combinatorial proof of Vandermonde's Identity
Combinatorial proof of summation of $\sum\limits_{k = 0}^n {n \choose k}^2= {2n \choose n}$
To debug a JACK/Pulseaudio issue, I want to understand when and why the pulseaudio daemon is started by systemd (on Fedora).  Using:  $ ps -o'pid,ppid,args' `pgrep pulse`   I see that the pulseaudio daemon is being started by systemd (pid=1)   PID  PPID COMMAND 2738     1 /usr/bin/pulseaudio --start   However, I was unable to find any unit-file on my system containing pulseaudio or even just pulse.  My specific questions are:  A) Is there a way to determine the systemd unit that caused the creation of a specific process (in my example output, process 2738, the PA daemon)?  B) Are there alternative approaches to find out which unit-dependency chain or other settings of systemd resulted in the invocation of /usr/bin/pulseaudio --start? What should be in the sources.list for Kali Rolling Linux? This is exact content from my sources.list:  #  # deb cdrom:[Debian GNU/Linux 2016.1 _Kali-rolling_ - Official Snapshot amd64 LIVE/INSTALL Binary 20160830-11:29]/ kali-rolling contrib main non-free #deb cdrom:[Debian GNU/Linux 2016.1 _Kali-rolling_ - Official Snapshot amd64 LIVE/INSTALL Binary 20160830-11:29]/ kali-rolling contrib main non-free   I couldn't install the Synaptic package manager and was told to modify sources.list.  I googled a lot and added some repos into it and ran update and upgrade command only to see Kali crash... I had to recover it.  Please mention the exact content that should be there to ensure that Kali Linux works. Best way to center a  on a page vertically and horizontally? Best way to center a &lt;div&gt; element on a page both vertically and horizontally?  I know that margin-left: auto; margin-right: auto; will center on the horizontal, but what is the best way to do it vertically, too? JavaScript: Bitwise shift of long long number I need to bitwise shift a value 64 times in JavaScript. But JavaScript starts rounding after 32.  For example:  for(var j = 0; j &lt; 64; j++) {     mask = mask &lt;&lt; 1;     console.log(mask); }   This prints value from 0 to 1073741824 but then rounds of and starts printing 0. How can we get "Developer" bugs logged with Technical Support? I recently found a bug, which I mentioned on , as well as submitting a case, and after a few days fumbling around with the case, they closed it by saying that they only provide this type of support for Premier Support customers.  Is there a better channel for reporting bugs than by submitting a case, or is there something "special" we have to do? I realize this question is similar to , except that I've already gone the route of , as suggested by the answer there (i.e. submitting a case), to no avail.  How can I make sure that a bug like this can find its way to someone capable of at least confirming that the behavior was expected? This behavior isn't documented anywhere, and so I assume that it's a bug that needs to be addressed. Calculating $\lim_{n\to\infty}\sqrt{n}\sin(\sin...(\sin(x)..)$ CPU clock speed versus CPU core count - higher GHz, or more cores for SQL Server? We are beginning to provision a set of physical servers for a virtual cluster of SQL Server 2016 nodes within VMware. We will be utilizing Enterprise Edition licenses.  We plan on setting up 6 nodes, but there is a bit of a debate on what the ideal way to provision the physical servers with regards to CPU clock speed versus CPU core count.  I know this is largely dependent on transaction volume and number of databases stored among other software-specific factors, but is there a general rule of thumb that is advised?  