Number of vertices is one more than the number of edges
How many edges does an undirected tree with $n$ nodes have?
How to get contact from stolen Android phone? I already read . And that answer did not work for me. I have explained it on the question.  My android phone was stolen. I need to get all contact of my phone. I had enable sync in that phone.  When I visit , it has only a very few contacts (seems like below 10% and all them has an email too). How can I get all contacts? Adobe Reader and xcolor within a table breaks the line render I'm using the xcolor package to colour the top row in a table. This is displayed properly using PDF viewers such as Preview or Skim (Mac). But using Adobe Reader 10.1.3 (either Windows or Mac) doesn't display the table line properly.   But removing the colouring for the table fixes it. Here is the example that works as expected:  \begin{table} \caption{Table showing Feature 5's Functional Tests} \hspace*{-0.5in} \begin{tabular}[h!]{|p{0.6cm}|p{4cm}|p{5cm}|p{4cm}|p{1.5cm}|} \hline {\bf Test ID } &amp; {\bf Task } &amp; {\bf Expected} &amp; {\bf Actual} &amp; {\bf Outcome} \\ \hline    5.1 &amp; Something &amp;  Something &amp; Something &amp;  Success \\ \hline    5.2 &amp; Something &amp;  Something &amp; Something &amp;  Success \\ \hline    5.3 &amp; Something &amp;  Something &amp; Something &amp;  Success \\ \hline    5.4 &amp; Something &amp;  Something &amp; Something &amp;  Success \\ \hline \end{tabular}  \end{table}   And here is the example with a coloured row:  \begin{table} \caption{Table showing Feature 5's Functional Tests} \hspace*{-0.5in} \begin{tabular}[h!]{|p{0.6cm}|p{4cm}|p{5cm}|p{4cm}|p{1.5cm}|} \hline \rowcolor{gray!25} {\bf Test ID } &amp; {\bf Task } &amp; {\bf Expected} &amp; {\bf Actual} &amp; {\bf Outcome} \\ \hline    5.1 &amp; Something &amp;  Something &amp; Something &amp;  Success \\ \hline    5.2 &amp; Something &amp;  Something &amp; Something &amp;  Success \\ \hline    5.3 &amp; Something &amp;  Something &amp; Something &amp;  Success \\ \hline    5.4 &amp; Something &amp;  Something &amp; Something &amp;  Success \\ \hline \end{tabular}  \end{table}   Removing the \rowcolor{} command replacing it with \cellcolor{} creates the same view. Any idea how to fix this?  I've tried printing it out of Adobe and the lines print fine, but as this will be read on as a soft copy it's pretty important. .  Any idea how to fix this? I extracted simple example:  require 'pp' x = 1..3 pp x.map do |i| {:value =&gt; i, :double =&gt; (i*2)} end pp x.map { |i| {:value =&gt; i, :double =&gt; (i*2)} }  pp(x.map do |i| {:value =&gt; i, :double =&gt; (i*2)} end) pp(x.map { |i| {:value =&gt; i, :double =&gt; (i*2)} })   I am wondering why first pp produces:  [1, 2, 3]   While all the oders are giving:  [{:value=&gt;1, :double=&gt;2}, {:value=&gt;2, :double=&gt;4}, {:value=&gt;3, :double=&gt;6}]   I assume it has something to do with operator precedence. Where can I find good explanation? How can i change formatting of the table of contents to avoid section numbering and section title to overlap?  Document:  \documentclass[10pt,oneside,notitlepage]{book}  % Use Heuristica Font \usepackage{heuristica} \usepackage[heuristica,vvarbb,bigdelims]{newtxmath} \renewcommand*\oldstylenums[1]{\textosf{#1}}   \begin{document}  \tableofcontents  \part{Part one}  \chapter{Chapter Oh} \chapter{Chapter Oh} \chapter{Chapter Oh} \chapter{Chapter Oh} \chapter{Chapter Oh} \chapter{Chapter Oh} \chapter{Chapter Oh} \chapter{Chapter Oh} \chapter{Chapter Oh} \chapter{Chapter Oh} \chapter{Chapter Oh} \chapter{Chapter Oh} \chapter{Chapter Oh} \chapter{Chapter Oh} \chapter{Chapter Oh} \chapter{Chapter Oh} \chapter{Chapter Oh} \chapter{Chapter Oh} \chapter{Chapter Oh} \chapter{Chapter Oh}   \section{Chapter Hi} \section{Chapter Hi} \section{Chapter Hi} \section{Chapter Hi} \section{Chapter Hi} \section{Chapter Hi} \section{Chapter Hi} \section{Chapter Hi} \section{Chapter Hi} \section{Chapter Hi}  \end{document}   Example output: How to prove whether a matrix has rank $1$ If $u$ $∈ \mathbb R^{m \times 1}$ and $v ∈ \mathbb R^{n \times 1}$ how do you show that the $(m \times  n)$ matrix $uv^T$ has rank $1$? Would providing an example be sufficient to prove it? Suppose $X$ and $Y$ are independent positive random variables with probability density functions $f_X$ and $f_Y$ respectively. Show that $Z=X/Y$ is absolutely continuous and find its probability density function.   $\textbf{My Thought:}$ In order to show that $Z$ is absolutely continuous, I need to show that it has a commulative distribution function. Then I can also differentiate the latter to obtain the probability density function of $Z$. I have the following calculation \begin{align*} P(X/Y\leq z)&amp;=P(X\ge zY,Y&lt;0)+P(X\le zY,Y&gt;0)\\ &amp;=\int_{-\infty}^{0}\left(\int_{yz}^{\infty}f_{X}(x)dx\right)f_{Y}(y)dy+\int_{0}^{\infty}\left(\int_{-\infty}^{yz}f_{X}(x)dx\right)f_{Y}(y)dy\\ &amp;=\int_{0}^{\infty}\left(\int_{0}^{yz}f_{X}(x)dx\right)f_{Y}(y)dy. \end{align*} Also, if $z&lt;0$, then we have that $F_Z(z)=0,$ since $X,Y$ are positive random variables. Therefore the random variable $Z$ is continuous, and we can differentiate it to obtain the probability distribution function of $Z$. We have that $f_Z(z)=0$ if $z\leq 0,$ and otherwise we have $$f_Z(z)=\int_{0}^{\infty}yf_{X}(yz)f_Y(y)dy.