Proving the Invariance of a Set.
How to prove that a set is invariant?
Multiple object instances represented by two slightly shifted rectangles In a software diagram, I'd like to be able to represent multiple instances of an object by two rectangles with one of them slightly shifted. I recently learned about the pic style so I'm using it with two nodes to achieve what I want.  The following example does work, but I'd like to know if I can do the same without using the fill=white. My hack kind of assumes the entire picture background is white and that may not always hold. I'd like to be able to crop the unwanted picture part out.  \documentclass[crop,border=3pt]{standalone}  \usepackage{pgf,tikz} \usetikzlibrary{positioning}  \begin{document} \begin{tikzpicture}[     every node/.style={font=\sffamily\small},     block/.style={         rectangle, draw, align=center, minimum width=2cm,         minimum height=1.4cm, font=\sffamily\small     },     multinode/.pic={         \node [block] at (0.1,0.1) {};         \node [block, \tikzpictextoptions, fill=white] at (0,0) {\tikzpictext};% &lt;-- FILL WHITE     } ]     \node [block] (STG) at (2,5) {single\\instance};     \pic [pic text={multi\\instance}] at (5,5) {multinode}; \end{tikzpicture} \end{document}   The following images show first one version with the fill white hack, and then one version without it. How do we prove $n^n \mid m^m \Rightarrow n \mid m$? I'm not sure I've got this right. When proving $a^n \mid b^n \Rightarrow a \mid b$, can we do this indirectly? In short,  "Suppose $a$ does not divide $b$, this implies that $a^n$ does not divide $b^n$. But $a^n \mid b^n$, hence $a$ divides $b$."  How about $n^n \mid m^m \Rightarrow n \mid m$? Can we do this the same way? Let$\ p_n$ be the$\ n$-th prime. Can you give me a proof for$\ \prod_{i=1}^\infty \frac{p_i-1}{p_i}=P\approx \frac{1}{11.0453}$? I found$\ \prod_{i=1}^\infty \frac{p_i}{p_i-1}\approx 11.0453$ on . Moreover, writing a paper, should one provide a proof or it is trivial? Thanks in advance. What is the difference in this specific breadboard wiring? Theorem 13.9 in these  is the polynomial spectral mapping theorem:     Theorem 13.9 For a polynomial $p$ we have $\sigma(p(T)) = p(\sigma(T))$.     My questions:   First direction: Why would $q(T) (p(T) - p(\lambda) I)$ be an inverse for $T - \lambda I$? Update: @RobertIsrael said that instead $(p(T) - p(\lambda) I)^{-1} q(T)$ should be $(p(T) - p(\lambda) I)^{-1}$ be? According to the lines in the proof above, we have $$ (p(T) - p(\lambda) I)^{-1} = ((T - \lambda I) q(T))^{-1} \overset{?!}{=} q(T)^{-1} (T - \lambda I)^{-1}, $$ but as $\lambda \in \sigma(T)$ by assumption, $(T - \lambda I)^{-1}$ doesn't exist, right? For the second direction: Why does $x_i \in \sigma(T)$ imply that $p(x_i) - \mu = 0$? I thought that $x_i \in \sigma(T)$ only implies that $T - x_i I$ is not boundedly invertible. Since $\mu \in \sigma(p(T))$ we have that $p(T) - \mu I = c \prod_{k = 1}^{n} (T - x_i I)$ is not boundedly invertible but how does that help?   In both statements there seem to be a notion of "a product of operators is not invertible iff all factors aren't invertible", but how can that be true? Consider i.e. shift operators on a sequence spaces such that $ST = I$, i.e. the left and right shift operators, then the above mentioned notion doesn't hold, right? Expanding an event group shows more events than labeled when there are new events What's the difference between using a 50mm f/1.8G and a 50mm f/1.8D with a Nikon D80? Recently, Nikon announced their , as revamp of the old 50mm f/1.8D. :     The main difference between this new   "nifty fifty" and the almost   decade-old ƒ1.8D is that it has an   autofocus motor inside. This means you   can use it with any current or recent   Nikon SLR. The older lens lacks this   motor and is instead driven by one in   the camera, which cheaper bodies don't   have.   I currently use a Nikon D80 (and I don't plan on changing that any time soon), and I've been considering getting such a lens. Is there any difference between the D and the G lenses when used on a D80? I believe that the D one works just as well, since D80's have internal autofocus motors - is that right?  Besides the autofocus motor aspect, is there any other big difference between them? Since the D is priced at around $120 and the G at $220, should I just get the D? Using the expression "the same" for a previously mentioned item I frequently use &quot;the same&quot; as in the following sentence.  I need the details of XYZ project. Could you please send me the same as soon as possible?  