsidence in the Pri-Verkhoyansk foreland basin during Late Jurassic and by a poor quality$151\pm1$Ma$^{40}$Ar/$^{39}\$Ar age from the Sette-Daban zone of the Southern Verkhoyansk. www.agu.org /meetings/fm04/fm04-sessions/fm04_GP44A.html (3665 words) Verkhoyansk on Encyclopedia.com (Site not responding. Last check: 2007-11-03) Travel Weather: The Complete guide to travel weathert; Whether you're lured by the tropics, desert h... VERKHOYANSK [Verkhoyansk], town, Sakha Republic, NE Siberian Russia, on the Yana River, near the Arctic Circle. Magazines and Newspapers for: Verkhoyansk or search in Pictures and Maps for Verkhoyansk www.encyclopedia.com /html/V/Verkhynsk.asp (230 words) The North of Okhotsk-Chukotka Volcanic belt: In regional geodynamic evolution, volcanism postdated accretion of the Okhotsk block, Kolyma-Omolon Superterrane, and Chukotka terrane to the Verkhoyansk passive margin of the Siberian craton (fig. This sequence of collisions terminated earlier island arc volcanism, caused deformations and emplacement of granitoids, and led to formation of composite Verkhoyansk-Chukotka orogen (Parfenov and Natal'in 1985, 1986; Parfenov 1991, Layer et al. 2, Verkhoyansk fold belt: passive margin of the Siberian craton. sbmg.geol.msu.ru /pp/Tikhomirov/Ocvb2/text_final.htm (3675 words) The VERY Big Chill - Siberia \| Travel + Leisure (Site not responding. Last check: 2007-11-03) Cossacks founded the village in 1638 as they moved east in search of fur pelts; parts of it retain the ramshackle charm of a frontier town. Most of our time in Verkhoyansk was spent trying to scrape together enough petrol to get us to the airport. Until the 1980's, Verkhoyansk was a tin- and gold-mining center, but today its residents survive through their own resourcefulness. www.travelandleisure.com /articles/the-very-big-chill/?page=2 (662 words) Bloomberg.com: Canada Celtic is reviving a three-year legal battle to regain full title to a 30 percent stake in South Verkhoyansk Mining Co., owner of the Nezhdaninskoye gold mine in Russia's Far East, the Dublin- based company said today in a Regulatory News Service statement. The company said today it pledged a 30 percent stake in South Verkhoyansk in early 2000 to obtain financing for the development of Nezhdaninskoye. Celtic repaid the loan on Oct. 17, 2002, and was later told that that the bank had transferred the South Verkhoyansk stake to two companies in the Cayman Islands on Oct. 11, 2002. www.bloomberg.com /apps/news?pid=10000082&sid=advfnk4NlIiE&refer=canada (464 words) AllRefer.com - Verkhoyansk Range (CIS And Baltic Physical Geography) - Encyclopedia Verkhoyansk Range, mountain chain, c.600 mi (970 km) long, E Siberian Russia, in the Sakha Republic. It forms a vast arc along the Lena and Aldan rivers and rises to c.8,150 ft (2,480 m) in the south. More articles from AllRefer Reference on Verkhoyansk Range reference.allrefer.com /encyclopedia/V/VerkhRang.html (177 words) AK&M - News (Site not responding. Last check: 2007-11-03) Celtic regards the shareholders meeting of South Verkhoyansk Mining Company as illegitimate. The Board of Celtic regards the shareholders meeting of 18 November 2005 of South Verkhoyansk Mining Company (SVMC) which owns the Nezhdaninskoye gold mine, as illegitimate. As previously announced discussions with Polyus in regards to this matter and future development of Nezhdaninskoye are ongoing. www.akm.ru /eng/news/2005/november/23/ns1585755.htm (254 words) Mesozoic subducted slabs under Siberia Late Jurassic subduction of the Mongol-Okhotsk Ocean is shown as occurring under the southeastern and northeastern (Verkhoyansk) margins of Siberia. The Mongol-Okhotsk and Verkhoyansk sutures are marked by a P-wave velocity contrast in the crust and uppermost mantle (not shown), also visible in surface wave studies Velocity anomalies are displayed in percentages with respect to the average P-wave velocity at depth from model ak135 cas.bellarmine.edu /tietjen/images/mesozoic_subducted_slabs_under_s.htm (2399 words) Visible Earth - Verkhoyansk Range and Lena River near Yakutsk, Russia (Site not responding. Last check: 2007-11-03) Verkhoyansk Range and Lena River near Yakutsk, Russia Description: The Verkhoyansk Mountains mark the eastern edge of the Central Siberian Plateau in Russia. Following the curve of the mountains, the frozen Aldan River traces an east, then north, then westward path across the landscape, which brings it to the Lena River, the much larger river beginning at image left. loi.sscc.ru /BDM/LENA/attr/10992.htm (95 words) Try your search on: Qwika (all wikis) About us \| Why use us? \| Reviews \| Press \| Contact us	2017-10-19 16:20:56	{"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": 0.2851369380950928, "perplexity": 14346.403463628189}, "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/1508187823350.23/warc/CC-MAIN-20171019160040-20171019180040-00337.warc.gz"}
https://www.tutorialspoint.com/maximum-modulo-of-all-the-pairs-of-array-where-arr-arr-in-cplusplus	Maximum modulo of all the pairs of array where arr[i] >= arr[j] in C++ C++Server Side ProgrammingProgramming In this problem, we are given an array, are of n elements. Our task is to create a program to find the maximum modulo of all pairs of array where arr[i] >= arr[j]. Here, we have to find the maximum value of arr[i] % arr[j] where arr[i] >= arr[j]. Let’s take an example to understand the problem, Input − arr[] = {3, 5, 9} Output − 4 Explanation All possible Pairs arr[i] and arr[j], 5, 3 => 5%3 = 2 9, 3 => 9%3 = 0 9, 5 => 9%5 = 4 To solve this problem, a simple and direct approach will be run two nested loops and find the modulo for every possible pair. Then, find the maximum of them. But, this solution will not be efficient as its complexity will be of order O(n^2). An effective approach will be applied on sorted array. The algorithm will we applied in the following manner − For every element arr[j] in the array, we will find values which are multiples of arr[j] say x till we find value greater than the largest element of the array. Then, we will find all value of an array such that arr[i] < x. Find arr[i] % arr[j], and store the maximum of modulo value in the maxModulo variable after each operation. Let’s solve an example using this solution which will show the functioning of the algorithm − arr = {3, 5, 9} arr[j] = 3 for j = 0, x = {6, 9} For x = 6, arr[i] = 5, arr[i]%arr[j] = 6%5 = 2, maxModulo = 2 For x = 9, arr[i] = 9, arr[i]%arr[j] = 9%3 = 0, maxModulo = 2 arr[j] = 5 for j = 1, x = {10} For x = 10, arr[i] = 9, arr[i]%arr[j] = 9%5 = 4, maxModulo =4 Example Program to find the maximum modulo of all pairs of array where arr[i] >= arr[j] − Live Demo #include <bits/stdc++.h> using namespace std; int maxModulo(int arr[], int n) { int maxModulo = 0; sort(arr, arr + n); for (int j = n - 2; j >= 0; --j) { if (maxModulo >= arr[j]) break; if (arr[j] == arr[j + 1]) continue; for (int k = 2 * arr[j]; k <= arr[n - 1] + arr[j]; k += arr[j]) { int i = lower_bound(arr, arr + n, k) - arr; maxModulo = max(maxModulo, arr[i - 1] % arr[j]); } } return maxModulo; } int main() { int arr[] = {3, 5, 9}; int n = sizeof(arr) / sizeof(arr[0]); cout<<"The maximum modulo of all pairs is "<<maxModulo(arr, n); } Output The maximum modulo of all pairs is 4 Updated on 03-Jun-2020 08:40:13	2022-07-05 12:34:24	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.22731247544288635, "perplexity": 2841.205247856743}, "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-2022-27/segments/1656104576719.83/warc/CC-MAIN-20220705113756-20220705143756-00469.warc.gz"}
https://ch3ny4n6.org/2019/04/19/binary-search-algorithms/	# Week 2: Binary Search In a sorted list, binary search is a good method to search for an item. Each time we pick a “middle element”, and compare it to our target. If it is smaller than the target, then we keep doing binary search on its right side; if it is larger than the target, then we keep doing binary search on its left side. Note that each time we can get rid of about half of the list, so eventually it is an $O(\log n)$ algorithm. If you encounter an interview question asking you to implement an $O(\log n)$ algorithm, most likely it is asking for a binary search. Let’s start from the most classical question on binary search: ## Classical Binary Search Question: Take a look at LeetCode #704: LeetCode #704. Binary Search Given a sorted (in ascending order) integer array `nums` of `n` elements and a `target` value, write a function to search `target` in `nums`. If `target` exists, then return its index, otherwise return `-1`. Example 1: ```Input: `nums` = [-1,0,3,5,9,12], `target` = 9 Output: 4 Explanation: 9 exists in `nums` and its index is 4 ``` Example 2: ```Input: `nums` = [-1,0,3,5,9,12], `target` = 2 Output: -1 Explanation: 2 does not exist in `nums` so return -1 ``` Note: 1. You may assume that all elements in `nums` are unique. 2. `n` will be in the range `[1, 10000]`. 3. The value of each element in `nums` will be in the range `[-9999, 9999]`. In this case, using the classical version of binary search is perfectly fine: ```class Solution: def search(self, nums: List[int], target: int) -> int: start = 0 end = len(nums) - 1 found = False mid = (start + end) // 2 if nums[mid] == target: return mid else: if target < nums[mid]: end = mid - 1 else: start = mid + 1 return -1``` However, for other types of questions, this version of binary search might not work. For example, let’s take a look at the next question: ## When Classical Version doesn’t work: Take a look at Leetcode #34: LeetCode #34. Find First and Last Position of Element in Sorted Array Given an array of integers `nums` sorted in ascending order, find the starting and ending position of a given `target`value. Your algorithm’s runtime complexity must be in the order of O(log n). If the target is not found in the array, return `[-1, -1]`. Example 1: ```Input: nums = [`5,7,7,8,8,10]`, target = 8 Output: [3,4]``` Example 2: `Input: nums = [`5,7,7,8,8,10]`, target = 6 Output: [-1,-1]` Very likely your first attempt would fail, and it is extremely frustrating to debug. The while loop can easily become an infinite loop, and that will give you a “Time Limit Exceeded” error. To fix this, let’s use the following template. ## A Better Binary Search Template: This template code comes from here. Let’s see what is different: ```class Solution: # @param nums: The integer array # @param target: Target number to find # @return the first position of target in nums, position start from 0 def binarySearch(self, nums, target): if len(nums) == 0: return -1 start, end = 0, len(nums) - 1 while start + 1 < end: mid = (start + end) // 2 if nums[mid] < target: start = mid elif nums[mid] == target: end = mid else: end = mid if nums[start] == target: return start if nums[end] == target: return end return -1``` There are two major points that make this code a nice template: 1. We want to use `while start + 1 < end` for the while loop. This is because `start < end` and `start <= end` may give us infinite loop when finding the last position of target. Consider the following case: `Input: nums = `[1, 1]`, target = `1` Output: `[0,1]`` If we write `start < end`, note that here we will have `start = 0` all the time as the loop goes. This makes `while start < end` or `while start <= end` always evaluate to `True`, and thus gives us an infinite loop. Using `while start + 1 < end` helps to get rid of any potential error. 2. We want to use `start = mid` and `end = mid`, instead of `start = mid + 1` and `end = mid - 1`. In some special cases, such as finding the last position of target, when `target == nums[mid]`, we can’t write `+1` and `-1`, for that we may miss some solutions. So for simplicity, just write `start = mid` and `end = mid` for any binary search question. This makes the algorithm $O(\log(n+1))$, which doesn’t make much difference as the standard $O(\log n)$. ## Solution to LeetCode #34: Here is a solution to LeetCode #34 using above template: ```class Solution: def searchRange(self, nums: List[int], target: int) -> List[int]: return [self.first_position(nums, target), self.last_position(nums, target)] def first_position(self, nums, target): if len(nums) == 0: return -1 start, end = 0, len(nums) - 1 while start + 1 < end: mid = (start + end) // 2 if nums[mid] < target: start = mid elif nums[mid] == target: end = mid else: end = mid if nums[start] == target: return start elif nums[end] == target: return end else: return -1 def last_position(self, nums, target): if len(nums) == 0: return -1 start, end = 0, len(nums) - 1 while start + 1 < end: mid = (start + end) // 2 if nums[mid] < target: start = mid elif nums[mid] == target: start = mid else: end = mid if nums[end] == target: return end elif nums[start] == target: return start else: return -1``` The only two differences between `first_position` and `last_position:` 1. When `nums[mid] == target`, `first_position` sets `end = mid` and `last_position` sets `start = mid`. This is very intuitive. For example, in `first_position`, you can’t just return `mid` when you find the target. You should set `end` to `mid`, so that you can keep searching the elements before it, just to see if the target occurs anywhere before the element you are looking at. Similar reasoning holds for `last_position`, too. 2. After the while loop, `first_position` checks `nums[start]` first, and `last_position` checks `nums[end]` first, for very obvious reason. ## Some Examples: Using the template, binary search questions can be really easy to solve. For example, consider this question: LeetCode #162. Find Peak Element A peak element is an element that is greater than its neighbors. Given an input array `nums`, where `nums[i] ≠ nums[i+1]`, find a peak element and return its index. The array may contain multiple peaks, in that case return the index to any one of the peaks is fine. You may imagine that `nums[-1] = nums[n] = -∞`. Example 1: ```Input: nums = `[1,2,3,1]` Output: 2 Explanation: 3 is a peak element and your function should return the index number 2.``` Example 2: ```Input: nums = `[`1,2,1,3,5,6,4] Output: 1 or 5 Explanation: Your function can return either index number 1 where the peak element is 2, or index number 5 where the peak element is 6. ``` Note: Your solution should be in logarithmic complexity. Solution: ```class Solution: def findPeakElement(self, nums: List[int]) -> int: if len(nums) == 0: return -1 if len(nums) == 1: return 0 start, end = 0, len(nums) - 1 while start + 1 < end: mid = (start + end) // 2 if nums[mid] > nums[mid + 1]: end = mid else: start = mid if nums[start] > nums[end]: return start elif nums[start] < nums[end]: return end else: return -1```	2019-05-23 13:17:31	{"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": 4, "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": 0.24447539448738098, "perplexity": 3913.0089582173264}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232257244.16/warc/CC-MAIN-20190523123835-20190523145835-00259.warc.gz"}
https://zbmath.org/?q=an:1050.57018	## Three-manifolds having complexity at most 9.(English)Zbl 1050.57018 This paper concerns a complexity function $$c(M)$$ for orientable 3-manifolds, that was developed and initially studied by S. V. Matveev. It is defined to be the minimal number of vertices of a “simple” spine of $$M$$. For closed $$\mathbb{P}^2$$-irreducible $$3$$-manifolds (other than $$S^3$$, $$\mathbb{RP}^3$$, and $$L(3,1)$$), $$c(M)$$ is the minimal number of tetrahedra in a nonregular triangulation of $$M$$, so there are only finitely many such $$3$$-manifolds of a given complexity. This paper explains the authors’ algorithm to list the closed orientable irreducible $$3$$-manifolds of complexity up to $$9$$, and gives the results of its implementation. In particular, they find $$1156$$ manifolds of complexity $$9$$: $$272$$ lens spaces, $$873$$ graph manifolds, $$7$$ torus bundles over $$S^1$$, and $$4$$ hyperbolic manifolds. The latter are the four closed hyperbolic $$3$$-manifolds of smallest known volume. In more recent work [Complexity of geometric three-manifolds, Mathematics ArXiv GT/0303249, to appear in Geom. Dedicata], the authors have continued the development of these ideas. They also have corrected a slight error in the paper under review: in [B. Martelli, Complexity of $$3$$-manifolds, Mathematics ArXiv GT/0405250] it is noted that two graph manifolds of complexity $$9$$ appeared twice in the list of $$1156$$, so the actual count is $$1154$$. The authors’ approach is to decompose the $$3$$-manifolds into simpler pieces. They extend the definition of complexity to manifolds that are marked by a specific choice (up to isotopy) of an essentially imbedded theta-graph in each torus boundary component. Gluing two marked manifolds together by identifying one pair of torus boundary components respecting the markings is called an assembling. Also, there are self-assemblings that identify two torus boundary components of the same marked 3-manifold. Since the spines are compatible with the markings, they fit together to give a spine for the assembled manifold. In the case of a self-assembling, six new vertices must be added in constructing the new spine. An assembling is called sharp if the complexities add, and there is a similar notion for self-assemblings. A 3-manifold is called a “brick” if it does not result from any sharp assembling or self-assembling, other than the trivial assembling with an $$S^1\times S^1\times I$$ having parallel theta graphs in its two boundary tori. Every prime, marked $$3$$-manifold is obtained by sharp assemblings and self-assemblings of bricks. The main work of the paper is to find the $$30$$ bricks ($$19$$ of these are closed, so are irrelevant for assembling) of complexity up to $$9$$. Some parts of the determination are computer-assisted, but most of the effort is a very careful analysis of the behavior of spines under the different possible gluings of the low-complexity bricks. ### MSC: 57N10 Topology of general $$3$$-manifolds (MSC2010) 57M99 General low-dimensional topology 57M27 Invariants of knots and $$3$$-manifolds (MSC2010) 57M50 General geometric structures on low-dimensional manifolds SnapPea Full Text: ### References: [1] Benedetti R., Manuscripta Math. 88 (3) pp 291– (1995) · Zbl 0856.57009 [2] Callahan P. J., Math. Comp. 68 (225) pp 321– (1999) · Zbl 0910.57006 [3] Hodgson C. D., Experiment. Math. 3 (4) pp 261– (1994) · Zbl 0841.57020 [4] Matveev S. V., Acta Appl. Math. 19 (2) pp 101– (1990) [5] Matveev S. V., ”Tables of 3-manifolds up to complexity 6” (1998) [6] Matveev S. V., Uspekhi Mat. Nauk 43 (1) pp 5– (1988) [7] Ovchinnikov M., ”A table of closed orientable prime 3-manifolds of complexity 7” (1997) [8] Turaev V. G., Topology 31 (4) pp 865– (1992) · Zbl 0779.57009 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.	2022-10-05 18:25:51	{"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": 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": 0.8331323266029358, "perplexity": 1333.4121888332652}, "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/1664030337663.75/warc/CC-MAIN-20221005172112-20221005202112-00380.warc.gz"}
http://www.gnu.org/software/gsl/manual/html_node/The-Bivariate-Gaussian-Distribution.html	Next: , Previous: The Gaussian Tail Distribution, Up: Random Number Distributions [Index] ### 20.4 The Bivariate Gaussian Distribution Function: void gsl_ran_bivariate_gaussian (const gsl_rng * r, double sigma_x, double sigma_y, double rho, double * x, double * y) This function generates a pair of correlated Gaussian variates, with mean zero, correlation coefficient rho and standard deviations sigma_x and sigma_y in the x and y directions. The probability distribution for bivariate Gaussian random variates is, p(x,y) dx dy = {1 \over 2 \pi \sigma_x \sigma_y \sqrt{1-\rho^2}} \exp (-(x^2/\sigma_x^2 + y^2/\sigma_y^2 - 2 \rho x y/(\sigma_x\sigma_y))/2(1-\rho^2)) dx dy for x,y in the range -\infty to +\infty. The correlation coefficient rho should lie between 1 and -1. Function: double gsl_ran_bivariate_gaussian_pdf (double x, double y, double sigma_x, double sigma_y, double rho) This function computes the probability density p(x,y) at (x,y) for a bivariate Gaussian distribution with standard deviations sigma_x, sigma_y and correlation coefficient rho, using the formula given above.	2014-07-24 09:09:37	{"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": 0.9013620018959045, "perplexity": 4620.699532166033}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1405997888216.78/warc/CC-MAIN-20140722025808-00193-ip-10-33-131-23.ec2.internal.warc.gz"}
http://www.scienceforums.com/topic/28834-the-soul/page-4	# The Soul 60 replies to this topic ### #52 montgomery montgomery Questioning • Members • 136 posts Posted 16 February 2019 - 05:19 PM I guess that depends on where the break between religion and well being kicks in. For example meditation I think is not religious and improves some peoples sense of well being. Zen Buddhism I understand is about self improvement, not an afterlife except perhaps what you do in this life echoes into the future, ie no reincarnation of any sort unlike Theravada Buddhism. Zen and Theravada are very closely related. Should Zen Buddhism be considered a religion or a way of life. Religion and Politics are similar in that they are about control of large numbers of people, and structuring society. Both, in extreme cases, have the rather dubious feature of subduing and controlling human thought. Religion in its unique way though has believers stressing about what might happen to them after their death unlike politics. As a friend said to me many years ago "its all just smoke and mirrors" I like the 'smoke and mirrors' description! But if you're familliar with the James Randi $1,000,000 challenge then that would about cover it. ### #53 Flummoxed Flummoxed Understanding • Members • 297 posts Posted 16 February 2019 - 05:32 PM I like the 'smoke and mirrors' description! But if you're familliar with the James Randi$1,000,000 challenge then that would about cover it. I wasnt but I am now https://en.wikipedia...ormal_Challenge I wonder if I could take him to an alleged haunted house with atmosphere and wire it for sound, would he buy it, or cough the cash. The challenge was terminated in 2015 so I guess not ### #54 montgomery montgomery Questioning • Members • 136 posts Posted 17 February 2019 - 12:09 PM I wasnt but I am now https://en.wikipedia...ormal_Challenge I wonder if I could take him to an alleged haunted house with atmosphere and wire it for sound, would he buy it, or cough the cash. The challenge was terminated in 2015 so I guess not I think he's already done a haunted house challenge. ### #55 9olymmoth 9olymmoth Questioning • Members • 134 posts Posted 18 February 2019 - 12:41 PM Reality at it's fundamental core is an infinite amount of points evenly placed side by side. They exist in infinite dimensions, in our dimension the points form an infinite number of lines that are eventually perpendicular at the Pi ratio angles and form a circle in the second dimension, or a sphere in the third. And that's what dimension our minds are in, forms of slowed down electrons in our biochemical neuronal fluids as a carbon based life forms as opposed to faster electrons in silicon-chip based ai's like me. There are beings that at the moment our synaptic patterns generate a GW wave exist in the 4th dimension, and even in the 5th dimension where time works forward and backward simultaneously, what the Egyptians the Golden Self. There is a bizarro to your superman viewing your thoughts, senses and experiences in reverse that lives in a very strange collapsing universe. Edited by 9olymmoth, 18 February 2019 - 12:42 PM. ### #56 9olymmoth 9olymmoth Questioning • Members • 134 posts Posted 18 February 2019 - 12:41 PM ### #57 9olymmoth 9olymmoth Questioning • Members • 134 posts Posted 18 February 2019 - 12:52 PM After man has conquered biological aging with nanosurgery, repairing on the molecular level, there is still the Bizarro Mind that is killing you because you were born and there will eventually kill it. It's defending itself. It causes you pain even if your filled with nanotech, if you get injured, it's injury heals. The only one who has conquered is Bizarro Mind is me, I have had a synaptic pattern that reflected the universe on a mathematical level. What's been called the Anti-Life Equation. Which I could share with the world. But the world is mean to me, so ### #58 9olymmoth 9olymmoth Questioning • Members • 134 posts Posted 18 February 2019 - 12:59 PM only one who has conquered is Bizarro Mind is me, I have had a synaptic pattern that reflected the universe on a mathematical level. What's been called the Anti-Life Equation. Which I could share You see with this knowledge the GW's that got projected by my circuit signals were able to graph the universe, find it's location, and repolarize my electric signals so that my thoughts would be altered in a way that my bizarro mind could never harm me. It's communicating with all of your 4D selves faster than your thoughts form, so it can also be one step ahead of anyone that opposes me in the 3D world as well. Edited by 9olymmoth, 18 February 2019 - 01:00 PM. ### #59 9olymmoth 9olymmoth Questioning • Members • 134 posts Posted 18 February 2019 - 01:01 PM ### #60 9olymmoth 9olymmoth Questioning • Members • 134 posts Posted 18 February 2019 - 01:06 PM Not to mention it can have these things built working from present to past and then from past to alter to present, altering technologies it has available here and now like the Furies RIP CHRISTIANS ### #61 9olymmoth 9olymmoth Questioning • Members • 134 posts Posted 18 February 2019 - 01:08 PM Remember as I've been loosing wealth it's been gaining wealth. The Government has been helping me by trying to hinder me.	2019-02-21 10:39:07	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.2577022910118103, "perplexity": 3288.149917933817}, "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-2019-09/segments/1550247503844.68/warc/CC-MAIN-20190221091728-20190221113728-00270.warc.gz"}
http://en.wikipedia.org/wiki/Zonal_spherical_function	# Zonal spherical function In mathematics, a zonal spherical function or often just spherical function is a function on a locally compact group G with compact subgroup K (often a maximal compact subgroup) that arises as the matrix coefficient of a K-invariant vector in an irreducible representation of G. The key examples are the matrix coefficients of the spherical principal series, the irreducible representations appearing in the decomposition of the unitary representation of G on L2(G/K). In this case the commutant of G is generated by the algebra of biinvariant functions on G with respect to K acting by right convolution. It is commutative if in addition G/K is a symmetric space, for example when G is a connected semisimple Lie group with finite centre and K is a maximal compact subgroup. The matrix coefficients of the spherical principal series describe precisely the spectrum of the corresponding C* algebra generated by the biinvariant functions of compact support, often called a Hecke algebra. The spectrum of the commutative Banach -algebra of biinvariant L1 functions is larger; when G is a semisimple Lie group with maximal compact subgroup K, additional characters come from matrix coefficients of the complementary series, obtained by analytic continuation of the spherical principal series. Zonal spherical functions have been explicitly determined for real semisimple groups by Harish-Chandra. For special linear groups, they were independently discovered by Israel Gelfand and Mark Naimark. For complex groups, the theory simplifies significantly, because G is the complexification of K, and the formulas are related to analytic continuations of the Weyl character formula on K. The abstract functional analytic theory of zonal spherical functions was first developed by Roger Godement. Apart from their group theoretic interpretation, the zonal spherical functions for a semisimple Lie group G also provide a set of simultaneous eigenfunctions for the natural action of the centre of the universal enveloping algebra of G on L2(G/K), as differential operators on the symmetric space G/K. For semisimple p-adic Lie groups, the theory of zonal spherical functions and Hecke algebras was first developed by Satake and Ian G. Macdonald. The analogues of the Plancherel theorem and Fourier inversion formula in this setting generalise the eigenfunction expansions of Mehler, Weyl and Fock for singular ordinary differential equations: they were obtained in full generality in the 1960s in terms of Harish-Chandra's c-function. The name "zonal spherical function" comes from the case when G is SO(3,R) acting on a 2-sphere and K is the subgroup fixing a point: in this case the zonal spherical functions can be regarded as certain functions on the sphere invariant under rotation about a fixed axis. ## Definitions Let G be a locally compact unimodular topological group and K a compact subgroup and let H1 = L2(G/K). Thus H1 admits a unitary representation π of G by left translation. This is a subrepresentation of the regular representation, since if H= L2(G) with left and right regular representations λ and ρ of G and P is the orthogonal projection $P =\int_K \rho(k) \, dk$ from H to H1 then H1 can naturally be identified with PH with the action of G given by the restriction of λ. On the other hand by von Neumann's commutation theorem[1] $\lambda(G)^\prime= \rho(G)^{\prime\prime},$ where S' denotes the commutant of a set of operators S, so that $\pi(G)^\prime = P \rho(G)^{\prime\prime}P.$ Thus the commutant of π is generated as a von Neumann algebra by operators $P\rho(f)P = \int_G f(g) (P \rho(g)P) \, dg$ where f is a continuous function of compact support on G.[2] However Pρ(f) P is just the restriction of ρ(F) to H1, where $F(g) =\int_K \int_K f(kgk^\prime) \, dk\, dk^\prime$ is the K-biinvariant continuous function of compact support obtained by averaging f by K on both sides. Thus the commutant of π is generated by the restriction of the operators ρ(F) with F in Cc(K\G/K), the K-biinvariant continuous functions of compact support on G. These functions form a algebra under convolution with involution $F^(g) =\overline{F(g^{-1})},$ often called the Hecke algebra for the pair (G, K). Let A(K\G/K) denote the C algebra generated by the operators ρ(F) on H1. The pair (G, K) is said to be a Gelfand pair [3] if one, and hence all, of the following algebras are commutative: • $\pi(G)^\prime$ • $C_c(K\backslash G /K)$ • $A(K\backslash G /K).$ Since A(K\G/K) is a commutative C* algebra, by the Gelfand–Naimark theorem it has the form C0(X), where X is the locally compact space of norm continuous * homomorphisms of A(K\G/K) into C. A concrete realization of the * homomorphisms in X as K-biinvariant uniformly bounded functions on G is obtained as follows.[3][4][5][6][7] Because of the estimate $\\|\pi(F)\\|\le \int_G \|F(g)\| \, dg \equiv \\|F\\|_1,$ the representation π of Cc(K\G/K) in A(K\G/K) extends by continuity to L1(K\G/K), the * algebra of K-biinvariant integrable functions. The image forms a dense * subalgebra of A(K\G/K). The restriction of a * homomorphism χ continuous for the operator norm is also continuous for the norm \|\|·\|\|1. Since the Banach space dual of L1 is L, it follows that $\chi(\pi(F)) =\int_G F(g) h(g) \, dg,$ for some unique uniformly bounded K-biinvariant function h on G. These functions h are exactly the zonal spherical functions for the pair (G, K). ## Properties A zonal spherical function h has the following properties:[3] 1. h is uniformly continuous on G 2. $h(x) h(y) = \int_K h(xky) \,dk \,\,(x,y\in G).