I saw somewhere that the above integral is equal to $\pi/4$ for all real number $m$.  This seems to be surprising. Does anyone have a nice proof?
Let $a$ be a non-zero real number. Is it true that the value of $$\int\limits_0^\infty \frac{\mathrm{d} x}{(1+x^2)(1+x^a)}$$  is independent on $a$?
Remove Sender name ( not email address) from outgoing Mac Mail Prove $\sin(x)\tan(x) > x^2$ for $x \in ( \,0, \frac{\pi}{2}) \,$ Prove $\sin(x)\tan(x) &gt; x^2$ for $x \in ( \,0, \frac{\pi}{2}) \,$  So I did the following: Let $f(x) = \sin(x)\tan(x) - x^2$.  Then of course $f(0) = 0$ and I want to show that for $x \in ( \,0, \frac{\pi}{2}) \,$ $f'(x) &gt; 0$ but I don't really know where to go from here as I can't get anything reasonable done with the derivative.  I was thinking of approximating it using the following inequality $\sin(x) &lt; x &lt; \tan(x)$.   Is that a good direction? Minor grid in pgfplot not showing when specifying ticks manualy Firstly I would like a minor tick between every major tick in my pgfplots graph (and minor grids). I wrote minor tick num=1 but it doesn't work. I am not using a logarithmic axis so it is unrelated to other questions.  Secondly, why is the tick label "1" missing for each axis? How do I get that to display?     \documentclass{standalone} \usepackage{pgfplots} \begin{document} \begin{tikzpicture}  \begin{axis}[% footnotesize, width=4in, height=4in, axis equal, scale only axis, separate axis lines, grid=both, every outer x axis line/.append style={black}, every x tick label/.append style={font=\color{black}}, xmin=0, xmax=1, xlabel={X}, every outer y axis line/.append style={black}, every y tick label/.append style={font=\color{black}}, ymin=0, ymax=1, ylabel={Y}, minor tick num=1, xtick={0,0.1,...,1}, ytick={0,0.1,...,1}, axis background/.style={fill=white}, legend style={at={(0.97,0.03)},anchor=south east,legend cell align=left,align=left,fill=white} ]  \end{axis}  \end{tikzpicture}% \end{document} How do I execute the following shell command using the Python  module?  echo "input data" | awk -f script.awk | sort &gt; outfile.txt   The input data will come from a string, so I don't actually need echo.  I've got this far, can anyone explain how I get it to pipe through sort too?  p_awk = subprocess.Popen(["awk","-f","script.awk"],                           stdin=subprocess.PIPE,                           stdout=file("outfile.txt", "w")) p_awk.communicate( "input data" )   UPDATE:  Note that while the accepted answer below doesn't actually answer the question as asked, I believe S.Lott is right and it's better to avoid having to solve that problem in the first place! How do you tell if a D&D book is 3.0 or 3.5? My lecturer has used some notation that I've never seen before: it is a (matrix) product symbol with a left-to-right arrow over the top. Does anybody know what this means?  Thanks in advance.  Edit: It looks like this: Should postdocs work only on the projects they are hired for, or can they work with other people in the same research group on other projects? Would the latter be an issue with the funding body, when the project proposal and budget only covers the expenses for hiring one postdoc? Note: it is the supervisors who got the grants, not the postdocs themselves. Which browsers are officially supported, and what else do I need? Which browsers are officially supported by Stack Exchange?  Also, what else is needed in order to use every feature of the site? In chapter 2 section 7 (pg 151) of Hartshorne's algebraic geometry there is an example given that talks about automorphisms of $\mathbb{P}_k^n$. In that example Hartshorne states that $\mathcal{O}(-1)$ has no global sections. However, we know that $\mathcal{O}(1)$ is generated by global sections. This is stated at the first of the section that if $\mathbb{P}_k^n = Proj k[x_0,...,x_n]$ then the $x_0,...,x_n$ give rise to global sections $x_0,...,x_n\in\Gamma(\mathbb{P}_k^n,\mathcal{O}(1))$.  I guess I don't understand this twisted structure sheaf very well, or to be honest I don't think I understand structure sheaves in general as well as I would like. The twisting part seems simple at first- you shift the grading of Proj over and then take the structure sheaf.  Admittedly I don't feel comfortable using the structure sheaf other than using the basic facts about it that Hartshorne gives when it is introduced. If anyone could give some insight as to what's going on here or how I might be able to understand this better it would be much appreciated. Thanks. Short story with spaceship and humanity not realising they should be on it The plot is about:   A spaceship travelling through the universe Earth is a rogue planet whose inhabitants have been cut off and don't realise that their true purpose is to be part of the spaceship   The reason earth was in turmoil (wars etc.) was that they didn't realise that their true role was to fit into the ship  The ship contained 3 or 4 beings that made the thing work Planets throughout the universe are populated with the various beings which are components of the space craft   I seem to remember like an eye shaped being Each of them fulfil their role Humans are part of that as well.  