Fractional Laplacian on the torus
Fractional Laplacian on the torus
Where to place 'only' relative to prepositions? I know that questions about the placement of 'only', are often asked here; accordingly, I searched for an answer to my question before posting it.  Question  Where are focusing adverbs placed relative to prepositions?  For example,   For his presentation, he needed to talk only about tomatoes. For his presentation, he needed to talk about only tomatoes.   My Attempt to Answer the Question  In both examples, it seems to me that 'only' is placed close enough to what it is intended to modify (the topic of the subject's discourse: tomatoes) that I cannot see any justifiable misinterpretations.  One of the examples sounds better to me, but at this point in my studies, I am more interested in what the traditional rules of grammar require, than in how the wordings sound. I want to ask about two sentences, and what would be the correct way to form a question:   It's very important that we eat healthy food, ...........?  Which would be the correct question tag: (isn't it) or (don't we)? It's a must that all applicants follow the rules of the competition, ..........?  Which would the correct question tag: (don't they) or (mustn't they)?   Please could you tell me the correct answers, and, more importantly, why those are correct? Prove that $\lim_{N\rightarrow\infty}(1/N)\sum_{n=1}^N f(nx)=\int_{0}^1f(t)dt$ Suppose $f$ is continuous and periodic on the reals with period 1. Prove that if $x\in[0,1]$ is an irrational number, then  $$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^N f(nx)=\int_{0}^1f(t)dt$$   Suggestion: First consider $f(t) = e^{2\pi(ikt)}$ where k is an integer.   I can see that this is a limit of a weighted average, but the suggestion throws me off. I've seen the suggestion in fourier transforms but it's not clicking at the moment. Any help would be welcome. Magic 8 Ball Problem This problem is probably simple enough to have an analogous problem, I just don't know the name so I'm going to describe it and hopefully somebody can point me in the right direction.  The problem is this: estimating the number of sides of a .   Let's say you perform the following process:   Shake the "Magic 8 Ball" and mark down the result If the result has not previously been seen, add it to the set of possible results and label the trial "N" (for "new") If the result has been previously seen, label the trial "O" (for "old") Repeat all steps   I imagine the following kind of sequence occurring: NNNNONNOONOONOOONOOOONONNOONOOONNOONOOOOOONOOONOOOOOOONOOOOOOOONOOOOOOOO...  Now imagine we don't know there are twenty sides to a Magic 8 Ball. Or imagine that you have a Magic 8 Ball with 1000 sides. As the number of "new" trials approaches the actual number of sides, we'll get increasingly more "old" trials showing up in the mix. Once we've seen all of the possible results, we'll always get "old" results. But we're never 100% sure we've seen every single possible result.  So here are the questions I'm interested in:   As we proceed with trials, can we estimate the total number of "sides" of the Magic 8 ball based on the number of "new" and "old" trials up to this point? Can we calculate a probability that our current estimate is correct, or a probability that some bounded estimate is correct? I have created a C# application which uses a SQL Server database. I have other computers connected to me and to each other in a workgroup. I have shared my C# application with others. When they open the application they get the error      A network related or instance-specific error occured while establishing a connection to SQL Server. the server was not found or was not accessible   But the application is working fine on my PC. The connection string I am using is   Data Source=ASHISHPC1\SQLEXPRESS;Initial Catalog=ACW;User ID=ash159;Password=ashish159   which is stored in a .config file.   The application is working fine on my PC. What must I do? I have enabled the TCP/IP in the server but the same error persists. Some change in connection string or something else? What is a NumberFormatException and how can I fix it? Indian passport threading coming off My Indian passport threading is coming off i.e. the cover of the passport is coming off, there are only 2 threads remaining.   Will it be a problem at the airport when going abroad ? Also anyone who has experience of getting a replacement passport fairly quick ? Because I need to get a passport in like 15 days. How do I pass the forest stage on Candy Box 2? I cannot pass the forest stage on Candy Box 2. The dogs and forest mosquitos kill me fast.  My stats and equipment are:   220 health Polished silver sword (7 damage, also have the trolls bludgeon; 12-16 damage) Leather boots and gloves (no stat bonus) Basic and advanced spells from the witch across the bridge   On another save I also increased my health to 1000 and still could not pass the forest stage, any advice?  Thanks. How to prove sum of powers property of roots of unity? I have multiple cameras in a scene. What I want to do is to change which one is active. This way I can view the scene from different angles. How would I do this? Change bend angle and swap node position on arrow find the number of injections/surjections I've been banging my head on this problem for some time now, and could really use help. Bear in mind I'm not very good at this sort of thing and am struggling to get by in class.  