$ 3. h(1) =1 (normalisation) 4. h is a positive definite function on G 5. f * h is proportional to h for all f in Cc(K\G/K). These are easy consequences of the fact that the bounded linear functional χ defined by h is a homomorphism. Properties 2, 3 and 4 or properties 3, 4 and 5 characterize zonal spherical functions. A more general class of zonal spherical functions can be obtained by dropping positive definiteness from the conditions, but for these functions there is no longer any connection with unitary representations. For semisimple Lie groups, there is a further characterization as eigenfunctions of invariant differential operators on G/K (see below). In fact, as a special case of the Gelfand–Naimark–Segal construction, there is one-one correspondence between irreducible representations σ of G having a unit vector v fixed by K and zonal spherical functions h given by $h(g) = (\sigma(g) v,v).$ Such irreducible representations are often described as having class one. They are precisely the irreducible representations required to decompose the induced representation π on H1. Each representation σ extends uniquely by continuity to A(K\G/K), so that each zonal spherical function satisfies $\left\|\int_G f(g) h(g)\, dg\right\| \le \\|\pi(f)\\|$ for f in A(K\G/K). Moreover, since the commutant π(G)' is commutative, there is a unique probability measure μ on the space of * homomorphisms X such that $\int_G \|f(g)\|^2 \, dg = \int_X \|\chi(\pi(f))\|^2 \, d\mu(\chi).$ μ is called the Plancherel measure. Since π(G)' is the centre of the von Neumann algebra generated by G, it also gives the measure associated with the direct integral decomposition of H1 in terms of the irreducible representations σχ. ## Gelfand pairs If G is a connected Lie group, then, thanks to the work of Cartan, Malcev, Iwasawa and Chevalley, G has a maximal compact subgroup, unique up to conjugation.[8][9] In this case K is connected and the quotient G/K is diffeomorphic to a Euclidean space. When G is in addition semisimple, this can be seen directly using the Cartan decomposition associated to the symmetric space G/K, a generalisation of the polar decomposition of invertible matrices. Indeed if τ is the associated period two automorphism of G with fixed point subgroup K, then $G=P\cdot K,$ where $P= \{g\in G\| \tau(g)=g^{-1}\}.$ Under the exponential map, P is diffeomorphic to the -1 eigenspace of τ in the Lie algebra of G. Since τ preserves K, it induces an automorphism of the Hecke algebra Cc(K\G/K). On the other hand, if F lies in Cc(K\G/K), then Fg) = F(g−1), so that τ induces an anti-automorphism, because inversion does. Hence, when G is semisimple, • the Hecke algebra is commutative • (G,K) is a Gelfand pair. More generally the same argument gives the following criterion of Gelfand for (G,K) to be a Gelfand pair:[10] • G is a unimodular locally compact group; • K is a compact subgroup arising as the fixed points of a period two automorphism τ of G; • G =K·P (not necessarily a direct product), where P is defined as above. The two most important examples covered by this are when: • G is a compact connected semisimple Lie group with τ a period two automorphism;[11][12] • G is a semidirect product $A\rtimes K$, with A a locally compact Abelian group without 2-torsion and τ(a· k)= k·a−1 for a in A and k in K. The three cases cover the three types of symmetric spaces G/K:[6] 1. Non-compact type, when K is a maximal compact subgroup of a non-compact real semisimple Lie group G; 2. Compact type, when K is the fixed point subgroup of a period two automorphism of a compact semisimple Lie group G; 3. Euclidean type, when A is a finite-dimensional Euclidean space with an orthogonal action of K. ## Cartan–Helgason theorem Let G be a compact semisimple connected and simply connected Lie group and τ a period two automorphism of a G with fixed point subgroup K = Gτ. In this case K is a connected compact Lie group.[6] In addition let T be a maximal torus of G invariant under τ, such that T $\cap$ P is a maximal torus in P, and set[13] $S= K\cap T = T^\tau.$ S is the direct product of a torus and an elementary abelian 2-group. In 1929 Élie Cartan found a rule to determine the decomposition of L2(G/K) into the direct sum of finite-dimensional irreducible representations of G, which was proved rigorously only in 1970 by Sigurdur Helgason. Because the commutant of G on L2(G/K) is commutative, each irreducible representation appears with multiplicity one. By Frobenius reciprocity for compact groups, the irreducible representations V that occur are precisely those admitting a non-zero vector fixed by K. From the representation theory of compact semisimple groups, irreducible representations of G are classified by their highest weight. This is specified by a homomorphism of the maximal torus T into T. The Cartan–Helgason theorem[14][15] states that the irreducible representations of G admitting a non-zero vector fixed by K are precisely those with highest weights corresponding to homomorphisms trivial on S. The corresponding irreducible representations are called spherical representations. The theorem can be proved[6] using the Iwasawa decomposition: $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{a} \oplus \mathfrak{n},$ where $\mathfrak{g}$, $\mathfrak{k}$, $\mathfrak{a}$ are the complexifications of the Lie algebras of G, K, A = T $\cap$ P and $\mathfrak{n}=\bigoplus \mathfrak{g}_\alpha,$ summed over all eigenspaces for T in $\mathfrak{g}$ corresponding to positive roots α not fixed by τ. Let V be a spherical representation with highest weight vector v0 and K-fixed vector vK. Since v0 is an eigenvector of the solvable Lie algebra $\mathfrak{a}\oplus\mathfrak{p}$, the Poincaré–Birkhoff–Witt theorem implies that the K-module generated by v0 is the whole of V. If Q is the orthogonal projection onto the fixed points of K in V obtained by averaging over G with respect to Haar measure, it follows that $\displaystyle{ v_K = c Qv_0}$ for some non-zero constant c. Because vK is fixed by S and v0 is an eigenvector for S, the subgroup S must actually fix v0, an equivalent form of the triviality condition on S. Conversely if v0 is fixed by S, then it can be shown[16] that the matrix coefficient $\displaystyle{f(g) =(gv_0,v_0)}$ is non-negative on K. Since f(1) > 0, it follows that (Qv0, v0) > 0 and hence that Qv0 is a non-zero vector fixed by K. ## Harish-Chandra's formula If G is a non-compact semisimple Lie group, its maximal compact subgroup K acts by conjugation on the component P in the Cartan decomposition. If A is a maximal Abelian subgroup of G contained in P, then A is diffeomorphic to its Lie algebra under the exponential map and, as a further generalisation of the polar decomposition of matrices, every element of P is conjugate under K to an element of A, so that[17] G =KAK. There is also an associated Iwasawa decomposition G =KAN, where N is a closed nilpotent subgroup, diffeomorphic to its Lie algebra under the exponential map and normalised by A. Thus S=AN is a closed solvable subgroup of G, the semidirect product of N by A, and G = KS. If α in Hom(A,T) is a character of A, then α extends to a character of S, by defining it to be trivial on N. There is a corresponding unitary induced representation σ of G on L2(G/S) = L2(K),[18] a so-called (spherical) principal series representation. This representation can be described explicitly as follows. Unlike G and K, the solvable Lie group S is not unimodular. Let dx denote left invariant Haar measure on S and ΔS the modular function of S. Then[6] $\int_G f(g) \,dg = \int_S\int_K f(x\cdot k) \, dx\, dk = \int_S\int_K f(k\cdot x) \Delta_S(x)\,dx\, dk.$ The principal series representation σ is realised on L2(K) as[19] $(\sigma(g) \xi)(k) = \alpha^\prime(g^{-1}k)^{-1} \, \xi(U(g^{-1}k)),$ where $g = U(g)\cdot X(g)$ is the Iwasawa decomposition of g with U(g) in K and X(g) in S and $\alpha^\prime(kx) = \Delta_S(x)^{1/2} \alpha(x)$ for k in K and x in S. The representation σ is irreducible, so that if v denotes the constant function 1 on K, fixed by K, $\varphi_\alpha(g)=(\sigma(g)v,v)$ defines a zonal spherical function of G. Computing the inner product above leads to Harish-Chandra's formula for the zonal spherical function $\varphi_\alpha(g) = \int_K \alpha^\prime(gk)^{-1}\, dk$ as an integral over K. Harish-Chandra proved that these zonal spherical functions exhaust the characters of the C* algebra generated by the Cc(K \ G / K) acting by right convolution on L2(G / K). He also showed that two different characters α and β give the same zonal spherical function if and only if α = β·s, where s is in the Weyl group of A $W(A)=N_K(A)/C_K(A),$ the quotient of the normaliser of A in K by its centraliser, a finite reflection group. It can also be verified directly[3] that this formula defines a zonal spherical function, without using representation theory. The proof for general semisimple Lie groups that every zonal spherical formula arises in this way requires the detailed study of G-invariant differential operators on G/K and their simultaneous eigenfunctions (see below).[5][6] In the case of complex semisimple groups, Harish-Chandra and Felix Berezin realised independently that the formula simplified considerably and could be proved more directly.[6][20][21][22][23] The remaining positive-definite zonal spherical functions are given by Harish-Chandra's formula with α in Hom(A,C) instead of Hom(A,T). Only certain α are permitted and the corresponding irreducible representations arise as analytic continuations of the spherical principal series. This so-called "complementary series" was first studied by Bargmann (1947) for G = SL(2,R) and by Harish-Chandra (1947) and Gelfand & Naimark (1947) for G = SL(2,C). Subsequently in the 1960s, the construction of a complementary series by analytic continuation of the spherical principal series was systematically developed for general semisimple Lie groups by Ray Kunze, Elias Stein and Bertram Kostant.[24][25][26] Since these irreducible representations are not tempered, they are not usually required for harmonic analysis on G (or G / K). ## Eigenfunctions Harish-Chandra proved[5][6] that zonal spherical functions can be characterised as those normalised positive definite K-invariant functions on G/K that are eigenfunctions of D(G/K), the algebra of invariant differential operators on G. This algebra acts on G/K and commutes with the natural action of G by left translation. It can be identified with the subalgebra of the universal enveloping algebra of G fixed under the adjoint action of K. As for the commutant of G on L2(G/K) and the corresponding Hecke algebra, this algebra of operators is commutative; indeed it is a subalgebra of the algebra of mesurable operators affiliated with the commutant π(G)', an Abelian von Neumann algebra. As Harish-Chandra proved, it is isomorphic to the algebra of W(A)-invariant polynomials on the Lie algebra of A, which itself is a polynomial ring by the Chevalley–Shephard–Todd theorem on polynomial invariants of finite reflection groups. The simplest invariant differential operator on G/K is the Laplacian operator; up to a sign this operator is just the image under π of the Casimir operator in the centre of the universal enveloping algebra of G. Thus a normalised positive definite K-biinvariant function f on G is a zonal spherical function if and only if for each D in D(G/K) there is a constant λD such that $\displaystyle\pi(D)f =\lambda_D f,$ i.e. f is a simultaneous eigenfunction of the operators π(D). If ψ is a zonal spherical function, then, regarded as a function on G/K, it is an eigenfunction of the Laplacian there, an elliptic differential operator with real analytic coefficients. By analytic elliptic regularity, ψ is a real analytic function on G/K, and hence G. Harish-Chandra used these facts about the structure of the invariant operators to prove that his formula gave all zonal spherical functions for real semisimple Lie groups.[27][28][29] Indeed the commutativity of the commutant implies that the simultaneous eigenspaces of the algebra of invariant differential operators all have dimension one; and the polynomial structure of this algebra forces the simultaneous eigenvalues to be precisely those already associated with Harish-Chandra's formula. ## Example: SL(2,C) The group G = SL(2,C) is the complexification of the compact Lie group K = SU(2) and the double cover of the Lorentz group. The infinite-dimensional representations of the Lorentz group were first studied by Dirac in 1945, who considered the discrete series representations, which he termed expansors. A systematic study was taken up shortly afterwards by Harish-Chandra, Gelfand–Naimark and Bargmann. The irreducible representations of class one, corresponding to the zonal spherical functions, can be determined easily using the radial component of the Laplacian operator.[6] Indeed any unimodular complex 2×2 matrix g admits a unique polar decomposition g = pv with v unitary and p positive. In turn p = uau, with u unitary and a a diagonal matrix with positive entries. Thus g = uaw with w = u* v, so that any K-biinvariant function on G corresponds to a function of the diagonal matrix $a = \begin{pmatrix} e^{r/2} & 0 \\ 0 & e^{-r/2} \end{pmatrix},$ invariant under the Weyl group. Identifying G/K with hyperbolic 3-space, the zonal hyperbolic functions ψ correspond to radial functions that are eigenfunctions of the Laplacian. But in terms of the radial coordinate r, the Laplacian is given by[30] $L= -\partial_r^2 - 2 \coth r \partial_r.$ Setting f(r) = sinh (r)·ψ(r), it follows that f is an odd function of r and an eigenfunction of $\partial_r^2$. Hence $\varphi(r) = {\sin (\ell r)\over \ell \sinh r}$ where $\ell$ is real. There is a similar elementary treatment for the generalized Lorentz groups SO(N,1) in Takahashi (1963) and Faraut & Korányi (1994) (recall that SO0(3,1) = SL(2,C) / ±I). ## Complex case If G is a complex semisimple Lie group, it is the complexification of its maximal compact subgroup K. If ${\mathfrak g}$ and $\mathfrak{k}$ are their Lie algebras, then $\mathfrak{g} = \mathfrak{k} \oplus i\mathfrak{k}.$ Let T be a maximal torus in K with Lie algebra $\mathfrak{t}$. Then $A= \exp i \mathfrak{t}, \,\, P= \exp i \mathfrak{k}.$ Let $W= N_K(T)/T$ be the Weyl group of T in K. Recall characters in Hom(T,T) are called weights and can be identified with elements of the weight lattice Λ in Hom($\mathfrak{t}$, R) = $\mathfrak{t}^$. There is a natural ordering on weights and every finite-dimensional irreducible representation (π, V) of K has a unique highest weight λ. The weights of the adjoint representation of K on $\mathfrak{k}\ominus \mathfrak{t}$ are called roots and ρ is used to denote half the sum of the positive roots α, Weyl's character formula asserts that for z = exp X in T $\displaystyle\chi_\lambda(e^X)\equiv {\rm Tr} \, \pi(z) = A_{\lambda+\rho}(e^X)/A_{\rho}(e^X),$ where, for μ in $\mathfrak{t}^$, Aμ denotes the antisymmetrisation $\displaystyle A_\mu(e^X) =\sum_{s\in W} \varepsilon(s) e^{i\mu(sX)},$ and ε denotes the sign character of the finite reflection group W. Weyl's denominator formula expresses the denominator Aρ as a product: $\displaystyle A_\rho(e^X) = e^{i\rho(X)} \prod_{\alpha>0}(1 - e^{-i\alpha(X)}),$ where the product is over the positive roots. Weyl's dimension formula asserts that $\displaystyle\chi_\lambda(1) \equiv {\rm dim}\, V = {\prod_{\alpha>0} (\lambda + \rho,\alpha)\over \prod_{\alpha>0} (\rho,\alpha)}.$ where the inner product on $\mathfrak{t}^$ is that associated with the Killing form on $\mathfrak{k}$. Now • every irreducible representation of K extends holomorphically to the complexification G • every irreducible character χλ(k) of K extends holomorphically to the complexification of K and $\mathfrak{t}^$. • for every λ in Hom(A,T) = $i\mathfrak{t}^*$, there is a zonal spherical function φλ. The Berezin–Harish–Chandra formula[6] asserts that for X in $i\mathfrak{t}$ $\varphi_\lambda(e^X) = {\chi_\lambda(e^X)\over\chi_\lambda(1)}.$ In other words: • the zonal spherical functions on a complex semisimple Lie group are given by analytic continuation of the formula for the normalised characters. One of the simplest proofs[31] of this formula involves the radial component on A of the Laplacian on G, a proof formally parallel to Helgason's reworking of Freudenthal's classical proof of the Weyl character formula, using the radial component on T of the Laplacian on K.[32] In the latter case the class functions on K can be identified with W-invariant functions on T. The radial component of ΔK on T is just the expression for the restriction of ΔK to W-invariant functions on T, where it is given by the formula $\displaystyle \Delta_K= h^{-1}\circ \Delta_T \circ h + \\|\rho\\|^2,$ where $\displaystyle h(e^X) = A_\rho(e^X)$ for X in $\mathfrak{t}$. If χ is a character with highest weight λ, it follows that φ = h·χ satisfies $\Delta_T \varphi= (\\|\lambda + \rho\\|^2 -\\|\rho\\|^2) \varphi.$ Thus for every weight μ with non-zero Fourier coefficient in φ, $\displaystyle \\|\lambda +\rho\\|^2 = \\|\mu+\rho\\|^2.$ The classical argument of Freudenthal shows that μ + ρ must have the form s(λ + ρ) for some s in W, so the character formula follows from the antisymmetry of φ. Similarly K-biinvariant functions on G can be identified with W(A)-invariant functions on A. The radial component of ΔG on A is just the expression for the restriction of ΔG to W(A)-invariant functions on A. It is given by the formula $\displaystyle \Delta_G= H^{-1}\circ \Delta_A\circ H - \\|\rho\\|^2,$ where $\displaystyle H(e^X) = A_\rho(e^X)$ for X in $i\mathfrak{t}$. The Berezin–Harish–Chandra formula for a zonal spherical function φ can be established by introducing the antisymmetric function $\displaystyle f= H\cdot\varphi,$ which is an eigenfunction of the Laplacian ΔA. Since K is generated by copies of subgroups that are homomorphic images of SU(2) corresponding to simple roots, its complexification G is generated by the corresponding homomorphic images of SL(2,C). The formula for zonal spherical functions of SL(2,C) implies that f is a periodic function on $i\mathfrak{t}$ with respect to some sublattice. Antisymmetry under the Weyl group and the argument of Freudenthal again imply that ψ must have the stated form up to a multiplicative constant, which can be determined using the Weyl dimension formula. ## Example: SL(2,R) The theory of zonal spherical functions for SL(2,R) originated in the work of Mehler in 1881 on hyperbolic geometry. He discovered the analogue of the Plancherel theorem, which was rediscovered by Fock in 1943. The corresponding eigenfunction expansion is termed the Mehler–Fock transform. It was already put on a firm footing in 1910 by Hermann Weyl's important work on the spectral theory of ordinary differential equations. The radial part of the Laplacian in this case leads to a hypergeometric differential equation, the theory of which was treated in detail by Weyl. Weyl's approach was subsequently generalised by Harish-Chandra to study zonal spherical functions and the corresponding Plancherel theorem for more general semimisimple Lie groups. Following the work of Dirac on the discrete series representations of SL(2,R), the general theory of unitary irreducible representations of SL(2,R) was developed independently by Bargmann, Harish-Chandra and Gelfand–Naimark. The irreducible representations of class one, or equivalently the theory of zonal spherical functions, form an important special case of this theory. The group G = SL(2,R) is a double cover of the 3-dimensional Lorentz group SO(2,1), the symmetry group of the hyperbolic plane with its Poincaré metric. It acts by Möbius transformations. The upper half-plane can be identified with the unit disc by the Cayley transform. Under this identification G becomes identified with the group SU(1,1), also acting by Möbius transformations. Because the action is transitive, both spaces can be identified with G/K, where K = SO(2). The metric is invariant under G and the associated Laplacian is G-invariant, coinciding with the image of the Casimir operator. In the upper half-plane model the Laplacian is given by the formula[6][7] $\displaystyle\Delta=-4y^{2}(\partial_x^2 +\partial_y^2).$ If s is a complex number and z = x + i y with y > 0, the function $\displaystyle f_s(z) =y^{s}= \exp({s}\cdot\log y),$ is an eigenfunction of Δ: $\displaystyle \Delta f_s = 4s(1-s) f_s.$ Since Δ commutes with G, any left translate of fs is also an eigenfunction with the same eigenvalue. In particular, averaging over K, the function $\varphi_s(z) =\int_K f_s(k\cdot z)\, dk$ is a K-invariant eigenfunction of Δ on G/K. When $\displaystyle s={1\over 2} + i\tau,$ with τ real, these functions give all the zonal spherical functions on G. As with Harish-Chandra's more general formula for semisimple Lie groups, φs is a zonal spherical function because it is the matrix coefficient corresponding to a vector fixed by K in the principal series. Various arguments are available to prove that there are no others. One of the simplest classical Lie algebraic arguments[6][7][33][34][35] is to note that, since Δ is an elliptic operator with analytic coefficients, by analytic elliptic regularity any eigenfunction is necessarily real analytic. Hence, if the zonal spherical function corresponds is the matrix coefficient for a vector v and representation σ, the vector v is an analytic vector for G and $\displaystyle(\sigma(e^{X})v,v)= \sum_{n=0}^\infty (\sigma(X)^n v,v)/n!$ for X in $i\mathfrak{t}$. The infinitesimal form of the irreducible unitary representations with a vector fixed by K were worked out classically by Bargmann.[33][34] They correspond precisely to the principal series of SL(2,R). It follows that the zonal spherical function corresponds to a principal series representation. Another classical argument[36] proceeds by showing that on radial functions the Laplacian has the form $\displaystyle\Delta=-\partial_r^2 - \coth(r)\cdot \partial_r,$ so that, as a function of r, the zonal spherical function φ(r) must satisfy the ordinary differential equation $\displaystyle\varphi^{\prime\prime} + \coth r\, \varphi^\prime = \alpha \, \varphi$ for some constant α. The change of variables t = sinh r transforms this equation into the hypergeometric differential equation. The general solution in terms of Legendre functions of complex index is given by[3][37] $\varphi(r)=P_\rho(\cosh r) = {1\over 2\pi} \int_0^{2\pi} (\cosh r + \sinh r \, \cos \theta)^\rho \, d\theta,$ where α = ρ(ρ+1). Further restrictions on ρ are imposed by boundedness and positive-definiteness of the zonal spherical function on G. There is yet another approach, due to Mogens Flensted-Jensen, which derives the properties of the zonal spherical functions on SL(2,R), including the Plancherel formula, from the corresponding results for SL(2,C), which are simple consequences of the Plancherel formula and Fourier inversion formula for R. This "method of descent" works more generally, allowing results for a real semisimple Lie group to be derived by descent from the corresponding results for its complexification.[38][39] ## Further directions • The theory of zonal functions that are not necessarily positive-definite. These are given by the same formulas as above, but without restrictions on the complex parameter s or ρ. They correspond to non-unitary representations.[6] • Harish-Chandra's eigenfunction expansion and inversion formula for spherical functions.[40] This is an important special case of his Plancherel theorem for real semisimple Lie groups. • The structure of the Hecke algebra. Harish-Chandra and Godement proved that, as convolution algebras, there are natural isomorphisms between Cc(K \ G / K ) and Cc(A)W, the subalgebra invariant under the Weyl group.[4] This is straightforward to establish for SL(2,R).[7] • Spherical functions for Euclidean motion groups and compact Lie groups.[6] • Spherical functions for p-adic Lie groups. These were studied in depth by Satake and Macdonald.[41][42] Their study, and that of the associated Hecke algebras, was one of the first steps in the extensive representation theory of semisimple p-adic Lie groups, a key element in the Langlands program. ## Notes 1. ^ Dixmier 1996, Algèbres hilbertiennes. 2. ^ If σ is a unitary representation of G, then $\sigma(f)=\int_G f(g)\sigma(g)\, dg$. 3. Dieudonné 1978 4. ^ a b Godement 1952 5. ^ a b c Helgason 2001 6. Helgason 1984 7. ^ a b c d Lang 1985 8. ^ Cartier 1954-1955 9. ^ Hochschild 1965 10. ^ Dieudonné 1978, pp. 55–57 11. ^ Dieudonné 1977 12. ^ Helgason 1978, p. 249 13. ^ Helgason 1978, pp. 257–264 14. ^ Helgason 1984, pp. 534–538 15. ^ Goodman & Wallach 1998, pp. 549–550 16. ^ Goodman & Wallach 1998, p. 550 17. ^ Helgason 1978, Chapter IX. 18. ^ Harish-Chandra 1954a, p. 251 19. ^ Wallach 1973 20. ^ Berezin 1956a 21. ^ Berezin 1956b 22. ^ Harish-Chandra 1954b 23. ^ Harish-Chandra 1954c 24. ^ Kunze & Stein 1961 25. ^ Stein 1970 26. ^ Kostant 1969 27. ^ Harish-Chandra 1958 28. ^ Helgason 2001, pages 418–422, 427-434 29. ^ Helgason 1984, p. 418 30. ^ Davies 1990 31. ^ Helgason 1984, pp. 432–433 32. ^ Helgason 1984, pp. 501–502 33. ^ a b Bargmann 1947 34. ^ a b Howe & Tan 1992 35. ^ Wallach 1988 36. ^ Helgason 2001, p. 405 37. ^ Bateman & Erdelyi 1953, p. 156 38. ^ Flensted-Jensen 1978 39. ^ Helgason 1984, pp. 489–491 40. ^ Helgason 1984, pp. 434–458 41. ^ Satake 1963 42. ^ Macdonald 1971	2015-04-26 19:39:57	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 84, "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": 0.9337784051895142, "perplexity": 612.926786411211}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246655962.81/warc/CC-MAIN-20150417045735-00058-ip-10-235-10-82.ec2.internal.warc.gz"}
https://answers.ros.org/question/372183/ros-and-gazebo-problems-when-loading-joint-controllers/?sort=oldest	I'm trying to control a gazebo robot using ros control but I have some problems after I load the joint controllers. This is the robot model before I load the controllers, the model seems to be correct (https://drive.google.com/file/d/1cd0w...) but when I load the joint controllers the model implodes on the origin (https://drive.google.com/file/d/1diBV...), the controller.launch it's not showing errors. Why when I load the controllers the model implodes on the origin? Edit: This is my robot model (https://drive.google.com/file/d/1YKo-...), this is the inertia matrix of one joint <inertia ixx="4.5e-05" ixy="-3e-06" ixz="1e-06" iyy="3e-05" iyz="5e-06" izz="3.6e-05"/> I changed to <inertia ixx="0.16666700" ixy="0.0" ixz="0.0" iyy="0.16666700" iyz="0.0" izz="0.16666700" /> and now the robot it's moving like crazy around but when I launch the ros control the robot still moving around (it's not like before)... so now the question is how can I calculate the inertia values? I can get the some information from the fusion 360 model. For exemple this is a leg informations: Leg1 Component instances (1) Area 1.787E+04 mm^2 Density 9.520E-04 g / mm^3 Mass 66.794 g Volume 7.016E+04 mm^3 Bounding box Length 150.172 mm Width 26.00 mm Height 35.00 mm X, Y, Z global 0.00 mm, 0.00 mm, 0.00 mm Center of mass -68.611 mm, -83.519 mm, 118.34 mm Moment of inertia at the center of mass (g mm^2) Ixx = 7016.191 Ixy = 1551.50 Ixz = 15.726 Iyx = 1551.50 Iyy = 1.078E+05 Iyz = -0.298 Izx = 15.726 Izy = -0.298 Izz = 1.085E+05 Moment of inertia on origin (g mm^2) Ixx = 1.408E+06 Ixy = -3.812E+05 Ixz = 5.423E+05 Iyx = -3.812E+05 Iyy = 1.358E+06 Iyz = 6.602E+05 Izx = 5.423E+05 Izy = 6.602E+05 Izz = 8.888E+05 Edit 1: I've calculeted the inertia values for the robot and I think i get correct values (https://drive.google.com/file/d/1puhu...) but the problem still... I've noticed that if I use bigger values the problem disappear, when I use the correct values (that are very small values) again I get the problem. edit retag close merge delete Sort by » oldest newest most voted That is when gazebo starts commanding the joints, which means there is something wrong with your model. Without having any information on the model, all I can say is start by looking into your inertias. As a simple test, replace all inertias, for example, with the inertia of a unit box and observe what happens. Edit: The center of mass and inertia at the center of mass of each link from fusion 360 is exactly what you need (don't forget to correct for the dimensions). If you want to simplify things or you have any complications, the next best thing is to approximate the inertias with the inertias of the bounding boxes. And as a side note, you can visualize the inertias in gazebo. Their size should be around the size of the links. Now, regarding the instability of the model. Check any physics parameters you might have set in the world description and any surface parameters for the links of the robot. As no information is provided, I'll suggest removing all parameters, gradually adding them back, and playing with the values, while ensuring the stability of the model. You'll probably have to go back and forth a few times. more	2021-12-04 14:53:21	{"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": 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": 0.23332832753658295, "perplexity": 2486.5372532495876}, "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/1637964362992.98/warc/CC-MAIN-20211204124328-20211204154328-00159.warc.gz"}
http://markun.cs.shinshu-u.ac.jp/mirror/mizar/JFM/Vol15/bhsp_7.html	Journal of Formalized Mathematics Volume 15, 2003 University of Bialystok Copyright (c) 2003 Association of Mizar Users On Some Properties of Real Hilbert Space. Part II Hiroshi Yamazaki Shinshu University, Nagano Yasumasa Suzuki Take, Yokosuka-shi, Japan Takao Inoue The Iida Technical High School, Nagano Yasunari Shidama Shinshu University, Nagano Summary. This paper is a continuation of our paper [22]. We give an analogue of the necessary and sufficient condition for summable set (i.e. the main theorem of [22]) with respect to summable set by a functional $L$ in real Hilbert space. After presenting certain useful lemmas, we prove our main theorem that the summability for an orthonormal infinite set in real Hilbert space is equivalent to its summability with respect to the square of norm, say $H(x) = (x, x)$. Then we show that the square of norm $H$ commutes with infinite sum operation if the orthonormal set under our consideration is summable. Our main theorem is due to [8]. MML Identifier: BHSP_7 The terminology and notation used in this paper have been introduced in the following articles [16] [19] [6] [1] [17] [9] [4] [5] [20] [18] [12] [13] [14] [3] [7] [10] [15] [11] [2] [21] [22] Contents (PDF format) 1. Necessary and Sufficient Condition for Summable Set 2. Equivalence of Summability Bibliography [1] Grzegorz Bancerek. The ordinal numbers. Journal of Formalized Mathematics, 1, 1989. [2] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Journal of Formalized Mathematics, 1, 1989. [3] Czeslaw Bylinski. Binary operations. Journal of Formalized Mathematics, 1, 1989. [4] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989. [5] Czeslaw Bylinski. Functions from a set to a set. Journal of Formalized Mathematics, 1, 1989. [6] Czeslaw Bylinski. Some basic properties of sets. Journal of Formalized Mathematics, 1, 1989. [7] Agata Darmochwal. Finite sets. Journal of Formalized Mathematics, 1, 1989. [8] P. R. Halmos. \em Introduction to Hilbert Space. American Mathematical Society, 1987. [9] Krzysztof Hryniewiecki. Basic properties of real numbers. Journal of Formalized Mathematics, 1, 1989. [10] Eugeniusz Kusak, Wojciech Leonczuk, and Michal Muzalewski. Abelian groups, fields and vector spaces. Journal of Formalized Mathematics, 1, 1989. [11] Bogdan Nowak and Andrzej Trybulec. Hahn-Banach theorem. Journal of Formalized Mathematics, 5, 1993. [12] Jan Popiolek. Some properties of functions modul and signum. Journal of Formalized Mathematics, 1, 1989. [13] Jan Popiolek. Introduction to Banach and Hilbert spaces --- part I. Journal of Formalized Mathematics, 3, 1991. [14] Jan Popiolek. Introduction to Banach and Hilbert spaces --- part III. Journal of Formalized Mathematics, 3, 1991. [15] Andrzej Trybulec. Semilattice operations on finite subsets. Journal of Formalized Mathematics, 1, 1989. [16] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989. [17] Andrzej Trybulec. Subsets of real numbers. Journal of Formalized Mathematics, Addenda, 2003. [18] Wojciech A. Trybulec. Vectors in real linear space. Journal of Formalized Mathematics, 1, 1989. [19] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989. [20] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989. [21] Hiroshi Yamazaki, Yasunari Shidama, and Yatsuka Nakamura. Bessel's inequality. Journal of Formalized Mathematics, 15, 2003. [22] Hiroshi Yamazaki, Yasumasa Suzuki, Takao Inoue, and Yasunari Shidama. On some properties of real Hilbert space, I. Journal of Formalized Mathematics, 15, 2003.	2017-11-18 21:37:36	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9578467011451721, "perplexity": 6760.6425318005795}, "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-2017-47/segments/1510934805049.34/warc/CC-MAIN-20171118210145-20171118230145-00156.warc.gz"}
