Help with Polar Decomposition Proof
polar decomposition of a square A when A is not full rank/invertible
Can you flash any android device with "fastbood flash" command (of the adb)? I'm wondering if there's any log which stores information about a particular program such as when the program has started and closed manually on Windows such as MS Word, or Chrome. Any ideas? How to output HTML from JSP  block? I just started learning JSP technology, and came across a wall.  How do you output HTML from a method in &lt;%! ... %> JSP declaration block?  This doesn't work:  &lt;%!  void someOutput() {     out.println("Some Output"); } %&gt; ... &lt;% someOutput(); %&gt;   Server says there's no &ldquo;out&rdquo;.  U: I do know how to rewrite code with this method returning a string, but is there a way to do this inside &lt;%! void () {  } %> ? Though it may be non-optimal, it's still interesting. Label PGFPlot table points with coordinates I would like to label each plotted point with its x and y coordinate. I am able to label with a single one using the MWE below. I must avoid manually labeling each point since my actual table size is larger and will be changed out regularly. \documentclass{standalone}   \usepackage{tikz} \usepackage{pgfplots} \pgfplotsset{compat=1.16}  \begin{document}  \pgfplotstableread[col sep=&amp;,row sep=\\]{  x     &amp;  y     \\  0.940 &amp;  0.992 \\  0.992 &amp; -0.940 \\ -0.992 &amp;  0.940 \\ -0.940 &amp; -0.992 \\ }\mytable  \begin{tikzpicture}  \begin{axis}     \addplot+[only marks, nodes near coords]table[x=x,y=y,meta=x]{\mytable}; \end{axis}  \end{tikzpicture} \end{document}  which creates the following figure  The desired output is for each point to be labelled with both coordinates. For specific example the top left point would be labelled with &quot;(0.94, 0.992)&quot;. What language is the Ubuntu Software Center written in? Just been wondering, since elementary AppCenter is written with Vala and Gtk+3.  Also, what GUI wrapper does it use? This question led to a new package:      I'm giving my students an old exam and barely remembered to change the date at the top. A colleague and I joked about how long we could reuse the same exam, and it made me wonder if I could use LaTeX to create an exam that looked like it was written in 1963 on a typewriter.  Here's what I'd like to see:   Typewriter font even in math mode. (Several ways to accomplish this; is there a "best" way?) Slight inconsistencies in the printing of each letter. I attempt this below adapting code taken from . Each word has a random vertical shift, though I'd like to see it reduced to each letter having its own vertical shift. Other typewriter-like irregularities would be great. It would be great if math-mode delimeters \left(, \right}, etc., can be created in a piece-meal fashion as they did in days of old. Fraction bars, tops of square-root symbols -- any horizontal line, really -- is made up of a series of en-dashes that don't quite match up.   Here is what I've come up with so far:  \documentclass[10pt]{article}% This is a document class providing more font size options \usepackage{graphicx} \usepackage{ifthen} \usepackage[textwidth=7.5in,textheight=9.5in]{geometry} \pagestyle{empty}  % thanks to Bruno Le Floch: https://tex.stackexchange.com/q/9331/4012 % and in his comments to https://tex.stackexchange.com/a/29458/4012 \usepackage[first=-1,last=1]{lcg}% you can play around with these values \makeatletter \newcommand{\globalrand}{\rand\global\cr@nd\cr@nd} \makeatother  \newcommand{\randomvshift}[1]{\globalrand\raisebox{\value{rand}pt}{#1}}  %%% thanks to Martin Scharrer: https://tex.stackexchange.com/q/11598/4012 \makeatletter \def\typewriter#1{%     \@typewriter#1 \@empty } \def\@typewriter#1 #2{%    \randomvshift{#1}\space    \ifx #2\@empty\else     \expandafter\@typewriter    \fi    #2% } \makeatother \begin{document}\tt  \typewriter{Math 101 Fall 1963}  \begin{enumerate} \item   \typewriter{$\frac{\texttt{d}}{\texttt{dx}}\texttt{(x)}^2$}  \item \typewriter{State the Fundamental Theorem of Calculus.}  \end{enumerate}  \end{document} Show that $\left| \frac{1}{x}-\frac{1}{z} \right| \le \left|\frac{1}{x} - \frac{1}{y} \right| + \left|\frac{1}{y} - \frac{1}{z} \right|$. I'm trying to prove that  $$ \left| \frac{1}{x}-\frac{1}{y} \right| $$  is a metric on $X = \mathbb R^+$.  The properties of non-negativity and symmetry are obvious (do I actually even need to prove them?), but the tricky one is, as usual the inequality property, i.e.  $$ d(x,z) \le d(x,y) + d(y,z) \text{ for any } z \in X. $$  So, ok, I have that  $$ d(x,z) \le d(x,y) + d(y,z) \\ \iff \left| \frac{1}{x}-\frac{1}{z} \right| \le \left|\frac{1}{x} - \frac{1}{y} \right| + \left|\frac{1}{y} - \frac{1}{z} \right| \\ \iff \left| \frac{1}{x} \right|- \left| \frac{1}{z} \right| \le \left| \frac{1}{x}-\frac{1}{z} \right| \le \left|\frac{1}{x} - \frac{1}{y} \right| + \left|\frac{1}{y} - \frac{1}{z} \right| $$  ...and that's about as far as I've gotten–which is to say, not very far. I also tried multiplying both sides by $ \left| \frac{1}{x}-\frac{1}{z} \right| $, but that wasn't very fruitful. If someone could point me in the right direction, it'd be greatly appreciated. Thanks. How could I retrieve the exact duplicate value from conditionally formatted list that highlight duplicates? I have imported a webpage table in to Google Sheets. It frequently being updated. A series of calculations has been conducted on the sheet named Shawn Of the link spreadsheet.  