Assign variable to solve
Assign the results from a Solve to variable(s)
How to show that if $\phi(n)$ equals to n itself, then n must be 1? Help us test the alpha version of our Android app We are in the early stages of development of the official Stack Exchange Android app and we need your help with testing. To set a few expectations: there will be bugs, crashes, and missing features, all of which will be fixed in .    We are initially targeting Android devices running  (Android 4.1 or higher) starting with phones (up to ~5 inch) and eventually moving our way to tablets.    To register for alpha testing .  After you register we will send you an invite to a private Google community where you can find a link to download our app directly from the  store. The invitations will be sent out in waves, so if you don’t get an invite immediately, be patient.  Crash reports will be automatically reported back to us using . Bugs or feature requests should be reported here on meta and tagged . Please do not use the Google community to report bugs.  As always, we welcome your feedback and are committed to building the best mobile app possible for browsing the Stack Exchange sites. Why would turning off a sink faucet stop the shower? Solving functional equation $f(x+y)+f(x-y)=2f(x)\cos y$? How can I solve this functional equation, where $x,y$ are any real numbers and $f:\mathbb{R}\to \mathbb R$ is a function such that : $$f(x+y)+f(x-y)=2f(x)\cos y$$    I tried substituting $x=0$ to get $f(y)+f(-y)=2f(0)\cos y$ . Taking $x=y$ gives $f(2x)+f(0)=2f(x)\cos x$. I similar get some more relations like that, but its not really helping me in finding anything useful. I also think the function must either be $\sin$ or $\cos$ looking at the product to sum formulae. Can anyone tell me how should I solve this? I encountered this particular combinatorial identity :$$\begin{align}&amp;1\dbinom{n-1}{r-1}+2\dbinom{n-2}{r-1}+\cdots+(n+1-r)\dbinom{r-1}{r-1}\\=&amp;\dbinom{n}{r}+\dbinom{n-1}{r}+\cdots+\dbinom{r}{r}\\=&amp; \dbinom{n+1}{r+1}\end{align}$$How does one figure it out? Export File Dialog in Blender 2.80 How can I get bring up a popup file browser in Blender 2.80 invoked by a button click in my addon, to define the name and location of a file to be exported?  In Blender 2.7x, I did this via the context window manager in the invoke function, but it is very possible that this hasn't been the preferred method for years, as I'm not finding useful documentation when I search for how to update this.  class MYTOOL_OT_write_data(bpy.types.Operator):     bl_idname = "mytool.write_data"     bl_label = "Write data file"      directory: bpy.props.StringProperty(subtype="FILE_PATH")     filename: bpy.props.StringProperty(subtype="FILE_NAME")      def invoke(self, context, event):  # 2.7x method         WindowManager = context.window_manager         context.window_manager.FILEBROWSERect_add(self)  # &lt;-- error here         return{"RUNNING_MODAL"}      def execute(self, context):         directory = self.directory         filename = self.filename         # Write csv data file using normal python functionality         return{'FINISHED'}   In Blender 2.80, I am getting the error "AttributeError: 'WindowManager' object has no attribute 'FILEBROWSERect_add'".  Thanks for your help.    UPDATED SOLUTION based on brockmann's solution below.  The trick is to use the ExportHelper util (which existed long before Blender 2.80):  from bpy_extras.io_utils import ExportHelper  class MYTOOL_OT_write_data(bpy.types.Operator, ExportHelper):     bl_idname = "mytool.write_data"     bl_label = "Write data file"      filename_ext = ".csv"  # ExportHelper mixin class uses this     def execute(self, context):         filepath = self.filepath         # f = open(filepath, 'w')         # f.write(stuff)         # f.close()         return{'FINISHED'} Let $R$ be a commutative ring with $1$. Suppose that all ideals $I \neq R$ are prime. Prove that $R$ is a field.   