Dealing with downvote trolling
A better serial voting trap
Batch convert 4 PDF pages to one PDF page Evaluating $\int_0^{\tfrac{\pi}{4}}\ln(\cos x-\sin x)\ln(\cos x) dx - \int_0^{\tfrac{\pi}{4}}\ln(\cos x+\sin x)\ln(\sin x)dx=\frac{G\ln 2}{2}$ In order to compute, in an elementary way,  $\displaystyle \int_0^1 \frac{x \arctan x \log \left( 1-x^2\right)}{1+x^2}dx$  (see  )  i need to show, in a simple way, that:  $\displaystyle \int_0^{\tfrac{\pi}{4}}\ln(\cos x-\sin x)\ln(\cos x) dx - \int_0^{\tfrac{\pi}{4}}\ln(\cos x+\sin x)\ln(\sin x)dx=\dfrac{G\ln 2}{2}$  $G$ being the Catalan constant.  The reason of my interest for this question is, if i am right, that this formula permits to find out a relation between integrals:  $\displaystyle \int_0^1 \dfrac{\ln(1+x)\ln(1+x^2)}{1+x^2}dx$  $\displaystyle \int_0^1 \dfrac{\ln(1+x)^2}{1+x^2}dx$  $\displaystyle \int_0^{\tfrac{\pi}{4}}\big(\ln(\cos x)\big)^2 \dfrac{}{}dx$  and some constants. Install suspension forks on non-suspension bike I added a comment to  but something strange is happening. The end code reads "); but SO decided to change that to ";);.    I did not add a ; in front of the ).  To further explain what's happening here, the following:  "http://example.com/." http://example.com/. "http://example.com." http://example.com. "http://example.com" http://example.com   renders as: $x^3-9=y^2$ find integral solutions Find all integral solutions  $x^3-9=y^2$  I tried many times but still no idea how to solve it. I will be grateful for any help. Is it possible to use the Lagrangian formalism involving physical terms to answer the question of why all planets are approximately spherical?  Let's assume that a planet is 'born' when lots of particles of uniform density are piled together, and constrained only by the Newtonian gravity of the $N$ particles with no external gravity.    My question is: Is it possible to use the variational calculus starting from this situation to prove that planets should be spherical? Is it permissible to use somebody else's words (without quotations) to describe a concept once their work is cited? Quite often one comes across a really succinct and precise explanation of a concept. I'm wondering if it is permissible to use this explanation verbatim, if the source work is cited in your paper (next to the explanation)? What's in question here is at most 3 sentences. Here is the proof I tried to weave while trying to prove this theorem:     Theorem. Every infinite subset of a countable set is countable.      Proof. Let $A$ be a countable set and $E\subset A$ be infinite. Then $A\thicksim\mathbb{N}$. This implies that there is a sequence $\left\{x_{n}\right\}_{n\in\mathbb{N}}$ where $x_{n}\in A$. Construct a sequence $\left\{n_{k}\right\}_{k\in\mathbb{N}}$ where $x_{n_{k}}\in A$, $n_{1}$ is the smallest positive integer such that $x_{n_{1}}\in A$, and $n_{k+1}&gt;n_{k}$. Then      $$ E=\bigcup_{k=1}^{\infty}x_{n_{k}}, $$      and $E$ is countable, because $E\thicksim\mathbb{N}$. $\blacksquare$   Is it convincing? How to enable sleep/standby on Windows Server 2012 Hyper V Proof for alternative definition of the derivative I read somewhere recently that you can define the derivative as follows:  $$f'(x) = \lim_{h, k \to 0^+} \frac{f(x +h) - f(x - k)}{h + k}$$   I have been trying to prove this for about 2 hours, and can't seem to get it done. How should I proceed?  Edit: Assume $f$ is differentiable for all $x \in (x - \delta, x + \delta)$ for some $\delta &gt; 0$. The following question is from Artin's Algebra.      If $A$ and $B$ are two square matrices with real entries, show that $AB-BA=I$ has no solutions.    I have no idea on how to tackle this question. I tried block multiplication, but it didn't appear to work. Virtualbox Server/Client topology What's the frequency of filesystem check on boot? What is the maximum number of times the root filesystem on this computer can be mounted before it is automatically checked? What is the string concatenation operator in Oracle? What is the string concatenation operator in Oracle SQL?    Are there any "interesting" features I should be careful of?    (This seems obvious, but I couldn't find a previous question asking it). Return a value when using jQuery.each()? The other day I was looking at Stack Overflow on my Android phone. All is made very nice and all, but I could not find the Review button/link. Is it being implemented, did I miss it, or it is not available? How can I determine which process owns a hotkey in Windows? Given a hotkey, how can I find which program owns it? Unity Performance: How many Materials per Object? I'm reading Linear Algebra by Hoffman, Kunze where the authors explained that a $n\times n$ matrix $A$ being invertible is equivalent to the fact that $A$ is row-equivalent to $n\times n$ matrix $R$ which is an identity matrix.  In the proof of the theorem, they wrote:     $$R= E_k\ldots E_2E_1 A$$ where $E_1,\ldots,E_k$ are elementary matrices. Each $E_j$ is invertible, and so $$A = E_1^{-1}\ldots E_k^{-1}~R\,. $$  ... Since, $R$ is a (square) row-reduced echelon matrix, $R$ is invertible if and only if $R=I\,.$  [...]   I couldn't get the conclusion, since any row of $R$ can't be zero, it has to be identity matrix $I\,.$ Why is it so?  Isn't there any other row-reduced echelon matrix other than the identity matrix having no zero row and invertible? Why is it so? While I applaud the excellent work of , the question has come up:     How long will be the elected moderator’s “mandate”? Is it for life, or are we going to have “moderator elections” regularly?   Of course, any community moderator who wishes to recuse him or herself can certainly do that at any time.   Should we have term limits that require community moderators be cycled out periodically, so others get a chance to moderate? Should we hold regular (say, yearly) moderator elections? Is it better to have a larger pool of community moderators, and let the normal ebb and flow of turnover get us "natural" variability in moderation?   What are your thoughts on this?