$G$ is a simple graph that consists of a vertex set $V(G) = \{v_1, v_2, ..., v_n\}$ and an edge set $E(G) = \{e_1, e_2, ..., e_m\}$ where each edge is an ordered pair of vertices. The edge $\{u,v\}$ is denoted $uv$.  A walk of length $k$ is a sequence $v_0,e_1,v_1,e_2,...,e_k,v_k$ of vertices and edges such that $e_i=v_{i-1}v_i$ for all $i$. A path is a walk with no repeated vertex.    Prove:  If $G$ is a finite simple graph in which every vertex has degree at least $k$, then $G$ contains a path of length at least $k$.  I believe this has something to do with the longest path but I can't come up with a reasoning/proof that is of length of at least $k$.
Prove there's a simple path of length $k$ in a simple graph $G$ where all the vertices have degree of at least $k$.   Relevant definitions:  $G$ is a simple graph that consists of a vertex set $V(G) = \{v_1, v_2, ..., v_n\}$ and an edge set $E(G) = \{e_1, e_2, ..., e_m\}$ where each edge is an ordered pair of vertices. The edge $\{u,v\}$ is denoted $uv$.  A walk of length $k$ is a sequence $v_0,e_1,v_1,e_2,...,e_k,v_k$ of vertices and edges such that $e_i=v_{i-1}v_i$ for all $i$. A path is a walk with no repeated vertex.   My attempt:  Induction, for $k=1$ it's obvious.  Suppose for $k-1$ and we'll prove for $k$.  Let $G$ be a simple graph such that $\forall v \in V : d(v)\ge k$.  We can assume that there are at least $k+1$ vertices since there are $k$ neighbours to every vertex.  Let $v_0$ be some vertex, it has $k$ neighbours, we'll move to one of its neighbours say $v_1$ and remove $v_0$. So now there are $k$ vertices with a degree of at least $k-1$ and from the induction hypothesis we'll have a path of length $k$.  Is using only $k$ vertices from $n$ right? Does it keep the generality when using only all the neighbours of $v_0$?
Walking up/down a level road/street Is walking up/down the road/street applicable if the street is not sloped?  I think I heard something about "with/against the numbering of houses" but I'm not sure if it's correct.  Also, if that were correct, why is walking down the street  than walking up the street? How can I speed up D&D 3.5e combat? Non-trivial cyclic group with unique generator Let $G$ be a non-trivial cyclic group with an unique generator. Show that $G$ has exactly two elements.   Let $g$ be a generator of $G$. My first idea to approach the problem was to show that if $|G|=n &gt; 2$ then there is an integer $k$ between 1 and $n$ such that $\langle g^k \rangle = G$ or, equivalently, $g \in \langle g^k \rangle$. My second idea was to analyse the problem in 2 separate cases. For the first one, if $n$ is prime, let $k$ be a positive integer then, by Lagrange's Theorem, $|g^k| \mid n$ and, therefore, $\langle g^k \rangle = n$. Hence, we must conclude that, if $n &gt; 2$ the generator cannot be unique. For the second case, where $n$ is composite, I was hoping to get to a contradiction, but could not advance any further. Is there a way to develop either of these ideas or is it necessary a new approach? Why Does Harry Have a Scar When Avada Kedavra Leaves No Trace? Why does Harry have a scar when Avada Kedavra leaves no trace?     ‘The Muggle authorities were perplexed. As far as I am aware, they do not know to this day how the Riddles died, for the Avada Kedavra Curse does not usually leave any sign of damage ... the exception sits before me,’ Dumbledore added, with a nod to Harry’s scar.   Half-Blood Prince - page 343 - Bloomsbury - chapter 17, A Sluggish Memory   What caused Harry to retain a scar from Avada Kedavra, when no one else in know history showed any signs of cause of death when Avada Kedavra was used?  What caused Harry's scar?  I'd love an answer based in canon, such as the books, interviews with J.K. Rowling, or Pottermore. Identity for convolution of central binomial coefficients: $\sum\limits_{k=0}^n \binom{2k}{k}\binom{2(n-k)}{n-k}=2^{2n}$ It's not difficult to show that  $$(1-z^2)^{-1/2}=\sum_{n=0}^\infty \binom{2n}{n}2^{-2n}z^{2n}$$  On the other hand, we have $(1-z^2)^{-1}=\sum z^{2n}$.  Squaring the first power series and comparing terms gives us  $$\sum_{k=0}^n \binom{2k}{k}\binom{2(n-k)}{n-k}2^{-2n}=1$$  that is,  $$\sum_{k=0}^n \binom{2k}{k}\binom{2(n-k)}{n-k}=2^{2n}$$  My question: is there a more direct, combinatorial proof of this identity?  I've been racking my brains trying to come up with one but I'm not having much success. Avoid trailing zeroes in printf() I keep stumbling on the format specifiers for the printf() family of functions.  What I want is to be able to print a double (or float) with a maximum given number of digits after the decimal point.  If I use:  printf("%1.3f", 359.01335); printf("%1.3f", 359.00999);   I get  359.013 359.010   Instead of the desired  359.013 359.01   Can anybody help me? Fatou's lemma and measurable sets I don't know how can I imply Fatou's lemma for any measurable sets $A_k$, that is, $$\lambda(\liminf A_k)\le\liminf\lambda(A_k).