I keep reading that \begin{align} is preferred to \begin{eqnarray}.  Unfortunately, I can't get align to justify rcl. I'll change between align and eqnarray, and the rightmost column is right justified. So, align for me looks like rcr. Does anyone know what I could be doing? I want the old rcl justification.  Thanks, Aaron
Motivation  I want to include a list of related equations, say, for a proof, in my LaTeX document. As far as I know, I have two good options, eqnarray and align.   Question  What is the difference between eqnarray and align, and how do I know which I should be using? Or does it matter at all?
Find $n$ such that $x^2 + x + 1$ is a factor of $(x+1)^n - x^n - 1$. I have to find the form of n i.e. whether n is even or odd and whether it is multiple of 2 or 3 such that:  $x^2 + x + 1$ is a factor of $(x+1)^n - x^n - 1$  What I tried:  $$x^2 + x + 1 = (x + 1)^2 - x $$  also,  $$(x+1)^n-x^n-1= \binom{n}{o}x^n + \binom{n}{1}x^{n-1} + ... + \binom{n}{n-1}x + \binom{n}{n}.1 - x^n - 1$$  $$=\binom{n}{1}x^{n-1} + ... + \binom{n}{n-1}x$$  after getting this far I tried with taking x and n common but still couldn't able to find the right direction.  I need extra hints to solve this. Elementary proof for $\lim\limits_{n \to\infty}\frac{n!e^n}{n^n} = +\infty$ As seen in  question, the series $\sum\limits_{n=1}^{\infty} \dfrac{n!e^n}{n^n}$ diverges. (One way to see this is by noting that the terms of the sum are greater than $1$ and, therefore, don't converge to zero.)  However, more is true: not only are the terms greater than $1$, they blow up to $+\infty$! To see this, one can apply Stirling's approximation and get $\dfrac{n!e^n}{n^n} \sim \sqrt{2\pi n}$.  My question is: is there an elementary proof for $\lim_{n \to\infty}\dfrac{n!e^n}{n^n} = +\infty$?  The reason I'm asking this is that I'm taking a Calculus course and I was assigned the exercise of proving the divergence of the series above and I would feel more satisfied if I could prove that its terms go to $+\infty$ (as opposed to merely proving the terms are greater than $1$.) Obviously, things like Stirling's approximation are outside the scope of the course, so I can't really use them.  (To be clear, you can use anything one learns in $4$ semesters of standard Calculus courses. I hope this restriction is not too obscure.) Version Control: Entire site or just pieces? I've been using git to track changes in the plugins and themes I've developed for different sites. Each plugin or theme has their own repository in whichever local install of WordPress I happen to be using for that project. This works fine when I'm just working on one solitary plugin or theme.  But when I have a local version of an entire site which is a collection of different plugins, themes, WordPress core, etc. Should I be keeping that entire directory under version control?  Is there a best practice to always have a collection of repositories for each theme/plugin, or for each client site to have one monolithic repository that includes all the code used in the project (including the latest copy of WordPress via svn.) SSH login without password with kerberos How to create a shortcut that executes an xdotool command to simulate a key press? I'm trying to simulate a media key press using a custom shortcut.  So far, I've been able to achieve the simulation I need using the command:  xdotool key XF86AudioPlay   It works perfectly, it pauses or starts the music player every time it is run.  The problem comes when trying to run it as a shortcut.  First I tried to run an alias with a custom shortcut, but it did not work.  As explained on :     The commands run by keyboard bindings are not parsed via a shell. Your best bet is to create a directory bin in your homedir.   Next I created the suggested script and named it simplay:  #!/bin/sh xdotool key XF86AudioPlay   It is located inside a bin folder which is part of the PATH, this works fine and can be run from any location.  When trying to create the Custom Shortcut, I was unable to get it to execute, the configuration is the following:  Name: test Command: simplay  Shortcut: Ctrl+Alt+R   After searching a bit more I found another way to execute the shortcut:  gnome-terminal -e simplay   I tried it on a terminal and noticed that when executing it it would rapidly open a new terminal window, execute the child process, pausing the music and immediately close the window.  I changed the command on the shortcut and when hitting it I noticed the same behavior opening a terminal and immediately being closed but this time the child process did not pause the music, which makes me think that the script has been called on all my previous attempts but it is somehow not being able to execute.  How can I solve this problem according to your knowledge? Kill process spawned by ssh when ssh dies I use a Windows 10 machine remotely at work. Both the local and remote machine are in the same ISP network (not same local network, but same ISP). I've been using it like that for almost a year now and I had never experienced absolutely any issues.  Until recently. I don't know if anything specific triggered this behavior, but I feel that it started happening after I upgraded the remote to W10 1903. I later upgraded the local one as well and I feel that it only made the problem worse (but I don't really have any data).  From time to time, the session will freeze - the image stops updating. However, what I type and mouse clicks are still being sent. No error message is shown whatsoever, the image simply freezes. To make it work again, I have to close and re-open the session. Everything that I had typed and/or clicked while it was frozen is there.  