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Unix Truncate Column in csv file Where's the error in this $2=1$ fake proof? I'm reading Spivak's Calculus:  2 What's wrong with the following &quot;proof&quot;? Let $x=y$. Then $$x^2=xy\tag{1}$$ $$x^2-y^2=xy-y^2\tag{2}$$ $$(x+y)(x-y)=y(x-y)\tag{3}$$ $$x+y=y\tag{4}$$ $$2y=y\tag{5}$$ $$2=1\tag{6}$$  I guess the problem  is in $(3)$, it seems he tried to divide both sides by $(x-y)$. The operation would be acceptable in an example such as:  $$12x=12\tag{1}$$ $$\frac{12x}{12}=\frac{12}{12}\tag{2}$$ $$x=1\tag{3}$$  I'm lost at what should be causing this, my naive exploration in the nature of both examples came to the following: In the case of $12x=12$, we have an imbalance: We have $x$ in only one side then operations and dividing both sides by $12$ make sense. Also, In  $\color{red}{12}\color{green}{x}=12$ we have a $\color{red}{coefficient}$ and a $\color{green}{variable}$, the nature of those seems to differ from the nature of $$\color{green}{(x+y)}\color{red}{(x-y)}=y(x-y)$$ It's like: It's okay to do the thing in $12x=12$, but for doing it on $(x+y)(x-y)=y(x-y)$ we need first to simplify $(x+y)(x-y)$ to $x^2-y^2$. How can I make a peer-to-peer multiplayer game? How can I make a p2p multiplayer game? I would like to have a server-less multiplayer game. But then, how all the clients know each other?  Why the p2p-protocol is so famous in file transfer but not in multiplayer games? Test if there are files matching a pattern in order to execute a script clerics have the ability  (PHB, p. 62):     When a creature within 5 feet of you that you can see hits you with an attack, you can use your reaction to cause the creature to make a Dexterity saving throw. The creature takes 2d8 lightning or thunder damage (your choice) on a failed saving throw, and half as much damage on a successful one.   It doesn't specify what the DC of the saving throw is.   I assume it uses the spell save DC, even though it technically isn't a spell?     Spell save DC = 8 + your proficiency bonus + your Wisdom modifier The entire  site is blank right now.  The header and footer are shown, but no questions. If $(X, \mathcal{T})$ has a countable subbasis, then it has a countable basis If $E$ is a connected space, then so is $\overline{E}$. I have to prove that if $E$ is a connected space, then so is $\overline{E}$ (the closure of $E$) a connected space. I tried to prove the contrapositive. So suppose that $\overline{E}$ is not connected in a metric space $X$. This implies there are $A\subset X$ and $B\subset X$ such that $A\cap B=\emptyset$, $A\cap \overline{E}\neq \emptyset$, $B\cap \overline{E}\neq \emptyset$ and $\overline{E}\subset A\cup B$. I will try to prove that this $A$ and $B$ will also work for $E$. Suppose that $A\cap E=\emptyset$ and $B\cap E=\emptyset$, this implies that $A\cup B$ is a subset of the limit points of $E$, because $A\cap \overline{E}\neq \emptyset$ and $B\cap \overline{E}\neq \emptyset$. This contradicts the fact that $\overline{E}\subset A\cup B$, so we have to conclude that $A\cap E\neq \emptyset\neq B\cap E$. Since $E\subset\overline{E}\subset A\cup B$, we have that $E$ is also not connected.  My question is whether my proof is correct, because I have a little bit doubt. If it's not correct, what is the best I can do? Thanks in advance!   Edit1: The definition we use for a connected space $E$ in a metric space $X$ is that if $E$ is connected, then there are no open $A,B\subset X$ such that $A\cap B=\emptyset$, $A\cap E\neq \emptyset\neq B\cap E$ and $E\subset A\cup B$.  Edit2: $A$ and $B$ must be open. querying Recently account viewed How to start itemize on same line as text? I've created a new environment called 'Example' like this:  \newenvironment{example}     {\textbf{Example:} \begin{itemize}}     {\end{itemize}}   The environment works fine, but the items are starting after the Example: text. Is it possible to start with the first item on the same line as the Example: text and vertically aline the rest of the items with the first item? Show that the LU decomposition of matrices of the form $\left[\begin{smallmatrix}0& x\\0 & y\end{smallmatrix}\right]$ is not unique How can I show that every matrix of the form $\begin{bmatrix}0&amp; x\\0 &amp; y\end{bmatrix}$ has an $LU$ factorization and that even if $L$ is unit lower triangular there is not a unique factorization?  I am completely stuck. Any ideas would be greatly appreciated Knowing which java.exe process to kill on a windows machine When a java based application starts to misbehave on a windows machine, you want to be able to kill the process in the task manager if you can't quit the application normally. Most of the time, there's more than one java based application running on my machine. Is there a better way than just randomly killing java.exe processes in hope that you'll hit the correct application eventually?  