Prove that $S_5$ has no elements of order $120$.
The Maximum possible order for an element $S_n$
How do you do load testing and capacity planning for web sites? This is a  about capacity planning for web sites. Related:      What are some recommended tools and methods of capacity planning for web sites and web-applications? Please feel free to describe different tools and techniques for different web-servers, frameworks, etc., as well as best-practices that apply to web servers in general. Proving a sequence converges by defining the sequence recursively. Let $a &gt; 0$. Choose $x_1 &gt;$ $\sqrt a$. Define a sequence {$x_n$} recursively as $x_{n+1} = 1/2(x_n + a/x_n)$ for $n &gt; 1$.   Prove that lim$x_n = \sqrt a$.  I think I first want to prove that {$x_n$} is bounded below by $\sqrt a$. But, I am having trouble with this. Could someone help me out a little? Why the first moment is standardized before computing higher moments, but higher moments are not? stab(S) is isomorphic to $ S_k \times S_{n-k} $ Show that the group $S_n$ (right) acts on the set of subsets of $\{1,2,\ldots n\}$ by $\rho(S,\sigma)=S\sigma$. Show that there are $n+1$ orbits, one for each possible value of $|S|$. Show that if $|S|=k$ then $\mathrm{Stab}(S)\cong S_k\times S_{n-k}$.  I have shown that there are n+1 orbits but now I am stuck with the last bit. my initial thought is that if I let $S =\{1,2,3,...,k\}$ then $\mathrm{Stab}(S)$ would involve $(1)(2)(3)...(k)$ i.e. mapping to itself and then $k+1$ to $n$ mapped in any ways ($S_{n-k}$) I feel like $\mathrm{Stab}$ is isomorphic to $S_{n-k}$ what am I misunderstanding here? How would I prove the last part? Thanks a lot How to break out of jQuery each Loop MacBook Pro: How to disable discrete GPU permanently from EFI? Let $A_n$ denote an $n \times n$ tridiagonal matrix.  $$A_n=\begin{pmatrix}2 &amp; -1 &amp; &amp; &amp; 0 \\ -1 &amp; 2 &amp; -1 &amp; &amp; \\ &amp; \ddots &amp; \ddots &amp; \ddots &amp; \\ &amp; &amp; -1 &amp; 2 &amp; -1 \\ 0 &amp; &amp; &amp; -1 &amp; 2 \end{pmatrix} \quad\text{for }n \ge 2$$  Set $D_n = \det(A_n)$  Prove that $D_n = 2D_{n-1} - D_{n-2}$ for $n \ge 4$. Numbering in algorithmicx I have simple code:  \documentclass{article}  \usepackage{algorithm}  \usepackage{algpseudocode}  \begin{document}  \begin{algorithm}   \caption{Examples}\label{alg:Examples}   \begin{algorithmic}[1]   \State $X=45$   \State $X=45$   \State $X=45$   \State $X=45$   \State $X=45$    \end{algorithmic} \end{algorithm}          \end{document}     How to put a point after the number of the algorithm? I use chapters in report:  \documentclass[a4paper,12pt]{report}  \usepackage{algorithm}  \usepackage{algpseudocode}  \begin{document}  \chapter{Chapter 1}  \begin{equation} X_2=4 \end{equation}  \begin{algorithm}   \caption{Examples}\label{alg:Examples}   \begin{algorithmic}[1]   \State $X=45$   \State $X=45$   \State $X=45$   \State $X=45$   \State $X=45$    \end{algorithmic} \end{algorithm}  \end{document}     How to change the numbering to take into account the chapters? Preloading images with jQuery Is there a way to delete all stack exchange accounts at the same time?  I also want to remove all traces of my posts on the site. Is that possible?   Will I have to delete all of my posts manually?  Can I use the stack exchange API for this? Better explanation of the common general relativity illustration (stretched sheet of fabric) Is there any way to debug bash script without using echo and logging?  I'm talking about using breakpoints and stuff like that. findstr /v "black"  File1.txt   Above DOS command will display content of 'File1.txt' which are not matching string "black".  How to modify this command , if I need to filter words "black" and "white" ? I am developing a set of classes that implement a common interface. A consumer of my library shall expect each of these classes to implement a certain set of static functions. Is there anyway that I can decorate these class so that the compiler will catch the case where one of the functions is not implemented.  