Determine all the permutations $\sigma \in S_5$ that commute with the permutation $\alpha = (1 \space 2)(3 \space 4 \space 5).$
This is a homework problem and I'm having trouble just getting a basic understanding.  I understand that $S_6$ is a symmetric group of degree $6$.  I'm not sure how to start looking for elements.
How can you find and replace text in a file using the Windows command-line environment? Environment variables are not passed to sub scripts while running on crontab I was trying to run one of the scripts (for example, wrapper.sh) which internally executes other scripts using crontab. It works fine without any issues.  When I have created a environment file (which stores the passwords-DB and paths) and trying to execute the main script (wrapper.sh) the cron entry doesn't work on those scripts which are called internally in the wrapper.  NOTE: I tried changing PATH, added env file location before the script in cron but it still didn't work:  9 9 * * * . /$SOMEPATH/some.env; /$SOMEPATH/wrapper.sh   The environment script and wrapper.sh are in same path. when the cron runs it passes the env variables to wrapper but not to the scripts which are called internally in wrapper. Also a mail is received on /var/mail/$user saying the env file as "No such file or Directory". I've gone through many posts but couldn't figure out the resolution. cat file : prints prior to "user :/" prompt Travel grant for undergraduate researcher for attending conference I am an undergraduate student doing independent research. My research work is selected for presentation at a very reputed conference in the US. I am from India. I cannot afford to attend the conference at all. How should I fund my trip? The travel grant for students available from the conference is very low. It can hardly buy me a one way ticket. What are my options? How to add the word "Figure" to the list of figures? quick selection of a random row from a large table in mysql Put two figures side by side There are already several threads that deal with this: they suggest having one figure and putting putting minipages inside, or using the subcaption package and having subfigure environments. However, I've tried both of these approaches and I'm still going wrong:  \begin{figure}[h]  \begin{minipage}[b]{0.2\linewidth}  \begin{tikzpicture}[sibling distance=0pt] \Tree [.TP         [.DP \edge[roof]; {a 'we'} ]         [.T$'$             [.T \node(li){li 'eat$_i$'}; ]             [.VP                 [.Spec t ]                 [.V$'$                      [.DP \edge[roof]; {saka 'rice'} ]                     [.V \node(t){t$_i$}; ] ] ] ] ] \draw [-&gt;] (t) .. controls +(south:3) and +(south:3) .. (li); \end{tikzpicture}  \end{minipage}\hfill  \begin{minipage}[b]{0.2\linewidth}  \begin{tikzpicture}[sibling distance=0pt] \Tree [.TP         [.DP \edge[roof]; {a 'we'} ]         [.T$'$             [.T \node(li){la 'have'}; ]             [.VP                 [.Spec t ]                 [.V$'$                      [.DP \edge[roof]; {saka 'rice'} ]                     [.V \node(t){li 'eaten'}; ] ] ] ] ] \draw [red,-&gt;] (t) .. controls +(south:3) and +(south:3) .. (li); \end{tikzpicture}  \end{minipage}  \end{figure} iOS crash log catch, debug info.. Catch and send via email to the Dev team I've searched the net for such information and found different command lines, like these ones:  sudo apt-get remove application sudo apt-get remove application*  sudo apt-get remove --purge application sudo apt-get remove --purge application*  sudo apt-get purge application sudo apt-get purge application*   So, what is the correct way? Is it necessary to use that "*"?  After that, I also found these commands:  sudo updatedb sudo locate application sudo rm -rf (file/folder name) Ubuntu in VirtualBox wants to install on every boot I have installed Ubuntu 11.10 using Virtual Box in my Windows XP. I have use Ubuntu 11.10 iso to install it. First time when I installed every thing was fine but when I shutdown the VirtualBox and start Ubuntu again, it asks me to install Ubuntu and try Ubuntu without installing.  Do I need to do some extra settings? When app is moved to SD card, where is the app data stored on SD card? How did Stannis's army get north of the wall? If Mance Rayder had to attack Castle Black to use the tunnel to get through the Wall and get out of the north, how did Stannis get his army across the Wall into the north without anyone knowing? If there was another way over the Wall wouldn't the wildings go that way too? Prove $\sup(f+g) \le \sup f + \sup g$ Suppose $D$ is a nonempty bounded subset of reals. Let $f:D \to \mathbb R$ and $g:D \to \mathbb R$. Define $(f+g)(x)=f(x)+g(x)$. Prove $\sup(f+g)(D) \le \sup f(D) + \sup g(D)$ (also prove that $\sup (f+g)$ exists).   