Binomial Sum Formula
Prove that $\binom{m+n}{m}=\sum\limits_{i=0}^m \binom{m}{i}\binom{n}{i}$
Difference between "right of=" and "right=of" in PGF/TikZ In the PGF/TikZ manual, sometimes I see the option right of=somenode instead of right=of somenode. They look very similar, but the effects are different.  The distance between nodes positioned with the latter option is boundary-to-boundary, as stated in the manual. However, with the first option, it seems that the distance is shorter. I couldn't find any explanation of the first option in the manual.  My question is: what is the difference between the two options? Is there any explanation in the manual that I missed? Probability for roots of quadratic equation to be real, with coefficients being dice rolls. I really need help with this question.     The coefficients $a,b,c$ of the quadratic equation $ax^2+bx+c=0$ are determined by throwing $3$ dice and reading off the value shown on the uppermost face of each die, so that the first die gives $a$, the second $b$ and and third $c$. Find the probabilities that the roots the equations are real, complex and equal.   I was thinking about using the fundamental formula but i'm not sure how to go about doing it. Help would be greatly appreciated. How can I make a plane's edges taper inwards? How can I deform one edge or multiple edges of a plane or cube so they are tapered in and rounded equally? Where do I get a SIS mirage 672 vga driver? How to Edit PDFs? unable to use synaptic via launcher When I open synaptic via launcher on gnome-panel, it opens and asks me for the password but if I enter the password, it says its wrong even though the password is correct and it used to work OK few days ago. If I open synaptic via terminal using sudo synaptic then it opens OK without any problem. Colored item names in Minecraft? I came across a map and when I was playing I found some coloured and bolded item names. I thought it isn't possible as at most you can only use the custom name tag to rename things! Is there a way to colour and bold item names in vanilla Minecraft? External programs are OK. Oh, and the wiki page of formatting codes doesn't really do much. In swimming, is there a competitive advantage to being in one lane or another? Upon observing swimming competitions in the Olympics, it is not clear to me whether one of the eight lanes is more advantageous versus another. I've seen vague claims but nothing backed up with science.  In competitive swimming, is there an advantage to being in one lane or another? Furthermore, does the stroke (ie: breast, fly, freestyle, etc.) affect this effect? I can show this best by a picture:    This cross also shows when I export it as raster or pdf. What is it and how can I get rid of it?  I am using Qgis 2.4 and the map was done with the help of the Grass plugin AniMove on Linux Mint. What is god for religious people? In order to discuss with religious friends about god, I would like to know if there's some accepted objective definitions of God for main religions, even if belief is subjective.   For example, is he "a person who..."? In that case I would address features of a person, as capabilities and limitations. Is he "an entity..."? In that case I would use systems theory.   I feel lost on this, being agnostic from a religious family, where I was taught that God is a mess of things, including a beautiful bearded face, omnipotence and a mix of human defects and metaphysical powers. What do repeating '^@' symbols in a logfile mean? A few times I have come across this in a log file:  Sep 17 09:35:57 00187d378926 ovpn-00187d378926[1701]: /sbin/ifconfig tun0 0.0.0.0 Sep 17 09:35:57 00187d378926 kernel: Kernel logging (proc) stopped. Sep 17 09:35:57 00187d378926 rsyslogd: [origin software="rsyslogd" swVersion="4.6.4" x-pid="1707" x-info="http://www.rsyslog.com"] exiting on signal 15. ^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@ ^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@ ^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@   What does it mean?  Sometimes it is a single line, other times it is lots of lines.  I usually see it happen on Debian 6 machines. Find the determinant of the following matrix Uniform convergence of derivatives, Tao 14.2.7. This is ex. 14.2.7. from Terence Tao's Analysis II book.   Let $I:=[a,b]$ be an interval and $f_n:I \rightarrow \mathbb R$ differentiable functions with $f_n'$ converges uniform to a function $g:I \rightarrow \mathbb R$. Suppose $\exists x_0 \in I: \lim \limits_{n \rightarrow \infty} f_n(x_0) = L \in \mathbb R$. Then the $f_n$ converge uniformly to a differentiable function $f:I \rightarrow \mathbb R$ with $f' = g$.  We are not given that the $f_n'$ are continuous but he gives the hint that $$ d_{\infty}(f_n',f_m') \leq \epsilon \Rightarrow |(f_n(x)-f_m(x))-(f_n(x_0)-f_m(x_0))| \leq \epsilon |x-x_0| $$ This can be shown by the mean value theorem. My question is : How does this help me to prove the theorem ? Log of mounted/unmounted devices in Unix How do you know when to wean? What signs can a baby show to indicate he/she is ready for weaning?  What does a parent look for to determine the right timing for this for baby, mom and family? Matching text using grep or awk pkgloader is not finding l3regex I would like to try a package which seems interesting, but it some dependencies are not satisfied.   Here is my minimal example:  \documentclass{article}  \usepackage{pkgloader} \LoadPackagesNow  \begin{document}     pckloader trial \end{document}   And I get the following error: ! LaTeX Error: File 'l3regex.sty' not found.. Therefore, I tried to update my distribution with the linux commands  tlmgr update --self tlmgr update --all   But the package l3regex is still not foundable. Plus, my computer is not lying since l3regex.sty is not in my folder /texmf-dist/tex/latex/*.  So, is it something normal ? How can I workaround this problem ? I know from reading about the gravitational waves detected by Ligo, that when an object has angular acceleration, it produces gravitational waves.  I'm wondering if an object creates gravitational waves when only accelerating in one direction, however?   I'm also curious as to how the nature of the waves would differ in this case. Show that the range of $p(z)e^{q(z)}$ for nonconstant polynomials $p$ and $q$ is all of $\mathbb{C}$. How many inventions did Fred and George have? Fred and George Weasley are known for their trouble-making and general &quot;solemnly swearing they are up to no good&quot;. Their antics/adventures give something to smile about and show us not all wizards are as serious as Professor McGonagall and Hermione. This gives them a reputation of being less intelligent and indeed they only achieved three OWLs. However, we also see the twins have quite a brilliant side! Being able to open and successfully run a shop in Diagon Alley without having completed their Hogwarts education is no easy take (IMO, anyway). Their creations helped students get out of class, our heroes out of tight spot, and even stop our enemies execute an evil plot. Ginny may have said it best:  &quot;We’ve been hearing explosions out of their room for ages, but we never thought they were actually making things,&quot; said Ginny. &quot;We thought they just liked the noise.&quot; Harry Potter and the Goblet of Fire  Do we know how many things Fred and George invented?