Given $n \times n$ matrices, $A$ and $B$, show that ${\rm rank}(AB) \leq {\rm rank}(A)$
How to prove and interpret $\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B))$?
How can particles be emitted from particles? In a scene with two particle system configured as boids and boids brain set to fight, one swarm attacks the other. I would like to emit child particles as lines (laser or something) to visualize some fighting action. I couldn't find a way to setup the line emitting particle system, either they are created from the plane or further cones will be added. Is there a way to do this as I saw on a  but I didn't get the trick? Blend file   At blendersushi  the author says:  PARTICLE INSTANCE MODIFIER &gt; PARTICLES With this stacking thing, we can now pretty much emit Particles from emitted Particles, &gt; if you know how to set it up. Not very obvious, but basically you Instance some meshes &gt; (using Instance Modifier) and then add Particle System to the object with Instance &gt; Modifier. Make sure you have &quot;Use Modifier Stack&quot; turned on.  But I can't get this to work. In France the higher education is like this:   3 years of "Licence" 2 years of "Master" 3 years of "Doctorat"   But what about in the United States?  I heard that they have things called "undergraduate", "graduate" and "postgraduate". How many years do each of those take? And what are the equivalencies between these and the "Licence - Master - Doctorat"?  Note: If the answer to this question depends on the field of studies, then the field is physics. SSH login without password with kerberos De Rham Cohomology of the tangent bundle of a manifold I would like to compute the de Rham cohomology of the tangent bundle $TM$ of a manifold $M$. It seems to me that we can just homotopy each fibre to a point, and that this would give a homotopy equivalence between $TM$ and $M$. Since de Rham cohomology is a homotopy invariant, this would imply that $H^*_{\text{dR}}(TM) \cong H^*_{\text{dR}}(M)$.     Is it always true that the tangent bundle of a manifold is homotopic to the manifold itself by the homotopy of each fibre with a point? I cant use lhopital's rule here is my attempt:  Limit of $(\frac{n^{2}-2n+1}{n^{2}-4n+2})^{n}$ =$e^{\lim_{n\to \infty} n\times \ln(\frac{1 -2-n +\frac{1}{n^{2}}}{1 -\frac{4}{n} +\frac{2}{n^{2}}})}$ I got stuck here Easy way to use variables of enum types as string in C? Here's what I am trying to do:    typedef enum { ONE, TWO, THREE } Numbers;   I am trying to write a function that would do a switch case similar to the following:   char num_str[10]; int process_numbers_str(Numbers num) {   switch(num) {     case ONE:     case TWO:     case THREE:     {       strcpy(num_str, num); //some way to get the symbolic constant name in here?     } break;     default:       return 0; //no match   return 1; }   Instead of defining at every case, is there a way to set it using the enum variable like I am trying to do above? QM and Renormalization (layman) I was reading Michio Kaku's Beyond Einstein. In it, I think, he explains that when physicsts treat a particle as a geometric point they end up with infinity when calculating the strength of the particle's field as you approach the particle.  First, did I get that part right?  Second, how does (in very simple terms) renormalization try to fix this in Quantum Mechanics? If you can include some type of word picture that would be great. Accepted answer will go to the clearest explanation.  Update  I got a little further in the book (lol) and Kaku talks about using symmetry to remove the divergences in the math. I'd appreciate an answer that incorporates this also. Thanks guys! At least two questions (, ) on this site indicate that in more than a few isolated cases at least, letters of recommendation are in fact written by the applicant and only signed off (or perhaps rephrased somewhat) by the recommender, who would actually be responsible for writing the letter.  