CMD `::` autoformats badly
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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. Why didn't the Emperor realize that torturing Luke to death in front of Vader might be a bad idea? Multiple returns from a function Find a recurrence relation for the number of ternary strings of length n that do not contain two consecutive 0s or two consecutive 1s. Basic question, how do I resume interrupted / failed downloads in Firefox? Why is $\frac{1}{\infty } \approx 0 $ and $ \frac{1}{0} = {\infty}$? First I have checked the search option but found nothing relevant to my problem and also level of math. I just started learning the language of mathematics, on my own and I have trouble understanding why  $  \frac{1}{\infty } \approx 0 $ and  why $\frac{1}{0} = {\infty}$?  I want to know how exactly do I get to these answers. It might sound strange but it's hard for me as a beginner to picture things like this on my own. How do I simplify and evaluate the limit of $(\sqrt x - 1)/(\sqrt[3] x - 1)$ as $x\to 1$? Consider this limit:  $$ \lim_{x \to 1} \frac{\sqrt x -  1}{ \sqrt[3] x -  1}  $$  The answer is given to be 2 in the textbook.  Our math professor skipped this question telling us it is not in our syllabus, but how can it be solved? With $x+y\ge z$ $(x,y,z\ge0)$, prove that: $$\frac{x}{1+x}+\frac{y}{1+y}\ge\frac{z}{1+z}$$  I'm aware that using analytic view this is easy since $f(x)=\frac{x}{1+x}$ is concave in $[0,\infty)$. However I want to prove it using merely algebraic techniques. Is that possible? Is there any way of doing something like,  &lt;script runat=server&gt;     Platform.Load("core","1");     //initialize DE     var _Sent = DataExtension.Init("_Sent");     //retrieve data     var _Sent_data = _Sent.Rows.Retrieve();     Write("Length: " + _Sent_data.length); //returns 0 &lt;/script&gt;   What would the external key for a system dataview be? Using VHDL code to design a JK Flip Flop I'm using quartus II to design a JK Flip Flop. However, my results show unknown output. Why is it?  Intended design circuit:    VHDL code:  library IEEE;  use IEEE.STD_LOGIC_1164.ALL;  entity JKFlipFlopGate is     port(         J,K,Clk : in std_logic;             --JK Flip-Flop gate input 1 &amp; 2         Q,Qbar : out std_logic              --JK Flip-Flop gate output     ); end JKFlipFlopGate;  architecture result of JKFlipFlopGate is         signal out1,out2,out3,out4 : std_logic;  begin     out1    &lt;= NOT(J AND Clk AND out4);                      out2    &lt;= NOT(K AND Clk AND out3);      out3    &lt;= out1 NAND out4;     out4    &lt;= out2 NAND out3;      Q       &lt;= out3;     Qbar    &lt;= out4;     end result; Why does the beta distribution become U shaped when $\alpha$ and $\beta$ <1? In the Beta distribution (used to model Bernoulli probabilities), the $\alpha$ and $\beta$ parameters can be interpreted as the number of heads$+1$ and the number of tails$+1$ seen. So, if they were both $2$, it would lean towards the coin being fair and have a maximum at $0.5$. If they are both $20$, the distribution would become even surer we're dealing with a fair coin and peak even more at $p=0.5$.  What I don't get is its behavior when $\alpha$ and $\beta$ both become $&lt;1$.   In that case, it becomes U-shaped and the density peaks at $p=0$ and $p=1$. Meaning the coin is likely to be two-sided. I know there is an intuition for this since I think I had an idea about it a long time ago. However, I've been trying to recollect all day and can't piece it together. Does anyone have an intuition? Why does $x$ divided by zero not equal $x$? CLI path to ftp network connection I've setup a connectio to an FTP server in Nautilus / Nemo / Caja, the file manager. Now I can browse the contents via this file manager.  How can I browse these files via the command line? What is the path to this network connection? Is this possible, is this a system connection, or specific to the file manager, and do I have to setup a separate FTP connection to do that? If I open this connection in Caja, I see it in Nemo as well, so it seems like a system connection. What is the formula for Real vs Apparent Depth, when not looking perpendicular to the interface?   I know the formula $$\frac{\text{real depth}}{\text{apparent depth}}=n$$   but I'm not looking for this. Structure of a C++ Object in Memory Vs a Struct If I have a class as follows     class Example_Class     {        private:          int x;           int y;         public:           Example_Class()           {               x = 8;              y = 9;          }        ~Example_Class()         { }     };   And a struct as follows  struct {    int x;    int y; } example_struct;   Is the structure in memory of the  example_struct simmilar to that in Example_Class  for example if I do the following  struct example_struct foo_struct; Example_Class foo_class = Example_Class();  memcpy(&amp;foo_struct, &amp;foo_class, sizeof(foo_struct));   will foo_struct.x = 8 and foo_struct.y = 9 (ie: the same values as the x,y values in the foo_class) ?  The reason I ask is I have a C++ library (don't want to change it) that is sharing an object with C code and I want to use a struct to represent the object coming from the C++ library. I'm only interested in the attributes of the object.  I know the ideal situation would be to have Example_class wrap arround a common structure between the C and C++ code but it is not going to be easy to change the C++ library in use. how to read xml file from url using php If humans were able to catch all sun energy reaching the earth for their use, will the climate change? I guess that energy will be used up and, at the end, will contribute to heat the earth, so I see no big differences...  please explain your point of view. Can not move polygons but only vertices of an object by bmesh! Is there a way to programmatically determine if a font file has a specific Unicode Glyph? I'm working on a project that generates PDFs that can contain fairly complex math and science formulas.  The text is rendered in Times New Roman, which has pretty good Unicode coverage, but not complete. We have a system in place to swap in a more Unicode complete font for code points that don't have a glyph in TNR (like most of the "stranger" math symbols,) but I can't seem to find a way to query the *.ttf file to see if a given glyph is present.  So far, I've just hard-coded a lookup table of which code points are present, but I'd much prefer an automatic solution.  I'm using VB.Net in a web system under ASP.net, but solutions in any programming language/environment would be appreciated.  Edit: The win32 solution looks excellent, but the specific case I'm trying to solve is in an ASP.Net web system.  Is there a way to do this without including the windows API DLLs into my web site? How to configure axis on pgfplots? How to configure axis on pgfplots? Similar for this.  \documentclass{article} \usepackage{tikz,pgfplots} \pagestyle{empty} \begin{document} \begin{tikzpicture}[&gt;=latex] %x axis \draw[-&gt;] (-5,0) -- (5,0) node[below] {$x$}; \foreach \x in {-4,...,-1,1,2,...,4} \draw[shift={(\x,0)}] (0pt,2pt) -- (0pt,-2pt) node[below] {\footnotesize $\x$}; %y axis \draw[-&gt;] (0,-5) -- (0,5) node[left] {$y$}; \foreach \y in {-4,...,-1,1,2,...,4} \draw[shift={(0,\y)}] (2pt,0pt) -- (-2pt,0pt) node[left] {\footnotesize $\y$}; \node[below left] at (0,0) {\footnotesize $0$}; \end{tikzpicture} \end{document}