In the math competition  of my university, it was asked to prove this result, I have been trying to do it using polar coordinates, but this path has not led me to something concrete $$\int_{-\infty}^{\infty} e^{-\xi^2t}e^{i\xi x}d\xi=\sqrt{\frac{\pi}{t}}e^{-x^2/4t} \quad \forall \ x\in \mathbb{R},t&gt;0$$
I would like to work out the Fourier transform of the Gaussian function   $$f(x) = \exp \left(-n^2(x-m)^2 \right)$$  It seems likely that I will need to use differentiation and the shift rule at some point, but I can't seem to get the calculation to work. Does anyone have any advice?  By the way, I am using   $$\mbox{FT}(f)(k)=\int_{-\infty}^\infty f(x)e^{-ikx}\,\mathrm d x$$  as my definition of a Fourier transform. Many thanks.
How to know if Google Sheets IMPORTDATA, IMPORTFEED, IMPORTHTML or IMPORTXML functions are able to get data from a resource hosted on a website? Vertically align text next to an image? Why won't vertical-align: middle work?  And yet, vertical-align: top does work.   span{   vertical-align: middle; } &lt;div&gt;   &lt;img src="https://via.placeholder.com/30" alt="small img" /&gt;   &lt;span&gt;Doesn't work.&lt;/span&gt; &lt;/div&gt; What are these rays that appear in photograph of sun? In many images of light emitting objects we see such rays. Why do they appear ? What is the math behind their number and direction? I have proved in earlier exercises of this book that $\sqrt 2$ and $\sqrt 3$ are irrational. Then, the sum of two irrational numbers is an irrational number. Thus, $\sqrt 2 + \sqrt 3$ is irrational. My first question is, is this reasoning correct?  Secondly, the book wants me to use the fact that if $n$ is an integer that is not a perfect square, then $\sqrt n$ is irrational. This means that $\sqrt 6$ is irrational. How are we to use this fact? Can we reason as follows:  $\sqrt 6$ is irrational  $\Rightarrow \sqrt{2 \cdot 3}$ is irrational.  $\Rightarrow \sqrt 2 \cdot \sqrt 3$ is irrational  $\Rightarrow \sqrt 2$ or $\sqrt 3$ or both are irrational.  $\Rightarrow \sqrt 2 + \sqrt 3$ is irrational.  Is this way of reasoning correct? Theoretical proof of the constant of speed of light $c$ in vacuum in all frames of references Is there any theoretical proof for the constant speed of light in all frames of references? I know it is experimentally proven but just curious. Some improvements to blockquotes Update  Sorry for the delay! I’ve been thinking about this feedback in between my other projects, and I’ve landed where I’ve started 😕  I’ve made sure to continue highlighting spoilers in gray. The differentiation from normal blockquotes is super important since their behavior is so different. I explored keeping spoilers yellow, but that still has the same theming issues throughout the network. Red seemed too powerful, with its error connotations.  I understand completely that folks are using blockquotes to highlight images. However, I don’t think that’s a  we want to formally pave. I’m ok with being opinionated here—blockquotes should be used for quoting, not as a way to highlight images. I understand that this will be frustrating to Machavity who made excellent points about differentiation in his answer, but I don’t think image highlighting ought to dictate how blockquotes are styled.  I’m also cautious of adding borders to images as explored by user1306322 for the same reasons and for the additional fear of being too prescriptive in how we display images there.  It’s a tough one! I went back and forth with the design team and I still think this is the best path forward.  Original post  We’re making a few changes to blockquotes soon. Historically, we had a bit of feedback that the yellow backgrounds were overpowering, reducing the contrast of the quote itself. The yellow border also lacks contrast with the background. Others have indicated that there should be  and between .  