With the standard notations, what's $|Z(Sym(n))|$ equal to, in terms of $n$?
Find the center of the symmetry group $S_n$.   Attempt:  By definition, the center is $Z(S_n) = \{ a \in S_n : ag = ga \forall\  g \in S_n\}$. Then we know the identity $e$ is in $S_n$ since there is always the trivial permutation. Suppose $a$ is in $S_n$, but not equal to identity. Now we can imagine the permutation as bijective function that maps from $\{1,2,\dotsc,n\}$ to $\{1,2,\dotsc,n\}$.  So suppose $p$ is a permutation map. Then $p$ maps from a location $i$ to a location $j$. Take $p(i) = j$ where $i\neq j$. Let $k$ be in $\{1,2,\dotsc,n\}$, where $k$, $i$ and $j$ are all different elements. The cycle $r = (jk)$, then we will see if this commutes.  $rp(i) = rj$    Can someone please help me, I am stuck?  Thank you.
Let$\ p_n$ be the$\ n$-th prime. Can you give me a proof for$\ \prod_{i=1}^\infty \frac{p_i-1}{p_i}=P\approx \frac{1}{11.0453}$? I found$\ \prod_{i=1}^\infty \frac{p_i}{p_i-1}\approx 11.0453$ on . Moreover, writing a paper, should one provide a proof or it is trivial? Thanks in advance. Variation of the metric with respect to the metric For a variation of the metric $g^{\mu\nu}$ with respect to $g^{\alpha\beta}$ you might expect the result (at least I did):  \begin{equation} \frac{\delta g^{\mu\nu}}{\delta g^{\alpha\beta}}= \delta^\mu_\alpha\delta^\nu_\beta. \end{equation}  but then to preserve the fact that $g^{\mu\nu}$ is symmetric under interchange of $\mu$ and $\nu$ we should probably symmetrise the right hand side like this:  \begin{equation}\frac{\delta g^{\mu\nu}}{\delta g^{\alpha\beta}}= \delta^\mu_\alpha\delta^\nu_\beta + \delta^\mu_\beta\delta^\nu_\alpha.\end{equation}  Is this reasonable/correct? If not, why not?  It seems that I can derive some weird results if this is right (or maybe I'm just making other mistakes). How to theme the maintenance mode page? What would be the best practice for theming the maintenance mode page in Drupal 7? Quantifiers "most" vs. "most of" Is there a way to get the raw SQL string executed when calling PDOStatement::execute() on a prepared statement? For debugging purposes this would be extremely useful. Can we denied permission for synchronized data extension in Salesforce Marketing Cloud, My requirement is, I don't want to allow users to access synchronized data extension either in contact builder or in Subscribers. Maybe the underlying question is what the Voltage-Current curve looks like. Can I drive it from a voltage source (like you drive a heater) or from a current source (like you drive an LED)? Or even different than those two options?  ADDITIONAL1: Say (hypothetically) I have two commercially available identical Peltier's, they are spec'd 6V/3A. Can I connect these in series to a 12Vdc power supply without any worries?  CONCLUSION1: Current/Voltage load curve is reasonably linear, both driving from current or voltage source will do fine as long as the device is operated within its spec. (Olin Lathrop, Russell McMahon)  CONCLUSION2: Don't drive a Peltier from PWM, power loss due to current increase, grows more rapidly than cooling power. (Olin Lathrop)  CONCLUSION3: Beware the mechanical wear of the device with continues cycling. Eg. don't use a thermostat on/off controller. (Russell McMahon) How to know where shell variables and functions are set? When I type the set command in my system I've got this extract out :  __colormgr_commandlist='     create-device     create-profile     delete-device     delete-profile     device-add-profile     device-get-default-profile     device-get-profile-for-qualifier     device-inhibit     device-make-profile-default     device-set-kind     device-set-model     device-set-serial     device-set-vendor     find-device     find-device-by-property     find-profile     find-profile-by-filename     get-devices     get-devices-by-kind     get-profiles     get-sensor-reading     get-sensors     get-standard-space     profile-set-filename     profile-set-qualifier     sensor-lock     sensor-set-options     ' __grub_script_check_program=grub-script-check _backup_glob='@(#*#|*@(~|.