Explanation of two slit experiment Whilst going through 'a brief history of time' ,I faced difficulty in understanding the two slit experiment. How can individual electrons cause fringes?
How does the two slit experiment work? Whilst going through 'A brief history of time', I faced difficulty in understanding the . How can an individual electron cause fringes on screen? I was unable to understand it? Please explain it simply.
A locally constant sheaf on a locally connected space is a covering space; Proof? Reciprocal of Holder's inequality Does anyone know how to do this exercise? I couldn't do it.  Let $(X,S,\mu)$ be a $\sigma$ -finite measure space and $g\in M(X,S)$ such that $gs\in\mathcal{L}_1$ for any simple function $s\in\mathcal{L}_p$, $p\in(1,\infty)$. Suppose there is $A\geq0$ such that $\left|\int gs\,d\mu\right|\leq A||s||_p$ for all $s$. Show that $g\in\mathcal{L}_p$ with $\frac1p+\frac1q=1$ and $||g||_q\leq A$. Do accepted answers still serve a useful purpose? In the Stack Exchange voting system, we have two classes of votes: standard up / down votes that anyone can cast, and special  that only the asker can.   Although accepting an answer does not always pin it at the top of the answer list, that green check mark does  about the correctness of the answer. This gives a single person much greater weight than the rest of the community. That green check mark can seem confusing to visitors, who see  or  answers marked as being correct. I've heard this from several friends recently who were surprised when they came across factually wrong answers that had been downvoted, yet still had that check mark next to them.  Moderators regularly field flags from even more experienced users who demand that the accepted answer be changed from one that is incorrect or even harmful to another. We decline these, because we're not supposed to be making technical judgments like that and because we have no way of accepting answers on behalf of the asker. However, these flags do show that some people have problems with the presentation of accepted answers vs. the community voting on all the others.  There's also all the fun that came with arguments over accept rate, and people badgering askers to accept their answers.   Questions on the Stack Exchange network are intended to be broadly useful, to help out many others in the future beyond the person asking the question. If that's our goal, why should the person asking the question have any more say than the rest of the world? If it's a broadly applicable question, why should one person be able to highlight something as correct simply because they asked it first?  I guess what I'm asking is: if we did away with accepted answers, converting them to upvotes where appropriate, and only let community voting sort out the best answers, would we lose something important? I thought I'd ask after reading  to , because I wasn't sure how I felt about accepted answers anymore. "Please do not repeat the mistake" and "Please do not repeat the mistake again" Please do not repeat the mistake. Please do not repeat the mistake again.  Should the again be omitted? “Repeat” means “do again”. So to “repeat again” means “to do again again”. Is the second sentence grammatically wrong? You should choose your Display Name when signing up Is there a better way to do optional function parameters in JavaScript? I've always handled optional parameters in JavaScript like this:  function myFunc(requiredArg, optionalArg){   optionalArg = optionalArg || 'defaultValue';    // Do stuff }   Is there a better way to do it?  Are there any cases where using || like that is going to fail? I'm planning to buy a new Android phone and I'd like to be able to transfer the progress I have made at several games to the new device, so I won't have to play all the levels again from scratch.  I currently have an HTC Desire and will buy an HTC Evo 3D.  There's probably no simple/general solution to this.  But I was wondering if anyone did something similar and could provide some advice (e.g. where to look for files that might hold game progress information). How can I rename many files at once? I need to rename a bunch of files in order to fix their sort order, therefore I need to be able to do a sort of "find and replace" so I can replace chunks of the filenames at a time.  What tools should I use? I prefer GUI but a command line tool recommendation would also be fine. Baby Rudin claim: $1+\frac{1}{3}-\frac{1}{2}+\frac{1}{5}+\frac{1}{7}-\frac{1}{4}+\frac{1}{9}+\frac{1}{11}-\frac{1}{6}...$ converges This sequence is a rearrangement of the series $1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}...$. Note that at this point in the text we do not have any theorem about the convergence of rearrangements.  Let $\{s_n\}$ be the sequence of partials sums of the series then for $n \ge 0$  $s_{3(n+1)} = \sum ^n _ {k=0} \frac{1}{4k+1} + \frac{1}{4k+3} - \frac{2}{4k+4}$  We can view it as the sequence(on $n$) of partials sums of  $\sum_0 a_n = \sum_0 \frac{1}{4n+1} + \frac{1}{4n+3} - \frac{2}{4n+4}$  Where $|a_n| = a_n = \frac{1}{4n+4}\{\frac{3}{4n+1}+\frac{1}{4n+3}\} \le \frac{1}{4n^2}$.  By the comparison test $s_{3(n+1)}$ converges to some real $\alpha$.  