Prove that $Dom(S \circ R )= Dom (R)$
If $\operatorname{ran} F \subseteq \operatorname{dom} G$, then $\operatorname{dom}(G \circ F) = \operatorname{dom} F$.
Matrices - Conditions for $AB+BA=0$ The Problem Let $A$ be the matrix $\bigl(\begin{smallmatrix}a&amp;b\\c&amp;d\end{smallmatrix} \bigr)$, where no one of $a,b,c,d$ is $0$. Let $B$ be a $2\times 2$ matrix such that $AB+BA=\bigl(\begin{smallmatrix} 0&amp;0\\ 0&amp;0 \end{smallmatrix} \bigr)$. Show that either   $a+d=0$, in which case the general solution for $B$ depends on 2 parameters, or $ad-bc=0$, in which case the general solution for $B$ depends on one parameter.   (this is question 22 of the last matrix exercise of Further Pure Mathematics by Bostock et al.)  Comments Writing $B=\bigl(\begin{smallmatrix} e&amp;f\\ g&amp;h \end{smallmatrix} \bigr)$ and multiplying out I get that    $(a+d)(f+g)+(b+c)(e+h)=0$ $ae+bg+cf+dh=0$   but I am unable to get the required restrictions on $a,b,c,d$. Is there a quick way of doing the problem that doesn't require manual computation? I thought of considering invertible and non-invertible cases but couldn't get anywhere. Help would be much appreciated. If $x_1,x_2,\ldots,x_n$ are real numbers larger than $1$, prove that $$\dfrac{1}{1+x_1}+\dfrac{1}{1+x_2}+\cdots+\dfrac{1}{1+x_n} \geq \dfrac{n}{\sqrt[n]{x_1x_2\cdots x_n}+1}$$   Attempt  AM-GM doesn't work here since we will get an upper bound. I don't see Cauchy-Schwarz working either. Thus, I think a substitution might work, but I am unsure of which one to use. Zero to the zero power – is $0^0=1$? Could someone provide me with a good explanation of why $0^0=1$?  My train of thought:  $x&gt;0$  $0^x=0^{x-0}=0^x/0^0$, so  $0^0=0^x/0^x=\,?$  Possible answers:   $0^0\cdot0^x=1\cdot0^0$, so $0^0=1$ $0^0=0^x/0^x=0/0$, which is undefined   PS. I've read the  on mathforum.org, but it isn't clear to me. Installing Cygwin packages from the command-line Why are my render times 30+ minutes, but other people's are ~3 minutes? I'm giving Deus Ex a quick playthrough as a refresher before jumping into Human Revolution. In the interest of expediency, I've decided to cheat. ^.^  Normally, cheats must be re-enabled for each level when playing the original DX. I've encountered instructions on several sites (word-for-word identical instructions, so clearly copy/pasted) for permanently enabling cheats, but they aren't working for me:     Because you will need to redo this for every level, you can open up 'user.ini' and add 'bCheatsEnabled True' in the [DeusEx.DeusExGameInfo] section of the file to allow cheats to be activated forever.   Not only is that not working for me, it doesn't even look right compared to the rest of the file (space instead of =). With increasing desperation, I've tried adding that line (both verbatim and switching to an equal sign) to every section of both User.ini and DeusEx.ini which looks even remotely plausible — even creating a [DeusEx.JCDentonMale] section, just in case.  Does anyone know how to get this working? I'm running DX:GOTY version 1.112fm from Steam on Windows 7. $f$ strictly increasing does not imply $f'>0$ We know that a function $f: [a,b] \to \mathbb{R}$ continuous on $[a,b]$ and differentiable on $(a,b)$, and if $f'&gt;0 \mbox{ on} (a,b)$ , f is strictly increasing on $[a,b]$. Is there any counterexample that shows the converse fails?   I have been trying to come up with simple examples but they all involve functions that are discontinuous or has derivative $f'=0$ which does not agree with the hypothesis hmmm Not to Mention ≈ [Let Alone ≈ Much Less ≈ Still Less] For brevity, I symbolise synonymity with ≈.  So  X ≈ Y  means  X and Y are synonyms.  From : let alone ≈ not to mention  From :  let alone ≈ much less ≈ still less.  Are these all synonyms? What are the similarities and differences between each? Selection of logical connectives {¬,∨,∧,⇒,⇔} in set theory? If $f,g:X \to Y$ are continuous and $Y$ is $T_2$, then $\{x \in X\,|\,f(x)=g(x)\}$ is closed I'd like to know if the following proof is valid.  The only thing I'm not sure about (though I can't see why it's invalid if it is) is if we can always use the Hausdorfness of $Y$ to separate an open set from $f(C)=g(C)$.  "Let $X$ be a space, $Y$ a $T_2$-space, and $f,g:X \to Y$ continuous functions.  Prove that $C:=\{x \in \,|\,f(x)=g(x)\}$ is a closed subset of $X$."  Let $x \in X$ be such that $f(x) \ne g(x)$.  Since $Y$ is $T_2$, there are open sets $U_\alpha \ni f(x)$ and $V_\alpha \ni g(x)$ such that $U_\alpha \cap O_\alpha=V_\alpha \cap O_\alpha=\varnothing$ for all $\alpha \in A$ where $A$ is the set indexing the points of $f(C)=g(C)$, and $O_\alpha$ is an open set containing $x_\alpha \in f(C)=g(C)$.  Let $U:=\displaystyle\bigcap_{\alpha \in A} U_\alpha$ and $V:=\displaystyle\bigcap_{\alpha \in A} V_\alpha$.  Then $U \cap f(C)=V \cap f(C)=\varnothing$.  Therefore, since $f,g$ are continuous, $f^{-1}(U)$ and $g^{-1}(V)$ are open sets in $X$ disjoint from $C$.  Since $X\setminus A$ is the union of such open sets, it is open itself and therefore $C$ is closed.  Thanks. What are the assumptions for applying a quantile regression model? The question has been asked (one time) on CV before, but the answer is really imprecise and does not really answer the question in my opinion.  So: What are the assumptions for estimating a linear regression model via quantile regression?  