Is there any way I can debug this? Looking for story title - younger gen waters down age-slowing drink of older gen, crammed into small apartment I remember a short story in which multiple generations of people being crammed into a small apartment in a city. The magical drink that either reduced or completely stopped aging was something like a soda that came in a six pack. The story focused on visual aspects of the drink at some point. The younger family members complained that their elders were living too long and apparently decided to water down the soda, I think that conclusion was ultimately left up to the reader. Many thanks for any help tracking down this story, might be one of Kurt Vonnegut's perhaps? Confused between Nested Quantifiers I am reading nested quantifiers. I am confused in between two cases,   1. Existential Quantifier before Universal Quantifier  2. Universal Quantifier before Existential Quantifier   I would be very thankful if someone highlights the difference between them and also give an example.   Thanks in advance. Time-Stream Crossing and Forgetfulness In The Day of the Doctor, it's explained that the Doctor doesn't remember having interacted with himself, and if he interacts with a future version of himself, the younger version forgets.  But in Time Crash, 10th solves the problem of the TARDISes occupying the same space because he remembers himself doing it from when he was 5th.  If this is true, 5th wouldn't have wondered whether or not he was going to regenerate when he did.  Does the TARDIS occupying the same space with itself re-instate those memories? Or is it just possible that I'm asking a ridiculous question that will never have an answer because Doctor Who.  :D I finally got dragonrend and defeated Alduin, but I'm still too weak for fighting ancient dragons. Any certain weapon for killing them.  (I've got lvl 20 one handed weapons and lvl 46 for archery, so the Dragonbane is out of the question) Prove if nonempty $K \subset\mathbb{R}$ is compact, then both sup $K$ and inf $K$ exist and are elements of $K$. Prove if nonempty $K \subset \mathbb{R}$ is compact, then both sup $K$ and inf $K$ exist and are elements of $K$.   I manage to prove the existence of sup $K$ by using the Axiom of Completeness. But I failed to prove that $\sup K \in K$. Can anyone guide me ? How prove that $\lim\limits_{x\to+\infty}f(x)=\lim\limits_{x\to+\infty}f'(x)=0$ if $\lim\limits_{x\to+\infty}([f'(x)]^2+f^3(x))=0$? Question:     Let $f$ be differentiable on $[0,+\infty)$, such as$$\lim_{x\to+\infty}\left([f'(x)]^2+f^3(x)\right)=0$$show that $$\lim_{x\to\infty}f(x)=\lim_{x\to\infty}f'(x)=0$$   I think this problem is similar to  and (the  background)  For my problem I guess that if  $$\lim_{x\to+\infty}\left([f'(x)]^{2m}+f^{2n+1}(x)\right)=0,m,n\in N^{+},n,m&gt;1$$ then we have $\lim_{x\to\infty}f(x)=\lim_{x\to\infty}f'(x)=0$. I will be attending a summer school which will last 10 days.  I want to know: for applying for a Schengen visa, is the purpose of my trip considered as touristic or study? I've seen lots of examples of making an entire regular expression case-insensitive.  What I'm wondering about is having just part of the expression be case-insensitive.  For example, let's say I have a string like this:  fooFOOfOoFoOBARBARbarbarbAr   What if I want to match all occurrences of "foo" regardless of case but I only want to match the upper-case "BAR"s?  The ideal solution would be something that works across regex flavors but I'm interested in hearing language-specific ones as well (Thanks )  Edit  The link Espo provided was very helpful.  There's a good example in there about turning modifiers on and off within the expression.  For my contrived example, I can do something like this:  (?i)foo*(?-i)|BAR   which makes the match case-insensitive for just the foo portion of the match.  That seemed to work in most regex implementations except Javascript, Python, and a few others (as Espo mentioned).    The big ones that I was wondering about (Perl, PHP, .NET) all support inline mode changes. The repository 'http://dl.google.com/linux/chrome/deb stable Release' does not have a Release file After executing this command  sudo apt-get update &amp;&amp; sudo apt-get -y upgrade   I get this error  W: The repository 'http://dl.google.com/linux/chrome/deb stable Release' does not have a Release file. N: Data from such a repository can't be authenticated and is therefore potentially dangerous to use. N: See apt-secure(8) manpage for repository creation and user configuration details.   I've seen past inquiries on same issue but the answers given are specific to other repositories.  Any help will be appreciated. Prove: $$x^\alpha-\alpha x \le 1- \alpha  \\ \forall x\ge 0 \ ,  \ 0&lt;\alpha &lt;1$$   It does resamble Lagrange's MVT, in order to get to the RHS, say $\alpha\in[0,1]\Rightarrow \frac{1-\alpha-0^\alpha-\alpha 0}{1-0}=1-\alpha$.  But I'm not sure what to do about the LHS. I always get a negative value in the LHS and we know that the RHS will always be postive but that doesn't really help with proving it...  Edit: maybe define a function: $g(x) = x^\alpha-\alpha x - 1+ \alpha$ Derive it: $g'(x) = \alpha x^{\alpha-1}-\alpha $ but then I get to $x^{\alpha-1}=0$    Also, I had two questions when trying to figure this out:   Can we use deriviation of both sides of an inequality to prove it ? Can we use limit of both sides of an inequaltiy to prove it ?   Thanks. Geomagnetic reversal? List sorting by custom field I dont know how to sort a list of SObject Case.  I have a List&lt;Case&gt; newCases which i iterate over in for cycle, where i save Cases to my new List&lt;Case&gt; casesList.  for (Case newCase : newCases) {             if(newCase.Case_Content__c == '111' &amp;&amp; newCase.Status == '24'){                 String cuid = newCase.CUID__c;                 cuidSet.add(cuid);                 casesList.add(newCase);                 casesList.sort(); //TODO sorting by CUID__c             }         }   What i need to do is, to sort this new list by field CUID__c. So the Cases with same CUID are together.  Thank you for any help :) I have to teach the Bayes formula to some undergraduates, in the form:  $$ P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{P(B|A)P(A)}{P(B)} $$  I was wondering if anyone had any really thoughtful or alternative ways of teaching it instead of the standard approach.