$\binom{74}{37}-2$ is divisible by :  a) $1369$  b) $38$  c) $36$   d) $none$  $of$ $ these$  I have no idea how to solve this...I tried writing $\binom{74}{37}$ in some useful form but its not helping...any clues?? Thanks in advance!!
how to prove $\forall p$ prime :   $\binom{2p}{p} \equiv 2 \pmod p$ we have:  $\binom{2p}{p} = \frac{2p (2p-1)(2p-3)...1}{p!p!}$ but how to continue?
Is the sorting algorithm used by .NET's Array.Sort() method a  algorithm? In PCA, when the number of dimensions $d$ is greater than (or even equal to) the number of samples $N$, why is it that you will have at most $N-1$ non-zero eigenvectors? In other words, the rank of the covariance matrix amongst the $d\ge N$ dimensions is $N-1$.  Example: Your samples are vectorized images, which are of dimension $d = 640\times480 = 307\,200$, but you only have $N=10$ images. Change \textwidth and \textheight in mid-document I set my \textwidth and \textheight in the preamble and would like to change both for some pages. How can this be done? E.g., the following example for \textwidth appears to work for the page number, but not for the text:  \documentclass{article}  \textwidth=5cm \textheight=5cm \begin{document} A \hfill B \eject \textwidth=10cm A \hfill B \end{document}   Output (true to scale): Find all polynomials $P(x)$ such that $P(x^2)=P(x)^2$ Find all polynomials $P:\mathbb{C}\rightarrow\mathbb{C}$ such that $$P(x^2)=P(x)^2 .$$   Here is what I tried:  First, it is easy to see the constant solutions, namely $P\equiv 0,P\equiv 1$.  Let $r$ be a root of $P$ (i.e. $P(r)=0$). It follows that all terms in the infinite sequence $r,r^2,r^4,\dots,r^{2n},\dots$ are roots of $P$. To avoid a polynomial with infinite roots, we have to have the sequence be periodic. So it turns out that either $r=0$, or the roots of $P$ are roots of unity of degree $2n$ for some $n\in\mathbb{N}$.  In the first case, we get $P(x)=xQ(x)$. Plugging in, we get $$x^2Q(x^2)=x^2Q(x)^2\implies Q(x^2)=Q(x)^2 ,$$ i.e. that $Q$ satisfies the same condition as $P$.  In the second case, we have $P(x)=(x^{2n}-1)Q(x)$. Plugging in, we get $(x^{2n}-1)Q(x^2)=(x^{2n}-2x^n+1)Q(x)^2$. I was not sure how to proceed from here. Are C++ enums signed or unsigned? And by extension is it safe to validate an input by checking that it is &lt;= your max value, and leave out >= your min value (assuming you started at 0 and incremented by 1)? C# difference between == and Equals() Pretty printing XML with javascript I have a string that represents a non indented XML that I would like to pretty-print. For example:  &lt;root&gt;&lt;node/&gt;&lt;/root&gt;   should become:  &lt;root&gt;   &lt;node/&gt; &lt;/root&gt;   Syntax highlighting is not a requirement. To tackle the problem I first transform the XML to add carriage returns and white spaces and then use a  tag to output the XML. To add new lines and white spaces I wrote the following function:   function formatXml(xml) {     var formatted = '';     var reg = /(&gt;)(&lt;)(\/*)/g;     xml = xml.replace(reg, '$1\r\n$2$3');     var pad = 0;     jQuery.each(xml.split('\r\n'), function(index, node) {         var indent = 0;         if (node.match( /.+&lt;\/\w[^&gt;]*&gt;$/ )) {             indent = 0;         } else if (node.match( /^&lt;\/\w/ )) {             if (pad != 0) {                 pad -= 1;             }         } else if (node.match( /^&lt;\w[^&gt;]*[^\/]&gt;.*$/ )) {             indent = 1;         } else {             indent = 0;         }          var padding = '';         for (var i = 0; i &lt; pad; i++) {             padding += '  ';         }          formatted += padding + node + '\r\n';         pad += indent;     });      return formatted; }   I then call the function like this:  jQuery('pre.formatted-xml').text(formatXml('&lt;root&gt;&lt;node1/&gt;&lt;/root&gt;'));   This works perfectly fine for me but while I was writing the previous function I thought that there must be a better way. So my question is do you know of any better way given an XML string to pretty-print it in an html page? Any javascript frameworks and/or plugins that could do the job are welcome. My only requirement is this to be done on the client side. How to parse hex values into a uint? uint color;  bool parsedhex = uint.TryParse(TextBox1.Text, out color);  //where Text is of the form 0xFF0000 if(parsedhex)    //...   doesn't work. What am i doing wrong? What is this shell/Bash syntax: someVariable=someValue someCommand If $\sum a_n$ converges and $b_n=\sum\limits_{k=n}^{\infty}a_k $, prove that $\sum \frac{a_n}{b_n}$ diverges Let $\displaystyle \sum a_n$ be  convergent series of positive terms and set  $\displaystyle b_n=\sum_{k=n}^{\infty}a_k$ , then prove that $\displaystyle\sum \frac{a_n}{b_n}$ diverges.   I could see that $\{b_n\}$ is monotonically decreasing sequence converging to $0$ and I can write $\displaystyle\sum \frac{a_n}{b_n}=\sum\frac{b_n-b_{n+1}}{b_n}$, how shall I proceed further? I have a DB Server running CentOS 6.4 and MySQL 5.5 from RPMs (not source).  