I understand why this is the case, just not how to prove it. Left side is pretty much $\sup  (f(x)+g(x))$ and right side is $\sup (f(x) + g(y))$ for $x,\,y \in D$. Basically $f+g$ has to use the same variable and $f(D)+g(D)$ use different ones. But I don't know how to go about proving this.     The second part of the question is to find a specific example where strict inequality holds.    Let $D=\{a,b\}$ and $f: a \to 1,\, b\to 0, \,g: a \to 0,\, b\to 1$. $\sup f(D) = 1,\, \sup g(D) = 1,\, \sup f(D) + \sup g(D) = 2.$ $\sup (f+g)(D) = 1$ (if we choose a, $f+g = 1+0,\, b,\, f+g=0+1$). How do you express that your understanding of something is completely changed? When something blows your mind and changes your understanding of something that you've believed in for years, how do you express it (formally or informally)? "Yesterday" time specification shown when it was actually the day before yesterday Looking at , I see "asked yesterday" (which I really hate; I'd like to see an exact time specification rather than this vague one, especially in the list of questions). But it was not asked yesterday! When I hover over it to show the exact time, it shows it was actually the day before yesterday!  This snapshot was taken at 2012-03-31 15:00 UTC (plus or minus a few minutes):    Whatever different opinions you may have on the implementation, yesterday is a word with a certain meaning. Redefining it is not a good idea. At least rephrase this as "1 day ago" or something. Fluent and Query Expression — Is there any benefit(s) of one over other? LINQ is one of the greatest improvements to .NET since generics and it saves me tons of time, and lines of code.  However, the fluent syntax seems to come much more natural to me than the query expression syntax.  var title = entries.Where(e =&gt; e.Approved)     .OrderBy(e =&gt; e.Rating).Select(e =&gt; e.Title)     .FirstOrDefault();  var query = (from e in entries              where e.Approved              orderby e.Rating              select e.Title).FirstOrDefault();   Is there any difference between the two or is there any particular benefit of one over other? just wondering if anyone has ever tried embedding and actually integrating any js engine into the .net environment. I could find and actually use (after a LOT of pain and effort, since it's pretty outdated and not quite finished) spidermonkey-dotnet project. Anyone with experience in this area? Engines like SquirrelFish, V8..   Not that I'm not satisfied with Mozilla's Spidermonkey (using it for Rails-like miniframework for custom components inside the core ASP.NET application), but I'd still love to explore a bit further with the options. The command-line solutions are not what I'd need, I cannot rely on anything else than CLR, I need to call methods from/to JavaScript/C# objects.  // c# class public class A {     public string Hello(string msg)     {         return msg + " whatewer";     } }  // js snippet var a = new A(); console.log(a.Hello('Call me')); // i have a console.log implemented, don't worry, it's not a client-side code :)   Just to clarify - I'm not trying to actually program the application itself in server-side javascript. It's used solely for writing custom user subapplications (can be seen as some sort of DSL). It's much easier (and safer) to allow normal people programming in js than C#. Simplest code for array intersection in javascript Let $f(x)$ a irreducible polynomial of degree $n$ over a field $F$. Let $g(x)$ be a polynomial in $F[x]$. Prove that every irreducible factor of the composition $f(g(x))$ has a degree which is divisible by $n$.  I don't know even how to begin. I really need help. Thanks From what time I've spent with threads in Java, I've found these two ways to write threads:  With implements Runnable:  public class MyRunnable implements Runnable {     public void run() {         //Code     } } //Started with a "new Thread(new MyRunnable()).start()" call   Or, with extends Thread:  public class MyThread extends Thread {     public MyThread() {         super("MyThread");     }     public void run() {         //Code     } } //Started with a "new MyThread().start()" call   Is there any significant difference in these two blocks of code ?