Let $p > 2$ be prime. Show that there exist integers $a,b \geq 0 $ satisfying the congruence $a^2 + b^2 \equiv -1 $(mod $p$).
Show that for any prime $ p $, there are integers $ x $ and $ y $ such that $ p|(x^{2} + y^{2} + 1) $.
How can I clear event subscriptions in C#? Take the following C# class:  c1 {  event EventHandler someEvent; }   If there are a lot of subscriptions to c1's someEvent event and I want to clear them all, what is the best way to achieve this? Also consider that subscriptions to this event could be/are lambdas/anonymous delegates.  Currently my solution is to add a ResetSubscriptions() method to c1 that sets someEvent to null. I don't know if this has any unseen consequences. Types of circular acceleration? To my knowledge there are three types of acceleration when a body (e.g. a rod) is moving in a circle about an axis. These are:   Angular acceleration : this is the rate of change of angular velocity. Tangential acceleration : this is the linear acceleration of the system in a tangential direction to the circle and equals the radius times the angular acceleration. Radial/centripetal acceleration : this is the linear acceleration of the system that is directed inwards towards the center of the circle.   I also think there are two types of velocity:   Linear velocity: this is its velocity in the tangential direction and is constantly changing Angular velocity or angular frequency: this is the rate of change of angle.   Is the above correct? And have I missed anything? Change \headheight for a \pagestyle I have a document class that auto generates header from some user input. The problem is the first page should have a different page header than the others and the headline can be multi line. So \headheight is too small. Here is a demonstration of the problem:  \documentclass[twoside]{scrartcl} \newcommand{\longfirstpagehead}{% \begin{minipage}{10cm} some\\long\\text\\text\\text\\text\\text\\text \end{minipage}} \newcommand{\normalhead}{normal} \usepackage{scrpage2}  \newpagestyle{firstpage}{%     {\longfirstpagehead\hfill} %twoside links     {\longfirstpagehead\hfill} %twoside rechts     {\longfirstpagehead\hfill} %oneside     (\textwidth,1pt) }{%     {\pagemark\hfill} %twoside links     {\hfill\pagemark} %twoside rechts     {\hfill\pagemark} %oneside } %  \newpagestyle{normal}{%     {\normalhead\hfill} %twoside links     {\normalhead\hfill} %twoside rechts     {\normalhead\hfill} %oneside     (\textwidth,1pt) }{%     {\pagemark\hfill} %twoside links     {\hfill\pagemark} %twoside rechts     {\hfill\pagemark} %oneside } %    \pagestyle{normal} \AtBeginDocument{     \thispagestyle{firstpage}}  \usepackage{blindtext} \begin{document}     %\vspace*{1cm}     \blindtext      \blindtext     \pagebreak \end{document}   I tried to read the height with \settototalheight and:   add a \vspace at beginning of document (unnice hack):  still the bad box warning (of cause) first section has different alignment  manually change \headheight in pagestyle  text height and page foot is wrong (can't change this because some scoping?)  use Komascript in pagestyle  \recalctypearea warning  geometry in pagestyle  pretty much the same problem like the manual attemp    So I would like to change the \headheight for the pagestyle firstpage, but the page should still have the same page margins, etc. The problem is that this has to work with \thispagestyle. (Better Ideas are also welcome) Something like \thispagegeometry in geometry would be great. ;) The ending I got is :      the genocide   When I reset my file, my past choices still take effect. I want to start a clean run.  I don't want to buy the game again, what do I do? Actual memory usage of a process The following are the memory usage of mysql and apache respectively on my server. As per the output of pmap say, mysql is using about 379M and apache is using 277M.   [root@server ~]# pmap 10436 | grep total  total           379564K  [root@server ~]# pmap 10515 | grep total  total           277588K   Comparing this with the output of top, I see the values are almost matching.    PID USER      PR  NI  VIRT  RES  SHR S %CPU %MEM    TIME+  COMMAND 10515 apache    20   0  271m  32m 3132 S  0.0  6.6   0:00.73 /usr/sbin/httpd 10436 mysql     20   0  370m  21m 6188 S  0.0  4.3   0:06.07 /usr/libexec/mysqld --basedir=....   Now these values definitely are not the current memory usage of those two processes, since if it were it would've exceeded the 512M ram on my system and I understand the fact that these are the size of the pages assigned to these two processes and not really the size of the memory actively used by them. Now, when we use pmap -x, I see an extra coloumn Dirty which shows far less memory usage for the process. As seen in the example show below, the Dirty coloumn shows 15M as opposed to 379M in the first coloumn. My question is: Is the value under coloumn Dirty is the 'real' amount of memory actively used by that process? If its not, then how can we find out the real memory usage of a process? Not ps and top for the same reasons above. Do we have anything under /proc that will give this info?  [root@server ~]# pmap -x 10436 | grep total total kB          379564   21528   15340 [root@server ~]#   [root@server ~]# free -m              total       used       free     shared    buffers     cached Mem:           489        447         41          0         52        214 -/+ buffers/cache:        180        308 Swap:         1023          0       1023 [root@server ~]# Let $L,M$ two subspaces of the vector space, $V$ such that both $L,M \ne V$.   Prove: $L\cup M \ne V$.     I think this is a case of a proof by contradiction. Lets assume $L \cup M = V$.   Hence, $$\dim(V) = \dim(L) + \dim(M) - \dim(L\cap M)$$  How to proceed? Let f : A → B and g : B → C be bijections. Prove that g◦f : A → C is a bijection  Can someone show me the steps I should take to solve this problem? How to trick a command into thinking its output is going to a terminal Given a command that changes its behaviour when its output is going to a terminal (e.g. produce coloured output), how can that output be redirected in a pipeline while preserving the changed behaviour?  There must be a utility for that, which I am not aware of.  Some commands, like grep --color=always, have option flags to force the behaviour, but the question is how to work around programs that rely solely on testing their output file descriptor.  If it matters, my shell is bash on Linux. I had a look at Knuth's The Art of Computer Programming, book 1. In chapter 1, section 1.2.2, exercise 25, he presents the following algorithm for calculating logarithm:  given $x\in[1,2)$, do the following:  L1: [Initialize] Set $y\leftarrow0$, $z\leftarrow x/2$, $k\leftarrow1$.  L2: [Test for end] If $x=1$, stop.  L3: [Compare] If $x-z&lt;1$, go to L5.  L4: [Reduce values] Set $x\leftarrow x-z$, $z\leftarrow x/2^k$, $y\leftarrow y+\log_b(2^k/(2^k-1))$, and go to L2.  L5: [Shift] Set $z\leftarrow z/2$, $k\leftarrow k+1$, and go to L2.  (The algorithm needs an auxiliary table which stores $\log_b2$, $\log_b(4/3)$, and so on, to as many values as the precision of the computer.  Then Knuth concludes: this exercise is to explain why the above algorithm will terminate and why it computes an approximation of $y=\log_b x$.  Well, I see that it works, but I am not able to explain why... I am using Blender to mix the video from my camera with the audio from my external recording setup.  It works great mostly and I am very pleased with it.  But the problem is that I can't start the audio recorder an even number of thirtieths of a second from when I started the camera.  And Blender seems to only let me move the audio track in the VSE by whole frames (1/30 of a second).  Here are a couple screenshots:   The track on top is the external audio and the one on the bottom is the camera's audio.  Between the two screenshots I have only moved the audio track by one frame.  But as you can see, I need to move it by about 1/2 frame to line up the audio properly with the video.  So can I move a track in the VSE by fractions of a frame? Map of with object references as keys? How do I add .e00 data into my QGIS project? Is There Any Way to get the template Name By full URL for example  If the full url is  www.example.com/hello-world   I need a function that return the template name (page.php)  I know that WordPress matches every url with regular expression to determine the query and decide which template to include.  That's exactly what I need, a built in or (custom made) function that takes the URL and returns back the corresponding template Redirecting stdout to a file you don't have write permission on How to load JAR files dynamically at Runtime? Why is it so hard to do this in Java? If you want to have any kind of module system you need to be able to load JAR files dynamically. I'm told there's a way of doing it by writing your own ClassLoader, but that's a lot of work for something that should (in my mind at least) be as easy as calling a method with a JAR file as its argument.  Any suggestions for simple code that does this? My variables are   LD_LIBRARY_PATH=/usr/lib/oracle/11.2/client64/lib ORACLE_HOME=/usr/lib/oracle/11.2/client64   How to save these variables permanently ? How to exclude/delete files only? Is there a way to exclude all files, in all subfolders, and keep the folder structure? Can you get a Windows (AD) username in PHP? I have a PHP web application on an intranet that can extract the IP and host name of the current user on that page, but I was wondering if there is a way to get/extract their Active Directory/Windows username as well. Is this possible? How can I set up dual monitors with nVidia? Converting a number with comma as decimal point to float