How to run program on Unix server and then close SSH?
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Simplify $\arctan (\frac{1}{2}\tan (2A)) + \arctan (\cot (A)) + \arctan (\cot ^{3}(A)) $ How to simplify $$\arctan \left(\frac{1}{2}\tan (2A)\right) +  \arctan (\cot (A)) + \arctan (\cot ^{3}(A)) $$ for  $0&lt; A&lt; \pi /4$?  This is one of the problems in a book I'm using. It is actually an objective question , with 4 options given , so i just put $A=\pi /4$ (even though technically its disallowed as $0&lt; A&lt; \pi /4$) and got the answer as $\pi $ which was one of the options , so that must be the answer (and it is weirdly written in options as $4 \arctan (1) $ ).  Still , I'm not able to actually solve this problem. I know the formula for sum of three arctans , but it gets just too messy and looks hard to simplify and it is not obvious that the answer will be constant for all $0&lt; A&lt; \pi /4$. And I don't know of any other way to approach such problems. Axiom of Choice: Where does my argument for proving the axiom of choice fail? Help me understand why this is an axiom, and not a theorem. In terms of purely set theory, the axiom of choice says that for any set $A$, its power set (with empty set removed) has a choice function, i.e. there exists a function $f\colon \mathcal{P}^*(A)\rightarrow A$ such that for any subset $S$ of $A$, $f(S)\in S.$ Is this correct?  My question then is about proving this fact, so that we do not need to put it as an axiom. Now as per the research done on this single object- Axiom of Choice, I believe here that there should be some falsity in my argument. I do not find the mistake.  For any $S\in \mathcal{P}^*(A)$, since $S\neq \emptyset$, $\exists s\in S$. Define $f(S)=s$. Then $f$ is a choice function.  This was showing me that the axiom of choice is proved, but then why it had been put as an axiom? For example, in  book, the author asserts that      It is a metatheorem of mathematical logic that it is impossible to specify the function that assigns to each non-empty subset of $\mathbb{R}$, an element of itself.     There are several notes and books on axiom of choice, but here I am trying to understand through doing some argument for some problem, where problem actually arises. Capitalisation of "Section" and "Chapter" in a Ph.D. Thesis I am submitting a Ph.D. thesis fairly soon and my supervisor has flagged my use of capitalisation in "Section" and "Chapter" as possibly incorrect. I have googled about a bit and I see mixed opinions.  So my question is, when writing a computer science Ph.D. thesis, what is the correct way to capitalise "Section", "Chapter", "Appendix", "Figure", "Table", ... ?  For example, what is the correct capitalisation for the following:   "In Chapter 3, it was shown that..." "In the previous Section, a method was presented to..." "The graph in Figure 3 shows..." Value of $\frac{x^n -a^n}{ x-a}$? How can I iterate over each file in a directory using a for loop?  And how could I tell if a certain entry is a directory or if it's just a file? biblatex ieee style same authors I tried to search for similar questions, as I always do, but was not able to find one. I just created a new  on Data.Stack Exchange with my existing Google OpenID and it seems to be not associated with my regular Stack Exchange account. Is this by design? and if so, why? How can I see the history of stories I've read on Apple News? Sometimes I'll be interrupted in the middle of reading a story on Apple News and later when I open the app, it didn't save the story I was previously reading. I want to be able to resume reading the article, how can I find it again? Uniform continuity of $f(x)=x^{2/3}\log x$ on $[0, \infty)$ Let $f$ be a real valued function on $[0,\infty)$ defined by  $$f(x)= \begin{cases} x^{{2}/{3}}\log x &amp; \text{ if } x&gt;0 \\  0 &amp; \text{ if } x= 0 \end{cases}$$  Then which of the following is true?  A. $f$ is discontinuous at $x=0$.  B. $f$ is continuous on $[0,\infty)$, but not uniformly continuous on $[0,\infty)$.  