Let $(\Omega, \mathcal{M}, \mu)$ be a measure space, $E \subset \Omega$ and $f: \Omega \to \mathbb{R}$ are measurable. Let $E_t= \{x \in E \space | \space |f(x)| \geq t \}$ for $p&gt;0$. I already showed that for $p&gt;0$: $$\mu(E_t) \leq \frac{1}{t^p} \int_E |f|^pd \mu \space .$$ Now I want to show that  $$\lim_{n \to \infty} \frac{1}{\sqrt{n}} \int_{[0,1]^n} |x|dx= \frac{1}{\sqrt3}\space .$$ $[0,1]^n \subset \mathbb{R}^n$ is the unit-cube, $|x|$ is the euclidean norm and the integration is with the Lebesgue measure in $\mathbb{R}^n$.  I was told that the first step is to show $$\lambda(E_t) \leq \frac{1}{n} \frac{A}{t^2}$$ with using the Lebesgue measure in $\mathbb{R}^n$, with $t&gt;0$, $p=2$, $E=[0,1]^n$ and $f(x)=\frac{|x|^2}{n}-\frac{1}{3}=\frac{1}{n}\sum_{k=1}^n(x^2_k-\frac{1}{3})$ for $A&gt;0$. How do I do that? $i \neq j: \int_E (x_i^2-\frac{1}{3})(x_j^2-\frac{1}{3})dx$ and $|\int_E \frac{|x|}{\sqrt{n}}dx-\frac{1}{\sqrt{3}}| \leq \sqrt{3} \int_E |\frac{|x|^2}{n}-\frac{1}{3}|dx \leq \sqrt{3}(\lambda(E_t)+t\lambda(E$ \ $E_t))$ are also tips we've been given but I don't really know how to apply them.
I am trying to show $$\lim_{n\rightarrow \infty} \int_{[0,1]^n} \frac{|x|}{\sqrt{n}}dx=\frac{1}{\sqrt{3}}$$ where $[0,1]^n$ is the unit cube in $\mathbb{R}^n$. I would appreciate any hint.
Recently I found a claim saying that  $$  \int_0^\infty \left( \frac{\sin ax}{x}\right)\left( \frac{\sin bx}{x}\right) \mathrm{d}x= \pi \min(a,b)/2  $$ from what I can see this seems to be true. I already know that $\int_{0}^\infty \operatorname{sinc}xy\,\mathrm{d}y = \pi/2$, and so independant of $y$. My suspicion is that this is closely related to the integral above.   Can someone give me some suggestions for evaluating the integral above? Also are there any generalizations for the integral? Eg $$ \int_{0}^{\infty} \left( \prod_{k=1}^N \frac{\sin (a_k \cdot x)}{x} \right) \,\mathrm{d}x $$ Where $a_k, \cdots, a_N$ are arbitrary positive constants. It seems related to the , but there are some subtle differences. Could I use a 1/2TB USB 2/3 external HDD with an XBox 360? Keeping tables/figures close to where they are mentioned Let $H$ be a subgroup of $G$ with $[G:H]=n$. Let $N$ be the kernel of the left-multiplication action on the cosets of $H$. Show $[G:N]$ divides $n!$ Let $H$ be a subgroup of $G$ with $[G:H]=n$ and $G$ finite.  Let $G$ act on left cosets of $H$ by left multiplication.  Let $N$ be the kernel of the action.  Show that $[G:N]$ divides $n!$.  Ok so I have a theorem that says that if $G$ is simple then it embeds into $S_n$.  So if $G$ is simple than I'm done because $G$ divides $n!$ and thus $N$ divides $n!$ and hence $[G:N]$ divides $n!$.  Now I also know that $N$ is the largest normal subgroup of $G$ contained in $H$.  Thus if $G$ isn't simple then it contains some non-trivial normal subgroup $K$ and thus $K\cap H$ is normal in $H$ and by the 2nd Iso Theorem $H/K\cap H\cong KH/H$...    As you can see I'm not sure where to go from here. I was hoping to prove that $K\cap H$ wasn't trivial and thus that $N$ couldn't be trivial since it must contain $K\cap H$, but I'm not sure I can, and even if I could I'm not sure that would help me in showing that $[G:N]$ is small enough to divide $n!$.  Can anyone give some guidance?  Thanks. While trying some problems along with my friends we had difficulty in this question.   True or False: The value of the infinite product $$\prod\limits_{n=2}^{\infty} \biggl(1-\frac{1}{n^{2}}\biggr)$$ is $1$.   I couldn't do it and my friend justified it by saying that since the terms in the product have values less than $1$, so the value of the product can never be $1$. I don't know whether this justification is correct or not. But i referred to Tom Apostol's Mathematical Analysis book and found a theorem which states, that    The infinite product $\prod(1-a_{n})$ converges if the series $\sum a_{n}$ converges.   This assures that the above product converges. Could anyone help me in finding out where it converges to? And,   Does there exist a function $f$ in $\mathbb{N}$ ( like $n^{2}$, $n^{3}$) such that $\displaystyle \prod\limits_{n=1}^{\infty} \Bigl(1-\frac{1}{f(n)}\Bigr)$ has the value $1$? Clarify an idiom in a political text Please carefully read the text below:     On 24 November, 1993, a meeting of Leftist intellectuals occurred in   London under the auspices of the Institute for Public Policy   Research (IPPR), which is a Labour-leaning think-tank.  A short document   was circulated in advance of the said meeting to clarify its   purpose.        Among other things, the document declared that the task of the   IPPR was: "To do what the Right did in the seventies, namely to break   through the prevailing parameters of debate and offer a new   perspective on contemporary British politics."    The explanatory document also said, "Our concern is not to engage in a   philosophical debate about foundations of socialism."      If this meant that those foundations were not the appropriate thing to   talk about at the 24 November meeting, then that might have been   right; not every thing has to be discussed at every meeting.  But if   what was meant was that discussion of philosophical foundations is not   what the Left now needs, then I disagree, and if that indeed is what   was meant, then it is curious that the breakthrough by the Right should   have been invoked as an achievement for the left to emulate, for if   there is a lesson for the Left in the Right's breakthrough, it is that   the Left must repossess itself of its traditional foundations on   pain of continuing along its present, politically feeble, reactive   course.  