Source: If $f \in L^2$ , then $\hat{f} \in L^2$ and $2\pi ||f||_{L^2} = ||\hat{f}||_{L^2}$, where $\hat{f}$ is the fourier transform
Step in the Stein's proof of Plancherel Theorem
Product of an exponential function of IDD variables if $x_i$ are IID  continuous random variables, with $E[x_i]=μ$, is the following correct?   $E\left[\prod_{i=1}^n(1+⍺\space x_i)^i\right] = \prod_{i=1}^nE\left[(1+⍺\space x_i)^i\right] =  \prod_{i=1}^n(1+⍺\space E\left[x_i\right])^i = (1+⍺\space μ)^n$  thanks! Automatic reboot after ubuntu freeze? How Can I Do It? 74190 counter problem So, I got my 74190 counter shows number like this  9-8-7-6-5-4-3-2-1-0-5-4-3-2-1-0-5-4-3-2-1-0-5-4-3-2-1-0 and so on.  My questions:   What can I do to change the output count down into 5-4-3-2-1-0-5-4-3-2-1-0 and so on? When it changes 0 into 5 it moves so fast. How could I fix this one? There was a reason why I constantly failed physics at school and university, and that reason was, apart from the fact I was immensely lazy, that I mentally refused to &quot;believe&quot; more advanced stuff until I understand the fundamentals (which I, eventually, never did). As such, one of the most fundamental things in physics that I still don't understand (a year after dropping out from the university) is the concept of . No one cared to explain what a field actually is, they just used to throw in a bunch of formulas and everyone was content. The school textbook definition for a field (electromagnetic in this particular case, but they were similar), as I remember it, goes like:  An electromagnetic field is a special kind of substance by which charged moving particles or physical bodies with a magnetic moment interact.  A special kind of substance, are they for real? This sounds like the authors themselves didn't quite understand what a field is so they decided to throw in a bunch of buzzwords to make it sounds right. I'm fine with the second half but a special kind of substance really bugs me, so I'd like to focus on that. Is a field material? Apparently, it isn't. It doesn't consist of particles like my laptop or even the light. If it isn't material, is it real or is it just a concept that helps to explain our observations? While this is prone to speculations, I think we can agree that in scope of this discussion particles actually do exist and laws of physics don't (the latter are nothing but human ideas so I suspect Universe doesn't &quot;know&quot; a thing about them, at least if we're talking raw matter and don't take it on metalevel where human knowledge, being a part of the Universe, makes the Universe contain laws of physics). Any laws are only a product of human thinking while the stars are likely to exist without us homo sapiens messing around. Or am I wrong here too? I hope you already see why I hate physics. Is a field not material but still real? Can something &quot;not touchable&quot; by definition be considered part of our Universe by physicists? I used to imagine that a &quot;snapshot&quot; of our Universe in time would contain information about each particle and its position, and this would've been enough to &quot;de&quot; it but I guess my programmer metaphors are largely off the track. (Oh, and I know that the uncertainty principle makes such (de)serialization impossible — I only mean that I thought the Universe can be &quot;defined&quot; as the set of all material objects in it). Is such assumption false? At this point, if fields indeed are not material but are part of the Universe, I don't really see how they are different from  except for perhaps a more geeky flavor. When I talked about this with the teacher who helped me to prepare for the exams (which I did pass, by the way, it was before I dropped out), she said to me that, if I wanted hardcore definitions,  a field is a function that returns a value for a point in space.  Now this finally makes a hell lot of sense to me but I still don't understand how mathematical functions can be a part of the Universe and shape the reality. Eigenvectors of real symmetric matrices are orthogonal Can someone point me to a paper, or show here, why symmetric matrices have orthogonal eigenvectors? In particular, I'd like to see proof that for a symmetric matrix $A$ there exists decomposition $A = Q\Lambda Q^{-1} = Q\Lambda Q^{T}$ where $\Lambda$ is diagonal. Setting the editing style on a UITableViewCell Vertical spaces between lines in cases environment \documentclass{article} \usepackage{amsmath} \begin{document} \begin{equation*}   X_i =    \begin{cases}     \left[\begin{smallmatrix}       0 \\ 1 \\ 2     \end{smallmatrix}\right]     &amp;     \text{if $i$ is even;}     \\     \left[\begin{smallmatrix}       2 \\ 1 \\ 0     \end{smallmatrix}\right]     &amp;     \text{otherwise.