I would like to prove the following, $$\sum_{k=0}^{n-1} \text{cos}\left(\frac{2\pi k}{n} \right) = \sum_{k=0}^{n-1} \text{sin}\left(\frac{2\pi k}{n} \right) = 0$$ This is equivalent to showing that if we have a regular n-gon, all of the vectors in $\mathbb{R}^2$ pointing to every vertex cancel eachother out to $\textbf{0}$. It seems obvious enough and i've tried a number of examples for small $n$.  I am not sure how to start a proof though.
Is there anybody who can help me show the following?      $$ \sum_{x=0}^{n-1} \cos\left(k +x{2\pi\over n}\right) =0  \qquad\hbox{and}\qquad  \sum_{x=0}^{n-1} \sin\left(k +x{2\pi\over n}\right) =0 $$   I can only prove this claim for the sine expression when $k=0$, by grouping the output values such that they all cancel out (for both $n$ even and odd). I have thought about using complex numbers in exponential form to prove it, but I can't do it ... :P   I would really appreciate an algerbraic proof.
Finding an angle within an 80-80-20 isosceles triangle Possible Duplicate:          I am a little bit confused about asking a question:     Can you please tell me my next work?   or     Could you please tell me my next work?   Which is more correct? Does $\pi$ contain all possible number combinations? I came across the :    Which states:     $\pi$ Pi      Pi is an infinite, nonrepeating $($sic$)$ decimal - meaning that   every possible number combination exists somewhere in pi.  Converted   into ASCII text, somewhere in that infinite string of digits is the   name of every person you will ever love, the date, time and manner    of your death, and the answers to all the great questions of    the universe.   Is this true? Does it make any sense ? Filtering points in GeoServer layer The product of all the elements of a finite abelian group I'm trying to prove the following statements.  Let $G$ be a finite abelian group $G = \{a_{1}, a_{2}, ..., a_{n}\}$.   If there is no element $x \neq e$ in $G$ such that $x = x^{-1}$, then $a_{1}a_{2} \cdot \cdot \cdot a_{n} = e$.   Since the only element in $G$ that is an inverse of itself is the identity element $e$, for every other element $k$, it must have an inverse $a_{k}^{-1} = a_{j}$ where $k \neq j$. Thus $a_{1}a_{1}^{-1}a_{2}a_{2}^{-1} \cdot \cdot \cdot a_{n}a_{n}^{-1} = e$.   If there is exactly one $x \neq e$ in $G$ such that $x = x^{-1}$, then $a_{1}a_{2} \cdot \cdot \cdot a_{n} = x$.   This is stating that $x$ is not the identity element but is its own inverse. Then every other element $p$ must also have an inverse $a_{p}^{-1} = a_{w}$ where $p \neq w$. Similarly to the first question, a rearrangement can be done: $a_{1}a_{1}^{-1}a_{2}a_{2}^{-1} \cdot \cdot \cdot xx^{-1} \cdot \cdot \cdot  a_{n}a_{n}^{-1} = xx^{-1} = e$. And this is where I am stuck since I proved another statement.    Any comments would be appreciated for both problems. How can I view the time of a text message in iOS 7? In iOS 7, I've noticed that some SMS/MMS messages have a date and time stamp displayed above the message.  Although this is useful, not all text's in the list receive such a date and time stamp.  Is there a way that I can view the date and time for ever text message? Android - Call log/ history : Retrieve a dialed number when there was no network coverage I dialed a number when I had no network support but it does not appear in the call history of my android phone.  Is there any way to get it ? (system logs or whatever)  THanks a lot in advance, Jack Why is feed-forward mechanism used in hash functions? The compression function of SHA-1 when used in Davies-Meyer mode adds its input to the chaining values at the final step. For the first message block, the IV is used as the input and in the next step, the previous block's output (as specified by Merkle-Damgård) is used. These values are fed to the last round of the compression function.  Why would removing the feed-forward part (i.e.,the final step where the input value is added to the result to get the final output) make the hash function insecure? Apart from the fact that the compression function would not satisfy desired randomness properties (being a permutation for each fixed message). Who wrote about an energy free space elevator? Am I Running as a Service I am currently writing a little bootstrap code for a service that can be run in the console. It essentially boils down to calling the OnStart() method instead of using the ServiceBase to start and stop the service (because it doesn't run the application if it isn't installed as a service and makes debugging a nightmare).  