Evaluate $\int_0^{2\pi}\ln(1+a^2-2a\cos(x))dx\text{ ,with }a>0$
Computing $\int_{0}^{\pi}\ln\left(1-2a\cos x+a^2\right) \, dx$
Did spacetime start with the Big bang? Did spacetime start with the Big Bang?  I mean, was there any presence of this spacetime we are experiencing now before big bang? And could there be a presence/existence of any other space-time before the big bang? What is the limit of $\frac{\prod\mathrm{Odd}}{\prod\mathrm{Even}}?$ What is this limit   $$   \frac{1\times3\times5\times\cdots}{2\times4\times6\times8\times\cdots} = \lim_{n \rightarrow \infty}\prod_{i=1}^{n}\frac{(2i-1)}{2i} $$   I remember that it was something involving $\pi$.  How can I compute it?    In addition; how can I compute it's sum series limit? How can you profile a Python script? Captured variable in a loop in C# I met an interesting issue about C#. I have code like below.  List&lt;Func&lt;int&gt;&gt; actions = new List&lt;Func&lt;int&gt;&gt;();  int variable = 0; while (variable &lt; 5) {     actions.Add(() =&gt; variable * 2);     ++ variable; }  foreach (var act in actions) {     Console.WriteLine(act.Invoke()); }   I expect it to output 0, 2, 4, 6, 8. However, it actually outputs five 10s.  It seems that it is due to all actions referring to one captured variable. As a result, when they get invoked, they all have same output.  Is there a way to work round this limit to have each action instance have its own captured variable? Convergence in Probability Was this an oversight?  Or is it to do with the JVM? Define the Lie algebra $\mathfrak{g}$ of a Lie group G to be the set of all left-invariant vector fields of $G$. I want to prove that   $$          f: \mathfrak{g} \rightarrow T_eG \\              X \mapsto X(e) $$ is a linear isomorphism.  The only thing that I have left to prove is surjectivity.Letting $v\in T_eG$, we set $X\in \mathcal{X}(G)$ such that $\forall x\in G,\,X(x)=L_{x_{*,e}}(v)$ (I still have to show that such an $X$ is indeed a vector field on $G$). We get that  $$ f(X)=X(e)=L_{e_{*,e}}(v)=v $$  where the last equality holds because  $$       L_{e_{*,e}}(v)f = v(f \circ L_e) = v(f \circ id_G) = v(f) $$ for any $f\in C^\infty(G)$. (I am thinking of the tangent vectors as derivations).  Now all we have left to do is prove that such an $X$ is left-invariant, and it is therefore an element of $\mathfrak{g}$, but I do not know how. I tried to prove that  $$            \forall f\in C^\infty(G),\, L_{{x}_*,y}(X(y))f=X(L_x(y))f $$ but with no success. Shell with better editing capabilities Shells (at least what I know) have really bad command editing capabilities.  For example:    I can't point the cursor to any point of the command. I can't select and backspace/delete/replace text in the command.   Currently these are so distracting, I use gedit to edit commands before pasting them to the terminal.  Is there any solution that's good for these? Why does the new review page sometimes shows questions to vote to close without anybody having voted to close it? I was reviewing  for , and noticed that nobody voted to close it, even though the page reports a duplicate.      Why does this happen? How to hide terminal output when executing a command? There is an exercise in Humphreys's An Introduction to Lie Algebras and Representation Theory: "Any nilpotent Lie algebra contains a codimension 1 ideal".  The proof I am thinking of is the following.  Suppose the Lie algebra $L$ satisfies $L\neq[L,L]$. Then $\pi\colon L\to L/[L,L]$ is a projection onto an abelian algebra. Choose a codimension 1 subspace $I\subset L/[L,L]$. Then $I$ is an ideal and its preimage $\pi^{-1}(I)$ is necessarily an ideal in $L$, because any subspace containing the derived algebra is an ideal. Moreover, $\pi^{-1}(I)$ has codimension 1. $\square$  The only condition on $L$ is that $L\neq[L,L]$, which holds for solvable Lie algebras as well. Is this proof correct? How come Humphreys didn't set the exercise with "nilpotent" replaced by "solvable"? Why are heatmaps on qgis 2.8.1 produced in the wrong CRS? Monty hall problem extended. Why does the odd part of the function integrate to $0$? When evaluating the integral $$\int^1_{-1}\frac{\cos(x)}{1+e^{1/x}}dx,$$ the author says that the odd part of the function integrates to $0$ over $-1$ to $1$. I have calculated the even part of the function ($\frac{1}{2}\cos(x)$), so the integral is equal to $\sin(1)$. But why exactly does the odd part integrate to $0$? Free way to clone HDD to SSD? Is there a free way to clone a full Windows 7 installation from HDD to SSD? (The free part is important, I know I can pay for Acronis True Image.)      SSDs are usually much smaller than HDDs which complicates matters. For instance, I have about 50GB data, my new SSD is 80GB but I still can't use Windows clone/restore system if my HDD is something like 300GB.  Therefore, I'm looking for a third party tool that will be able to:   Create a clone of my C: partition Store it onto an external USB drive Boot from some rescue / restore CD Load the image onto the new SSD   Acronis True Image seems like a tool that should support this but it's a paid-for software and the trial version cannot create clones. Acronis Migrate Easy is a fully functional tria for 15 days but need both the HDD and SSD connected at the same time which is a bit of a problem for me as a laptop user.  P.S. I almost got there combining Windows image backup combined with Acronis True Image restore CD (which can load .vhd's) but was not able to boot as some boot information were probably lost and could not be repaired using the Win7 installation DVD. Given $|z|&lt;1$ with $z \in \mathbb{C}$ prove $\lim_{n \to \infty} z^n=0$.  My attempt given $\epsilon &gt;0$ we wish to show that there exists $\delta &gt;0$ such that $n&gt; \delta$ means $|z^n-0| &lt; \epsilon$.  For $n \in \mathbb{N}$. We have $|z^n|=|z|^n$. We wish $|z|^n &lt; \epsilon$. This means,  $$n \ln |z| &lt; \ln \epsilon$$  But because $\ln |z|&lt;\ln 1=0$ dividing both sides by $\ln |z|$ gives:  $n&gt; \frac{\ln \epsilon}{\ln |z|}$  So that's what we choose for $\delta$.   My question is is how can we get away from the assumption that $n \in \mathbb{N}$. Or is this correct even with the assumption? When can we speak of collinearity In linear models we need to check if a relationship exists among the explanatory variables. If they correlate too much then there is collinearity (i.e., the variables partly explain each other). I am currently just looking at the pairwise correlation between each of the explanatory variables.  Question 1: What classifies as too much correlation? For example, is a Pearson correlation of 0.5 too much?  Question 2: Can we fully determine whether there is collinearity between two variables based on the correlation coefficient or does it depend on other factors?  Question 3: Does a graphical check of the scatterplot of the two variables add anything to what the correlation coefficient indicates? Unexpected behavior of rule matching a pattern Email SMTP validator I need to send hundreds of newsletters, but would like to check first if email exists on server. It's called  validation, at least I think so, based on my research on Internet.    There's several libraries that can do that, and also a page with open-source code in  (), but I have hard time reading ASP Classic, and it seems that it uses some third-party library...    Is there some code for SMTP validation in C#, and/or general explanation of how it works? How can we prove charge invariance under Lorentz Transformation? We have gravitational force between two massive particles and we have electromagnetic force between two charged particles. When special relativity suggests that mass is not an invariant quantity why we have electric charge as an invariant quantity ?