Evaluating $\tan^{-1}\left(\frac{1}{2}\right)+\tan^{-1}\left(\frac{1}{5}\right)+\tan^{-1}\left(\frac{1}{8}\right)$
What is the value of $\arctan(1/2)+\arctan(1/5)+\arctan(1/8)$?
How permanent magnets work?Are they working with motion of electrons? You're too clever a man to imagine this.   The above sentence was , a man of excellent rhetorical skills.   Since he said it, I doubt it's wrong, grammatically. But, I wonder if there is an explanation for this. Because adjectives always come after the article not before it. e.g. You're a clever man.   How could this be, grammatically? Detecting an undefined object property How to reset KDE desktop to default on Ubuntu 14.04? KDE and Gnome Desktops were looking awesome on my Ubuntu 14.04 LTS, today everything seemed ugly. Slim long font, arabic letters in Chromium/Firefox are barely readable no matter how big I made them on system font settings. How can I get back to my previous looks settings, without compromising other settings and tweaks?  Here are my versions:  kde4-config -v:  Qt: 4.8.6 KDE Development Platform: 4.13.3 kde4-config: 1.0   kded4 --version:  Qt: 4.8.6 KDE Development Platform: 4.13.3 KDE Daemon: 4.13.3 How can I make a plane's edges taper inwards? How can I deform one edge or multiple edges of a plane or cube so they are tapered in and rounded equally? Using package nomencl I want to shade the area of overlap for adjacent normal curves, shown in the lower graph. I can get shading between the first two curves but not for subsequent curves. I need shading at overlapping areas indicated by the arrows. I applied the "logic" of the first fill for the subsequent curves but that does not work.   The arrows and letters in the example image are not part of the MWE.   \documentclass{article}  \usepackage{pgfplots} \pgfplotsset{ticks=none} \pgfplotsset{compat=1.7}  \usepgfplotslibrary{fillbetween}  \pgfmathdeclarefunction{gauss}{2}{%normal distribution where #1 = mean and #2 = sd}     \pgfmathparse{exp(-((x-#1)^2)/(2*#2^2))}% }  \pgfplotsset{baseplot/.style={%     no markers,      domain=1:4.5,      samples=100,     smooth,     axis lines*=left,     height=5cm, width=12cm,     enlargelimits=upper, clip=false, axis on top,     xlabel = near ticks,     xlabel={Resource} }}  \begin{document}  \begin{tikzpicture}   \begin{axis}[baseplot]   \addplot+[name path=A]{gauss(2.7,0.4)};   \addplot+[name path=B]{gauss(3.2,0.4)};   \fill[gray!20, intersection segments ={of= B and A}]; \end{axis} \end{tikzpicture}  \vspace*{3\baselineskip}  \begin{tikzpicture}   \begin{axis}[baseplot]   \addplot+[name path=A]{gauss(1.7,0.15)};   \addplot+[name path=B]{gauss(2.2, 0.15)};   \addplot+[name path=C]{gauss(2.7, 0.15)};   \addplot+[name path=D]{gauss(3.2, 0.15)};   \addplot+[name path=E]{gauss(3.7, 0.15)};   \fill[gray!20, intersection segments ={of= B and A}];   \fill[gray!20, intersection segments ={of= C and B}];   \fill[gray!20, intersection segments ={of= D and C}];   \fill[gray!20, intersection segments ={of= E and D}]; \end{axis} \end{tikzpicture}  \end{document} I know that atan2 gives me the absolute angle of any vector. But it doesn't give a value from 0 to 360 degrees. Instead, it gives a value (if I'm not mistaken), between 0 and 180, or between 0 and -180. For example, instead of giving angle 270, it will say -90.  This is a little confusing for me. Let me try to describe how I think this works, and please tell me if I'm correct. Actually, I'll draw it.    Is this correct? I'm having a little trouble understanding this. Thanks Print to standard printer from Python? We work heavily with serialization and having to specify Serializable tag on every object we use is kind of a burden. Especially when it's a 3rd-party class that we can't really change.  The question is: since Serializable is an empty interface and Java provides robust serialization once you add  implements Serializable - why didn't they make everything serializable and that's it?    