I know rank AB is not same as rank BA.    I take here particular case.  Is this particular case true?
How do we prove that  $\operatorname{rank}(A) = \operatorname{rank}(AA^T) = \operatorname{rank}(A^TA)$ ?  Is it always true?
I'm looking to cite a number of web pages using bibtex and I was wondering if there was a specific template of the form @&lt;template name here&gt; for doing that. If you could use the following website as an example that would be great Here is what I tried doing:  #!/bin/sh res=`df | awk 'FNR == 2 { print $5 }'` res2=$((res+0))   Here is my error:     -bash: 53%: syntax error: operand expected (error token is "%")   I Googled 'string to int bash' which showed me $((res+0)) but it doesn't work due to the % in the string (53%). How do I convert it in such a way I can do an if statement such as the one below (but that works)?  if [ "$res2" -ge 50 ] then     echo "Your disk &gt;=50% full!" fi For a given class I would like to have tracing functionality i.e. I would like to log every method call (method signature and actual parameter values) and every method exit (just the method signature).      How do I accomplish this assuming that:     I don't want to use any 3rd party AOP libraries for C#, I don't want to add duplicate code to all the methods that I want to trace,   I don't want to change the public API of the class - users of the class should be able to call all the methods in exactly the same way.     To make the question more concrete let's assume there are 3 classes:     public class Caller   {      public static void Call()       {          Traced traced = new Traced();          traced.Method1();          traced.Method2();       }  }   public class Traced   {      public void Method1(String name, Int32 value) { }       public void Method2(Object object) { }  }   public class Logger  {      public static void LogStart(MethodInfo method, Object[] parameterValues);       public static void LogEnd(MethodInfo method);  }   How do I invoke Logger.LogStart and Logger.LogEnd for every call to Method1 and Method2 without modifying the Caller.Call method and without adding the calls explicitly to Traced.Method1 and Traced.Method2?  Edit:  What would be the solution if I'm allowed to slightly change the Call method? Is there any intuition or meaning regarding Cauchy-Riemann equations? Holomorphic functions are one of the most beautiful objects in mathematics. However, Cauchy-Riemann equations are a little bit of a mystery for me. What does that mean intuitively when a function satisfyes those equations? How to remove enamel from wire? I want to repair my computer mouse cable which has become disconnected from the PCB. The wire is insulated with enamel and I have to remove it before soldering it back onto the board.  How should I remove the enamel? I need a regular expression to select all the text between two outer brackets.  Example: some text(text here(possible text)text(possible text(more text)))end text  Result: (text here(possible text)text(possible text(more text))) I have two different list. In the first list are some people picker columns.  Is it possible to create a lookup column for list 2 to add the user from the first list? I know it isn't supported in Sharepoint, but i hope someone have an idea to solve my problem. Does a non-abelian semigroup without identity exist? Why can't I sort questions by "votes" or "unanswered" without logging in? Can the manual and tome magic items that increase stats be used multiple times by bypassing the century wait time via the spell Sequester? Assume we have a Wizard who is at least level 13 and can reliably cast  many times (either through acquired spell scrolls, or has acquired enough material components for it to be a non-issue), has a safe place (through the spell ) to be under the influence of Sequester for many centuries, and has access to the Manual of Bodily Health, Manual of Gainful Exercise, Manual of Quickness of Action, Tome of Clear Thought, Tome of Leadership and Influence, and Tome of Understanding.  Per the item descriptions in the DMG:     ...your [STAT] score increases by 2, as does your maximum for that score. The manual then loses its magic, but regains it in a century.   Now, normally this would be a once-in-a-lifetime use, or maybe twice (due to the lifespan of the races), but if a wizard used Sequester on themself, many years can pass without growing older:     ...[willing target creature] falls into a state of suspended animation. Time ceases to flow for it, and it doesn't grow older. You can set a condition for the spell to end [before the spell is dispelled by the caster].   So, assuming that time continues to flow for the magic items, but not for the sequestered wizard, can they study these books and take the necessary long rests, Sequester for a hundred years,  “wake” from the Sequester, and then repeat an arbitrary-but-finite number of times for an arbitrarily high (but finite) improvement to their stats?  To end the cycle and escape the demiplane, the wizard would simply Plane Shift out:     ...You can specify a target destination in general terms ... and you appear in or near that destination. Beamer -- not indenting itemize in a single environment The output in the MWE posted at the end of this post is shown below. Notice how the default indentation from itemize gives an asymmetric slide w.r.t the spacing to the left and right of the itemize environment. How can I push the bullet points to the left so that the spaces to the left and right are the same? Such a solution would ideally be:   implementable on a one-off basis, so that it is applied when necessary (since some itemize environments should be given the default indentation) independent of the margin size, so that the solution doesn't need to be changed when the margin size is changed     \documentclass{beamer}   \begin{document}  \begin{frame}{A title} \begin{itemize} \item Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod \item tempor incididunt ut labore et dolore magna aliqua. \item Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi \item ut aliquip ex ea commodo consequat. \begin{itemize} \item Duis aute irure dolor in reprehenderit in voluptate velit esse \item cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat \end{itemize} \end{itemize}    \end{frame} \end{document} How to add thickness to a curve? How can I add thickness to a curve? I already tried with the extrude and depth options in the curve settings, but it doesn't work properly. The Chinese government has a really specific list of requirements for visa photos:    I had an old unused photo from last time I applied for a visa or passport, and compared it to the requirements (decided not to use it because it was damaged). It seemed to match most of the requirements, except that it seems that they want a photo which isn't square: The required width is less than the required height.  Assuming most drugstores are still providing square passport photos, the question becomes: Should I cut the photo to match the specified width?   What adds to the confusion is if you look at the  the box for the photo is nearly square, not the oblong shape in the requirements. And I have seen other advice that you need to make sure it fully fits the box.  Any advice about what to do here? $ d_1,d_2\mid n\iff {\rm lcm}(d_1,d_2)\mid n\ $ [LCM Universal Property] Selecting ARIMA orders based on ACF-PACF vs. auto.arima I've allocated an "array" of mystruct of size n like this:  if (NULL == (p = calloc(sizeof(struct mystruct) * n,1))) {  /* handle error */ }   Later on, I only have access to p, and no longer have n. Is there a way to determine the length of the array given just the pointer p?  I figure it must be possible, since free(p) does just that. I know malloc() keeps track of how much memory it has allocated, and that's why it knows the length; perhaps there is a way to query for this information? Something like...  int length = askMallocLibraryHowMuchMemoryWasAlloced(p) / sizeof(mystruct)   I know I should just rework the code so that I know n, but I'd rather not if possible. Any ideas? How to upgrade Git on Ubuntu Hardy? HP PSC-2100 can't seem to get xsane and HPLIP (printing working) How to check which package created a user? It is suspected that some packages have scripts run at install time which create user accounts in /etc/passwd. How can I check which package created a specific user? Insert eps graphics with graphicx (file not found)