I know it will eventually be caught when building the consuming code. And I also know how to get around this problem using a kind of factory class.  Just curious to know if there is any syntax/attributes out there for requiring static functions on a class.  Ed Removed the word 'interface' to avoid confusion. Too much whitespace in "Thanks for the feedback" While attempting to downvote something on a site I don't have an account on, I see this message, which has too much whitespace:    The same thing happens with upvotes (and プラス票が):    (I think it's safe to say that this is a network-wide bug, since I took these all on different sites.)  However, it looks more reasonable when you are not logged into the network at all (e.g. in private browsing):    An earlier version of this issue was reported at , but the difference is that there is no newsletter UI on the right anymore. (The extra space could have been justified before given the height of the newsletter; now it's definitely a bug.)  Also, is newsletter supposed to be gone? Infinite Cartesian product of countable sets is uncountable Let $\{E_n\}_{n\in\mathbb{N}}$ be a sequence of countable sets and let $S=E_1\times\cdots\times E_n\times\cdots $. Show $S$ is uncountable. Prove that the same statement holds if each $E_n=\{0,1\}$.  By the definition of Cartesian product of sets,   $$\displaystyle S=\Pi_{n\in\mathbb{N}} \{f\colon\mathbb{N}\rightarrow\cup_{n\in\mathbb{N}}E_n\mid\forall n, f(n)\in E_n\}$$  If $E_n=\{ 0,1\}$, then  $$\displaystyle S_{01}=\Pi_{n\in\mathbb{N}}\{0,1\}=E^{\mathbb{N}}$$, where $E=\{0,1\}$.  By a theorem, $\cup_{n\in\mathbb{N}} E_n$ is countable since the sequence is countable.  I'm not sure how to go on from here to show $S$ is uncountable. Can we say anything about the function $f$ that maps a countable $\mathbb{N}$ to another countable union of sequence of countable sets? I found this problem on a well-known problem solving website. It is apparently from Ramanujan.     With   $$\LARGE{a = \frac{1}{1 + \frac{e^{-2\pi}}{1 + \frac{e^{-4\pi}}{1 + \ddots}}}},$$   what is $a$?   The basic methods I know for questions of this type were not helpful to me.  I do not require a full solution, but feel free to post one if you would like. Hints or lecture notes on how to solve problems of this nature would be appreciated.   Keep in mind that I am at an undergraduate level in mathematics. I am new to Linux and Ubuntu. When I don't know how to do something with this OS I search on the web and Ask Ubuntu and always find the answer.  But sometimes it's not a great answer nor complete tutorial. I follow the instructions anyway, but I don't know what all the commands mean.  So my question is:   When I follow a tutorial, am I 100% safe? If not, then how can I tell I am in danger? How to include PHP files that require an absolute path? I have a directory structure like the following;     script.php      inc/include1.php   inc/include2.php      objects/object1.php   objects/object2.php      soap/soap.php   Now, I use those objects in both script.php and /soap/soap.php, I could move them, but I want the directory structure like that for a specific reason. When executing script.php the include path is inc/include.php and when executing /soap/soap.php it's ../inc, absolute paths work, /mnt/webdev/[project name]/inc/include1.php... But it's an ugly solution if I ever want to move the directory to a different location.  So is there a way to use relative paths, or a way to programmatically generate the "/mnt/webdev/[project name]/"? How can I test whether my camera is working after it was dropped? Issue : How can I evaluate my cameras performance synthetically?  Scenario : Bought a Nikon D3100 an year back. Photos looked good enough. The camera fell, zoom lens effected.  But works. Cleaned lens using soap, water etc.  Carefully. Now photos look much worse.  How do I know if Photos are really bad?  Is there a tool?    Sample Photo  Now (Bad)   Original (Good)   Please help me realize whether my camera has gone bad.  Can you tell your personal opinion also?