I have an arbitrary graph and want to draw an envelope around/behind a certain subset of the graphs nodes. To clarify what I want to achieve, I have made a sketch in Inkscape:    Any ideas how to realize something like that?
To make it simple, I want to highlight a group of tikz nodes like in the following picture    It was borrowed from the . I would like to know which Tikz commands I can use to do the job in order to automatize this task for several trees I have to make. It is important to keep the blending effect of colors like in node g. Here is the MWE also borrowed from :  \documentclass{article}  \usepackage{tikz} \usetikzlibrary{arrows,backgrounds,calc,trees}  \begin{document} \thispagestyle{empty}  \begin{tikzpicture} \node (f) {f}     child { node (g) {g}       child { node (a) {a}     }       child { node (b) {b}     }   }     child { node (h) {h}       child { node (c) {c}     }   }; \end{tikzpicture}  \end{document}
Show that the operator norm is submultiplicative We had in our lecture on numerical analysis the following: Let $\mathrm{Lin}(X,Y)$ be the set of all linear maps $X\rightarrow Y$. Let $A\in\mathrm{Lin}(\mathbb R^l,\mathbb R^n)$ and $B\in\mathrm{Lin}(\mathbb R^n,\mathbb R^m)$ and $\|C\|_{op}:=\max_{\|x\|\leq1}\|C(x)\|$. Then our lecturer followed $\|A\circ B\|_{op}\leq\|A\|_{op}\cdot\|B\|_{op}$. So he didn't prove it and so I've tried it by my own. My attempt: $$\begin{aligned} \|A\circ B\|_{\mathrm{op}} &amp;= \max_{\|x\| &lt; 1}\|(A \circ B)(x)\| \\ &amp; \leq \max_{\|x\| &lt; 1} \|A\|_{\mathrm{op}} \|B(x)\| \\ &amp;= \|A\|_{\mathrm{op}} \max_{\|x\|\leq1}\|B(x)\|\\ &amp;= \|A\|_{\mathrm{op}}\|B\|_{\mathrm{op}} \end{aligned}$$ But this seems too easy. I am really interested in a nice proof so anybody could help? Thanks a lot! Prove that the sum of digits of $(999...9)^{3}$ (cube of integer with $n$ digits $9$) is $18n$ Someone had posted a question on this site as to what would be sum of digits of $999999999999^3$ (twelve $9s$ ) equal to?   I did some computation and found the pattern that sum of digits of  $9^3 = 18$,  $99^3 = 36$,  $999^3 = 54$ and so on. So, I had replied that sum of digits of  $999999999999^3 = 12 \cdot 18 = 216$.  Can anybody help me prove this, that sum of digits of  $(\underbrace{999\dots9}_{n\text{ times}})^3 = 18n.$ Covering estate tax on US assets for non-resident alien I am a non-resident alien holding a brokerage account in the USA but residing outside the USA. My country does not have an estate tax treaty with the USA. So my estate is liable for a significant estate tax up to 40%. Is there any way to avoid or to cover this tax? I thought of buying term insurance for myself, but I am not sure whether US companies will sell term insurance to a non-resident. I want to buy insurance from a US company because I want the death benefit to be in USD. Can someone suggest as to what the best way to approach this issue is? I'm wondering where a new path has to be added to the PATH environment variable. I know this can be accomplished by editing .bashrc (for example), but it's not clear how to do this.  This way:  export PATH=~/opt/bin:$PATH   or this?    export PATH=$PATH:~/opt/bin For some reason, I can run halt, poweroff, shutdown, and reboot as a regular user without root. How can I fix this? Is it something to do with user groups, or is it a seting somewhere? In a P.I.D., if $a^m = b^m$ and $a^n = b^n$ for $m, n \in \mathbb{N}$ with $\gcd(m,n) = 1$, then $a=b$ Let $R$ be a principal ideal domain, and $a, b \in R$ with $a^m = b^m$ and $a^n = b^n$ for $m, n \in \mathbb{N}$, and $\gcd(m, n) = 1$. I now want to show that we then already have $a = b$.   I think the statement is easily seen to be true if $a = 0$ (or $b = 0$), since if $a = 0$ and $a^m = b^m$, then $a = b$ since $R$ doesn't contain any nontrivial zero divisors, so $b ≠ 0$ already implicates $b^m = b \cdot b^{m-1} ≠ 0$.  I don't really know how to prove the statement for the general case, where I can't directly apply the fact that $R$ doesn't have any zero divisors except $0$. I most likely have to use the fact at some point that $gcd(a, b) = 1$, but I'm not sure how. Migrated questions with a nonexistent author should still show the author's name in the questions list I just noticed . Its author doesn't have an account on this site, which makes it show up like this on the homepage:        It looks like a simple thing - but it gets in the way of the 'flow' of the page:        Couldn't we somehow get the author's name, so that migrated questions with authors that don't have an account here still show up, like this:        Much better. Please? Why does my iPhone power off when the battery is down to 30%? Are C++ Templates just Macros in disguise? Why is voltage added when batteries are connected in series but not in parallel? As I understand, the different voltage between two object is caused by the total different charged between two object. Thus, the higher extra electron between two opposite charged object, the higher voltage between them, isn't it? So if i have two battery or capacitor connect in parallel, the negative connect to negative, the positive connect to the positive. Caused the charge from each plate to be double. So that's as my expected, it must be double voltage, but ironically why it's out of my expected and the reality is vice-versa? Build a basic Python iterator If $m,n\in N$ Prove that there is such a positive integer k, such that $(\sqrt{m}+\sqrt{m+1})^n=\sqrt{k}+\sqrt{k+1}$ If $m,n\in N$ Prove that there is such a positive integer k, such that $(\sqrt{m}+\sqrt{m+1})^n=\sqrt{k}+\sqrt{k+1}$ I attempted to solve this question using binomial coefficients, saying: $(\sqrt{m}+\sqrt{m+1})^n=\sqrt{k}+\sqrt{k+1}=\sum\limits_{a=0}^n {n\choose a}*\sqrt{m}^{n-a}*\sqrt{m+1}^a$ and from here I was thinking that I had to do something with $\sqrt{m}^{n-a}$ with $\sqrt{k}$ and do something else with $\sqrt{k+1}$ and $\sqrt{m+1}^a$. Unfortunately I couldn't think of what to do with this circumstance. I then thought of maybe taking $(\sqrt{k}+\sqrt{k+1})^2=(\sum\limits_{a=0}^n {n\choose a}*\sqrt{m}^{n-a}*\sqrt{m+1}^a)^2$, but this immediately over-complicated the question. Could you please explain to me how to solve this question and how to solve similar questions in the future? Multiply cited reference in footnotes I am a Saint Lucian and will be traveling to Manchester, UK soon about which I am a bit nervous.  