The physical meaning of ${\rm Tr}()$ in quantum mechanics
What information does the trace of a matrix give?
Probability of $p(x_1<x_2<\max(x_3,x_4))$ How Can I prove (a+b) mod m = (a mod m) + (b mod m)) mod m? modular arithmetic  How Can I prove (a+b) mod m = (a mod m) + (b mod m)) mod m ? Intuition behind $k$-algebra, $k$-algebra morphisms? What's the fastest connection speed / shortest elapsed time using Migration Assistant? Does the shebang determine the shell which runs the script? How to make text aligned left/center/right in the same line? I am trying to align some text to left/center/right in the same line. For example, I want to put my phone number on the left, my name in the center, and my email on the right, how do I do that? Reading the Result of Traceroute When you use tracert www.example.com in the command prompt, what exactly does the returned information tell you, and how does it go about getting that information. Up until now I have only used traceroute to get the I.P. address of a website without really thinking about the bigger purpose of traceroute. Are the endings "-zation" and "-sation" interchangeable? What is with words that have forms that end both in -zation and -sation, such as localization and localisation?  Many spell checkers recommend -zation. Get variable from PHP to JavaScript I want to use a PHP variable in JavaScript. How is it possible? Why isn't an upgrade to 20.04 from 18.04 available yet? I want to use do-release-upgrade to upgrade from 18.04 to 20.04, but I can't do that do-release-upgrade Checking for a new Ubuntu release There is no development version of an LTS available.  It seems that  is used, and it still points to 18.04. Do you know when (or how?) can I upgrade to 20.04? How to randomize (shuffle) a JavaScript array? I have a High poly mesh and I have decided to make the low poly by merging as it is easiest for me. Would it be possible to change the W Hotkey , so that it merges to the center automatically? It is time consuming pressing W > Merge > Center. What is the biggest display of power through the Force in Star Wars? "Reality" of EM waves vs. wavefunction of individual photons - why not treat the wave function as equally "Real"? In thinking how to ask this question (somewhat) succinctly, I keep coming back to a Microwave Oven.  A Microwave Oven has a grid of holes over the window specifically designed to be smaller in diameter than the wavelength of the microwaves it produces, yet larger than the wavelengths of the visible light spectrum - this is so you can watch your food being heated without getting an eye full of microwaves.  The "realness" of electromagnetic waves seems indisputable - both from the microwave example above, and also because if I want to broadcast a radio wave with a certain wavelength, then I need to make sure I have an antenna of corresponding length to produce the wave I'm looking for.  Furthermore, we discuss and treat these waves as real, measurable "objects" that exist and can be manipulated.     Now, if I want to describe the behavior of my Microwave Oven in the framework of QM (let's pretend my oven is going to only produce 1 photon of energy corresponding to a wavelength of the microwave spectrum for simplicity) then I'll describe the behavior of that photon as a wavefunction that evolves over time and gives a probability distribution within my microwave that similarly does not allow the photon to pass through the safety grid and exit the oven cavity giving me a retinal burn.  The difference is, the wavefunction is never treated as something "real" in this description.  When the safety grid is described as working to protect you because it has holes smaller than the wavelength of the classical waves it's blocking, this is a useful description that seems to describe "real" objects/be-ables.  While it is possible to describe why an individual photon has low probability of passing through the same grid but extended physical properties such as wavelength (in space) are treated as non-real because we're dealing with a point particle and with behavior described by something we also treat as non-real (the wavefunction); it seems unclear to me why we insist this wave function which predicts behavior of physical measurements so well is somehow "non-real."    Put another way, if we have no problem treating EM waves as "Real," then why do we insist on treating the wavefunction that describes the same behavior as "unreal?"  I understand there is recent research (Eric Cavalcanti and his group for one) trying to argue this point, but as every respectable physics professor, I've ever encountered, has treated the wavefunction as an indisputably non-real mathematical tool, I needed to ask this community for an answer. Virtual Host not working as expected when I'm using .dev domain extension I'm doing a Laravel project and I need to set up a virtual host example.dev in my system. For that I've created a copy of 000-default.conf and named it as example.dev.conf and placed in /etc/apache2/sites-available. The contents of the file is given below (comments removed):  &lt;VirtualHost *:80&gt;     &lt;Directory /var/www/html/example/public/&gt;         Options Indexes FollowSymLinks MultiViews         AllowOverride All         Require all granted     &lt;/Directory&gt;      ServerName example.dev     ServerAlias www.example.dev      ServerAdmin admin@example.dev     DocumentRoot /var/www/html/example/public      ErrorLog ${APACHE_LOG_DIR}/error.log     CustomLog ${APACHE_LOG_DIR}/access.log combined &lt;/VirtualHost&gt;   Then I've enabled the site using a2ensite command and disabled 000-default.conf using a2dissite command. Then it will open my project when I try localhost (/var/www/html) in my browser and not example.dev. The virtual host example.dev always produce "The site can't be reached" error in browsers.  Any suggestions? Effective marketing strategies for independent game projects I would really like to hear how other independents generally approach marketing their game with limited resources. I know how critical marketing is since I've seen it first hand on several fronts. I've been part of teams that didn't put an ounce of marketing into their project and didn't come close to meeting expectations. I've also poured a lot of time into visiting forums to discuss our previous title which resulted in minor bumps over time.  I really think  is doing some amazing grass roots marketing with the way they are managing their community around their Blog. They've massed a following that is willing to pay (far in advance) them for their game before they finish it to help support them. That's awesome.  But how many people are really willing to divulge such information that early? Should that be a concern? Should I start talking about my next project before I really have anything implemented?   What has worked for you? I would be particularly interested to hear how people approach Apple's iTunes store. I believe we're looking at around 33,000 games on the store these days. Show default image when no uploaded image Let $\psi: C([0,1]; \mathbb{R}) \mapsto C([0,1]; \mathbb{R})  $, be such that if $x \in [0,1]$ then $$\psi(f)(x) = \int\limits_{0}^{x} f(t) dt $$  I am asked to show that $\psi$ is not a contraction but $\psi^{2}$ is. And then conclude that $\psi$ has a unique fixed point.   I´ve managed to prove a statment that helps me conclude the last part but I´m having trouble proving that the function is not a contraction. This is what I've got.   If $f,g \in  C([0,1]; \mathbb{R})$  then $$ |\psi (f) - \psi(g)| \leq \int\limits_{0}^{x} |f(t)-g(t)| dt \leq \sup\limits_{x\in [0,1]} |f(x)-g(x)| \int\limits_{0}^{x} 1 dt = d(f,g) x  $$ Then $|\psi (f) - \psi(g)| \leq d(f,g) x $,  and taking supremum on the left hand side allows me to conclude that $d(\psi (f),\psi (g)) \leq d(f,g) x$  Is there any problem with this procedure? Does anybody has an insight on how to procede with the $\psi^{2}$ part?   Many Thanks! I want to have a vector with a superscript:  \vec{o}^t   But this gives very bad results: the t is overlapping with the vector arrow. How do I fix this (and why does this give bad results in the first place) ? Show that $f(z)=0$. Suppose that $f:\mathbb{C}\rightarrow\mathbb{C}$ is analytic on the open unit disc and continuous on the closed unit disc. Assume that $f(z)=0$ on an arc of the circle $\{z\in\mathbb{C}:|z|=1\}$. Show that $f(z)=0$.