Is $\text{rank} (AA^*)=\text{rank}(A)$ for all nonsquare matrices?
Prove $\operatorname{rank}A^TA=\operatorname{rank}A$ for any $A\in M_{m \times n}$
Is there a command that outputs ONLY the packages explicitly installed by the user? (ubuntu/debian) I have tried searching for this but there seems to be no command that can output a list of packages (ideally in Ubuntu) that I have installed, not including any dependencies. Is there a simple way to fill in a grungy font to make it solid? I'm looking to make the entire letter solid. I want to complete the grungy white space and fill it out with solid black. Is there a simple way to do this? Why do electrons around nucleus radiate light according to classical physics As I navigate through physics stackexchange, I noticed .  Electrons radiate light when revolving around nucleus? Why is it so obvious?  Note that I do not know anything about quantum mechanics - and I just want classical stories. The paradox:    We have a circle and there is another circle with smaller radius. They are co-centeric.  If circle make full turn without sliding, both smaller and bigger circle make full turn too. If we assume that the passed road is equal to the circumference of circles. We have got smaller circle's radius is equal to bigger one's.  Unsatisfying Solutions I found:    "Do not assume that smaller circle's circumference is equal to passed road since the surface that contacts to the ground is bigger one. " // Okey but it does not explain the paradox, it explains just what is the wrong assumption (even does not explain why it is a wrong assumption.) It is undeniable that every point on both smaller and bigger circle will contact exactly one and only one point on their path. Therefore we can think that this is a bijective maps and smaller circle is isomorphic to bigger one. (Okey but ....)   Question: What is the true answer? What is wrong with the definition of circumference of a circle and relationship with its taken path. Connecting subplots Is there a way to draw a sort of connection between subplots in TiKz/pgfplot, like in this ? I'm looking to achieve something like this:    But I would need to draw the "x 2000" line between the plots.  EDIT: So right now I have something like this:    Coming from my main.tex:  \input{aipcheck}  \documentclass[final]{aipproc}  \layoutstyle{8x11single} \usepackage[utf8]{inputenc} \usepackage{algorithm2e} \usepackage{color} \usepackage[usenames,dvipsnames]{xcolor} \usepackage{cleveref} \usepackage{pgfplots}  \pgfplotsset{ tick label style={font=\tiny}, %axis numbers label style={font=\small}, %axis labels legend style={font=\tiny} }  \begin{document}  (...)  \begin{figure}     \centering     \newlength\figureheight      \newlength\figurewidth      \setlength\figureheight{5cm}      \setlength\figurewidth{5cm}     \input{figs/_ddmp2d.tikz} \label{fig2D} \end{figure}   with _ddmp2d.tikz, originally created from matlab2tikz, but already modified, containing:  % This file was created by matlab2tikz v0.3.3. % Copyright (c) 2008--2013, Nico Schlömer &lt;nico.schloemer@gmail.com&gt; % All rights reserved. %  % The latest updates can be retrieved from %   http://www.mathworks.com/matlabcentral/fileexchange/22022-matlab2tikz % where you can also make suggestions and rate matlab2tikz. %  %  %   \begin{tikzpicture}  \begin{axis}[% width=\figurewidth, height=\figureheight, scale only axis, xmin=0, xmax=15, xlabel={Number of iterations}, ymin=0, ymax=16000000000000, name=plot1, after end axis/.append code={\coordinate (a) at (axis description cs:1,0);}]  \addplot [ thick, color=red, dash pattern=on 1pt off 3pt on 3pt off 3pt, forget plot ] table[row sep=crcr]{ 1 15841000000000\\ (...) 15 25160000000\\ }; \addplot [ thick, color=black, dashed, forget plot ] table[row sep=crcr]{ 1 15841000000000\\ (...) 15 52501000000\\ }; \addplot [ thick, color=blue, dotted, forget plot ] table[row sep=crcr]{ 1 15841000000000\\ (...) 15 5324200000\\ }; \addplot [ thick, color=green, solid, forget plot ] table[row sep=crcr]{ 1 15841000000000\\ (...) 15 551920000\\ }; \addplot [ thick, color=green, solid, forget plot ] table[row sep=crcr]{ 3 0\\ 3 16000000000000\\ };  \addplot [ thick, color=blue, dotted, forget plot ] table[row sep=crcr]{ 4 0\\ 4 16000000000000\\ }; \addplot [ thick, color=black, dashed, forget plot ] table[row sep=crcr]{ 5 0\\ 5 16000000000000\\ }; \addplot [ thick, color=red, dash pattern=on 1pt off 3pt on 3pt off 3pt, forget plot ] table[row sep=crcr]{ 8 0\\ 8 16000000000000\\ }; \end{axis}  \begin{scope}[xshift=5cm] \begin{axis}[% width=0.35\figurewidth, height=\figureheight, scale only axis, xmin=995, xmax=1000, xtick={997,998,999,1000}, xticklabels={$997$,$998$,$999$,$1000$}, ymin=0, ymax=8000000000, at=(plot1.right of south east), anchor=left of south west, yticklabel pos=right, axis x discontinuity=crunch, after end axis/.append code={\coordinate (b) at (axis description cs:0,1);} ] \addplot [ thick, color=red, dash pattern=on 1pt off 3pt on 3pt off 3pt ] table[row