These yellows also compete with theming in our communities, especially those with backgrounds that aren’t perfectly white like English Language &amp; Usage and Role-Playing Games.   We’re proposing we switch to a single gray bar in our blockquotes. This should fix those theming issues and make it easier to distinguish between quotes, spoilers and code.  This will also improve blockquote nesting, since our current nesting is a mess because the darker yellow bar is barely noticeable on the paler yellow background.  Spoilers should pretty much behave exactly the same way. More room for later improvements there.  Here’s how our current blockquotes look. You can click to expand:    Here’s what we propose:    There is a lot of prior art out there for this approach. GitHub has styled theirs this way for a long time. So has Slack.     Let me know what y’all think. I think these changes will be an improvement, and I hope you do too.  Huge thanks to Catija for gathering some meta posts on blockquotes and creating an awesome blockquote playground post for testing. Continued fractions Have the Linux kernel bugs that caused data loss when using a Samsung 850 Pro SSD been resolved? I just got a brand new Samsung 850 Pro SSD, and I would like to install Ubuntu 14.04 onto it.  I have read some pretty ugly stories of data loss with the 850 series SSDs when using Linux due to bugs in the kernel, as well as bugs in the TRIM operation.  I have done my research, but most of what I'm finding is very outdated. For example, see , and , as well as .  My question is, what is the current status? Have all major the bugs been worked out yet? I'm hand-maintaining an HTML document, and I'm looking for a way to automatically insert a link around text in a table. Let me illustrate:  &lt;table&gt;&lt;tr&gt;&lt;td class="case"&gt;123456&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;   I would like to automatically make every text in a TD with class "case" a link to that case in our bug tracking system (which, incidentally, is FogBugz).  So I'd like that "123456" to be changed to a link of this form:  &lt;a href="http://bugs.example.com/fogbugz/default.php?123456"&gt;123456&lt;/a&gt;   Is that possible? I've played with the :before and :after pseudo-elements, but there doesn't seem to be a way to repeat the case number. Environment.TickCount vs DateTime.Now I am trying to solve an integral in $k$-space where I want to calculate the ratio of the average energy of an arbitrary electronic band ($E(k)=k^n)$ to the square of average group velocity of electrons($v_g   = k^{n-1})$ approximately in the form of this expression given by $$L=\frac{\frac{\sum_{\mathbf{k}}k^{n}}{\sum_{\mathbf{k}}1}}{\left(\frac{\sum_{\mathbf{k}}k^{n-1}}{\sum_{\mathbf{k}}1}\right)^2}\;.$$  Now, I want to do this for $d$-dimensions, so I convert this sum to integral.  $$L=\frac{\dfrac{\int_{0}^{k_f}k^{n} d^d k}{\int_{0}^{k_f}1d^d k}}{\left(\dfrac{\int_{0}^{k_f}k^{n-1}d^d k}{\int_{0}^{k_f}1d^d k}\right)^2}$$ where $d^dk  = \dfrac{d \pi^{d/2}}{\Gamma(\frac{d}{2}+1)}k^{d-1}dk$ is the volume element in $d$-dimensional space.  I want to get a simple expression for $L$ and how it depends with $k_f,n$ and $d$. any physical insight into this integral could also be helpful. Let $x,y,z>0$ and $x+y+z=1$, then find the least value of ${{x}\over {2-x}}+{{y}\over {2-y}}+{{z}\over {2-z}}$ Let $x,y,z&gt;0$ and $x+y+z=1$, then find the least value of  $${{x}\over {2-x}}+{{y}\over {2-y}}+{{z}\over {2-z}}$$  I tried various ways of rearranging and using AM > GM inequality. But I couldn't get it. I am not good at inequalities. Please help me.  