What functions satisfy $f \circ g = g \circ f\ne \text{id}$?
When functions commute under composition
Add an alert when answering a question that has at least a close vote as duplicate I have one feature-request with the objective of having duplicate questions with no answers:   To (try to) avoid people answering duplicate questions, add an alert that says the question has a probable duplicate if it has a close vote as Exact duplicate, asking the answerer to go to the original and answer there.   What do you think? My objective is letting the duplicate question exist for search purposes, but the answers centralized in the root question. Can we get x comment votes per y minutes, rather than one vote per 5 seconds? When we define the S-matrix, what are "in" and "out" states? I have seen the scattering matrix defined using initial ("in") and final ("out") eigenstates of the free hamiltonian, with  $$\left| \vec{p}_1 \cdots \vec{p}_n \; \text{out} \right\rangle = S^{-1} \left| \vec{p}_1 \cdots \vec{p}_n \; \text{in} \right\rangle$$  so that  $$\left\langle \vec{p}_1 \cdots \vec{p}_n \; \text{out} \mid \vec{q}_1 \cdots \vec{q}_m \; \text{in} \right\rangle = \left\langle \vec{p}_1 \cdots \vec{p}_n \; \text{in} \mid S \mid \vec{q}_1 \cdots \vec{q}_m \; \text{in} \right\rangle.$$  1) What are "in" and "out" states?   2) Are they Fock states?   3) In Schrödinger or Heisenberg or interaction representation?   4) How are they related? (I believe that I see what they handwavily represent physically, but not formally.)  My main issue is that, if "in" and "out" states are one-particle eigenstates of the free hamiltonian, i.e. if $\left| \vec{p}_1 \text{out} \right\rangle$ describes a free particle with momentum $\vec{p}_1$, and $\left| \vec{p}_1 \text{in} \right\rangle$ also describes a free particle with momentum $\vec{p}_1$, then $\left| \vec{p}_1 \text{out} \right\rangle = \left| \vec{p}_1 \text{in} \right\rangle$ ... which is false. Still, books (some at least) describe these "in" and "out" states like that.  Moreover, I have seen (e.g. in Wikipedia, but also ) that the scattering matrix is a map between two different Fock states, and I don't understand that.   5) Do states of the interacting system live in the same Fock space that asymptotic free states?   6) And if not, where do they live?  Understandable references would be appreciated. I think this story dates from the 1950s, plus or minus a decade - I read it about 30-35 years ago in a (then) fairly old story collection. It involves explorers on a planet falling into a gelatinous object that dissolves most of their bodies except for their brains and nervous systems. About four of them fall in to the same creature, and they discover that by thinking about it they can grow limbs and other body parts. At first they try to fight each other, by growing arms so they can hit each other's brains - and one of them realises that growing a skull would be a good idea - but later they agree to grow legs and move the creature around co-operatively. I would love to identify this story. Sakurai QM section 5.8 If anyone is familiar with Sakurai's book, specifically section 5.8 on energy shift and decay width, I am stuck and could use some help. I can't see how he derives 5.8.9 (in the revised edition). He starts with (3) (given below), and derives (1) and (2). The equation is as follows:$$(1){\space}{\space}\space{\space}\space{\space}{\dot c_i\over c_i}\approx{{-i\over \hbar}V_{ii}+{\left(-i \over \hbar\right)}^2{{{\left|V_{ii}\right|}^2}\over \eta}+\left({-i\over \hbar}\right) {\sum_{m\not=i}}{{{\left|V_{mi}\right|}^2}\over {(E_i-E_m+i\hbar\eta)}} \over 1-{i \over \hbar}{V_{ii} \over \eta}} $$ $$(2)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\approx{-i\over \hbar}V_{ii}+\left({-i\over \hbar}\right) {\sum_{m\not=i}}{{{\left|V_{mi}\right|}^2}\over {(E_i-E_m+i\hbar\eta)}}$$ where $$(3)\,\,\,\,\,\,\,\,\,c_i(t)=\lim_{\eta\to 0}\left({1-{i\over \hbar\eta}V_{ii}}{e^{{\eta}t}}+{{\left({-i\over \hbar}\right)}^2}{{\left|V_{ii}\right|}^2}{e^{2{\eta}t}\over 2{\eta}^2}+\left({-i\over \hbar}\right){\sum_{m\not=i}{{{{\left|V_{mi}\right|}^2}}\over 2\eta(E_i-E_m+i{\hbar}\eta}}\right)$$First of all, he discards all kinds of terms in the denominator of the first expression, and I can't see how he justifies this, because as ${\eta}\to0$, every term in the expression for $c_i(t)$ blows up (besides the 1 of course). Then he seems to multiply top and bottom by the complex conjugate of the denominator in (1), arriving at (2). He seems to discard a term from the denominator here, which is of second order in V, and says the equation is formally correct up to second order. This doesn't make sense to me! Also, soon after, he has this strange expression with $$\lim_{x\to 0} {1\over(x+iε)}=Pr.