How to find matching word in array
How do I check if an array includes a value in JavaScript?
Change the display scaling on the fly I'm looking for a keyboard shortcut or a CLI instruction to change the display scale on the fly. Seems there's nothing in compiz-config manager to handle this? Show that $\ker \hat{T} = \text{ann}(\text{range } T)$ This is an old exam problem: Let $V$ and $W$ be finite dimensional vector spaces over a field $F$ and let $T: V \to W$ be a linear transformation. Define $\hat{T}: W^* \to V^*$ by $(\hat{T}(f))(v)=f(T(v))$. Here, $U^*$ denotes the dual vector space $U$. Since $\hat{T}$ is a linear transformation prove $\ker \hat{T} = \text{ann}(\text{range } T)$ where $\text{ann}(Y) = \{g \in U^* : g(y)=0 \text{ for all } y \in Y \}$ for a subspace $Y$ of vector space $U$.  I'm really not sure how to start this problem. Any advice would be great! Setting chain length My chain broke and in a moment of silliness (I was cold wet and slightly annoyed)  I threw the broken one away before thinking about measuring the new one.  Is there a guide to getting the chain length/tension correct ?  It's an 8 speed road bike (Shimano Sora back, double ring front) I've notice that the business to business site AliBaba and it's more consumer friendly AliExpress website have a number of bike components for sale. They tend to be complete component groupsets and the like but are some quite reasonable deals, especially if you are building a bike from scratch.  I've used AliExpress before to purchase electronic components, which have been good quality and are what they say on the tin.   Has anyone used AliExpress to buy bike components and have they had any trouble? (quality, counterfeit etc.)  One benefit is that your money is held in escrow until you receive the item and aprove the payment so some safety. Are token creatures considerd as a creature spell? Are token creatures considerd creature spells? For instince when I use  tap effect combined with  would it allow me to draw a card? Finding the limit $\displaystyle\lim_{x\to 0+} \left(\frac{\sin x}x\right)^{1/{x^2}}$ Find the following limit: $$\displaystyle\lim_{x\to 0+} \left(\frac{\sin x}x\right)^{1/{x^2}}$$   Well I tried to do the $\exp\left(\frac{ \ln\frac{\sin x}{x}}{x^2}\right)$ then apply LHR but I seem to get to endless dervivations...   There's got to be a more simple approach. How is the classical twin paradox resolved? I read a lot about the classical twin paradox recently. What confuses me is that some authors claim that it can be resolved within , others say that you need . Now, what is true (and why)? For closed subsets $A,B \subseteq X$ with $X = A \cup B$ show that $f \colon X \to Y$ is continuous iff $f|_A$ and $f|_B$ are continuous. I was just trying to solve an exercise in Munkres' Analysis on Manifolds to get better at continuous functions and topology in $\mathbb{R}^{n}$, and I got stuck. Here it is:      Let $X = A \cup B$, $A$ and $B$ are subspaces of $X$. Let $f:X \rightarrow Y.$ Suppose that the restricted functions $f_{|A}:A \rightarrow Y$ and $f_{|B}:B \rightarrow Y$ are continuous. Show that if both $A$ and $B$ are closed in $X$, then $f$ is continuous.    Our definition of continuity is:  $f$ is continuous at $x \in X$ if for each open set $V$ of $Y$ containing $f(x)$, there is an open set $U$ of $X$ containing $x$ such that $f(U) \subset V$. $f$ is continuous if it is continuous at each point of its domain.  What I've tried so far:  Suppose $A$ and $B$ are closed in $X$. Suppose $A \cap B = \emptyset$. Then $X -A$ is open in $X$. But $X - A = B$ which is closed in $X$, which is a contradiction. Hence, $A \cap B \ne  \emptyset$. I'm not sure if this will be useful in the problem or not. If $x \in X$, then it's either in $A$ or in $B$. Suppose without loss of generality that it's in $A$. Then for all open sets $V$ of $Y$ s.t. $f(x) \in V$, $\exists$ an open set $U_{1}$ of $A$ s.t. $U_{1} \subset A, a \in U_{1}$ and $f(U_{1}) \subset V$. This is almost what we want, except that we want to find an open set $U$ of $X$ s.t. $U \subset A, a \in U$ and $f(U) \subset V$. This is because $U_{1}$ being open in $A$ does not necessarily mean that it's open in $X$ because $A$ is