Suppose that $f\in L^1(\mathbb{R})$ and that $f(x)=0$ if $x\notin[0,1]$. For $n=1,2,\ldots,$ let $f_n(x)=f\left(x+\frac{1}{n}\right)$. Prove that $\|f_n-f\|_1\to 0$ as $n\to\infty$.
Let $f \in L^1(\mathbb{R})$, for $a\in \mathbb{R}$ let $f_a(x)=f(x-a)$, prove that: $$\lim_{a\rightarrow 0}\|f_a -f \|_1=0$$  I know that there exists $g\in C(\mathbb{R})$ s.t $\|f-g\|_1 \leq \epsilon$, this is also true for $f_a$ and $g_a$.  Now I have the next estimation:  \begin{align*} \|f_a-f\|_1 &amp;= \int_{\mathbb{R}} |f(x)-f(x-a)| dx \\ &amp;= \int |f(x)-g(x)+g(x)-g(x-a)+g(x-a)-f(x-a)| \\ &amp;\leq \|f-g\|_1 + \|f_a-g_a\|_1 + \int |g(x)-g(x-a)| \end{align*} my question is: I can argue that $\lim_{a\to0} \int |g(x)-g(x-a)| = \lim_{a\to0} \lim_{T\to\infty} |g(x_0)-g(x_0-a)| 2T$, now I think that I can change the order of the limits here, but I am not sure why?  P.s $x_0$ is some point in $[-T,T]$, and the above is valid from the intermediate integral theorem for continuous functions, right?  Thanks.
Exit of "> " mode in Unix shell In sh (not Bash), how would we abort execution of command when the prompt is in &gt; mode?  For example, when entering a string with quotes only at the beginning, it makes the prompt look like &gt;, without the ability to quit it normally, unless hitting Ctrl + D. Example:  root@MyPC:~# echo "hello &gt; I am &gt; (How do I exit of this mode?)   In case that I don't know which character delimits the string (" or ' or simply escaping a newline/spaces with backslash), is there a way to let Bash know that I want to abort the execution of the current command? Is there any parameter (in terms of physical quantities such as mass, length, charge...) which can be used to decide when to treat a system quantum mechanically and not classically? Check if a user has a role How can I check if a user has a certain role?  I found  but it is for Drupal 6. Well, I'm trying to find the Thevenin equivalent of this circuit:    As you can see, the only current source is a dependent source. However, the control variable is in a wire where, if nothing is connected between A and B, is \$i_\Delta = 0\$ A, and that makes the source 0 A. However, if I connect an impedance \$Z\$ to the circuit, I will have current, and a Thevenin equivalent. I don't know how to solve this problem. I've bee searching here for a while, and I found this:   . I can see how is solved that case. However, I still don't know how to apply that method to my circuit.  Any help about finding the Thevenin equivalent is appreciated n.n I know that like is a preposition but why not using it as conjunction?  Examples:        It's as if I'm walking on air   It's like I'm walking on air      What is the difference? Academic dishonest: Possession of unauthorized materials during an exam that you didn't even know existed So unfortunately during an exam, I had the misfortune of using an eraser (with cover) that was not mine and had a small amount of words inside it unknown to me. I did not know this was the case, as I don't have a habit of unraveling erasers or looking at them closely?  However, now I face the problem in trying to defend myself against the accusation of me cheating.  I'm specifically concerned on my university's blank statement of "possession of unauthorized materials during an exam" is considered academic dishonesty. Seems like student conduct has all the power in trying to prove academic dishonesty, despite none of my test containing any of what to me is  unreadable, small notes. How to use a custom time in browser to test for client vs server time difference Based on the following interchange between Harry and Dumbledore in the pivotal &quot;King's Cross&quot; chapter of Deathly Hallows, I was wondering when exactly Dumbledore put together that Harry would have to die in order to bring about Voldemort's ultimate demise:  &quot;And you knew this? You knew--all along?&quot; &quot;I guessed. But my guesses have usually been good,&quot; said Dumbledore (Deathly Hallows, 599, hardcover).  The conversation that spurred Harry's question includes material dating back to Halloween '81, but also specifically mentions the events of the graveyard from GoF. So, &quot;knowing all along&quot; could refer to everything (i.e., Harry must die, Voldemort ignorantly tethered him to life, etc.) or just that Voldemort tethered him to life. Other important information pertaining to this question can be found in OOTP, when Dumbledore confirms (if he didn't already know) that Harry contains part of Voldemort's soul. He uses the silver instrument to find out that Harry is just like Nagini, having the ability to share mind and soul with Voldemort. He also says the following:  &quot;I guessed, nearly fifteen years ago,&quot; said Dumbledore, &quot;when I saw the scar upon your forehead, what it might mean. I guessed that it might be the sign of a connection forged between you and Voldemort&quot; (Order of the Phoenix, 826-827, hardcover).  While confirming that he guessed that Harry might share a connection with Voldemort from the very beginning, Dumbledore does not confirm with this statement that he knew Harry would ultimately have to give up his life. In Philosopher's Stone, when Dumbledore drops off Harry on the Dursley's doorstep, we know he is not entirely truthful with McGonagall.  McGonagall asks: &quot;How in the name of heaven did Harry survive?&quot; &quot;We can only guess,&quot; said Dumbledore (Philosopher's Stone, 12, hardcover).  Of course, this was a lie. Dumbledore already knew how Harry survived, otherwise he wouldn't be delivering Harry to Lily's sister to extend the blood sacrifice. By proxy, he also knows that Voldemort will return b/c of Horcruxes, otherwise he wouldn't be so worried about Harry. Finally, when asked in the same scene if he could remove the scar, Dumbledore says:  &quot;Even if I could, I wouldn't. Scars can come in handy&quot; (Philosopher's Stone, 15, hardcover).  So, did Dumbledore know at this very moment that Harry would have to die? Did he at least guess it to be true during this scene? We know he gains perfect confirmation in OOTP, but did he know it &quot;all along,&quot; as a particular reading of the King's Cross interchange in DH would have us believe? Check Constraint with Case Statement I want to add a check constraint on a column if entered value for column is greater than 3 than it should be saved as 1 else entered value. For that scenario I am using below query but on insertion of data it shows an error message.  