What is the difference between :after and ::after?
Should I use single or double colon notation for pseudo-elements?
Given a sample of n units out of a population of N, population median can be estimated by the sample median.  How can we get the variance of this estimator? Given two sets$$A=\{1,3,5\}\quad,\quad B=\{1,3,8\}.$$ Then I compute the $A\setminus B=\{5\}.$   But my book$^\dagger$said: ... . Also, observe that $x\notin A\setminus B$ does not mean that $x\notin A\lor x\in B$. Why?  I don't know how to explain the Why, please give me some example.    $^\dagger$, second edition, page $4$.  EDIT: My intuition tells me that it does mean that. I guess there is some special case about null-set or about the restriction of the universal set. If $\lim_{n\rightarrow\infty} f(n+1) - f(n) = L$, prove that $\lim_{n\rightarrow\infty} f(n)/n = L$ Here's the problem in full. I've stared at it for hours and can't get anywhere, so a hint would be nice.  Suppose that $f:\mathbb N\rightarrow\mathbb R$. If $$\lim_{n\rightarrow\infty}f(n+1)-f(n) = L$$ prove that $\lim_{n\rightarrow\infty}f(n)/n$ exists and equals $L$.  This problem is marked with the word "Cauchy", so I'm guessing Cauchy sequences or something related to them will be necessary, but I can't figure out what.  Edit: It appears as if the Stolz-Cesaro Theorem makes this quite easy. Unfortunately though, we have not discussed this, so I won't be able to use this. What are Thumbs.db files for? How can I prevent them from being created? If a charge travels in a circle it must accelerate, thereby producing EM.  However, a wire in a circular loop is analogous to many charges moving in a circle. So, why don't circular currents produce EM? (I have not found any evidence that circular currents can produce EM).  A similar question is: . However, my question asks about circular wires and direct current. Thanks. I have two Xbox 360's, one WiFi and the other wired through the same Linksys E3000 router with DD-WRT (v24-sp2 (04/13/11) std-usb-nas firmware).   Mine (the wired one, though hers has been wired in the past and it didn't appear to help) always seems to be open (green), and hers is either moderate or closed (oj/red)  Even with the original router firmware I've never been able to get both consoles into "open nat" mode, and it makes playing certain games together almost impossible for me and my wife  because we can't seem to get into the same game.   I have uPnP enabled and the DMZ disabled, and some minor port forwarding to allow for a Minecraft server (Ports 25565 and 25566 for a testing copy of the server), but other than that the settings are mainly at their default, on the xbox's and the router.   Are there any settings I can configure to allow this to work properly? Show that if $u$ is harmonic in $\mathbb R^n$ and $u=o(|x|)$, then $u$ is a constant.(Hint: use the solid version of the mean value property $u(x)=\frac{1}{\omega R^n} \int _{B_{R(x)}} u(y)dy$, and estimate $|u(x)-u(x')|$.)  Can anyone tell me how to use this hint? When two accounts are merged, all the posts/votes are moved over to the "master" account, and the other account ("slave"?) is deleted. The merge is propagated network-wide, so a similar thing happens on each site.  But, this does not happen to the chat user. The "slave" chat user is kept, with a 404 parent profile link. I feel that the chat user should be merged as well, re attributing all the chat messages.  Thoughts? Installing ChatJax bookmark on the iPad and iPhone Compute volume of tetrahedron using a triple integral I'm trying to compute the volume of a tetrahedron with the vertices (0, 0, 0), (0, 0, 1), (2, 0, 0), (0, 2, 0). It needs to be done using a triple integral. Not allowed to use "det" or other functions.  I think first I need to find a plane equation using those 4 points right? But how do I do that? Eigenvalues of Householder matrix What would be the eigenvalues for a Householder matrix defined as: $H = I - 2 u u^T$? Can someone explain it  to me intuitively or with a simple proof? Let $R$ be a sub-ring of a ring $S$. Let $f,g$ be non-zero polynomials in $R[x]$ and assume that the leading coefficient of $f$ is a unit in $R$. If $f$ divides $g$ in $S[x]$, show that $f$ divides $g$ in $R[x]$.   