how to evaluate this definite integral $\int_0^\infty\frac{\sin^2(x)}{x^2}dx$?
Integral of Sinc Function Squared Over The Real Line
Is there an unsubdivide function in Blender? Ubuntu 18.04 opening system setting logs me out I've recently installed Ubuntu 18.04 in a dual-boot configuration with Windows 10. I noticed today that, when trying to open system settings, Ubuntu seems to log me out to the lock screen. When I log back in, all previously-running applications are now closed. Very unsure what to do from here, any help much appreciated. Windows 8 turns program sound volume down randomly I recently upgraded to Windows 8 from Win 7 64 bit and have started to encounter a strange issue. When I have any program running which plays sound, such as a game or spotify, after a random interval (usually around 5-10 minutes) the program sound will set itself really low.    If I Alt-Tab to the desktop and check the Mixer I can see that the Application volume has automatically set itself really low - the other volume levels (i.e. the Device speakers) stay the same.  I'm able to just set the application volume back to a reasonable level and it stays that way for another 5 - 10 minutes then goes quiet again.  My sound is Realtek High Definition Audio and my machine is an Acer Aspire 5738G if that helps in any way.  Any help much appreciated! What's the function like sum() but for multiplication? product()? Python's  function returns the sum of numbers in an iterable.  sum([3,4,5]) == 3 + 4 + 5 == 12   I'm looking for the function that returns the product instead.  somelib.somefunc([3,4,5]) == 3 * 4 * 5 == 60   I'm pretty sure such a function exists, but I can't find it. How do I animate time? What is the number of strings at size $n$ that is constructed from ${a,b,c,d}$ and there is an even num of $a$ What is the number of strings at size $n$ that is constructed from ${a,b,c,d}$ and there is an even num of $a$?  I have tried to answer it as recursion formula in the following logic:  In order to build $ A(n) $, I will split into several cases:   if the string starts with $a$, then I will need another $a$ to make an even number of $a's$. so we get $aa[A(n-2)]$ meaning 2 $a's$ at the start + a valid string at the size of $n-2$. if it starts with $b,c,d$ we get $b[A(n-1)]$ + $c[A(n-1)]$ +$d[A(n-1)]$, meaning a one of the letters + a valid string at the size of $n-1$.   So we will get a recursion formula that looks like that:  $A(n) = A(n-2) + 3A(n-1)$  I'm not 100% on my answer, am I right to approach it in that direction? Is it possible to decompose a hermitian and unitrary matrix $A$ into the sum of the Pauli matrix Kronecker products?   For example, I have a matrix 16x16 and want it to be decomposed into something like $$A =\sum_{i,j,k,l}h_{ijkl}\sigma_i\otimes\sigma_j\otimes\sigma_k\otimes\sigma_l$$ I can't find any clear explanation how it can be done. I found only this for two-dimentional case: Frobenius Inequality Rank I was looking for an answer for this problem in terms of matrices, but I really don't know how to prove this result. The proposition says that:  Let $A\in M_{m\times k}(\mathbb{C})$, $B\in M_{k\times p}(\mathbb{C})$ and $C\in M_{p\times n}(\mathbb{C})$, then $\textrm{rank}(AB)+\textrm{rank}(BC)\leq \textrm{rank}(B)+\textrm{rank}(ABC)$ How to line up the pieces at the coast near the shipwreck? In The Witness, I was walking along the coast, and I came upon this:    Now, since I've learned      that mazes can be embedded in the environment,   I see the puzzle here, but I just can't get the pieces to line up well enough.  Standing on the couch and walking all around it doesn't seem to get me a       functional maze.   Where do I need to stand to activate the puzzle successfully? Explain what is a transistor in plain English! closed form for $\binom{n}{0}+\binom{n}{3}+\binom{n}{6}+...+\binom{n}{n}$ Is the Ratio of Associative Binary Operations to All Binary Operations on a Set of $n$ Elements Generally Small? I started thinking about the number of associative (binary) operations on a set with $n$ elements today.  Looking online I found  which indicates only $113$ of the possible $19,683$ operations on a three-element set satisfy association.  So, $50\%$ of binary operations on a two-element set satisfy association, and less than $0.6\%$ of all binary operations on a three element set satisfy association.  For an $n$-element set, where $n$ denotes a natural number, is the ratio $R_{n}=A_{n}/B_{n}$, of the number of associative binary operations $A_{n}$ to the number of all binary operations $B_{n}$, in general small?  