$$ How can I prove it? And is there any example in $\mathbb R$ of sequence of measurable sets $A_k$ such that $A_k\subset[0,1]$, $\lim\lambda(A_k)=1$, but $\liminf A_k=\varnothing$ ? Thanks for your help! Why aren't logarithms defined for negative $x$? We have duplicates in our all contacts in Marketing Cloud. To solve this, we want to delete all contacts (audience builder).   We use the Marketing Cloud connector, and send emails to synchronized data extensions. Because of this connector the all contacts will be filled with fresh data after 15 minutes right?  Is there any reason not to do this? What can go wrong? I don't think I've seen any answers to this question, but I'm curious as to why, when you have a 64-bit OS installed (ex: Vista x64), IE defaults to the 32-bit version? Are there drawbacks to using the x64 version of IE (IE8, specifically)? Title of short story: God granting immortality to others to find a way to kill himself 'const int' vs. 'int const' as function parameters in C++ and C Consider:  int testfunc1 (const int a) {   return a; }  int testfunc2 (int const a) {   return a; }   Are these two functions the same in every aspect or is there a difference?  I'm interested in an answer for the C language, but if there is something interesting in the C++ language, I'd like to know as well. Verb agreement with "what" This came up when reviewing a written English exercise, and I realized that I don't entirely trust my intuition on the answer.  Which of the following is correct?  Or are both valid?  (Or is it strongly advisable to rewrite the sentence in some other way?)   "I see what appears to be a man and his son." "I see what appear to be a man and his son." I have Windows 8 (final, MSDN) on my Macbook Pro Retina.  Windows 8's ongoing, continuous tantrum of insisting that it's on a tablet device is causing it to always have a touch keyboard 'icon' on  the taskbar.  The 'icon' is actually a toolbar and I can temporarily disable it by right clicking the taskbar and DE-selecting it. However, upon rebooting the system, it always insists on coming back and activating itself.    How can I permanently disable the touch keyboard toolbar in the taskbar? How do I add CSS files? Here is an example of my *.info file.      Why the page cannot load the CSS that I add myself? C non-blocking keyboard input I'm trying to write a program in C (on Linux) that loops until the user presses a key, but shouldn't require a keypress to continue each loop.  Is there a simple way to do this? I figure I could possibly do it with select() but that seems like a lot of work.  Alternatively, is there a way to catch a ctrl-c keypress to do cleanup before the program closes instead of non-blocking io? How does the speed of electrons change around a circuit? I have been thinking about ways of teaching electronics and I'm wondering if the following is true...  For starters, when we talk about voltage as energy per unit charge, is this energy manifest simply as the kinetic energy of the electron?  So, all other things being equal, will the electrons coming out of a 4 volt battery will have twice the velocity of electrons coming out of a 1 volt battery because of ${1 \over 2}mv^2$?  This would seem to imply that the voltage drop across different parts of the circuit is essentially a comparison of the electron speeds at those points.  Furthermore doesn't this imply that the electrons are nearly halted by the time they reach the positive terminal of the battery? This makes intuitive sense to me, it seems pedagogically sound, and it provides an explanation for how electrons "know" to give up their energy across the circuit -- electrostatic repulsion communicates later resistances to earlier parts of the circuit.  How much of the above is correct? The one things that makes me suspicious is that, in this picture, electron density at the positive terminal of the battery is very high, and repulsive forces would prohibit this. On the other hand, if the Joules per Coulomb are not coming from kinetic energy, then what? The field? But wouldn't that just dissipate as light?  Something is wrong but I don't know what. SQLiteException while inserting values in the database in Android I am having trouble getting a basic example of a matrix of notes to show up in my beamer presentation. I am using the code copied from the pgf manual. Here is a complete working example. The error I get upon compilation is:  "Package pgfbasematrix Error: Single ampersand used with wrong catcode."  \documentclass{beamer} \usepackage{tikz} \usetikzlibrary{arrows,decorations.pathmorphing,backgrounds,fit,petri,positioning,matrix}  \begin{document}  \begin{frame}  \begin{tikzpicture} \matrix (magic) [matrix of nodes] { 8 &amp; 1 &amp; 6 \\ 3 &amp; 5 &amp; 7 \\ 4 &amp; 9 &amp; 2 \\ }; \draw[thick,red,-&gt;] (magic-1-1) |- (magic-2-3); \end{tikzpicture}  \end{frame} \end{document} Combining two covariance matrices