Proof with chromatic index.
Prove $\chi(G)\chi(\bar{G}) \geq n$ for chromatic number of graph and its complement
Difference in sentence meaning between present continuous and simple present constructions Is there offside when the ball deflects off an opponent? I was refereeing a 10 year old's junior game today and this situation happened. The home team's defender attempted to  pass from their own half to their forward who was in an offside position beyond the halfway line.  I was going to blow the whistle but the ball deflected off an away defender and then went to the originally offside attacker who put the ball wide.  I viewed this as onside at the time because the ball struck the away team player, however now I think it was offside because at the time of pass, the home player was offside.  Is this offside or not? What is the size of this microcontroller? Having trouble reading a data sheet Extract all numbers from a single string in R Evaluating $\int_{0}^{\frac{\pi}{2}}\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}\, \mathrm{d}x$ How to get coloured terminal over ssh? When I connect over ssh to remote hosts everything is just a single font/color. I would like to have colours like I do locally e.g. green for executable and blue for symlinks etc. And such that when I run $ git diff on the ssh host it shows me diff with colours =) Intersection and union of simply connected domain I have some basics concept that, in $\mathbb{C}$, a "simply connected domain $D$" is a region in $\mathbb{C}$ with no holes. I am not sure whether it has various formal definition , but the one used in my complex analysis class looks very complicated. I barely understand what it means :  $\textbf{Definition}$ Let $\gamma_0, \gamma_1$ be two continuous closed path in a domain $D \subseteq \mathbb{C}$ parameterized by $I = [0,1]$;that is, $\gamma_i(0) = \gamma_i(1)$ for all $i = 0, 1.$ We say $\gamma_0, \gamma_1$ are $\textbf{homotopic as closed paths on D}$ if there exists function $\delta : I \times I \rightarrow D$ such that   $1) \ \delta(t,0) = \gamma_0(t) \ \forall t \in I \ \ $    $2) \ \delta(t,1) = \gamma_1(t) \ \forall t \in I \ \ $     $3) \ \delta(0,u) = \delta(1,u) \ \forall u \in I.$  A continuous closed path is $\textbf{homotopic to a point}$ if it is homotopic to a constant path(as a closed path).  $\textbf{Definition}$ A region $D \subseteq \mathbb{C}$ is called $\textbf{simply conected}$ if every continuous closed path in $D$ is homotopic to a point in $D$.  $\textbf{Problem :}$ 1) Let $D_1, D_2$ be simply connected plane domains whose intersection is nonempty and connected. Prove that their intersection and union are both simply connected.  2) Let $P, Q$ be smooth functions on a domain $D \subseteq \mathbb{C}$, Find necessary and sufficient condition for the form $P dz + Q d\bar{z}$ to be closed. I'm more of a math guy than a physics guy so bear with me....  In fractal geometry, fractals are considered to have fractional dimension. For instance an object such as the Koch curve has a fractal dimension of around $1.2$. You can also define a concept called fractal extent (which is the experimental version of hausdorff measure). This extent is a measure of how large a fractal is. It's even been used to more accurately model the distribution of coastline bird nesting.   Even though fractals work as an interpretation of geometrical units, I don't think they would describe things such as $kg^{1/2}$ very well. In addition, I've heard about white Gaussian noise that has square root time units.   What is the physical meaning of fractional units such as the above? (Since it obviously has little geometric meaning) Normally, LaTeX will only create references for the BibTeX entries cited in the text. Is there a way of extracting these entries into a different .bib file automatically? For example if I have a .bib file with two entries, and only one is cited in a particular text, I need a way of creating a new .bib file with just that reference. Adding multiple CKEditor instances in jquery I gzip directories very often at work. What I normally do is  tar -zcvf file.tar.gz /path/to/directory   Is there a way to specify the compression level here? I want to use the best compression possible even if it takes more time to compress. Every finitely generated algebra over a field is a Jacobson ring Knowing that the polynomial ring in $n$ variables over a field $k$ is a Jacobson ring, how can we prove from it that every finitely generated $k$-algebra is a Jacobson ring?   EDIT: We define a ring to be Jacobson if every prime ideal is an intersection of maximal ideals. Query works on Oracle 11g but fails on Oracle 8i Compact space and Hausdorff space A continuous map from a compact space to a Hausdorff space is closed. Why this is true?  Help me please I want to learn why this is correct. Do we keep hats from year to year? This is the first year that I pay attention to hats (I love hats!) and I wonder if I will keep my hats for next year?  Or will I start over with zero hats next year? Printer not recognized on Ubuntu Libre Office I am running 14.04 and have a HP Deskjet 2545 wireless printer which is supported by Ubuntu. I have run HpLIP, the latest version and installed the printer.  The printer is recognised by Ubuntu, printer and network, but LibreOffice do not see the printer. Only a Generic printer.  I don't know what to do anymore and would like to get some good advice "Temporarily unreachable" when using fetch as Googlebot from Webmasters Tools The page I'm trying to fetch is public.  The server logs show no request.  Google reports that any page on my site is "Temporarily unreachable". It won't even fetch the verification file, a static HTML file that was successfully used to verify with the webmaster tool application itself.   My site is running on the .  What is causing this? I have a registered StackOverflow account which I normally log into using OpenID.  Recently I logged out in order to post a question anonymously (supplying a fake name and of my secondary email addresses).  The problem now is that I'm logged in to StackOverflow as this "dummy" user, unable to see a way to log out.  The top menu normally has a "logout" link, but when you aren't fully registered this link is replaced with "register".  There is no "logout" link for me. :(  So..   When logged out there is a "login" link. When logged in there is a "logout" link. When "logged in" with an unregistered account, there is only a "register" link (no "logout"). I'm creating an application which lets you define events with a time frame.  I want to automatically fill in the end date when the user selects or changes the start date.  I can't quite figure out, however, how to get the difference between the two times, and then how to create a new end Date using that difference. Different ways to prove there are infinitely many primes?