But $s_{3(n+1)+1} = s_{3(n+1)}+ \frac{1}{4n+5}$ and $s_{3(n+1)+2} = s_{3(n+1)}+ \frac{1}{4n+5}+\frac{1}{4n+7} $hence we have a partition of $\{s_n\}$ into subsequences which tend to $\alpha$ and this implies $s_n \rightarrow \alpha$.  Is my proof correct? Any alternative solutions are appreciated. Let us view our deleted comments Consider following structure:  . |_s |_s  |_ss  |_ss+   |_sss  |_ss |_s |_s   How can I demote a subsection to subsubsection without changing markups ?  . |_s |_s  |_ss   |_sss+    |_p  |_ss |_s |_s   How can I promote a subsection to section without changing markups ?  . |_s |_s  |_ss |_s+  |_ss  |_ss |_s |_s   A wrapper command would be excellent  \begin[+1]{level} \section{} \end{level} Moyal Product in Non Commutative Quantum Mechanics Can someone please explain me what is a ?  Also, how does putting $$X_a(\psi) ~=~ x_a\star\psi$$ realise $$[X_a,X_b]=i\theta_{ab}{\bf 1}?$$   Ref: How does the simulation hypothesis relate to the ethics of simulated violence? Say, for the sake of this question,  is real. Are we then hurting "persons"/conscious beings in "our simulations" - video games - when we fight them? Why (not)? If so, is this unethical? Think for example about shooting an alien in the game No Man's Sky.  Edit/answer on the comment of Joseph Weissman: My reasoning is like this:   It is unethical to harm a person (with a person being everything that has ). By the simulation hypothesis (assuming it's true), everything around us (everything that we know of), is simulated. By 1 and 2, all persons we know of are simulated. So it's unethical to harm the simulated persons around us. A game is a simulation just like we are in a simulation, so the game has simulated persons. Therefore -  - it's unethical to harm the persons in a video game.   I know that the premises - besides the simulation hypothesis itself - are debatable, but that's the reason I asked the question in the first place.  I was also thinking that the characters in the video game differ because they have far less to no intelligence, that much of it is scripted. But I neither have a proof that we ourselves aren't scripted, with free will being an illusion. Also, how intelligent does a character in a simulation/video game has to be to have personhood then? I think this is a case of the .  Just to be sure, please note that this question is not about whether the simulation hypothesis is true or how likely it is to be true. It is about (one of the) consequences if the simulation hypothesis is true. It's also not a question about violence in games in general. Is there a simple, constructive, 1-1 mapping between the reals and the irrationals? Is there a simple, constructive, 1-1 mapping between the reals and the irrationals?  I know that the Cantor–Bernstein–Schroeder theorem implies the existence of a 1-1 mapping between the reals and the irrationals, but the proofs of this theorem are nonconstructive.  I wondered if a simple (not involving an infinite set of mappings) constructive (so the mapping is straightforwardly specified) mapping existed.  I have considered things like mapping the rationals to the rationals plus a fixed irrational, but then I could not figure out how to prevent an infinite (possible uncountably infinite) regression. Is it possible to change the linkage of Nether portals? I have 3 portals that link together, but I want to remove at least the middle one.  Edit for clarification:    The portals I created are 1, 4 and 5, the others were spawned by the game as the "other end". I want to remove the link between 5 and 4 and create a link between 5 and (the yet non-existent) 6.  The goal is to be able to travel to 5 in the route: 1, 2, 6, 5. I don't really need 3 and 4 now. I am trying to send an email using a Visualforce template and I want that template to include an icon. Is this possible? Simple programs with tutorials for beginners in graphic design? W.K.T, any user(with more than 2k rep) can review 20 suggested edits per day.  Today I saw the newly appointed moderator  reviewed more than 20.    Is it possible for moderators to review more than the daily limited count?  If yes, can any moderators can do it at any time?  I didn't see this before. I dont understand the following very simple statement:  If $\tau \in S_n$ has order $m$, then $\sigma \tau\sigma^{-1}$ has also order $m$.  The proof is:  Suppose $\tau$ has order $m$.   $(\sigma \tau \sigma^{-1})^{m} = \sigma \tau^{m}\sigma^{-1}= (1)$  Then (using  exercise 17) they say that the order of $\sigma\tau\sigma^{-1}$ cannot be less than $n$ (remember we are in $S_n$), since $(\sigma\tau\sigma^{-1})^k=1 $ implies $\sigma\tau^k\sigma^{-1} =(1)$, and then $\tau^k=\sigma^{-1} \sigma = (1)$  I don't understand the last part.... Who can help me? "Shall" versus "will" for the first person When I studied English, I was taught that shall is used to express the future tense for the first person (singular, or plural), and will is used for the same purpose in the other cases.  When I look for shall on the NOAD, I read the following sentence, in a note about the usage of shall:     In practice, however, shall and will are today used more or less interchangeably in statements (although not in questions).   Does that mean shall and will are used with almost the same meaning?  Does I shall return have the same meaning of I will return?