There are loads of planets with humans on but earth has gotten overlooked so it never realised that's what they were designed for The story concludes with the ships picking up someone from Earth   and with the "human" realising that this is what they should really be doing  Humans have a different name that explains their role in the ship.   I probably read it around 30 years ago in English in a sci-fi collection book. Differentiation operator and eigenvalues Let $V = \{p(x) \in F[x] \ | \ \deg(p(x)) \le n\}$. Let $T : V \to V$ be given by differentiation, in essence $$T(p(x)) = p'(x)$$ It seems to me that the only eigenvalue that can exist is $\lambda =0$, since the given operator lowers the degree of any given polynomial and hence makes it impossible for a constant multiple of a polynomial to be equal to the polynomial's derivative. Is my claim correct? Orthogonal complement examples I am looking for an example such that in a pre-Hilbert space $H$ we have for a subspace $U$ that  (i) $\bar{U} \oplus U^\perp \neq H$  (ii) $ \bar{U} \neq U^{\perp \perp}$  Since finite and closed subspaces will probably not do it, I am not good at inventing nice examples. Does anybody here have a few at hand? Question about $\lim _{q \rightarrow \infty}\|f\|_{q}=\|f\|_{\infty}$ Let $(X,B,\mu)$ be a complete measure space,Show that $$\lim _{q \rightarrow \infty}\|f\|_{q}=\|f\|_{\infty}, \quad \forall f \in \bigcup_{p} \bigcap_{p \leqslant q&lt;\infty} L^{q}$$ So,$\lim _{q \rightarrow \infty}\|f\|_{q}$ , $\|f\|_{\infty}$ are equal-norm with space $ L^{\infty} \cap\left(\bigcup_{p} \bigcap_{p \leqslant q} L^{q}\right)$. Case 1: $m(X)&lt;\infty $.It's easy to prove that. Case 2: $m(X)=\infty $. I have no idea about it,And I started to doubt the correctness of this conclusion. Can somebody give me a hint for this problem or just give an example to prove that this is a wrong conclusion when $m(X)=\infty $. Thanks in advance. There is this puzzling thing that is called : paradoxically, warm (35°C) water freezes faster than cold (5°C) water.  As a physisist, I've been asked about it several times already. And I have no definite answer to that, apart from the standard: "there are many things that can influence it".  So, does anyone knows about the status or progress on that effect? Any recent reviews, publications or other references? Sum of a power series $n x^n$ I would like to know: How come that  $$\sum_{n=1}^\infty n x^n=\frac{x}{(x-1)^2}$$  Why isn't it infinity? Rules when using a Joker I had on my rack among other tiles, a black 9, 1 x blue 10, and a black 12.  On the table was a set of 3 x 10s including a Joker(no blue 10), and other sets of tiles including an available black 11.  Can I replace the joker with my blue 10 and then set out :  my black 9, the joker(representing a black 10), a black 11 from the table, and my black 12? When and what cause Voldemort to look the way he does, vs how we've seen him as a teen? As the title says, when and what made Voldemort look the way he does (pale, snake-like etc) as opposed to how we've seen him in his earlier years as Tom Riddle? Is it just a side effect of his unnatural resurrection? Before he died attacking Harry as a baby did he look like a normal guy? Where is the line drawn with domain names which include a trademark? A search on google for "iphone developer" turns up loads of websites which have "iphone" in them, a trademarked name by Apple.  So I'm led to believe that having a domain such as iphonedeveloper.com is ok? Well, you're still using Apple's trademark, but it would be hard to brand yourself otherwise. You're an IPHONE DEVELOPER... right?  Well, what if I want to provide a website where users pay to get a list of the best offers from Ebay? I might have a domain like ebaydeals.com (I don't... i'm just speculating!).  Now I've heard that places like Ebay are really hot on the trigger and fire out emails to people who register domains like that straight away. But whats the difference?  In both cases I'm making money from the trademark, effectively, so is it just down to how lenient the company who owns the trademark is? Or are there rules? Is there a specific "line" you don't cross?  Thanks I try to create a 2D game for Android. It looks like Cut the rope (it isn't a clone but it have same level representation) if it has a value. My problem that I can't understand that optimal sprite sizes I should use. I.e. what target screen resolution should I use to sprites didn't have anomalies on HD-devices (like Samsung Galaxy S7 and etc., phones and tablets)?    To make my question is clearer I ask about this 1 unit = X pixels for xxhdpi-screens. What is X?    P.S. Sorry if it's a duplicate I will be grateful for link on this post cuz I couldn't find it. Java Strings: "String s = new String("silly");"