Proof of equation with binomial coefficients: $\sum\limits_{k=1}^{n} (k+1) \binom{n}{k} = 2^{n-1} \cdot (n+2)-1$
How to prove this binomial identity $\sum_{r=0}^n {r {n \choose r}} = n2^{n-1}$?
Proving the sum of two independent Cauchy Random Variables is Cauchy Is there any method to show that the sum of two independent Cauchy random variables is Cauchy? I know that it can be derived using Characteristic Functions, but the point is, I have not yet learnt Characteristic Functions. I do not know anything about Complex Analysis, Residue Theorem, etc.   I would want to prove the statement only using Real Calculus. Feel free to use Double Integrals if you please.   On searching, I found . However, I was wondering if I could get some help directly on the convolution formula:  $$f_Z(z)=\int_{-\infty}^\infty f_X(x)f_Y(z-x)\,dx=\int_{-\infty}^\infty\frac{1}{\pi^2}.\frac{1}{1+x^2}.\frac{1}{1+(z-x)^2}dx\tag{1}$$  Here I have supposed that $X,Y$ are Independent Standard Cauchy. But I think the general formula can be derived easily after some substitutions. I need some help on how to proceed from $(1)$.  EDIT: Just as what the hint in the hyperlink said, I got the answer using that hint. However, I am not quite sure that the hint is algebraically correct. Maybe there has been some typing mistake in the book. How can I make a logo in Adobe Illustrator like the AntiParticle logo below, from ?   It is a large number circles of different, varying sizes, which form the shape of a text character. How to output a counter with leading zeros? I can output the value of a counter in a LaTeX document via:  \setcounter{mycounter}{1} \arabic{mycounter}   Which inserts a 1. But how do I format the output by, for example, adding a leading zero if mycounter is less than 10 in a way similar to what  lets you do by specifying %02d as your format string? Using multiple wildcard parameters with arcpy.ListFeatureClasses I know there has to be a way to do this but I'm having trouble figuring out how to do so. I would like to create a list of feature classes to perform operations on however, I want to have the list contain feature classes which have 2 separate and distinct file name identifiers. The code would look something like:  FCS = arcpy.ListFeatureClasses("X_*" OR "*_Y")   I know OR is incorrect however, is there some way to do this without having to re write code to work with the second wildcard. Thanks for helping gout, i know there is a selection method in the Select By Attributes which allows for multiple wildcard use using the OR statement. I got pretty confused at that question... Can anyone help me understand this question better and/or drop some hints please? Thank you! Subgroups of $U(n)$ isomorphic to a direct sum of cyclic groups Let $U(n)$ to be the set of all positive integers less then $n$ and relatively prime to $n$. Then $U(n)$ is a group under multiplication modulo $n$.  A. Find integer $n$ such that $U(n)$ contains subgroup isomorphic to $Z_5\oplus Z_5$.  B. Show that there is an $U(n)$ containing a subgroup isomorphic to $Z_3 \oplus  Z_3$, but not isomorphic to $Z_4 \oplus  Z_4$. How to send variable to an inline shell-script? How to make unit selection circles merge? I would like to know how to make this effect of merged circle selection. Here are images to illustrate:     Basically I'm looking for this effect:    How the merge effect of the circles can be achieved ? I didn't found any explanation concerning this effect. I know that to project those texture I can develop a decal system but I don't know how to create the merging effect.  If possible, I'm looking for purely shaders solution. Why isn't Jon executed on sight? So, to get spoilers out of the way -       Jon Snow is too honorable to break his oath and leave the Night's Watch.  However, he is stabbed a bunch, dies, and due to a line in the Night's Watch oath - "It shall not end until my death" - is able to leave without his usual reservation.   Now, this is fine from Jon's perspective.  However, there is a fairly rigid law about the Wall's deserters being punished by death. Which makes me wonder: why isn't Jon having more trouble with this?   He knows what happened, but others don't.  