Invariance of maximally entangled state
Invariance of a maximally entangled state under unitary operation $U \otimes U^\dagger$
Repeat String - Javascript What is the best or most concise method for returning a string repeated an arbitrary amount of times?  The following is my best shot so far:  function repeat(s, n){     var a = [];     while(a.length &lt; n){         a.push(s);     }     return a.join(''); } I might want to answer my own question because:  I arrived at a solution before there were any other correct answers or I like mine better, can I still answer my own question?  I have a folder with lists of code snippets for things I always forget or get wrong, such as regular expressions for processing XML, common WinForms control idioms, etc. Is it considered poor etiquette for Stack Overflow to ask a question I already know the answer to, and then self answer, just so it gets recorded on the website?   Related:          For more information, see &quot;&quot; in the . How do I show that $a^p + b^p > (a + b)^p$ for $a,b>0$ and $0<p<1$? A question on my math homework asks us to show that if $0 &lt; p &lt; 1$ and $a, b &gt; 0$, then $a^p + b^p &gt; (a + b)^p$. I have no idea how to do this, any pointers? How do you refer to a hyponym that is the same word as the hypernym? What word (or how do you phrase things) do you use when the ostensible word for the class is the same as the word for a subset of the class?  For example, in the United States, there are many brands of sugared, carbonated soft drink that one can buy: Coca-Cola (or Coke), Pepsi, RC Cola, Sprite, 7-Up, A&amp;W or Mug Root Beer, etc). The first two are kinds of 'coke' or 'cola' (but no one says 'cola').  , one asks for any type of such drink as 'coke'. How does one then ask for the particular kind of 'coke' that is 'Coke' (pronounced the same)? Does one reduplicate and say a 'Coke coke', does one repeat with emphasis (or without), does one use a different word like 'coca-cola' (co-cola), or what?  The (perceived) difficulty (surely those in the South are able to get the drink they want somehow) isn't limited to there. Where the drink is called 'soda', if you ask for a 'coke', you'll still sometimes have the choice among Coca-Cola, Pepsi, etc. "Can I have a Coke?" the response might be "Is Pepsi OK?". "Sure, Pepsi, Coke, whatever." You don't use 'cola', you just ask for the particular kind and hope the listener will help differentiate.   Given these multiple strategies for disambiguating, what do you do say to get a 'Coke' in the southern US? are there any other examples of a hypernym being the same as the hyponym, and so creating a disambiguation problem? Should multiverse theories be rebranded as meta physics? I answered a question here with a quote with the following quote from Cosmologist George Ellis.     “Similar claims [about a multiverse] have been made since antiquity by many cultures. What is new is the assertion that the   multiverse is a scientific theory, with all that implies about being   mathematically rigorous and experimentally testable. I am skeptical   about this claim. I do not believe the existence of those other   universes has been proved—or ever could be. Proponents of the   multiverse, as well as greatly enlarging our conception of physical   reality, are implicitly redefining what is meant by ‘science.’”—pg 39      “For a cosmologist, the basic problem with all multiverse proposals is the presence of a cosmic visual horizon. The horizon is   the limit to how far away we can see, because signals traveling toward   us at the speed of light (which is finite) have not had time since the   beginning of the universe to reach us from farther out. All the   parallel universes lie outside our horizon and remain beyond our   capacity to see, now or ever, no matter how technology evolves. In   fact, they are to far away to have had any influence on our universe   whatsoever. That is why none of the claims made by multiverse   enthusiasts can be directly substantiated.”