Dealing with social syntactic sugar
Should 'Hi', 'thanks', taglines, and salutations be removed from posts?
$x^y+y^x>1$ for all $(x, y)\in \mathbb{R_+^2}$ Prove that $x^y+y^x&gt;1$ for all $(x, y)\in \mathbb{R_+^2}$. Is there a correct gender-neutral singular pronoun ("his" vs. "her" vs. "their")? Is there a pronoun I can use as a gender-neutral pronoun when referring back to a singular noun phrase?  Each student should save his questions until the end. Each student should save her questions until the end. Why would turning off a sink faucet stop the shower? Verification that $\lim_{x \to a}\frac{f(x)-f(a)}{x-a}$ and $\lim_{h\to 0}\frac{f(a+h) -f(a)}{h}$ are equivalent definitions of the derivative. I wanted to verify that for definition of the derivative it is true that:  $$ \lim_{x \to a}\frac{f(x)-f(a)}{x-a}= \lim_{h\to 0}\frac{f(a+h) -f(a)}{h}$$  If I denote $h=x-a$,  we can  let $x\to a$, this means that $h \to 0$. Also notice that $f(x)=f(a+h)$ $$ \lim_{x \to a}\frac{f(x)-f(a)}{x-a}=\lim_{h \to 0}\frac{f(a+h)-f(a)}{h}$$ As desired. Is the main idea here really just to use a variable substitution? I suppose there is some formal theorem I also use here that when $f(x) \to y$  we can let $g(f(x)) \to g(y)$ if the function $g$ is continuous. How can I update youtube-dl? If $(X_n)$ is i.i.d. and $ \frac1n\sum\limits_{k=1}^{n} {X_k}\to Y$ almost surely then $X_1$ is integrable (converse of SLLN) Let $(\Omega,\mathcal F,P)$ be a finite measure space.  Let $X_n:\Omega \rightarrow \mathbb R$ be a sequence of iid r.v's  I need to prove that if:  $ n^{-1}\sum _{k=1}^{n} {X_k} $ converges almost surely to $Y$ then all $X_k$ have expectation.  If I understand correctly then $X_k$ has expectations means $X_k$ is in $\mathcal L^1(\Omega)$.  And I know that on finite measure space Converging in expectations is converging in $\mathcal L^1(\Omega)$ and it's stronger than almost sure convergence.  And I know that from linearity of expectation even if one of the sequence is not in $\mathcal L^1(\Omega)$ then $Y$ is not in $\mathcal L^1(\Omega)$.  How do I continue? How can you programmatically tell an HTML SELECT to drop down (for example, due to mouseover)? @Id @GeneratedValue but set own ID value How to prove that limit doesn't exist using epsilon-delta definition? It is easy to prove the limit exists, all we have to show is there exists a relationship between $\delta$ and $\epsilon$. But how are we supposed to prove limit doesn't exists? The problem is when we are proving for a limit we already know what the limit is and with that, algebra is all that's needed.   Please show through an example (you may show that $\lim_{x\rightarrow0} \frac{1}{x}$ doesn't exist)  If possible please use the explanation scheme that is used by this answer Alignment issue for the last modified information in the "Top Questions" page In the  (homepage), the asked/modified X mins ago text was left-aligned next to the tags. Previously the text was right-aligned. Seems some alignment issue introduced on that page recently. Screenshot for reference:      Configuration Version     Operating System Windows 10 Pro   Mozilla Firefox 89.0 (64-bit)   Google Chrome 91.0.4472.101 (Official Build) (64-bit)     Update: I see the same alignment issue in all other Stack Exchange site's &quot;Top Questions&quot; page too. Meaning of backtick in floating-point literal Does an unpublished document you upload to ResearchGate count as prior publication when submitting to a journal? On ResearchGate, if you upload something like a thesis, and then it says publish resources, does this mean the thesis becomes properly published in the sense of the usual requirements of journals that there be no prior publication of the submitted work?   I ask because I’m drafting thesis chapters into journal articles and did not choose to publish the thesis as a whole work in order to do so, so I want to make sure that by clicking publish resources I am not actually publishing the copy I’ve uploaded, I’m just making it publicly available? Does this mean it becomes published despite it not technically being a published work? (If so, I’ll just remove the copy I uploaded). Is three hours enough for an international-domestic transfer at Delhi? My International flight will be reaching New Delhi IGI at 2:45 AM and I am planning to book a domestic flight at 5:45 AM.  Is three hours sufficient for me to move from International to Domestic?  I believe there must be a shuttle service inside the airport to transfer me from International to Domestic but still I don't know if Customs time at IGI could cause any delay or not. If $f^2$ is Riemann Integrable is $f$ always Riemann Integrable? This week I've read two news stories about two different camera lenses that can magnify things that are very far away.  The first is the , which is a point-and-shoot camera that costs $600, has an 83x optical superzoom lens ("the same field of view as a 2,000mm lens on a full-frame camera"), and lets you see the craters of the moon.  The second is the , which weighs 36 pounds, costs $180,000, and can take portraits of people a half mile away.  A non-photographer might ask, "Why would somebody spend their life savings on a gigantic telescope of a lens when you could take far away pictures with a handheld camera for 1/300th the price?"  