Physical intuition behind level repulsion?
Why is the second order perturbative correction to the ground state energy always down?
Does each season of Game of Thrones roughly equal a book? Does each current Game of Thrones season roughly equal one A Song of Ice and Fire book? And do we know which future seasons will cover which books?  I am planning to read the first book after the TV show has finished depicting the events in the first book. I would like to know if after each TV season ends I can expect to be able to read a book covering the events depicted on screen? Prove that, for any $n+1$ integers $a_{1},a_{2},....,a_{n+1}$, there exist two of the integers $a_{i}$ and $a_{j}$ with $i \neq j$, such that $a_{i} - a_{j}$ is divisible by $n$.  Please help me about this problem,I know there is lots of answers can find from Google, but I am not really understand them. I hope someone can give me more details explanation. I would really appreciate your help. Will yearly moderation statistics be published on all sites this year, and future years? Can every non-simple group $G$ be written as a semidirect product? When people want to classify a group with certain (small) order, they seem to find a normal subgroup $H$ and a subgroup $K$, and then consider how many distinct $H \rtimes_\varphi K$ are there.  My question is: For any group $G$, does it always exist normal subgroup $H$, subgroup $K$ and $\phi : K \rightarrow Aut(H)$ s.t. $G\cong H \rtimes_\varphi K $?  Sorry if this question is too basic...  Thanks the comment and the answers. Then if the $G$ is not simple, can we have can $G\cong H\rtimes_\varphi K$ for proper subgroup $H$, $K$, with $H$ normal?  EDIT:  If we have a group $G$, that is not simple (then $G$ has at least one normal subgroup), are there always a non-trivial normal subgroup $H$ and a proper subgroup $K$ s.t. $G \cong H \rtimes K$. (Note that if $G$ is simple this is clearly not possible).  But the quaternion group and $Z_4$ given in answers have shown that we cannot always have it. While CtrlX works fine in vim under windows, CtrlA selects all (duh).  Is there a way to increment a number with a keystroke under windows? I am running the latest version of Linux Mint with Cinnamon. I'm trying to map Caps Lock to Ctrl, but I cannot figure out how to do it. All web searches I've done have led me to older versions of Linux Mint (there is no keyboard layout option in my settings). How can I do this? Moment of inertia of a hollow sphere wrt the centre? I've been trying to compute the moment of inertia of a uniform hollow sphere (thin walled) wrt the centre, but I'm not quite sure what was wrong with my initial attempt (I've come to the correct answer now with a different method). Ok, here was my first method:  Consider a uniform hollow sphere of radius $R$ and mass $M$. On the hollow sphere, consider a concentric ring of radius $r$ and thickness $\text{d}x$. The mass of the ring is therefore $\text{d}m = \frac{M}{4\pi R^2}\cdot 2\pi r\cdot\text{d}x$. Now, use $r^2 = R^2 - x^2:$ $$\text{d}m = \frac{M}{4\pi R^2}\cdot 2\pi \left(R^2 - x^2 \right)^{1/2}\text{d}x$$ and the moment of inertia of a ring wrt the centre is $I = MR^2$, therefore: $$\text{d}I = \text{d}m\cdot r^2 = \frac{M}{4\pi R^2}\cdot 2\pi\left(R^2 - x^2\right)^{3/2}\text{d}x $$ Integrating to get the total moment of inertia: $$I = \int_{-R}^{R} \frac{M}{4\pi R^2} \cdot 2\pi\cdot \left(R^2 - x^2\right)^{3/2}\ \text{d}x = \frac{3MR^2 \pi}{16}$$  which obviously isn't correct as the real moment of inertia wrt the centre is $\frac{2MR^2}{3}$.   What was wrong with this method? Was it how I constructed the element? Any help would be appreciated, thanks very much. How do I obtain the "final" standard deviation from a series of values containing individual (also null) SD values? I have a list of values, say:  AVGprice SDprice count year 10          3      5   1999 20          2      3   2000 30          8      20  2001  40        undef    1   2002   where AVGprice is the average of a certain price and SD its respective standard deviation for that particular year. Count represents the number of occurrences.  