T prove that a statement holds for any integer
Using congruences, show $\frac{1}{5}n^5 + \frac{1}{3}n^3 + \frac{7}{15}n$ is integer for every $n$
I need to compile between two and four times before I see changes I make in the editor (TeXstudio) and each time I compile, a new .tex file is created with random letters and numbers at the end.   For example, If I create the file hello.tex and compile two to four times I'll get the file,   hello.tex.QU6664      then I make some small changes in the editor and compile two to four times I'll get the additional file,   hello.tex.Ak6664   After continuing to do this (each time compiling between two to four times before I see the changes I made), I now have many files that look like  hello.tex.QU6664, hello.tex.Ak6664, hello.tex.YX6664, hello.tex.sN6664, hello.tex.Ss6664, ...   A little bit of additional information (in case it's relevant): Everything was working fine up until a few days ago, i.e. only needed to compile once and only the file hello.tex was created. I still only get one .aux,  .synctex and .pdf files. I use TeXstudio to write and compile. I don't think it has anything to do with the bibliography (I tried removing bibliography altogether and still having the same problem).  How do I prevent these additional files from appearing and only compiling once like I used to? Has anyone had a similar problem? Inequality $\sum\limits_{1\le k\le n}\frac{\sin kx}{k}\ge 0$ (Fejer-Jackson) Show the following inequality for any $x\in [0, \pi]$ and $n\in \mathbb{N}^*$, $$ \sum_{1\le k\le n}\frac{\sin kx}{k}\ge 0. $$  I have this question a very long time ago from a book or magazine but I cannot solve it by myself and did not know how to solve it until today.   My try: for $n=1, 2, 3$, one can check this by hand. Query to collect the recent TimeSampled row using inner joins in MS SQL How to compute $\lim _{x\to 0}\frac{x\bigl(\sqrt{3e^x+e^{3x^2}}-2\bigr)}{4-(\cos x+1)^2}$? I have a problem with this limit, I don't know what method to use. I have no idea how to compute it. Is it possible to compute this limit with the McLaurin expansion? Can you explain the method and the steps used? Thanks. (I prefer to avoid to use L'Hospital's rule.)  $$\lim _{x\to 0}\frac{x\bigl(\sqrt{3e^x+e^{3x^2}}-2\bigr)}{4-(\cos x+1)^2}$$ If I have a Type C visa from the Norwegian embassy valid for February 9-28, and my Type D national visa from the German embassy valid from March 1 - August 31, 2017, do I need to leave/fly out of the Schengen area on the 28th and fly back on the 1st day of March? Evaluation order of Button's action How do you select a particular option in a SELECT element in jQuery? Draw Lewis structures like a book I want to draw a Lewis structure as in the picture. I tried to do with chemfig but does not perform the separation of the points and a minus in the superscript. What is the unit of the coordinate system in PSTricks? If I use the following code, there two different kinds of units. One for the coordinate system used to declare \begin{pspicture} with (0,0) to (20,10), and the centimeter used in 4.2cm.  \begin{pspicture}(0,0)(20,10) \rput[lt](0.5,9){\psframebox[linewidth=1pt,framearc=0]{\parbox[c]{4.2cm}{\raggedright Text Goes Here...}}} \end{pspicture}   Can I use a unified dimension for both PSTricks objects and \parbox? Possible Duplicate:          On Windows I am able to switch between keyboard layouts by pressing Alt+Shift. Is there a similar keyboard shortcut in Ubuntu? How do I add a .sty file to my MacTeX/TeXShop installation? I've searched online, but I have not found any clear instructions how to permanently add .sty files to my TeXShop installation. I'm using a Mac with OSX (TeXShop is a mac only program).  What is the best way of doing this? The one comprehensive set of instructions I found .   I've got as far as extracting the files (running LaTeX on the .ins and . dtx files) but after that I don't know what to do. I also don't understand how to update the index of my distribution, and whether there are particular OSX specifics (the instructions are for Windows). Dynamically reformatting man pages on terminal dimension changes One common workflow of mine is to open a manual page in a terminal, then another terminal in which to test things. The man page is formatted to the initial dimensions of the first terminal. When I now resize my windows (or have my WM do that for me automatically), there is either a gap to the right of the preformatted page, or lines wrap. At this point I usually q(uit) and !! (run again), which loses my position in the page.  I assume the formatting process is quite CPU intensive, or maybe it stems from ancient times of fixed terminal sizes. The less pager dynamically reacts to terminal resize events, so it should be possible in theory.  