Evaluate $\sum_{n=1}^{\infty}\arctan (\frac{2}{n^2})$
Showing that $ \sum_{n=1}^{\infty} \arctan \left( \frac{2}{n^2} \right) =\frac{3\pi}{4}$
How many points in RSA, such that $m^e = m \bmod n$ For every RSA-cryptosystem, there exist some messages $m$, for which it holds that $m^e \equiv m \pmod n$ As to the question, how many such messages exist, this question has already been asked and answered here:  However, I do only understand the answer up to a certain point Suppose that $n$ is only 1 prime. Then it holds that the number of unconcealable messages is $\gcd(e-1, p-1)+1$ This is because in $m^e \equiv m \pmod  p$ , either $m = 0$ or $m \not= 0$. In the latter case $m$ can be expressed as $g^t$, where $g$ is a Generator in $\mathbb{Z_p}$ $g^{t*e} \equiv g^t \pmod p$ $g^{t(e-1) \bmod p -1 } \equiv g^{0 \bmod p -1} \pmod p$ $t(e-1) \equiv 0 \pmod {p-1} $ has exactly $\gcd(e-1, p-1)$  solutions, so the total number of unconcealable messages is, together with $m=0$ , $\gcd(e-1, p-1)+1$ When we now look at real RSA, which has two prime factors in its modulus, it holds that the number of unconcealable messages is $(\gcd(e-1, p-1)+1)\cdot(\gcd(e-1, q-1)+1)$ Why is that so? I suppose it has to do with CRT, but I just cant understand why. I have the two PGF nodes foo and bar positioned in a row.  \documentclass{article} \usepackage{tikz} \usetikzlibrary{positioning} \begin{document}   \begin{tikzpicture}[node distance=0.2cm,mynode/.style={rectangle,draw}]     \node[mynode] (foo)                {foo};     \node[mynode] (bar) [right=of foo] {bar};   \end{tikzpicture} \end{document}   How can I create a third node below those two nodes with the width of foo, bar and the node distance between them? Is there a way to do this with the fit library or is this the wrong approach? How can I install Ubuntu encrypted with LUKS with dual-boot? The Ubuntu installation disk has an option to install Ubuntu encrypted using LUKS. However, there is no option to perform an encrypted installation along-side existing partitions for a dual-boot scenario.  How can I install Ubuntu encrypted alongside another partition from the live disk? Can the jQuery UI Datepicker be made to disable Saturdays and Sundays (and holidays)? I use a datepicker for choosing an appointment day.  I already set the date range to be only for the next month.  That works fine.  I want to exclude Saturdays and Sundays from the available choices.  Can this be done?  If so, how? Question: Integrate$$\int\limits_0^1dx\,\frac {\log(1-x)\log^2(1+x)}x=-\frac {\pi^4}{240}$$   I'm curious as to if there is a way to integrate this. I've tried using integration by parts to get$$I=-\frac {\pi^2}6\log^22+2\int\limits_0^1dx\,\frac {\operatorname{Li}_2(x)\log(1+x)}{1+x}$$However, I'm not sure how to continue even further. The polylog in the second integrand seems a bit intimidating and I don't see how the first term even helps. Change style of labels of Tikz nodes of a given style Use induction to prove the following: $1! + 2! + .... + n! \le (n + 1)!$ How can I remove the qed symbols from all lemma proofs? the following code will only make the qed not appear in the second proof. However, it it possible to make the qed symbol never appear at the end of the proofs without having to specify it locally every time?  \documentclass{article} \usepackage{amsthm} \begin{document} \begin{proof} This has the QED symbol. \end{proof} \begin{proof}\renewcommand{\qedsymbol}{} This hasn't. \end{proof} \begin{proof} And this has it again. \end{proof} \end{document} Switch command output language from native language to english I was reading through some of the questions and answers here on the Rings of Power and I just wondered why the ring effects, affected different races in a slightly different way.  I know other questions ask and answer some of these things, but I just wondered why the Rings of Power caused greed in Dwarves for ever more amounts of treasure.  