create table tblTestCheckConstraint  ( id int, NewColumn int )   alter table tblTestCheckConstraint add constraint chk_tblTestCheckConstraint_NewColumn1 CHECK   (     CASE         WHEN NewColumn &gt;4 THEN 1         ELSE NewColumn     END = 1 )   Insert Statement:   insert into tblTestCheckConstraint values ( 1,5) insert into tblTestCheckConstraint values ( 1,2)   Error Message:   Msg 547, Level 16, State 0, Line 1 The INSERT statement conflicted with the CHECK constraint "chk_tblTestCheckConstraint_NewColumn1". The conflict occurred in database "DBName" , table "dbo.tblTestCheckConstraint", column 'NewColumn'. The statement has been terminated. What evidence is there supporting an alternative parentage for Jon Snow? I've heard speculation that Jon Snow's actual parents may be Rhaegar and Lyanna Stark, but in reading the books I'm not sure what evidence could be used to support this conclusion.  I thought there might have been something in the conversation Davos had before being smuggled into White Harbor, but I couldn't stretch anything that far.  What evidence do proponents of this theory cite?  Does any of it involve stuff in the texts or just sweeping generalizations of the characters' personalities (e.g. "Ned Stark is too honorable to father a bastard"). Why does subsurf modifier cause weird deformation at the end of a cylinder? My problem is simple. I think that the strange deformations on the top are there because there are so many vertices in one face. How do I fix this while maintaining nice topology?  Although I triangulated the top polygon, the problem still persists. While browsing Stack Overflow, I noticed that I lost a lot of reputation:        Is this serial downvoting? Will it be corrected by a rep-recalculation? How often are those done?    Here's a second instance: Generating correlated binomial random variables I was wondering if it might be possible to generate correlated random binomial variables following a linear transformation approach?  Below, I tried something simple in R and it produces some correlation. But I was wondering if there is a principled way to do this?  X1 = rbinom(1e4, 6, .5) ; X2 = rbinom(1e4, 6, .5) ;  X3 = rbinom(1e4, 6, .5) ; a = .5  Y1 = X1 + (a*X2) ; Y2 = X2 + (a*X3) ## Y1 and Y2 are supposed to be correlated  cor(Y1, Y2) "ignorance" can also mean a "willful disregard of something"? I try to use the 2nd or 3rd definition of words (to slow down speed readers). However, I am not sure if "ignorance" has a 2nd definition.  (common meaning)  is "a lack of knowledge". With this meaning, no syllable is stressed.   My ignorance of hardware means I cannot build my own pc.   (2nd meaning?) If pronounced differently, I think that "ignorance" can also mean "a willful and careless disregard of something". The "nor" syllable in "ig-NOR-ance" must be stressed.     The French were aware of the strategic weakness of the Maginot Line, and their ignorance of it enabled the German blitz through the Low Countries.   My ignorance of my girlfriend's hourly text messages quickly caused our break-up.   In a dictionary, "ignorance" is only defined as "lack of knowledge". But, I am not talking about that "ignorance". I am talking about the "ignorance" that is derived from "to ignore" and thus pronounced just like "to ignore", and thus has the meaning of "to ignore". There is precedence for this:  "to accept" --- derive ---> "acceptance" "to ignore" --- derive ---> "ignorance" ....  Anyone agree with me? Why won't any faces show up in the UV Image Editor? Can you help me with my software licensing issue? This is a  about Licensing.    Questions on licensing are off-topic on Server Fault.  If your question has been closed as a duplicate of this question, then this is because we want to help you understand why licence questions are off topic rather than just telling you "it just is". In all likelihood, this question will not address your question directly, it was not meant to.    I have a question regarding software licensing. Can the Server Fault community please help with the following:   How many licenses do I need? Is this licensing configuration valid? What CALs do I need to be properly licensed? Can I run this product in a virtual environment? Can I downgrade this product to an earlier version? Am I entitled to feature X with license Y? Fatal error: Class 'SelectQuery' not found in includes/database/database.inc on line 813 Fatal error: Class 'SelectQuery' not found in /home/content/71/11241671/html/drupal/includes/database/database.inc on line 813   We built a site in a subfolder called /drupal while building the site, and then moved the files to the root directory when we were finished but it gave the above error on a WSOD.  I have tried:    flushing the cache before moving  tried the registry rebuild truncating and replacing the registry and registry_file tables with a fresh install copy, and rebuilding the registry again rewrite base in htaccess setting a new base url in settings.php   But nothing has worked.  The second we put the files back into the /drupal folder, it works again. It also looks like the error mentions the file in the /drupal directory, which isnt there anymore.   This is on 7.37   Anyone have any ideas? Why is shape with shrinkwrap modifier flat? Right-Click menu awareness on Web Application The web application I am building has a custom right-click menu. I'm concerned that the user might not find this menu as normally right-clicking will only display the default browser's menu. Is there a standard way to make the user aware of custom right-click menus? Is there another way to show this information? I have a sample composed by 2500 count data values. I've plotted in R the corresponding histogram and ecdf. I've run the One-Sample Kolmogorov-Smirnov test to check if the distribution is either exponential or poisson, but I got negative results.  Is it possible that the One-Sample Kolmogorov-Smirnov test, it's not the correct test to work with this count data.  What is the best way to identify a distribution for this kind of data?