We now have a sequence of functions \[ f_n = \frac{\sin{nx}\sin{x}}{x^2}, \] and we need to show that $\lVert f_n \rVert_1 \to \infty$ as $n \to \infty$, which means \[ \lim_{n \to \infty}\int_{-\infty}^{\infty}|f_n(x)|dx = 0. \]  How could we do that? It's easy to see that $\int_{-\infty}^{\infty}f_ndx=\pi$, but there is an absolute value function. I managed to compute \[ J_n = \int_{0}^{2\pi}|\sin{nx}\sin{x}|dx=\frac{2n}{n^2-1}\frac{\sin\frac{(n+1)\pi}{2n}}{\sin\frac{\pi}{2n}} \] and wanted to multiply it with $\sum_{j=1}^{\infty}\frac{1}{(2\pi j)^2}$, but the sequence $\{J_n\}$ converges (with respect to $n$), so if my calculation is not wrong, this estimation is too floopy. For those who are interested about the source of this problem, you can find it on W. Rudin's Real and Complex Analysis, Chapter 9, exercise 2. It is the last step of proving the range of the Fourier transform is a proper subset of $C_0$. Define $g_n = \chi_{[-n,n]}$, we can show that $g_n \ast g_1$ is the Fourier transform of $f_n$. It remains to prove that $\lVert f_n \rVert_1 \to \infty$ as $n \to \infty$. And this is where I stuck. Update: think I find the solution in another question: . So I'm OK now. But I think random people in the future may search this function so I think this title may help people. Still if you have good method it's a good idea to put it down.
Let $f_n = \mathbb 1_{[-n,n]}$ for all $n \in \mathbb{N}$   Compute explicitly $f_n \star f_1$ for all $n \in \mathbb{N}$.  Show that $f_n \star f_1$ is the Fourier transform of $g_n = \frac{  \sin{(2\pi x)} \sin{(2 \pi nx)}}{\pi ^2 x^2}$  Show that $\| g_n \|_1 \to \infty$  Deduce that the Fourier transform is not surjective from $\mathcal{L}^1(\mathbb{R})$ to $C_0( \mathbb{R})$    Here, $C_0(\mathbb{R})$ is the space of continuous functions $f : \mathbb{R} \to \mathbb{R}$ that tends to $0$ at $-\infty$ and $+\infty$. I managed to compute 1) and proved 2). But I don't know how to prove 3). Moreover, I don't see how can 1), 2) and 3) prove 4). Can anyone help?
Negative Mass and gravitation If two limits don't exist, are we sure the limit of sum doesn't exist? This is a true/false question.     If the limits $$\lim_ {x\to a}\ f(x)\quad \text{ and }\quad  \lim_ {x\to a}\ g(x)$$ don't exist, limits $$\lim_ {x\to a}\ (f(x)+g(x))\quad \text{ and }\quad \lim_ {x\to a}\ (f(x)g(x))$$ don't exist also.   My idea is, that, because $$\lim_ {x\to a}\ (f(x)+g(x))=\lim_ {x\to a}\ f(x)+\lim_ {x\to a} g(x)$$ then the limits doesn't exist, correct? One server with multiple desktops "heads" with VNC I am managing a system of kiosks.  Each kiosk currently is running a web browser with the application for the kiosk running in the browser.  Each kiosk needs to be able to display separate content.  At times, the application running in the web browser freezes.  Thus, I have to go out to the site to refresh the page.  I want to see if there is a way to have one central server that has multiple browser heads.  Then each kiosk would run a program like VNC to display one of the heads.  This way when the program freezes, I just have to login to the central server and refresh the page.  Getting VNC or another remote desktop software installed on the clients is no problem.  What I am looking for is a way to have VNC remote into a specific head of a head of a web browser.  Does such a thing exist?  Or do I have to run a VM for each kiosk to remote into?  Any advice, pointers, or solutions would be helpful. Somewhere on Stack Exchange I saw the equation     $$\gcd(2^m-1,2^n-1)=2^{\gcd(m,n)}-1.$$   I had never seen this before, so I started trying to prove it. Without success...  Can anyone explain me (so actually prove) why this equation is true?  And can we say the same when replacing the '$2$' by any integer number '$a$'? Normal Subgroup Counterexample Im having trouble with the second part of this question,  Let $H$ be a normal subgroup of $G$ with $|G:H| = n$,  i) Prove $g^n \in H$ $\forall g \in G$ (which i have done)  ii) Give an example to show that this all false when $H$ is not normal in $G$.(which I am having trouble with showing)  Any suggestions? I uploaded a pre-print of an article on arXiv 1 year ago. The article was accepted in SCI journal and currently available online. However, I found a typo mistake in one equation and want to add more discussion related one figure (On researchgate I am receiving a lot of queries regarding that figure). Should i upload a new version of pre-print on arXiv? (How citations will work in this case? as arxiv version is more advanced compare to Journal version) OrI may upload a short document on arXiv as an errata/correction with extra discussion?Or I may address errata and discussion in online page of journal? Actually this journal allows the registered users to comment and start discussion related to published article. What is that "1 view" which automatically appears after submitting a question? Are the questions submitted on SO and other SE sites initialized with 1 view, or does the first view belong to the user who posted it?  