How do magnets attract (QED)?
Feynman diagram for attractive forces
Which site is more reasonable for questions about R, SO or stats.stackexchange.com? I've been using SO for a little over a year now, and recently I've noticed a shift of users between SO and stats.stackexchange.com.   I'm wondering if there has been any discussion on where questions about R, a statistical environment, best fits. Would you say it's better to ask R questions on SO or would it be better to ask over at stats.stackexchange.com? What, if any, determinants would you use to decide to post between the two? What about cross posting? Automatic left and right commands The use of imaginary and complex values comes up in many physics and engineering derivations. I have a question about that: Is the use of complex numbers simply to make the process of derivation easier, or is it an essential ingredient, without which it would be impossible to derive some results?   I can identify two different settings where the answer may be different:   It doesn't look like it is mandatory for results in Newtonian mechanics, General Relativity and classical electrodynamics. Can we say the same thing about quantum mechanics either way for sure?   Could this be a difference in quantum mechanics over the classical picture? I run Windows XP and the wallpaper I used to have got lost. I have backups of the system itself, so if I knew where to look, I could find a copy of the old wallpaper.  I have some idea of the original filename (searching now), and I recall it saves it as a bitmap, but an exact location would be helpful.  It was a photo/JPEG that used to be on the filesystem, and I don't have backups, so the Firefox, and Internet Explorer wallpaper locations are unlikely. I checked %SYSTEMROOT%\system32 as well - so I'm assuming it's somewhere else. The answer is, according to the book: $(m-1)!\cdot \binom{m+r-1}{r}$ I get why this is true. You arrange $m$ people in their seats in $(m-1)!$ ways and then you put $r$ empty spots inbetween them.I was thinking - choose $m$ seats out of $m+r$ to put the people on => $\binom{m+r}{m}$ Arrange them once you've chosen the seats $\to (m-1)!$ So in total: $\binom{m+r}{m} \cdot (m-1)!$ This is obviously not true. What's wrong with the way I'm thinking? Log4Net: Programmatically specify multiple loggers (with multiple file appenders) How to (programmatically, without xml config) configure multiple loggers with Log4Net? I need them to write to different files. Resizing Controls in MFC I am writing a program which has two panes (via CSplitter), however I am having problems figuring out out to resize the controls in each frame. For simplicity, can someone tell me how I would do it for a basic frame with a single CEdit control?   I'm fairly sure it is to do with the CEdit::OnSize() function... But I'm not really getting anywhere...  Thanks! :) I am currently studying  answer on the Moore-Penrose pseudoinverse and the Euclidean norm by the user &quot;Etienne dM&quot;. The first point of their answer proceeds as follows:  Let $x$ be $A^+y$.  Let me begin by the second point. For all $z$, we have: \begin{align} \lVert Az-b \rVert_2^2 &amp;= \lVert Ax-b \rVert_2^2 + \lVert A(z-x) \rVert_2^2 + 2 (z-x)^TA^T(Ax-b)\\ &amp; \geq  \lVert Ax-b \rVert_2^2 + 2 (z-x)^TA^T(Ax-b) \end{align} Moreover, because $(AA^+)^T = AA^+$, $$ A^T(Ax-b) = ((AA^+)A)^Tb - A^Tb = 0$$ Thus, we prove that for all $z$, $\rVert Az-b \lVert_2^2 \geq\rVert Ax-b \lVert_2^2$, that is to say $A^+b$ is as close as possible to $y$ in term of the Euclidian norm $\lVert Ax-b\rVert_2$.   I realise that $x = A^+ y$, but I don't understand how any of this implies that &quot;$A^+ b$ is as close as possible to $y$ in terms of the Euclidean norm $\lVert Ax-b\rVert_2$&quot;. I would greatly appreciate it if people would please take the time to explain this to me. I am trying to solve an integral in $k$-space where I want to calculate the ratio of the average energy of an arbitrary electronic band ($E(k)=k^n)$ to the square of average group velocity of electrons($v_g   = k^{n-1})$ approximately in the form of this expression given by $$L=\frac{\frac{\sum_{\mathbf{k}}k^{n}}{\sum_{\mathbf{k}}1}}{\left(\frac{\sum_{\mathbf{k}}k^{n-1}}{\sum_{\mathbf{k}}1}\right)^2}\;.$$  Now, I want to do this for $d$-dimensions, so I convert this sum to integral.  $$L=\frac{\dfrac{\int_{0}^{k_f}k^{n} d^d k}{\int_{0}^{k_f}1d^d k}}{\left(\dfrac{\int_{0}^{k_f}k^{n-1}d^d k}{\int_{0}^{k_f}1d^d k}\right)^2}$$ where $d^dk  = \dfrac{d \pi^{d/2}}{\Gamma(\frac{d}{2}+1)}k^{d-1}dk$ is the volume element in $d$-dimensional space.  I want to get a simple expression for $L$ and how it depends with $k_f,n$ and $d$. any physical insight into this integral could also be helpful. Can Wubi use the ISO I already have? The Windows Installer for Ubuntu desktop instructions say that an ISO image for Ubuntu is downloaded during the installation process.  