Right now I am using Debugger.IsAttached to determine if I should use ServiceBase.Run or [service].OnStart, but I know that isn't the best idea because some times end users want to run the service in a console (to see the output etc. realtime).  Any ideas on how I could determine if the Windows service controller started 'me', or if the user started 'me' in the console? Apparantly  is not the answer. I thought about using commandline args, but that seems 'dirty'.  I could always see about a try-catch statement around ServiceBase.Run, but that seems dirty. Edit: Try catch doesn't work.  I have a solution: putting it up here for all the other interested stackers:      public void Run()     {         if (Debugger.IsAttached || Environment.GetCommandLineArgs().Contains&lt;string&gt;("-console"))         {             RunAllServices();         }         else         {             try             {                 string temp = Console.Title;                 ServiceBase.Run((ServiceBase[])ComponentsToRun);             }             catch             {                 RunAllServices();             }         }     } // void Run      private void RunAllServices()     {         foreach (ConsoleService component in ComponentsToRun)         {             component.Start();         }         WaitForCTRLC();         foreach (ConsoleService component in ComponentsToRun)         {             component.Stop();         }     }   EDIT: There was another question on StackOverflow where the guy had problems with the Environment.CurrentDirectory being "C:\Windows\System32" looks like that may be the answer. I will test today. Current situation: I'm required to cite "normative" documents (read: standards and manuals) separately from articles and books etc. Using biblatex and biber with the numeric style I'm doing that by issuing multiple printbibliography commands like so:  \usepackage[style=numeric,backend=biber,bibencoding=utf8]{biblatex} %... lots of stuff \printbibliography[type=standard,title=Standards \&amp; Manuals] \vskip\baselineskip \printbibliography[type=manual,heading=none] \printbibliography[nottype=standard,nottype=manual]   which gives a heading Standards &amp; Manuals, prints a list of manuals, leaves a bit of space then the list of standards and finally, under a new heading Bibliography gives books and articles etc. Thus far everything is fine, enter:  The Problem: I'm using the shorthand field in my database for the standards and manuals which works perfect as far as output within the text is concerned and sets the shorthand as the label in the bibliography and prints the details to the right of it - problem is: my shorthands are considerably longer than the labels generated by the numeric style, but even though they're in three different \printbibliography commands, everything gets indented to the maximum width of the label, making the part for books and articles look quite ridiculous.  The following shows the problem with only two separate \printbibliography (and an obviously exaggeratedly long shorthand):   Now  what I would obviously want is for each of the \printbibliography commands to only use the indentation actually necessary to accommodate the largest label appearing within, i.e. rendering the Manuals section as above, but the Bibliography section like follows:   Can anyone help me to achieve this? Or give me some pointer where to look?  The above was produced with the following MWE on an up-to-date TeXLive 2014:  \documentclass[10pt,british]{scrbook}  \usepackage{babel} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern}  \usepackage{filecontents} \begin{filecontents*}{\jobname.bib} @manual{HowTo,     title = {How to Make the World Go Round},     shorthand = {HowToMakeTheWorlGoRound},     publisher = {The Publisher},     year = {2000} }  @article{Article,     author = {Author, A. and Buthor, B.},     journal = {Journal Title},     pages = {113--126},     title = {The Article},     volume = {65},     year = {1900} } \end{filecontents*}  \usepackage[style=numeric,backend=biber,bibencoding=utf8]{biblatex} \addbibresource{\jobname.bib}  \usepackage[hidelinks]{hyperref}  \begin{document} \let\cleardoublepage\relax \chapter{The Document} Nothing to see here, but we refer to the \parencite{HowTo} based on a principle discovered by \textcite{Article}.  \printbibliography[type=manual,title=Manuals] \printbibliography[nottype=manual] \end{document} What is the cause of this sudden drop in CV queue size? At Sat, 30 Nov 2013 14:51:01 GMT, there was a sudden drop in the close votes review queue size from 98932 to 98314. This is about 600 questions over the course of one minute.  It doesn't seem like a routine cleanup script. If it was, this is a very odd time for it to run. Also, the drop occurred over the span of a minute, so it's highly unlikely there was simply increased review activity. So, I ask: What was its cause?         (yes please don't shoot me I know my freehand circle is blue) (it's actually a dark teal look again) (the source for this script I  (slightly modified), found through  post)  Raw data:  Epoch time  CV Size 1385822821  98932 1385822881  98932 1385822941  98932 1385823001  98932 *** 1385823061  98932 *** 1385823121  98314     [author's note: the difference between iterations is 60 seconds] 1385823181  98314 1385823241  98314 1385823301  98314 1385823361  98314 How to compare Unicode strings in Javascript? Force applied off center on an object Assume there is a rigid body in deep space with mass $m$ and moment of inertia $I$. A force that varies with time, $F(t)$, is applied to the body off-center at a distance $r$ from its center of mass. How do I calculate the instantaneous acceleration, rotational acceleration, and trajectory of this object, assuming it starts from rest? Number Theory: Find all solutions of $\phi(n)=16$ and $\phi(n)=24$ Conditional Expectation of Composite Function Preliminaries  Let $(\Omega, \mathcal{G}, \mathbb{P})$ be a complete probability space.  Let $D$ be a complete, separable, metrizable topological space with Borel $\sigma$-algebra $\mathcal{B}(D)$ (such as $D = \mathbb{R}^q$ with $\sigma$-algebra $\mathcal{B}(D) = \mathcal{B}(\mathbb{R}^d)$).  Let $\mathbb{R}$ be equipped with its canonical Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R})$.  Let $g: \Omega \times D \rightarrow \mathbb{R}$ be a bounded $(\mathcal{G} \otimes \mathcal{B}(D) ) / \mathcal{B}(\mathbb{R})$-measurable function.  Let $\Pi: \Omega \rightarrow D$ be a $\mathcal{G}/\mathcal{B}(D)$-measurable random variable.  Let $H : \Omega \rightarrow \mathbb{R}$ be a $\mathcal{G}/\mathcal{B}(\mathbb{R})$-measurable random variable, defined by $$ H(\omega) := g(\omega, \Pi(\omega)).$$ Note, that, since $g$ is bounded, we have $H \in \mathcal{L}^2(\Omega, \mathcal{G}, \mathbb{P})$.  Let $j: D \rightarrow \mathcal{L}^2(\Omega, \mathcal{G}, \mathbb{P}) $ be defined by $$ j(\pi)(\omega) := g(\omega, \pi)  $$  For all $\pi \in D$, let $j(\pi)$ be independent of $\Pi$.  Question  I am interested in the conditional expectation $$\mathbb{E}[H \mid \Pi] :\Omega \rightarrow \mathbb{R}$$ of $H$ with respect to $\Pi$. More specifically, I suspect that (a $\mathbb{P}$-unique version of) this condititional expectation is given by  $$ \mathbb{E}[H \mid \Pi] (\omega) = \mathbb{E}[j(\Pi(\omega))], \quad (\dagger)  $$ whereby $\mathbb{E}[j(\Pi(\omega))]$ can of course also be written as $$\mathbb{E}[j(\Pi(\omega))] = \int_{\Omega} j(\Pi(\omega))(\tilde{\omega}) d\mathbb{P}(\tilde{\omega}) . $$  How can I prove, that $(\dagger)$ is the case? I have tried, tracking the definition of conditional expectation and using Fubini, but with little success so far.  Thanks for any advice! How to create a DEB file manually? I want to create a DEB file manually. I would like to just provide a folder which contains data to install, and a script to be executed after installation.  Is this possible? Convert a comma delimited list of numbers to a hyphenated list or range Is there any formula for number of divisors of $a \times b$? Let $a$ and $b$ be two numbers,    Number of divisors of $a$ is $n_1$; Number of divisors of $b$ is $n_2$;    How to find the number of divisors $N$ of product $a \times b$ by using known number of divisors of factors $n_1$ and $n_2$?  If $a = 20$ and $b = 21$, then $n_1 = 6$ (number of divisors of $a$), $n_2 = 4$ (number of divisors of $b$. Then $a \times b = 420$ and $N = 32$ (number of divisors of $a \times b$).  Is there a formula to find $N$ in terms of $n_1$ and $n_2$? How to carry audio over SSH? Is there a way to carry audio over ssh?   Say after connecting to remote machine through ssh -X I opened a browser and want to watch a youtube video. Currently I could only able to watch the video without any audio. But I want to hear the audio too. How to do that?