Help me some hints.  Thanks a lot. On really old Ubuntu 6.06 - How to upgrade I'm on an old Ubuntu version (Ubuntu 6.06.2 LTS Server) which isn't really supported any more. So the automatic updates don't work. I have ISPConfig installed to manage a few web sites.  What's the best course of action? I'd like to upgrade to something current, but I can't figure out how to go about doing this, given that the apt-get scripts don't really work any longer. How to show that $7\mid a^2+b^2$ implies $7\mid a$ and $7\mid b$? For my proof I distinguished the two possible cases which derive from $7 \mid a^2+b^2$:   Case one: $7\mid a^2$ and $7 \mid b^2$ Case two (which (I think) is not possible): $7$ does not divide $a^2$ and $7$ does not divide $b^2$, but their sum.  I've shown that case $1$ implies $7\mid a$ and $7\mid b$, so I just have show that case 2 isn't possible - I'll be happy for any input.  MfG, Karl Are these plural or singular? I happen, time and again, to read and hear people – among which people working in cinema – regretting the fact that film (in the sense of actual reels of plastic strip) is nowadays very rarely used when shooting movies. Is there some ground to such nostalgia?  Of course I am not looking for opinions for or against film vs. digital media, but sourced, objective reasons. Is there anything that could be done with film but not with digital supports? Are there measurable differences in effects on viewers? In Minecraft, there are a lot of cool sounds. I decided to find there location and move them to my desktop. I have Ubuntu, and if when I remove them from that folder will Minecraft be unable to play them or is it smart enough to figure out that they are gone and download them again. Before minecraft was installed, it had to download the files. I even moved my favorite song (creative 6) from the sounds. The sounds are not labeled and have seemingly random names like b0b12118a97ddd733a50e9382e91a21652568641 and how will I know what to call them that will be recognized without digging through the index if Minecraft fails to download them? Or do I not have to worry because it will download them again upon launch?  Note: Why does it not label them and call them random names? I am trying to mimic our institute's PPt-template for presentation foils using the beamer-class. The corporate quidelines prescribe the use of Helvetica as the main font. However, package helvet does not provide a good mathematical font. Therefore I want to replace it with the math font from the cmbright package:  \documentclass{beamer}  \usepackage{lmodern} \usepackage[T1]{fontenc}  %main font Helevtica \renewcommand{\familydefault}{\sfdefault} \usepackage{helvet}  %prevent beamer from meddling with math fonts \usefonttheme{professionalfonts}  %following lines borrowed from cmbright.sty \DeclareSymbolFont      {operators} {OT1}{cmbr}{m}{n} \DeclareSymbolFont        {letters} {OML}{cmbrm}{m}{it} \SetSymbolFont      {letters}{bold} {OML}{cmbrm}{b}{it} \DeclareSymbolFont        {symbols} {OMS}{cmbrs}{m}{n} \DeclareMathAlphabet{\mathit} {OT1}{cmbr}{m}{sl} \DeclareMathAlphabet{\mathbf} {OT1}{cmbr}{bx}{n} \DeclareMathAlphabet{\mathtt} {OT1}{cmtl}{m}{n} \DeclareMathAlphabet{\mathbold}{OML}{cmbrm}{b}{it} \DeclareMathSymbol{\alpha}{\mathalpha}{letters}{11} \DeclareMathSymbol{\beta}{\mathalpha}{letters}{12} \DeclareMathSymbol{\gamma}{\mathalpha}{letters}{13} \DeclareMathSymbol{\delta}{\mathalpha}{letters}{14} \DeclareMathSymbol{\epsilon}{\mathalpha}{letters}{15} \DeclareMathSymbol{\zeta}{\mathalpha}{letters}{16} \DeclareMathSymbol{\eta}{\mathalpha}{letters}{17} \DeclareMathSymbol{\theta}{\mathalpha}{letters}{18} \DeclareMathSymbol{\iota}{\mathalpha}{letters}{19} \DeclareMathSymbol{\kappa}{\mathalpha}{