For instance, is a dual 8-core, 3.2 GHz physical server (16 cores) more preferential to a dual 16-core, 2.6 GHz server (32 cores)?  Has anyone come across a white paper that further delves into this type of topic? As I have seen in other posts, i.stack.imgur.com is blocked by numerous companies' firewalls including my own. I have started the process to get the restriction lifted but to increase the likelihood of it being granted, have only requested that i.stack.imgur.com be unblocked.  Staying within the context of SO, imgur and the hosting arrangement SO has with them, are there any other sub-domains/domains that I should be requesting be unblocked? i.e. p.stack.imgur.com or *.stack.imgur.com?  I'm not sure if images can still be embedded/hotlinked in posts from other image hosts or not but if they can I'm not interested in unblocking those hosts, just the addresses used by SO for imgur.  Cheers :) How can I convert String to Int? Do sets, whose power sets have the same cardinality, have the same cardinality? Is it generally true that if $|P(A)|=|P(B)|$ then $|A|=|B|$?  Why? Thanks. Showing that a group of order $pq$ is cyclic if it has normal subgroups of order $p$ and $q$ This is question 2.9 #9 from Topics in Algebra by Herstein:     If $o(G)$ is $pq$ where $p$ and $q$ are distinct prime numbers and if $G$ has a normal subgroup of order $p$ and a normal subgroup of order $q$, prove that $G$ is cyclic.   Let $N$ denote the normal subgroup of order $p$ and $M$ denote the normal subgroup of order $q$. Here are a couple things I noted while exploring the problem:   $N$ and $M$ are cyclic, since they have prime order. $G = NM$ [1].   Any hints where to look next?    [1] Since $N,M$ are normal, $NM \le G$. Then $o(NM) \mid o(G)$, so $o(NM)$ is either $1$, $p$, $q$, or $pq$. The first three don't work because they force one of $N$ or $M$ to be $(e)$, so we have $o(NM) = pq$. Therefore, $NM = G$. I just updated from Ubuntu 10.10 to 11.04 and have an issue regarding the screen resolution. I have Intel integrated gfx chip and my monitor supports resolutions larger than 1024x768. (in 10.10 I've been using 1280x1024) But as soon as I upgraded, I'm stuck with 1024x768 resolution and seems I can't change it.  running xrandr In terminal yields the following results,   Screen 0: minimum 320 x 200, current 1024 x 768, maximum 4096 x 4096 LVDS1 connected 1024x768+0+0 (normal left inverted right x axis y axis) 0mm x 0mm    1280x800       58.1 +    1024x768       60.0*     800x600        60.3     56.2      640x480        59.9   VGA1 connected 1024x768+0+0 (normal left inverted right x axis y axis) 344mm x 194mm    1366x768       59.9 +    1360x768       60.0      1024x768       75.1     72.0     70.1     60.0*     832x624        74.6      800x600        72.2     75.0     60.3     56.2      640x480        72.8     75.0     66.7     60.0      720x400        70.1     1280x1024_60.00 (0xce)  109.0MHz         h: width  1280 start 1368 end 1496 total 1712 skew    0 clock   63.7KHz         v: height 1024 start 1027 end 1034 total 1063           clock   59.9Hz   What maybe the problem? Is it a bug? What kind of steps I should take in order to get a higher resolution? (changing xorg.conf maybe?) Any insight is highly appreciated. Thanks in advance.   