I found this expression for the time-evolution operator: $$\begin{split} U(t) &amp; = T_{\leftarrow}\exp\left[-i\int_0^t ds H(s)\right] \\ &amp;= \exp\left[-\frac{1}{2}\int_0^t ds\int_0^t ds' [H(s),H(s')]\theta(s-s')\right]\exp\left[-i\int_0^t ds H(s)\right]\end{split}$$ with $H$ Hamiltonian and $\theta(s-s')$ the Heaviside step function. The commutator of the Hamiltonian at two different times is a c-number function, I don't know if it's important. Does anyone know how to get the last part?
I have a question concerning the time ordering operator. Let's suppose we have a time evolution generated by some Hamiltonian $H(t)$ given by $$ U(t)=T_\leftarrow\exp\left(-\mathrm{i}\int_0^t\mathrm{d}s\,H(s)\right)\tag{1}. $$ In Breuer and Petruccione, it's said that if the commutator of the Hamiltonian at some time $t$ with itself at some other time $t^\prime$ is a $c$-number function/ a complex function, i.e. $$\left[H(t),\,H(t^\prime)\right]=f(t,t'),$$ then the time evolution is given by $$ U(t)=\exp\left(-\frac{1}{2}\int_0^t\mathrm{d}s\int_0^t\mathrm{d}s^\prime\,\left[H(s),\,H(s^\prime)\right]\Theta(s-s^\prime)\right)$$ $$\times \exp\left(-\mathrm{i}\int_0^t\mathrm{ds}\,H(s)\right),\tag{2} $$ where $\Theta(s-s^\prime)$ is the Heaviside function. No proof or reference is given and I could not find any explanation anywhere, this is why I ask here. Any help would be very appreciated.
A short story about finding an inhabitable planet and meeting aliens So the story goes, humans in a spaceship search for an inhabitable planet, being surprised they found none in the vicinity of Earth. They find one, and it seems uninhabited. At the same time are observed by the aliens, who go down on the planet to meet them. The aliens sit them down and tell them a story about war and conflict that ruled the galaxy, and how all the sentient races in the galaxy got together and got rid of the race that tried to conquer all by blowing up their planets.  In the end you realise man is this terrible conqueror the aliens are talking about. MacBook Pro Trackpad Suddenly Stopped Working Completely The trackpad on my Macbook Pro of 2 years has suddenly stopped working out of blue.  The trackpad doesn't move, respond to any gestures nor do any clicks work. 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Bash scripts ran from from gnome/nautilus don't have environment variables In my ~/.bashrc and ~/.profile I have some variables set to some filepaths:  export MY_VAR1="/path/to/somewhere"   In my ~/bin/ (which is added to my PATH) I have a few scripts that rely on those variables being set.  If I run those commands from the terminal, they work as expected.  However, if I try to double click on a script in Nautilus and run it, it runs as if those variables were not set.  Same thing happens if I add one of the scripts to Gnome's "Startup Applications" using the GUI utility or if I add them to the Gnome menu using the "Main Menu" GUI utility.  I would've thought that since I included these variables in my ~/.profile, which is run at login that gnome and nautilus and processes spawned by them would have access to those variables.  Both Gnome and Nautilus are run under my login user. All my scripts in ~/bin have the bash shebang at the top.  I am new to bash scripting, sorry if I am missing something obvious. I am using Debian Wheezy and Gnome3. Does a creature with Lifelink provide lifegain equal to its power, or the defender's toughness? I have recently began playing a deck that has a number of creatures with lifelink in it.  I want to be certain I understand the mechanic and the following question occurred to me.  Which damage total do you use to determine the life gain from lifelink?  I had assumed that if  a 3/3 creature with lifelink was blocked by a 2/2 creature that the life gain total would be 2, the maximum that the blocker can absorb before dying.  This seems to be incorrect, as when I read the example attached to CR 119.4c      [...] attacks with a 3/3 creature with wither and lifelink. It’s blocked by a 2/2 creature, [...]. The damage event starts out as [3 damage is dealt to the 2/2 creature, 2 damage is dealt to the 3/3 creature]   Have I been short changing myself? I'm using a PIN code to log on my computer. However, Num ⇩ is always turned off, even if on before shutting down my computer. This is quite annoying as I always have to retype my PIN.  I've tried the suggestions  (same as ). But with no result. If I don't enable num lock on log in screen and use the numbers above the normal keys, num lock turns on after log on. Then it's too late, unfortunately.  So, how can I make num lock enabled by default when logging in? Thanks.  Edit: While I still want to solve this, I've figured it is not such a big problem. I will need to press some key for the "enter PIN screen" to show, and using Num ⇩ works and enables num lock. But then, still annoying if the computer was only locked (with num lock on) and I turn it off, hehe. Number agreement between subject and object How do I re-authorize Quicktime Player to use the microphone when once it has been denied? I wanted to use a USB audio input to record a video, and got something I've never seen before having done this 100 times; it asked "Authorize Quicktime to use the microphone?" I clicked no, because I didn't want it to use the internal mic, I wanted it to use a USB input.  