But what if I already have an ISO image for Ubuntu. Can I make the Windows Installer use that, so it doesn't have to download an ISO image again?  12.04 LTS and 12.10 are the versions of Ubuntu I'm interested in installing, in case that matters. bcmwl-kernel-source broken on kernel: 5.8.0-34-generic The security guard problem I have windows 10 home and from time to time I see a command line window popping up for less than a second. I want to find out which process is causing that. It's not very often that this window pops up so I need some tool that can run for a long time.  Does anyone have a good idea?  Regards Tobias A diffeomorphism whose tangent map preserves dot products is an isometry. I'm having trouble solving the following problem.  If $F:\mathbb{R^3} \to \mathbb{R^3}$ is a diffeomorphism such that $F\ast$(the tangent map of $F$) preserves dot products, show that $F$ is an isometry. (Hint: Show that $F$ preserves lengths of curve segments and deduce that $F^{-1}$ does also.)  I'm trying to proceed as in the hint, but I'm stuck on the first step, showing the $F$ preserves lengths of curve segments. Moreover, how does showing that both $F$ and its inverse preserves length of curve segments show that it is an isometry. And how do I characterize the curve segments. I would appreciate any solutions, hints or suggestions. If I take a sponge and place it in a shallow dish of water (i.e. water level is lower than height of sponge), it absorbs water until the sponge is wet, including a portion of the sponge above the water level. In other words, it seems the sponge pulls some water from the bath up into itself, doing work, and the water gains some gravitational potential energy.  Where does the energy required to do this work come from? My suspicion is that the answer involves physical and/or chemical bonds between the water and the sponge, or possibly the change in the surface area to volume ratio of the water. Given that not all employees have a diamond:     All Stack Overflow employees aren't given a moderator diamond; the Community team has tightened up internal policies regarding who gets one and who doesn't. [...] —      can we have some visual indication that they are an employee?  This was  back in 2011, but declined. However, the situation appears to have changed now with fewer employees being given a diamond.  That way when they post on a meta site we know that it's somewhat "official".  Failing that please ensure that everyone includes the fact they're an employee in their profile given that  to add employee to their profile pages was also declined. If $A$ is an $n \times n$ matrix and $ A^2 = 0$, then $\text{rank}(A)\le n/2$. Was revising for a Linear Algebra when I came across this question.     "Given a Matrix $A$ over $\mathbb{R}^{n \times n}$, and $A^2 = 0$,   show that rank($A$) $\leq n/2$"   My attempt:  I was thinking of the theorem from Cayley-Hamilton but the matrix is a $n \times n$ matrix hence that's probably invalid.  Is there perhaps a significance of $A^2 = 0$? apart from a special case of the nilpotent matrix? In JavaScript can I make a "click" event fire programmatically for a file input element? I'd like to make a click event fire on an &lt;input type="file"&gt; tag programmatically.  Just calling click() doesn't seem to do anything or at least it doesn't pop up a file selection dialog.  I've been experimenting with capturing events using listeners and redirecting the event, but I haven't been able to get that to actually perform the event like someone clicked on it. In my javascript I have this      loopDeLoop:         while (foo !== bar) {             switch (fubar) {                 case reallyFubar:                     if (anotherFoo == anotherBar) {                         break loopDeLoop;                     }                     break;                 default:                     break;             }         }   But JSLint says... lint warning: use of label  Here's the notes from        Labels   JavaScript allows any statement to have a label, and labels have a separate name space. JSLint is more strict.      JSLint expects labels only on statements that interact with break: switch, while, do, and for. JSLint expects that labels will be distinct from vars and parameters.   How do I construct the above to get rid of the warning?  Thanks, Greg The problem is this:      Suppose $I \subseteq R$ is a nilpotent ideal and there is $r \in R$ with $r \equiv r^2 \pmod I$. Show $r \equiv e \pmod I$ for some $e \in R$ idempotent.   I have spent a few hours rolling around in abstracta with no destination. I believe that if I could write down a concrete example of this, the example could guide me through the abstract definitions and show me where to look for an idempotent in $R$. I have looked at 2x2 matrices and could not find any such examples. Might anybody have one?