I tried perusing man pages, searching the Web, asking on IRC -- the whole lot -- but couldn't come up with anything.   Can I trigger reformatting from within or outside of the man utility? Is there a version of the man utility that resizes the page dynamically?  Is there way to customize some part of the formatting/display process to make it update on SIGWINCH? Cartesian Product of Two Countable Sets is Countable I was going through some articles about "African American Vernacular English". . . These articles give some examples, but they do not clearly specify hard rules to be followed in AAVE.  First of all, I don't even have strong skills in Standard English, so understanding AAVE seems a lot more difficult for me. I did, however, craft a few sentences in both SE and AAVE to validate what understanding I do have. Are the following conversions correct? You don't have to validate each and every item, but rather guide me with rules that I should follow, but didn't, in the following examples.   What the hell are you all doing here?  —  What the hell y'all doin' here? Hey you! What the hell are you doing man? Stop that nonsense now. Do you get it?  — Hey you! What the hell ya doin’ man? Stop that nonsense now. D'ya get it? Come on, can't you open your mouths?   — Come on, ya can't open your mouths??  Hey, who's out there?  —  Hey, who out there? I know, you're listening to us   — I know, ya listenin’ to us Somebody is out there man. — Somebody out there man. Wow, it's delicious.  — Wow, it delicious. Holy crap, who're these people? — Holy crap, who these people? At which embassy should I apply for a Schengen visa when staying a few days in several countries? I'm planning to go to Germany on Sept 6-8, then Netherlands on Sept 8-11 and exit Switzerland on Sept 11-14.  My question is:  Which embassy do I need to apply for the Schengen visa? The number of days are the same for Netherlands and Switzerland, but I'm planning to apply in Netherlands since it will be my second point of entry. Detecting structured data in Google Webmaster Tools What is the Physical Meaning of Commutation of Two Operators? I understand the mathematics of commutation relations and anti-commutation relations, but what does it physically mean for an observable (self-adjoint operator) to commute with another observable (self-adjoint operator) in quantum mechanics?   E.g. an operator $A$ with the Hamiltonian $H$? macOS Sierra restarting download instead of resuming Is it "I" or "me" in "Keep Tom and I/me updated"? I'm looking for the fastest way to determine if a long value is a perfect square (i.e. its square root is another integer):    I've done it the easy way, by using the built-in  function, but I'm wondering if there is a way to do it faster by restricting yourself to integer-only domain. Maintaining a lookup table is impractical (since there are about 231.5 integers whose square is less than 263).   Here is the very simple and straightforward way I'm doing it now:  public final static boolean isPerfectSquare(long n) {   if (n &lt; 0)     return false;    long tst = (long)(Math.sqrt(n) + 0.5);   return tst*tst == n; }   Note: I'm using this function in many  problems.  So no one else will ever have to maintain this code.  And this kind of micro-optimization could actually make a difference, since part of the challenge is to do every algorithm in less than a minute, and this function will need to be called millions of times in some problems.    I've tried the different solutions to the problem:   After exhaustive testing, I found that adding 0.5 to the result of Math.sqrt() is not necessary, at least not on my machine. The  was faster, but it gave incorrect results for n >= 410881.  However, as suggested by , we can use the FISR hack for n &lt; 410881. Newton's method was a good bit slower than Math.sqrt().  This is probably because Math.sqrt() uses something similar to Newton's Method, but implemented in the hardware so it's much faster than in Java.  Also, Newton's Method still required use of doubles. A modified Newton's method, which used a few tricks so that only integer math was involved, required some hacks to avoid overflow (I want this function to work with all positive 64-bit signed integers), and it was still slower than Math.sqrt(). Binary chop was even slower.  This makes sense because the binary chop will on average require 16 passes to find the square root of a 64-bit number. According to John's tests, using or statements is faster in C++ than using a switch, but in Java and C# there appears to be no difference between or and switch. I also tried making a lookup table (as a private static array of 64 boolean values).  Then instead of either switch or or statement, I would just say if(lookup[(int)(n&amp;0x3F)]) { test } else return false;.  To my surprise, this was (just slightly) slower. This is because .