UPDATE    Screenshot after running xrandr --addmode VGA1 1360x768  As you can see, side bar is not completely visible and Ubuntu logo at the task bar is missing. Also when you open an application, the Task bar of the application (where it should go to the top panel) is missing as well.. Uploading video to youtube in right format Let $E$ be a finite dimensional vector space over field $\mathbb R$ with $E^*$ as the dual space of $E$.($\dim E=n$)   $\Omega^P(E):=\{ \alpha\colon \overbrace{E\times\cdots\times E}^{p- times}\rightarrow \mathbb R\  \ , \alpha \text{ is alternating multilinear map}\}$ $\Omega^P(E^*):=\{ u\colon \overbrace{E^*\times\cdots\times E^*}^{p- times}\rightarrow \mathbb R\  \ , u \text{ is alternating multilinear map}\}$  I wish to prove that there exist an $\textbf{unique}$ bilinear form $B\colon \Omega^p(E^*)\times\Omega^p(E)\rightarrow\mathbb R$   such that for all $ f_i\in E^* \ , u_j\in E^{**}\simeq E \qquad  i,j=1,\cdots,p$, it has the following rule:  $B(u_1\wedge\cdots\wedge u_p,f_1\wedge\cdots\wedge f_p)=\det[f_i(u_j)]$.  $\wedge$ is wedge product between alternating maps i.e. $\alpha\wedge\beta(v_1,\cdots,v_{p+p})=\dfrac{1}{p! p!}\displaystyle\sum_{\sigma\in S_{p+p}}sign(\sigma)\alpha(v_{\sigma(1)},\cdots,v_{\sigma(p)})\beta(v_{\sigma(p+1)},\cdots,v_{\sigma(p+p)})$  How can I do this? Make Launchpad Appear on screen with mouse If that's not possible, then a way to trigger the Dock on the screen with mouse without moving my cursor to the bottom of the screen is also an acceptable solution (I know that the launchpad appears on the screen with Dock)  Also i did see the  about this. Drush 6 works great of corse on a Drupal 6 website, but will I have all of the Drush 6 features when working on a Drupal 7 website? Should I use Drush 7 when working on a drupal 7 website?  I found this post that says you can switch between the two version easily, is this correct?  switch to drush6 command: composer global require drush/drush:6.*  switch to drush7 command: composer global require drush/drush:dev-master Pre-generate Review audits to avoid the lag? real memory usage Another problem from quora.  What can be said about $v =\prod_{s=2}^{\infty} \zeta(s) $?  Wolfy says that $v \approx 2.294856591673313794183 $.  The Inverse Symbolic Calculator () finds nothing.  Here are the close values:  $2294856391585950 = (0064) sum((7/6*n^3-3*n^2+65/6*n-3)/(Fibo(n)+1),n=1..inf)\\ 2294856397653578 = (0001) GAM(5/6)*BesI(1,2)^{GAM(7/12)}\\ 2294856469349552 = (0006) 1/4357571\\ 2294856473388934 = (0324) 1/10*(10+14^{1/2}*10^{1/4})^{1/2}*10^{3/4}\\ 2294856488490537 = (0001) BesJ(0,1)^{GAM(1/3)/GAM(5/12)}\\ 2294856524215021 = (0314) sin(Pi*8/45)-sin(Pi*14/51)\\ 2294856591673313 \text{ would be here}\\ 2294856740350177 = (0011) sum((2*n^3-5*n^2+22*n-14)/(n!+2),n=1..inf)\\ 2294856991319600 = (0324) 23^{1/2}/(12^{3/4}-3^{2/3})^{1/2}\\ 2294857017919943 = (0001) Feig2*Li4(1/2)/GAM(5/6)\\ 2294857089879260 = (0248) F(11/26;27/50;1) \\ 2294857493303450 = (0404) Psi(1/21)+Psi(19/21)+Psi(13/14)\\ $  I tried using Euler's zeta product and partition sum, but this didn't seem to help. Here is what resulted:  $\zeta(s) =\dfrac1{\prod_p (1-p^{-s})} $.  $\begin{array}\\ v &amp;=\prod_{s=2}^{\infty} \zeta(s)\\ &amp;=\prod_{s=2}^{\infty} \dfrac1{\prod_p (1-p^{-s})}\\ &amp;=\prod_{s=2}^{\infty} \prod_p\dfrac1{ (1-p^{-s})}\\ &amp;= \prod_p\prod_{s=2}^{\infty}\dfrac1{ (1-p^{-s})}\\ \end{array} $  Euler's product identity states that $\prod_{n=1}^{\infty}\dfrac1{1-zx^n} =\sum_{n=1}^{\infty} \dfrac{x^nz^n}{\prod_{k=1}^n(1-x^k)} $. Putting $x=z=\dfrac1{p} $, $\prod_{n=2}^{\infty}\dfrac1{1-p^{-n}} =\sum_{n=1}^{\infty} \dfrac{p^{-2n}}{\prod_{k=1}^n(1-p^{-k})} $.  That's as far as I can go. Is there a way to select the children of a parent object?