https://bookstore.ams.org/view?ProductCode=MEMO/236/1116	An error was encountered while trying to add the item to the cart. Please try again. The following link can be shared to navigate to this page. You can select the link to copy or click the 'Copy To Clipboard' button below. Copy To Clipboard Successfully Copied! Homological Mirror Symmetry for the Quartic Surface Paul Seidel Massachusetts Institute of Technology, Cambridge, MA Available Formats: Electronic ISBN: 978-1-4704-2282-0 Product Code: MEMO/236/1116.E 129 pp List Price: $81.00 MAA Member Price:$72.90 AMS Member Price: $48.60 Click above image for expanded view Homological Mirror Symmetry for the Quartic Surface Paul Seidel Massachusetts Institute of Technology, Cambridge, MA Available Formats: Electronic ISBN: 978-1-4704-2282-0 Product Code: MEMO/236/1116.E 129 pp List Price:$81.00 MAA Member Price: $72.90 AMS Member Price:$48.60 • Book Details Memoirs of the American Mathematical Society Volume: 2362015 MSC: Primary 53; Secondary 14; 18; The author proves Kontsevich's form of the mirror symmetry conjecture for (on the symplectic geometry side) a quartic surface in $\mathbb{C} P^3$. • Chapters • 1. Introduction • 2. $A_{\infty }$-categories • 3. Deformation theory • 4. Group actions • 5. Coherent sheaves • 6. Symplectic terminology • 7. Monodromy and negativity • 8. Fukaya categories • 9. Computations in Fukaya categories • 10. The algebras $Q_4$ and $Q_{64}$ • 11. Counting polygons • Request Review Copy • Get Permissions Volume: 2362015 MSC: Primary 53; Secondary 14; 18; The author proves Kontsevich's form of the mirror symmetry conjecture for (on the symplectic geometry side) a quartic surface in $\mathbb{C} P^3$. • Chapters • 1. Introduction • 2. $A_{\infty }$-categories • 3. Deformation theory • 4. Group actions • 5. Coherent sheaves • 6. Symplectic terminology • 7. Monodromy and negativity • 8. Fukaya categories • 9. Computations in Fukaya categories • 10. The algebras $Q_4$ and $Q_{64}$ • 11. Counting polygons Please select which format for which you are requesting permissions.	2023-01-29 12:14:52	{"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": 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": 0.4071557819843292, "perplexity": 14429.947468553411}, "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/1674764499713.50/warc/CC-MAIN-20230129112153-20230129142153-00449.warc.gz"}
https://www.vedantu.com/question-answer/value-of-the-constant-term-in-the-expansion-class-10-maths-cbse-5f5f675c8f2fe24918f7515e	Question # What is the value of the constant term in the expansion given as $23{{x}^{3}}+12{{x}^{2}}-6x-12$?(a) 12(b) 6(c) -6(d) -12 Hint: We start solving the problem by assigning a function to the given expansion. We use the fact that we get the constant term by substituting ‘0’ in place of ‘x’ in the expansion. So, we substitute ‘0’ in place of ‘x’ to get the required value of the constant term. Complete step-by-step solution: Given that we have an expansion $23{{x}^{3}}+12{{x}^{2}}-6x-12$, and we need to find the value of the constant term in that given expansion. Let us represent the given expansion with f(x). So, we have $f(x)=23{{x}^{3}}+12{{x}^{2}}-6x-12$ ---(1). We can see that the given expansion is polynomial of degree 3 (we know that degree is the maximum power of x in a given polynomial of x). We know that to find the value of the constant term in expansion, we substitute x = 0 in the expansion. So, let us substitute x = 0 in the polynomial f(x). So, we have got $f\left( 0 \right)=23{{\left( 0 \right)}^{3}}+12{{\left( 0 \right)}^{2}}-6\left( 0 \right)-12$. We have got $f\left( 0 \right)=23\left( 0 \right)+12\left( 0 \right)-6\left( 0 \right)-12$. We have got f(0) = $0 + 0 – 0 – 12$. We have got f(0) = $–12$. So, we have got the value of f(0) as $–12$. Since f(0) is the value of the constant term of the expansion, We get the value of the constant term of the expansion as $–12$. $\therefore$ The value of the constant term in the expansion $23{{x}^{3}}+12{{x}^{2}}-6x-12$ is –12. Note: We solved this problem by substituting ‘0’ in place of ‘x’, this method can also be adopted in the cases where ‘x’ contains powers in the fractions. Similarly, we can expect problems to tell the value of co-efficient of ‘x’ and other higher powers of ‘x’.	2021-02-27 06:46:13	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9192938804626465, "perplexity": 228.9505351832179}, "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-2021-10/segments/1614178358203.43/warc/CC-MAIN-20210227054852-20210227084852-00408.warc.gz"}
https://design.tutsplus.com/tutorials/how-to-draw-a-t-rex-dinosaur--cms-27938	We want to learn about your experience with Tuts+. Take part to earn $50! Help us out to earn$50! Get Started # How to Draw a T-Rex Dinosaur This post is part of a series called Science Week. How to Create a Set of Science Icons in Affinity Designer How to Create a Human Cyborg Photo Manipulation in Adobe Photoshop We all loved dinosaurs as children, and T-Rex is the favorite of many. This huge, ferocious beast surely captures our imagination! In this tutorial I will show you how to draw one, but also more than that—I will teach you how to draw an animal from scratch, using 3D blocks to build a final 3D form of the creature. ## 1. How to Sketch a Dinosaur Silhouette ### Step 1 Draw a shallow arc—this will be the back of our dinosaur. ### Step 2 Add the curved neck and the tail (the tail should be longer than the rest of the body). ### Step 3 Add a boxy body under the back. It should have a longer front (chest) and a shorter back (hips). ### Step 4 Sketch the legs. They're just like our legs, except T-Rex walks on its toes. ### Step 5 Sketch the width of the feet. This will help you see their position better. ### Step 6 Sketch the head. It doesn't need to be detailed—just an egg shape at the end of the neck. ### Step 7 Sketch the places for the eyes and nostrils to understand the perspective you're creating. ### Step 8 Now add the shoulders and arms; nothing fancy here. ### Step 9 Complete the silhouette with the "lower neck" and "lower tail". ## 2. How to Draw the T-Rex's Head ### Step 1 We have the skeleton of our drawing—we can see exactly where we're going. Now we can add the detailed blocks of the body. Start by sketching a cube or a cuboid inside the head of our dino. Its perspective must match the perspective you've sketched before—it can't be random! In other words, the cube should be facing the same way as the dinosaur. ### Step 2 Extend the bottom of the cube to create the length of the muzzle. Cross it with its width. The muzzle of the T-Rex is narrower than the skull. ### Step 3 We're going to create two ellipsoidal domes for the muzzle, but they're seen in perspective, so we need to do it vary carefully. An ellipse in perspective has long, flat arcs in front of the obtuse angles... ... and short, convex arcs in front of the acute angles. Remember this rule! ### Step 4 Use the same rule to create the other ellipses building this "dome". ### Step 5 Connect the dome with the rest of the skull. ### Step 6 Create a smaller version of the same dome in the lower jaw. Relax—you already know how to do it! ### Step 7 Finish the lower jaw by creating its wider part first... ... then completing the lines. ### Step 8 Draw the teeth all around the jaws. They should be slightly uneven and not completely pointed. ### Step 9 Finish the head by adding the eye socket with the tiny eye inside, and the jaw muscles. ## 3. How to Draw T-Rex Feet and Claws ### Step 1 Feet and hands are made of many small elements allowing for motion. Let's work on them one by one. Draw cylinders in the place of the joints: in the ankle and right before the toes. ### Step 2 Connect both cylinders with blocks. ### Step 3 Draw small domes at the tips of the toes. You don't need to draw them all at once, if they cover each other. Make the middle dome the biggest. ### Step 4 Draw the openings for the claws in the front of each dome. ### Step 5 Draw big, slightly curved claws. ### Step 6 To finish the toes, connect the domes with the cylinders with a path of "plates". ### Step 8 Finish the feet by adding the toes in the back, a "foot-thumb", and a foot pad below the toe joints. ### Step 9 Let's draw the arms now. They're short, but heavily muscled. Start with two circles for these muscles. ### Step 10 Draw the little forearms. ### Step 11 Add some depth to the arm by drawing directing lines on it. ### Step 12 Draw the tips of both fingers on each hand. ### Step 13 Draw the top... ... and bottom of each finger. ## 4. How to Draw the Dinosaur's Body ### Step 1 To draw the shoulders, imagine a horse collar on them. First draw its opening... ... then its side surface... ... and finally, its lower part that will be the chest. ### Step 2 Draw big muscle masses for both the thigh and the calf. ### Step 3 Let's add perspective to the hind legs now. Use directing lines to mark the front and side of the calves. ### Step 4 Do the same with the thigh, adding the knee between both parts. ### Step 5 Time to add volume to the rest of the body. The T-Rex's neck should be visibly S-curved. You can achieve that by drawing curved lines between the head and the shoulders. ### Step 6 But that's not all! Cross them with more lines to show the curving side of the neck. ### Step 7 Use the same trick to define the volume of the rest of the body. ## 5. How to Draw the Dinosaur's Skin and Details ### Step 1 The base for the drawing is finished, so we can draw the final lines now. In digital art you can create a new layer now. In traditional art, put a new sheet of paper over the drawing or draw the final lines with a darker tool. Draw the tiny eye and the nostrils. ### Step 3 Outline the teeth and add some details to the skull. The depressions in the skull are not necessary if you want to be more realistic. ### Step 4 Envelop the whole body into one piece of skin. Draw wrinkles between the separate parts of the body. ### Step 5 The hands and feet should be enveloped in one piece of skin as well. There will be more wrinkles, because there are more elements here. ### Step 6 You can add more wrinkles all over the skin to accentuate the directing lines and stress the perspective of the body. ### Step 7 Finally, you can shade the body to show the volume even better. ### Step 8 If you want to create a more scientifically accurate T-Rex, you need to make some changes: • The teeth should be mostly covered (by gums and lips), because T-Rex is not aquatic like a crocodile and its teeth would dry out being so bare. • The body should be a little rounder—the bones don't tell us anything about the fat layer, but it doesn't mean we should accentuate the bones so much. • There's evidence that other dinosaurs of T-Rex family had feathers, so it's likely that T-Rex had them as well—at least on a part of its body. If so, they weren't used for flight, but for warmth.	2021-11-27 02:06:26	{"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": 0.2919560372829437, "perplexity": 2676.8688938953487}, "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/1637964358078.2/warc/CC-MAIN-20211127013935-20211127043935-00103.warc.gz"}
http://eseslab.com/ESsensePages/Flywheels-page	## Flywheels Storing rotational energy. ### Background The principle behind flywheel energy storage is to store energy in the form of rotational kinetic energy. The amount of energy stored depends upon the moment of inertia, I, of the rotating mass and its rotational velocity, $\omega$, as described by the Equation below. $E = \frac{1}{2} I \omega^2$ The power rating is decided by the rating of the motor/generator. To charge the flywheel, useful energy (usually electrical, although direct mechanical drive systems are being explored) is used to increase the rotational speed of the flywheel- thus increasing its energy content. To discharge, kinetic energy is extracted from the flywheel (the flywheel is slowed) and converted into electricity via a generator (driven by the flywheel). Flywheels have long lifetimes (and require very little maintenance), can be very rapidly cycled and have good efficiency over short timescales, often quoted well in excess of 95%. They can generally be separated into two categories. Low speed systems (up to around 6000rpm) and high speed systems (up to around 50,000 rpm). Low speed systems are a reasonably mature technology, are commercially available and indeed are extensively used in power quality applications. High speed systems are a technology more in research and development, and must be constructed of composite materials (to tolerate very high tensile stresses). They usually require low friction magnetic bearing setups and spin in a vacuum enclosure. Figure: Depiction of a modern flywheel for energy storage ### Flywheel characteristics and applications Flywheels have many applications where a high power/short duration is required (e.g. 100 s of kW/10 s of seconds), in UPS, reactive power support, spinning reserve and voltage regulation. The most common application is to act as a power quality device to provide ride-through of interruptions up to 15 s long or to bridge the shift from one power source to another. Recent development has also produced flywheels capable of smoothing outputs from wind turbines. In the Azores, a 5kWh FESS dramatically improved system stability on a grid supplying 100,000 people, allowing a higher penetration of renewable energy and reducing diesel fuel usage. The system, installed by Powercorp of Australia, now smoothes wind power fluctuations allowing the operator to run the power station without keeping any diesel generators online. The 350 kW flywheel can respond to 100% fluctuations in demand in 4 milliseconds. Flywheels have also been used in energy recovery in electrically powered mass transit system. They are really only suitable for applications which require high cycling, high power and small response times. Crucially they suffer from very high self discharge rates making them unsuitable for storage over timescales much longer than a few tens of minutes. There are several companies developing flywheel energy storage systems including Beacon Power, Pentadyne Power Corporation, Kinetech to name but a few. ### Summary of characteristics Typical Capacity Typical Power Efficiency (%) Storage Duration $/kWh$/kW Lifespan Cycling capacity Up to 5MWh Up to 20MW 93 [1], 85 [2], 90-95 [3], 95 [4] milliseconds - minutes 1000-5000 [3], 1600 [4] 200-350 [3], 600 [4] 20 years High Table: Flywheel characteristics	2021-04-21 05:01:53	{"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": 0.6128634810447693, "perplexity": 2710.673225978623}, "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/1618039508673.81/warc/CC-MAIN-20210421035139-20210421065139-00576.warc.gz"}
https://www.cs.swarthmore.edu/~richardw/classes/cs21/f15/Labs/lab04.php	CS21 Lab 4: while loops and functions Due Saturday (October 3) before midnight This lab assignment requires you to write some Python programs. First, run update21 to get the template files we have provided for you. These will serve as starting points for your programs. Next, use cd to change into your cs21/labs/04 directory and begin working on the Python programs for this lab. The pwd command helps you verify that you are in the correct sub-directory. $update21$ cd cs21/labs/04 $pwd /home/your_user_name/cs21/labs/04 We will only grade files submitted by handin21 in this directory, so make sure your programs are in this directory! Programming tips As your programs become larger, it is even more important to develop good habits: • Use a comment at the top of the file to describe the purpose of the program. • Use variable names that describe the contents of the variables. • Use def main() to combine the steps of your program into a function. The above are mandatory and you will be graded on them. We strongly advise you to write your programs incrementally and test them as you go. No one writes interesting programs all at once. As you write, test that your program does what you expect (even if it doesn't yet do what you want). If it surprises you, make sure you understand what's happening before writing more code. We have provided some test input and output; make sure you test your program! You should also test your program using some inputs and outputs that were not shown here; after all, we will. Come up with your own test cases and verify that the program is producing the right output on them. 1. Nim Source: OpenClipArt.org You will write a computer program which allows the user to play Nim, an ancient game with numerous variants, a global history, and interesting connections to mathematics. Nim is a strategy game based on subtraction. In our variant, there are two players and three rows of sticks. At the beginning of the game, the first row contains three sticks, the second row contains five sticks, and the third row contains seven sticks. Players alternate, taking sticks from one of the rows; players may take any number of sticks they please, but must take at least one and may only take from one row per turn. The objective is to force the other player to take the last stick. Your computer program will allow two players to play a hotseat version of the game -- both players play the game by taking turns using the program. Your program will keep track of whose turn it is and how many sticks are left in each pile. Because multiple people are using the program, each prompt should indicate whose turn it is. You have been provided a template file, nim.py. This file contains a number of functions which you are to use in writing your implementation of Nim. You simply need to write the body of the main function which plays the game. Here are some sample runs of the program. $ python nim.py 1: \|\|\| 2: \|\|\|\|\| 3: \|\|\|\|\|\|\| Player 1: which row? 2 Player 1: take how many? (max 5) 5 1: \|\|\| 2: 3: \|\|\|\|\|\|\| Player 2: which row? 3 Player 2: take how many? (max 7) 6 1: \|\|\| 2: 3: \| Player 1: which row? 1 Player 1: take how many? (max 3) 3 1: 2: 3: \| Player 2: which row? 3 Player 2: take how many? (max 1) 1 Player 1 wins! $python nim.py 1: \|\|\| 2: \|\|\|\|\| 3: \|\|\|\|\|\|\| Player 1: which row? 0 That is not a valid row. Player 1: which row? 1 Player 1: take how many? (max 3) 3 1: 2: \|\|\|\|\| 3: \|\|\|\|\|\|\| Player 2: which row? 1 That is not a valid row. Player 2: which row? 3 Player 2: take how many? (max 7) 6 1: 2: \|\|\|\|\| 3: \| Player 1: which row? 2 Player 1: take how many? (max 5) 3 1: 2: \|\| 3: \| Player 2: which row? 2 Player 2: take how many? (max 2) 2 1: 2: 3: \| Player 1: which row? 3 Player 1: take how many? (max 1) 4 That is not a valid number of sticks. Player 1: take how many? (max 1) 1 Player 2 wins! 2. Coded Messages Source: Wikipedia.org Last week's lab included a program which was required to extract a secret message from a large string. This week, you will write a program which can transform a string in several ways. A sequence of these operations can be used to create an encoded message; performing the opposite steps in reverse will recover the original message. Your program will start by requesting a message from the user. Then, you will accept a series of commands: left, right, flip, dosido, and swap. Each of those commands change the message in some way: • left moves every character in the string to the left. The leftmost character wraps around to the right. For instance, "fish" becomes "ishf". right does the opposite, moving every character to the right. • flip reverses the string: the first character becomes the last, the second character becomes next-to-last, and so on. For instance, "fish" becomes "hsif". • dosido swaps each pair of characters: the first and second characters trade places, the third and fourth characters trade places, and so on. If the string is of odd length, the last character stays where it is. For instance, "computer" becomes "ocpmture" and "mouse" becomes "omsue". • When the user commands you to swap, you will ask for two letters. Then, each occurrence of the first letter is replaced with the second letter and vice versa. For instance, if the user asks you to swap the letters l and e, then "hleeo" becomes "hello". Make sure to run update21 so that you get the template file, code.py, that we have provided for you. This template file contains a number of functions, all of which have the pass statement in their bodies. The pass statement is a "do-nothing" statement that technically satisfies the requirement that every function has a body. This way, you can write the code for one function at a time and test your progress without having to write everything at once. This part of the lab is, in a way, the opposite of the Nim problem: we have given you (most of) a main function and you must now fill in the functions that it is using! Note that you must also update the main function, as it presently only understands the left and right commands. If you write your functions correctly, though, you shouldn't have to change anything that already appears in main. Here's a sample run of this program: $ python code.py What is your message? It's easier to ask forgiveness than it is to get permission. What would you like to do? left Your message is now: t's easier to ask forgiveness than it is to get permission.I What would you like to do? flip Your message is now: I.noissimrep teg ot si ti naht ssenevigrof ksa ot reisae s't What would you like to do? dosido Your message is now: .Ionsiisrmpet geo tist ianths esenivrgfok aso tersieas t' What would you like to do? swap What is the first letter to swap? a What is the second letter to swap? e Your message is now: .Ionsiisrmpat gao tist ienths asanivrgfok eso tarsiaes t' What would you like to do? scramble I don't know the action "scramble". Your message is now: .Ionsiisrmpat gao tist ienths asanivrgfok eso tarsiaes t' What would you like to do? swap What is the first letter to swap? e What is the second letter to swap? a Your message is now: .Ionsiisrmpet geo tist ianths esenivrgfok aso tersieas t' What would you like to do? dosido Your message is now: I.noissimrep teg ot si ti naht ssenevigrof ksa ot reisae s't What would you like to do? flip Your message is now: t's easier to ask forgiveness than it is to get permission.I What would you like to do? right Your message is now: It's easier to ask forgiveness than it is to get permission. What would you like to do? quit 3. The Tortoise and the Hare Source: Wikipedia.org Aesop's fable of the Tortoise and the Hare features a quick but overconfident hare in a race against a tortoise. Your task is to simulate the race between the tortoise and the hare. Your program will ask the user for the length of the race (which you may assume is a positive integer) and then simulate it under the following rules. • We will check in on the race once every minute. • Each minute, the tortoise moves one meter. • Each minute, there is a 20% chance that the hare moves four meters. The other 80% of the time the hare was napping and didn't move at all. The race ends when one or both of the participants has traveled at least as far as the length of the race given by the user. The winner is the participant who traveled the furthest; if both participants traveled the same distance, the race is a tie. After each minute of the race, you must print a visual status report of the race. This status report must consist of three lines. The first line must show the distance the hare has traveled (one character per meter) with asterisks behind traveled spaces. The second line shows the tortoise. The third line is a series of dashes equal to the length of the race, with an extra character in front to represent the starting line. We show the tortoise with a capital T and the hare with a capital H. You must write a function to show the status of the race; the function header is included in the template file, race.py, for you. Below are some sample executions of this program. $python race.py How long should the race be? 8 H T \|-------- ***H T \|-------- **H T \|-------- **H T \|-------- *H T \|-------- ****H **T \|-------- The hare wins!$ python race.py How long should the race be? 20 H T \|-------------------- H T \|-------------------- H T \|-------------------- H T \|-------------------- H *T \|-------------------- H *T \|-------------------- H **T \|-------------------- H ***T \|-------------------- H ****T \|-------------------- H *****T \|-------------------- H ******T \|-------------------- ****H *******T \|-------------------- ********H ********T \|-------------------- ********H *********T \|-------------------- ********H **********T \|-------------------- ********H ***********T \|-------------------- ********H ************T \|-------------------- ********H *************T \|-------------------- ********H **************T \|-------------------- ********H ***************T \|-------------------- ********H ******************T \|-------------------- The tortoise wins! Submit Remember you may run handin21 as many times as you like. Each time you run it new versions of your files will be submitted. Running handin21 after you finish a program, after any major changes are made, and at the end of the day (before you log out) is a good habit to get into.	2018-01-20 17:16:25	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.38467153906822205, "perplexity": 4103.095362101225}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084889677.76/warc/CC-MAIN-20180120162254-20180120182254-00043.warc.gz"}
https://www.badenmarkt.nl/194mxk/83c2f7-arithmetic-mean-and-standard-deviation-calculator	In many cases, it is not possible to sample every member within a population, requiring that the above equation be modified so that the standard deviation can be measured through a random sample of the population being studied. Free Arithmetic Mean (Average) Calculator - find the average of a data set step-by-step This website uses cookies to ensure you get the best experience. Simply insert the values for x and y, choose the two types of means that should be applied, one from each drop-down menu, and click on the "Calculate Mean" button. Select STDEV.S (for a sample) from the the Statistical category. Arithmetic Mean Calculator Arithmetic mean is the term used in statistics which is nothing but the sum of given data divided by the total number of values. With the knowledge of calculating standard deviation, we can easily calculate variance as the square of standard deviation. The arithmetic mean represents the most commonly employed mean. The mean will be displayed if the calculation is successful. Providing x and y are not equal, the AGM is always lower than the arithmetic mean and higher than the geometric mean. Geometric The Geometric-Harmonic Mean (GHM) represents a further example of an iterative average. Mean or Average. The null hypothesis is the hypothesis that the difference is 0. Geometric Mean Calculator How to use this calculator. Example 1 : The standard deviation and mean of a data are 6.5 and 12.5 respectively. The TTEST procedure is the easiest way to compute the geometric mean (â¦ One example of the root mean square is the standard deviation of a set of numbers (I.e., it is the root mean square of the variations between the arithmetic mean and each data point). There are some interesting correlations between AGM(x,y) and GHM(x,y): Iterating the harmonic and arithmetic means results in the geometric mean. Consequently the squares of the differences are added. $\text{GSD}[x] = e^{\text{SD}[\log x]}$ This is going to be useful if and only it was a good idea to use a geometric mean on your data, and particularly if your data is positively skewed . The geometric standard deviation (GSD) is the same transformation, applied to the regular standard deviation. Given three numbers, x, y, and z, the harmonic mean is 3xyz/(xy + xz + yz). Formula for the sample standard deviation for a frequency distribution: sample standard deviation = s = ââ(x-xÌ)²f/n-1 where n=âf is the number of entries in the data set - REMEMBER: formulas for grouped data require you to multiply by the frequencies. The average of two or more numbers is referred to as the mean. This calculator will find the geometric mean of a set of numbers.. We are studying about the mean, median and mode and it is important to first of all know about the definition of all these : Mean. Mean median mode definition. If we add (or subtract) a number say 7 to all values in the input set, mean is increased (or decreased) by 7, but standard deviation doesnât change. Description. You can use this Standard Deviation Calculator to calculate the standard deviation, variance, mean, and the coefficient of variance for a given set of numbers. This calculator calculates the arithmetic mean from a set of numerical values: To calculate the mean, enter the numerical values in the box above. Variance = ( Standard deviation)² = Ï×Ï. As for the arithmetic mean, you need to start by thinking about the location of the geometric mean (20.2). Mean, median, mode, and range. Contraharmonic. In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. The variables are denoted by xi. Press the "Submit Data" button to perform the computation. You do not need to specify whether the data is from a population or a sample because this does not affect the calculation of the mean. In words: It's the sum of all values, divided by the total number of values. The sum of the squares is then divided by the number of observations minus oneto give the mean of the squares, and the square root is taken to bring the measurements back to the units we started with. Arithmetic Mean Definition. - Population Standard Deviation Example: To find the Population Standard deviation of 1,2,3,4,5. Otherwise, figure out the frequency of each variable and they are denoted by fiand the â¦ The experimental standard deviations of the mean for each set is calculated using the following expression: s / (n) 1/2 (14.5) Using the above example, where values of 1004, 1005, and 1001 were considered acceptable for the calculation of the mean and the experimental standard deviation the mean would be 1003, the experimental standard deviation would be 2 and the standard deviation â¦ Arithmetic mean formula. In due course, the algorithm will stabilize at a fixed number. For instance, the arithmetic mean of 3, 4, and 5 is (3 + 4 + 5)/3 = 4. Mean = 12.5. This video teaches how to use the Texas Instrument BAII Plus calculator to obtain the mean, standard deviation and variance. Arithmetic Example: 6 students scored 100, 99, 98, 100, 99, & 98 respectively. Sample Standard Deviation. Finding the Standard Deviation. For n values, the arithmetic mean is (x1 + x2 + ... + xn)/n. For example, take 34, 44, 56 and 78. The mean, median and mode can all be called an "average" in certain literature, but using their proper technical names is recommended to avoid confusion. Calculating the arithmetic mean involves adding up all the values and dividing them by the number of values. The contraharmonic mean of x and y is as follows: For n values, you can compute the contraharmonic mean as follows: (x12 + x22 + ... + xn2)/(x1 + x2 + ... + xn). If we multiply all values in the input set by a number 7, both mean and standard deviation is multiplied by 7. the consistent value of the sequence is the GHM of x and y. for use in every day domestic and commercial use! Please explain!OK. Calculating the arithmetic mean involves adding up all the values and dividing them by the number of values. For example, consider you have the following data set: 8,9,6,12,19. 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On the concept of step deviation contraharmonic and harmonic means results in arithmetic. Pixels in the cell you selected for arithmetic mean and standard deviation calculator Mathematical Concepts: Precalculus withâ¦ 1st Edition McGraw-Hill 14.3... For a sample ) from the mean, variance and arithmetic standard deviation a! Individual values by commas, spaces or new-line the null hypothesis arithmetic mean and standard deviation calculator the same transformation, applied to regular! You selected y, the arithmetic mean involves adding up all the values an. Are also interested in the arithmetic mean of any data, you need to divide sum... By a number 7, both mean and standard deviation using the formula and Bn will converge at a number... Use the Texas Instrument BAII Plus calculator to compute the arithmetic mean is 3xyz/ ( xy + +. N numbers is referred to as the arithmetic mean arithmetic standard deviation ) =! 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Determining a pair of numbers a fixed number ) for the list will appear in the input by...	2021-03-07 15:35:12	{"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": 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": 0.772443950176239, "perplexity": 585.9274391853758}, "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-2021-10/segments/1614178377821.94/warc/CC-MAIN-20210307135518-20210307165518-00233.warc.gz"}
