I am working on some presentation with beamer. However, I need a specific frame size (not the default values of aspect ratios). What I need is to crop the frame around its content and control the padding size. For example:  \documentclass{beamer}  \beamertemplatenavigationsymbolsempty  \begin{document} \begin{frame} \begin{tikzpicture}  \draw (0,0) circle(3cm);  \end{tikzpicture} \end{frame} \end{document}   And here is what I got:    What I am looking for is to get:
I'm programming a presentation in LaTeX using the beamer class.  However, it defaults to 4:3 aspect ratio slides, while everything I use is 16:9. Is there an easy way to change this using a command or two?
Show that  $$t(n) = 1 + \sum_{ j=0}^{n-1} t(j)$$  is the same as  $$t(n) = 2^n$$  Initial condition $t(0) = 1$ I have recently started reading and watching Naruto. I was wondering in what aspect does the clones made by different techniques differ? They all have the same potential and capabilities. How do I make Thunderbird open links in Chromium? I want to be able to click links in Thunderbird emails and have them open in Chromium (package name chromium-browser). Instead, they open in Firefox.  I have tried the following:   xfce preferred applications (Chromium is not listed, so I had to hunt down the executable) -- next time I run chromium, it complains that it is no longer the default browser. Telling Chromium to set itself as the default browser. Thunderbird opens links in Firefox. update-alternatives. It's already listed as an alternative at the same priority as Firefox. The "open web browser" app menu item opens Chromium. Thunderbird opens links in Firefox.   I have looked through the entire preferences gui in Thunderbird (but not the advanced config editor yet) and seen nothing that mentions what browser to open links in.  Please note that this is not Google Chrome. Integer squares of the form $x^4+x^3+x^2+x+1$ I want to solve ${y^2}=x^4+x^3+x^2+x+1$ in $\mathbb{Z}$.    I can find four solutions. Is there another solution? I know that there are six solutions. $$(-1, \pm 1)\;,\;(0, \pm 1).$$  My try: \begin{align}   &amp; x^4+x^3+x^2+x+1=0\quad \Rightarrow \underbrace{\left( x+\frac{1}{x} \right)^2}_{t^2}+\underbrace{\left( x+\frac{1}{x} \right)}_{t}-1=0\,\Rightarrow t\notin \mathbb{Z} \\   &amp; x^4+x^3+x^2+x+1={\pm 1}^2\quad\Rightarrow \,x=-1\,\,,\,x=0.\\  \end{align}   Thanks. What is the purpose of the first test in an inductive proof? Learning about proof by induction, I take it that the first step is always something like "test if the proposition holds for $n = \textrm{[the minimum value]}$"  Like this:       Prove that $1+2+3+...+n = \frac{n(n + 1)}{2}$ for all $n \ge 1$.   Test it for $n = 1$:  $$n = \frac{n(n + 1)}{2} \implies 1 = \frac{2}{2}\textrm{, which holds.}$$  * The rest of the proof goes here *    So, I do it all the time (like a standard). But I never really thought about why. Why do I do such test? I can see that if the test does not hold, it cannot be proven by induction, I guess. But is there another reason we do this? cron ignores variables defined in ".bashrc" and ".bash_profile" I have defined "SHELL" variable in /etc/crontab file:  [martin@martin ~]$ grep SHELL /etc/crontab  SHELL=/usr/local/bin/bash [martin@martin ~]$ file /usr/local/bin/bash /usr/local/bin/bash: ELF 32-bit LSB executable, Intel 80386, version 1 (FreeBSD), dynamically linked (uses shared libs), for FreeBSD 8.0 (800107), stripped [martin@martin ~]$    In addition, all my scripts in /etc/crontab file are started under user "martin". However /home/martin/.bash_profile(for login shell) and /home/martin/.bashrc(for non-logging shell) contain some variables which are ignored in case of cron job, but are used in case I log into machine over SSH or open new bash session. Why cron ignores those variables? Isn't cron simply executing "/usr/local/bin/bash my-script.sh" with permissions for user "martin"? How to track the number of tag wikis I edited? I want to see the number of tag wiki that I edited till now. Also I want to see which are accepted and which are rejected with the detail of all those tags. How can I know that? Also I want to know the progress for Research Assisstant badge. I used to use a lot ctrl+alt+numpad to move windows, how can I use this feature on ubuntu 18 like it was default before?  Thank you in advance I want to prove that $xe^{-x}$ is bounded by some constant for all the $x \in [0, \infty)$. So I take derivative of this guy, and found that on $x \in [0, \infty)$, its derivative is positive from $[0, 1)$, $0$ at $x=1$, and negative from (1, in infinity). So  $xe^{-x}$ achieves its maximum in $[0, \infty)$ at $x=1$, which is $1/e$. However, to prove that it's bounded by $1/e$, I need to prove that as $x$ goes to infinity, $xe^{-x}$ converges to $0$. Because otherwise, if it went to some value below $(-1/e)$, $xe^{-x}$ would not be bounded by $1/e$. How can I prove that $xe^{-x}$ converges to $0$? I mean, as $x$ goes to infinity, $e^{-x}$ converges to zero, but at the same time, x diverges to infinity. How can I formally prove that the product term converges to $0$? Do I need to use the epsilon definition of convergence to prove it?  Could you just show me the formal proof of $xe^{-x}$ converges to $0$? Or is there's any other way to approach this question? My professor suggested that I should use the power series for $e^{-x}$...?   Your help is greatly appreciated! Thank you so much!  I hope you enjoy your Saturday! JSP in /WEB-INF returns "HTTP Status 404 The requested resource is not available" Nested Mathematical Functions This question has bugged me for a while. I want to write a nested function, f(g(x)). If I naively enter it:  \begin{align}     f(g(x))=y \end{align}     The problem is that each and every parenthesis is the same size as all of the others. Shouldn't the outer parentheses be larger? But if I do this:  \begin{align}     f\left(g(x)\right)=y \end{align}     Now it looks even less like the functions are nested. What's the proper way to do this? "print()" is producing undesired output in Visual Studio Code How to check iOS version? Assigning CRS to Lat/Lon in R giving error: "Geographical CRS given to non-conformant data" Why and how do "bigger pixels" make a better picture? This is going off of today's iPhone 5S announcement, when Phil Schiller said that "bigger pixels" is the key to a better picture.      What does this mean? Is it true or just a marketing ploy? Truncated 200GB table but disk space not released I have only 2GB left, so I need to remove this history table. This table now is empty but the database disk space not released. And the database file is 320GB. How can I find infinity focus on a kit lens with no markers? I have a kit lens which has no markers. What would be a recommended way to find infinity focus on this lens?  I plan to shoot objects at night -- not necessarily stars or the moon. So, adjusting the focus using the viewfinder may not be an option. Servlet returns "HTTP Status 404 The requested resource (/servlet) is not available" Reset password in Drupal 6? how to crop image in to pieces programmatically