I wrote $x$ as $1-(y+z)$ and I took $x+y$ as $a$ and the others as $b$ and $c$. And I am trying. Prove by induction that $(x+1)^n - nx - 1$ is divisible by $x^2$ Basis step has already been completed. I've started off with the inductive step as just $n=k+1$, trying to involve $f(k)$ into it so that the left over parts can be deducible to be divisible by $x^2$ but getting stuck on this inductive step. Filter log file entries based on date range Checking incoming call in iphone Playing system sound without importing your own I'm using a class I've derived from DocumentPaginator (see below) to print simple (text only) reports from a WPF application. I've got it so that everything prints correctly, But how do I get it to do a print preview before printing? I have a feeling I need to use a DocumentViewer but I can't figure out how.  Here's my Paginator Class:  public class RowPaginator : DocumentPaginator {     private int rows;     private Size pageSize;     private int rowsPerPage;      public RowPaginator(int rows)     {         this.rows = rows;     }      public override DocumentPage GetPage(int pageNumber)     {         int currentRow = rowsPerPage * pageNumber;         int rowsToPrint = Math.Min(rowsPerPage, rows - (rowsPerPage * pageNumber - 1));         var page = new PageElementRenderer(pageNumber + 1, PageCount, currentRow, rowsToPrint)                        {                            Width = PageSize.Width,                            Height = PageSize.Height                        };         page.Measure(PageSize);         page.Arrange(new Rect(new Point(0, 0), PageSize));         return new DocumentPage(page);     }      public override bool IsPageCountValid { get { return true; } }      public override int PageCount { get { return (int)Math.Ceiling(this.rows / (double)this.rowsPerPage); } }      public override Size PageSize     {         get { return this.pageSize; }         set         {             this.pageSize = value;             this.rowsPerPage = PageElementRenderer.RowsPerPage(this.pageSize.Height);             if (rowsPerPage &lt;= 0)                 throw new InvalidOperationException("Page can't fit any rows!");         }     }      public override IDocumentPaginatorSource Source { get { return null; } } }   The PageElementRenderer is just a simple UserControl that displays the data (at the moment just a list of rows).  Here's how I use my Row Paginator  PrintDialog dialog = new PrintDialog(); if (dialog.ShowDialog() == true) {     var paginator = new RowPaginator(rowsToPrint) { PageSize = new Size(dialog.PrintableAreaWidth, dialog.PrintableAreaHeight) };      dialog.PrintDocument(paginator, "Rows Document"); }   Sorry for the code dump, but I didn't want to miss something relevant. ii) Show that $u$ is not a real part of the function which analytic on $\mathbb{C} \backslash \lbrace 0 \rbrace$ Suppose $u(x,y)=\ln(x^2+y^2)$  i) Show that $u$ is harmonic on $\mathbb{C} \backslash \lbrace 0 \rbrace$  ii) Show that $u$ is not the real part of a function which analytic on  $\mathbb{C} \backslash \lbrace 0 \rbrace$  I manage to show the first part. For the second part, note that $u(x,y)=\ln(x^2+y^2)=\ln(|z|^2)=2 \Re \log(z)$  But this only show that $u$ is not a real part of $\log(z)$. I don know how to show $u$ cannot be the real part of a function which analytic on $\mathbb{C} \backslash \lbrace 0 \rbrace$  Can anyone guide me? Shiny locked Legendaries in ORAS Serebii says that (probably) the only  in ORAS are Kyogre, Groudon, Rayquaza and Deoxys. This means that Reshiram and Zekrom, that were shiny locked in BW/B2W2, now can be found shiny.    Can anyone prove this? I'd like to know if it's true before waste hours in softresetting. Integrate : $\int \frac{x^2}{(x\cos x -\sin x)(x\sin x +\cos x)}dx$ $$\int \frac{x^2}{(x\cos x -\sin x)(x\sin x +\cos x)}\ dx$$   My approach :   Dividing the denominator by $\cos^2x$ we get $\dfrac{x^2\sec^2x }{(x -\tan x)(x\tan x +1)}$ then  $$\int \frac{x^2\sec^2x}{x^2\tan x -x\tan^2x+x-\tan x}\ dx=\int \frac{x^2(1+\tan^2x)}{x^2\tan x -x\tan^2x+x-\tan x}dx$$  But I am not getting any relation between numerator and denominator so that I will take any substitution and solve further please suggest whether it is correct and how to proceed in this. Thanks.