Show that the square of any prime number is the IONOF of an integer.
Show that the square of any prime number is the Factof of some integer
Value of having done applied math research when applying to PhD programs in pure math I will be a second year math under-grad in September, and currently doing an internship at applied math research institute. Most probably, I will come up with a research paper at the end of the internship.  However, my interest after my bachelors is to do PhD in pure math from a top math school.My question is this:     Would the admission committee consider my current research since it is in applied math?    In addition, if anyone can suggest me some ideas on how can I bridge the gap between my paper in applied math and my interest in analysis, I would be extremely grateful.  P.S.:  is a related but different question also of mine. Given $a,b,c,d \in \mathbb{Z}$, there is $x,y$ such that  $$(a^2 + b^2)(c^2 + d^2) = x^2 + y^2$$  One can show this by considering the complex number $a + bi$ and $c+ di$, using complex properties to deduce that $x = ac - bd, y = ad + bc$ is a solution.   However, given that either $a^2, b^2$ are distinct or $c^2,d^2$ are distinct, then how can one can find nonzero solutions $x,y$?  Sorry, I am assuming $a,b,c,d$ are nonzero. At  I've read that being $\sin(x)=\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)!}x^{2n+1}$ by using  we get $\sin'(x)=\sum_{n=0}^{\infty}(2n+1) \frac{(-1)^n}{(2n+1)!}x^{2n}=\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n)!}x^{2n}=\cos(x)$ but the page  says that if $f(x)=\sum_{n=0}^{\infty}c_n(x-a)^n$ then inside the radius of convergence of the power series we have $f'(x)=\sum_{n=1}^{\infty}nc_n(x-a)^{n-1}$ so shouldn't it be $\sin'(x)=\sum_{n=1}^{\infty}(2n+1)\frac{(-1)^n}{(2n+1)!}x^{2n+1-1}=\sum_{n=1}^{\infty}\frac{(-1)^n}{(2n)!}x^{2n}$?  Now I understand that this last result has a missing term due to the index of the series being $1$ and not $0$ but being the $1$ mandatory because of the above theorem and since trying to shift index doesn't help (we get $\sin'(x)=\sum_{n=0}^{\infty}\frac{(-1)^{n+1}}{(2n+2)!}x^{2n+2}$) how can the result that ProofWiki shows be rigorously justified?   (I mean, I think we can't just say "oh, there's a missing term!" after having applied the theorem and magically shift the index of the series to obtain the desired result) I wonder a generalization of Jensen's inequality: let $\mathbf{X,Y}$ be two positive definite matrices, can we obtain the following Jensen like inequality $$(1-\lambda)\mathbf{X}^{-1}+\lambda\mathbf{Y}^{-1} \succeq((1-\lambda)\mathbf{X}+\lambda\mathbf{Y})^{-1}$$ for any $\lambda \in (0,1)$, where the notation $\mathbf{A}\succeq \mathbf{B}$ represents matrix $(\mathbf{A-B})$ positive semi-definite. I was rewatching the Breaking Bad episode S04E08 (Hermanos) and I didn't quite understand why exactly Don Eladio killed Gus's partner, but he let Gus live. He says:     The only reason you're alive and he's not, is because I know who you are.   But I still have yet to fully understand. Understanding minipages - aligning at top I would like to be able to align two minipage at the top, without worrying about what is in the minipage.  I am aware of the optional [t] parameter, but still have some confusion as to how this all works.  I tend to think of a minipage as a regular page in that whatever is placed on it starts at the top and fills the page going across and then down the page, but this does not appear to be the case.  In the MWE example below, I have color coded the sections for reference purposes.   The first case (black) works just as I would expect. The two minipages are aligned exactly as I want with the [t] specifier.  No question here.  The next case (red) only works if the inner minipage also has the [t] specifier. Why is that necessary?  I would have thought that any content would be placed at the top of this page, and since both the outer minipage had a [t] things should have aligned at the top. I would like to place a graphic in the second mini page, and still have the two aligned at the top.   The simple case (blue) of using \includegraphics directly does not result in the content being aligned at the top.  So, then I attempt to put this in a \parbox (green) and this too does not yield the desired results.  I also try to place the graphic inside an inner minipage (brown) and specify the [t] alignment, and even though this worked in the red section, it does not here.  The only difference between the red and brown being that the red section had text, whereas the brown section has a graphic.  I am aware of question regarding  which suggests to place \vspace{0pt} at the top of the minipage, and that does provide the desired alignment (cyan), but adds vertical space at the top.  So, my second question is why is the \vspace{0pt} necessary when a graphic is included, but not when it is text.  This appears to be a hack and wondering if there is a replacement for minipage that works better without having to worry about exactly what is in the minipage?  