EDIT: Thank you to all the people who pointed me to Sysinternal's Process Explorer - Exactly what I'm looking for! How to find all the subclasses of a class given its name? Proving the existence of a sequence of functions converging to 0 with an added property I want to calculate $$ \int_{0}^{\infty} \frac{\arctan{2t}}{e^{2\pi t}+1} \, {\rm d}t $$ and am wondering if there at all is an analytical expression for the result. I haven't tried much so far, but expressing the denominator as the geometric series and then substituting $z=2\pi t (n+1)$. Equivalently I get $$ \sum_{n=0}^{\infty} \frac{{\rm si}\left(\pi(n+1)\right)}{2\pi (n+1)} $$ where $$ {\rm si}(x) = -\int_{x}^{\infty} \frac{\sin(t)}{t} \, {\rm d}t $$ is the shifted ${\rm Si}(x)$ function. The value is approximately $0.03618247146$, in fact I know from comparison with something else that it is exactly $\frac{\ln \pi -1}{4}$, but there should be some (direct) derivation, or?  Thanks for your input if you have any.  Edit: I tried to follow your instructions, but am now stuck. So I wrote $$ {\cal L}^{-1} \left(\frac{\psi(t)-\psi(1)}{t-1}\right)(s) =\sum_{n=0}^{\infty} \frac{e^{-sn}}{n+1} = e^s \int_s^\infty \frac{{\rm d}x}{e^{x}-1} $$ and taking the Laplace-transform \begin{align} \psi(t) &amp;= \psi(1) + (t-1)\int_0^\infty {\rm d}s \, e^{-s(t-1)} \int_s^\infty \frac{{\rm d}x}{e^{x}-1} \\ &amp;= \psi(1) + (t-1) \int_0^\infty {\rm d}x \int_{0}^{x} {\rm d}s \, \frac{e^{-s(t-1)}}{e^{x}-1} \\ &amp;= \psi(1) + \int_{0}^{\infty} {\rm d}x \, \frac{1-e^{-x(t-1)}}{e^x-1} \tag{1} \\ &amp;= -\frac{1}{t} + \psi(1) + \int_0^\infty {\rm d}x \, \frac{1-e^{-tx}}{e^x-1} \tag{2} \\ &amp;= \dots \\ &amp;=-\sum_{k=0}^{n} \frac{1}{t+k} + \psi(1) + \int_0^\infty {\rm d}x \, \frac{1-e^{-(t+n)x}}{e^x-1} \\ &amp;\stackrel{(\rm{2})}{=} -\frac{1}{2t} + \ln(t) - \int_0^\infty {\rm d}x \, \left\{ \frac{1}{2} - \frac{1}{x} + \frac{1}{e^x-1}\right\} e^{-tx} \tag{Binet 1} \end{align} where $\psi(1)=-\gamma$.  Now we integrate over $t$ from $1$ to $y$ yielding $$ \ln \Gamma(y) \stackrel{({\rm 1})}{=} \psi(1)\,(y-1) + \int_0^\infty {\rm d}x \, \frac{y-1+\frac{e^{-x(y-1)}-1}{x}}{e^x-1} $$ but what now? Substitute? Number of permutations of $n$ elements where no number $i$ is in position $i$ In what format does piped output get sent and received from one command/program to another? POSIX compliant way to redirect stdout and stderr to a file I am trying to write a script which is POSIX compliant, so it can run on any *nix system (Debian, Fedora, CentOS, AIX, Arch... ALL of them). When it comes to redirection, I am rather confused on what is supported by POSIX and what is not.  If I redirect ls -l file missingfile &amp;&gt; out.txt this works perfectly in bash. The line from stderr complaining that missingfile doesn't exist, and the permissions output from file, are both in out.txt. However, I believe this is only working because these shells support more than just POSIX standard operators. Upon researching this question I've come across conflicting answers.  , for example, seems to imply that &amp;&gt;, &gt;&amp;, &gt;&gt;&amp; and &amp;&gt;&gt; are non-standard operations. However,  explicitly states that some-program &gt; some_file 2&gt;&amp;1 is POSIX compliant. Does this mean that the &gt;&amp; operator (using digits for stderr &amp; stdout) is POSIX compliant whereas using &amp;&gt; to auto redirect both stderr and stdout to a file is not POSIX compliant? Or are both &amp;&gt; and &gt;&amp; not POSIX compliant / POSIX compliant (one of these two people are wrong)?  I considered just avoiding the &amp; sign completely and using ls -l file missingfile &gt;out.txt 2&gt;out.txt however this comes with its own issue. Running the ls command this way causes the shell to open two file handles for out.txt, both pointing to offset 0 in the file. So when ls looks for these two files, one of the messages gets clobbered.  What one would expect ls -l file missingfile &gt;out.txt 2&gt;out.txt to output:  -rw-r--r-- 1 blah blah 0 Jun  3 13:18 file ls: cannot access 'missingfile': No such file or directory   What is actually output:  -rw-r--r-- 1 blah blah 0 Jun  3 13:18 file le or directory   What is the best way to redirect both stdout and stderr to a file in a POSIX compliant fashion? Comma with 'who' in this specific sentence Can you shed light into this?  The coach came to the defence of the player who has been sacked from the team following his differences with the captain.  OR  The coach came to the defence of the player, who has been sacked from the team following his differences with the captain.  Isn't the subordinate clause that begins with 'who', non-restrictive? Unresolved external symbol on static class members Very simply put:  I have a class that consists mostly of static public members, so I can group similar functions together that still have to be called from other classes/functions.  Anyway, I have defined two static unsigned char variables in my class public scope, when I try to modify these values in the same class' constructor, I am getting an "unresolved external symbol" error at compilation.  class test  { public:     static unsigned char X;     static unsigned char Y;      ...      test(); };  test::test()  {     X = 1;     Y = 2; }   I'm new to C++ so go easy on me. Why can't I do this?