Now, it won't let me get rid of the denial.  I've quit QT, quit &amp; reopened it twice and it still won't let me choose ANY input source. It still says it doesn't have access to the microphone.  Is this a new Mojave 10.14.2 bug or what?  How the heck do I get out of this?  Thanks! I just tried to access a folder like so:  \\somecomputeronmynetwork\somelocation$   When going to this location I'm prompted for a user name and password.  I put one in, and it let me in fine.  Now I need to remove that login, so I can try a different user name and password.  What's the easiest way to do this? Find all polynomials $P(x)$ such that $P(x^2)=P(x)^2$ Find all polynomials $P:\mathbb{C}\rightarrow\mathbb{C}$ such that $$P(x^2)=P(x)^2 .$$   Here is what I tried:  First, it is easy to see the constant solutions, namely $P\equiv 0,P\equiv 1$.  Let $r$ be a root of $P$ (i.e. $P(r)=0$). It follows that all terms in the infinite sequence $r,r^2,r^4,\dots,r^{2n},\dots$ are roots of $P$. To avoid a polynomial with infinite roots, we have to have the sequence be periodic. So it turns out that either $r=0$, or the roots of $P$ are roots of unity of degree $2n$ for some $n\in\mathbb{N}$.  In the first case, we get $P(x)=xQ(x)$. Plugging in, we get $$x^2Q(x^2)=x^2Q(x)^2\implies Q(x^2)=Q(x)^2 ,$$ i.e. that $Q$ satisfies the same condition as $P$.  In the second case, we have $P(x)=(x^{2n}-1)Q(x)$. Plugging in, we get $(x^{2n}-1)Q(x^2)=(x^{2n}-2x^n+1)Q(x)^2$. I was not sure how to proceed from here. Roots of $\,ax^2+bx+c\,$ are irrational if $a,b,c$ are odd integers How would I show the following.  For all odd integers a,b, and c if z is a solution of $ax^2+bx+c=0$ the z is not rational.  How would I show this? This is what I did  Let a be an odd integer then for a integer k it has the form $a=2k+1$  then $b=2t+1$ for if c integer.  then $c=2r+1$ if s is an integer then I do  $(2k+1)(x)^2+2t+1(x)+(2r+1)=0$  But I find my self stuck on what to do any hint are appreciated. How can I check whether a variable is defined in JavaScript? How to check whether a JavaScript variable defined in cross-browser way?  I ran into this problem when writing some JavaScript utilizing FireBug logging. I wrote some code like below:  function profileRun(f) {     // f: functions to be profiled     console.profile(f.constructor);     f();     console.profileEnd(f.constructor); }   It works fine in FireFox/FireBug, but it reports error in IE8 RC1. So, I'd like to do some checking whether console variable exists in the execution environment.  Below code works fine in FireFox, but not in IE8 RC1.   function profileRun(f) {     if (console != undefined) {         console.profile(f.constructor);     }      f();      if (console != undefined) {         console.profileEnd(f.constructor);     } }   However, if I do it this way. It works in IE8 RC1. Why?   function profileRun(f) {     if (window.console != undefined) {         console.profile(f.constructor);     }      f();      if (window.console != undefined) {         console.profileEnd(f.constructor);     } }   Is there any cross-browser way to check it? I've used recursion quite a lot on my many years of programming to solve simple problems, but I'm fully aware that sometimes you need iteration due to memory/speed problems.  So, sometime in the very far past I went to try and find if there existed any "pattern" or text-book way of transforming a common recursion approach to iteration and found nothing. Or at least nothing that I can remember it would help.   Are there general rules? Is there a "pattern"? Do I need a transit visa for a layover in the United Arab Emirates? Do I need a transit visa for a layover via airports in the UAE, such as Dubai International Airport or Abu Dhabi International Airport? If so, what are the requirements for such a visa? Can I get the transit visa on arrival? Learning special relativity and working with Minkowski diagrams for a while now i am still trying to get my head around some oddity.   From my understanding time can often be seen as the dependent variable in those diagrams, so why put it to the vertical axis? What confuses me even more is the apparent convention to denote a point in space time $(t,x)$ even though $x$ is on the horizontal and $t$ is on the vertical axis, so the usual notation would be to refer to that vector as $(x, t)$.  Are there mostly historical reasons or why is it beneficial to use this notation and to violate usual conventions? How can I explain to people that the phrase “off of ” is grammatically incorrect? How can I explain to people that the phrase off of is grammatically incorrect?  I‘ve heard this phrase used a lot, especially by Americans (though they aren't the only ones).    In my understanding, off of should usually be replaced by off, as in,     I took the book off the shelf   as opposed to     I took the book off of the shelf   Am I wrong?  Or is there perhaps some simple way I can explain this to most people? Can someone give me an example of a function $f: I\to \mathbb R$ which is uniform continuous but not Hölder continuous (for any $\alpha$)? 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Understanding the formula for the Confidence Interval of the difference between two sample proportions