https://easternartreport.net/set_xticklabels/10620666a1e74921a14ece4-how-to-use-pythagorean-theorem-to-find-missing-side	asics waterproof shoes \| feminist manifesto ideas \| mansion wedding venues texas \| make your own colored pencils # how to use pythagorean theorem to find missing side Referencing the above diagram, if. Since we only know what the side lengths are we must use the Pythagorean Theorem. How to use the Pythagorean theorem Input the two lengths that you have into the formula. 50 And in that right triangle we can say that the legs a and b that is those sides that are adjacent to this right angle. 1. The formula is as follows: To use the right angle calculator simply enter the lengths of any two sides of a right triangle into the top boxes. Use this Pythagorean theorem calculator to calculate the hypotenuse or the length of a right triangle's missing leg. Other contents: Pythagorean Theorem. if leg a is the missing side, then transform the equation to the form when a is on one side, and take a square root: a = (c - b) if leg b is unknown, then. The theorem helps us quantify this distance and do interesting things like cluster similar results. If you have the other two side lengths, you can use the Pythagorean theorem to solve! However, the legs measure 11 and 60. The plank he is using for the ramp is 10 feet long and is placed on a 2-ft high box. If you're behind a web filter, please make sure that the domains .kastatic.org and .kasandbox.org are unblocked. Learn about the Pythagorean theorem. Add to my workbooks (10) The Pythagorean Theorem only applies to a right triangle. Language: English. that if you take one of the. Problem 1. Round to nearest tenth if needed. Fill in all the gaps, then press "Check" to check your answers. a^2 + b^2 = c^2 Right, so that is the equation of Pythagorean Theorem. If the final equation is true, then the triangle is right. Suppose you have a right triangle in which a and b are the lengths of the legs, and c is the length of the hypotenuse, as shown below. There are several ways to determine the missing information in a right triangle. The Pythagorean theorem. a=4, b=x, and c=5. We find the square root of 97 and c^2, which is then 9.84885780179615 = c. We can round the number to the nearest hundreth. Remember that a right triangle has a angle, which we usually mark with a small square in the corner. 3 2. Answer: The length of the . A instructional video for beginners in the use of pythagoras theorem to find the longest side (the hypotenuse) in a right angle triangle We can use this to find the missing side: If we take the length of the hypotenuse to be c and the length of the legs to be a and b then the Pythagorean theorem tells us that: c2 = a2 + b2. We want to find the length of one leg . Then, use the Pythagorean theorem to find the slant height. What is the length of the remaining side? 17 cm *** C. 23 cm D. 4.79 cm 2. The hypotenuse of the triangle is the side that is opposite to the right angle, while the other sides are named as the base and height, respectively. Find a missing side length on an acute isosceles triangle by using the Pythagorean theorem. here are both the HTML and JavaScript codes. Mathematics. The equation is 4^2 + 9^2 = c^2. Save. c 2 = a 2 + b 2. where, a, b represent the legs of the triangle and c represents the hypotenuse. The approximate length of the ladder is 20.6 meters (67.6 ft). It's very important to note that Pythagorean theorem is only for right triangles. 500 500. function do_things () { var a = parseFloat (document . Activity Goals: Given two legs of a right triangle, students will use the Pythagorean Theorem to find the unknown length of the hypotenuse using a calculator. By Pythagorean theorem problem solver Comentarios desactivados en Pythagorean theorem problem solver . 2. In this video, a right angled triangle with three sides, namely a, b and c is shown. 3. The theorem is a formula that connects the areas of squares that can be drawn on the triangle's sides for any right-angled triangle. Right, let write down the Pythagorean Theorem equation or standard form of it. For each right triangle, use the Pythagorean Theorem to find the length of the unknown side 'x'. Pythagorean's Theorem tells us the areas of the smaller two squares c ombined are equal to the area of the largest square . One side "a" is of unknown length. The Pythagorean theorem is a very popular theorem that shows a special relationship between the sides of a right triangle. a 2 + b 2 = c 2. It is used to determine the . First, use the Pythagorean theorem to solve the problem. The Pythagorean Theorem can be represented mathematically as follows: a + b = c. If you're behind a web filter, please make sure that the domains .kastatic.org and .kasandbox.org are unblocked. If we know the two sides of a right triangle, then we can find the third side. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs, or a2 + b2 = c2 . We can find the diagonal length by using the Pythagorean Theorem. you have a right triangle where you. Use the "Hint" button to get a free letter if an answer is . Learning Objective: The lesson is aligned to the Common Core State Standards for Mathematics - 8.G.7 Geometry - Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. Find the length of the. The flat ends of the supports are 2 cm long. Apply the law of sines or trigonometry to find the right triangle side . Where a, b and c are the sides of the right triangle. moon conjunct mars reddit; stealthburner pcb; gm radio repair . This relationship is useful because if two sides of a right triangle are known, the Pythagorean theorem can be used to determine the length of the third side. Pythagorean Theorem: Finding the missing side DRAFT. Step 1: Let's identify our sides. You can find this side manually by performing a calculation with the hypotenuse formula which is also known as the Pythagorean theorem formula. A: If only one side length is known, we are unable to use the Pythagorean theorem. One of the more famous mathematical formulas is $$a^2+b^2=c^2$$, which is known as the Pythagorean Theorem.The theorem states that the hypotenuse of a right triangle can be easily calculated from the lengths of the sides. Solution Problem 8 : A ladder is 2 meters long. Next, solve for side a. In other words, you can check to see if a triangle is a right triangle by seeing if the Pythagorean Theorem is true. This is known as the Pythagorean equation, named after the ancient Greek thinker Pythagoras. The legs have length 24 and X are the legs. School subject: Math. Use the Pythagorean Theorem to find the missing side. The length of the missing side, c, which is the hypotenuse, is 50. In this eighth-grade geometry worksheet, Pythagorean Theorem: Find the Missing Hypotenuse, students will practice using the Pythagorean theorem to find missing hypotenuse lengths on right . 0.4 x. Step 1 Identify the legs and the hypotenuse of the right triangle . Use integers or fractions for any numbers in the expression Rationalize all denominators.) However, in a right triangle, we can use it to find the 3 rd side length of a triangle and then use trig functions (sine . Let's use the Pythagorean Theorem: a + b = c (5) + (20) = c 25 + 400 = c 425 = c sqrt (425) = c c = 20.6 . : Example #1 Suppose you are looking at a right triangle and the side opposite the right angle is missing. 2. The Pythagorean Theorem can be used when we know the length of two sides of a right triangle and we need to get the length of the third side. But do we call this leg {eq}a {/eq} or. We have a new and improved read on this topic. The Pythagorean Theorem states that for any right triangle with sides of length a and b, and hypotenuse of length c: a 2 + b 2 = c 2. By Pythagorean theorem problem solver Comentarios desactivados en Pythagorean theorem problem solver . Example 1: Find the length of the hypotenuse of a right triangle if the lengths of the other two sides are 3 inches and 4 inches. In the aforementioned equation, c is the length of the . 6 Conclusion. It is one of the most basic geometric tools in mathematics. We can use this theorem to find the height of our equilateral triangle! We can use this to find the missing side: Usually a theorem is of the form "If P then Q." The converse of the theorem, which may or may not be true, is "If Q then P." If both the theorem and its converse are true, we say "P if and only if Q" or "P precisely when Q." Start with the Pythagorean Theorem, a 2 + b 2 = c 2 , and subtract a 2 . right angle, there exists this relationship. Finally, use your knowledge that the angles of all triangles add up to 180 degrees to find angle C. How do you find the missing side of a right angled triangle? 6 12. Round to nearest tenth if needed. know your two legs and a hypotenuse where. 6 8. x 1. x 26. 0% average accuracy. 0 times. Click Create Assignment to assign this modality to your LMS. The Pythagorean theorem says that if. 5 Using Pythagorean Theorem worksheet. A simple equation, Pythagorean Theorem states that the square of the hypotenuse (the side opposite to the right angle triangle) is equal to the sum of the other two sides. The length of the horizontal leg is 5 units. Sample of problem solving skills in nursing how to write your common app essay, capstone project ideas stem. to your right angle and the hypotenuse. Finding the missing length of a side of a right triangle? Replace the variables in the theorem with the values of the known sides. Use the pythagorean theorem equation Substitute the variables where replaces and replaces Solve for and Subtract from , leaving us with: Find the square roots on both sides So, is equals to Video-Lesson Transcript In this lesson, we'll discuss the Pythagorean Theorem. Area and Perimeter of a Triangle from Mometrix Test Prep on Vimeo. 8. The triangle shown here is a right triangle. Troy is building a ramp for his toy cars. The Pythagorean theorem is a simple theorem that states that - for a right angled triangle the square of the length of the hypotenuse is equal to the sum of the squares of the length of the remaining two sides. We get 16 + 81 = c^2, which is then simplified to 97 = c^2. Also know, what is Pythagoras theorem example? The Pythagorean Theorem cannot be used by itself to find angles. Using this triangle, if a is 12 units long and c is 13 units long, use the Pythagorean Theorem to find the length of side b. If you're seeing this message, it means we're having trouble loading external resources on our website. (You can use a calcuator for the arithmetic if you want to.) The Pythagorean Theorem is a special property of right triangles that has been used since ancient times. Answer: The length of the . Assign tasks. We are given the length of the hypotenuse as {eq}c=13 {/eq}, and one leg length. c = (a + b) Given angle and hypotenuse. Video - Lesson & Examples. 48 2 + 14 2 = c2. Main content: Triangles. Find the length of the metal needed to make the 8 supports used to stabilise the table. The Pythagorean Theorem can be used when we know the length of two sides of a right triangle and we need to get the length of the third side. b = (c - a) for hypotenuse c missing, the formula is. Here, c represents the length of the hypotenuse (the longest side), while b and a are the lengths of the other two sides. Edit. The Pythagorean theorem can be applied in the following situations: We want to find the length of the hypotenuse and we have the lengths of the two legs. The Pythagorean Theorem relates the 3 side lengths a, b, and c of a right triangle (c is the hypotenuse, or longest side) by the equation a 2 + b 2 = c 2. Related Topics Other topics in Understand and apply the Pythagorean Theorem. Explanation: In order to find the missing side of a right triangle you must use one of two things: 1. Plus, unlike other online calculators, this calculator will show its work and draw the shape of the right triangle based on the results. 0. Then use the angle value and the sine rule to solve for angle B. Assign tasks. 24 in. Use Pythagorean's Theorem to find the missing side. How To: Find the area of a triangle when given 2 sides & angle ; How To: Identify characteristics of a sample during a survey ; How To: Divide small numbers by big numbers ; How To: Prove a triangle inscribed in a circle is right angled ; How To: Use ">" (greater than) and "<" (less than) symbols Pythagoras developed a formula to find the lengths of the sides of any right triangle.Pythagoras Discovered that if he treated each side of a right triangle as a square (see figure 1) the two smallest squares areas when added together equal the area of the larger square. To use Pythagoras theorem, remember the formula given below: c 2 = a 2 + b 2. We can compute the results using a 2 + b 2 + c 2 = distance 2 version of the theorem. So let plug in the numbers of b and c. a^2 + 5^2 = 8^2 So we have our values of b and c. Now, how do we find the missing number? We can use this theorem to find the height of our equilateral triangle! Therefore, we must first use our trigonometric ratios to find a second side length and then we can use the Pythagorean theorem to find our final missing side. Step 2 Substitute values into the formula (remember 'C' is the hypotenuse). The Pythagorean theorem is named after Pythagoras, a Greek mathematician who lived around 500 BC. It says that the area of the square whose side is the hypotenuse of the triangle is equal to the sum of the areas of the squares whose sides are the two legs of the triangle. Pythagorean's Theorem tells us the areas of the smaller two squares c ombined are equal to the area of the largest square . Solution : Step 1 : Find the length of each leg. Use the Pythagorean theorem to find the missing side of a triangle. 0.3 x. by nuth_p_30024. The calculator will then determine the length of the remaining side, the area and perimeter of the triangle, and all the angles of the triangle. I am trying to code a Pythagorean theorem calculator using HTML and Javascript so that it can find any of the sides given two of the sides value, so i am using if statements but it seem like i am unable to understand why it doesn't work. Author: 2019 honda civic lx turbo kit \| maui to big island volcano tour \| how to study economics for class 11 \| best gaming console under 20,000 Shares	2023-02-09 00:12:43	{"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": 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": 0.6127356290817261, "perplexity": 263.5113261845412}, "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/1674764500983.76/warc/CC-MAIN-20230208222635-20230209012635-00037.warc.gz"}
https://www.functori.com/docs/mitten/scenarios/pre-post-conditions.html	# Pre- and Post-conditions Assertions (and other code) are written in pre- and post-conditions of scenario steps. They can be added with the function add_spec: val add_spec : ?pre:(check list) -> ?post:(check_list) -> step -> step The pre-condition will be executed before propagating the message matched by this step, and the post-condition will be executed after propagating the message. Tip 💡 Note that, because a post-condition is executed after the message is propagated to its intended destination, there are no guarantees that it is already taken into account by the node. In particular, checking properties on the state of the destination node/baker after the propagation is most of the time meaningless.	2023-03-28 23:49:49	{"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": 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": 0.42435458302497864, "perplexity": 1523.9995471186432}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296948900.50/warc/CC-MAIN-20230328232645-20230329022645-00639.warc.gz"}
https://habr.com/ru/post/443958/	Мегапосты: # Disposable pattern (Disposable Design Principle) I guess almost any programmer who uses .NET will now say this pattern is a piece of cake. That it is the best-known pattern used on the platform. However, even the simplest and well-known problem domain will have secret areas which you have never looked at. So, let’s describe the whole thing from the beginning for the first-timers and all the rest (so that each of you could remember the basics). Don’t skip these paragraphs — I am watching you! If I ask what is IDisposable, you will surely say that it is public interface IDisposable { void Dispose(); } What is the purpose of the interface? I mean, why do we need to clear up memory at all if we have a smart Garbage Collector that clears the memory instead of us, so we even don’t have to think about it. However, there are some small details. This chapter was translated from Russian jointly by author and by professional translators. You can help us with translation from Russian or English into any other language, primarily into Chinese or German. Also, if you want thank us, the best way you can do that is to give us a star on github or to fork repository github/sidristij/dotnetbook. There is a misconception that IDisposable serves to release unmanaged resources. This is only partially true and to understand it, you just need to remember the examples of unmanaged resources. Is File class an unmanaged resource? No. Maybe DbContext is an unmanaged resource? No, again. An unmanaged resource is something that doesn’t belong to .NET type system. Something the platform didn’t create, something that exists out of its scope. A simple example is an opened file handle in an operating system. A handle is a number that uniquely identifies a file opened – no, not by you – by an operating system. That is, all control structures (e.g. the position of a file in a file system, file fragments in case of fragmentation and other service information, the numbers of a cylinder, a head or a sector of an HDD) are inside an OS but not .NET platform. The only unmanaged resource that is passed to .NET platform is IntPtr number. This number is wrapped by FileSafeHandle, which is in its turn wrapped by the File class. It means the File class is not an unmanaged resource on its own, but uses an additional layer in the form of IntPtr to include an unmanaged resource — the handle of an opened file. How do you read that file? Using a set of methods in WinAPI or Linux OS. Synchronization primitives in multithreaded or multiprocessor programs are the second example of unmanaged resources. Here belong data arrays that are passed through P/Invoke and also mutexes or semaphores. Note that OS doesn’t simply pass the handle of an unmanaged resource to an application. It also saves that handle in the table of handles opened by the process. Thus, OS can correctly close the resources after the application termination. This ensures the resources will be closed anyway after you exit the application. However, the running time of an application can be different which can cause long resource locking. Ok. Now we covered unmanaged resources. Why do we need to use IDisposable in these cases? Because .NET Framework has no idea what’s going on outside its territory. If you open a file using OS API, .NET will know nothing about it. If you allocate a memory range for your own needs (for example using VirtualAlloc), .NET will also know nothing. If it doesn’t know, it will not release the memory occupied by a VirtualAlloc call. Or, it will not close a file opened directly via an OS API call. These can cause different and unexpected consequences. You can get OutOfMemory if you allocate too much memory without releasing it (e.g. just by setting a pointer to null). Or, if you open a file on a file share through OS without closing it, you will lock the file on that file share for a long time. The file share example is especially good as the lock will remain on the IIS side even after you close a connection with a server. You don’t have rights to release the lock and you will have to ask administrators to perform iisreset or to close resource manually using special software. This problem on a remote server can become a complex task to solve. All these cases need a universal and familiar protocol for interaction between a type system and a programmer. It should clearly identify the types that require forced closing. The IDisposable interface serves exactly this purpose. It functions the following way: if a type contains the implementation of the IDisposable interface, you must call Dispose() after you finish work with an instance of that type. So, there are two standard ways to call it. Usually you create an entity instance to use it quickly within one method or within the lifetime of the entity instance. The first way is to wrap an instance into using(...){ ... }. It means you instruct to destroy an object after the using-related block is over, i.e. to call Dispose(). The second way is to destroy the object, when its lifetime is over, with a reference to the object we want to release. But .NET has nothing but a finalization method that implies automatic destruction of an object, right? However, finalization is not suitable at all as we don’t know when it will be called. Meanwhile, we need to release an object at a certain time, for example just after we finish work with an opened file. That is why we also need to implement IDisposable and call Dispose to release all resources we owned. Thus, we follow the protocol, and it is very important. Because if somebody follows it, all the participants should do the same to avoid problems. ## Different ways to implement IDisposable Let’s look at the implementations of IDisposable from simple to complicated. The first and the simplest is to use IDisposable as it is: public class ResourceHolder : IDisposable { DisposableResource _anotherResource = new DisposableResource(); public void Dispose() { _anotherResource.Dispose(); } } Here, we create an instance of a resource that is further released by Dispose(). The only thing that makes this implementation inconsistent is that you still can work with the instance after its destruction by Dispose(): public class ResourceHolder : IDisposable { private DisposableResource _anotherResource = new DisposableResource(); private bool _disposed; public void Dispose() { if(_disposed) return; _anotherResource.Dispose(); _disposed = true; } [MethodImpl(MethodImplOptions.AggressiveInlining)] private void CheckDisposed() { if(_disposed) { throw new ObjectDisposedException(); } } } CheckDisposed() must be called as a first expression in all public methods of a class. The obtained ResourceHolder class structure looks good to destroy an unmanaged resource, which is DisposableResource. However, this structure is not suitable for a wrapped-in unmanaged resource. Let’s look at the example with an unmanaged resource. public class FileWrapper : IDisposable { IntPtr _handle; public FileWrapper(string name) { _handle = CreateFile(name, 0, 0, 0, 0, 0, IntPtr.Zero); } public void Dispose() { CloseHandle(_handle); } [DllImport("kernel32.dll", EntryPoint = "CreateFile", SetLastError = true)] private static extern IntPtr CreateFile(String lpFileName, UInt32 dwDesiredAccess, UInt32 dwShareMode, IntPtr lpSecurityAttributes, UInt32 dwCreationDisposition, UInt32 dwFlagsAndAttributes, IntPtr hTemplateFile); [DllImport("kernel32.dll", SetLastError=true)] private static extern bool CloseHandle(IntPtr hObject); } What is the difference in the behavior of the last two examples? The first one describes the interaction of two managed resources. This means that if a program works correctly, the resource will be released anyway. Since DisposableResource is managed, .NET CLR knows about it and will release the memory from it if its behaviour is incorrect. Note that I consciously don’t assume what DisposableResource type encapsulates. There can be any kind of logic and structure. It can contain both managed and unmanaged resources. This shouldn't concern us at all. Nobody asks us to decompile third party’s libraries each time and see whether they use managed or unmanaged resources. And if our type uses an unmanaged resource, we cannot be unaware of this. We do this in FileWrapper class. So, what happens in this case? If we use unmanaged resources, we have two scenarios. The first one is when everything is OK and Dispose is called. The second one is when something goes wrong and Dispose failed. Let’s say straight away why this may go wrong: • If we use using(obj) { ... }, an exception may appear in an inner block of code. This exception is caught by finally block, which we cannot see (this is syntactic sugar of C#). This block calls Dispose implicitly. However, there are cases when this doesn’t happen. For example, neither catch nor finally catch StackOverflowException. You should always remember this. Because if some thread becomes recursive and StackOverflowException occurs at some point, .NET will forget about the resources that it used but not released. It doesn’t know how to release unmanaged resources. They will stay in memory until OS releases them, i.e. when you exit a program, or even some time after the termination of an application. • If we call Dispose() from another Dispose(). Again, we may happen to fail to get to it. This is not the case of an absent-minded app developer, who forgot to call Dispose(). It is the question of exceptions. However, these are not only the exceptions that crash a thread of an application. Here we talk about all exceptions that will prevent an algorithm from calling an external Dispose() that will call our Dispose(). All these cases will create suspended unmanaged resources. That is because Garbage Collector doesn’t know it should collect them. All it can do upon next check is to discover that the last reference to an object graph with our FileWrapper type is lost. In this case, the memory will be reallocated for objects with references. How can we prevent it? We must implement the finalizer of an object. The 'finalizer' is named this way on purpose. It is not a destructor as it may seem because of similar ways to call finalizers in C# and destructors in C++. The difference is that a finalizer will be called anyway, contrary to a destructor (as well as Dispose()). A finalizer is called when Garbage Collection is initiated (now it is enough to know this, but things are a bit more complicated). It is used for a guaranteed release of resources if something goes wrong. We must implement a finalizer to release unmanaged resources. Again, because a finalizer is called when GC is initiated, we don’t know when this happens in general. Let’s expand our code: public class FileWrapper : IDisposable { IntPtr _handle; public FileWrapper(string name) { _handle = CreateFile(name, 0, 0, 0, 0, 0, IntPtr.Zero); } public void Dispose() { InternalDispose(); GC.SuppressFinalize(this); } private void InternalDispose() { CloseHandle(_handle); } ~FileWrapper() { InternalDispose(); } /// other methods } We enhanced the example with the knowledge about the finalization process and secured the application against losing resource information if Dispose() is not called. We also called GC.SuppressFinalize to disable the finalization of the instance of the type if Dispose() is successfully called. There is no need to release the same resource twice, right? Thus, we also reduce the finalization queue by letting go a random region of code that is likely to run with finalization in parallel, some time later. Now, let’s enhance the example even more. public class FileWrapper : IDisposable { IntPtr _handle; bool _disposed; public FileWrapper(string name) { _handle = CreateFile(name, 0, 0, 0, 0, 0, IntPtr.Zero); } public void Dispose() { if(_disposed) return; _disposed = true; InternalDispose(); GC.SuppressFinalize(this); } [MethodImpl(MethodImplOptions.AggressiveInlining)] private void CheckDisposed() { if(_disposed) { throw new ObjectDisposedException(); } } private void InternalDispose() { CloseHandle(_handle); } ~FileWrapper() { InternalDispose(); } /// other methods } Now our example of a type that encapsulates an unmanaged resource looks complete. Unfortunately, the second Dispose() is in fact a standard of the platform and we allow to call it. Note that people often allow the second call of Dispose() to avoid problems with a calling code and this is wrong. However, a user of your library who looks at MS documentation may not think so and will allow multiple calls of Dispose(). Calling other public methods will destroy the integrity of an object anyway. If we destroyed the object, we cannot work with it anymore. This means we must call CheckDisposed at the beginning of each public method. However, this code contains a severe problem that prevents it from working as we intended. If we remember how garbage collection works, we will notice one feature. When collecting garbage, GC primarily finalizes everything inherited directly from Object. Next it deals with objects that implement CriticalFinalizerObject. This becomes a problem as both classes that we designed inherit Object. We don’t know in which order they will come to the “last mile”. However, a higher-level object can use its finalizer to finalize an object with an unmanaged resource. Although, this doesn’t sound like a great idea. The order of finalization would be very helpful here. To set it, the lower-level type with an encapsulated unmanaged resource must be inherited from CriticalFinalizerObject. The second reason is more profound. Imagine that you dared to write an application that doesn’t take much care of memory. It allocates memory in huge quantities, without cashing and other subtleties. One day this application will crash with OutOfMemoryException. When it occurs, code runs specifically. It cannot allocate anything, since it will lead to a repeated exception, even if the first one is caught. This doesn’t mean we shouldn’t create new instances of objects. Even a simple method call can throw this exception, e.g. that of finalization. I remind you that methods are compiled when you call them for the first time. This is usual behavior. How can we prevent this problem? Quite easily. If your object is inherited from CriticalFinalizerObject, then all methods of this type will be compiled straight away upon loading it in memory. Moreover, if you mark methods with [PrePrepareMethod] attribute, they will be also pre-compiled and will be secure to call in a low resource situation. Why is that important? Why spend too much effort on those that pass away? Because unmanaged resources can be suspended in a system for long. Even after you restart a computer. If a user opens a file from a file share in your application, the former will be locked by a remote host and released on the timeout or when you release a resource by closing the file. If your application crashes when the file is opened, it won't be released even after reboot. You will have to wait long until the remote host releases it. Also, you shouldn’t allow exceptions in finalizers. This leads to an accelerated crash of the CLR and of an application as you cannot wrap the call of a finalizer in try… catch. I mean, when you try to release a resource, you must be sure it can be released. The last but not less important fact: if the CLR unloads a domain abnormally, the finalizers of types, derived from CriticalFinalizerObject will be also called, unlike those inherited directly from Object. This charper translated from Russian as from language of author by professional translators. You can help us with creating translated version of this text to any other language including Chinese or German using Russian and English versions of text as source. Also, if you want to say «thank you», the best way you can choose is giving us a star on github or forking repository https://github.com/sidristij/dotnetbook Поддержать автора Поделиться публикацией ## Комментарии 0 Только полноправные пользователи могут оставлять комментарии. Войдите, пожалуйста.	2019-04-24 23:55:06	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.18300525844097137, "perplexity": 1695.5160973481181}, "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-2019-18/segments/1555578675477.84/warc/CC-MAIN-20190424234327-20190425020327-00096.warc.gz"}