Run Down:   I own a small business in my country (in the Caribbean), I have a certificate mentioning the details of the business from my Government. I pay rent in my country, but I do not have any loans (I hate owing). I pay life insurance in my country. I have a US B-1/B-2 visa (Visitor &amp; Business). I have visited the US once (last year November) on my US B-1/B-2 visa.   What I will be doing in the UK:   I plan to spend three months out of the six months which would be granted if I get leave to enter stamp. I plan to skydive in the first week and spend a few nights at an Airbnb. I plan to visit the Manchester Football Club stadium, I'm not the biggest of football fans but since I'm in Manchester, why not! I plan to do some more sightseeing and enjoy my time there. After the first week I will be staying with my female friend who owns an apartment for the rest of the time showing me around and helping me experience the UK lifestyle until June.   Why I'm leaving in June:   I am a WordPress enthusiast (I'm a Programmer) and WordCamp EU will be happening in France this year so I plan to fly from the UK to Paris where I will spend three days then take another flight from there to my country.   How my flights will be:   I will be taking a one-way Thomas Cook flight from my country to Manchester but will be purchasing my ticket for Manchester to France ahead of time so I could show Thomas Cook as well as Immigration that I don't plan to stay.   My concerns, could someone address them point by point please?:   I have a bank account but I just use it for collecting easy payments from my clients and pay for expenses. I store my savings in a credit union because of higher interest rates paid than the bank. When I applied for my US visa I showed a credit union statement showing my total balance. I'm seeing online that I should provide statements but these statements would look very bad since I my bank account is just for expenses... I mean it sometimes gets as low as $30 then, when my monthly earnings come in from clients, I withdraw savings to the credit union and leave what I need in the account for expenses. I read another post and it said people with bank statements showing poor spending might have difficulty but it's simply because the account is used for expenses not savings. For the later stages of my visit (after the first week) I will be staying with my friend. I will simply be staying with her to save on cost of Airbnbs for the next two months three weeks, even though they could be cheap, free is better right? I will be booking the Airbnb, skydiving and tours prior to traveling to the UK, will that help? Is my stay too long for staying with someone? I won't be depending on her for anything since I have my own funds...it's just convenient that she has a place where I could stay. If it is then I don't have a problem shortening my stay to 1-2 weeks and just head back to my country and take a separate flight to Paris in June.   I am nervous because I really want to visit the UK and see my friend, I am also not sure if the minimum amount of time that could be given to a visitor is six months? What if I get a round trip for two weeks and the Immigration officer gives me exactly two weeks stay, is that possible? Or is the minimum they could give six months?  Your help is very much appreciated, I want soon to try backpacking (nomad life) but I don't think my poor heart could take the stress of worrying about immigration :(  Update  Since people in the comments are asking about my relationship with the friend I plan to stay with even though she's from my country originally but now a UK citizens and we are simply friends. How to make equally sized windows in corners of the screen? Simple Power Series Expansion for Problems similar to $f = (1 + \epsilon \,x)^{1/\epsilon}$ I was flicking through a book on perturbation methods and saw a simple question asking the reader to expand the following expression for $f$ in a power series (up to the first 2 terms):  $f = (1 + \epsilon \,x)^{1/\epsilon}$, where $\epsilon$ is a small parameter. I'm sure this is very simple, but I wasn't certain about the best way to approach this. A quick look at mathematica tells me the solution is $e^x - \frac{1}{2} (e^x x^2) \,\epsilon + ...$. How would I go about getting this answer - and more importantly, how would I systematically find series expansions for problems similar to this one? Is the distance function in a metric space (uniformly) continuous? Let $(X, d)$ be a metric space. Is the function $x\mapsto d(x, z)$ continuous? Is it uniformly continuous? Proper use of passive and active voices? AMT stock option scenario Is it possible to have the dock in different positions in different desktop? Is it possible to have the dock in different positions in different desktops? In particular I'd like to change the settings between a normal desktop and a desktop where there's an app at full screen. In this way I'd keep the dock in lower side of the desktop when there's an app at full screen and vertically on the left side during normal use.  I have this problem that, sicne I usually have the dock vertically on the left, it pops out everytime i go to the far left in an app I'm using with full screen, and using this app I have to move the mouse to the left quite often.