Prove that for $G = (V,E) $ $$ \chi(G) \chi(\overline{G}) \ge |V| $$ Please for some advices. Thanks in advance.
Let us denote by $\chi(G)$ the chromatic number, which is the smallest number of colours needed to colour the graph $G$ with $n$ vertices. Let $\bar{G}$ be the complement of $G$. Show that  (a) $\chi(G)+\chi(\bar{G}) \leq n+1$  (b) $\chi(G)\chi(\bar{G}) \geq n$  I was able to prove (a) using induction. Any hints on proving (b)?
Resolve $\lim_{x \to 0} \frac {\tan 3x}{\tan 5x}$ I started resolving this limit:  $$\lim_{x \to 0} \dfrac{\tan 3x}{\tan 5x}$$  so:  $$\begin{align}\lim_{x \to 0} \dfrac{\tan 3x}{ \tan 5x} &amp; = \lim_{x \to 0} \dfrac{\frac{\sin 3x}{\cos 3x }}{\frac{\sin 5x}{\cos 5x}} \\ \\ &amp; = \lim_{x \to 0} \dfrac{ \sin (3x) \cdot \cos (5x)}{ \sin(5x) \cdot \cos (3x)}\\ \\ &amp; = \quad?\end{align}$$  Could someone help me please In an exercise in Atiyah-Macdonald it asks to prove that the prime spectrum $\operatorname{Spec}(A)$ of a commutative ring $A$ as a topological space $X$ (with the Zariski Topology) is compact.   Now because the basic open sets $X_f = \{\mathfrak{p} \in \operatorname{Spec} (A) : \{f\} \not\subseteq \mathfrak{p} \}$ form a basis for the Zariski Topology it suffices to consider the case when   $$X = \bigcup_{i \in I} X_{f_i}$$   where $I$ is some index set. Then taking the complement on both sides we get that   $$\emptyset = \bigcap_{i \in I} X_{f_i}^c$$  so there is no prime ideal $\mathfrak{p}$ of $A$ such that all the $f_i$'s are in $\mathfrak{p}$. Now from here I am able to show that the ideal generated by the $f_i$'s is the whole ring as follows. Since there is no prime ideal $\mathfrak{p}$ such that all the $f_i \in \mathfrak{p}$, it is clear that there is no $\mathfrak{p}$ such that $(f_i) \subseteq \mathfrak{p}$ for all $i \in I.$ Taking a sum over all the $i$ then gives $$\sum_{i \in I} (f_i) = (1).$$  Now here's the problem:          How do I show from here that there is an equation of the form $1 = \sum_{i \in J} f_ig_i,$ where $g_i \in A$ and $J$ some finite subset of $I$?       This part has been giving me a headache. I am not sure if the finiteness bit has to do with algebra, topology or the fact that we are dealing with prime ideals.  This is not a homework problem but rather for self-study.  $\textbf{Edit:}$ I have posted my answer below after the discussion with Dylan and Pierre. Set an alias on Bourne shell I am using a scrtipt with instruction 'SH -c command', where command is an alias from bash, this alias can not be interpreted into Bourne shell. ¿Do you have any idea for set this alias into Bourne shell and getting that it works in the new shell? Thanks everyone ! How do I  prevent .xsession-errors from eating disk space? Is there any way to completely disable the .xsession-errors file?  I have it as a symlink to /tmp so that my laptop hard disk can (hopefully) go to sleep for once, but at least 95% of the stuff in the file (it fills up at about 500k an hour) is...  (nautilus:1618): GLib-GObject-CRITICAL **: g_value_get_object: assertion `G_VALUE_HOLDS_OBJECT (value)' failed  which is total garbage to me.  I have tried doing a symlink to /dev/null but that does not work (the resulting link is overwritten), and I also do not want the activity so that hopefully my laptop can go to sleep for once.  I am using Ubuntu 11.04, with no special add-ons to Nautilus. Is there any way to run file explorer as administrator under Windows 10 After install Windows 10, it seems that I lost the ability to run explorer.exe as administrator with a user in administrator group.   The reason I want to do that is, in our team's development environment, we are using a C# exe to do some environment configuration which will start command prompt under administrator mode and subst a drive. As a result those substed drives are not visible within the file explorer since it is run not as administrator. That will be kind of inconvenience and sometimes make mistakes.   I was able to run explorer.exe from task manager with the option "create with privileges" checked and then I can see all the drives in explorer. But now this is not working anymore.   I knew there are other options to workaround this workflow, but just want to make sure that if it is now totally impossible under Windows 10?  