And the explanation that he actually did die doesn't seem like one the lands' lords and authorities would take on faith.  I know I wouldn't.  His story isn't quite like Sam's, who was sent by the Night's Watch to become a maester.  Jon is famous enough that many would know who he is, and that he has taken the black.  But a story like him dying and coming back to life is probably one that takes quite a but of time to stick - if it ever does.  A possible explanation might be that people know Jon is honorable and trust him. Or that they don't care about the law, and think that they can somehow benefit from not executing him - although the law seems to be a strict one, and not everyone seems to be psyched to help Jon out.  Or maybe by that time people coming back to life is a common enough thing that people believe it that easily.  Speculation aside, however, I am curious whether this was directly addressed from the higher-ups - the makers of the show, or the big boss himself.  Why isn't Jon executed on sight? Cannot find Underground Jungle Temple On my 1.2 Terraria Map I have been mining from left to right in my large underground jungle using my rocket launcher. Despite this, I have had no luck on finding the Underground Jungle Lihzahrd Temple. I have been searching the layer above where lava constantly spawns. Have I simply spawned in a world without one or am I searching in the wrong place? I have also considered using the clentimator with Dark Blue ammo to provide light, should I do that? Adding tags to files in OS X: How to use, where are tags stored, and will they work on a network? I've heard it is possible to add tags to a file's meta data in OS X, and have Spotlight search through such tags.   In a nutshell, how would one access this feature in Finder &ndash; not Spotlight (Command+Space) &ndash; but the adding of tags themselves? Where are these tags stored?  Do they go in .DS_Store (I tend to delete those!) or elsewhere in the filesystem? Does this tagging mechanism work on removable drives?  What are the requirements? Does this tagging mechanism work over a network share?  I have some Windows SMB network shares at home that I frequently store files on.  Would it work on those or is another configuration required? Aside from Spotlight, are there any command-line tools for searching through or accessing the tag data for files?   Thank you! Homeomorphism(topological spaces) version of Cantor–Bernstein–Schroeder theorem TeX Live 2016: minted doesn't work with LuaLaTeX (bug fixed in June 2016) I've installed TeX Live 2016 as it was released today (thanks to the developers for that).  Now I have the following MWE:  \documentclass{article} \usepackage{minted} \begin{document}  \begin{minted}{ruby} class Foo   def init     pi = Math::PI     @var = "Pi is approx. #{pi}"   end end \end{minted}  \end{document}   When running pdflatex --shell-escape test.tex it runs through the file and produces the PDF as expected. Doing the same with lualatex --shell-escape test.tex fails with  This is LuaTeX, Version 0.95.0 (TeX Live 2016)   system commands enabled. (./test.tex LaTeX2e &lt;2016/03/31&gt; Babel &lt;3.9r&gt; and hyphenation patterns for 1 language(s) loaded. (/opt/texlive/2016/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/opt/texlive/2016/texmf-dist/tex/latex/base/size10.clo)) (/opt/texlive/2016/texmf-dist/tex/latex/minted/minted.sty (/opt/texlive/2016/texmf-dist/tex/latex/graphics/keyval.sty) (/opt/texlive/2016/texmf-dist/tex/latex/oberdiek/kvoptions.sty (/opt/texlive/2016/texmf-dist/tex/generic/oberdiek/ltxcmds.sty) (/opt/texlive/2016/texmf-dist/tex/generic/oberdiek/kvsetkeys.sty (/opt/texlive/2016/texmf-dist/tex/generic/oberdiek/infwarerr.sty) (/opt/texlive/2016/texmf-dist/tex/generic/oberdiek/etexcmds.sty (/opt/texlive/2016/texmf-dist/tex/generic/oberdiek/ifluatex.sty)))) (/opt/texlive/2016/texmf-dist/tex/latex/fancyvrb/fancyvrb.sty Style option: `fancyvrb' v2.7a, with DG/SPQR fixes, and firstline=lastline fix  &lt;2008/02/07&gt; (tvz)) (/opt/texlive/2016/texmf-dist/tex/latex/float/float.sty) (/opt/texlive/2016/texmf-dist/tex/latex/base/ifthen.sty) (/opt/texlive/2016/texmf-dist/tex/latex/tools/calc.sty) (/opt/texlive/2016/texmf-dist/tex/latex/ifplatform/ifplatform.sty (/opt/texlive/2016/texmf-dist/tex/generic/oberdiek/pdftexcmds.sty (/opt/texlive/2016/texmf-dist/tex/generic/oberdiek/ifpdf.sty) (/opt/texlive/2016/texmf-dist/tex/generic/oberdiek/luatex-loader.sty (/opt/texlive/2016/texmf-dist/scripts/oberdiek/oberdiek.luatex.lua))) (/opt/texlive/2016/texmf-dist/tex/generic/oberdiek/catchfile.sty) uname -s &gt; "test.w18"  ! Package catchfile Error: File `test.w18' not found.  