Well, I am basically a non-photographer, and I would like to ask that question. What does the big one give you that the little one doesn't provide? I understand that small changes in lenses can make a big difference in size and price, but this seems to be of a different scale entirely.   What is going on with the optics of the lenses that makes the small one able to be so small and the big one have to be so...big?   I'm trying to get a basic intuition for what makes lenses different, and I realize I don't understand enough about how lenses work to even know how to compare these apples and watermelons. Cauchy-Schwarz inequality proof (but not the usual one) Before you downvote/vote-to-close, I am not asking for a proof of: $$\sum^n_{i=1}a_ib_i\le\sqrt{\sum_{i=1}^na_i^2}\sqrt{\sum^n_{i=1}b_i^2 } $$ Which is what EVERY link I've found assumes is the inequality (for a proof of that see:  )    I am reading a book that claims the Cauchy-Schwarz inequality is actually: $$\vert\langle x,y\rangle\vert\le\Vert x\Vert\Vert y\Vert$$ where $\Vert x\Vert :=\sqrt{\langle x,x\rangle}$  with the additional claim: equality holds $\iff\ x,y$ are linearly dependent  I cannot find a proof of this claim (only proofs for the dot product inner product). My question is: what is the simplest way to prove this (I define simplest to mean "using as few definitions outside of the inner product itself as possible" - so that's not including Gramm-Schmitt orthonormalisation, for example.)  I've just found some possible answers in the links on the right (annoyingly) but fortunately I have a second, albeit softer, question:  What inequalities that are common for say $\mathbb{R}^n$ are actually based on the inner product?   (In future I will check this site first, I was targeting lecture notes and such in my search)  Addendum: I want to use this on not-finite dimensional vector spaces. I've found one proof that relies on finite-ness. Beamer Bibliography Icon I want to list my references in a beamer presentation using thebibliography environment, with the nice little    book icon (\beamertemplatebookbibitems) and  article icon (\beamertemplatearticlebibitems)   in front of each source.  Is there an icon for websites too? something like \beamertemplatewebsitebibitems? If not, is there a way to implement this easily ? Should I submit bank statements when applying for a UK Visa? What do they say about me? The UK Visas and Immigration Directorate publishes  that provides helpful tips that can result in successful visa applications. Section 2 of this guidance opens is entitled &quot;other documents you may want to provide&quot; (emphasis mine).  Section 2: other documents you may want to provide – all visitors This section provides guidance on the types of documents that you may want to provide to help us consider your application against the Immigration Rules. Previous travel documents/passports, which show previous travel. Financial documents showing that you have sufficient funds available.  These must clearly show that you have access to the funds, such as:  bank statements building society book proof of earnings such as a letter from employer confirming employment details (start date of employment, salary, role, company contact details) where a third party (who is either in the UK or will be legally in the UK at the time of your visit) is providing financial support to you e.g. a business, a friend or a relative, documents to show that they have sufficient resources to support you in addition to themselves and any dependant family should be provided   The first item is &quot;bank statements&quot; and this ties in with the , which says...  evidence that you can support yourself during your trip, eg bank statements or payslips from the last 6 months  (Note 'eg' is a Latin abbreviation meaning &quot;for example&quot;; this appears to be optional rather than a fixed requirement.) Question: How should someone interpret this such that the chances of success are maximized? How many statements should I submit? What are they looking for and what do my bank statements tell about me? Minecraft crashes on launch with EXCEPTION_ACCESS_VIOLATION, Problematic frame: ig4dev32.dll or ig4dev64.dll or ig4icd32.dll or ig4icd64.dll When I try to launch Minecraft 1.7.4 or newer, the game crashes on startup, and the game output screen says this:  #  EXCEPTION_ACCESS_VIOLATION (0xc0000005) at pc=*, pid=*, tid=* # # JRE version: Java™ SE Runtime Environment (7.0_51-b13) (build 1.7.0_51-b13) # Java VM: Java HotSpot™ Client VM (* mixed mode, sharing windows-* ) # Problematic frame: # C  [ig4dev32.dll+0x3e88] OR # C  [ig4dev64.dll+0x55ec] OR # C  [ig4icd32.dll+0x104a77] OR  # C  [ig4icd64.dll+0x84930a]   This happens with both Java 7 and 8.  What causes this error, and what can I do to fix the issue? $127$ has an interesting property: It is the $31$st prime number and its rank ($31$) is also a prime. $31$ is the $11$th prime so its rank is also a prime. $11$ is also a prime number with a rank ($5$) that is also a prime. $5$ is the 3rd prime number and and so its rank ($3$) is also a prime. And finally $3$ is the $2$nd prime so its rank is also a prime... Is there a name for primes whose rank (index in the prime series) is also a prime? That is:     If $a_i$ is the $i$th prime number, then $i$ is a also a prime.   How about numbers like $127$ where going down ranks of ranks always produce prime numbers (down to rank $2$ of course)? The first $11$ :-) primes in this (infinite) series would be:  $3, 5, 11, 31, 127, 709, 5381, 52711, 648391, 9737333, 174440041, \dotsc$  Many thanks!