$\lim_{ n\to \infty} \frac{n!}{n^n}$ via L'Hospital's rule
What's the limit of the sequence $\lim\limits_{n \to\infty} \frac{n!}{n^n}$?
In naive set theory, I believe ∅ = {∅} = {{∅}} is correct, but just wanted to make sure that I understood this correctly.   ∅ is an empty set, so having an empty set as an element of a set that contains nothing else is pretty much the same thing as e.g. marking the number 2 as 2.000000, right? Doesn't the symbol ∅ intrinsically imply that this is an empty set which contains it self? Do you have to take the second leg of a domestic flight? This is the command I entered in the command block: testfor @p {Inventory:[{id:minecraft:diamond,Count:10}]}  But it doesn't seem to work. What am I doing wrong? Executing multiple commands from a Windows cmd script I'm trying to write a Windows cmd script to perform several tasks in series. However, it always stops after the first command in the script.  The command it stops after is a maven build (not sure if that's relevant).  How do I make it carry on and run each task in turn please?  Installing any software or configuring the registry etc is completely out of the question - it has to work on a vanilla Windows XP installation I'm afraid.  Ideally I'd like the script to abort if any of the commands failed, but that's a "nice to have", not essential.  Thanks. Show that there exists a unique $v_0 \in H^1(0,1)$ such that $u(0)=\int_0^1(u'v_0'+uv_0), \forall u \in H^1(0,1)$ Show that there exists a unique $v_0 \in H^1(0,1)$ such that $u(0)=\int_0^1(u'v_0'+uv_0), \forall u \in H^1(0,1)$. Further Show that $v_0$ is the solution of some differential equation with appropriate boundary conditions. Compute $v_0$ explicitly.   Let $f: H^1(0,1) \to \mathbb{R}$ be defined by $f(u)=u(0)$. Then I showed that $f$ is linear and continuous. Hence there exists $v_0 \in H^1(0,1)$ such that $u(0)=\int_{0}^1u'v_0'+\int_{0}^1uv_0$  The differential equation is given by $f=-u''+u$ with the conditions $u'(0)=u'(1)=0$.   How do I compute $v_0$ explicitly?? Make a file pointer read/write to an in-memory location What is the difference between "du -sh *" and "du -sh ./*"? What's the difference between du -sh * and du -sh ./* ?  Note: What interests me is the * and ./* parts. What is the black thing on the circuit What is the black thing on the below circuit?   Is it used to hide the IC (or other components) that is used on the circuit?   What material is it? prove inequality through mean value theorem Suppose we have a sample of size n= 9, where the sum equals 27, and the sum of the squaredvalues equals 113. Sample mean is 27/9=3 and mean of squared values is 113/9 = 12.56 Can we then say that the sample variance $S^2 = E(\overline X^2) - E(\overline X)^2 = 12.55 - 9 = 3.56$ ? Difference between $ and # in Linux environment In a  a doubt was voiced about my answer, which claims GCM to requires $2^{128}$ for a successful forgery. The doubt was that the square root needs to be taken meaning the security would be $2^{64}$.  So of course I immediately checked the relevant security theorem ():  $$\mathbf{Adv}^{\text{auth}}_{\operatorname{GCM}[\operatorname{Perm}(n),\tau]}(\mathcal A)\leq \frac{0.5(\sigma+q+q'+1)^2}{2^n}+\frac{q'(\ell_A+1)}{2^\tau}$$  with $\sigma$ being the total plaintext size in blocks, $q$ being the total number of encryption queries, $q'$ being the total number of decryption queries, $n$ being the size in bits of the underlying permutation, $\ell_A$ being maximum authenticated input length in blocks and $\tau$ being the tag size in bits.  Now clearly from this we can see that performing $2^{n/2}=2^{64}$ queries yields a strong enough advantage for a break.  Now my question is: When talking about "n-bits of security" or "needs $2^n$ operations to break", do we usually talk about "online" security, ie queries to an oracle can be done and cost $1$ or "offline" security where no oracles are available or do we need to make this decision depending on the situation?  TL;DR: Does GCM offer 64 or 128 bit security? lstlisting directly after section start removes margin before next section If I place an lstlisting directly after the start of a (sub)section, the margin before the following (sub)section (title) disappears.  Here is a an example document showing the issue:  \documentclass[twocolumn]{article} \usepackage{listings} \begin{document}  \section{Lorem ipsum} Consectetur adipiscing elit.  \begin{lstlisting} Lorem \end{lstlisting}  Sed quis ante nec mi volutpat congue eu id leo.  \section{Dolor sit amet} Suspendisse at lorem mattis, semper massa id, euismod risus.  \newpage  \section{Lorem ipsum} \begin{lstlisting} Lorem \end{lstlisting}  Sed quis ante nec mi volutpat congue eu id leo.  \section{Dolor sit amet} Suspendisse at lorem mattis, semper massa id, euismod risus.  \end{document}   And you can see the (annotated) result here:    Why is there no margin before the fourth section (title)? Is this a bug in my LaTeX environment (MacTeX-2013), or is this the expected behavior? Natural Sounding Text to Speech? I need to make Windows 7 STOP disabling the wifi adapter when the wired LAN is connected Will an object always fall at an infinite speed in a black hole? What is the longest word you can come up with that the letters are all in alphabetical order? It's Friday again, how about some fun to get us into the weekend?  What is the longest word you can come up with for which all the letters in that word are in alphabetical order?   Rules:   English words only    Can't be a name of a place, person or other proper noun.   if it contains the same letter twice in a row, that does not disqualify it.   No fair looking it up on Google!   Update: One more rule to help you guys out.   The word can be in either ascending or descending alphabetical order. Let $H$, $K$ be subgroups of $G$. Prove that $o(HK) = \frac{o(H)o(K)}{o(H \cap K)}$.  I need this theorem to prove something. SF Short Story: alien apes invade during nuclear war TSQL Script Execution Through Power shell, Special Characters Rendering Problem We have a simple T-SQL script file (.sql), which we execute through Powershell. Everything works fine, except the script portion, where we are using some special characters as string (example: 'ô' ).   The same script works fine, if I just execute it through SQL Server Management Studio, but through Powershell it's not rendering these special characters properly.  Here how we access the script file to execute on all databases on instance  Add-PSSnapin SqlServerCmdletSnapin100 Add-PSSnapin SqlServerProviderSnapin100 invoke-Sqlcmd -InputFile "C:\DailyPatch\run.sql" | Out-File -filePath "C:\DailyPatch\dbname.txt"  $text=Get-Content C:\DailyPatch\dbname.txt $a = $text.length $a=$a-3  $logFile = 'C:\DailyPatch\dblog.txt'  for ($i=3 ; $i -le $a ;$i++) {        write-host $text[$i]     $error.clear()      Invoke-sqlcmd -inputfile "C:\DailyPatch\Script4AllDBs.sql"-database $text[$i]-OutputSqlErrors $true|Out-File -filePath "C:\DailyPatch\dbname.txt"      foreach ($rr in $error) {         $text[$i] &gt;&gt; $logFile         $rr &gt;&gt; $logFile     } }   How to set collation in Powershell to avoid such kind of issues?  ps. Unicode thing is already handled on script side  Added: Special characters are being changed to � (Replacement Character)