(1/x)-iπδ(x)$$ What is this Pr symbol? Any help would be much appreciated! A commutative ring is a field iff the only ideals are $(0)$ and $(1)$ Usually, when we see two letters put together it is to define the pronunciation or to differentiate synonyms or just its foreign origins. But some words don't seem to have any reason to double up.     Why does vacuum have two of the letter U?  Why does aardvark have two of the letter A?  Why does llama have two of the letter L? When building console applications that take parameters, you can use the arguments passed to Main(string[] args).   In the past I've simply indexed/looped that array and done a few regular expressions to extract the values. However, when the commands get more complicated, the parsing can get pretty ugly.    So I'm interested in:   Libraries that you use Patterns that you use   Assume the commands always adhere to common standards such as . How to use San Francisco font on Windows? I just proved that any finite group of order $p^2$ for $p$ a prime is abelian. The author now asks to show that there are only two such groups up to isomorphism. The first group I can think of is $G=\Bbb Z/p\Bbb Z\oplus \Bbb Z/p\Bbb Z$. This is abelian and has order $p^2$. I think the other is $\Bbb Z/p^2 \Bbb Z$.   Now, it should follow from the fact that there is only one cyclic group of order $n$ up to isomorphism that these two are unique up to isomorphism. All I need to show is these two are in fact not isomorphic. It suffices to show that $G$ as before is not cyclic. But this is easy to see, since we cannot generate any $(x,y)$ with $x\neq y$ by repeated addition of some $(z,z)$.  Now, it suffices to show that any other group of order $p^2$ is isomorphic to either one of these two groups. If the group is cyclic, we're done, so assume it is not cyclic. One can see that $G=\langle (1,0) ,(0,1)\rangle$. How can I move on? How to get rid of OTA software update notification? I upgraded my Nexus 4 to Lollipop a few days ago but I don't like it. I'll leave out my reasons because they're not relevant here. I wiped my phone and installed a clean stock 4.2.2 version instead. I'm happy with this.  Not long after, Android automatically downloaded the 4.3 upgrade and notifies me with the choices "Later" and "Install". I don't want to do this upgrade.  How do I:   get rid of the notification, and  remove the downloaded update (to free up precious storage space).     I have already tried choosing "Later" but the same notification immediately reappears.    Based on Google search results, I also tried going to Settings &gt; Apps &gt; All &gt; Google Services Framework and unticked "show notifications" but it still appears. Force stop or disable also doesn't seem to help, so I've re-enabled the service including notifications. I didn't see any other apps in this list that sounded relevant - did I just not see it? When you are writing Selenium/WebDriver automation scripts you will probably run into the exception NoSuchElementException. You think your selector is working, but somehow the element cannot be found.  What strategy should I take to analyse different reasons that this might be the case?   What should I research and how? Good answers would contain example situations and how to resolve them. (e.g. verify selectors, frames, waiting, etc) I'm still a newbie with Linux but I'm trying to learn. I have two USBs, one is 4GB and has the ISO install of Ubuntu (I think it's called live, the one with maximum of 4GB of permanent storage) and got a 64gb usb to make a permanent full install on it. I installed it following  tutorial and everything looks fine (I had to create the EFI partition with the Terminal though) but it just won't boot, the live install boots without problems.   I'm using the school's computers which have the UEFI menu password protected so I can't access that and I won't have a laptop for like 3 weeks (main reason why I'm doing this). I tested it with a friend's laptop disabling the Secure boot but it still wouldn't boot. I also saw that some tutorials don't add the EFI partition so I tried without that one and still won't work. And I know it's not the 64GB usb not booting because I had a live install on it too before and it booted fine.  