Is this grammatically correct? I have not been corrected till now, but I have not seen the usage from native speakers very much either. Update: Seems like my example sentence was ambiguous. Hope the following sentence is better.  I need the documents for the meeting. Could you please send me the same as soon as possible? Why are site stats graph for beta sites not available to public? Currently, on all beta sites, the current site stats is placed prominently on the front page, which is great for knowing how the site is currently doing.  However, it doesn't say anything about the trend of the numbers - whether the site is sailing or it is sinking compared to the last month. According to Madara Uchiha, our moderator on Anime &amp; Manga SE, . Disregarding data which are not already public, , I can't see any reason why we are not allowed access to graphical representation of public site statistics. In fact, there is already a third-party app that scraps the information from the home page and . Can we have a beautiful visualization on how our beta sites evolve over time, from pony to unicorn?   Images stolen borrowed from Bulbapedia Can Button[] generate output without using Print[]? Given an InputStream called in which contains audio data in a compressed format (such as MP3 or OGG), I wish to create a byte array containing a WAV conversion of the input data. Unfortunately, if you try to do this, JavaSound hands you the following error:  java.io.IOException: stream length not specified   I managed to get it to work by writing the wav to a temporary file, then reading it back in, as shown below:  AudioInputStream source = AudioSystem.getAudioInputStream(new BufferedInputStream(in, 1024)); AudioInputStream pcm = AudioSystem.getAudioInputStream(AudioFormat.Encoding.PCM_SIGNED, source); AudioInputStream ulaw = AudioSystem.getAudioInputStream(AudioFormat.Encoding.ULAW, pcm); File tempFile = File.createTempFile("wav", "tmp"); AudioSystem.write(ulaw, AudioFileFormat.Type.WAVE, tempFile); // The fileToByteArray() method reads the file // into a byte array; omitted for brevity byte[] bytes = fileToByteArray(tempFile); tempFile.delete(); return bytes;   This is obviously less desirable. Is there a better way? What is the domain of $x^x$ as a real valued function? Consider the function $f(x) = x^x$.  Wolfram alpha tells me that this function's domain is $x : x&gt;0$, $x \in \mathbb{R}$. I can't see why it cannot be defined for a number like $(-2)$. I mean $(-2)^{-2}=0.25$, the same Wolfram Alpha told me. I realize that fractional powers for negative numbers may cause problems, but it could be defined for integers. Thanks for any help. This question led to a new package:      I want to reference to citations from other file. Similar to what xr package does to labels, but I want to apply it for the bibliography.  If I'm able to do \cite{someone} in the main file, and it creates let say the number [4], I want to do the same \cite{someone} in a second file, and also create the same number [4]. Similar to what xr does with the labels.  By the way, I tried to add the main.bbl file from the main file. However, that adds all the references to the end of the second file. I want to avoid this.  Is there any package that accomplish this? How do I choose grid and block dimensions for CUDA kernels? How to prove the following product? $$\frac{\sin(x)}{x}= \left(1+\frac{x}{\pi}\right) \left(1-\frac{x}{\pi}\right) \left(1+\frac{x}{2\pi}\right) \left(1-\frac{x}{2\pi}\right) \left(1+\frac{x}{3\pi}\right) \left(1-\frac{x}{3\pi}\right)\cdots$$ How do you normalize a zero vector In Problems in Mathematical Analysis, Volume 2 by Kaczor and Nowak, Problem 2.3.34 is stated as follow.   Find this limit using L`Hôpital rule  $$\lim\limits_{x\to\infty}x((1+\frac{1}{x})^x-e).$$  I have tried by using L'Hospital rule five times but have no result.  I can compute this limit by using Taylor expansion. How to stop event propagation with inline onclick attribute? Consider the following:  &lt;div onclick="alert('you clicked the header')" class="header"&gt;   &lt;span onclick="alert('you clicked inside the header');"&gt;something inside the header&lt;/span&gt; &lt;/div&gt;   How can I make it so that when the user clicks the span, it does not fire the div's click event? dpkg: error: dpkg status database is locked by another process I just wondered if somebody could tell me what this means:  E: dpkg was interrupted, you must manually run 'sudo dpkg --configure -a' to correct the problem. reidsr@ubuntu:~$ sudo dpkg --configure -a dpkg: error: dpkg status database is locked by another process reidsr@ubuntu:~$    How do I run sudo dpkg --configure -a  manually? GUID COMB strategy in EF