I don't mean the following questions as equivalent, but since they seem more concretely answerable in principle, does   $$ \lim_{n \to +\infty} \frac{A_{n}}{B_{n}} = 0 ? $$       Also, is $F : n \to R_{n}$ a monotonically decreasing function?  Some Background of the Question: In his Linear Algebra Problem Book 1995 on p. 6 Paul Halmos writes "The commonly accepted attitudes toward the commutative law and the associative law are different.  Many real life operations fail to commute; the mathematical community has learned to live with that fact and even to enjoy it.  Violations of the associative law, on the other hand, are usually considered by specialists only."  For all I know, Halmos might only have written that to motivate the study of Linear Algebra and doesn't quite literally mean what he appears to say.  But, if he means what he appears to say, and if $F : n \to R_{n}$ is monotonically decreasing, or $R_{n}$ is generally small, I think there's something amiss with what Halmos says, since non-associative operations seem so common that one may as well enjoy them. How do I install Ubuntu Server (step-by-step)? I am able to parse strings containing date/time with time.strptime  &gt;&gt;&gt; import time &gt;&gt;&gt; time.strptime('30/03/09 16:31:32', '%d/%m/%y %H:%M:%S') (2009, 3, 30, 16, 31, 32, 0, 89, -1)   How can I parse a time string that contains milliseconds?  &gt;&gt;&gt; time.strptime('30/03/09 16:31:32.123', '%d/%m/%y %H:%M:%S') Traceback (most recent call last):   File "&lt;stdin&gt;", line 1, in &lt;module&gt;   File "/usr/lib/python2.5/_strptime.py", line 333, in strptime     data_string[found.end():]) ValueError: unconverted data remains: .123 One click to site meta? Right now from a main site, it takes TWO CLICKS to get to http://meta.SITE.stackexchange.com/ - ONE!  TWO!    Why not ONE click on the &quot;Hot Meta Posts&quot; &amp; &quot;Featured On Meta&quot; - What ticket reservation do I need for a Schengen visa? I have a visa appointment this Friday 19 at the Italian Consulate in Philadelphia and I have a question about the flight reservation requirement: According to the Visa requirements "Confirmed flight reservation for a round trip showing complete itinerary."  Does this mean I have to reserve the tickets, pay and print the itinerary?  In this case if my visa gets denied then I will lose the money I paid or can I just show the planned itinerary (written not booked)? in this case the travel agency would not give me the Travel Insurance coverage until I book and pay for the reservation.   I am a US green card holder and applying from North Carolina. How do I center the text horizontally and vertically in a TextView, so that it appears exactly in the middle of the TextView in Android? Multiple authors in one row with refmark, multiple affiliations in separated columns in another row I wonder how we can list author names in the following format in LaTeX.   I have 5 authors with 3 different affiliations. I am trying to list all author names with refmark in one row, and 3 affiliations with refmark in another row, but each in one separated column. It should look like this:          author1{1} author2{2} author3{1} author4{2} author5{3}            {1}department      {2}department        {3}department               school              school              school               email                email              email   I did saw some papers that have the above format, but cannot search for any guideline. IEEE guideline suggests another way to list authors, but sadly it does not look that neat          author1{1} author2{2} author3{1} author4{2} author5{3}                         {1} department, school                               email                         {2} department, school                               email                         {3} department, school                               email Let $a\ge 2$ be an integer. Show that for positive integers $m,n$, we have $a^n - 1$ divides $a^m - 1$ if and only if $n$ divides $m$.   I am having trouble showing this. I've seen a similar problem on here but it only shows one direction for the proof. I would like to see both directions.  This is what I have so far: Let $m = nq + r$, then $a^m - 1 = a^{nq + r}- 1 = a^{nq}a^r - 1 = a^{nq}a^r - a^r + a^r - 1 = a^r(a^{nq} - 1) + a^r - 1$. Since $0\le r\lt n$, $r = 0$ then $a^m - 1$ = $a^{nq} - 1$ so $a^m$ = $a^{nq}$ . Thus $m = nq$ hence $n\mid m$. How do I properly install a system app given its .apk? I removed a system app (com.android.mms) and I have the .apk needed to restore it, however it won't install through the standard channels (running the .apk gives me "application not installed"). What's the proper way to install a system app's .apk?