See the catchfile package documentation for explanation. Type  H &lt;return&gt;  for immediate help.  ...                                                l.93     \CatchFileDef\@tempa{\ip@file}{}  ?    As mentioned, it's a freshly installed TeX Live 2016; I've also checked the installed Pygmentize version, which is 2.1.3 (installed via pip). Testing with TeX Live 2015, the above code, however, works without problems with both pdflatex and lualatex.  Does anybody have a hint for me on how to fix this, or is it a bug in lualatex or minted (or whatever other package) that should be reported?  Update: With minted version 2.2 (2016-06-08), the package loads the shellesc package automatically, as mentioned by @David Carlisle and @egreg in the comments; therefore, the problem should not occur any more, if you have this version installed. Maximum total distance between points on a sphere What is the configuration (set of locations) of $n$ points on the surface of a sphere such that the sum of distances is maximum for $n=1,2,3,...$?  The sum of distances is measured by summing the lengths of every straight line (through the sphere) connecting every possible combination of $2$ points.  All the points are on a single sphere of radius $R$.  Here's a visualization:   Acknowledgements: Based on  Physics S.E. question. Image from . How to prove $\lim\limits_{n \to \infty} (1+\frac1n)^n = e$? Following are other questions which I think I need to know:   From a non X Session? (meaning root isn't logged into X) If multiple people were logged in on X, could I auto-detect who was on which screen, and thus programmatically detect which screen I need to launch the app on? Can I launch the app as the user? ( ok I'm 99.999% sure this is a yes ) Can I detect if users of group X are logged in to X? Isn't finding the inverse of $a$, that is, $a'$ in $aa'\equiv1\pmod{m}$ equivalent to solving the diophantine equation $aa'-mb=1$, where the unknowns are $a'$ and $b$? I have seem some answers on this site (where the extended Euclidean Algorithm is mentioned mainly) as well as looked up some books but there is no mention of this. Am I going wrong somewhere or is this a correct method of finding modular inverses? Also can't we find the Bézout's coefficients by solving the corresponding diophantine equation instead of using the extended Euclidean Algorithm? Meaning of Conditional sentences with 'should' and 'happen to' Where and why do I have to put the "template" and "typename" keywords? In templates, where and why do I have to put typename and template on dependent names? What exactly are dependent names anyway?    I have the following code:  template &lt;typename T, typename Tail&gt; // Tail will be a UnionNode too. struct UnionNode : public Tail {     // ...     template&lt;typename U&gt; struct inUnion {         // Q: where to add typename/template here?         typedef Tail::inUnion&lt;U&gt; dummy;      };     template&lt; &gt; struct inUnion&lt;T&gt; {     }; }; template &lt;typename T&gt; // For the last node Tn. struct UnionNode&lt;T, void&gt; {     // ...     template&lt;typename U&gt; struct inUnion {         char fail[ -2 + (sizeof(U)%2) ]; // Cannot be instantiated for any U     };     template&lt; &gt; struct inUnion&lt;T&gt; {     }; };   The problem I have is in the typedef Tail::inUnion&lt;U&gt; dummy line. I'm fairly certain that inUnion is a dependent name, and VC++ is quite right in choking on it. I also know that I should be able to add template somewhere to tell the compiler that inUnion is a template-id. But where exactly? And should it then assume that inUnion is a class template, i.e. inUnion&lt;U&gt; names a type and not a function? How to get the client IP address in PHP