Problem:     Given $ \alpha : A \to B       \;\;\;\;\;   |A|=p \;\;\;\;|B|=q $      Find:         Number of Injections: $A \to B$   Number of Surjections: $ A \to B$    Prove that ∃ a bijection $A \to B$       $\Rightarrow$ p = q   find number of bijections: $A \to B$         I apologize for dropping such a big problem on you all, but I just can't get anywhere with this. Help?  EDIT: These answers are really helpful and I believe I've got down the number of Injections, but  we haven't gone through permutations or combinatorics in class so my understanding of it is really hamstrung. I'm unclear on the Stirling theorem as discussed here: , specifically the part where inclusion-exclusion comes into play. I'm not sure how to apply that, since right now I'm letting p = 4 and q = 3, so I don't know where inclusion-exclusion would be used (if at all!) since I don't know how elements would be missing. Understanding a particular evaluation of $\prod\limits_{n=2}^{\infty}\left(1-\frac{1}{n^3}\right)$ I'm having a hard time understanding the following evaluation of the infinite product  $$ \prod_{n=2}^{\infty}\left(1-\frac{1}{n^3}\right).$$ In particular, I don't understand how you go from line 2 to line 3. Here $\omega = -\frac{1}{2}+ i \frac{\sqrt{3}}{2}$, which is a primitive third root of unity.  $$\begin{align}\prod_{n=2}^{\infty}\left(1-\frac{1}{n^3}\right)  &amp;=\prod_{n=2}^{\infty}\frac{(n-1)(n^2+n+1)}{n^3} \\  &amp;=\lim_{m\to\infty}\frac{1}{m}\prod_{n=2}^m\frac{(n-\omega)(n-\omega^2)}{n^2} \\  &amp;=\lim_{m\to\infty}\frac{\Gamma(m+1-\omega)\Gamma(m+1-\omega^2)}{m(m!)^2\Gamma(-\omega)\Gamma(-\omega^2)(1-\omega)(1-\omega^2)(-\omega)(-\omega^2)} \\  &amp;=\frac{1}{3\Gamma(-\omega)\Gamma(-\omega^2)} \\ &amp;=\frac{\sin{\pi(-\omega)}}{3\pi} \\  &amp;=\frac{\cosh (\frac{\sqrt{3}\pi}{2})}{3\pi } \end{align}$$ EDIT: I think I'm starting to make sense out of this by writing out the terms. $$ \begin{align} \prod_{n=2}^{m} \frac{(n- \omega)(n-\omega^{2})}{n^{2}} &amp;= \frac{(2-\omega)(3-\omega) \cdots (m- \omega)(2-\omega^{2})(3-\omega^{2})\cdots (m- \omega^{2})}{2^{2} \cdot 3^{2} \cdot \cdots  \cdot m^{2}}  \\ &amp;= \frac{1}{(m!)^{2}} \frac{\Gamma(m+1-\omega)}{(1-\omega)(-\omega)\Gamma(-\omega)} \frac{\Gamma(m+1-\omega^{2})}{(1-\omega^{2})(-\omega^{2})\Gamma(-\omega^{2})} \end{align}$$ Now I have to figure out how to take the limit as $m \to \infty$. Occasionally, I've come across a webpage that tries to pop open a new window (for user input, or something important), but the popup blocker prevents this from happening.  What methods can the calling window use to make sure the new window launched properly? As far as I know, those are the only two.  They should be pronounced Soogher and Soor, shouldn't they?  I looked them up on Dictionary.com, and their etymologies reveal no trace of an SH, except where the listing for sugar had:     Middle English sugre, sucre (noun) &lt; Middle French sucre  &lt; Medieval Latin succārum  &lt; Italian zucchero  &lt; Arabic sukkar;  obscurely akin to Persian shakar, Greek sákcharon   I see an obscure kinship to shakar, but the word morphed so many times since then that the SH disappeared pretty much completely.  And sure is even worse, with no sign of SH:     Middle English sur ( e ) &lt; Middle French sur, Old French seur  &lt; Latin sēcūrus   Why are these the only two like this? Text gets stretched or disappears when scrolling When scrolling in the Stack Exchange Android app, the text gets stretched vertically, code disappears, and sometimes the entire post content disappears.  This doesn't make the app completely useless, but it's pretty close.   Motorola Razr MAXX, Android 4.0.4. Add an image directly from the clipboard I've tried to find other questions covering this, but (surprisingly) I did not find any.  When I want to add a screenshot to a post, as far I know the only way to do it is to save it as a file and then upload the file, using Stack Overflow's image hosting. OK, much better than finding a hosting site yourself, but sometimes I wish I could just paste the image from the clipboard.  It's not as straightforward as it may seem, I know. , and it needs a Java applet to get it done. As I understand it, electromagnetic waves have two components which are the result of each other, i.e., when a moving electric charge creates a changing magnetic field at point X then a changing electric field is created at point Y and this repeating process is what creates EM waves, so therefore, it requires no medium. Is my understanding correct?  One thing that I'm surprised to know is that light is also called an electromagnetic wave.  Does this include light of any kind, for example: light from a bulb, a tube, and also from the Sun? How do they contain electric and magnetic fields? Live USB for installation does not boot, black error screen followed by visual artifacts on a purple screen I'm trying to install Ubuntu 18.04 desktop with Rufus on a computer with the following specs:  CPU: AMD Ryzen 5 2600 MOBO: ASUS Prime B450M-A/CSM RAM: G-Skill Ripjaws DDR4 F4-2400C15D-16GVB GPU: Nvidia GeForce GTX750 PSU: Apexgaming AG-650M SSHD: Seagate Firecuda Compute mATX: Rosewill SCM-01  Basically it boots from the USB (FAT32) and brings up a black error screen with words and numbers. Then it transitions to the purple splash screen with &quot;ubuntu&quot; and 5 dots, however the screen shows various visual glitches depending on which USB port I plug into. When trying to boot the computer, it boots to a black screen with the following error message: ACPI Error: Needed [Integer/String/Buffer], found [Region]    that shows what I mean. How is an instance initializer different from a constructor?