undirected graphs and Hamiltonian cycles
Edges on an even number of Hamilton cycles
When do mobs despawn in Minecraft? I understand monsters only spawn in a 9 x 9 grid of chunks centered on the player, and I assume they stay around while they remain within that distance of you. But do they despawn as soon as you move away from them? Or do they remain? If so what (if anything) causes them to despawn? Are the rules any different for passive mobs?  Will running far away from an area and then coming right back clear it of monsters? I wanted to show that $X' / U^\perp \cong U'$, for $U$ being a closed subspace of the Banach space $X$.  Therefore I looked at $l: X' / U^\perp \cong U' , x' + U^\perp=[x']  \mapsto x'|_U$.  It is clear that this map is onto, as we just take a $u' \in U'$. Then I can use Hahn-Banach to get an appropriate $x'$ such that $x'|_U = u'$ and in particular $||x'||=||u'||$.  Now I need to show that is isometric: since we always have $\|[x']\|\le \|x'\|$ and by Hahn-Banach we have $\|x'\|=\|x'_{|U}\|$, this shows $\|[x′]\|\le \|x'_{|D}\|$ and it should be always true that $\forall y' \in U^\perp: \|x'-y'\| \ge \|x'_{|D}\|$, hence $\|[x']\|\ge \|x'_{|D}\|$. Is this reasoning correct? The isometric property follows. So I've been trying to figure out how to prove the following.     Let $(a_n)$ be a sequence of positive numbers such that $\sum\limits_{n=1}^\infty a_n =\infty$, and define $s_n=\sum\limits_{i=1}^n a_i$. Then $\sum\limits_{n=1}^\infty\frac{a_n}{s_n} =\infty$ as well.   I can prove it by comparing it to $\int_1^\infty \frac{1}{x} \, dx $ if the sequence $a_n$ is bounded by some $M$, but that's as far as I've been able to get. System file windows open behind Unity launcher I'm on Ubuntu 16.04.1 at the moment, and system windows like the Files window, Terminal or System Software open behind the Unity launcher, which means they are partially hidden behind it. I can move them with Alt+F7, but I'd like a more permanent solution to this problem. Is Ghost Adventures real or fake? Is there an option to create and Opt-in a user in the same API call? QR factorization of complex matrix If you have two complex numbers $a,b$ how can you find the QR factorization of   $ M = \begin{bmatrix}        aI_n\\          bI_n \end{bmatrix} $,  I can't seem to be able to do it. I tried an earlier trick, but I now know it's not necessary true for complex numbers. Is there anyway to do this?  I think if I use a Givens algorithm I should be able to reduce it to a QR factorization. Size commands do not work in mathmode MWE:  \documentclass{article} \begin{document} $\wedge{\small\wedge}{\tiny\wedge}$ \end{document}   All the \wedges have the same size... Why does the zoom sometimes stop at a point? I often find that I can zoom in to a point, but then cannot zoom any more, even when there is a lot of room to zoom to the meshes on the scene.   This is particularly annoying in the following scenario:  I often land up left clicking on the scene and the 3D cursor shifts there. I use Shift+C to shift the 3D cursor back to the origin. However, this also centers the view on the cursor and shifts the view to show all objects, and sometimes this zooms things way out &ndash; at this point, if I try to zoom in, I can zoom in to a point (using the pinch/zoom on trackpad), but then it stops zooming in.   How do I zoom in after that? Is there anything else I am doing wrong in this case? Vandermonde Determinant This is an exercise from Ian Stewart's Galois Theory, $3^{rd}$ edition:     If $z_1,z_2,\ldots,z_n$ are distinct complex numbers, show that the determinant $$D=\left|\begin{array}[cccc]  11&amp;1&amp;\cdots&amp;1\\ z_1&amp;z_2&amp;\cdots&amp;z_n\\ z_1^2&amp;z_2^2&amp;\cdots&amp;z_n^2\\ \vdots&amp;\vdots&amp;\ddots&amp;\vdots\\ z_1^{n-1}&amp;z_2^{n-1}&amp;\cdots&amp;z_n^{n-1} \end{array}\right|$$   is nonzero.       Hint: Consider the $z_j$ as independent indeterminates over $\Bbb C$. Then $D$ is a polynomial in the $z_j$, of total degree $0+1+\cdots+(n-1)=\frac{1}{2}n(n-1)$. Moreover, $D$ vanishes whenever $z_j=z_k$ for $k\neq j$, as it then has two identical rows. Therefore $D$ is divisible by $z_j-z_k$ for all $j\neq k$, hence it is divisible by $\prod_{j&lt;k}(z_j-z_k)$. Now compare degrees.   