—pg 40-41      “A remarkable fact about our universe is that physical constants have just the right values needed to allow for complex structures,   including living things. Steven Weinberg, Martin Rees, Leonard   Susskind and others contend that an exotic multiverse provides a tidy   explanation for this apparent coincidence: if all possible values   occur in a large enough collection of universes, then viable ones for   life will surely be found somewhere. This reasoning has been applied,   in particular, to explanation the density of the dark energy that is   speeding up the expansion of the universe today. I agree that the   multiverse is a possible valid explanation for the value of this   density; arguably, it is the only scientifically based option we have   right now. But we have no hope of testing it observationally.”—pg 42      “All in all, the case for the multiverse is inconclusive. The basic reason is the extreme flexibility of the proposal: it is more a   concept than well-defined theory. Most proposals involve a patchwork   of different ideas rather than a coherent whole. The basic mechanism   for eternal inflation does not itself cause physics to be different in   each domain in a multiverse; for that, it needs to be coupled to   another speculative theory. Although they can be fitted together,   there is nothing inevitable about it. … Nothing is wrong with   scientifically based philosophical speculation, which is what   multiverse proposals are. But we should name it for what it is.”—pg 43   Now with this in mind if these theories cannot be independently verified should their truth not lie in  method outside physics or what is also known as meta physics? Where can I ask Android related questions in Stack Exchange? I would like to know where I can ask Android apps and other mobile related questions. Is there a Stack Exchange site for such questions? Proving that the sequence $F_{n}(x)=\sum\limits_{k=1}^{n} \frac{\sin{kx}}{k}$ is boundedly convergent on $\mathbb{R}$ Here is an exercise, on analysis which i am stuck.    How do I prove that if $F_{n}(x)=\sum\limits_{k=1}^{n} \frac{\sin{kx}}{k}$, then the sequence $\{F_{n}(x)\}$ is boundedly convergent on $\mathbb{R}$? Let $\kappa, \lambda,\theta$ be infinite cardinals of $M$ where $M$ is a transitive model of ZFC. Let $\mathcal{P}=Fn(\kappa\times \omega,2)$ (finite functions from $\kappa\times \omega$ to 2) and G is a (M,P)-generic set. I want to show $(\lambda^\theta)^{M[G]}=(max(\kappa,\lambda)^\theta)^M$. One direction is easy $(\lambda^\theta)^M \leq (\lambda^\theta)^{M[G]}$ and $(\kappa^\theta)^M\leq (\kappa^\theta)^{M[G]}\leq ((2^\omega)^\theta)^{M[G]}=(2^\theta)^{M[G]}\leq (\lambda^\theta)^{M[G]}$ where $(\kappa \leq 2^\omega)^{M[G]}$ is by the forcing construction. But now I am not quite sure how to get the other direction. Any thoughts? The Frobenius Coin Problem If the laws of random variables weakly converge to a point mass at $c$, then the random variables converge in probability to $c$. This is question 10.3.1 from Rosenthal's First Look at Rigorous Probability.  Suppose $\mathcal{L}(X_n) \Rightarrow \delta_c$ for some $c \in \mathbb{R}$. Prove that $\{X_n\}$ converges to $c$ in probability.  Here $\mathcal{L}(X_n)$ refers to the distribution of $X_n$.  Is the following proof correct?  Is a simpler proof available?  Is there a simple proof using Skorohod's theorem?  I attempted to prove the contrapositive. Suppose $X_n \not \to c$ in probability.  This means that there is an $\varepsilon &gt; 0$ so that for all $N$, there exists an $m &gt; N$ such that  \begin{equation}   P(|X_m - c| \geq \varepsilon) &gt; \eta, \text{ for some $\eta &gt; 0$}. \end{equation}  Consider  \begin{equation}   B := \{x \in \mathbb{R} : |x - c| \leq \varepsilon\} \in \mathcal{B} \text{ the Borel sets on $\mathbb{R}$}. \end{equation}  Then $B^c \in \mathcal{B}$ and denoting the distribution of $X_n$ as $\mu_n$ we have  \begin{equation}   \mu_n(B^c) = P(X_n \in B^c) = P(|X_n - c| \geq \varepsilon) &gt; \eta. \end{equation}  For all $N$, there exists an $m &gt; N$ such that the cumulative distribution function of $X_m$ is not the same as the cumulative distribution function of $\delta_c$; in particular it's mass changes somewhere at least $\varepsilon$ distance from $c$. Finally because $\delta_c(\{x\}) = 0$ for all $x \neq c$ this completes the proof. Highlight \verb in inline text Simplest Light triggered relay I am building a large array of Light Switched relays, and due to the size of the array i want to keep the part count down.  