closed in $X$. (i know how to expand this argument out if needed) My idea now is to try to find a subset $U$ of $U_{1}$ such that $U$ is open in $X$. We have two cases:  (1) $U_{1}$ does not intersect the boundary of $A$. (2) $U_{1}$ intersects the boundary of $A$. Now I'm not sure what to do. I'm wondering if it would help to investigate the relationship between Bd(A) and Bd(B).. In any case (no pun intended), any help/advice would be greatly appreciated. :)  Sincerely,  Vien Break a loop animation (Make Cyclic) to start a new one I've used the 'Make Cyclic' to loop these keyframes (image below). So, I want to break the loop at a certain time to do a new animation (insert another kf to do a different thing than the loop does). Any help how to do this?    Thanks in advance! Where should shell tools be installed? Are there Apple guidelines for where user shell tools in Mountain Lion should be installed? Some (e.g. TeX, or Apple's Git) end up in /usr, and others (e.g. X11) seem to end up in /opt (sometimes with a symlink from /usr), while others (e.g., Git) end up in /user/local, and I wonder what the difference is, and whether one is preferred over the other. IF/ELSE script in sender profile I'm trying to create a dynamic sender profile where the from name and from email change if a value is present in a specific field of the data extension. I want the Lead First Name value and Lead Last Name values to appear. I'm trying to keep all the logic in the sender profile. %%=IIF(EMPTY(AttributeValue(&quot;X&quot;)), &quot;Bill&quot;, AttributeValue(&quot;LeadFirstName&quot;) AttributeValue(&quot;LeadLastName&quot;))=%%  Not sure why, but doing the test, the script shows up as the sender and isn't resolving. Linux/Bash: Create relative soft links to files in directory tree? How can I install the Google Apps Package (Play Store, ...) on my Android device? No, this is not a , nor is it a solicitation on  to backup or  and only a little bit about Where to backup and When to do it.  This is basically a strategy question: what, where and when all together. "How" is not really relevant: there are tons of back-up programs out there and they all do the same basic thing: make copies of data.  But the real question is: are these back-up programs making copies of the data important to you?  How should you install&amp;run Ubuntu while safeguarding your data?  There is no "one size fits all" when it comes to computers, Ubuntu versions or a backup strategy, therefore I've split this up into 5 basic user types with each their own answer:       (or you're running in UEFI mode!)   And as the above does not fit everyone, one for the impatient:   !    So click one of the above links and go to your user type!  P.S. If you're running a server farm and you've read this question so far, I'm sorry: this is about a single PC, not a network, nor a server farm. This picture is a part of a table i am using, the terms subcases are vertically written but i cant center them vertically also in the next columns terms like A, B, etc are also not vertically centered. I made this table using tabulary package a. My code is  \begin{table}[H]  \centering  \caption{Saturation Table}  \label{tab:5}%  \begin{tabulary}{1\linewidth}{J J J J J J J J J J J J J}  \toprule  &amp; \textbf{Method} &amp; \multicolumn{1}{c}{\textbf{Parameters}} &amp; \textbf{Step 0} &amp; \textbf{Step 1} &amp; \textbf{Step 2} &amp; \textbf{Step 3} &amp; \textbf{Step 4} &amp; \textbf{Step 5} &amp; \textbf{Step 6} &amp; \textbf{Step 7} &amp; \textbf{Step 8} &amp; \textbf{Step 9} \\  \midrule  \multirow{4}[4]{*}{\begin{sideways}\textbf{Subcase A}\end{sideways}} &amp;  \multirow{2}[2]{*}{\textbf{A}} &amp; \textbf{No} &amp; 1 &amp; 2 &amp; 3 &amp; 4 &amp; 5 &amp; \textbf{6} &amp; &amp; &amp; &amp; \\[1mm]  &amp; &amp; \textbf{Maximum Saturation {(\%)}} &amp; 1 &amp; 2 &amp; 3 &amp; 4 &amp; 4 &amp; \textbf{5} &amp; &amp; &amp; &amp; \\  \cmidrule{2-13} &amp; \multirow{2}[2]{*}{\textbf{B}} &amp; \textbf{No} &amp; 1 &amp; 2 &amp; 3 &amp; 4 &amp; 5 &amp; 6 &amp; 7 &amp; \textbf{8} &amp; &amp; \\[1mm]  &amp; &amp; \textbf{Maximum Saturation {(\%)}} &amp; 6 &amp; 6 &amp; 7 &amp; 8 &amp; 9 &amp; 9 &amp; 19 &amp; \textbf{8} &amp; &amp; \\  \bottomrule  \end{tabulary}%  \end{table}% How to rank a million images with a crowdsourced sort I'd like to rank a collection of landscape images by making a game whereby site visitors can rate them, in order to find out which images people find the  most appealing.  What would be a good method of doing that?   Hot-or-Not style? I.e. show a single image, ask the user to rank it from 1-10. As I see it, this allows me to average the scores, and I would just need to ensure that I get an even distribution of votes across all the images. Fairly simple to implement. Pick A-or-B? I.e. show two images, ask user to pick the better one. This is appealing as there is no numerical ranking, it's just a comparison. But how would I implement it? My first thought was to do it as a quicksort, with the comparison operations being provided by humans, and once completed, simply repeat the sort ad-infinitum.   How would you do it?  If you need numbers, I'm talking about one million images, on a site with 20,000 daily visits. I'd imagine a small proportion might play the game, for the sake of argument, lets say I can generate 2,000 human sort operations a day! It's a non-profit website, and the terminally curious will find it through my profile :) I am going through Shankar's Principles of Quantum Mechanics and am having trouble finding the free particle propagator $U(t)$ that satisfies $$\lvert\psi (t)\rangle = U(t)\lvert\psi (0)\rangle$$ due to the degeneracy of the $E$ eigenkets. Shankar says that if the Hamiltonian has degenerate eigenvalues, we change the propagator equation from $$U(t) = \sum_{E} \lvert E\rangle\langle E\rvert e^{-iEt/{\hbar}}\tag{4.3.13}$$ to $$U(t) = \sum_{E} \sum_\alpha \lvert E, \alpha\rangle\langle E, \alpha\rvert e^{-iEt/\hbar} $$ where $\lvert E, \alpha \rangle $ are the orthonormal eigenkets for the $E$ eigenspace. This makes sense to me, because it seems like whenever we deal with degeneracy to use a projection. He also says for a Hamiltonian with no degeneracy to change the sum into an integral, so I imagine that for a Hamiltonian with a continuous, degenerate spectrum we would have this propagator: $$ U(t) = \sum_\alpha\int_\Re dE\lvert E, \alpha\rangle\langle E, \alpha\rvert e^{-iEt/\hbar}$$  I tried applying this to the free particle and had some trouble. In the book, after solving for the eigenkets and eigenvalues, $\lvert E\rangle = \lvert p\rangle$ and $p = \pm\sqrt{2mE}$, he chooses the propagator as a function of the momentum eigenvalues $p$ so that $$U(t) = \int_\Re dp\lvert p\rangle\langle p\rvert \exp(-ip^2t/2m).\tag{5.1.9}$$ I understand how he got to this, but I'm having some trouble because this doesn't appear to be the same expression that I have above. Splitting the integral up, we have $$U(t) = \int_{-\infty}^0dp\lvert p\rangle\langle p\rvert \exp(-ip^2t/2m) + \int_0^\infty dp\lvert p\rangle\langle p\rvert \exp(-ip^2t/2m)$$ The left integral can changed to an integral over $-p$, and the bounds switch from $0$ to $\infty$ such that: $$U(t) =  \int_{0}^\infty d(-p)\lvert -p\rangle\langle -p\rvert \exp(-ip^2t/2m) + \int_0^\infty dp\lvert p\rangle\langle p\rvert \exp(-ip^2t/2m)$$ Now we can substitute and interchange $\lvert p\rangle = \lvert E, + \rangle$,  $\lvert -p\rangle = \lvert E, - \rangle$, $E = p^2/2m$, and $dp = \pm mdE/\sqrt{2mE}$ to get: $$U(t) = \sum_{\alpha = \pm} \int_0^\infty dEm/\sqrt{2mE}\lvert E, \alpha\rangle\langle E, \alpha\rvert e^{-iEt/\hbar}. $$ This is the result that we are asked to prove in Exercise 5.1.1. Why is this not the same as the first expression for the propagator? Where did the extra $ m/\sqrt{2mE} $ come from? Am I just wrong in assuming that that is the general expression for a Hamiltonian with continuous and degenerate eigenvalues? Prove $(1-a)(1-b)(1-c)(1-d)&gt;1-a-b-c-d$ and $ a,b,c,d$ are real and between 0 and 1.    I can do this with $$(1-a)(1-b)&gt;1-a-b  \\ 1-a-b+ab&gt;1-a-b  \\ ab&gt;0  $$  But with $c$ and $d$, this becomes much more difficult. Can someone nudge me in the right direction? I want to monitor private message and want to check and store at admin side. so admin can see who doing message to whom and make the policy for to sending the message. Get OS-level system information