}   \end{cases} \end{equation*} \end{document}   This is the LaTeX code that I wrote based on a cases environment. There's no space between the matrices on the right. What's the best way to add some vertical spaces to separate them?    p.s. This is a simplified form of my original writing, so I cannot just accept suggestions like 'use a matrix transpose to transform the column vector into a row vector.' Dropping one of the columns when using one-hot encoding Lost credit card replaced with new card and new numbers. Credit score affected? I recently left my credit card at a bar so I reported it as lost to my bank, and my bank said I will get a replacement within a few days. For what it's worth, I picked my card up from the bar a few days later and made sure to cut it up.  The new credit card has arrived and has new numbers. Will this negatively affect my credit score?  VISA, Chase Bank, USA. Solutions of $p!q! = r!$ The title says it all, more or less. Obviously, there are infinitely many "trivial" integral solutions of the form $p=n, q=(n!-1), r= n!$. How many non-trivial solutions are there?  I came across this about ten years ago; as far as I can tell, it hasn't appeared here before, so I thought that it might be of interest. I'm actually most interested in finding whether there was any progress made since Proving prime $p$ divides $\binom{p}{k}$ for $k\in\{1,\ldots,p-1\}$ Prove if $p$ is a prime then $p \mid \binom pk$ for $k\in\{1,\ldots,p-1\}$  I don't really know where to begin with this one.   I can see that I have to use the fact that $p$ is prime somewhere - the same is not true for composite numbers, for example $4\nmid 6=\binom42$.  I have checked that this is true for the first few primes:    $3$ divides $\binom 31=\binom32=3$ $5$ divides both $\binom 51=\binom 54=5$ and $\binom 52=\binom 53=10$ $7$ divides $\binom71=\binom76=7$, $\binom72=\binom75=21$ and $\binom73=\binom74=35$. list all suggested packages for currently installed packages How can I calculate the age of a person in year, month, days? How to Mount Guest DOS OS with HOST OS.? I am newbie to VM concept. actually i have installed DOS6.22 under virtualbox and using 32bit windows 7 OS as HOST Operating System, now i want to copy some folders/files from HOST OS to DOS6.22 Guest OS. how can i copy.? Find all solutions to $x^2\equiv 1\pmod {91},\ 91 = 7\cdot 13$ How to understand Cauchy's proof of AM-GM inequality(the last step) The AM-GM inequality: $$a_1a_2\cdots a_n\leq\left(\frac{a_1+\cdots + a_n}{n}\right)^n$$  the trivial case: $a_1a_2 \leq \left(\frac{a_1+a_2}{2}\right)^2 $ is self-evident.  then cauchy use this fact repeatedly. He get: $$a_1a_2\cdots a_{2^m} \leq \left(\frac{a_1+a_2+\cdots +a_{2^m}}{2^m}\right)^{2^m}.\tag{$\ast$}$$  This is easy to understand, and natural, but next step:  he let  $$b_1 = a_1,\ b_2=a_2,\ \ldots,\  b_n=a_n,\  b_{n+1}=\cdots=b_{2^m}=\frac{a_1+a_2+\cdots a_n}{n}$$  replace $a_1, a_2,\ldots ,a_{2^m}$ with $b_1, b_2, \ldots b_{2^m}$ in $(\ast)$ and simplified both left and right. He get the answer. I can repeat his proof, but I don't know why Cauchy can got this idea. How does he got it? Starting yesterday, I have a very odd problem on my Windows 10 desktop-- my CPU is constantly running at 100% (according to Task Manager), and it isn't throttling down. Because of this, it's constantly running at 3.3 GHz and the temperatures are rising to the point where the fans kick in at full force. If I go into the Windows power settings screen and manually change the processor power maximum to 50%, then it's capped at 50% frequency (~1.5 GHz), but it never goes below the maximum possible frequency.  The weird thing is, when I look at the process list, there isn't really anything obvious that's triggering it. None of the processes are abnormal, and though the process CPU percentages do add up to 100%, it doesn't make sense to me that each individual process is taking up as much CPU as Task Manager says it is. Everything seems to be "using" a significantly higher amount of CPU resources than it normally does.  As an attempt to diagnose the problem, I've also tried using Process Explorer. The really weird thing is that Process Explorer shows only 1-2% usage when idle, with the System Idle Process taking up the remaining ~98% as I would expect. Again, nothing looks out of the ordinary in the process list, but the processes are showing the levels of activity I would expect in Process Explorer.  I've already run both Windows Defender and Malware Bytes (a virus was my first guess), but nothing has turned up yet.  This same issue also occurs in Safe Mode, so I'm guessing that it's more along the lines of something to do with Windows itself...  So, to recap:   Windows 10 Desktop Intel 970, 3.3 GHz, 6 cores Task manager and resource monitor graphs shows 100% usage total No individual process is taking up significant resources in the resource monitor list CPU is not throttling down automatically based on the dynamic load, causing high temperatures and the fan to kick in Process Explorer is only showing 1-2% CPU usage   Before I go uninstalling everything, how would I figure out what's causing the issue if nothing weird is showing up in Task Manager or Process Explorer? How to fix AppArmor `could not open` error when installing php+mysql? How to install Ubuntu 18.04 alongside Windows 10 I'm looking for an answer considering both technical limitations and user experience.   Does it need a reload from a technical standpoint?