I am studying $L^p$ spaces and I ve found this exercise I could not solve.  Let $f$ $L^p(\mathbb{R}^n), 1 \le p &lt; \infty$ then $\|f(x + h) - f(x)\|_{L^p} \to 0 \Leftarrow |h| \to 0.$  It supposed to be easy since the continuos functions with compact support are dense in $L^p(\mathbb{R}^n)$ but I don't how to use this to conclude the result.  I tried to do:  Given $f \in L^p(\mathbb{R}^n)$ there existis a sequence $(f_n) \in \mathcal{C}_c(\mathbb{R}^n)$ such that $\|f_n - f\|_{L^p} \to 0 \Leftarrow n\to \infty.$  $\|f(x + h) - f(x)\|_{L^p} \le \|f(x+h)-f_n(x)\|_{L^P} + \|f(x) - f_n(x)\|_{L^p}.$  So, I can work with the second right term, but I have no idea what to do with the first.  Thanks
I am having a hard time with the following real analysis qual problem.  Any help would be awesome.  Suppose that $f \in L^p(\mathbb{R})$, where $1\leq p&lt; + \infty$. Let $T_r(f)(t)=f(t−r)$. Show that $\lim_{r \to 0} \|T_rf−f\|_{L_p} =0$, that is $$ \lim_{r\to 0} \left( \int_{\mathbb R} |f(t-r) - f(t)|^p \mathrm d t\right)^{1/p} =0.$$
SQL query that concatenates values from duplicate rows in a single table Let's say I have a table of employees, like this:  Name     Sex     Role  Bob      M       Developer Joe      M       QA   Now, I have a problem with duplicated rows in this table. I will fix it sometime but it's not the issue. What I currently need is something that queries such a table:  Name     Sex     Role  Bob      M       Developer Bob      M       Janitor Joe      M       QA Joe      M       CEO   And will output the following rows:  Name     Role  Bob      Developer, Janitor Joe      QA, CEO   It doesn't matter to me if the "Role" will be split to separate columns or be a single column with multiple values.  Using SQL Server 2008 if it matters. I have a "term reference" field, in a form that uses an auto-complete widget. I want to limit user input to terms in the vocabulary only, and output a message when a user tries to enter a non existing term.  I found the  module. However this does not work with forms that contain a field collection or a field group.  There is also a contributed patch . But I was wondering whether there was a simpler/cleaner solution like a module or maybe I missed an already existing way of doing it with latest drupal and taxonomy versions. Is a totally ordered set well-ordered, provided that its countable subsets are? Let $(X,⪯$) be totally ordered set, prove that if every non empty countable subset of $X$ is well ordered then X is well ordered.   It does seem obvious that any subset should have a minimum but I am not sure of how to prove it. Any help would be appreciated, and sorry I put up the same problem earlier but forgot to include the 'countable' subset. On the importance of the i.i.d. assumption in statistical learning Are all stormtroopers as poor soldiers as the ones in the movies? In the Star Wars films we consistently witness just how poor soldiers stormtroopers are. They show very sub-par marksmanship, crappy house searching tactics, and are easily fooled and defeated by Ewoks and Rebels.   That being said, I remember reading a character guide that defined stormtroopers as the very best of the best (best marksman, above average intelligence, etc.). So, in the realm of the Expanded Universe, does the mold that the movies set continue to hold true or are they just the exceptions to the rule? Passive with "to get" In spoken English (at least in America), passives are often formed with "to get" rather than "to be;" I would wager the majority of the time:   he got hit by a car I get driven to work I've gotten called for jury duty eight times   Are there specific verbs that prefer "be" to "get," or vice-versa, or semantic differences between these forms? I cannot think of any. Why does TeX prefer an overfull box to hyphenating here? Capture stdout and stderr into different variables Why doesn't an Environment texture appear when the world has a Volume Scatter shader? I'm working on lighting a scene in which I need to have both an Environment texture for the background and Volume Scatter for light cones, but when I check the volume section in World setting and choose the Volume Scatter, the environment becomes black background (but the lighting works as I want).  I'm using Blender 2.72  How can this be fixed? "An error occurred when reviewing this item" On approving an edit, I just get this pop up since morning. At first I thought there might be some internet connection issue, but this is now the 3rd to 4th time I got the same error; also, the review is accepted because on refresh I can see my name with approved text beside, but it still throws this error.    