Also, why did wrapping the figure in a parbox (green) or another minipage (brown) not solve this problem?   Here is the MWE:  \documentclass{article} \usepackage[demo]{graphicx} \usepackage{lipsum} \usepackage{xcolor}  \begin{document} \section{This Works - Aligned at top as desired} \begin{minipage}[t]{0.6\linewidth}     \lipsum[1] \end{minipage} \hfill \begin{minipage}[t]{0.3\linewidth}     \lipsum[2] \end{minipage}  \color{red} \section{Why does inner mini page required the [t]?} \label{Minipage-in-minipage} \begin{minipage}[t]{0.6\linewidth}     \lipsum[3] \end{minipage} \hfill \begin{minipage}[t]{0.3\linewidth}     \begin{minipage}[t]{\linewidth}% Why is [t] required here         \lipsum[4]     \end{minipage} \end{minipage}  \color{blue} \section{Why are These not aligned at top?} \begin{minipage}[t]{0.6\linewidth}     \lipsum[3] \end{minipage} \hfill \begin{minipage}[t]{0.3\linewidth}     \includegraphics[width=\linewidth]{foo} \end{minipage}  \color{green} \section{Why are These not aligned at top?} \begin{minipage}[t]{0.6\linewidth}     \lipsum[3] \end{minipage} \hfill \begin{minipage}[t]{0.3\linewidth}     \parbox{\linewidth}{\includegraphics[width=\linewidth]{foo}} \end{minipage}  \newpage \color{brown} \section{Why does this not work with the [t] as it did in red section?} This worked fine in Section \ref{Minipage-in-minipage}  \begin{minipage}[t]{0.6\linewidth}     \lipsum[3] \end{minipage} \hfill \begin{minipage}[t]{0.3\linewidth}     \begin{minipage}[t]{\linewidth}% Even with [t] here, this does not work         \parbox{\linewidth}{\includegraphics[width=\linewidth]{foo}}     \end{minipage} \end{minipage}   \color{cyan} \section{Adding a \textbackslash{}vspace\{0pt\} works} \begin{minipage}[t]{0.6\linewidth}     \vspace{0pt}% Hack!! But does do the job     \lipsum[3] \end{minipage} \hfill \begin{minipage}[t]{0.3\linewidth}     \vspace{0pt}% Hack!! But does do the job     \begin{minipage}[t]{\linewidth}% Why is [t] required here         \parbox{\linewidth}{\includegraphics[width=\linewidth]{foo}}     \end{minipage} \end{minipage} \end{document} When using sudo powers and calling gedit the top level menu with File Edit View Search Tools Documents Help is missing.  If I create user ID root and sign in once and call gedit will it create the necessary user configuration files that might fix this?  Will this also make the annoying error messages go away whenever I use sudo, gksu or pkexec as a regular user with elevated privileges togedit ?  Will there be other benefits with Nautilus and other Gnome derivative Ubuntu applications?  NOTE: by sudo powers you can google pkexec gedit 3.5k hits, gksu gedit 40k hits or sudo gedit 500k hits. I'm in the minority using the first method but I believe it will become the standard in Ubuntu 17.04.    June 12, 2017 Update Here are a list of errors you get when using pkexec gedit:  (gedit:13003): Gtk-WARNING **: Calling Inhibit failed: GDBus.Error:org.freedesktop.DBus.Error.ServiceUnknown: The name org.gnome.SessionManager was not provided by any .service files  ** (gedit:13003): WARNING **: Set document metadata failed: Setting attribute metadata::gedit-spell-enabled not supported  ** (gedit:13003): WARNING **: Set document metadata failed: Setting attribute metadata::gedit-encoding not supported  ** (gedit:13003): WARNING **: Set document metadata failed: Setting attribute metadata::gedit-position not supported   On this  there is a commitment to fix these distracting messages as of June 5, 2017. No indication is given for when the fix will be upstream nor if the functionality will be implemented or the error messages will simply disappear.  Note the bug report was filed over a year ago and over that time it only effects 18 people. Bold name in bib when having a prefix in last name There is a socket related function call in my code, that function is from another module thus out of my control, the problem is that it blocks for hours occasionally, which is totally unacceptable, How can I limit the function execution time from my code? I guess the solution must utilize another thread. How do I capture artificial light sources better? I am quite new to photography and was looking into how I can take better indoor photographs, particularly around night time.  I'm currently in a hotel room for the night and decided to try and take a picture of my room to test out my tripod which I haven't used before. However I noticed no matter what I set my camera settings to the artificial lights (particularly the lamp on the left, image attached) just appeared very washed out. You can't tell any detail about the lamp shade all you can see is just some sort of light but not much to it.  I tried a variety of fstop/shutter speed settings and tried a variety of ISO settings but they all appeared pretty much the same unless I massively underexposed the image.  Given that I wondered whether the solution would be to take a HDR image or take a normal exposure and an underexposed image and blend them later? I understand that artificial light like this can be very hard and harsh but I wondered if there's a way of improving the result with looking away from it?  