letters}{20} \DeclareMathSymbol{\lambda}{\mathalpha}{letters}{21} \DeclareMathSymbol{\mu}{\mathalpha}{letters}{22} \DeclareMathSymbol{\nu}{\mathalpha}{letters}{23} \DeclareMathSymbol{\xi}{\mathalpha}{letters}{24} \DeclareMathSymbol{\pi}{\mathalpha}{letters}{25} \DeclareMathSymbol{\rho}{\mathalpha}{letters}{26} \DeclareMathSymbol{\sigma}{\mathalpha}{letters}{27} \DeclareMathSymbol{\tau}{\mathalpha}{letters}{28} \DeclareMathSymbol{\upsilon}{\mathalpha}{letters}{29} \DeclareMathSymbol{\phi}{\mathalpha}{letters}{30} \DeclareMathSymbol{\chi}{\mathalpha}{letters}{31} \DeclareMathSymbol{\psi}{\mathalpha}{letters}{32} \DeclareMathSymbol{\omega}{\mathalpha}{letters}{33} \DeclareMathSymbol{\varepsilon}{\mathalpha}{letters}{34} \DeclareMathSymbol{\vartheta}{\mathalpha}{letters}{35} \DeclareMathSymbol{\varpi}{\mathalpha}{letters}{36} \DeclareMathSymbol{\varrho}{\mathalpha}{letters}{37} \DeclareMathSymbol{\varsigma}{\mathalpha}{letters}{38} \DeclareMathSymbol{\varphi}{\mathalpha}{letters}{39} \let\upDelta\Delta \let\upOmega\Omega \let\upGamma\Gamma \let\upTheta\Theta \let\upLambda\Lambda \let\upXi\Xi \let\upPi\Pi \let\upSigma\Sigma \let\upUpsilon\Upsilon \let\upPhi\Phi \let\upPsi\Psi  %doesn't seem to have an effect %\DeclareMathSizes{12}{12.5}{10}{10}  \begin{document} \begin{frame}  Some Math E$E$ \end{frame} \end{document}   This works quite well, except that the math letters are somewhat smaller than the normal Helvetica text. Is there a way to scale the fontsize, say by 1.1, if the typesetter is in math mode? I played around with it for a while, but I simply can't figure it out.  I made a tank that fires missiles, and when the missiles hit the walls, I want them to bounce off, but I want them to bounce off to the right angle.  Right now I haven't got any obstacles, the missiles just bounce off when they get outside the viewportRectangle I made.  Is the solution I'm looking for quite advanced?  Is there a relativly simple way to do it? Convert ArrayList to String[] array Can't find the reason why root directory usage is so large One adjective for two or more nouns Could we use one adjective for two nouns? Is the following sentence correct?     It has low speed and voltage. Equivalence of conditions for Convergence of Non-negative Random Series Question  Let $X_n\geq 0$ be independent for $n\geq 1$. The following are equivalent.  $\sum_{n=1}^\infty X_n&lt;\infty$ a.s. $\sum_{n=1}^\infty[P(X_n&gt;1)+E(X_nI(X_n\leq 1)]&lt;\infty$ $\sum_{n=1}^\infty E(X_n/(1+X_n))&lt;\infty$   My attempt I was able to prove $(1)$ implies $(2)$ and $(2)$ implies $(3)$ but not $(3)$ implies $(1)$. For $(1)$ implies $(2)$, observe that by Borel-Cantelli $\sum_{n=1}^\infty P(X_n&gt;1)&lt;\infty$, else $X_n&gt;1$ infinitely often w.p.1 so that $(1)$ is violated. To see that $\sum_{n=1}^\infty E(X_nI(X_n\leq 1)]&lt;\infty$, apply Kolmogorov three-series theorem (I think). For $(2)$ implies $(3)$ we can write that $$ \begin{align} \frac{X_n}{1+X_n}&amp; =\frac{X_n}{1+X_n}I(X_n\leq 1)+\frac{X_n}{1+X_n}I(X_n&gt;1)\\ &amp;\leq X_nI(X_n\leq 1)+ I(X_n&gt;1). \end{align} $$ Now take the expectation of both sides and sum on $n$ using $(2)$ to conclude $(3)$. For the last implication $(3)$ implies $(1)$, I tried to use Kolmogorov three series theorem. To this end put $Y_n=X_nI(X_n\leq 1)$. First note that $\sum_{n=1}^\infty P(X_n&gt;1)&lt;\infty$ since $$ P(X_n&gt;1)\leq P\left(\frac{X_n}{1+X_n}&gt;\frac{1}{2}\right)\leq 2E\frac{X_n}{1+X_n} $$ by Markov's inequality. But I am unable to show that $\sum EY_n&lt;\infty$ and $\sum\text{Var}(Y_n)&lt;\infty$. Any help is appreciated. What is butterfly lighting, and when do I use it? I often hear the term 'Butterfly Lighting' used when discussing high-fashion and glamor photography. How do I set up Butterfly Lighting, and when do I use it? Is frequency quantized in the black body spectrum?