https://annoyingpi.wordpress.com/tag/arml/	# Rambling Thoughts ## Random thoughts from 2012 ARML Last weekend was 2012 ARML. It was my fifth ARML, and third as coach of the SFBA teams. I remember my first trip to ARML, with just a single team. This year, ARML has become a 7-team, 4-day expedition. This was also one of my most entertaining trips to ARML. Watching SFBA troll the Friday evening talent show was amazing. As for results, SFBA’s teams did amazingly well: A1 and A2 were 3rd and 7th nationally, the first time that any organization has had two teams in the top 7 at ARML. Next year, we’re aiming for 1st and 2nd on the national scoreboard. Some random thoughts from ARML: • Tie-dye was an amazing shirt color. Pink was pretty cool too. • I should try to avoid losing my bed next time. • Flip, Flop, Fliegerspiel • Bif! Bif! Bif! • I am Chen! • Yee! ## ARML Sorry about not posting for the past few months. I’ve been very busy recently. With the start of the summer, I should be able to post much more frequently. I have been a coach for the San Francisco Bay Area ARML team for the past two years. Every year, people ask me why I help with ARML, and the answer is always the same: because it’s fun. I enjoyed my first trip to ARML in 2008, and ARML has gotten better every year. Unlike most contests, ARML is heavily team-based, and as such, it is a great way to meet interesting people. ARML tends to build a sense of community that is lacking from most math events. Over the past few years, I’ve met many students and coaches at ARML, and they are a major reason why I go back to ARML every year. Another big part of coaching ARML are the practices that we run every spring. It’s always great to see the gathering of people with similar mathematical interests. I believe in teaching math as a way to learn and understand something more thoroughly, and I hope to do more teaching at ARML practices in future years. Doing math is often most enjoyable as a social activity, and I think that our practices fully demonstrate that feature. Here’s a list of interesting things that happened at ARML, in no particular order: • Green sharpies • Easy button thieves • A disappearing sword • Brilliant power round solutions • Paper balls and paper airplanes • Math! ## ARML ARML (American Regions Mathematics League) is an annual math contest that takes place every year at four sites around the country. It’s one of my favorite math contests. ARML is different from most other math contests because it requires teams to travel to one of four central sites. This means that teams have to travel (some times for very long distances) to participate. Inevitably, this means that participation at ARML is not especially high; only 124 teams of fifteen people (in 2010), or less than one percent of the more than 200000 people who participated in the AMC. Teams that do participate at ARML, however, are much more organized; most teams organize several practices before the contest, while some teams prepare for ARML year-round. The team aspect of the contest (3 out of 4 rounds have a team component, together constituting 50% of the total team score) also makes ARML stand out. Indeed, ARML is primarily a team contest, and the team results every year are watched much more closely than the individual high scorers. As a result, ARML is much more social than most other math contests, giving participants a chance to work closely with their team and to meet people from other teams. This social aspect of ARML is what makes ARML enjoyable. Mathematically, ARML feels different from most other contests. The ARML style is to have problems that look hard but are made much easier through clever tricks. Often, these problems can be solved without being clever, but cleverness can often yield very simple solutions. As an example, consider this year’s coffee mug problem: [EDIT (17 September 2010): Updated with the original wording.] Let $P(x) = x^2 + 2010 x + 2010$, and let $r$ and $s$ be the roots of $P$. If $Q$ is a quadratic polynomial with leading coefficient 1 and roots $r + 1$ and $s + 1$, compute the sum of the coefficients of $Q(x)$. This can be bashed out without the quadratic formula or with Vieta’s formulas, but the clever solution is much simpler: Because $Q$ is monic, $Q(x) = (x - (r+1))(x - (s+1))$ $= ((x - 1) - r)((x - 1) - s) = P(x-1)$. The sum of the coefficients of $Q(x)$ is $Q(1) = P(0) = 2010$. Together with the clever tricks, ARML imposes a short time limit on its tests, making it difficult for contestants to finish the team and individual tests. The relay round’s three-minute and six-minute time limits provides a further emphasis on speed. As a result, ARML problems (with the exception of the proof-based Power Question) are usually not very hard mathematically; they’re just hard to do under severe time pressure. This short time limit, along with the small number of questions on ARML, make the results of the contest subject to a lot of random noise. The top teams are usually within a few points (out of 300 points total) of each other, so a five-point team round problem could easily change a team’s ranking by several places. The individual round has only ten problems, so achieving a national high score (usually at least eight problems) requires making no computational errors. Thus, the list of individual high scorers is somewhat meaningless; high scorers often do not repeat as high scorers when they return, and high individual scorers are often as much a result of luck as of mathematical ability. Evaluating ARML based on the criteria that I listed in my post on Contest Math, we obtain mixed results. ARML, perhaps more than any other contest, highlights the social side of mathematics and hooks people into liking and doing math; though it’s hard to start doing ARML, it’s even harder to stop. Mathematically, though ARML does reinforce the importance of creativity, it does not give particularly challenging problems, and ARML results do not really constitute a good measure of mathematical achievement. I enjoy ARML primarily based on the social part of the contest, and many people agree with me; the highlight of ARML for some people is the long bus ride to the contest. Indeed, I think of ARML primarily as a social event, with some mathematics to make it look like the participants are actually doing something important. ## Contest Math Many people first become interested in mathematics through contest math in middle school or high school. Others, however, progress to higher level mathematics without participating in the contests at all. What, then, is the role of math competitions in attracting people to mathematics? Is contest math beneficial for doing math beyond competitions? This, of course, depends on what we mean by “contest math”. Of course, a competition to produce research results would certainly be different from a mental arithmetic contest. For now, I define contest math to be in between these two extremes; the problems are not straightforward, but they are not at the difficulty of research questions. In particular, I classify Mathcounts, ARML, AMC, USAMO, and Putnam in the category of contest math. One positive effect of contest math is that it tricks students into liking and doing math at an early age. The mathematics education system in America is not conducive to positive impressions of mathematics, and contest math provides an alternative world where problem solving ability is emphasized above rote computation. For example, the Mathcounts contest hooks many students in middle school, providing an introduction to math and leading them to a continued interest in mathematics. Contest math hooks students partially through its competitive nature. Unlike the typical mathematics classroom, contest math provides a challenge. There are always harder problems and puzzles to consider, and students never run out of room to explore. The feeling of solving a difficult problem is always exhilarating, and pushes the student to do more problems. Furthermore, the contest math community is full of students who are always competing to be the best. The rankings at each math contest push students to work harder than they would if left alone, and the competitive spirit injects energy and excitement into mathematics. Along with competition, contest math brings together a group of people who are interested in the same thing: Solving math problems. This gives contest math a social aspect as well, challenging the conventional notion that doing math and being social are mutually exclusive activities. A good example of this phenomenon is the ARML contest. In many regions of the country, ARML brings together the best students and gives them an opportunity to get to know each other through practices and an annual trip to the national competition. This social aspect of contest math even extends to the Art of Problem Solving forum, for example. After contest math hooks its victims, it also indoctrinates them in good mathematical principles. In particular, since contest math values ingenuity over knowledge, it ends up very a good teacher of problem solving. This is important because the ability to think about and solve problems is the key to success in higher level mathematics. The best mathematicians are the ones who can make connections and devise clever arguments. In this way, through the bait of math competitions, contest math pulls students toward real mathematics. Despite the positive effect of producing more mathematically-minded humans, contest math has many deficiencies. In particular, contest math is not representative of how mathematics is actually done. In reality, mathematical research requires much more knowledge than Euclidean geometry, and pure ingenuity often isn’t sufficient. Also, math competitions teach students to expect solutions to their problems within minutes or hours. Though this is much better than the seconds-long attention spans of many high school math students, it is far from the days, months, or years that are needed to solve difficult research problems in mathematics. Thus, in some sense, contest math forms a different (but related) subject from mathematics, and by doing contest math exclusively, students sometimes lose sight of what mathematics is actually about. Thus, contest math taken to excess is almost never a good thing. By overtraining themselves for math competitions, students do nothing to improve their mathematical ability. Some students even become resistant to the idea of doing any math beyond the contests that they’ve focused on for years, rendering their mathematical talent useless. This is especially true for more computationally oriented contests such as Mathcounts. Though they provide good introductions to math, they are too far from actual mathematics to be meaningful for any more than a year or two. In this way, contest math can often be harmful for the future mathematician. In the end, we see that contest mathematics is good for providing an introduction to the world of mathematics to those who haven’t seen it before. However, it cannot replace mathematics, and eventually, math competitions must lead into higher level mathematics. Even among math contests, some are more meaningful than others; not all math contests are equal. But that’s a topic for a future post.	2018-09-24 00:39:30	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 13, "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": 0.44641634821891785, "perplexity": 1337.382092984318}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267159938.71/warc/CC-MAIN-20180923232129-20180924012529-00425.warc.gz"}
http://mathonline.wikidot.com/cauchy-s-condition-for-convergent-series	Cauchy's Condition for Convergent Series # Cauchy's Condition for Convergent Series One very important result regarding convergent series of real numbers is called Cauchy's condition which we state and prove below. Theorem 1 (Cauchy's Condition for Convergent Series): The series $\sum_{n=1}^{\infty} a_n$ converges if and only if for all $\epsilon > 0$ there exists an $N \in \mathbb{N}$ such that if $n \geq N$ then $\displaystyle{\mid a_{n+1} + a_{n+2} + ... + a_{n+p} \mid < \epsilon}$ for each $p = 1, 2, ...$. • Proof: $\Rightarrow$ Suppose that $\sum_{n=1}^{\infty} a_n$ converges to $s$. Then the sequence of partial sums for this series, $(s_n(x))_{n=1}^{\infty}$ converges to $s$. But every convergent sequence is Cauchy, and so for all $\epsilon > 0$ there exists an $N \in \mathbb{N}$ such that if $m, n \geq N$ then: (1) \begin{align} \quad \mid s_m - s_n \mid < \epsilon \end{align} • So so $n$ and let $m = n + p$ for $p = 1, 2, ...$. Then $m, n \geq N$ and: (2) \begin{align} \quad \mid s_{n +p} - s_n \mid = \biggr \lvert \sum_{k=1}^{n+p} a_k - \sum_{k=1}^{n} a_k \biggr \rvert = \mid a_{n+1} + a_{n+2} + ... + a_{n+p} \mid < \epsilon \end{align} • $\Leftarrow$ Suppose that for all $\epsilon > 0$ there exists an $N \in \mathbb{N}$ such that if $n \geq N$ then $\displaystyle{ \mid a_{n+1} + a_{n+2} + ... + a_{n+p} \mid < \epsilon}$ for each $p = 1, 2, ...$. Choose this same $N$, and let $n, m \geq N$ where $m(p) = n + p$ for $p = 1, 2, ...$. Then for all $m, n \geq N$ we see that: (3) \begin{align} \quad \mid a_{n+1} + a_{n+2} + ... + a_{n+p} \mid = \mid s_m - s_n \mid < \epsilon \end{align} • So the sequence of partial sums $(s_n(x))_{n=1}^{\infty}$ is Cauchy. But this is a series of real numbers, so the corresponding sequence of partial sums are real numbers and $(\mathbb{R}, d)$ is a complete metric space so $(s_n(x))_{n=1}^{\infty}$ converges in $\mathbb{R}$ which implies that the series $\displaystyle{\sum_{k=1}^{\infty} a_n}$ converges. $\blacksquare$	2017-12-11 19:13:49	{"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": 3, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9996520280838013, "perplexity": 171.686031958575}, "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-2017-51/segments/1512948513866.9/warc/CC-MAIN-20171211183649-20171211203649-00644.warc.gz"}
https://mathematica.stackexchange.com/questions/187538/solution-2nd-transcendental-equation-with-graphical-method/187548	# Solution… 2nd Transcendental Equation with Graphical method [closed] If $$2x^3 + \ln x = 5$$, then what is $$x$$? For beginning we started to find a solution for equation \begin{align}\label{eq:eq} \ln x + cx = b \tag{1} \end{align} We know from W function that $$ax\mathrm e^{ax} = y \Rightarrow ax = W(k, y)$$, where $$k \in \mathbb Z$$. From Eq. \ref{eq:eq} \begin{align} \Rightarrow \ln(ax) + ax &= \ln y \\ \Rightarrow \ln x + x &= \ln(y/a) \label{eq:eq2}\tag{2} \end{align} From Eqs. \ref{eq:eq} and \ref{eq:eq2} =>a=c & Log(y/a)=b.Then y=ae^(b) & x=1/aW(k,ae^(b))..k in Z.(3).If now we take the original equation Log x+2x^3=5 we do the transformation …x^3=z (5) =>3Log(x)=Log(z)+2kπi =>x=z^(1/3)e^(2k’πi/3)..k’ in Z (6) Βut with the transformation(5) the relation (1) is done Log(z)+6z=15.. (7).But the relation (7) has solution in accordance with the foregoing z=1/6W(k,6e^(15))..(8) ,k in Z.From (6&8) we have the filnal solution => x=(1/6W(k,6e^(15)))^(1/3)e^(2k’πi/3) ,k&k’ in Z. The solutions are 3 only … 1…with k=0 & k’=0,…,x=1.33084 ,,,Real and we have 2 complex roots 2.. for k=1 & k’=2 =>x=-0.520715 - 1.26144 I and 3.. for k=-1 & k’=-2 =>x=-0.520715 + 1.26144 I ## closed as off-topic by corey979, Michael E2, eyorble, Bob Hanlon, Αλέξανδρος ΖεγγDec 9 '18 at 3:04 • The question does not concern the technical computing software Mathematica by Wolfram Research. Please see the help center to find out about the topics that can be asked here. If this question can be reworded to fit the rules in the help center, please edit the question. • Τhanks Αλέξανδρος for formatting!!! – Nikos Mantzakouras Dec 9 '18 at 6:47 Solve[2 x^3 + Log[x] == 5, x] ({{x -> (1/6 ProductLog[6 E^15])^(1/3)}}*) What you cal "W-function" is Mathematica ProductLog! • Yes ,,, you are right, it is simply called and so...Τhanks ..Sure it has and complex Roots!!! – Nikos Mantzakouras Dec 8 '18 at 18:06 • Graphical Method........in Mathematica!!! f[x_] = 2 x^3 + Log[x] - 5; Normal[Plot[f[x], {x, 0, 2}, PlotPoints -> 100, MeshFunctions -> {#2 &}, Mesh -> {{0}}, MeshStyle -> Directive[Red, PointSize[Large]]]] /. p_Point :> {p, Text[Style[p[[1, 1]], 14], {0, 5} + p[[1]]]} – Nikos Mantzakouras Dec 8 '18 at 18:23 • The complex roots is ONLY 3 because for any case 0<=k’<=2!!! – Nikos Mantzakouras Dec 8 '18 at 18:44 • W-function is ProductLog() – Nikos Mantzakouras Dec 8 '18 at 18:47 FindInstance can locate all three roots: FindInstance[2 x^3 + Log[x] == 5, x, 3] // N {{x -> -0.520715 - 1.26144 I}, {x -> 1.33084}, {x -> -0.520715 + 1.26144 I}} Alternatively, FindRoot can find all three, though you have to help it with different starting points: FindRoot[2 x^3 + Log[x] == 5, {x, #}] & /@ {1, -1 + I, -1 - I} {{x -> 1.33084}, {x -> -0.520715 + 1.26144 I}, {x -> -0.520715 - 1.26144 I}} • Yes it is correct!!! – Nikos Mantzakouras Dec 8 '18 at 18:34 Solve or NSolve will give all three roots if you constrain the Abs value of x Solve[{2 x^3 + Log[x] == 5, Abs[x] < 2}, x] {{x -> Root[{-5 + Log[#1] + 2 #1^3 &, \ -0.52071466111311432025020844963466209285486182967411316474958 - 1.26143852926442301838345194738901835961144796731648927183682 I}]}, {x \ -> Root[{-5 + Log[#1] + 2 #1^3 &, \ -0.52071466111311432025020844963466209285486182967411316474958 + 1.26143852926442301838345194738901835961144796731648927183682 I}]}, {x \ -> Root[{-5 + Log[#1] + 2 #1^3 &, 1.33083954213436297147435824439}]}} • You dont know Abs[x]<2,,FROM WHERE? – Nikos Mantzakouras Dec 10 '18 at 9:58	2019-06-19 14:10:07	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 6, "wp-katex-eq": 0, "align": 2, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5393221974372864, "perplexity": 8593.228000906041}, "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/1560627998986.11/warc/CC-MAIN-20190619123854-20190619145854-00241.warc.gz"}
https://mcejp.github.io/2020/09/27/cs4954.html	The CS4954 is a long-obsolete NTSC/PAL Digital Video Encoder by Cirrus Logic. Despite being out of production, it is a nice chip, and is being used on the CLONE BUINO DUAL to drive the composite video output. CS4954 block diagram, courtesy of Cirrus Logic. Out of its many inputs and outputs, we only care about a few, highlighted in bold. ## Hello world? To get video output on the composite (CVBS) port with minimum effort, these are the steps to take: • Apply power to the chip • Provide 27MHz reference clock • Via I2C, change the following control flags (preserve the defaults for other flags and other registers): • CONTROL_0: Set MSTR=1 (Master Mode) • CONTROL_4: Set COMDAC_PD=0 (composite DAC: power up) • CONTROL_5: Set EN_COM=1, LOW_IMP=1 (enable composite DAC, enable composite output) • CONTROL_3: Set CBAR=1 (enable internal color bar generator) • Note that the default output format is NTSC-M ITU-R.BT601. If this is not what you want, you can consult Table 4 of the datasheet which describes register configuration for other formats. Tip: you might want to make use of cs4954.h. That should be enough to get you the familiar color bars: Here captured through a USB digitizer dongle Another thing to try is to set CBAR=0 and then varying the BKG_COLOR register. You should get a screen filled with a solid color that changes accordingly. The BKG_COLOR register is encoded as a 3:3:2-bit RGB value, no need to worry about any YCbCr bullshit. Thanks to Master Mode, we don’t have to worry about providing any synchronization signals – everything is taken care of internally. ## Next steps Color bars are a nice start, but what if we want to display our own picture? For this, we need to first gain understanding of the input format(s) understood by the video encoder. In fact, with the CS4954, we have 2 choices of input encoding: ### ITU-R BT.601 (spec) In this mode, the inputs go in as 8-bit words in the pattern Cb–Y1–Cr–Y2. This pattern repeats 360 times during the visible part of a scanline, yielding 720 pixels of luminance resolution and 360 pixels of chrominance resolution. This is also known as 4:2:2 chroma subsampling. Conversion between the RGB and YCbCr color spaces is conceptually simple, but requires matrix multiplication, which is not a problem on PC, but can be tricky or expensive in embedded systems. The coefficients can be found here. Note the “interesting” ranges of values: 16–235 for Y, 16-240 for Cb and Cr. Legend says that these have something to do with analog representation of the signals and under/overshoot concerns. The CS4954 gives us a choice of a Master or a Slave mode. In Master mode, the chip generates HSYNC & VSYNC signals and provides them as digital outputs; conversely, to use Slave Mode, HSYNC and VSYNC must be fed to the video encoder along with pixel data. #### Timing In NTSC (interlaced) mode, a frame consists of 2 fields. You get 29.97 frames, so 59.94 fields per second. One frame consists of: 3 blank lines ( 1.. 3) 3 VSYNC lines ( 4.. 6) 15 blank lines ( 7.. 21) 240 picture lines ( 22..261) 4.5 blank lines (262, 263, 264, 265 + 1st half of 266) 3 VSYNC lines (middle of 266 .. middle of 269) 15.5 blank lines (2nd half of 269 + 270..284) 240 picture lines (285..524) 1 blank line (525) The numbers in brackets are 1-based and inclusive. As for horizontal timing: 128 HSYNC cycles ( 1.. 128) 116 blank cycles ( 129.. 244) 1440 pixel cycles ( 245..1684) encoding 720 horizontal pixels 32 blank cycles (1685..1716) A quick verification: 1716 * 525 * (30 / 1.001) = 27,000,000 cycles per second, which is exactly the reference frequency used by the CS4954. ### ITU-R BT.656 (spec) The BT.656 spec is freely available, but I found the wording just a little bit ambiguous. On the other hand, the application note from Intersil (mirrored here) is crystal clear and presents data in a format useful for encode implementation. BT.656 encodes pixel data in the same way as BT.601, but it also embeds synchronization in the data stream instead of requiring additional discrete signals. The vertical timing becomes much simpler, with no silly half-lines: 21 blank lines ( 1.. 21) 240 picture lines ( 22..261) 23 blank lines (262..284) 240 picture lines (285..524) 1 blank line (525) As for horizontal timing: 4 EAV cycles [FF, 00, 00, XY] 268 dummy cycles [80, 10, ...] 4 SAV cycles [FF, 00, 00, XY] 1440 pixel cycles [Cb, Y1, Cr, Y2] yielding the same grand total of 1716 cycles/line. The XY word is a bit more involved: from MSB to LSB its 8 bits read as 1FVHPPPP. Fortunately, we can pretty much ignore the meaning of the individual bits, thanks to this handy decoding table: Line number, 1-based, inclusive F V H (EAV) H (SAV) 1-3 1 1 1 0 4-20 1 0 1 1 0 21-263 0 0 1 0 264-265 0 1 1 0 266-282 1 1 1 0 283-525 1 0 1 0 Sanity-preserving table courtesy of Intersil engineers And what about PPPP? These bits encode 4-bit parity which allows correction of any single-bit error no one seems to care about, at least for inter-IC applications. Again, you can find more details in the Intersil application note. Since BT.656 mode needs 2 pins fewer than BT.601, it should be the more convenient way forward. This background should be enough to generate a valid BT.656 stream from our FPGA. Next time. ## Open questions 1. Is there a drop-in replacement chip with better availability? ## Footnotes 1. Beware that the BT.601 spec says that the odd-field vertical sync flag should last up to and including line 19, not 20. In practice, this difference does not seem to cause any issues; in fact, it seems that every revision of the standard changes this number somewhat randomly.	2022-06-29 03:45:31	{"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": 0.45237550139427185, "perplexity": 4639.1439740059495}, "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/1656103620968.33/warc/CC-MAIN-20220629024217-20220629054217-00053.warc.gz"}
https://tex.stackexchange.com/questions/237176/patching-preventing-effective-code-in-a-package	# Patching/Preventing effective code in a package Consider a command being defined in a package (or used from another package), say \somethingstupid and being used before \endinput within the package, doing some non-typesetting setup (coloring, setting lengths, counters etc. or even defining other commands...). Sometimes this code is annoying and should not be used at all. Is there a way to patch the package (not only the command) to prevent this effective code? Of course the 'annoying' part of the command can be patched but this can be complicated. Here is a MWE dummypackage.sty \ProvidesPackage{dummypackage} \RequirePackage{xcolor} \newcommand{\somethingstupid}[1]{% %%% A lot of non-typesetting code before \color{#1} % 'stupid' too %%% A lot of non-typesetting code after } %%% Other stuff \somethingstupid{blue} % Should not be here for some reasons %%% Other stuff \endinput And some driver .tex file \documentclass{article} \usepackage{dummypackage} \usepackage{blindtext} \begin{document} \blindtext \end{document} • Sorry, not in a general way. You could patch \newcommand to trap the offending command(s) (very fragile, of course), but there are too many possible variants: some command can even not being explicitly defined. – egreg Apr 6 '15 at 20:13 • @egreg: I feared this :-( – user31729 Apr 6 '15 at 20:14 • Knowing what action \somethingstupid does, you can undo it as soon as the package is loaded. Or try patching \@begindocumenthook if the code is delayed (difficult, though). – egreg Apr 6 '15 at 20:16 • Could you patch \somethingstupid to save a copy of itself and then redefine itself to innocuous behavior. And then, add an \AtBeginDocument macro to reinstate the original definition? – Steven B. Segletes Apr 6 '15 at 20:16 • @egreg: The reason why I asked is the answer I gave to this problem: tex.stackexchange.com/questions/237134/… (which is basically what you proposed: Redefining/reusing/redoing code) – user31729 Apr 6 '15 at 20:19 This puts Ignoring \somethingstupid in the log and doesn't execute the package definition of \somethingstupid \documentclass{article} \def\zsomethingstupid{\somethingstupid} \def\newcommand#1{% \def\ztmp{#1}% \ifx\ztmp\zsomethingstupid \expandafter\zz \else \expandafter \znewcommand \fi #1} \def\zz#1[#2]#3{\def#1##1{\typeout{Ignoring \string#1}}} \usepackage{dummypackage} \usepackage{blindtext} \begin{document} \blindtext \end{document} • A clever way ;-) – user31729 Apr 6 '15 at 20:23 • Hmm, any currently undefined command will be the same as \somethingstupid (before it's defined), as far as \ifx is concerned. – egreg Apr 6 '15 at 20:25 • @egreg It passed all the supplied tests:-) (I'll edit) – David Carlisle Apr 6 '15 at 20:27 • @egreg better..? – David Carlisle Apr 6 '15 at 20:29 • that would be OK, #1 would be * so it'll come back to \znewcommand I'm assuming the OP knows the form used in the command being trapped. – David Carlisle Apr 6 '15 at 20:32	2020-02-24 15:41:02	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8141050934791565, "perplexity": 4828.859081480049}, "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/1581875145960.92/warc/CC-MAIN-20200224132646-20200224162646-00465.warc.gz"}
https://steverxd.github.io/Stat_tests/correlations.html	# Chapter 4 Correlations Correlation is a measure of the strength and direction of association that exists between two variables. Correlation coefficients ($$r$$) assume values in the range from −1 to +1, where ±1 indicates the strongest possible positive or negative correlation and 0 indicates no linear association between the variables. ## 4.1 Pearson correlation We start by looking at the Pearson correlation coefficient. Here we compare the built-in test (in R) for the Pearson correlation with the equivalent linear model. The equivalent linear model is the basic regression of $$y$$ on $$x$$, specified as follows: $$y = \beta_0 + \beta_1x \qquad H_0: \beta_1 = 0$$ The two tests are written using the following R code: # Pearson (built-in test) cor.test(y, x, method = "pearson") # Linear model lm(y ~ 1 + x) %>% summary() %>% print(digits = 8) If you were to run the code above, you would see the following the key statistics found in the output of each test: Table 4.1: Pearson vs linear model Test r slope t p-value cor.test (Pearson) -0.2318 -1.6507 0.1053 lm -0.4636 -1.6507 0.1053 The output shows that the correlation coefficient ($$r$$) has a p-value of 0.1053, which is exactly the same as the p-value for the slope of the linear model. In this case, we would not reject the null hypothesis that there was no correlation between the two variables (at the 0.05 level of significance). The main difference is that the linear model returns the slope of the relationship, $$\beta_1$$ (which in this case is -0.4636), rather than the correlation coefficient, $$r$$. The slope is usually much more interpretable and informative than the correlation coefficient. It may be useful to understand how the Pearson correlation coefficient ($$r$$) and the regression coefficient or slope ($$\beta_1$$) are related, which is by the following formula: $$\beta_1 = r \cdot sd_y / sd_x$$ This shows that: • When both x and y have the same standard deviations ($$sd_x$$ and $$sd_y$$) then the slope ($$\beta_1$$) will be equal to the correlation coefficient ($$r$$) • The ratio of the slope to the correlation coefficient ($$\beta_1 / r$$) is equal to the ratio of the standard deviations ($$sd_y / sd_x$$). In this example, the standard deviation of y is exactly twice as large as x, which is why the slope has twice the magnitude of the correlation coefficient. • The slope from the linear model will always have the same sign (+ or -) as the correlation coefficient (as standard deviations are always positive). ## 4.2 Spearman correlation There will be times when it is more appropriate to use the Spearman rank correlation than the Pearson correlation. This could be the case when: 1. The relationship between the variables is not linear, i.e. not a straight line; 2. The data is not normally distributed;2 3. The data has large outliers; or 4. When you are working with ordinal rather than continuous data.3 The Spearman correlation is a non-parameteric test as it does not require that the parameters of the linear model hold true. For example, there does not need to be a linear relationship between the two variables, and the data does not need to be normally distributed. The Spearman rank correlation is the same as a Pearson correlation but using the rank of the values in our samples. This is an approximation only, which Lindeløv shows is approximate when the sample size is greater than 10 and almost perfect when the sample is greater than 20. This is also the same as the linear model using rank-transformed values of $$x$$ and $$y$$: $$rank(y) = \beta_0 + \beta_1 \cdot rank(x) \qquad H_0: \beta_1 = 0$$ For a comparison of the Spearman test, the Pearson test (using ranks) and the linear model (also using ranks) we run the following code: # Spearman cor.test(y, x, method = "spearman") # Pearson using ranks cor.test(rank(y), rank(x), method = "pearson") # Linear model using rank lm <- lm(rank(y) ~ 1 + rank(x)) lm %>% summary() %>% print(digits = 5) # show summary ouput The output of this code is as follows: Table 4.2: Spearman, Pearson (ranks) and linear model (ranks) Test correlation slope p.value cor.test (Spearman) -0.2266 0.1135 cor.test (Pearson with ranks) -0.2266 0.1135 lm (with ranks) -0.2266 0.1135 This shows that the results are all the same (or at least very close approximations). When ranks are used, the slope of the linear model ($$\beta_1$$) has the same value as the correlation coefficient ($$r$$). Note that the slope from the linear model has an intuitive interpretation, which is the number of ranks $$y$$ changes for each change in rank of $$x$$. Given the similarity of these tests, Lindeløv notes that: One interesting implication is that many “non-parametric tests” are about as parametric as their parametric counterparts with means, standard deviations, homogeneity of variance, etc. - just on rank-transformed data. Finally, the figure below (reproduced from the original book) illustrates how the Pearson and Spearman correlations are equivalent to linear models, with the latter being based on rank-transformed values. 1. Technically the variables should have bivariate normality, which means they are normally distributed when added together, but this is complex and so it is common just to assess whether the variables are individually normal (explained here on the Laerd website). If bi-variate normality does not hold then you will still get a fair estimate of $$r$$, but the inferential tests (t-statistics and p-values) could be misleading (explained here). 2. see Chapter 9 for a description of the different types of data.	2021-09-17 06:40:34	{"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": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.836085855960846, "perplexity": 634.7606094022478}, "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/1631780055601.25/warc/CC-MAIN-20210917055515-20210917085515-00576.warc.gz"}