Is there any evidence about the prevalence of this practice that goes beyond anecdotes? I realize that cross-country (cultural) difference may play a role. Mathematics Ph.D program foreign language requirement I will soon be attending a Ph.D program in mathematics. I do not know how common this practice is, but the department has a requirement that graduate students must learn a foreign language and be able to read and write mathematics in that language. The three choices are French, German, or Russian. I personally have no preference on which to learn, but I was wondering if there were other reasons that would make one language more advantageous over the others in terms of a general mathematical career. Thanks in advance in any response. Do ISTQB/ISEB Testing Certificates prove that someone can test? I have passed the , and to be honest, I'm pretty sure that anyone could pass that exam with a couple of days of studying. My question is are these certificates really of value to my career and should I put any effort into acquiring the ? What is different about the Mac ISO image? At that location there is an ISO image which is described as "This image is adjusted to work properly on Mac systems."  What specific problems on the Mac does this image address that the standard images don't? I ask because I need to know which ISO to recommend to Apple Mac users. I know  that uses $x={\sqrt{2}}^{\sqrt{2}}$ to prove that there must exist an irrational to an irrational power that evaluates to a rational. But I don't know if $x$ itself is known to be irrational or rational.  Could someone enlighten me? Thanks! Ruby block and unparenthesized arguments I extracted simple example:  require 'pp' x = 1..3 pp x.map do |i| {:value =&gt; i, :double =&gt; (i*2)} end pp x.map { |i| {:value =&gt; i, :double =&gt; (i*2)} }  pp(x.map do |i| {:value =&gt; i, :double =&gt; (i*2)} end) pp(x.map { |i| {:value =&gt; i, :double =&gt; (i*2)} })   I am wondering why first pp produces:  [1, 2, 3]   While all the oders are giving:  [{:value=&gt;1, :double=&gt;2}, {:value=&gt;2, :double=&gt;4}, {:value=&gt;3, :double=&gt;6}]   I assume it has something to do with operator precedence. Where can I find good explanation? Word for someone who pays attention to details Implementation of "show" for function How can I tell whether a circle and a rectangle intersect in 2D Euclidean space? (i.e. classic 2D geometry) Creating polygon from hole in QGIS? How to cleanly launch a GUI app via the Terminal? Some GUI apps launch cleanly via the Terminal command line, but some don't, and they cause the Terminal to wait for the app to terminate. Even then, some don't "release" the command line.  The mysterious ampersand &amp; suffix seems to cause the terminal to put the process into the background (but I'm not sure what happens there).  Is there a way to launch an app via the Terminal so that there is no "hang on" effect, just like launching something via Alt+F2?  I'd like to have the command line available again immediately, without something still in the background and printing in the terminal. Which SUSY models are affected by the recent LHCb result? The LHCb has recently  the observation of $B_s \rightarrow \mu^+ \mu^-$ with a branching ratio that agrees with the Standard Model (SM). There are many blog posts about it (See: ).  The result initiated a debate over its implication on Supersymmetry in general, which I don't want to discuss here. But it certainly affects some Supersymmetric models that may have a different prediction than that of the SM.   My questions are,   Which models in particular are affected (or variants of models)? What is the MSSM status? The NMSSM? Does the result affect some of the parameter space of a particular model, or does it rule out an entire model/or variant of a model? Save entire process for continuation after reboot I developed an algorithm for a fairly hard problem in mathematics which is likely to need several months to finish. As I have limited resources only, I started this on my Ubuntu 12.04 (x86) laptop. Now I want to install some updates and actually restart the laptop (the "please reboot" message is just annoying).  Is there a way to save an entire process including its allocated memory for continuation beyond a reboot?  Here is some information about the process you might need. Please feel free to ask for further information if needed.   I called the process in a terminal with the command "./binary &gt; ./somefile &amp;" or "time ./binary > ./somefile &amp;", I cannot really remember. It's printing some debug information to std::cerr (not very often). It's currently using roughly 600.0 kiB and even though this will increase, it's unlikely to increase rapidly. the process runs with normal priority the kernel is 3.2.0-26-generic-pae, the cpu is an AMD, the operating system is Ubuntu 12.04 x86. it runs since 9 days and 14 hours (so too long to cancel it ;-) )