What's the best way to calculate the final AVGprice and SDprice for the whole period 1999-2002? For the former, I'd say a weighted average will do, but I am not sure about the final standard deviation - specially because of the undefined value. I am trying to determine the size in bytes of the contents in a VARBINARY(MAX) field in SQL Server 2005, using SQL. As I doubt there is native support for this, could it be done using CLR integration? Any ideas would be greatly appreciated. Apt-get stuck at 0% [Working] Let $X,Y,Z$ be subspaces of $V$ so that $X$ is a subspace of $Y$. Prove that $Y\cap (X+Z)=X+(Y\cap Z)$ Let $X,Y,Z$ be subspaces of $V$ so that $X$ is a subspace of $Y$. Prove that $Y\cap (X+Z)=X+(Y\cap Z)$  I know that I need to prove that $Y\cap (X+Z)\subseteq X+(Y\cap Z)$ and $X+(Y\cap Z)\subseteq Y\cap (X+Z)$ but I´m having a hard time.  Can you lend me a hand please? I would really appreciate it The title of  of  is, "Is it posible [sic]?"  This doesn't provide any information at all. It should be mandatory for everyone asking questions on the Internet to read , which says:     Help your community learn from the question. When you field a good question, ask yourself "How would the relevant documentation or FAQ have to change so that nobody has to answer this again?" Can you explain briefly the main concepts and command line tools used to manage file permissions? HABTM dropdown items not selected Among other sources,  states that triple DES using three seperate keys (k1, k2, k3) is vulnerable to meet-in-the-middle-attacks, while triple DES using only two keys (k1, k2, k1) is not.  Why is this attack possible if all the keys are different, but not if the first and the third are the same? If a commutative ring is semiprime and its prime ideals are maximal then it is von Neumann regular (absolutely flat). If a commutative ring is semiprime and its prime ideals are maximal then it is von Neumann regular (absolutely flat).  The converse, although not immediately apparent, can be proven quite easily. But I've been stuck on this one for awhile.  I found that Bourbaki has a question in his treatise on commutative algebra, namely: if all points in Spec(A) are closed (ie. prime ideals are maximal) then A/R (where R is the nilradical) is von Neumann regular (absolutely flat). If this can be proven, the above statement follows immediately since the nilradical of a semiprime ring 0?  Ideas?   Ps: this is my first time posting a question so apologies for any formatting issues. If A is a finite set and B is an infinite set, then B\A is an infinite set Prove: If $A$ is a finite set and $B$ is an infinite set, then $B \setminus A$ is an infinite set.   I've tried proving the contrapositive, that if $B \setminus A$ is a finite set then $A$ is infinite or $B$ is finite, but I don't think this is the way to go. How to split a string with any whitespace chars as delimiters What regex pattern would need I to pass to  java.lang.String.split()  to split a String into an Array of substrings using all whitespace characters (' ', '\t', '\n', etc.) as delimiters? Relation between magnetic moment and angular momentum -- classic theory How do I prove the relation between the vectors of magnetic moment $\vec\mu$ and angular momentum $\vec L$, $$\vec\mu=\gamma\vec L$$  ?  Many text books and lecture notes about the principles of magnetism show the relation of $\mu$ and $L$ as scalars only and then just state that the relation holds also for the vectors. An example:   $$\mu=I\cdot A = \frac{q}{t}\pi r^2 = \frac{qv}{2\pi rm}m\pi r^2$$  ($I=q/t$: current, $A$: area of a loop, $q$ charge, $t=2\pi r/v$: time of 1 rotation, $v$: velocity of particle, $m$: mass )  The angular momentum is $\vec L = \vec r\times\vec p$ or $L=mrv$ and therefore $$\Rightarrow \mu = {\frac{q}{2m}} L = \gamma L.$$  Why is this also true for the vectors? Is there a general explication by classical physics without the need of quantum theory? How can I get this 'Version: 0.1.0 - 94" value from the  tag in selenium webdriver? How can I get this Version: 0.1.0 - 94 value from the &lt;div&gt; tag in selenium webdriver? and here is my code:  &lt;div _ngcontent-qbk-c3="" style="text-align: center;"&gt;Version: 0.1.0 - 94&lt;/div&gt;