I wasn't trying to take any particular special picture here but just test out my camera a bit more but this was the first thing I noticed and I didn't know how to fix it. Any tips would be greatly appreciated. What is a good, short, word to describe a software engineer? What should I call a person who write software, computer programs ? I know he/she is a software engineer, can I call him/her as "Coder"? Moment generating functions/ Characteristic functions of $X,Y$ factor implies $X,Y$ independent. This is solely a reference request. I have heard a few versions of the following theorem:     If the joint moment generating function $\mathbb{E}[e^{uX+vY}] = \mathbb{E}[e^{uX}]\mathbb{E}[e^{vY}]$ whenever the expectations are finite, then $X,Y$ are independent.   And there is a similar version for characteristic functions. Could anyone provide me a serious reference which proves one or both of these theorems? How to make generated classes contain Javadoc from XML Schema documentation What's the best way to deal with hot/stuck pixels in long exposure night photographs? I discovered red/blue/white pixels on my night photographs tonight. Trying to confirm this, I took a black image with the lens on and indeed I saw a whole lot of them (I didn't count them but at least a good fifty on the whole picture).   I'm not exactly sure if they are hot pixels, stuck pixels or anything else.  I quickly gave a check to my last exported JPEG files (I get RAW out of the camera) and couldn't find anything. After a quick search, I've seen that Lightroom is caring about those pixels after RAW conversion.  I've read that, with Canon DSLRs, there is a software process that could take care of that directly on my camera. I don't know if I can use it without getting things even more messy.   My question is: Considering my crazy pixels are not visible on my pictures, should I get too emotional about them? My camera is under warranty but is it really worth it to return it to Canon? If it's only to receive it back 1 month later after Canon just used the very same menu to map out the pixels, it's a bit ridiculous. Could this be a sign that my camera has a bigger problem?  It seems related to shutter speed. When shooting a black image at 1/250s I get nothing. When shooting the same setup but 1s, I get a few of them. The more I increase the shutter speed, the more crazy pixels I get. JSF convertDateTime renders the previous day Simple property of a valuation on a field I am trying to create an Apex Trigger that performs a Rollup from a custom object to Contacts.   Here is the error that I am receiving:      Didn't understand relationship 'Assignments__r' in FROM part of query   call. If you are attempting to use a custom relationship, be sure to   append the '__r' after the custom relationship name. Please reference   your WSDL or the describe call for the appropriate names.   The child relationship name is Assignments.  Below is the code:      trigger RollupActiveAssigmentLevelsToContact on pse__Assignment__c (after insert, after update, after delete) {      set&lt;Id&gt; set_id = new set&lt;Id&gt;();      List&lt;Contact&gt; con_list = new List&lt;Contact&gt;();      if(trigger.isInsert || trigger.isUpdate){         for(pse__Assignment__c myAssignment : trigger.New) {             set_id.add(myAssignment.pse__Resource__c);         }     }     else if(trigger.isDelete){         for(pse__Assignment__c myAssignment : trigger.Old){             set_id.add(myAssignment.pse__Resource__c);         }     }      if(trigger.isAfter &amp;&amp; (trigger.isInsert || trigger.isUpdate || trigger.isDelete)){         con_list = [SELECT id,Total_Levels__c, (SELECT id,name FROM Assignments__r WHERE Active_Assignment__c = TRUE) FROM Contact WHERE id IN :set_id];          for(Contact con : con_list){             if(con.Assignments__r.size()&gt;0)                 con.Total_Levels__c = con.Assignments__r.size();             else                 con.Total_Levels__c = 0;         }         if(!con_list.isEmpty())         update con_list;     } }   Below is a snap of the Child Relationship name: An alternative software to Live Writer? Prove that between any two roots of $f$ there exists at least one root of $g$ $$f(x)= 1 - e^x\sin(x)$$   $$g(x)= 1 + e^x\cos(x)$$   Prove that between any two roots of $f$ there exists at least one root of $g$.   I know Rolle's Theorem and the Intermediate Value Theorem (I think) need to be used. Can someone show a step by step proof for this? I'm designing a small module that breaks out a DIP SPI EEPROM to a 2x5 pin header socket(more info here: ). I want to mount the socket on the end of the board, with the pins lying parallel to the board.  I think two 5x1 surface-mount header footprints, on both sides of the board right near the end would do the job. Trouble is, KiCAD doesn't come with such a footprint(if it does, I haven't found it yet). Where might I find one? If one doesn't exist, that's OK; I'll have a shot at making my own.