What am I missing? Windows Subsystem for Linux (WSL): what can't I do with the Ubuntu application for Microsoft Windows? Proving that $\lim\limits_{x \to 0}\frac{e^x-1}{x} = 1$ I was messing around with the definition of the derivative, trying to work out the formulas for the common functions using limits. I hit a roadblock, however, while trying to find the derivative of $e^x$. The process went something like this:  $$\begin{align} (e^x)' &amp;= \lim_{h \to 0} \frac{e^{x+h}-e^x}{h} \\ &amp;= \lim_{h \to 0} \frac{e^xe^h-e^x}{h} \\ &amp;= \lim_{h \to 0} e^x\frac{e^{h}-1}{h} \\ &amp;= e^x \lim_{h \to 0}\frac{e^h-1}{h} \end{align} $$  I can show that $\lim_{h\to 0} \frac{e^h-1}{h} = 1$ using L'Hôpital's, but it kind of defeats the purpose of working out the derivative, so I want to prove it in some other way. I've been trying, but I can't work anything out. Could someone give a hint? I have a file with ANSI colors.  test.txt: \e[0;31mExample\e[0m  I would like to display the content of this file in a terminal, like cat does, but I would like to display the colors as well. Is the ideal $(2,X+1)\subset\Bbb{Z}[X]$ prime, maximal or neither? I'm currently considering the ideal $(2,X+1)\subset\Bbb{Z}[X]$. I'm trying to figure out if this is a prime ideal, a maximal ideal, or neither. So far I have attempted to look at the quotient ring $\Bbb{Z}[X]/(2,X+1)$. I know that if this quotient ring is a field, then our ideal is maximal, since $\Bbb{Z}[X]$ is commutative. However I'm stuck on how I should be viewing this quotient ring. Let $a,b$ be positive integers.  When $$k = \frac{a^2 + b^2}{ab+1}$$ is an integer, it is a square.    Proof 1: (Ngô Bảo Châu): Rearrange to get $a^2-akb+b^2-k=0$, as a quadratic in $a$ this has two values: $a$ and $kb - a = (b^2-k)/a$. (The second root is determined in two different ways from the expansion $(x-r_1)(x-r_2) = x^2 - (r_1 + r_2)x + r_1 r_2$.)  Now suppose we have $a,b$ such that $k$ is an integer but not a square, by the investigation about roots we have that the second root is a nonzero integer since $k,b,a$ are integers and $k \not = b^2$, futhermore it is positive which is easily seen from its defining equation.  WLOG assume $a \ge b$ so that the second root is strictly smaller than $a$. This leads to a decent, replacing $a$ with the second root.    Proof 2 (Don Zagier): Apply reduction theory (specifically, Sätze 1 and 2 of Section 13 of my book on quadratic fields) to the quadratic form $x^2 + kxy + y^2$, which is the unique reduced quadratic form in its equivalence class.    Note that Proof 2 is pretty much the same as Proof 1 when written out in explicit detail, but I could not read Zagier's book because I cannot read German.  I would like to know more approaches to this and other alternative proofs of this result if possible! Thanks in advance.  I would also be interested in related problems (especially easier ones of a similar nature) and texts which cover the reduction theory in English. What happens when you need to tear-down a hotel but no houses are in the bank in Monopoly? TeXShop and Mac preview crashing when opening latex/beamer built pdfs After updating to Sierra 10.12.4 last week, TeXShop crashes:   when opening an existing .tex document that has a pdf associated to it immediately after compiling a new document, i.e. as soon as a pdf is created.   Furthermore, preview crashes opening almost any pdf that has been created with TeXShop. There is virtually nothing on the internet about it, and the only solution seems to be a complete reinstall of MacOSX (so as to reinstall the supposedly broken preview), which I'd like to avoid if at all possible.  Someone else has described a similar issue, but has not received any responses ().  I would be very grateful for some help, as I can neither produce work, nor look at any at the moment. Many thanks! Shell scripting: Way to parse parameters from command line myscript [-a a-arg] [-c c-arg] [-b] [-e] somedirectory   Given that I want my shell script be invoked at the command line using the above parameters - where [these brackets] denote that they are optional - what is the best method to parse them? How can I modify the advanced search form when my hook runs too early? (Fitzpatrick Advanced Calculus 2e, Sec. 2.4 #12)  For $c \gt 0$, consider the quadratic equation $x^2 - x - c = 0, x &gt; 0$.  Define the sequence $\{x_n\}$ recursively by fixing $|x_1| \lt c$ and then, if $n$ is an index for which $x_n$ has been defined, defining  $$x_{n+1} = \sqrt{c+x_n}$$  Prove that the sequence $\{x_n\}$ converges monotonically to the solution of the above equation.  Note: The answers below might assume $x_1 \gt 0$, but they still work, as we have $x_3 \gt 0$.    This is being repurposed in an effort to cut down on duplicates, see here: .  and here: .