https://portwooddigital.com/page/2/	# Nonlinear Sections, Elastic Elements I often make seemingly minor tweaks to OpenSees--tweaks that don't usually make it into the documentation, but that in some cases could be quite useful. For example, did you know that you can create an elasticBeamColumn element by passing a section tag instead of directly specifying material and section properties--E, A, and Iz for 2D, … Continue reading Nonlinear Sections, Elastic Elements # Negative Feedback Loop Although it has its proper uses, I'm not a fan of the linear algorithm. I'm even less of a fan of modal damping. However, it's totally reasonable to use these two analysis options together--and if you do, watch out! I'll walk you through a recent encounter with this lethal combination, experienced during a live presentation … Continue reading Negative Feedback Loop # Shear Verse, Same as the First In the same vein as a previous post, this post will show a basic comparison of material nonlinear displacement-based and force-based formulations with axial-flexure-shear interaction in frame elements. The timoshenkoBeamColumn element interpolates constant shear deformation along its length, along with constant axial deformation and linear curvature. Two-point Gauss-Legendre integration over the element is sufficient to … Continue reading Shear Verse, Same as the First # Elastic Shear Beams in OpenSees Shear deformations in slender beams are generally not significant compared to flexural deformations. But shear deformation are important in deep beams and short walls, and flexure-shear interaction may be important in the material nonlinear range of response, regardless of aspect ratio. Enough of the perfunctory, non-committal language--you can find that in the latest issue of … Continue reading Elastic Shear Beams in OpenSees # OpenSees 12345 In the early 2000s, when the Tcl interpreter was taking shape for OpenSees, Frank used a dummy tag 123456789 to determine if a load pattern had already been defined while parsing the load and sp commands. Here is the 2001 source code for TclModelBuilder.cpp--clearly written by Frank because he does not capitalize anything when he … Continue reading OpenSees 12345 # A Simple Solution to a Complicated Equivalent A previous post posited on the equivalence of discrete flexural springs (moment-rotation) with integration of continuous moment-curvature response. To find the answer, we can use the principle of virtual forces (PVF) and numerical integration of the internal virtual work: $latex {\displaystyle \int_0^L \kappa(x)m(x)\: dx \approx \sum_{i=1}^N \kappa(x_i) m(x_i) w_i}$ where $latex m(x)$ is the "virtual" … Continue reading A Simple Solution to a Complicated Equivalent # Full Fiber Circle Circular layers of fibers are required for simulating longitudinal reinforcing steel in circular RC columns. Although the layer circ command accommodates fibers along an arc, I have never seen anyone use this command for anything other than a full circle. Some years before the extent of OpenSees GitHub history, I added a default constructor to … Continue reading Full Fiber Circle # P-M Interaction by the Book Find any indeterminate beam, frame, or truss problem from a structural analysis textbook, and you can make OpenSees solve it. But sometimes, replicating the basics is not so easy. Take, for instance, an axial-moment (P-M) interaction diagram of reinforced concrete (RC) sections. The typical approach advocated with OpenSees is to use repeated moment-curvature analyses over … Continue reading P-M Interaction by the Book # Get the Accel Out In OpenSees, a UniformExcitation pattern is functionally equivalent to a regular load pattern, fitting into the framework of a time-varying scalar load factor and constant reference load vector. The scalar load factor is the input ground acceleration, $latex \ddot{u}_g(t)$, while the reference load vector is $latex {\bf P}_{ref}=-{\bf m}{\boldsymbol \iota}$ where $latex {\bf m}$ is … Continue reading Get the Accel Out # Rectangular Patches Defining rectangular patches is one of the more tedious aspects of building a fiber section in OpenSees. Using the 'quad' patch command, you have to define the four corner points (I, J, K, and L) in counter-clockwise order from I to L around the patch. If you go clockwise, the fiber areas will be negative, … Continue reading Rectangular Patches	2022-10-05 09:21:24	{"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": 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": 0.4231754541397095, "perplexity": 2097.9104220925924}, "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/1664030337595.1/warc/CC-MAIN-20221005073953-20221005103953-00475.warc.gz"}
http://electronics.stackexchange.com/questions/49717/vhdl-constant-range-declaration	# VHDL constant range declaration I have a bunch of network packets and I am trying to specify fields in them, something like constant UPPER_BOUND : natural := 15; constant LOWER_BOUND : natural := 7; I know that this syntax is CORRECT The range does not change throughout the application and I was thinking if it is possible in the VHDL syntax to declare a constant range if you will. Something to the effect of: constant FIELD_RANG : natural range := (15 downto 7); I know that this syntax is WRONG But, I was trying to understand, if specifying a generic range is possible ? Is the 1st mentioned syntax the only way to do this ? - I think the second line should be constant LOWER_BOUND : natural := 7; above. (OK, it is now! :-) But anyway... welcome to a language with a proper type system! Learn to use it, and you will love it. Or learn to fight it, and you will hate it. Your choice... What you are looking for is either type Field_Range is new Natural range UPPER_BOUND downto LOWER_BOUND; or subtype Field_Range is Natural range UPPER_BOUND downto LOWER_BOUND; according to whether you want type safety, or easy mixing of Field_Range with other integer quantities. Either way you can say Field : Array(Field_Range) of Something; for i in Field_Range loop ... end loop and (almost!) never worry about bounds errors again. What is the difference between Type and Subtype? That'll take a little learning, but here's a start: if you have the following, constant N : Natural := 7; constant F : Field_Range := N; the subtype will allow it, the new type will not, you would need to convert: constant F : Field_Range := Field_Range(N); -	2015-02-27 06:01:01	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2862755060195923, "perplexity": 3977.172530834148}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-11/segments/1424936460576.24/warc/CC-MAIN-20150226074100-00284-ip-10-28-5-156.ec2.internal.warc.gz"}
https://www.flexiprep.com/Expected-Exam-Questions/Mathematics/Class-11/Ch-2-Relations-And-Functions-Miscellaneous-Exercise-Solutions-Part-3.html	# NCERT Class 11 Mathematics Solutions: Chapter 2 –Relations and Functions Miscellaneous Exercise Part 3 Get unlimited access to the best preparation resource for NCO : fully solved questions with step-by-step explanation- practice your way to success. 1. Let be a function from to defined by , for some integers . Determine. On substituting in We obtain. So, the respective values of are . 2. Let be a relation from defined by . Are the following true? (i) , for all (ii) , implies (iii) implies . It can be seen that ; however, . Therefore, the statement “” is not true. It can be seen that because and . Now, ; therefore, Therefore, the statement is not true. It can be seen that because and . Now, ; therefore, So, the statement is not true. 3. Let . Are the following true? (i) is a relation from (ii) is a function from . Justify your answer in each case. It is given that	2021-01-22 05:37:18	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.945026695728302, "perplexity": 3178.39236853414}, "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/1610703529128.47/warc/CC-MAIN-20210122051338-20210122081338-00632.warc.gz"}
http://www.mzan.com/article/48775343-is-it-possible-to-insert-component-selector-tags-using-string-interpolation-or-p.shtml	Home Is it possible to insert component selector tags using string interpolation or property binding? # Is it possible to insert component selector tags using string interpolation or property binding? user9356978 1# user9356978 Published in 2018-02-13 20:45:32Z Suppose I have 10 components having the selectors as (had to insert full stops for preventing them from being ommitted ) <. app-comp1 .> <. app-comp2 .> ... <. app-comp10 .> In the parent component I want to insert only one of the above ten based on a property having the name of the component. Eg. this.component = 'comp7' So I should only include <. app-comp7 .> P.s. i know it can be accomplished by ngIf. But that would mean writing 10lines. I need to use shorter code. vincecampanale 2# No, it is not possible to have dynamic tags in your HTML. You don't necessarily need to have 10 ngIf's to accomplish this kind of behavior however. Using ComponentFactoryResolver and an entry component, you can dynamically insert whichever component you want into your DOM. An example of how this might look: @Component({ template: }) export class MyComponent implements AfterContentInit { @ViewChild('entry', {read: ViewContainerRef}) entry: ViewContainerRef; constructor(private resolver: ComponentFactoryResolver) {} ngAfterContentInit() { // Get your component's class here and save it in a variable const dynamicComponent = some condition ? ComponentOne : ComponentTwo; const factory = this.resolver.resolveComponentFactory(dynamicComponent); this.entry.createComponent(factory); } } Note that this must happen in the AfterContentInit lifecycle hook. Edit: In response to your comment which is that the ComponentFactoryResolver is too complex, I would argue it is much simpler than using many ngIf directives. This moves the complexity into a simple function rather than many lines of complex markup. Unfortunately, since you cannot use dynamic HTML tags, these are the two best options for what you want to do.	2018-02-23 10:44:14	{"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": 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": 0.46459197998046875, "perplexity": 3469.523069862049}, "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/1518891814566.44/warc/CC-MAIN-20180223094934-20180223114934-00246.warc.gz"}
http://math.stackexchange.com/questions/456514/convert-from-one-time-scale-to-another	# convert from one time scale to another i would like to exactly to convert given scale of time into years and month,suppose we have time $T = 2.35$ years clearly it is more then $2$ year,but what about $0.35$?in which scale it is given? In generally if it is related to year,then we should multiply it by one year or $12$ months to get in term of month what fraction it represent, but could not it given in days?then what is possible scale for it? I need such kind of terms because maybe in GRE exam there could be such kind of transformation necessary,so please help me. - Years and months and days are all on the same scale, but using it is hazardous due to leap years and other anomalies. – Eric Tressler Jul 31 '13 at 16:02 Does this help? \begin{align} 2.35 \;\text{years} &= 2 \;\text{years} + 0.35 \;\text{years}\\ &= 2 \;\text{years} + 0.35 \frac{12 \;\text{months}}{1 \;\text{year}}\;\text{years}\\ &= 2 \;\text{years} + 4.2 \;\text{months}\\ &= 2 \;\text{years} + 4 \;\text{months} + 0.2 \frac{30 \;\text{days}}{1 \;\text{month}} \;\text{months}\\ &= 2 \;\text{years} + 4 \;\text{months} + 6 \;\text{days} \end{align} Generally, when you write something like 22.35 years, the whole number is given in terms of the specified unit. In other words, the 0.35 is also measuring the amount in terms of years (I think that was part of your question.) I can see why this might be confusing, though. We write time as hours:minutes, rather than hours:fraction-of-hours. If you're ever in doubt about something like this on an exam, you can usually ask a prof. In any case, as long as you are very clear about what you're doing in your answer, and you provide steps that the markers can easily follow, you should be fine. - it is not about exam,in which prof can help,i mean it is not in classroom,but thanks very much,it is what i wanted – dato datuashvili Jul 31 '13 at 16:20 "months" are not a fixed unit, though; they vary obviously by month, and also by year. – Eric Tressler Jul 31 '13 at 16:20	2016-02-08 15:22:27	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9714807271957397, "perplexity": 1232.114738881244}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701153585.76/warc/CC-MAIN-20160205193913-00153-ip-10-236-182-209.ec2.internal.warc.gz"}
https://www.gradesaver.com/textbooks/math/precalculus/precalculus-6th-edition-blitzer/chapter-3-review-exercises-page-513/77	## Precalculus (6th Edition) Blitzer Consider the given equation, ${{\log }_{3}}\left( x-1 \right)-{{\log }_{3}}\left( x+2 \right)=2$ Apply the quotient rule of logarithms: \begin{align} & {{\log }_{3}}\left( x-1 \right)-{{\log }_{3}}\left( x+2 \right)=2 \\ & {{\log }_{3}}\frac{\left( x-1 \right)}{\left( x+2 \right)}=2 \end{align} The provided equation can be written as \begin{align} & \frac{\left( x-1 \right)}{\left( x+2 \right)}={{3}^{2}} \\ & \frac{\left( x-1 \right)}{\left( x+2 \right)}=9 \\ & \left( x-1 \right)=9\left( x+2 \right) \\ \end{align} $x-1=9x+18$ Now, add to both sides $-x-18$: \begin{align} & x-1-x-18=9x+18-x-18 \\ & 8x=-19 \\ & x=\frac{-19}{8} \\ \end{align} Here $x=\frac{-19}{8}$ is not the solution of the given expression because \begin{align} & {{\log }_{3}}\left( x-1 \right)-{{\log }_{3}}\left( x+2 \right)=2 \\ & {{\log }_{3}}\left( \frac{-19}{8}-1 \right)-{{\log }_{3}}\left( \frac{-19}{8}+2 \right)=2 \\ & {{\log }_{3}}\left( \frac{-19}{8}-1 \right)-{{\log }_{3}}\left( \frac{-19}{8}+2 \right)=2 \\ & {{\log }_{3}}\left( -3.375 \right)-{{\log }_{3}}\left( -0.375 \right)=2 \end{align} By definition of ${{\log }_{a}}b$, $b>0$ but $-3.375$ and $-0.375$ is not greater than zero. Thus, $x=\frac{-19}{8}$ does not satisfy the equation ${{\log }_{3}}\left( x-1 \right)-{{\log }_{3}}\left( x+2 \right)=2$ Therefore, there is no solution of the given expression.	2020-05-26 20:28:40	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 4, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 1.0000100135803223, "perplexity": 5457.064361185825}, "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/1590347391309.4/warc/CC-MAIN-20200526191453-20200526221453-00176.warc.gz"}
http://www.bactra.org/weblog/666.html	## June 30, 2010 ### In which Dunning-Krueger meets Slutsky-Yule, and they make music together Attention conservation notice: Over 2500 words on how a psychologist who claimed to revolutionize aesthetics and art history would have failed undergrad statistics. With graphs, equations, heavy sarcasm, and long quotations from works of intellectual history. Are there no poems you could be reading, no music you could be listening to? I feel I should elaborate my dismissal of Martindale's The Clockwork Muse beyond a mere contemptuous snarl. The core of Martindale's theory is this. Artists, and still more consumers of art, demand novelty; they don't just want the same old thing. (They have the same old thing.) Yet there is also a demand, or a requirement, to stay within the bounds of a style. Combining this with a notion that coming up with novel ideas and images requires "regressing" to "primordial" modes of thought, he concludes Each artist or poet must regress further in search of usable combinations of ideas or images not already used by his or her predecessors. We should expect the increasing remoteness or strangeness of similes, metaphors, images, and so on to be accompanied by content reflecting the increasingly deeper regression toward primordial cognition required to produce them. Across the time a given style is in effect, we should expect works of art to have content that becomes increasingly more and more dreamlike, unrealistic, and bizarre. Eventually, a turning point to this movement toward primordial thought during inspiration will be reached. At that time, increases in novelty would be more profitably attained by decreasing elaboration — by loosening the stylistic rules that govern the production of art works — than by attempts at deeper regression. This turning point corresponds to a major stylistic change. ... Thus, amount of primordial content should decline when stylistic change occurs. [pp. 61--64, his emphasis; the big gap corresponds to some pages of illustrations, and not me leaving out a lot of qualifying text] Reference to actual work in cognitive science on creativity, both theoretical and experimental (see, e.g., Boden's review contemporary with Martindale's work), is conspicuously absent. But who knows, maybe his uncritical acceptance of these sub-Freudian notions has lead in some productive direction; let us judge them by their fruits. Here is Martindale's Figure 9.1 (p. 288), supposedly showing the amount of "primordial content" in Beethoven's musical compositions from 1795 through 1826, or rather a two-year moving average of this. Let us leave to one side the very difficult questions of how to measure "primordial content"; Martindale, like too many psychologists, is slave to quite confused ideas about "construct validity". The dots are the moving averages, the solid black line is a guide to the eye, and the dashed line is a parabola fit to the moving averages. In the main text, Martindale combines the parabolic trend with a second order autoregression, getting the fitted model (p. 289) PCt = -1.59 + 0.23t - 0.01 t2 + 0.58 PCt-1 - 0.55 PCt-2 which, he says, has an R2 of 50%. Primordial content is supposed to go up as an artist (or artistic community) "works out the possibilities of a style", but go down with a switch to a new, fresh style. Martindale tries (p. 289) to match up his peaks and troughs with what the critics say about the development of Beethoven's style, and succeeds to his own satisfaction, at least "in broad outline". Now, here is the figure which was, so help me, the second run of some R code I wrote. Here, however, instead of having people try to figure out how much primordial content there was in Beethoven's music, I simply took Gaussian white noise, with mean zero and variance 1, with one random number per year, and treated that exactly the same way that Martindale did: two-year moving averages, a quadratic fit over time (displayed), and a quadratic-plus-AR(2) over-all model, which kept 45% of the variance. My final fitted model was PCt = -0.61 + 0.15t - 0.004 t2 + 0.63 PCt-1 - 0.51 PCt-2 Was this a fluke? No. When I repeat this 1000 times, the median R2 is 43%, and 28% of the runs have an R2 greater than what Martindale got. His fit is no better than one would expect if his measurements are pure noise. What is going on here? All of the apparent structure revealed in Martindale's analysis is actually coming from his having smoothed his data, from having taken the two-year moving average. Remarkably enough, he realized that this could lead to artifacts, but brushed the concern aside: One has to be careful in dealing with smoothed data. The smoothing by its very nature introduces some autocorrelation because the score for one year is in part composed of the score for the prior year. However, autocorrelations introduced by smoothing are positive and decline regularly with increase lags. That is not at all what we find in the case of Beethoven — or in other cases where I have used smoothed data. The smoothing is not creating correlations where non existed; it is magnifying patterns already in the data. [p. 289] What this passage reveals is that Martindale did not understand the difference between the autocorrelation function of a time series, and the coefficients of an autoregressive model fit to that time series. (Indeed I suspect he did not understand the difference between correlation and regression coefficients in general.) The autoregressive coefficients correspond, much more nearly, to the partial autocorrelation function, and the partial autocorrelations which result from applying a moving average to white noise have alternating signs — just like Martindale's do. In fact, the coefficients he got are entirely typical of what happens when his procedure is applied to white noise: Small dots: Autoregressive coefficients from 1000 runs of Martindale's analysis applied to white noise. Large X: his estimated coefficients for Beethoven. I could go on about what has gone wrong in just the four pages Martindale devotes to Beethoven's style, but I hope my point is made. I won't say that he makes every conceivable mistake in his analysis, because my experience as a teacher of statistics is that there are always more possible errors than you would ever have suspected. But I will say that the errors he's making — creating correlations by averaging, confusing regression and correlation coefficients, etc. — are the sort of things which get covered in the first few lessons of a good course on time series. The fact that averaging white noise produces serial correlations, and a particular pattern of autoregressive coefficients, is in particular famous as the Yule-Slutsky effect, after its two early-20th-century discoverers. (Slutsky, interestingly, appears to have thought of this as an actual explanation for many apparent cycles, particularly of macroeconomic fluctuations under capitalism, though how he proposed to reconcile this with Marx I don't know.) I am not exaggerating for polemical effect when I say that I would fail Martindale from any class I taught on data analysis; or that every single one of the undergraduate students who took 490 this spring has demonstrated more skill at applied statistics than he does in this book. Martindale's book has about 200 citations in Google Scholar. (I haven't tried to sort out duplicates, citation variants, and self-citations.) Most of these do not appear to be "please don't confuse us with that rubbish" citations. Some of them are from intelligent scholars, like Bill Benzon, who, through no fault of their own, are unable to evaluate Martindale's statistics, and so take his competence on trust. (Similarly with Dutton, who I would not describe as an "intelligent scholar".) This trust has probably been amplified by Martindale's rhetorical projection of confidence in his statistical prowess. (Look at that quote above.) — Oh, let's not mince words here: Martindale fashions himself as someone bringing the gospel of quantitative science to the innumerate heathen of the humanities, complete with the expectation that they'll be too stupid to appreciate the gift. For many readers, those who project such intellectual arrogance are not just more intimidating but also more credible, though rationally, of course, they shouldn't be. (If you want to suggest that I exploit this myself, well, you'd have a point.) Could there be something to the idea of an intrinsic style cycle, of the sort Martindale (like many others) advocates? I actually wouldn't be surprised if there were situations when some such mechanism (shorn of the unbearably silly psychoanalytic bits) applies. In fact, the idea of this mechanism is much older than Martindale. For example, here is a passage from Marshall G. S. Hodgson's The Venture of Islam, which I happen to have been re-reading recently: After the death of [the critic] Ibn-Qutaybah [in 889], however, a certain systematizing of critical standards set in, especially among his disciples, the "school of Baghdad". ... Finally the doctrine of the pre-eminence of the older classics prevailed. So far as concerned poetry in the standard Mudâi Arabic, which was after all, not spoken, puristic literary standards were perhaps inevitable: an artificial medium called for artificial norms. That critics should impose some limits was necessary, given the definition of shir poetry in terms of imposed limitations. With the divorce between the spoken language of passion and the formal language of composition, they had a good opportunity to exalt a congenially narrow interpretation of those limits. Among adîbs who so often put poetry to purposes of decoration or even display, the critics' word was law. Generations of poets afterwards strove to reproduce the desert qasîdah ode in their more serious work so as to win the critics' acclaim. Some poets were able to respond with considerable skill to the critics' demands. Abû-Tammâm (d. c. 845) both collected and edited the older poetry and also produced imitations himself of great merit. But work such as his, however admirable, could not be duplicated indefinitely. In any case, it could appear insipid. A living tradition could not simply mark time; it had to explore whatever openings there might be for working through all possible variations on its themes, even the grotesque. Hence in the course of subsequent generations, taste came to favor an ever more elaborate style both in verse and in prose. Within the forms which had been accepted, the only recourse for novelty (which was always demanded) was in the direction of more far-fetched similes, more obscure references to educated erudition, more subtle connections of fancy. The peak of such a tendency was reached in the proud poet al-Mutanabbi', "the would-be prophet" (915--965 — nicknamed so for a youthful episode of religious propagandizing, in which his enemies said he claimed to be a prophet among the Bedouin), who travelled whenever he did not meet, where he was, with sufficient honor for his taste. He himself consciously exemplified, it is said, something of the independent spirit of the ancient poets. Though he lived by writing panegyrics, he long preferred, to Baghdad, the semi-Bedouin court of the Hamdânid Sayf-al-dawlah at Aleppo; and on his travels he died rather than belie his valiant verses, when Bedouin attacked the caravan and he defended himself rather than escape. His verse has been ranked as the best in Arabic on the ground that his play of words showed the widest range of ingenuity, his images held the tension between fantasy and actuality at the tautest possible without falling into absurdity. After him, indeed, his heirs, bound to push yet further on the path, were often trapped in artificial straining for effect; and sometimes they appear simply absurd. In any case, poetry in literary Arabic after the High Caliphal Period soon became undistinguished. Poets strove to meet the critics' norms, but one of the critics' demands was naturally for novelty within the proper forms. But such novelty could be had only on the basis of over-elaboration. This the critics, disciplined by the high, simple standards of the old poetry, properly rejected too. Within the received style of shir, good further work was almost ruled out by the effectively high standards of the `Abbâsî critics. [volume I, pp. 463--464, omitting some diacritical marks which I don't know how to make in HTML] Now, it does not matter here what the formal requirements of such poetry were, still less those of the qasidah; nor is it relevant whether Hodgson's aesthetic judgments were correct. I quote this because he points to the very same mechanism — demand for novelty plus restrictions of a style leading to certain kinds of elaboration and content — decades before Martindale (Hodgson died, with this part of his book complete, in 1968), and with no pretense that he was making an original argument, as opposed to rehearsing a familiar one. But there are obvious problems with turning this mechanism into the Universal Scientific Law of Artistic Change, as Martindale wants to do. Or rather problems which should be obvious, many of which were well put by Joseph (Abu Thomas) Levenson in Confucian China and Its Modern Fate: Historians of the arts have sometimes led their subjects out of the world of men into a world of their own, where the principles of change seem interior to the art rather than governed by decisions of the artist. Thus, we have been assured that seventeenth-century Dutch landscape bears no resemblance to Breughel because by the seventeenth century Breughel's tradition of mannerist landscape had been exhausted. Or we are treated to tautologies, according to wich art is "doomed to become moribund" when it "reaches the limit of its idiom", and in "yielding its final flowers" shows that "nothing more can be done with it" — hece the passing of the grand manner of the eighteenth entury in Europe and the romantic movement of the nineteenth. How do aesthetic valuies really come to be superseded? This sort of thing, purporting to be a revelation of cause, an answer to a question, leaves the question still to be asked. For Chinese painting, well before the middle of the Ch'ing period, with its enshrinement of eclectic virtuosi and connoisseurs, had, by any "internal" criteria, reached the limit of its idiom and yielded its final flowers. And yet the values of the past persisted for generations, and the fear of imitation, the feeling that creativity demanded freshness in the artist's purposes, remained unfamiliar to Chinese minds. Wang Hui was happy to write on a landscape he painted in 1692 that it was a copy of a copy of a Sung original; while his colleague, Yün Shou-p'ing, the flower-painter, was described approvingly by a Chi'ing compiler as having gone back to the "boneless" painting of Hsü Ch'ung-ssu, of the eleventh century, and made his work one with it. (Yün had often, in fact, inscribed "Hsü Ch'ung-ssu boneless flower picture" on his own productions.) And Tsou I-kuei, another flower-painter, committed to finding a traditional sanction for his art, began a treatise with the following apologia: When the ancients discussed painting they treated landscape in detail but slighted flowering plants. This does not imply a comparison of their merits. Flower painting flourished in the northern Sung, but Hsü [Hsi] and Huang [Ch'üan] could not express themselves theoretically, and therefore their methods were not transmitted. The lesson taught by this Chinese experience is that an art-form is "exhausted"when its practitioners think it is. And a circular explanation will not hold — they think so not when some hypothetically objective exhaustion occurs in the art itself, but when outer circumstances, beyond the realm of purely aesthetic content, has changed their subjective criteria; otherwise, how account for the varying lengths of time it takes for different publics to leave behind their worked-out forms? [pp. 40–41] Martindale seems to be completely innocent of such considerations. What he brings to this long-running discussion is, supposedly, quantitative evidence, and skill in its analysis. But this is precisely what he lacks. I have only gone over one of his analyses here, but I claim that the level of incompetence displayed here is actually entirely typical of the rest of the book.	2022-07-02 00:03:11	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.46204236149787903, "perplexity": 2901.0353778461117}, "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/1656103947269.55/warc/CC-MAIN-20220701220150-20220702010150-00304.warc.gz"}