@(bak|orig|rej|swp|dpkg*|rpm@(orig|new|save))))' _xspecs=([freeamp]="!*.@(mp3|ogg|pls|m3u)" [cdiff]="!*.@(dif?(f)|?(d)patch)?(.@([gx]z|bz2|lzma))" [bibtex]="!*.aux" [rgview]="*.@(o|so|so.!(conf|*/*)|a|[rs]pm|gif|jp?(e)g|mp3|mp?(e)g|avi|asf|ogg|class)" [oowriter]="!*.@(sxw|stw|sxg|sgl|doc?([mx])|dot?([mx])|rtf|txt|htm|html|?(f)odt|ott|odm)" [chromium-browser]="!*.@(?([xX]|[sS])[hH][tT][mM]?([lL]))" [tex]="!*.@(?(la)tex|texi|dtx|ins|ltx|dbj)" [netscape]="!*.@(?([xX]|[sS])[hH][tT][mM]?([lL]))" .../.. _xinetd_services ()  {      local xinetddir=/etc/xinetd.d;     if [[ -d $xinetddir ]]; then         local restore_nullglob=$(shopt -p nullglob);         shopt -s nullglob;         local -a svcs=($( printf '%s\n' $xinetddir/!($_backup_glob) ));         $restore_nullglob;         COMPREPLY+=($( compgen -W '${svcs[@]#$xinetddir/}' -- "$cur" ));     fi } dequote ()  {      eval printf %s "$1" 2&gt; /dev/null } quote ()  {      local quoted=${1//\'/\'\\\'\'};     printf "'%s'" "$quoted" } quote_readline ()  {      local quoted;     _quote_readline_by_ref "$1" ret;     printf %s "$ret" }   I checked all files in my knowledge as /etc/profile, /etc/environment, and/or ~/.bashrc. I didn't find any generation  script or code calling.   Do you have any advice where is come from ? I want to play a game in an older update, but can't see a way to choose a specific version. Is this possible? Beryllium Vacuum Sphere Boat/Aircraft Design simulates perfectly but won't work on FPGA Baker-Hausdorff Lemma from Sakurai's book I'd like to show that, given to hermitian operators $A,G$ on a Hilbert space $\mathscr{H}$, the following identity holds: $$ e^{iG\lambda}A e^{-iG\lambda} = A + i\lambda [G,A] + \frac{\left(i\lambda\right)^2}{2!}[G,[G,A]]+\ldots+\frac{(i\lambda)^n}{n!}\underbrace{[G,[G,[G,\ldots[G}_{n\ times},A]]]\ldots]+\ldots $$ where $\lambda$ denotes a real parameter and $[\ \!,]$ indicates the commutator.  This is a proof left to the reader by Sakurai in his books on Modern Quantum Mechanics. How to make dot match newline characters using regular expressions "Always on Top" Windows with Java In Java, is there a way to have a window that is "Always on top" regardless if the user switches focus to another application?  I've searched the web, and all of the solutions lean to some sort of JNI interface with native bindings.  Truly this can't be the only way to do it?.. or is it? Proof by Induction: For any integer $n > 0$, $3|(5^{2n} − 1)$. Line numbering paragraph oriented? I'm using the lineno package with the pagewise option but I need a way to get the numbering restart every paragraph break. I need a general solution to pass to the latex command, e.g.:  latex "\AtBeginDocument{\RequirePackage[pagewise]{lineno}\linenumbers}\input{mydocument.tex}"   I read the lineno manual but the package does not provide this feature. Is there any trick to do that? Complete residue system proof I was given a proof of this in class, but it does not make sense to me, can someone provide a better proof, or explain the one I have provided, thanks.  Suppose $\gcd(a,n) = 1$, where $a$ and $n$ are natural numbers. Prove that $0$, $a$, $2a$, $3a,\ldots, (n-1)a$, is complete residue system modulo $n$.  Proof given in class:  \begin{align*} ka &amp;\equiv ra \pmod{n}\\ &amp;\Rightarrow  k \equiv r \pmod{n}\\ &amp;\Rightarrow  \text{$k-r$ divides $n$}\\ &amp;\Rightarrow \text{$k=r$ because $0\leq k\lt n$, $0\leq r\lt n$} \end{align*} How to write an RSS feed with Java? I'm using Java, and need to generate a simple, standards-compliant RSS feed. How can I go about this? Image of the tensor product of strict maps of Banach spaces How to recursively download a folder via FTP on Linux I'm trying to ftp a folder using the command line ftp client, but so far I've only been able to use 'get' to get individual files.