Why is a question given a free first view? It just appears, as soon as the question is published. Can someone please help me thanks!  this is where I got the image from Several companies are commercializing tests for telomere length such as this one . I understand the basic mechanism for why telomeres shorten during DNA replication, but how good is the evidence that telomere length is a reliable indicator of healthspan/lifespan? I am trying to print a simple value for $AR1_p1 but the variable $i is not evaluating. for i in 1 2 3 4 do     AR1_p1=22     AR1_p2=23     AR1_p3=24     AR1_p3=25     echo $AR1_p$i done   It's like concatenating dynamically. Any suggestions on how to fix this?. So basically as it stands I have a laptop which has no cd/dvd drive, and I don't have a usb drive.  The laptop has windows 7 installed with ubuntu 11.04 installed through wubi.  What I want to do is remove windows completely, and make ubuntu the only OS installed on the system.  Is there a way to do this without re-installing ubuntu? (i.e. can I take my wubi install away from windows?)  or is there a way to from inside ubuntu have it run the ubuntu iso somehow so I can just wipe the system and install it fresh? (even if it means I need to have an e.g. 2gb partition just for the image to reside in). Clip tool gives ERROR 000732? Can ASP.NET MVC generate elements with lower case name and id attributes Simple way to understand what derivative is I have a question about notation specifically square brackets $[$ and round brackets $($.  My textbook doesn't explain any of this and I cannot find a reliable source online to confirm the difference.  So my question is: What is the difference between round brackets and square brackets in terms of notation in Field Theory? For example, I see $F(x)$ and $F[x]$ in my textbook and I've always assumed they were the same thing. But apparently they're not. Is there ever a time they're the same?  I wanted to know the difference, it may be a silly question but it's something I want to make sure I understand. I have two NTFS partitions, and I don't want to mount them manually every-time I start Ubuntu.   How can I do this?  Is there a tool or a code to use?   If so, is it safe to automount? specially when they are being used by another OS? Is there a way to detect the collapse in wave function of entangled particles instantaneously? Is there a machine or instrument which will notify us instantly when the wave-function has collapsed if we have access to only one of the entangled particles? Note that the wave function collapse will be triggered by observing the particle which we do not have. Or in other words is there a way to identify (instantaneously) if the entangled pair of a particle has been observed and the wave function has collapsed. Am I being scammed by a sugar daddy? I met a guy on a sugar daddy dating site and we talked, I told him my situation and he said he would help.   He asked for my bank account information and I told him you can use cash app or PayPal to go through. He said his account manager deals with his finances. So I set up an account with a bank that I don’t use so that my primary account isn’t used.  Eventually he was in the process of sending me money and we had agreed on a amount that wasn't crazy. But then he asked for my address and I asked why and he said just wondering. Then said he wanted to send me a surprise for being his baby. Which I didn’t give.  Then later on he sent the money and said he sent me a bonus and that I was supposed to use that to pay his mothers caregiver, which he didn’t mention until after.  I kept questioning him and he was saying that he works in construction and he’s trusting me to carry out the errand. Then he said don’t let him down and I asked him if he was threatening me and he kept saying I need this done but didn’t really answer my question.  Am I being scammed? I’m about to change my password but is this a scam? Get index of each capture in a JavaScript regex Can I share my usb keyboard between two computers I love my Filco keyboard, I use it as primary keyboard for my Lenovo thinkpad all time. Now Im moving into the mac world and wonder:  is it possible to split my USB keyboard between two USB ports, ie one at my lenovo and one at my new iMac.   I was thinking that maybe there exists some kind of box with a switch to decide which port should receive the input, or preferable it would be possible to switch between the two via a keyboard shortcut.