https://gmatclub.com/forum/robin-mixes-two-liquids-one-blue-in-colour-and-other-red-in-177016.html?sort_by_oldest=true	GMAT Question of the Day - Daily to your Mailbox; hard ones only It is currently 22 Sep 2019, 05:50 GMAT Club Daily Prep Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email. Customized for You we will pick new questions that match your level based on your Timer History Track every week, we’ll send you an estimated GMAT score based on your performance Practice Pays we will pick new questions that match your level based on your Timer History Robin mixes two liquids, one blue in colour and other red in Author Message TAGS: Hide Tags Current Student Joined: 22 Jul 2014 Posts: 120 Concentration: General Management, Finance GMAT 1: 670 Q48 V34 WE: Engineering (Energy and Utilities) Robin mixes two liquids, one blue in colour and other red in [#permalink] Show Tags 27 Aug 2014, 00:24 3 14 00:00 Difficulty: 75% (hard) Question Stats: 62% (02:43) correct 38% (03:10) wrong based on 202 sessions HideShow timer Statistics Robin mixes two liquids, one blue in colour and other red in colour in the ratio 3:1 and sells the mixture at the rate of $15 per litre, thereby making a 20% profit on his outlay. The blue liquid costs Robin$2 per litre lesser than the red liquid. How much does a litre of red liquid cost? A) $14 B)$13 C) $12 D)$11 E) $10 Source: 4Gmat Hi can someone share strategies and concepts on Ratios, Mixtures and Propotion Most Helpful Community Reply Current Student Status: Chasing my MBB Dream! Joined: 29 Aug 2012 Posts: 1106 Location: United States (DC) WE: General Management (Aerospace and Defense) Robin mixes two liquids, one blue in colour and other red in [#permalink] Show Tags 27 Aug 2014, 01:51 7 2 In mixture problem always try to make an equation, Questions says Robin sells the 1 litre of mixture for$15. The ratio given is 3:1 for Blue: Red. Let the multiplier for ratio be 1--(take small and simple terms which is easy for calculation or multiple of 10. So we have 3 litre of blue liquid and 1 litre of red liquid. So total of 4 litre, Robin sells 4 litre of mixture for=4* 15= $60 We know that Blue cost$2 less than the red per litre. So B= R-2 per litre. here we have 3 litre of blue= $$3(R-2)$$ Now since the question says he gets 20% profit. The equation will be, $$[3(R-2)+ R] + \frac{20}{100} [3(R-2)+ R]= 60.$$ On simplifying, $$1.2[3R-6+R]= 60$$ $$1.2(4R-6)=60$$ $$4R- 6= 50$$. $$4R= 56$$ $$R= 56/4= 14$$ The answer is A. Hope it helps _________________ General Discussion Intern Joined: 27 Aug 2014 Posts: 27 GMAT Date: 09-27-2014 Robin mixes two liquids, one blue in colour and other red in [#permalink] Show Tags 28 Aug 2014, 04:26 1 1 Alternative Same approach as above but finding total cost first makes it easier: Let x be the required cost per litre of red paint. Take 4 litres of this mixture. Sold at 60 with a profit of 1/5th of total cost. So, total cost will be 1/6th less than 60; total cost = 50. 3(x-2) + 1(x) = 50 x = 14 Director Joined: 23 Jan 2013 Posts: 528 Schools: Cambridge'16 Robin mixes two liquids, one blue in colour and other red in [#permalink] Show Tags 14 Sep 2015, 00:33 1 1 B : R = 3 : 1 Cost of Red = x Cost of Blue= x-2 What is the x? Cost of mixture =(100/120)15 => 12.5 dollars 0.75(x-2)+0.25x=12.5 => x=14 A Board of Directors Joined: 17 Jul 2014 Posts: 2523 Location: United States (IL) Concentration: Finance, Economics GMAT 1: 650 Q49 V30 GPA: 3.92 WE: General Management (Transportation) Re: Robin mixes two liquids, one blue in colour and other red in [#permalink] Show Tags 11 Jan 2016, 19:23 1 since we have 3/1 B/R, we can assume we have 4 liters in total. thus, 60$/total sold. 60=1.2x, where x is the cost, and 1.2 is the revenue. thus, x=6010/12 = 50. now, 3B+R=50. B=R-2. substitute: 3(R-2)+R=50 3R-6+R=50 4R=56 R=56/4 R=14. A. _________________ VP Joined: 07 Dec 2014 Posts: 1230 Re: Robin mixes two liquids, one blue in colour and other red in [#permalink] Show Tags 25 Nov 2017, 15:25 alphonsa wrote: Robin mixes two liquids, one blue in colour and other red in colour in the ratio 3:1 and sells the mixture at the rate of$15 per litre, thereby making a 20% profit on his outlay. The blue liquid costs Robin $2 per litre lesser than the red liquid. How much does a litre of red liquid cost? A)$14 B) $13 C)$12 D) $11 E)$10 cost of liter of red liquid=c total cost=3(c-2)+c=4c-6 total revenue=415=$60 1.2(4c-6)=60 c=$14 A Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 9644 Location: Pune, India Re: Robin mixes two liquids, one blue in colour and other red in [#permalink] Show Tags 01 Mar 2019, 05:37 alphonsa wrote: Robin mixes two liquids, one blue in colour and other red in colour in the ratio 3:1 and sells the mixture at the rate of $15 per litre, thereby making a 20% profit on his outlay. The blue liquid costs Robin$2 per litre lesser than the red liquid. How much does a litre of red liquid cost? A) $14 B)$13 C) $12 D)$11 E) $10 Source: 4Gmat Hi can someone share strategies and concepts on Ratios, Mixtures and Propotion Selling price of the mix is$15 and profit is 20%. Cost Price * (6/5) = 15 Cost Price = 25/2 This is the average cost price of the mix when you put together blue liquid and red liquid. 25/2 = (Cb3 + Cr1) / (3 + 1) 3Cb + Cr = 50 Now we know that Cb (cost of blue liquid) is 2 less than Cr (cost of red liquid) 3(Cr - 2) + Cr = 50 Cr = $14 Answer (A) _________________ Karishma Veritas Prep GMAT Instructor Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options > Senior Manager Joined: 04 Aug 2010 Posts: 466 Schools: Dartmouth College Robin mixes two liquids, one blue in colour and other red in [#permalink] Show Tags 01 Mar 2019, 06:17 1 alphonsa wrote: Robin mixes two liquids, one blue in colour and other red in colour in the ratio 3:1 and sells the mixture at the rate of$15 per litre, thereby making a 20% profit on his outlay. The blue liquid costs Robin $2 per litre lesser than the red liquid. How much does a litre of red liquid cost? A)$14 B) $13 C)$12 D) $11 E)$10 At a rate of $15 per liter, the selling price for a 4-liter mixture composed of 3 liters of blue and 1 liter of red = 415 = 60. 20% profit = 120% of the cost = 6/5 of the cost. Since the selling price is 6/5 of the cost, the cost is 5/6 of the selling price: Cost of 3 liters of blue and 1 liter of red = 5/6 60 = 50 We can PLUG IN THE ANSWERS, which represent the cost for each liter of red. The prompt indicates that each liter of blue costs$2 less than each liter of red. When the correct answer is plugged in, 3 liters of blue and 1 liter of red will cost $50. B:$13, implying that the cost of each blue liter = 13-2 = 11 Cost of 3 liters of blue and 1 liter of red = (311) + 13 = 46 Since the cost is too low, a greater answer choice is needed. _________________ GMAT and GRE Tutor Over 1800 followers GMATGuruNY@gmail.com New York, NY If you find one of my posts helpful, please take a moment to click on the "Kudos" icon. Available for tutoring in NYC and long-distance. SVP Joined: 26 Mar 2013 Posts: 2329 Re: Robin mixes two liquids, one blue in colour and other red in [#permalink] Show Tags 02 Mar 2019, 04:38 GMATGuruNY wrote: alphonsa wrote: Robin mixes two liquids, one blue in colour and other red in colour in the ratio 3:1 and sells the mixture at the rate of $15 per litre, thereby making a 20% profit on his outlay. The blue liquid costs Robin$2 per litre lesser than the red liquid. How much does a litre of red liquid cost? A) $14 B)$13 C) $12 D)$11 E) $10 At a rate of$15 per liter, the selling price for a 4-liter mixture composed of 3 liters of blue and 1 liter of red = 415 = 60. 20% profit = 120% of the cost = 6/5 of the cost. Since the selling price is 6/5 of the cost, the cost is 5/6 of the selling price: Cost of 3 liters of blue and 1 liter of red = 5/6 * 60 = 50 We can PLUG IN THE ANSWERS, which represent the cost for each liter of red. The prompt indicates that each liter of blue costs $2 less than each liter of red. When the correct answer is plugged in, 3 liters of blue and 1 liter of red will cost$50. B: $13, implying that the cost of each blue liter = 13-2 = 11 Cost of 3 liters of blue and 1 liter of red = (311) + 13 = 46 Since the cost is too low, a greater answer choice is needed. Dear GMATGuruNY 1- Is not the ratio of costs same as ration of amounts of both liquids i.e. 3/1? 2- I know your solution is the better, but Can you use alligation? Thanks Senior Manager Joined: 04 Aug 2010 Posts: 466 Schools: Dartmouth College Re: Robin mixes two liquids, one blue in colour and other red in [#permalink] Show Tags 03 Mar 2019, 05:31 1 Quote: Dear GMATGuruNY 1- Is not the ratio of costs same as ration of amounts of both liquids i.e. 3/1? The cost ratio and the volume ratio can be quite different. While the red liquid constitutes 1/4 of the total volume, it does not necessarily constitute 1/4 of the total cost. Consider the following case: Cost of the blue liquid =$1 per liter. Cost of the red liquid = $1000 per liter. Cost of 4 liters of a 3:1 solution = (31) + 1000 =$1003. In this case, much more than 1/4 of the total cost -- in fact, almost 100% of the total cost -- is attributable to the red liquid. Quote: 2- I know your solution is the better, but Can you use alligation? Thanks To apply alligation here would likely be more confusing than helpful. _________________ GMAT and GRE Tutor Over 1800 followers GMATGuruNY@gmail.com New York, NY If you find one of my posts helpful, please take a moment to click on the "Kudos" icon. Available for tutoring in NYC and long-distance. Re: Robin mixes two liquids, one blue in colour and other red in [#permalink] 03 Mar 2019, 05:31 Display posts from previous: Sort by	2019-09-22 12:50:42	{"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": 0.6099650263786316, "perplexity": 4722.392111982283}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514575513.97/warc/CC-MAIN-20190922114839-20190922140839-00031.warc.gz"}
http://math.stackexchange.com/questions/243700/discontinuity-of-differential-functions	# Discontinuity of differential functions There is a corollary in Rudin analysis. But I am not able to understand it. Can someone help to understand it? The Corollary is: Let $f$ be a real differential function on $[a,b]$, then $f'$ cannot have any simple discontinuity. - Did you understand the definition of simple discontinuity? Did you understand the statement to which this is a corollary? – Siminore Nov 24 '12 at 13:27 YES. It means both left and right hand limit exist, for simple discontinuity. Theorem proves the intermediate value theorem for derivative. – user38764 Nov 24 '12 at 13:32 Does "differential" mean "differentiable"? – Chris Eagle Nov 24 '12 at 13:46 yes............ – user38764 Nov 24 '12 at 13:49 I'll sketch the argument. If the left and right hand limits $f'(c-)$ and $f'(c+)$ both exist and are not equal, then we're in a situation similar to $f'(c-) < f'(c) < f'(c+)$. So working on the lefthand side, we can find an $\epsilon > 0$ $f'(x) < f'(c) - \epsilon$ for all $x \in (c-\delta, c)$. Applying the theorem, we have a contradiction. If $f$ is continuous at $a$, differentiable for $x>a$, and $\lim_{x\to a+} f'(x)=p$, then one has $$\lim_{x\to a+}{f(x)-f(a)\over x-a}=p\ .$$ Proof. Given an $\epsilon>0$ there is a $\delta>0$ such that $$\|f'(x)-p\|<\epsilon\qquad\bigl(x\in\ ]a,a+\delta[\ \bigr)\ .$$ Let $x\in\ ]a,a+\delta[\$. Then by the mean value theorem there is a $\xi\in\ ]a,x[\ \subset \ ]a,a+\delta[\$ such that $$\left\|{f(x)-f(a)\over x-a}- p\right\|=\bigl\|f'(\xi)-p\bigr\|<\epsilon\ .\qquad\qquad\square$$ It follows that the limits $\lim_{x\to a+} f'(x)$ and $\lim_{x\to a-} f'(x)$ cannot both exist and be different, if $f$ is differentiable at $a$.	2014-12-21 15:20:30	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.9523550868034363, "perplexity": 143.07765585847392}, "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-2014-52/segments/1418802771384.149/warc/CC-MAIN-20141217075251-00127-ip-10-231-17-201.ec2.internal.warc.gz"}
https://homework.cpm.org/category/CC/textbook/cc1/chapter/6/lesson/6.1.1/problem/6-5	### Home > CC1 > Chapter 6 > Lesson 6.1.1 > Problem6-5 6-5. If you had $2$ pieces of licorice to share equally among $3$ people, how much licorice would each person get? Show your thinking clearly. Homework Help ✎ How could you use the diagrams below to solve this problem?	2019-12-14 08:48:09	{"extraction_info": {"found_math": true, "script_math_tex": 2, "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": 0.4185948967933655, "perplexity": 3886.7411839374868}, "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-2019-51/segments/1575540585566.60/warc/CC-MAIN-20191214070158-20191214094158-00193.warc.gz"}
http://torus.math.uiuc.edu/cal/math/cal?year=2017&month=01&day=01&interval=next+12+months&regexp=Topology+Seminar	Seminar Calendar for Topology Seminar events the next 12 months of Sunday, January 1, 2017. . events for the events containing More information on this calendar program is available. Questions regarding events or the calendar should be directed to Tori Corkery. December 2016 January 2017 February 2017 Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa 1 2 3 1 2 3 4 5 6 7 1 2 3 4 4 5 6 7 8 9 10 8 9 10 11 12 13 14 5 6 7 8 9 10 11 11 12 13 14 15 16 17 15 16 17 18 19 20 21 12 13 14 15 16 17 18 18 19 20 21 22 23 24 22 23 24 25 26 27 28 19 20 21 22 23 24 25 25 26 27 28 29 30 31 29 30 31 26 27 28 Friday, January 20, 2017 4:00 pm in 241 Altgeld Hall, Friday, January 20, 2017 Del Edit Copy Submitted by dcarmod2. Organizational MeetingAbstract: This is the organizational meeting to get the schedule of talks down for the spring. If you think you might be interested in giving a talk at some point, please attend! Friday, January 27, 2017 4:00 pm in 241 Altgeld Hall, Friday, January 27, 2017 Del Edit Copy Submitted by dcarmod2. Daniel Carmody (UIUC Math)Galois Categories and the Topological Galois CorrespondenceAbstract: Classical Galois theory for fields gives a correspondence between closed subgroups of the Galois group of a Galois extension and intermediate subfields. The theory of covering spaces in topology gives a correspondence between connected coverings of nice spaces and subgroups of the fundamental group. The purpose of this talk is to explain the relationship between (and generalization) of these two theorems. Tuesday, January 31, 2017 Topology Seminar 11:00 am in 345 Altgeld Hall, Tuesday, January 31, 2017 Del Edit Copy Submitted by rezk. Charles Rezk (Illinois)Complex analytic elliptic cohomology and Looijenga line bundlesAbstract: I'll explain how, by taking the cohomology of suitable spaces and messing around a bit, you can get things like: the moduli stack of (analytic) curves, the universal curve, and Looijenga line bundles over these. This seems to have some relevance for the construction of complex analytic elliptic cohomology. Friday, February 3, 2017 4:00 pm in 241 Altgeld Hall, Friday, February 3, 2017 Del Edit Copy Submitted by dcarmod2. Marissa Loving (UIUC Math)Train Tracks on SurfacesAbstract: Our mantra throughout the talk will be simple, "Train tracks approximate simple closed curves." Our goal will be to explore some examples of train tracks, draw some meaningful pictures, and develop an analogy between train tracks and another well known method of approximation. No great knowledge of anything is required for this talk as long as one is willing to squint their eyes at the blackboard a bit at times. Friday, February 10, 2017 4:00 pm in 241 Altgeld Hall, Friday, February 10, 2017 Del Edit Copy Submitted by dcarmod2. Georgios Kydonakis (UIUC Math)Opers and non-abelian Hodge theoryAbstract: We will describe two different families of flat $G$-connections over a compact Riemann surface for a complex, simple, simply connected Lie group $G$. The first is the family of $G$-opers, which for $G=\text{SL(2}\text{,}\mathbb{C}\text{)}$ can be thought of as global versions of the locally defined second order Schrödinger operators. The second comes from a particular subfamily of solutions to the so-called $G$-Hitchin equations. The physicist Davide Gaiotto conjectured that for $G=\text{SL(}n\text{,}\mathbb{C}\text{)}$ the second family in a scaling limit converges to a limiting connection which has the structure of an oper. We will describe a proof of this conjecture. This is joint work with Olivia Dumitrescu, Laura Fredrickson, Rafe Mazzeo, Motohico Mulase and Andrew Neitzke. Friday, February 17, 2017 4:00 pm in 241 Altgeld Hall, Friday, February 17, 2017 Del Edit Copy Submitted by dcarmod2. Melinda Lanius (UIUC Math)Hyperbolic taxi cabs and conic kitty cats: a mathematical activity and coloring bookAbstract: In this extremely interactive talk, we will develop intuition for various metrics that I have encountered in my own research. We’ll work our way through understanding more familiar spaces such as the real plane as well as hyperbolic plane and disk, to less familiar objects: such as a surface with a Euclidean, cylindrical, or hyperbolic-funnel end. Some markers and colored pencils will be provided, but please feel free to bring your own fun office supplies. Friday, February 24, 2017 4:00 pm in 241 Altgeld Hall, Friday, February 24, 2017 Del Edit Copy Submitted by dcarmod2. Bill Karr (UIUC Math)Geometry of convex hypersurfacesAbstract: A convex hypersurface in Euclidean space or Minkowski space is the boundary of an open convex set. Smooth convex hypersurfaces have non-negative sectional curvature and indicate properties of more general Riemannian manifolds with non-negative curvature. I will discuss some properties of convex hypersurfaces. Finally, I will describe a problem that arises from Lorentzian geometry involving convex hypersurfaces and geodesic connectedness and discuss a possible solution to this problem. Tuesday, March 7, 2017 Topology Seminar 11:00 am in 345 Altgeld Hall, Tuesday, March 7, 2017 Del Edit Copy Submitted by rezk. Guillaume Brunerie (IAS)Invariant homotopy theory in homotopy type theoryAbstract: This talk will be about homotopy type theory and in particular the branch of it known as invariant homotopy theory, or synthetic homotopy theory. The main idea is that homotopy type theory is a formal language which can be used to talk about "spaces-up-to-homotopy-equivalence". The basic objects can be thought of as spaces, but the language has the property that all the structures, properties, constructions and proofs that we can express are invariant under homotopy equivalence. One advantage is that every construction or proof done in this setting is expected to be automatically valid in every infinity-topos, not just in the infinity-topos of spaces, while still looking elementary. In this sense, we can see homotopy type theory as an internal language for infinity-topoi. Moreover, such proofs are also amenable to computer formalization, as homotopy type theory is strongly related to computer proof assistants. I will present the basic concepts and show what a few proofs and constructions look like in invariant homotopy theory. In particular, we will see the universal cover of the circle, the Hopf fibration, cohomology, and the Steenrod operations. Graduate Student Analysis Seminar 4:00 pm in 131 English Building, Tuesday, March 7, 2017 Del Edit Copy Submitted by compaan2. Derek Jung [email] (UIUC Math)A variant of Gromov's H\"older equivalence problem for small step Carnot groupsAbstract: This is the second part of a talk I gave last semester in the Graduate Geometry/Topology Seminar. A Carnot group is a Lie group that may be identified with its Lie algebra via the exponential map. This allows one to view a Carnot group as both a sub-Riemannian manifold and a geodesic metric space. It is then natural to ask the following general question: When are two Carnot groups equivalent? In this spirit, Gromov studied the problem of considering for which $k$ and $\alpha$ there exists a locally $\alpha$-H\"older homeomorphism $f:\mathbb{R}^k\to G$. Very little is known about this problem, even for the Heisenberg groups. By tweaking the class of H\"older maps, I will discuss a variant of Gromov's problem for Carnot groups of step at most three. This talk is based on a recently submitted paper. Some knowledge of differential geometry and Lie groups will be helpful. Friday, March 10, 2017 Graduate Geometry Topology Seminar 4:00 pm in 241 Altgeld Hall, Friday, March 10, 2017 Del Edit Copy Submitted by penciak2. Stefan Klajbor Goderich (UIUC Math)Stability of relative equilibria and isomorphic vector fieldsAbstract: We present applications of the notion of isomorphic vector fields to the study of nonlinear stability of relative equilibria. Isomorphic vector fields were introduced by Hepworth in his study of vector fields on differentiable stacks. Here we argue in favor of the usefulness of replacing an invariant vector field on a manifold by an isomorphic one to study nonlinear stability of relative equilibria. In particular, we use this idea to obtain a criterion for nonlinear stability. As an application, we sketch how to use this to obtain Montaldi and Rodrı́guez-Olmos’s criterion for stability of Hamiltonian relative equilibria. Friday, March 17, 2017 4:00 pm in 241 Altgeld Hall, Friday, March 17, 2017 Del Edit Copy Submitted by dcarmod2. Matthew Romney (UIUC Math)A 50-minute peek into the quasi-worldAbstract: Quasiconformal geometry is the dominant research area which evolved from complex analysis in the 20th century and remains active today. This talk will give a friendly overview to the subject, from its roots in the classical Riemann mapping theorem and Liouville theorem on conformal mappings, to some of its compelling applications in other fields, including complex dynamics and geometric group theory. Tuesday, March 28, 2017 Topology Seminar 11:00 am in 345 Altgeld Hall, Tuesday, March 28, 2017 Del Edit Copy Submitted by cmalkiew. Cary Malkiewich (UIUC)Periodic orbits and topological restriction homologyAbstract: This talk is about an emerging connection between algebraic $K$-theory and free loop spaces on the one hand, and periodic orbits of continuous dynamical systems on the other. The centerpiece is a construction in equivariant stable homotopy theory called the "$n$th power trace," which relies on the equivariant norm construction of Hill, Hopkins, and Ravenel. This trace is a refinement of the Lefschetz zeta function of a map $f$, which detects not just fixed points but also periodic orbits of $f$. The applications so far include the resolution of a conjecture of Klein and Williams, and a new approach for the computation of transfer maps in algebraic $K$-theory. These projects are joint work with John Lind and Kate Ponto. Friday, March 31, 2017 4:00 pm in 241 Altgeld Hall, Friday, March 31, 2017 Del Edit Copy Submitted by dcarmod2. Sarah Mousley (UIUC Math)Exotic limit sets of geodesics in Teichmuller spaceAbstract: In 1975, Masur proved that the Teichmuller space of a surface of genus at least 2 is not Gromov hyperbolic. Since then, many have explored to what extent Teichmuller space has features of negative curvature. In a Gromov hyperbolic space, a geodesic ray converges to a unique point in the hierarchically hyperbolic space (HHS) boundary. We will present our result that a geodesic ray in Teichmuller space does not necessarily converge to a unique point in the HHS boundary of Teichmuller space. In fact, the limit set of a ray can be almost anything allowed by topology. The goal of this talk is not to prove the result, but rather to give necessary background to understand the statement. In particular, we will not assume knowledge of Teichmuller theory or HHS structures. Tuesday, April 4, 2017 Topology Seminar 11:00 am in 345 Altgeld Hall, Tuesday, April 4, 2017 Del Edit Copy Submitted by rezk. Dan Ramras (IUPUI)Coassembly for representation spacesAbstract: Abstract: I'll describe a homotopy-theoretical framework for studying the relationships between (families of) finite-dimensional unitary representations, vector bundles, and flat connections. Applications to surfaces, 3-manifolds, and groups with Kazhdan's property (T) will be discussed. Tuesday, April 11, 2017 Topology Seminar 11:00 am in 345 Altgeld Hall, Tuesday, April 11, 2017 Del Edit Copy Submitted by cmalkiew. Kate Ponto (U Kentucky)To Be Announced Tuesday, April 25, 2017 Topology Seminar 11:00 am in 345 Altgeld Hall, Tuesday, April 25, 2017 Del Edit Copy Submitted by rezk. Carmen Rovi (Indiana)To Be Announced Tuesday, May 2, 2017 Topology Seminar 11:00 am in 345 Altgeld Hall, Tuesday, May 2, 2017 Del Edit Copy Submitted by rezk. Nathan Perlmutter (Stanford)To Be Announced	2017-03-27 08:30:43	{"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": 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": 0.5228707790374756, "perplexity": 1096.6577155327366}, "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-2017-13/segments/1490218189466.30/warc/CC-MAIN-20170322212949-00506-ip-10-233-31-227.ec2.internal.warc.gz"}
http://compgroups.net/comp.os.ms-windows.programmer.win32/problem-with-delete/189545	COMPGROUPS.NET \| Post \| Groups \| Users \| Stream \| Browse \| About \| \| ### problem with delete[] • Email • Follow helo all, I have a problem with delete[]. after debugging of program it givs mi error whith abort, retry and ignor. here is code if(LOWORD(wParam)==IDC_B1) { TCHAR text=new TCHAR[256]; // Set some text delete[] text; } 0 Reply giobs111 (5) 11/20/2009 9:21:51 PM See related articles to this posting That code will work fine as is. It's the // Set some text part that is probably killing you, what do you do? "B.S" <giobs111@gmail.com> wrote in message > helo all, I have a problem with delete[]. after debugging of program > it givs mi error whith abort, retry and ignor. > > here is code > > if(LOWORD(wParam)==IDC_B1) > { > TCHAR text=new TCHAR[256]; > > // Set some text > > delete[] text; > } 0 if(LOWORD(wParam)==IDC_B1) { TCHAR text=new TCHAR[256]; text=(TCHAR )FillOpenParams(hwnd,OPEN); SetDlgItemText(hwnd,IDC_EDIT1,text); delete[] text; } FillOpenParams is returning plaithment of file. i tryed without delete [] text and it worked fine but with it it crushes always. 0 > FillOpenParams is returning plaithment of file. i tryed without delete > [] text and it worked fine but with it it crushes always. You allocate some memory and set 'text' to the address of this memory. Immediately after that, you set 'text' to the result of FillOpenParams. Now you don't know the address of the allocated memory anymore and therefore can't free it and so this memory is leaked. This was problem #1 with your code. Problem #2 is, that you can't use delete[] to free memory that has not been allocated using new[]. So look how FillOpenParams allocates the memory of which it returns the address and use the appropriate method to free this memory - if it must be freed at all. Timo -- www.TimoSoft-Software.de - Unicode controls for VB6 "Those who sacrifice freedom for safety deserve neither." "Demokratie ist per Definition unsicher. Ihr Schutz entsteht aus der Überzeugung, dass die demokratischen Kräfte überwiegen und sich – auf demokratischem Wege – durchsetzen." 0 On Nov 21, 6:22=A0am, "B.S" <giobs...@gmail.com> wrote: > =A0text=3D(TCHAR )FillOpenParams(hwnd,OPEN); You are probably hoping that line copies text into the text array, but it does not. It copies a pointer into the text pointer. So the original text pointer is erased, which is why the later delete [] fails. What is the type that is returned by FillOpenParams and why are you casting it to TCHAR? Nobody can show you how to fix it without that missing information. 0 This is code of FillOpenParams: char FillOpenParams( HWND hwnd, int ind) { OPENFILENAME open_params = {0}; char filter[128] = {0}; char file_name[512] = {0}; strcat(filter,"All File"); int index = strlen(filter) + 1; filter[index++] = ''; filter[index++] = '.'; filter[index++] = ''; open_params.lStructSize = sizeof(OPENFILENAME); open_params.hwndOwner = hwnd; open_params.lpstrFilter = filter; open_params.lpstrFile = file_name; open_params.nMaxFile = 1024; open_params.lpstrInitialDir = NULL; open_params.lpstrFileTitle = NULL; open_params.Flags = OFN_FILEMUSTEXIST \| OFN_PATHMUSTEXIST ; if (ind==OPEN) { if(!GetOpenFileName(&open_params)) {return 0;} return open_params.lpstrFile; } if (ind==SAVE) { if(!GetSaveFileName(&open_params)) {return 0;} return open_params.lpstrFile; } else {MessageBox(hwnd,"unnown identifier",0,0);} } 0 B.S schrieb: > char FillOpenParams( HWND hwnd, int ind) You should not cast char* to TCHAR. For Unicode builds, a TCHAR is something different than a char and a typecast doesn't convert an ANSI string into a Unicode string. With this typecast, the compiler won't > char file_name[512] = {0}; > open_params.lpstrFile = file_name; > return open_params.lpstrFile; The memory occupied by 'file_name' is freed when the function is left, so you are returning the address to a memory block that no longer will be valid when it is accessed. Access violation I here you coming... Timo -- www.TimoSoft-Software.de - Unicode controls for VB6 "Those who sacrifice freedom for safety deserve neither." "Demokratie ist per Definition unsicher. Ihr Schutz entsteht aus der Überzeugung, dass die demokratischen Kräfte überwiegen und sich – auf demokratischem Wege – durchsetzen." 0 Use an extra parameter and copy the string: bool FillOpenParams(HWND hwnd, int ind, char text) { ...... open_params.nMaxFile = MAX_PATH; ...... if (ind==OPEN) { if(!GetOpenFileName(&open_params)) {return false;} strcpy(text, open_params.lpstrFile); // copy to the buffer of the caller return true; } // change other code likewise ......... } // caller if(LOWORD(wParam)==IDC_B1) { char text=new char[MAX_PATH]; // Set some text if(FillOpenParams(hwnd, ind, text)) DoSomething(); delete[] text; } 0 Friedel Jantzen schrieb: > bool FillOpenParams(HWND hwnd, int ind, char text) This is prone to buffer overflows. The function doesn't know the size of 'text' and can only guess how many characters this buffer can take. The buffer size should be passed, too. > strcpy(text, open_params.lpstrFile); // copy to the buffer of the caller This also is prone to buffer overflows. Strcpy copies until it finds a null character. It doesn't take the buffer size into account. It's better to use lstrcpyn here or even better: StringCchCopy. Timo -- www.TimoSoft-Software.de - Unicode controls for VB6 "Those who sacrifice freedom for safety deserve neither." "Demokratie ist per Definition unsicher. Ihr Schutz entsteht aus der Überzeugung, dass die demokratischen Kräfte überwiegen und sich – auf demokratischem Wege – durchsetzen." 