Thoughts: If $f$ divides $g$ in $S[x]$, then  $g(x) = f(x)q(x)$ for some $q(x)$ in $S[x]$. From here I'm not sure what the question is asking. Do I have to show that $q(x)$ is an element of $R[x]$ ? Insights appreciated. Do strong and weak interactions have classical force fields as their limits? Electromagnetic interaction has classical electromagnetism as its classical limit. Is it possible to similarly describe strong and weak interactions classically? Is a finite group uniquely determined by the orders of its elements? I have my own AppleID ("Bob") but I don't use this email addy "Bob" anymore. I want to add my current, daily used email addy ("Jim") to my ID, or better yet, make it my only AppleID and delete "Bob". BUT... I already used "Jim" for an AppleID for an older friend who had no email address, so set it up for him on mine, "Jim".  My older friend now has an email address ("Ivan") so can I   add "Ivan" to his AppleID, make it the primary email address, and then delete "Jim" from it? Or... delete his AppleID and create a new one for him "Ivan" knowing he would lose his minor purchases   And most importantly, this would all be pointless unless: - will deleting/removing "Jim" as linked to my friend's AppleID then make "Jim" available for me and no longer registered to another account? Words with different meanings in American and British English This is similar to , but not quite the same.  There are quite a few words which have totally different meanings in American and British English and which are likely to cause confusion when heard by non-local speakers.  What are they?  We are looking for the same word with different meanings (for example jumper), and not words which are different between the two dialects (for example truck and lorry). Simplify a combinatorial sum Why is $\sum_{k=0}^n (_k^n) 2^k$ simplified as $3^n$ and not just $2^n$ ? The answer uses the binomial theorem as a solution: let $a=1$ and let $b=2$. So $1+2 = 3$. But since $1+2 = 3$ why do we not use $1+1 = 2$? Which close reason should I use for questions that do not demonstrate research effort? I regularily encounter questions on StackOverflow which are asked only because of the sole fact that the poster haven't made the effort to google for and the docs of a particular function/class/framework/whatever. My question is, to which category of the five proposed one should I vote when I vote to close the question? Is it: off-topic, not constructive, not a real question, exact dupe or too localized? I tend to be all for 'not a real question' - am I quite right? What is wrong with \ifnum#1=0.09\relax? The following code does not compile with the following error message     Missing = inserted for \ifnum   \documentclass[pstricks,border=4pt]{standalone} \usepackage{multido}  \psset{linewidth=.5\pslinewidth} \psset{xunit=2cm}  \newcommand\FixedScale[1]{%     \multips(0,0)(.1,0){\the\numexpr10*#1+1\relax}{\psline(0,0)(0,-3pt)}     \multips(0,0)(.5,0){\the\numexpr2*#1+1\relax}{\psline(0,0)(0,-6pt)}     \multips(0,0)(1,0){\the\numexpr#1+1\relax}{\psline(0,0)(0,-9pt)}     \multido{\i=0+1}{\the\numexpr#1+1\relax}{\uput[-90](\i,-5pt){$\i$}} }  \newcommand\VernierScale[2][0.09]{%     \rput(#2,0){%     \psset{xunit=#1,linecolor=red}%     \multips(0,0)(1,0){11}{\psline(0,0)(0,3pt)}%     \multips(5,0)(5,0){1}{\psline(0,0)(0,6pt)}%     \multips(0,0)(10,0){2}{\psline(0,0)(0,9pt)}%     \ifnum#1=0.11\relax         \multido{\i=0+5}{3}{\uput[90](\i,8pt){\tiny$\the\numexpr10-\i\relax$}}     \else         \multido{\i=0+5}{3}{\uput[90](\i,8pt){\tiny$\i$}}     \fi     } }  \begin{document} \multido{\n=0.00+0.01}{21}{% \begin{pspicture}(0,-.5)(2,.55) \FixedScale{2} \VernierScale[0.09]{\n} \end{pspicture}} \end{document}   What is wrong with \ifnum#1=0.09\relax? Decompose vector fields on product manifolds So, I know that tangent bundle of a product manifold $M \times N$ splits in a sum $$ T_{(x,y)}(M \times N) = T_xM \oplus T_yN, $$ so that is obvious that the sum $X \oplus Y$ of smooth vector fields $X \in \mathcal{T}(M)$ and $Y \in \mathcal{T}(N)$ is a smooth vector field of $M \times N$. I've been told that, though not every vector field in $\mathcal{T}(M \times N)$ is a sum, locally one can always find one such decomposition, which will in turn be unique due the fact the sum is a direct one.   How can I show that this decomposition exists locally? More than that, if $X = X_1 + X_2$ is the decomposition, is there a way the express the coordinate functions of $X_1$ and $X_2$ in terms of those of $X$?   First I thought about taking two frames that locally spam $TM$ and $TN$ and write down $X$ using them, but then the coordinate functions are of the form $X^i: M \times N \to \mathbb R$, and the vector field components in each subspace are not exactly fields of $M$ and $N$ because their coordinate functions don't have the right domains. Is there another better way to see this decomposition holds locally?