To my understanding (and as several CV users have mentioned), quantile regression does not assume any specific distribution of the error terms - does that mean that, in a time series model, autocorrelation and heteroscedasticity do not have to be accounted for?  What about the other Gauss-Markov assumptions? I would assume that the assumption of no perfect multicollinearity has to be met when applying quantile regression, but do the parameters have to be linear? The linearity assumption only has to hold for the specific quantile I would assume.  Anyways - I do not find any backup for any of my thoughts in the scientific literature and I would appreciate a comprehensive answer. Thank you! How do I find the package that provides a file? Windows of an application have minimum size and I can't find the window anymore I had an application (in case it matters: LibreOffice Calc] in full-size mode. I then clicked the middle of the three buttons of the top right to adjust the size. When I did that, the window became minimum size, literally a vertical line. I panicked and alt-tabbed, and now I can't even find the vertical line anymore. Opening new windows in that application opens all of them hidden to me, and closing/reopening or even restarting the OS doesn't help. I don't know how to resize my invisible windows anymore. Here's the preview of the windows in the application launcher: a vertical line (that I don't find on my screen anywhere if I click this). I'm trying out  with gnuplot to generate the data.  This works by having pgfplots write some stuff to an external file, run gnuplot on the file, and then use the generated data to plot the graph.  The problem I'm having is that I get different behaviour under xelatex as pdflatex or lualatex.  In xelatex, the data file is littered with ^^I whereas in the other two formats these are tabs.  It appears that the code ^^I is getting stored as a macro which then gets written out to the file, so possibly there's something funny going on with the ^^I syntax: xelatex is reading it literally.  Indeed, when I do \show\pgfplots@TAB (the macro in question) then for pdflatex and lualatex then I get a literal tab whereas for xelatex I get ^^I.  So how do I get a tab in xelatex?  And why isn't \gdef\pgfplots@TAB{^^I} working?  (The previous line in the code, by the way, is \catcode`\^^I=12 - I don't know if that has anything to do with it.) Differences between hosting options for SharePoint apps Please could someone tell me where can I download Ubuntu 32bit without using a torrent? Thanks Bjarne Stroustrup once mentioned in one of his bigthink interviews that one of the greatest problems most programmers have is not finding or using the right libraries.  On stackoverflow and programmers, I've observed a trend towards questions like "I want to do this, is there a [library | programming language] for that", or "I have these {n} [library | language] candidates for my project; which one would yield me the least amount of headache given these requirements".   Since questions like these aren't exactly programming questions in the sense of "this code should do this, but it does this, why?", they are often (though not always) being closed as "not constructive" because they do not have a definite answer (which is true, and according to the FAQ they would be off-topic on both stackoverflow and programmers).  Nonetheless, I think library questions are valid, and that it is helpful for any programmer to receive independent input from experienced programmers (at least it helps me a lot every time I browse through one of those closed questions, which I found through google). Does anyone agree that posting library choice questions should be explicitly allowed on either stackoverflow or programmers? Is it normal for BibTeX to replace similar author names with "------"? In my .bib file, I have two entries with exactly the same author names:  @article{Seshadrinathan2010A-Subjective-St,     Author = {K. Seshadrinathan and R. Soundararajan and A. C. Bovik and L. K. Cormack}, ....  @article{Seshadrinathan2009Study-of-Subjec,     Author = {K. Seshadrinathan and R. Soundararajan and A. C. Bovik and L. K. Cormack}, ....   When cited after each other, and typeset with the IEEEtran bibliography style, the .bbl file reads:  \bibitem{Seshadrinathan2010A-Subjective-St} K.~Seshadrinathan, R.~Soundararajan, A.~C. Bovik, and L.~K. Cormack, ``A   subjective study to evaluate video quality assessment algorithms,''   \emph{SPIE Proceedings Human Vision and Electronic Imaging}, 2010.  \bibitem{Seshadrinathan2009Study-of-Subjec} ------, ``Study of subjective and objective quality assessment of video,''   \emph{IEEE Transactions on Image Processing}, 2009.   As you can see, the author names have been replaced with ------. It looks like this in the final PDF:    Is this normal behavior? I want to create a scratch-card-like effect in new Unity UI's new Image tag.  I know it's possible with setPixels, but that's prohibitively expensive on mobile devices.  Something like this:    I'd like the user to be able to drag on any part of the image to "scratch off" the top layer and reveal the content underneath it.  How can I do this? Reading Excel files from C# Is there a free or open source library to read Excel files (.xls) directly from a C# program?   It does not need to be too fancy, just to select a worksheet and read the data as strings. So far, I've been using Export to Unicode text function of Excel, and parsing the resulting (tab-delimited) file, but I'd like to eliminate the manual step.