It's true that Dwarves loved mining and loved wealth and were built of sturdier stuff than men, but why did the Rings of Power not control Dwarves in the way the other Rings of Power controlled men and why did it produce greed and not invisibility? I moved a database from SQL Server 2005 to SQL Server 2012. Is there any way of restoring all the logins on the new server? Smeared creamy look effect i came across a photo where the gradient looks like it s smeared from cream. It is hard to explained by saying so here is the  Let say i create 2 similar shapes with the gradient pattern then adjust the opacity but the edges will still remain. But in the example the edges are appeared to be soft and hard randomly. Thanks in advance ! Recall the definition of the Weyl Chambers: A Weyl Chamber is a region of $V \setminus \bigcup_{\alpha \in \Phi} H_{\alpha}$, where $V$ is underlying Euclidean space, and $H_\alpha$ the hyperplane that is perpendicular to $\alpha$ the elements of  $\Phi$ in $V$.   I want to show that:   The Weyl chambers are open, convex and connected.   My attempts:   A single hyperplane $H_\alpha$ separates the space $V$ into two half-spaces: positive half-space $H_\alpha^+=\{x \in V \mid (\alpha,x)&gt;0\}$ and negative half-space $H_\alpha^-=\{x \in V \mid (\alpha,x)&lt;0\}$. It is clear that $V \setminus H_\alpha = H_\alpha^+\cup H^-_\alpha.$ As the result, the Weyl chambers can be regarded as components of  $$V \setminus \bigcup_{\alpha \in \Phi} H_{\alpha}= \bigcap_{\alpha \in \Phi} \left( V \setminus H_{\alpha} \right)= \bigcap_{\alpha \in \Phi} \left( H_\alpha^+\cup H_\alpha^- \right).$$ By the discussion above, a Weyl chamber is simply a $\textbf{intersection}$ of finite half-spaces. It is the case since one may write $$\left( H_\alpha^+\cup H_\alpha^- \right) \cap \left( H_\beta^+\cup H_\beta^- \right) =\left( H_\alpha^+\cap H_\beta^+ \right)  \cup\left( H_\alpha^+\cap H_\beta^- \right)  \cup \left( H_\alpha^-\cap H_\beta^+ \right)  \cup\left( H_\alpha^-\cap H_\beta^- \right).$$ Therefore openness and convexity of the Weyl chambers are straightforward since half-spaces are open and convex, so are their intersections. Moreover, we would replace $V$ by $\Bbb R^{\dim V}$ since they are isomorphic. Now we can prove that the Weyl chamber is connected by showing that every convex set in $\Bbb R^{\dim V}$ is connected. And I'm sure that proof of this statement existed. So am I done? How do I identify the C terminal on my HVAC? How do I identify the C terminal on an HVAC, furnace, or boiler? If the terminals don't have a screw labeled C what should I look for on the wiring diagram? What steps should I take if a wiring diagram cannot be found or if it indicates that a C terminal doesn't exist?  Note: This is an extension to  which asks about running the wires through the wall. This question is about where to connect the wire at the HVAC. Prove that if a function $f$ has a jump at an interior point of the interval $[a,b]$ then it cannot be the derivative of any function. Prove that if a function $f$ has a jump at an interior point of the interval $[a,b]$ then it cannot be the derivative of any function.  I know that for $f$ is differentiable in $(a,b)$ and that it has one-sided derivative $f_+' (a)≠f_-' (b)$ at the endpoints. If $C$ is a real number between $f_+' (a)$  and $f_-' (b)$, then there exists $c∈(a,b)$ such that $f' (c)=C $. How can I use this to prove the above? what does the ldapadd error: "additional info: objectClass: value #2invalid per syntax mean? Right at the beginning of the game, Meowth comes into your room and attempts to wake you up. At this point, I assume I have to press some button to get out of bed, but the D-Pad or A button doesn't do anything, nor any of the other buttons. How do I get out of bed to start the game? Should edits with only minor code improvements be accepted. As an example, I'm reviewing an edit that changes all double quotes to single quotes and renames a variable in some javascript. With the exception of the variable rename (which is not entirely necessary), the edit is frivolous. Should I accept on the merits of a debatable variable name change? If matrix $A$ is unitary and $B^2=A$, is $B$ necessarily unitary? How to compile a function whose arguments are lists containing strings?