I have filtered the my desired results into the Calculation sheet of that linked document... which is in columns A5:A. I then applied =ARRAYFORMULA(if(A5:A200=&quot;&quot;,,A5:A200*{12,13,37,101,111})) in columns B5. The Results of the formula extends from B5:F. This results from a5:F was subsequently conditionally formatted to highlighted duplicates in the aforementioned, i.e. a5:f using the lists of values in K5:K and I5:5. In Columns A5:F, is it possible to generate/place values in the range A5:f with matching name of states and aligned values in J5:K and H5:I? Let $(\Omega, \mathcal F,P)$ be a probability spaces and $H$ be a Hilbert space. By a random operator $A$ from $H$ to $H$ we mean a linear continuous mapping from $H$ into the Frechet space $L_0^H (\Omega, \mathcal F,P)$ of all $H$-valued random variables. My question is: How Can we define $|A|$ and the adjoint $A^*$? Why does the following hold:  \begin{equation*} \displaystyle \sum\limits_{n=0}^{\infty} 0.7^n=\frac{1}{1-0.7} = 10/3\quad ? \end{equation*}  Can we generalize the above to      $\displaystyle \sum_{n=0}^{\infty} x^n = \frac{1}{1-x}$ ?   Are there some values of $x$ for which the above formula is invalid?  What about if we take only a finite number of terms? Is there a simpler formula?     $\displaystyle \sum_{n=0}^{N} x^n$   Is there a name for such a sequence?    This is being repurposed in an effort to cut down on duplicates, see here: Coping with  duplicate questions.  and here: . Setup restarts the laptop and then restarts the installation all over and over again? Long site name hides the diamond in profile page Communities 'Search on Yammer' link in search center refiners I have a heavily branded site I'm working on and can't find where the "Search on Yammer" link is being populated from. It appears within the search refiner/filter web part in the left column but none of the display templates reference this.   I'd like to be able to customize the link, placement on the page, etc. without resorting to jQuery to move elements around the page.     And the markup coming through looks like this. The refiners themselves come through a series of display templates but the Yammer link is nowhere that I can see in any of them.   &lt;div webpartid="f09e98eb-6dc3-4a9b-8dbf-65d0fbc7b7c3" haspers="false" id="WebPartWPQ4" width="100%" class="ms-WPBody " allowdelete="false" style=""&gt;     &lt;div class="ms-ref-ctrl"&gt;         &lt;div class="ms-ref-refiner"&gt;             &lt;div id="Container"&gt;                 &lt;h2 class="ms-displayInline"&gt;                     &lt;a class="ms-ref-name ms-displayInlineBlock ms-srchnav-link-selected ms-link" href="javascript:void(window.open(                     'https://www.yammer.com/search?trk_event=o365_search&amp;amp;search='+                     encodeURIComponent(Srch.ScriptApplicationManager.get_current().queryGroups.                     Default.dataProvider.get_currentQueryState().k),'_blank'));" title="Open a new tab to search for this on Yammer"&gt;Search on Yammer&lt;/a&gt;                 &lt;/h2&gt;             &lt;/div&gt;         &lt;/div&gt;     &lt;/div&gt; &lt;/div&gt; &lt;div componentid="ctl00_SPWebPartManager1_g_f09e98eb_6dc3_4a9b_8dbf_65d0fbc7b7c3_csr" id="ctl00_SPWebPartManager1_g_f09e98eb_6dc3_4a9b_8dbf_65d0fbc7b7c3_csr"&gt;             ... refiners populated through display templates ... &lt;/div&gt; Changing the size of the mouse pointer in Ubuntu MATE 18.04 does not work I have Ubuntu 18.04 with the gnome-desktop  and parallel the Ubuntu MATE desktop installed. While I can easily change the size of the mouse pointer (DMZ-White) in the Gnome3 desktop , this does not work on the MATE desktop. I use for this the dconf-editor.  Maybe someone knows a solution?  Thanks a lot. Can you overload controller methods in ASP.NET MVC? I'm curious to see if you can overload controller methods in ASP.NET MVC.  Whenever I try, I get the error below.  The two methods accept different arguments.  Is this something that cannot be done?     The current request for action 'MyMethod' on controller type 'MyController' is ambiguous between the following action methods: What does Redis do when it runs out of memory? How to export / dump a MySql table into a text file including the field names (aka headers or column names) What else can you do in the turn you ready an action? Prove that $\Vert f(b)-f(a)-f'(a)(b-a) \Vert\leq \sup_{x\in [a,b]} \Vert f''(x)\Vert\Vert b-a\Vert^2$ Let $\Omega$ be an open set in $\Bbb{R}^n$ and $f:\Omega\to \Bbb{R}^m $ be of class $\Bbb{C}^2.$ Let $a,b$ be in $\Omega$ such that $[a,b]\subset \Omega.$ Prove that  \begin{align}\Vert f(b)-f(a)-f'(a)(b-a) \Vert\leq \sup_{x\in [a,b]} \Vert f''(x)\Vert\Vert b-a\Vert^2.\end{align} where $$f'(a)(b-a)=\big&lt;\nabla f(x),(b-a)\big&gt;=\sum^{n}_{k=1}\frac{\partial f_i}{\partial x_k}(a)(b_k-a_k).$$  I  think the idea of Mean Value Theorem for $C'$ functions in $\Bbb{R}^n$ () will help me in showing it but I don't know how to! Please, can anyone show me a proof or reference? Left and right subscript / superscript I am trying to put two subscripts at the left and right of a character. For example, something like:  _{t} p_{x}  where p is in the middle. How do you do this?