$$    Is my reasoning above correct?  Any feedback is much appreciated.  Thank you for your time. Setup IIS 7.5 with multiple website bindings and SSL? On IIS 7.5 I am trying to achieve this with two websites:  Default Web Site is bound to:  (blank host header port 80 - http) (blank host header port 443 - https) go.example.com www71.example.com the IP address of go.example.com  2nd web site "Beta" is bound to:  beta.example.com (blank host header port 443 - https) * using blank only because it doesn't seem to be possible to    bind https to a named host header   And both need to work with SSL.  But I have these problems:   When I type in beta.example.com, I see the go.example.com site instead I can not seem to add the SSL binding to both websites at once (I have a single *.example.com wildcard certificate).  The beta site will not even start if I add the https binding to it.   This is how I have set it up:    What is the correct way to set it up? PHP: How do I loop through every XML file in a directory? How do I get past the "do not enter" traffic sign that stops me from installing "Sierra" instead of "High Sierra"?  I booted into rescue mode and deleted the original "High Sierra" installation, but booting from the USB stick with the "Sierra" installer on it still gives me the "no". I'm wondering how it is correct to structure sentence and what punctuation should be used. In particular, is the next sentence correct:     I was wondering if there's any progress on the issue.    Or should I put a question mark in the end of it. Also, can it be restructured to:     I was wondering: is there any progress on the issue? How can I use an array of function pointers? How should I use array of function pointers in C?  How can I initialize them? Rounded caps on a Bezier Curve pipe This may be a simple question.  Essentially I am procedurally generating random 3D figures using a bezier curve and bezier circle bevel object, however, one of the only things I can't figure out how to do is how to round the end caps of the object.   I have tried using a taper object like suggested  but have had only limited success.  This is about as close as I've gotten:   I should note that all of this is done through Python and I'm not modeling any of this by hand. So any suggestions for how I can make the rounded caps programmatically would be greatly appreciated. How can I distinguish between words which have the -ing in a sentence that are nouns, verbs, or adjectives?    For example sometimes -ing come with word to give us a noun, and sometimes a verb or an adjective.       That interesting snake.   That crawling snake.   That barking dog keeps everyone awake.     Is there any rule to determine whether a word with -ing is a noun, verb, or adjective? I'm looking for some sample code that will sort the list items in an HTML list by alphabetical order.  Can anyone help?  Here is a sample list for people to work with:  &lt;ul class="alphaList"&gt;     &lt;li&gt;apples&lt;/li&gt;     &lt;li&gt;cats&lt;/li&gt;     &lt;li&gt;bears&lt;/li&gt; &lt;/ul&gt; Prove that for all integers $n$, $n \geq 1$,   $$1 + 3 + 5 + \cdots + (2n - 1) = n\cdot n$$   How would I go about proving this? What tense should I use to describe something in the past but probably is still true? The title may be a bit confusing. See the below example:     The movie (was/is) so good.   The simplest guideline on deciding which tense to use, past tense or present tense, to describe an object comes down to whether the description was only true that that time or is true even to the present point of time.  But now we talking about literary work(movies,books,etc) of which quality tend to stay the same over time, which tense should we use? Delete all but the most recent X files in bash Is there a simple way, in a pretty standard UNIX environment with bash, to run a command to delete all but the most recent X files from a directory?  To give a bit more of a concrete example, imagine some cron job writing out a file (say, a log file or a tar-ed up backup) to a directory every hour. I'd like a way to have another cron job running which would remove the oldest files in that directory until there are less than, say, 5.  And just to be clear, there's only one file present, it should never be deleted. How can I install Steam? How can I install and run  on Ubuntu? How would I solve the following problem involving elementary matrices? Let $A=\begin{pmatrix}0&amp;5\\ 7&amp;4\end{pmatrix}$.      1) Write $A$ as a product of $4$ elementary matrices.      2) Write $A^{-1}$ as a product of $4$ elementary matrices.   My work. I have managed to find $A^{-1}$, which came out to be this:  \begin{pmatrix}-\frac{4}{35}&amp;\frac{1}{7}\\ \frac{1}{5}&amp;0\end{pmatrix}  However, I am struggling to figure out how I would split each of these matrices into $4$ elementary matrices. Any help? Why is the gate charge curve (Miller plateau) of MOSFETs dependent on Vds?