I have dozens of databases but one of them is being hit quite hard with large selects which locks out other updates which in turn locks out other selects (at least I think this is what's happening!)  To alleviate this I would like to replicate one of the databases onto the same server so that the writes can write to the "master" and reads can be done on the "slave". I don't have the luxury of additional hardware.  I have found a lot of guides on the internet describing how to replicate to a different server, and one () describing how to replicate to the same server but it doesn't seem to be geared towards retrofitting this functionality to an existing database with 10s of GB of data in it already. It also talks about compiling MySQL from source.  Does anyone know how I can retrospectively convert my MySQL 5.5 config to replicate one of my databases on the same server?  My existing DB runs on port 3306 so I feel it should be as easy as firing up another instance of MySQL on port 3307 and making some changes to my.cnf but not sure what changes.  Thanks in anticipation. $$ 3x+6y+5z=7 $$ The general solution to this linear Diophantine equation is as described    $$ x = 5k+2l+14 $$ $$ y = -l $$ $$ z = -7-k $$ $$ k,l \in \mathbb{Z} $$  If I plug the original equation into  the solution is: $$ y = 5n+2x+2  $$ $$ z =-6n-3x-1 $$ $$ n \in \mathbb{Z} $$  I can rewrite this as:  $$ x = l $$ $$ y = 5k+2l+2 $$ $$ z = -6k-3l-1 $$ $$ k,l \in \mathbb{Z} $$  However now two equations depend on two variables ($k,l$) and one on one variable $l$. In the first solution one equation depends on two variables and two on one variable.  Questions:  How can I come from a representation like the one from wolfram alpha for the general solution to one where all equations depend on one distinct variable except one equation.  Is there always such a representation? I'm trying to find the derivative of $\sin^{-1}(x)$. I know the steps that lead to $\frac{1}{\sqrt{1-x^2}}$, however I don't understand why the following reasoning leads to a wrong answer.  Because  $$\frac{d}{dx}f^{-1}(x) = \frac{1}{f'(f^{-1}(x))} $$  If we plug in for $f(x) = \sin(x)$, and because $\frac{d}{dx}\sin(x) = \cos(x)$ we get  $$\frac{d}{dx}\sin^{-1}(x) = \frac{1}{\cos(\sin^{-1}(x))} $$  Since $\sin(x) = \cos(x-\frac{\pi}{2})$, we can state that $\sin^{-1}(x) = \cos^{-1}(x-\frac{\pi}{2})$. (I suspect this is what is wrong)  Thus,  $$\frac{d}{dx}\sin^{-1}(x) = \frac{1}{\cos(\cos^{-1}(x-\frac{\pi}{2}))}$$  Then, by the definition of inverse function, we have  $$\frac{d}{dx}\sin^{-1}(x) = \frac{1}{x - \frac{\pi}{2}}$$ Story where the hero ends up having his eyelids removed and being hooked up to some kind of virtual reality machine Trying to identify a short science fiction story where the protagonist is some kind of policeman or investigator, and at the end he chooses to live his life out hooked up to some kind of virtual reality machine. The ending describes him having his eyelids removed as he is attached to the machine, and then as the machine starts working he "becomes" a cowboy or some other western figure riding his horse downhill in a desert scene. onConfigurationChanged not getting called Why should the Standard Model be renormalizable? Effective theories like Little Higgs models or Nambu-Jona-Lasinio model are non-renormalizable and there is no problem with it, since an effective theory does not need to be renormalizable. These theories are valid until a scale $\Lambda$ (ultraviolet cutoff), after this scale an effective theory need a UV completion, a more "general" theory.  The Standard Model, as an effective theory of particle physics, needs a more general theory that addresses the phenomena beyond the Standard Model (an UV completion?). So, why should the Standard Model be renormalizable? I'm trying to migrate a MySQL-based app over to Microsoft SQL Server 2005 (not by choice, but that's life).  In the original app, we used almost entirely ANSI-SQL compliant statements, with one significant exception -- we used MySQL's group_concat function fairly frequently.  group_concat, by the way, does this: given a table of, say, employee names and projects...  SELECT empName, projID FROM project_members;   returns:  ANDY   |  A100 ANDY   |  B391 ANDY   |  X010 TOM    |  A100 TOM    |  A510   ... and here's what you get with group_concat:  SELECT      empName, group_concat(projID SEPARATOR ' / ')  FROM      project_members  GROUP BY      empName;   returns:  ANDY   |  A100 / B391 / X010 TOM    |  A100 / A510   So what I'd like to know is: Is it possible to write, say, a user-defined function in SQL Server which emulates the functionality of group_concat?   I have almost no experience using UDFs, stored procedures, or anything like that, just straight-up SQL, so please err on the side of too much explanation :) Is the use of "one of the" correct in the following context? Reproject GeoJSON from EPSG:3857 to EPSG:4326 Can you decide to counterspell after another counterspell has failed?