Any comment is appreciated. Why is CDC capture failing due to a missing dbo principal Can we omit parentheses when creating an object using the "new" operator? Raising a decimal to a power of decimal? The .net framework provides in the Math class a method for powering double. But by precision requirement I need to raise a decimal to a decimal power [ Pow(decimal a, decimal b) ]. Does the framework have such a function? Does anyone know of a library with this kind of function? Deleting comment removes "Add comment" link on Mobile If I delete a comment on the mobile version of the site (at least if it's the only comment on an answer), the "Add comment" link will vanish along with the comment, so I have to reload to leave another comment.     but got declined as , which is why I'm reposting this (as it may well be a new bug with the same symptoms).  I'm using Firefox 32.0.3 on Android. How do I create a shortcut for a script, and run it as a .desktop file? I have installed Teamspeak onto my Ubuntu 15.10 system. Teamspeak requires a script to be ran in-order to start. How can I create a symbolic link that is a .desktop so I can open the .desktop file, and have it run the script? I have looked at a few methods, but they all seem like hotfixes. Is there a proper way to do this? How can I really disable the Cmd to Win key mapping in Parallels Desktop 5? Removed it from System Preferences &gt; Keyboard. No go. Remapped ⌘ Cmd to be Ctrl. No go. Ripping out the ⌘ Cmd key shouldn't be the solution. Anyone?... Do tag edits count for Archaeologist? Since the final Black Hole (BH) had 3 solar masses less of mass than the original binary BH, it seems the 2 BHs lost mass, and with it event surface area, entropy, and information. If that came from inside the BHs it seems that it would be a contradiction of what a BH is --'nothing escapes its event horizon'. Some may say that came from the rotational energy of the pair, but the new BH now seems to have less than the two original ones summed. So, did some of that escape the horizons? Turn off autocomplete in PhpStorm What rules say that summoning undead is evil, and what rules say why? Help me to get command or ansibile Regex to fetch below details how to have a custom zsh auto_cd command Instead of the normal cd I've created my own function  function cd {     builtin cd "$@" &amp;&amp; ls -F }   This works great for when I use commands like cd someDir/subDir but not when I change directories via zsh's auto_cd someDir/subDir.  Is there a way to customize what command gets called for auto_cd? The limit of $((1+x)^{1/x} - e+ ex/2)/x^2$ as $x\to 0$ Let $a,b$ be positive integers.  When $$k = \frac{a^2 + b^2}{ab+1}$$ is an integer, it is a square.    Proof 1: (Ngô Bảo Châu): Rearrange to get $a^2-akb+b^2-k=0$, as a quadratic in $a$ this has two values: $a$ and $kb - a = (b^2-k)/a$. (The second root is determined in two different ways from the expansion $(x-r_1)(x-r_2) = x^2 - (r_1 + r_2)x + r_1 r_2$.)  Now suppose we have $a,b$ such that $k$ is an integer but not a square, by the investigation about roots we have that the second root is a nonzero integer since $k,b,a$ are integers and $k \not = b^2$, futhermore it is positive which is easily seen from its defining equation.  WLOG assume $a \ge b$ so that the second root is strictly smaller than $a$. This leads to a decent, replacing $a$ with the second root.    Proof 2 (Don Zagier): Apply reduction theory (specifically, Sätze 1 and 2 of Section 13 of my book on quadratic fields) to the quadratic form $x^2 + kxy + y^2$, which is the unique reduced quadratic form in its equivalence class.    Note that Proof 2 is pretty much the same as Proof 1 when written out in explicit detail, but I could not read Zagier's book because I cannot read German.  I would like to know more approaches to this and other alternative proofs of this result if possible! Thanks in advance.  I would also be interested in related problems (especially easier ones of a similar nature) and texts which cover the reduction theory in English. how to permanently disable ubuntu updates & hide install updates button pic included how to permanently disable ubuntu updates pop up &amp; hide or remove Install Updates button pic included see below: