The composition was defined as follow:  $(a,b)$ $\in$ $(R;S)$ $\Leftrightarrow$ $\exists c |  (a,c) \in  R \\  $&amp;$ (c,b)  \in S$ .  Also the composition was defined as the binary product of two matrix .  $SoR$ (relation of composition)$  = R * S $ (product of matrices) where the sum is the disjunction and the product is the conjunction.  If our two relations R and S are two convex polygon  Is there a geometric interpretation of the composition of two convex polygon ? for example :     The Green polygon was the searched result .
the composition was  defined as follow:  (a,b) \in (R;S) &lt;=&gt; there is c | (a,c) \in R  and (c,b) \in S .   If our two relations R and S are two convex polygon   Is there a geometric interpretation of the composition of two convex polygon ? for example :   the green polygon was the result of composition. How can i describe geometrically the result !!!
Automatic line breaking of long lines of text? Here, it is just a example. I have a line:  aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaa   Now, I want to let LaTeX automatically wrap the line according to the width of the text block/margins. Let $f(x)\in C^2:[0,1]\rightarrow\mathbb{R}$ satisfy $f(0)=0,f(1)=1,f'(0)=f'(1)=0$, prove that: $$\max_{x\in[0,1]}|f''(x)|\ge4$$  By using Taylor series I can prove that $\max_{x\in[0,1]}|f''(x)|\ge2$, but I cannot go further. Preloading images with jQuery I'm looking for a quick and easy way to preload images with JavaScript. I'm using jQuery if that's important.  I saw this here ():  function complexLoad(config, fileNames) {   for (var x = 0; x &lt; fileNames.length; x++) {     $("&lt;img&gt;").attr({       id: fileNames[x],       src: config.imgDir + fileNames[x] + config.imgFormat,       title: "The " + fileNames[x] + " nebula"     }).appendTo("#" + config.imgContainer).css({ display: "none" });   } };   But, it looks a bit over-the-top for what I want!  I know there are jQuery plugins out there that do this but they all seem a bit big (in size); I just need a quick, easy and short way of preloading images! I am having problems starting my Apache server, because port 443 is already in use.  It turns out, the system process (PID 4) uses the port 443. I don't have IIS installed, the services.msc shows (predictably) no Exchange server running, nor WWW-Services, nor IIS. I have no idea how to find out what service uses that port, short of just disabling each service one after the other, and I am not even sure that would help.  I would be grateful if someone could point me towards how I can get my SSL port back, thank you :)  P.S.: Of course "just switch Apache to another port for SSL" would solve the problem of not being able to start Apache. But I'd still like to know what is so insistent about hogging port 443. :)    I by now took the 'hard route' and disabled services one after the other. It turned out that the "Routing and RAS" service was the culprit.  Thank you all for the valuable input and the new tools in the combat against "WTF does my system do now?". Center text in color box I wanto to center a text in a colorbox which is scaled horicontal over the complete site.    How is this possible?  \documentclass[ ]{scrreprt}  \usepackage[ ]{scrpage2}  \usepackage[ngerman]{babel}                 %% Deutsches Sprachpaket \usepackage[utf8]{inputenc}                 %% Eingabezeichensatz UTF-8 \usepackage[T1]{fontenc}                    %% Kodierung für Umlaute \usepackage[ngerman]{translator}            %% Für Abkürzungsverzeichnis benötigt \usepackage{lmodern}                        %% Moderne Version von Computer Modern \usepackage{setspace}                       %% Zeileabstand anpassen \usepackage{xcolor}                         %% Definieren von Farben  \begin{document}  \chapter{} \definecolor{lightgray}{gray}{0.5} \newcommand \colbox [1]{\par\noindent\colorbox{gray}{\begin{minipage}{\textwidth}{ {#1}} \hfill \end{minipage}}}  \colbox{TEXT}  \end{document}   Moreover I want to avoid the overfull hbox. How to understand linux versioning I need to understand the Linux versioning system and distribution concepts. what are public, stable or final terms for versions? Why did Superman & Supergirl start to suffocate instantly when Luthor made our yellow Sun red? Spoiler Alert: If you haven't read all comics series &amp; story arc of New Krypton Saga viz. New Krypton, World of New Krypton, Last Stand of New Krypton, Superman: War of Supermen etc, drop this question right away!  While battling the evil alien Brainiac, Superman &amp; Supergirl discovered the long-lost bottled city of Kandor which was stolen from Krypton prior to its destruction. Superman re-enlarged the city on Earth, but it wasn't welcomed by Humans. Humans assaulted the city in which Kryptonian's leader Zor-El (Supergirl's father) was killed. Using highly evolved science, Alura (Zor-El's wife) created a new planet "New Krypton". Plus, she, as a new leader, freed General Zod from Phantom Zone to command new super military of New Krypton (every kryptonians had super powers due to yellow sun).  While New Krypton &amp; Earth were at cold war, Zod ordered his military to attack Earth. When military was in space, Sam Lane (Lois Lane's father) destroyed New Krypton by a trick making their race extinct again. Kryptonian military successfully crushed Earth's Mars defense facility &amp; they headed towards Earth.  When they were about to enter atmosphere of Earth:      Many Kryptonians died under the red sun because they couldn't survive in empty space. Its somewhat plausible because they weren't under yellow sun for long. But, what about Superman &amp; Supergirl? They did survive because they were under yellow sun for very long. But, why did they become very weak instantly? If the power draining rate by radiation of red sun was that high, how did Superman &amp; Supergirl survive?  