I am on a Windows 7 machine, running Firefox 24 (Icognito/Private Mode)    Also, this happens only when edit(1) is written on an question/answer, I don't go to review queue and approve, these reviews were directly loaded from the notification beneath the question/answer.  Martijn Pieters reproduced the problem from the review queue three times; each time the improve option was picked, accepting the suggested edit but editing the post some more. The network console shows that posting to https://stackoverflow.com/review/task-reviewed fails, the server responds with a 404 error:  Request headers: POST /review/task-reviewed HTTP/1.1 Host: stackoverflow.com Connection: keep-alive Content-Length: 137 Accept: application/json, text/javascript, */*; q=0.01 Origin: http://stackoverflow.com X-Requested-With: XMLHttpRequest User-Agent: Mozilla/5.0 (Macintosh; Intel Mac OS X 10_7_5) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/31.0.1650.63 Safari/537.36 Content-Type: application/x-www-form-urlencoded DNT: 1 Referer: https://stackoverflow.com/review/suggested-edits/3575396 Accept-Encoding: gzip,deflate,sdch Accept-Language: en,en-US;q=0.8,nl;q=0.6,nb;q=0.4,no;q=0.2,de;q=0.2,fr;q=0.2 Cookie: &lt;redacted&gt;  POST data: taskTypeId=1&amp;reviewTaskId=3575396&amp;taskResultTypeId=5&amp;fkey=f8791a329b445b658876f4d3afe7961e&amp;postCommentId=null&amp;taskResultSubtypeFlags=null  Response headers: HTTP/1.1 404 Not Found Cache-Control: private Pragma: no-cache Content-Type: application/json X-Frame-Options: SAMEORIGIN Date: Fri, 13 Dec 2013 12:04:53 GMT Content-Length: 0   On the  itself, no record is made of the accept; instead the suggested edit is recorded as approved by 3 people not including Martijn Pieters. In this case the improve review arrived too late.  In , improve again resulted in the error, but the Community account is shown rejecting the edit. How can I stop applications and services from running? When my phone starts up there are several application/services running in the background that I would rather not.  And when I press the Home button while using an app, that app usually goes into the background without exiting.  How can I stop these apps from running? Maximal ideals of polynomial rings in infinitely many variables Let $k$ be an algebraically closed field. Nullstellensatz states that the maximal ideals of the polynomial ring $R=k[X_1,\dots,X_n]$ are precisely those of the form $\langle X_1-a_1,\dots,X_n-a_n\rangle$, with $(a_1,\dots,a_n)\in k^n$.     What if we are working with infinitely many variables, say $\{X_i\}_{i\in I}$, being $I$ an infinite set? Again, the ideals of the form $\bigl\langle\{X_i-a_i: i\in I\}\bigr\rangle$, with $(a_i)_{i\in I}\subseteq k$ are maximal (even if $k$ is not algebraically closed), but what about the converse? How can I find the following limit :$$\lim_{n\to\infty}\left(1-\frac{1}{3}\right)^2\left(1-\frac{1}{6}\right)^2\left(1-\frac{1}{10}\right)^2\cdots\left(1-\frac{1}{\frac{n(n+1)}{2}}\right)^2$$There are four options $(a)\frac1{3}(b)\frac1{9}(c)\frac1{81}(d)0$ Thanks! Why doesn't my "Nikkor AF" lens autofocus on my Nikon D3300? Why are springs shaped the way they are? Why are springs coiled the way they are? Why not some other shape? Is the shape due to its elasticity or something? I am using the Service Console in Lightning Experience, and using the Chatter Feed component on the Case layout to display updates to the Case.  If I use the "Log a Call" Quick Action from the Chatter publisher, these calls are displayed in the Chatter feed.  This is the desired behaviour.  However, if I insert a Log a Call record related to the Case any other way - e.g. using the Developer Console, or my CTI system's custom components - the call does not display in the feed.  If I query the record, it looks exactly the same as the record that is created with the Quick Action.  It just seems that the Chatter post is never made in this situation.  I have checked my Feed Tracking settings:   Task has Enable Feed Tracking and Subject checked Case has All Related Objects checked   How can I enable calls created with a custom component (part of a CTI system) to display in the case feed? Which way is up? (electric outlet) 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. I want to enable gzip compression for acethehimalaya.com. Can anybody please tell me what is the best way of achieving compression in an Apache Webserver? Is there a BNF grammar of the TeX language? I'm looking for a BNF grammar of the TeX language, does it exist?  EDIT  For those of us who are not computer scientists, a BNF grammar is one kind of formal description of a CFG: .  For those who don't know their Chomsky, a CFG is a , which means (very roughly) that substitutions can be made independently of their use context.