C. $f$ is uniformly continuous on $[0,\infty)$.  D. $f$ is not uniformly continuous on $[0,\infty)$, but uniformly continuous on $[1,\infty)$.  I could see that $f(x)=x^{\frac{2}{3}}\log x$ is continous at $x=0$   But for uniform continuity I thought boundedness of derivative would help   But I found $f'(x)=\frac{2}{3}x^{-\frac{1}{3}}\log x + x^{-\frac{1}{3}}$   This $f'(x)$ is not bounded so i can not say any thing about uniform contunuity.  Please help me to clear this.   Thank you :)  P.S : This was already asked before ( some other person but it was not having enough details so it got closed. I tried to edit that question but someone said I should not edit OP to add more details. I was afraid that may be against the rules of MSE. So, I thought i should ask this separately with my own ideas. This may sound like a 'first world problem', but I am quite easily distracted by the 'Hot Network Questions' sidebar, as I find them quite amusing and some of them quite intellectual  But where the problem lies is that, even at work, if I google something up and find a solution on Stack Overflow, to some random problem I am working at, I (always) get distracted towards them (Hot Network Questions), and end up reading the whole Qs as well as As of that respective question, and this is adversely affecting my productivity.  Any suggestion would greatly help. Android: Making Https Request This question was asked in an entrance test for an undergraduate mathematics program today, held all over India.   Question: $f$ is a differentiable function in $[0,1]$ such that $f(f(x))=x$ and $f(0)=1$.  Find the value of $\int_{0}^{1} (x-f(x))^{2016} dx$.  I tried to solve it by substituting $f(f(x))$ in place of $x$, but could not proceed much further. Any hints or solutions will be appreciated. I need some advice as I am struggling with the following combinatorics exercise.  Let $k$ be a given positive integer. Show that any non-negative integer $N$ can be written uniquely in the form $$N=\binom{x_k}{k}+\binom{x_{k-1}}{k-1}+\dots+\binom{x_1}{1},$$ where $0 \leq x_1 &lt; \dots &lt; x_{k-1} &lt; x_k$.  The exercise is accompanied by this advice:  Let $x$ be such that $\binom{x}{k} \leq N &lt; \binom{x+1}{k}$. Then any possible representation has $x_k = x$. Now use induction and the fact that $N-\binom{x}{k} &lt; \binom{x}{k-1}$ to show the existence and uniqueness of the representation.  I am really stuck and thus grateful for the slightest hint on how to solve it. I have an iPhone 6 (with proper iTunes backup) where all of my contacts where, but I bought a new iPhone 7 and had to initialize it as new iPhone, due to several time-related constraints, so now I have a few but new contacts on my new phone, but none of the older ones, but would be the best practice oriented, for just recovering the old contacts from the iTunes backups?  PS: I still have the old iPhone 6, so it could be used as "temporary storage" if needed. Invert Color Ramp in QGIS i would like to know how to prove that if $a^2+b^2=c^2$ then   a) $3|a$ or $3|b$  b) $5|a$ or $5|b$ or $5|c$  c) $4|a$ or $4|b$   Also that $a^2+b^2+c^2+1$ can't be divisible by 8  Thank you very much! The arrival of Minecraft Beta 1.5 brought weather conditions. However, it can at times be inconvenient when you're trying to screenshot something and show it off (as it reduces lighting and obstructs your view slightly). Is there anyway to turn weather off?  Alternatively, is there a way to speed up the duration of weather or "skip" it? How to justify $\phi(n) \ge \sqrt{n}$ If $\phi(n)$ is the Euler-totient function, how can I show that $\phi(n) \ge \sqrt{n}$? I have updated my kubuntu 14.10 to kubuntu 15.04. And now after boot, when I enter my password on welcome screen, I will get black screen. It's looks like screen powered off. No cursor, no change brightness. Nothing. And when I try switch to text consoles (Ctrl+Alt+F1) nothing happening. If I doesn't login in welcome screen I can switch to text console. how do i change the color of typeface on photoshop?