My question is how to do this. I follow the hint and I'm thinking of $D$ as a cofactor expansion; is that the idea? Also, the fact that $D$ vanishes whenever $z_j=z_k$ for $k\neq j$ follows since it would have two identical columns, right? I don't see why $z_j=z_k$ would imply it has two identical rows, as the hint suggests. The fact that it is divisible by $(z_j-z_k)$ is easy since that would make $z_k$ a root for $k=j$, but I'm not quite sure I follow why the degrees would be different. I think the degree of the product would be $1+2+\cdots+(k-1)$, but then add these up for all $k&lt;n$?   I appreciate any clarification that you can provide! Aware abbreviating those commands need not always be helpful, I would like to know how to abbreviate them? The following is what I tried, which does not work:  \documentclass[11pt,handout]{beamer} \newcommand{\bframe}{\begin{frame}} \newcommand{\eframe}{\end{frame}} \newcommand{\ftitle}{\frametitle}   \author{Author} \title{Title}   \begin{document} \maketitle  \begin{frame} \frametitle{Outline}  \end{frame}   \bframe \ftitle{Introduction}  \eframe    \end{document}   At first I thought it might be the case that commands such as \bframe was already defined. But using \def instead still gives no desired result. Why can some holodeck matter leave the holodeck, while other holodeck matter (notably people) cannot? Which of $\{J_0,J_1,J_2\}$ and $\{J_0,J_+,J_-\}$ is a generator for the Lie algebra $\mathfrak{su}(2)$? I have just started learning about Lie algebra in the context of quantum mechanics and got confused with this: Some sources say the generators are $J_0,J_1$ and $J_2$ and some use $J_0,J_+$ and $J_−$. Which set is correct? Or if both are correct what key concept am I missing here? My understanding is that if have certain commutation relations then we know that Lie Algebra is such and such. But if we have two such choices then this understanding falls apart? How do I figure out what is $\mathfrak{su}(2)$ Lie Algebra in general? Maximum total distance between points on a sphere What is the configuration (set of locations) of $n$ points on the surface of a sphere such that the sum of distances is maximum for $n=1,2,3,...$?  The sum of distances is measured by summing the lengths of every straight line (through the sphere) connecting every possible combination of $2$ points.  All the points are on a single sphere of radius $R$.  Here's a visualization:   Acknowledgements: Based on  Physics S.E. question. Image from . I want to know where the Metro applications (Mail, People, Messages, ...) executable file path is.  Also, how can I run Mail application when my PC starts? $\limsup$ and cluster points Let $(x_n)$ be a sequence in $\mathbb R$. We call $y$ a cluster point of $(x_n)$ iff for every neighborhood $N$ of $y$ there are infinitely many $n$ such that $x_n \in N$. Let $C$ denote the set of cluster points of $(x_n)$.   By definition, $\limsup_{n \to \infty} x_n = \inf_n \sup_{k \geq n} x_k$.  Can you show me how to prove $\limsup_n x_n = \sup C$? Thanks. How to understand Cauchy's proof of AM-GM inequality(the last step) Grammar checking program for LibreOffice Is there a tool for checking grammar in LibreOffice? Add 1Password extensions support to the SE iOS app Please implement the URL scheme to call 1Password for password lookups on the iOS app.   a URL scheme for switching to 1Password to search item titles for a custom term, usually a service name like “twitter” or “evernote”. This speeds up the process for the user so they can copy something like their password and get back to your app  That will ease sign up speed using long, complicated or unmemorable passwords for many users of both apps. In the two years since the above blog was published there are now updated instructions for implementation and code on github: Linux command line. Move all files and directories in directory, except some files and directories I have a folder A which has files and directories, I want to move all those files and directories to another folder B, except file, file2, directory, and directory2.  How can this be done?