Is this a plot hole? Or, am I missing something? "He added" vs. "Did he add" (inversion after negative adverb) This is the third time I've tried to install Ubuntu Server 14.04.03 64-bit on the same laptop, and each time it caused different problems.  Each time I wrote the same iso image to a USB flash drive by using a different application and OS.   Universal USB Installer (pendrivelinux.com) from Windows WinImage writer from Windows built-in image writer from Linux Mint 17.2    The iso image is the same in all three cases, the USB flash drive is the same, and the laptop is the same. However each time the installation of Ubuntu gave me different installation options. For example, the last time I installed Ubuntu, it didn't show all the packages to choose while installing (Basic Ubuntu server, etc.). Stop the Ubuntu updater from nagging about Firefox English Language pack even after uninstalling Firefox I have run sudo apt purge firefox, and I have gone through all of the /.config files, etc., and Software Updater is still asking me daily if I want to install the &quot;English Language Pack for Firefox&quot;. I don't have Firefox installed. How can I stop the updater from asking to update something on a program that doesn't exist? Output of dpkg -l | grep firefox: ii  firefox-locale-en                          88.0+build2-0ubuntu0.18.04.2        amd64        English language pack for Firefox How do I log every command executed by a user? I want to monitor all user's activity in my server.  Even when the user executes a shell command from some editor like vim I want to see them in the log file.  I have checked the tool acct but it is not listing the complete commands. (Please correct me if I have missed some options which does already).  Which Linux tool I should be looking at to solve this problem? Where to find cross section data for $e^{-}$ + $p$ $\longrightarrow$  $p$ + $e^{-}$ ?  does not include it. I've used git clone to make a local copy of a GitHub repository containing a  I'd like to use. The local copy contains all of the source for the program; but as a newbie to Ubuntu, I've no idea of what tools or steps are involved in compiling and installing the driver.   I'd really appreciate if anyone could provide a step-by-step guide? What does 'condescending' exactly mean? I'm really confused about the meaning of the word 'condescending'. Online dictionaries define it as bossy/disdainful, which fits with...    But I find other usages where being condescending is obviously considered a good thing, perhaps more like being humble, or equal, rather than superior...    (and  of "condescending and friendly" in Google Books)  What's going on? Does this word really have two almost opposite meanings? Prove the inequality $n! \geq 2^n$ by induction I'm having difficulty solving an exercise in my course.  The question is:      Prove that $n!\geq 2^n$.   We have to do this by induction. I started like this:   The lowest natural number where the assumption is correct is $4$ as: $4! \geq 2^4 \iff 24 \ge 16$. The assumption is: $n! \ge 2^n$.   Now proof for $(n+1)$ which brings me to: $(n+1)! \ge 2^{(n+1)}$  I think I can rewrite it somehow like this:  $$ {(n+1)} \times {n!}  \stackrel{\text{(definition of factorial)}}{\ge} 2^n \times 2 $$  $$ (n+1) \times 2^n \ge 2^n \times 2 $$  Then I think I can eliminate the $2^n$ and have something like this: $n+1 \ge 2$, or $n \ge 1$.  But I think I'm wrong here some where and was hoping somebody has some advice on this. How can I prove the above assumption?  Any help would be appreciated, kind regards. How do I learn where the source file for a given Python module is installed? Is the method different on Windows than on Linux?  I'm trying to look for the source of the datetime module in particular, but I'm interested in a more general answer as well. There are $25$ horses. You can take $5$ horses at a time and race them. Each horse always finishes the race in the same amount of time, and there are no ties. The only information you get from each race is the order that the $5$ horses finished in. What is the smallest number of races you need to find the $3$ fastest horses, in order?   I found this problem on  and cannot figure out the best answer. Please help. How can I change the registered name in Windows 7 to my own? I am working on vector auto-regression (VARs) and impulse response function (IRFs) estimation based on panel data with 33 individuals over 77 quarters.  How should this type of situation be analyzed?  What algorithm's exist for this purpose?  I would prefer to conduct these analyses in R, so if anyone is familiar with R code or a package designed for this purpose that they could suggest, that would be especially helpful. cron ignores variables defined in ".bashrc" and ".bash_profile" I have defined "SHELL" variable in /etc/crontab file:  [martin@martin ~]$ grep SHELL /etc/crontab  SHELL=/usr/local/bin/bash [martin@martin ~]$ file /usr/local/bin/bash /usr/local/bin/bash: ELF 32-bit LSB executable, Intel 80386, version 1 (FreeBSD), dynamically linked (uses shared libs), for FreeBSD 8.0 (800107), stripped [martin@martin ~]$    In addition, all my scripts in /etc/crontab file are started under user "martin". However /home/martin/.bash_profile(for login shell) and /home/martin/.bashrc(for non-logging shell) contain some variables which are ignored in case of cron job, but are used in case I log into machine over SSH or open new bash session. Why cron ignores those variables? Isn't cron simply executing "/usr/local/bin/bash my-script.sh" with permissions for user "martin"?