Each 'Pixel module' has a 12V SPDT, bulb and an activate-on-dark circuit. There is a strong spotlight on the device and the viewers shadow is obscuring the light in their shadow so the swing is large.  I have seen the obvious LDR-and-Pot circuits with a junction out to a NPN Transistor, and the more stable comparator alternative  I was looking at "solar cell" types of Photodiodes (PiN Type), and I can't see why I cannot just lay one across a PNP transistor (Collector to Base?) or something so it applies a Base voltage when lit up.  Can someone briefly explain to me why I am wrong?  I figure it would be all over the net if it was that easy, and i cant find any examples!  Is this possibly the cheapest way to make a light switched 12V relay driver?  Note: I prefer solutions using discrete circuitry, as I have a thing for human readability, visual components etc :P Quad tree vs Grid based collision detection OpenLayers Plugin Error: latitude or longitude exceeded limits I'm aware the 'traceback' OpenLayers plugin error has been dealt with before, so apologies for raising the issue once again. Unfortunately for me, the solutions posted were way over my head, e.g. using workarounds and code posted on github, etc. I find it all mind-boggling.  If anyone could post a step by step answer for an absolute beginner like me I would be eternally grateful! Thank you. I hope that one day I can contribute something.  The error message reads:   An error has occured while executing Python code:  Traceback (most recent call last):   File "C:/Users/MCMA1580/.qgis//python/plugins\openlayers_plugin\openlayers_plugin.py", line 48, in addLayer     self.__plugin.addLayer(self)   File "C:/Users/MCMA1580/.qgis//python/plugins\openlayers_plugin\openlayers_plugin.py", line 190, in addLayer     self.__setMapSrsGoogle()   File "C:/Users/MCMA1580/.qgis//python/plugins\openlayers_plugin\openlayers_plugin.py", line 249, in __setMapSrsGoogle     extMap = coodTrans.transform(extMap, QgsCoordinateTransform.ForwardTransform) QgsCsException: forward transform of (0.884883, 2.549197) PROJ.4: +proj=longlat +datum=WGS84 +no_defs +to +proj=merc +a=6378137 +b=6378137 +lat_ts=0.0 +lon_0=0.0 +x_0=0.0 +y_0=0 +k=1.0 +units=m +nadgrids=@null +wktext  +no_defs Error: latitude or longitude exceeded limits With all the various races in the federation, how do the humans manage to dominate it? Wait until all jQuery Ajax requests are done? What are the transactional costs when buying and selling a house? From question . You have to use \r when replacing text for a newline, like this  :%s/%/\r/g   But when replacing end of lines and newlines for a character, you can do it like:  :%s/\n/%/g    What section of the manual documents these behaviors, and what's the reasoning behind them? Gamma Distribution out of sum of exponential random variables The script works fine if I run it with sh filename, but it ignores all the --exclude parameters if its run by cron:  #!/bin/sh TIMESTAMP=$(date +%u)  WEEK=`date +"%V"` if [ $(($WEEK%2)) -eq 0 ]; then     echo "even";     BACKUP_DIR="/mnt/backup/2_$TIMESTAMP"     mkdir -p "$BACKUP_DIR/www"  else     echo "odd";     BACKUP_DIR="/mnt/backup/1_$TIMESTAMP"     mkdir -p "$BACKUP_DIR/www"   fi  if [ $(date +%u) -gt 6 ]; then     zip -r -y $BACKUP_DIR/www/web.zip /var/www/ --exclude='var/www/\.opcache/*' --exclude='var/www/utils/phpThumb/cache/*' else     zip -r -y $BACKUP_DIR/www/web.zip /var/www/ --exclude='var/www/mycompany/files/*' --exclude='var/www/\.opcache/*' --exclude='var/www/mycompany/szamlazz/xml_szamlak/*' --exclude='var/www/mycompany/files_admin/szamlazz/*' --exclude='var/www/mycompany/files_admin/chatFiles/*' --exclude='var/www/mycompany/utils/phpThumb/cache/*' --exclude='*/backup-guard/*' --exclude='var/www/mycompany/mailer/*' --exclude='var/www/mycompany/attachments/*' fi  exit 0   The above script should create a larger backup on sunday, and a partial on weekdays, but it just zips everything.