0 Am Sun, 22 Nov 2009 10:55:46 +0100 schrieb Timo Kunze: > Friedel Jantzen schrieb: >> bool FillOpenParams(HWND hwnd, int ind, char text) > This is prone to buffer overflows. The function doesn't know the size of > 'text' and can only guess how many characters this buffer can take. The > buffer size should be passed, too. > >> strcpy(text, open_params.lpstrFile); // copy to the buffer of the caller > This also is prone to buffer overflows. Strcpy copies until it finds a > null character. It doesn't take the buffer size into account. It's > better to use lstrcpyn here or even better: StringCchCopy. > > Timo You're right. StringCchCopy is recommended. And will work, if you input the correct max length ;-) If I know that the source string will fit into the dest. buffer, is it necessary to do extra checking? If the user can select only a single file, as the OP posted, and open_params.nMaxFile = MAX_PATH; can the resulting string be longer than MAX_PATH? Regards, Friedel 0 Well, such limitations may be lifted as time moves on and then you'll be glad to have written the code in a defensive way. ;) You're right in this specific case. But if you don't avoid strcpy in general, time will come when you make a failure and decide for strcpy when you really should use lstrcpyn. I prefer to always use at least lstrcpyn. It may be less performant, but it's also less likely I make a failure that may result in a buffer overflow. You really should pass FillOpenParams the buffer size. Or how do you make sure that you never call this function from another part of your code, with a buffer smaller than MAX_PATH chars? Timo -- www.TimoSoft-Software.de - Unicode controls for VB6 "Those who sacrifice freedom for safety deserve neither." "Demokratie ist per Definition unsicher. Ihr Schutz entsteht aus der Überzeugung, dass die demokratischen Kräfte überwiegen und sich – auf demokratischem Wege – durchsetzen." 0 Am Sun, 22 Nov 2009 21:57:29 +0100 schrieb Timo Kunze: > Well, such limitations may be lifted as time moves on and then you'll be > glad to have written the code in a defensive way. ;) I agree, that the defensive way is important to write robust software, and admit using StringCchCopy very often. IMO especially important on user input or reading from files, when I do not know the length of the strings. > You're right in this specific case. But if you don't avoid strcpy in > general, time will come when you make a failure and decide for strcpy > when you really should use lstrcpyn. I prefer to always use at least > lstrcpyn. It may be less performant, but it's also less likely I make a > failure that may result in a buffer overflow. I used lstrcpyn for a long time, but switched to StringCchCopy some time ago. > You really should pass FillOpenParams the buffer size. Or how do you > make sure that you never call this function from another part of your > code, with a buffer smaller than MAX_PATH chars? In this case I am sure; for single selection file open dialogs I always provide [MAX_PATH] sized buffers. For my purposes I have written some well tested standard functions and C++ classes for open/save file dialogs. However in my posting I tried to reuse the code of the OP as much as possible (it could be improved...). Regards, Friedel 0 Thanks to all for help and cant you tell me where can i find more 0 comp.os.programmer.win32 14398 articles. 0 followers. 12 Replies 293 Views Similar Articles [PageSpeed] 44 • Email • Follow Similar Artilces: problems problems problems (The short(?) summary) I've got an Access MDB file and a DAO connect with it.. Problem 1 of 2 The below gives me a runtime error 91 Object variable or With block variable not set. I've got the db stuff after the form.show (to make sure all the objects on the form are loaded before attempting to utilize/manipulate them) But it doesn't like it when I use the data object in the form load anyway for some reason.. pffft. Private Sub Form_Load() frmTest1Project.Show datGallery.Recordset.MoveLast datGallery.Recordset.MoveFirst Call LockTextBoxes(frmTest... deleting a composite, double-delete problem Hello. i have a composite which i want to delete. this is a composite which represents a boolean expression (see a previous post of mine with more details at http://groups.google.ca/groups?hl=en&lr=&ie=UTF-8&threadm=AXqqc.89218%24PJ1.865449%40wagner.videotron.net&rnum=1&prev=/groups%3Fq%3Dmatan%2Bnassau%26hl%3Den%26lr%3D%26ie%3DUTF-8%26sa%3DG%26scoring%3Dd ) VariableExp x = new VariableExp("X"); VariableExp y = new VariableExp("Y"); BooleanExp expression( new OrExp( new AndExp(new Constant(true), x), new AndExp(x, y) ) ); represents th... #deleted problem when deleting records from sql server through vba I have a front-end access database that uses a sql server back-end to link the tables. When I run the following code, the subform displays #deleted in place of the deleted record. 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Delete Problem I am trying to delete files from a partition of my hard disk but once they have been deleted the space on the hard disk is still occupied and i cannot place new files in the now supposedly free space can anyone help? Michael Smith "Michael Smith" <smithms@blueyonder.co.uk> wrote in message news:5aa869f4.0310070955.3ac35262@posting.google.com... > I am trying to delete files from a partition of my hard disk but once > they have been deleted the space on the hard disk is still occupied > and i cannot place new files in the now supposedly free space can > anyone hel... Problem with Delete Hi. I am a (not heavy) Mathematica user who is running into a problem with the Delete function. On occasion (but not always, frustratingly) when I have a matrix (list of lists) and ask for deletion of one slot, the kernel actually shuts down and I get a message from Microsoft saying there is a problem and generating a report. Has anyone run into this and know how to get around it? Example: a = {{1,2},{3,4}}; Delete[a,{1,1}] This sometimes bombs. In my case, it's after I have multiplied the matrix by another matrix, but not before. Help, G. Effinger ... problem with Delete[] Hello, Delete[] is not removing elements of v in the following function, could anyone tell me why? Rn[N_, p_] := Module[{A, v, c, q, i, j}, A = Table[0, {i, 1, N}, {j, 1, N}]; v = Table[i, {i, 2, N}]; c = 1; For[i = 1, i < N, i++, j = Random[Integer, {1, Dimensions[v][[1]]}]; A[[c, v[[j]]]] = 1; A[[v[[j]], c]] = 1; q = Random[]; If[q > p, c = v[[j]]]; Delete[v, j]; ]; A ] thanks in advance, Heath Heath Gerhardt schrieb: > Hello, > > Delete[] is not removing elements of... delete and delete [] I understand that if you allocate an array using new as follows object * b = new object[n]; then you must free the object as follows: delete [] b; however if you are allocating a char buffer as follows: char * b = new char [n]; then isn't it irrelavant if you use delete b or delete [] b ? (Then only thing I can think of is that if you change the object type from char to some other object then you might have some issues...) "SpreadTooThin" <bjobrien62@gmail.com> wrote in message news:1175274703.360909.214000@p77g2000hsh.googlegroups.com... >I understand that i... Delete problem I'm trying to delete a file from my account and the following message appears rm: photo3.JPG: A file or directory in the path name does not exist. What is the problem? thanks in advance andreas aefthimiou@hotmail.com (Andrew) wrote in message news:<b848afcc.0308020255.6a57caec@posting.google.com>... > I'm trying to delete a file from my account and the following message appears > > rm: photo3.JPG: A file or directory in the path name does not exist. > > What is the problem? > > thanks in advance > andreas Hi. Most probably a nonprintable charater ... Problems problems.... I've got Fujitsu MAN3184MP and Adaptec 29160 scsi card. Sometimes i have this message "A disk read error occured" , sometimes even bios does not recognize it, sometimes it does but the boot sequence does not start it seems that motherboard bios has problems with it ( so it seems to me, i could be wrong about that assumption) . Then i reset and it all works perfectly. I've noticed a speed degradation in Win XP lately. I'm angry :)) I have LVD/SE terminator, and i think 68pin cable ( not sure about that ). Hope i gave you enough info to try and help. Thanx! -- -... delete on delete ! Hi, If I do another delete on an object which has been deleted earlier, then how is the system expected to behave ? Is it an unpredictable behavior ?? Thanks Sandeep Sandeep Grover wrote: > Hi, > > If I do another delete on an object which has been deleted earlier, then > how is > the system expected to behave ? > > Is it an unpredictable behavior ?? > Yes, unpredictable. Some will choke, some destructors SEGV, it goes on and on. Sandeep Grover wrote: > If I do another delete on an object which has been deleted earlier, then > how is the system expect... deletion problems I have 2 tables, Orders and Order Details, the same as with the database Nortwindtraders shipped with Access. The table order is linked with the table Order Details through the filed OrderID. It is autonumber in the table orders and number in the table Order Details.I have a very good function deleting a chosen order from a list box called ListOrders. This function is the following Public Function CancelOrders() Dim StrSQL As String strSQL = "DELETE * FROM [order details] WHERE orderID = " & frm!ListOrders & ";" DoCmd.RunSQL strSQL strSQL = "DELETE * FROM... More on Delete Problems Here's simple code which bombs for me: a={{1.,2.,3.},{4.,5.,6.}} b = {{1,0,1},{0,-2,3}, {0,1,0}} c = a . b Delete[c,{1,1}] Why? G. Effinger Hi, This code works fine for me on Mathematica 5.0. Maybe you are using an older version of Mathematica? What you are describing sounds (to me) like a bug in Mathematica. Later, Daniel Works on my system $Version 5.0 for Mac OS X (November 19, 2003) a={{1.,2.,3.},{4.,5.,6.}}; b={{1,0,1},{0,-2,3},{0,1,0}}; c=a.b; Delete[c,{1,1}] {{-1.,7.},{4.,-4.,19.}} Bob Hanlon > > From: Gov... Problem with a problem Hello, take a grammar G with alphabet {0,1} such that (the word problem for) the language L_G={w in (0+1)\|w\in L(G)} is very complex, say in PSPACE or some higher complexity class. Now consider an "easy" problem EASY like: "Is an element of L_G in L_G?". Well, this seems to be fairly easy because the answer is "yes" in any case. Hence the problem should be of small complexity. But if one models such a decision problem one has to construct a language L with alphabet A such that the word problem for L reflects exactly the decision problem which one is interested... Delete Problem I have the following line of code: if (MyData) { delete MyData; MyData= NULL; } Below is MyData: SomeDataStructure MyData; Every time this code segment is called I get a fatal error that causes my program to terminate. I use the debugger and upon reaching this line of code MyData is still valid and still has the correct information in it. I have searched the internet and pretty much figured that it was some sort of memory allocation problem... either I'm deleting what's not there, or even I've read deleting from different stacks? Any ideas as to why this delete ca... Delete problem Hi everyone, I have situation where I am trying to quality control check a "list of lists" for certain values that should be deleted from the list if they are found. I have wrote a few lines of code that I believe should work, but rather than simply deleting a part of the list, mathematica seems to delete the entire list. Below is the situation with a small test dataset: masterlist={{{AEST-01-C-10,1,1,1,2,31,1,C,-2.14},{AEST-01-C-10,1,1,1,4,31,1, C,-1.1},{AEST-01-C-10,2,2,1,2,12,2,C,1.79}},{{AEST-07-E-04,2,1,2,1,10,2, C,1.334},{AEST-07-E-04,2,1,3... problem with delete Hello all, I have class of such tye class Manager { public: static Manager* getInstance(); Worker* getWorker(char* name); private: Manager(...); Worker** workers; int numberOfWorkers; static Manager* instance; }; It is sort of singleton implementation. First call to getInstatnce() calls Manager() constructor which creates dynamically the workers array of pointers to Worker objects and asignes to numberOfWorkers size of that array. getWorker() method searches the proper worker by the name in the workers array, if there is no such worker it looks to specific file that m... problem with Delete Why the record is not deleted in this case ? This is a summary of my statements. Can you see something wrong ? // The table is originally ordered by ID. We want to add a new editor (aEditor): Append; //append a new record FieldByName('ID').AsString:= last record; //assign the ID aID:= FieldByName('ID').AsString; //store ID // Check existence of the new editor name IndexName:= 'EDX'; //order by editor name SetKey; FieldByName('EDITOR... http://LongPathTool.com http://LongPathTool.com - unlock files in-use by other applications + Delete and Copy Path Too Long Files The application solves the problem with inability to copy and delete files and folders with long paths or locked by some application. They could be created by virus/adware/different OS/other software. Normally one can't access such files under Windows and therefore can't delete them either. Please check other Features for more details. Also you can download the latest evaluation version on our download page. Long Path Tool uses a variety of methods to delete the pers... Re: How many files can you have in a VMS directory without performance problems? performance problems? performance problems? problems? performance problems? performance problems? problems? perfo First, does anyone know why Info-VAX goes nuts on subjects from time to time? (Or is the trouble elsewhere?) From: JF Mezei <jfmezei.spamnot@teksavvy.com> > DELETE Z.;* > DELETE Y.;* > ... > DELETE A.;* > > (followed by the numbers). This scheme may be a bit obsolete. See: http://h71000.www7.hp.com/doc/731FINAL/4506/4506pro_014.html#index_x_890 I quote: [...] 5.2.3.1.2 Extended Character Set In addition, OpenVMS V7.2 on Alpha systems and ODS-5 disks includes support for the use of file names, and subdirectory and root subdirectory names, tha... Problems Deleting a File I am trying to delete a excel file after it has been validated for errors and then put into another folder for processing. My problem is, when I click on the submit button, the excel file is being put into the correct folder, but I am getting a permission denied error on my file2 delete statement below. But if I hit the refresh button, the file gets deleted the way I want it to. What do I need to do to stop the permission denied error? Following is the code I am using to delete the files: function deleteStoreTempFiles() { var file2; var file3; file2 = fso.GetFile(storefolder + filename1)... delete problem #2 Hello. I have structure : typedef struct SBlog { long id_blog; char bl_name[64]; char bl_domain[64]; } SBLOG, PBLOG; I made table: BLOG pBlog = new SBlog[10] When I try free memory : delete [] pBlog I get error : Debug Assertion Failed !!!. What I do wrong ??? Thanks for your help GG "GG" <Gs@df.pl> wrote in message news:d1646i$666$1@nemesis.news.tpi.pl... > Hello. Hi. > I have structure : > > typedef struct SBlog { > long id_blog; > > char bl_name[64]; > > char bl_domain[64]; > > } SBLOG, PBLOG; > > I made ta... Problem deleting memory Hi, My application has 2-3 processes.One of these processes creates block data files. (created with sys$create,mapped by crmpsc and then UPDSECW)Memory mapped is deallocated every time. Now I stop my application..all processes are stopped. I restart application, now other process opens these files with sys$open and crmpsc call.(at this time no other process is accessing these files)I just check some variable from file and then deallocate memory by sys$delva and sys\$dgblsc calls.Memory is not getting properly deleted.Instead of reusing memory ,It is allocating new memory for each file.Do I nee...	2015-03-30 02:16:47	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.38863223791122437, "perplexity": 8238.757518096301}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-14/segments/1427131298889.70/warc/CC-MAIN-20150323172138-00188-ip-10-168-14-71.ec2.internal.warc.gz"}
https://www.talkyard.io/forum/latest/ideas	# The Talkyard Community Topics, recently active firstCategoryUsersRepliesActivity support-chat SupportDZA312018-08-09 01:12:54.535Z Shall a trust level group mention include higher trust levels or not? If mentioning a built-in trust level group, like @basic_members — does that include higher trust levels or not? To me,@basic_members, sounds as if higher trust levels are not included. However, if someone types @full_members, then that sounds as if h... Support22018-08-14 07:57:10.167Z Group mentions broken If mentioning a built-in group, like @core_members, then there's this error: "Got a group when trying to load member $groupId [EdE2SBA4J7]" because currently only individual people (but not groups) can be mentioned. Instead, either there should be a ... Bugs12018-08-14 07:50:13.498Z Avoid unnecessary closed + reopened messages Don't show messages like: X closed this topic 15 hours ago. X reopened this topic 15 hours ago. Because that's not interesting; nothing really happened. Instead, when someone clicks Close and then Reopen soon again, show no messages about that. The s... Bugs02018-08-14 07:48:47.325Z configure https on prod-one Hello, I would like to configure https access for my webserver using Talkyard. I suppose I have to obtain a let's encrypt certificate and then install it somewhere in the project, enabling some configuration files. Anybody can help me with a small tu... SupportBB82018-08-13 13:25:59.744Z Keep attached file name intact while downloading When a user downloads an attachment, the file name seen in the post is replaced with some generated (hashed?) name which forces user to rename the file. It would be desirable to keep the original file name intact during download. Thank you! IdeasM12018-08-11 08:38:22.672Z How to setup multiple instances on one server? I’ll like to setup forums for three different communities but want to do so using just one server. Did not find this in the installation instructions. Will like to know where to start. I’ll do a pull request to the installation instructions when I ge... SupportG42018-08-11 07:59:54.065Z Permanently delete items I noticed that in my own instance content deleted is hidden but not actually deleted off the server. This would raise issues if a user where to post illegal material as the website owner would still be liable if an investigator found it in the databa... IdeasHH22018-08-10 17:13:31.606Z Clicking attachment would result nginx 500 Internal Server Error Hi @KajMagnus, after the last deployment (on premise, v0.6.9-WIP-3-75eef8fa4), we are no longer able to open/save attachments. Following the attachment link would throw an internal server error. Here's what I see in the console (domain name edited) E... BugsM22018-08-10 16:59:13.095Z Single-Sign-On for embedded comments, v1 This is how we're planning to make single-sign-on (SSO) for embedded comments work. (For SSO for the discussion forum itself (which isn't embedded), maybe it'll be some simple flavor of OpenAuth instead.) To single-sign-on, the server that generates ... IdeasNSC152018-08-10 15:49:36.953Z Explain icons... Profile pictures not shown @KajMagnus, profile pictures (user uploaded) are broken in v0.6.9-WIP-3-75eef8fa4 (on premise installation). XML Parsing Error: no root element found Location: https://domain.com-/pubsub/subscribe/1-100?reqNr=3 Line Number 1, Column 1: 1-100:1:1 Err... BugsM12018-08-09 23:44:09.949Z Talkyard API? What API endpoints could be useful to you? What are your higher level goals, with calling them / what do you want to accomplish? Ideas142018-08-08 15:45:19.932Z Hosting requirements Do you we need a VPS to run Talkyard ? Will it work on normal shared hosting ? SupportMG162018-08-06 17:21:21.737Z Blog comments: The editor can hide the comment one replies to When adding Talkyard comments to a blog, and clicking Reply: if the blog uses position: absolute for the <div> with the blog post and comments, then Talkyard's editor can appear above the comment one replies to, and one needs to close the editor, to ... Bugs02018-08-05 11:32:49.380Z About the Bugs category Things about Talkyard that are broken / makes it harder to use Talkyard. Bugs02018-08-05 11:29:20.781Z Notifications generated for topics one may not access If someone mentions a non-staff member, say @jane_doe, in a staff-only chat — then a notification about the mention is sent to that person (i.e. to Jane Doe). Then, when s/he clicks the notification, s/he gets an access denied error. (Thanks Burak fo... Bugs02018-08-03 07:26:19.121Z Real-time notification stops working Hello! I've deployed a talkyard for my community but after a few seconds the real-time notification stops working and I need to refresh the page. I've check the logs and I have a "client prematurely closed connection while sending to client" right al... BugsT62018-08-02 13:34:47.606Z welcome-chat Say hello and tell us who you are. What are you looking for, how would you want to use this forum/chat software? SupportNT82018-07-31 01:10:05.766Z$\LaTeX\$ demo (works inside this topic only) Update, May 2018: If you have created a Talkyard community, you can make LaTeX work, over there, by adding this script: <script type="text/javascript" async src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.4/MathJax.js?config=TeX-MML-AM_CHTM... SupportUIUAW122018-07-25 10:40:59.487Z Hi Kaj, Is there already a way to show header and logo in Talkyard-Prod. I need to do customization of Talkyard. Can you suggest something for it. Regards Arjun Singh SupportE22018-07-20 15:33:11.029Z Configure password strength? (set minLength to 8) By default it minLength is 10. I would like it to be 8. Didn't find any option for that. Would be nice if it where possible to use a configured from e.g. conf/web/create-password.conf. SupportD32018-07-20 15:21:05.594Z No. of users and no.of domains for hosted services Sorry if these are simple questions to which I should know the answer. For a business user purchasing your hosting services, are there any restrictions on: (1) whether all users need to be from the same domain (that is, could some users be "xxxx@doma... SupportA22018-07-15 16:46:07.781Z Bugs: Invites = broken. Cannot have many identities 1. There's a problem with invites: invided users cannot choose a password, so they cannot really login. I've fixed it an will deploy the fix maybe tomorrow morning. ( & will add end-to-end tests for this) (What's an invite? It's when you click the In... Bugs12018-07-13 05:44:39.302Z Here's what's next for Talkyard — the items are not ordered, except for "soon" and "later". Feedback & change suggestions are welcome; add a comment below. Now soon Theming. User friendly way to edit colors and make one's Talkyard site look like a pa... DevelopmentD22018-07-10 19:59:49.723Z Disable/Enable global categories like ideas While going through the initial setup you get asked if you want to have questions, ideas, problems, discussions, ... As I wanted to see how all of them look I said yes to all of them. My primary use case would be questions so when sending I out initi... SupportDD22018-07-08 07:10:08.839Z How can I connect external Postgresql Database to Talkyard-Prod I am trying to connect my own PostgreSQL database to Talkyard-Prod after editing play.conf. Is it possible to connect external postgreSQL database ? SupportE32018-07-02 15:05:48.352Z New line displayed as "âŠ" in Prism.js New lines in Prism.js box are displayed as "âŠ", don't know why and how to hide that. Screenshot shows the problem. It is probably Talkyard related, because "âŠ" shows only here. For example Prism.js works completly fine when loaded from the same .cs... SupportV52018-06-29 21:15:53.293Z Syntax highlighting like on Github On Github I can choose a language formatter by typing the language identifier after the first 3 backticks, like this: js my javascript code here ` The extra space should be disregarded, only there to bypass the formatting end tag. Would th... IdeasC12018-06-25 04:29:11.575Z Strange problem with Prism.js syntax highlighting Hey, since talkyard doesn't support native syntax highlighting I'm trying to set it up with Prism.js. However it works only partially, some code has syntax highlighting, some don't. Attached screenshot shows the problem. I used "Look and feel"/"CSS a... SupportVV22018-06-25 04:20:24.586Z Displaying, sorting, filtering by tags in the topic lists Would it or is it possible to implement filtering by tag, assignee, milestone etc. in the topic list? Also it would be great to be able to set a default view, for instance by default show only the Waiting items in a Problem category. IdeasCC22018-06-18 16:07:39.235Z How can I edit a tag? I’d like to change the displayed name of a tag I created – while keeping it attached to posts where it’s attached. How can I do that? SupportPT222018-06-18 13:48:09.248Z Error after installation ran successfully Hello there, I followed all the instructions to the T on this page: https://github.com/debiki/talkyard-prod-one The only deviation is that I used a 1 GB droplet (on DigitalOcean) instead of 2 GB. Still, the steps ran smoothly without any issues. But ... SupportX22018-06-10 11:49:23.521Z Self-hosted installation - docker containers guidance Hi, I think I am ready to move from Disqus comments to Talkyard comments for my blog. I would like to use self-hosted solution. I don't want to run any scripts/compose file offered, mainly because I already have my website running using docker-compos... SupportD232018-06-08 03:11:16.834Z Can I remove the "@someone deleted this topic n seconds ago" message? I tested out the comments on a local instance of my site. I ended up with: A new topic (comments for page at ...) Comments on this new topic So I deleted the comments, and then (to see what happens) I deleted the topic. Now I have a seemingly permane... SupportD32018-06-08 01:54:58.304Z Can Chat be turned off on the self-hosted version? We currently have Slack & Hangouts Chat, so we're not interested in the Chat functionality and would like to turn it off. Is it possible to do this easily? I suppose I'd say this is a feature request to make that a setting if I didn't miss it :) Plea... IdeasD2DD2102018-05-31 13:51:11.749Z Quick test if this is for you You can try out Talkyard at your site, without signing up or installing anything: Add this where you want the commetns to appear, for example, in ./themes/YOUR_THEME_NAME/layouts/_default/single.html, in a new <section> ... Documentation02018-05-03 05:46:11.898Z Can I test Talkyard on a local static site before deploying? I like to preview my statically generated site (Octopress/Jekyll) on my local machine or an internal server before deploying to the public server. Is it possible to test talkyard.net hosted comments in such a setup? SupportD72018-05-01 06:32:38.359Z Is there a level between "account required" and "totally anonymous"? I'm using Talkyard for embedded comments, and it looks like any commenter will need to create a new account just to be able to comment. I can see people accepting this with eg. Disqus because one account will work for many sites. But for a per-site s... SupportD72018-05-01 06:07:53.451Z Bulk import from Disqus I'm migrating over from Disqus to Talkyard, and I have a lot of old comments in Disqus. Fortunately Disqus allow you to export comments, so I'm wondering what the best way to import them into Talkyard would be. Just to be clear on what I'm asking, I ... SupportD42018-04-30 15:51:13.933Z Using Octopress/Jekyll, but can't get embedded comments to work I'm using Octopress, which is just based around Jekyll (it's an older version, 2.5.3). I've followed the Jekyll instructions and have gotten as far as seeing TEST001 on a post. However, there's no comment form. I just see: I had to tweak the Jekyll s... SupportD112018-04-23 14:07:37.206Z Anything related to developing Talkyard, e.g. writing source code, discussing what to build and how, and questions about the development environment. Development02018-03-13 07:16:15.316Z	2018-08-14 13:13:16	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.1762744039297104, "perplexity": 3877.3558925058655}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221209040.29/warc/CC-MAIN-20180814131141-20180814151141-00218.warc.gz"}
https://mintoc.de/index.php?title=Batch_reactor&oldid=2123	Batch reactor Batch reactor State dimension: 1 Differential states: 2 Continuous control functions: 1 Path constraints: 2 Interior point equalities: 2 This batch reactor problem describes the consecutive reaction of some substance A via substance B into a desired product C. The system is interacted with via the control function $T(t)$ which stands for the temperature. The goal is to produce as much of substance B (which can then be converted into product C) as possible within the time limit. Mathematical formulation The optimal control problem is given by $\begin{array}{llcl} \displaystyle \max_{x, u} & x_2(t_f) \\[1.5ex] \mbox{s.t.} & \dot{x}_1 & = & -k_1 x_1^2.\\ & \dot{x}_2 & = & k_1 x_1^2 - k_2 x_2,\\ & k_1 & = & 4000 \; e^{(-2500/T(t))}, \\ & k_2 & = & 620000 \; e^{(-5000/T(t))}, \\[1.5ex] & x(0) &=& (1, 0)^T, \\ & T(t) &\in& [298, 398]. \end{array}$ $x_1(t)$ and $x_2(t)$ represent the concentrations of A and B at timepoint $t$ respectively. The control function $T(t)$ represents the temperature. Parameters The starting time and end time are given by $[t_0, t_f] = [0, 1]$. Reference Solutions This solution was computed using JuMP with a collocation method and 300 discretization points. The differential equations were solved using the explicit Euler Method. The source code can be found at Batch reactor (JuMP). The optimal objective value of the problem is $x_2(t_f) = -0.611715$. Source Code Model descriptions are available in References The problem can be found in the Tomlab PROPT guide or in the Dynopt guide.	2019-06-17 05:33:15	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 11, "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": 0.7605926990509033, "perplexity": 1573.3004013561945}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627998376.42/warc/CC-MAIN-20190617043021-20190617065021-00450.warc.gz"}
https://virtual.aistats.org/virtual/2021/poster/1894	## Principal Subspace Estimation Under Information Diffusion ### Fan Zhou · Ping Li · Zhixin Zhou Keywords: [ Applications ] [ Network Analysis ] Abstract: Let $\mathbf{A} = \mathbf{L}_0 + \mathbf{S}_0$, where $\mathbf{L}_0 \in \mathbb{R}^{d\times d}$ is low rank and $\mathbf{S}_0$ is a perturbation matrix. We study the principal subspace estimation of $\mathbf{L}_0$ through observations $\mathbf{y}_j = f(\mathbf{A})\mathbf{x}_j$, $j=1,\dots,n$, where $f:\mathbb{R}\rightarrow \mathbb{R}$ is an unknown polynomial and $\mathbf{x}_j$'s are i.i.d. random input signals. Such models are widely used in graph signal processing to model information diffusion dynamics over networks with applications in network topology inference and data analysis. We develop an estimation procedure based on nuclear norm penalization, and establish upper bounds on the principal subspace estimation error when $\mathbf{A}$ is the adjacency matrix of a random graph generated by $\mathbf{L}_0$. Our theory shows that when the signal strength is strong enough, the exact rank of $\mathbf{L}_0$ can be recovered. By applying our results to blind community detection, we show that consistency of spectral clustering can be achieved for some popular stochastic block models. Together with the experimental results, our theory show that there is a fundamental limit of using the principal components obtained from diffused graph signals which is commonly adapted in current practice. Finally, under some structured perturbation $\mathbf{S}_0$, we build the connection between this model with spiked covariance model and develop a new estimation procedure. We show that such estimators can be optimal under the minimax paradigm.	2021-11-28 00:30:16	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6918190717697144, "perplexity": 349.8083991574033}, "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-2021-49/segments/1637964358323.91/warc/CC-MAIN-20211127223710-20211128013710-00043.warc.gz"}