If the Left turns its back on its foundations, it will be   unable to make statements that are truly its own.   Since I've heard only one definition of the part identified in bold, I am at a dead end.  As you know, and according to Oxford's dictionary, the verb "repossess" means: To take back property or goods from SB who has arranged to buy them and can not pay.  Here, the object of verb is a thing not a person, but in the text, first,itself comes before of and its traditional foundations. As a result, I thought that this phrase might be an idiom and so has a different meaning with the original verb.  Please explain the meaning of this phrase. It used to be in settings under the friends tab. The friends tab no longer exists! Where do I find that infuriatingly missing setting now? How can I set up Samba shares to only be accessed by certain users? I have a RAID10 array mounted on Ubuntu Server 12.04. I have created a few folders within the mount point and want the following functionality.  There will be 4 users, 3 of them are windows users: 'one' 'two' &amp; 'three'. 'four' is a media streamer that only needs to access the MEDIA share. One Two and Three need to have full access to the media share and their own personal shares (for documents) which no other users but them can access.  Currently, User Four works perfectly (Has full access to the MEDIA folder and can't access folders owned by other users). The problem is, when logged in as the other user, I can't access either share; (tried using valid users =  and using chmod to add permissions to no avail).  TL;DR: I need to know how to configure Samba properly to restrict access to certain shares for certain users and allow all of them to access one communal folder (all files on a RAID10 mount). Inequality similar to the Schwarz-Pick theorem Given the function   $$ f(x) = \left\{\begin{array}{cc} e^{- \frac{1}{x^2}} &amp; x \neq 0 \\ 0 &amp; x = 0 \end{array}\right. $$  show that $\forall_{n\in \Bbb N} f^{(n)}(0) = 0$.  So I have to show that nth derivative is always equal to zero $0$. Now I guess that it is about finding some dependencies between the previous and next differential but I have yet to notice one. Could you be so kind to help me with that?  Thanks in advance! Intel® Core™2 Duo Processor E4300 - Is it a 64 Bit Processor? I have a PC with an Intel® Core™2 Duo Processor E4300 - I ran CPU-ID on it, and here are the results:    Is this capable of running Windows 64-bit? Needing help picturing the group idea. I have a question;   X and Y are subgroups of a group G.  If $|X|, |Y| &lt; \infty$, show that   $|XY| = \frac{|X||Y|}{|X \cap Y|}$  but I can't really picture what it is talking about to even get started.  Is it to do with looking for different ways so combine the elements {x, y, xy} ?  Thanks Not flying the first leg of the journey – Company cancels the second-leg ticket I've a friend that booked a round-trip journey. He booked A-to-B and B-to-A. Somehow he needed to be earlier at location B, so he took a train and did not take the flight from A to B.  When he went to the airport to take the "back home" flight (B-to-A), he was informed that his ticket was canceled because he did not take the A-to-B flight. They canceled the ticket and sold it to someone else.  Is this normal? The company said it was their right to do so, and he could not get the flight back. The company was Iberia. How would you explain why $e^{i\pi}+1=0$ to a middle school student? Hi I was asked by a friends child who is in middle school why $e^{i\pi}+1=0$. Now I couldn't think of a way to explain it so he would understand. Albert Einstein once said “If you can't explain it simply, you don't understand it well enough” so how would I explain it. *.xhtml url-pattern causes java.lang.NullPointerException at com.sun.faces.context.flash.ELFlash.loggingGetPhaseMapForReading How can I execute a function on pressing the enter key in an  field? My months and days in my File Manager are in German, how do I change that? On the photo you see 'Dez' : It means Dezember (German for December). I also saw one day Mittwoch (German for Wednesday). How do I change that German issue? This is only my File Manager and not completly but partially. Weird. My work has decided to issue their own  (CA) to handle different aspects of our work securely without paying for certificates.   Cryptographically sign emails Encrypt email contents Make access to things like the company  client-certificate based. Revoke the keys of former employees automatically   They sent me a .pem file, and I'm not sure how to add it to my Ubuntu install. The instructions sent were: "Double-clicking on it on a Mac should install it."&nbsp;  How do I proceed? Do I need to do something with  to create a .key, .csr, or .crt file? What spell almost killed Hermione in The Order of the Phoenix? The spell was cast without words, simply a flick of the Death Eater's wand, and a 'streak of purple flame' that lanced across her chest. It was later mentioned that the spell was weakened because of the lack of a verbal component, but still 'did enough damage as it was'. She was in the infirmary healing from the damage for some time. What spell was this? Upper bound and lower bound of an integral I have this problem and I don't know where to start. Can you give me some hint, please?  Suppose that $f:[a,b]\to [0,\infty]$ is a lebesgue integrable function. Let $B=\int_a^b fd\mu.$ Prove that $\sqrt{1+B^2}\leq \int_a^b\sqrt{1+f^2}d\mu\leq 1+B$.