sep=crcr]{ 995 6093100000\\ 996 6093100000\\ 997 6093100000\\ 998 6093100000\\ 999 6093100000\\ 1000 6093100000\\ };  \addplot [ thick, color=black, dashed ] table[row sep=crcr]{ 995 1577100000\\ 996 1598000000\\ 997 1598500000\\ 998 1628600000\\ 999 1631000000\\ 1000 1595900000\\ };  \addplot [ thick, color=blue, dotted ] table[row sep=crcr]{ 995 2592000000\\ 996 2591900000\\ 997 2591700000\\ 998 2591600000\\ 999 2591500000\\ 1000 2591400000\\ };  \addplot [ thick, color=green, solid ] table[row sep=crcr]{ 995 391810000\\ 996 391810000\\ 997 391810000\\ 998 391810000\\ 999 391810000\\ 1000 391810000\\ };  \end{axis} \draw[gray] (a) -- (b) node[pos=0.5,sloped,fill=white] {$\times$2000}; \end{scope}  \end{tikzpicture}% How are the file metadata stored in Windows? (I'm using Windows XP but I guess it's similar in all the recent versions of Windows.)  When you create for example a new empty text document, you'll find in its properties that it has size of 0 bytes. Zero bytes means no information. No data. But still, the file has some name, it can still carry the dates of last access, modification and creation. It carries the information whether it's hidden file or not, whether it's read-only or not...  So where are all the metadata stored? Keyword for the outer class from an anonymous inner class In the following snippet:  public class a {     public void otherMethod(){}     public void doStuff(String str, InnerClass b){}     public void method(a){         doStuff("asd",             new InnerClass(){                 public void innerMethod(){                     otherMethod();                 }             }         );     } }   Is there a keyword to refer to the outer class from the inner class?  Basically what I want to do is outer.otherMethod(), or something of the like, but can't seem to find anything. How to clone/copy all file/directory attributes onto different file/directory? I want to copy the attributes (ownership, group, ACL, extended attributes, etc.) of one directory to another but not the directory contents itself.  This does not work:  cp -v --attributes-only A B cp: omitting directory `A'              Note: It does not have to be cp. I know how to monitor the network interfaces on my local computer with Wireshark. But, how do I monitor the traffic flowing through my wireless access point from connected computers?  It seems like this might not be possible without some kind of special software on the wireless access point itself. Shorten title on each beamer slide In a beamer presentation the title is long but ok for the first slide. The problem is that when the title appears below on each slide it is very long and I would like to use here some sort of running or shorter title, how can this be done? How to make a drive NTFS from EXT4? I want to know how to make a drive ( non-system ) from EXT4 to NTFS. Can anyone help me out? I have an iPhone 6. How can I use two Bluetooth audio devices and the 3.5mm audio jack together? Me and my friends want to listen to music together. Two of us have Bluetooth Headphones (Beats Solo2 Wireless, Beats Studio Wireless) and one has a wired headset. Is there any way to use all three of these together? print python stack trace without exception being raised Is there a way to disable a specific JS script from a specific website? I want to disable a specific JS script from a specific website without disabling the javascript completely. Is there a way to do that ? How do I remotely control another Ubuntu desktop from Ubuntu? How do I remotely control another desktop (both are Ubuntu 12.04) from my client machine?  Is it possible to use Remmina Remote Desktop Client? Problem using atan in pgfplots I have a long-running Python server and would like to be able to upgrade a service without restarting the server. What's the best way do do this?  if foo.py has changed:     unimport foo  &lt;-- How do I do this?     import foo     myfoo = foo.Foo() Should I vote to close a duplicate question, even though it's much newer, and has more up to date answers? I have seen this many times, a question is being asked because a previous, duplicate question was asked and answered a few years ago and has no longer relevant answers.  The  is to edit the old question, or add new answers to it, but it is not always happening.  I was wondering, should I still mark it as duplicate and vote to close it? Or just leave it as is? Pass An Instantiated System.Type as a Type Parameter for a Generic Class The title is kind of obscure. What I want to know is if this is possible:  string typeName = &lt;read type name from somwhere&gt;; Type myType = Type.GetType(typeName);  MyGenericClass&lt;myType&gt; myGenericClass = new MyGenericClass&lt;myType&gt;();   Obviously, MyGenericClass is described as:  public class MyGenericClass&lt;T&gt;   Right now, the compiler complains that 'The type or namespace 'myType' could not be found." There has got to be a way to do this. What is the origin of "a cut above/below the rest"?