Please, I have been trying to make it work for two days now, any ideas? I can barely understand this stuff so if you could explain step by step that would be awesome. This poor creature is taken aboard the spaceship and the humans do not know that it is intelligent.  The alien can repeat what it hears, e,g, "pass the sugar".  It is able to escape at night and it communicates messages to its fellow aliens.  It directs that they turn up the heat of the atmosphere (somehow they can do this) so as to force the humans to leave.  The alien protagonist states that it will die w/ the humans or leave with them. Sum of the series $1+\frac{1\cdot 3}{6}+\frac{1\cdot3\cdot5}{6\cdot8}+\cdots$ Is there a word for people who always act too friendly in order to curry favor with their acquaintances? Is there a single noun or adjective for a person who is always flattering friends and acquaintances, not only to be liked and accepted but, at other times, to curry favor with them as well ?  I'm not looking for regionalisms but a word that will be understood in any English speaking country. Sneak Attack has the same type , this is easy, unless there are multiple damage types.  A Flame Tongue does slashing (or piercing) damage, and fire. Can I chose the Sneak Attack damage in this case to be fire too? Compiling LaTeX document empties .bbl file I'm trying out biblatex for the document I'm compiling, using the biber backend on my miktex distribution. Everything, including biber, should be up to date, assuming the MiKTeX repository currently holds the latest versions of everything.  This is what the biblatex documentation (p. 101) says about how often I have to run biber to update my .bbl file:     Whenever a reference to a work which has not been cited before is   added, this procedure must be repeated. This is also the case if the   last reference to a work which has been cited before is removed   because some citation labels may change in this case.   So if I make changes to my document (which is a very common activity in the process of writing a document), but leave the citations unchanged, I should not have to re-run biber before compiling the changes. However, every time I compile my document, regardless of the compiler or documentclass, the .bbl file is emptied. Consequently, the next time I compile the document without running biber, the bibliography is removed (seeing as the .bbl file contains nothing), and all my citations are broken. For this reason, I have to run biber before every time I compile the document, which largely counters the benefit of externalizing the bibliography in the first place.  How can I make the compilers stop overwriting my .bbl file and make everything behave as described in the biblatex manual?  Upon investigating the contents of the auxiliary files, I found that the .bbl file (after running biber, before compiling) begins with the following lines:  % $ biblatex auxiliary file $ % $ biblatex bbl format version 2.3 $   Does the biblatex version 2.8 still mark the .bbl files as version 2.3? If not, could this be the problem? And if so, why does my 2.8 version of biblatex (partly) behave as version 2.3?  Any help solving this problem would be much appreciated. Progressive Draw 2 and Draw 4 Uno When playing with progressive Draw 2 and Draw 4 and (say that there are 2 players playing) and    Player 1 goes out with a Draw 2  Player 2 puts down a Draw 2    Does player 1 have to pick up 4 cards? Which spells or abilities are affected by things that care about targeting? Note: this is intended to be a canonical question about whether or not a spell or ability targets so there are multiple sub-questions.   Can Hexproof or Shroud save my creature from ? What about a  that was cast for it's Overload cost? Does shroud prevent me from using the Monstrosity ability of a creature, or from giving it more counters with Proliferate? Can I cast an Aura Enchantment on one of my creatures with Shroud, or an opponent's creature with Hexproof? Or can I use the Equip ability to attach an equipment to one of my creatures with Shroud? Will 's ability trigger when my opponent plays ? And will I have to sacrifice my  if my opponent casts ? Will Shroud or Hexproof prevent an opponent from choosing me with 's ability? Will Hexproof keep one of my creatures from being affected by ? If an Aura or Equipment is attached to a creature with Hexproof, does the Hexproof protect the attached permanent?