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prove that a connected graph with $n$ vertices has at least $n-1$ edges Show that every connected graph with $n$ vertices has at least $n − 1$ edges.  How can I prove this?  Conceptually, I understand that the following graph has 3 vertices, and two edges:  a-----b-----c  with $a$, $b$ and $c$ being vertices, and $\{a,b\}$, $\{b,c\}$ being edges.  Is there some way to prove this logically?  --UPDATE--  Does this look correct?  Any advice on how to improve this proof would be appreciated.  Thank you. git-upload-pack: command not found, when cloning remote Git repo I have been using git to keep two copies of my project in sync, one is my local box, the other the test server. This is an issue which occurs when I log onto our remote development server using ssh;  git clone me@me.mydevbox.com:/home/chris/myproject Initialized empty Git repository in /tmp/myproject/.git/ Password: bash: git-upload-pack: command not found fatal: The remote end hung up unexpectedly fetch-pack from 'me@me.mydevbox.com:/home/chris/myproject' failed.   (the file-names have been changed to protect the guilty... !)   Both boxes run Solaris 10 AMD. I have done some digging,  if I add --upload-pack=$(which git-upload-pack) the command works, (and proves that $PATH contains the path to 'git-upload-pack' as per the RTFM solution)  but this is really annoying, plus 'git push' doesn't work, because I don't think there is a --unpack= option.    Incidentally, all the git commands work fine from my local box, it is the same version of the software (1.5.4.2), installed on the same NFS mount at /usr/local/bin.    Can anybody help? What is the exact difference between fluid and responsive design? "At least one" - singular or plural subject? Can I use javascript to dynamically change a video's source? Android 4.1 (Jelly Bean) was announced at Google I|O 2012 on June 27, 2012.  (You can review  at the official Android Developer's site.)  Of course, not all devices will get this update right away and others won't get it at all. Each manufacturer and carrier often choose to add their custom modifications which tend to delay the official release.  When will each device get Android 4.1, either officially or unofficially, if ever?  See also: How to Mark NTFS filesystem dirty from *NIX/Linux? Recommendation for mobile prepaid SIM card for smartphone + home calls in Germany What SIM card would you recommend for a smartphone user for travel in Germany?  It would be probably the only or the most important connection to the Internet, so large amounts of data and relatively fast transfer is an issue.  I need to stay in touch with friends and family, so the calls to other EU countries should be as cheap as possible.   For to the first criteria, I found the T-Mobile offer interesting. The internet costs 10 Euros/month and after reaching the transfer limit it stays relatively fast. According to tests (Galileo from Pro7, AFAIR) it has also the best coverage. However, calls to other countries are very expensive.  For to second criteria, Lebara and Mobilka are interesting. The calls are very cheap, but there are problems with signal strength in many locations, where T-Mobile has no problem. The voice quality, at least in Lebara, is also limited. They both have mobile internet with unlimited transfer, but after reaching the high speed limit, the transfer is reduced to 64 kbps, which is too slow.  I've found out that O2 has foreign calls at moderate prices, but I haven't yet checked for details. There are so many other options, making it is hard to compare them all without spending the whole weekend on it. So, if someone has found a better option than having 2 SIM cards and constantly switching them, this answer would be greatly appreciated. verb or adjective in "The blue page is *stapled* to the red page"? Trouble Viewing Whole HDR Image as cycles world When I load an HDR image into an environment texture using Cycles Render, only a small fraction of the image appears. I have a feeling it's an easy fix, but I can not find what needs to be changed. How disk space is used on Android device? I have critical low storage level and can't understand what takes much of it and how that's possible that the installed applications can use 7 GB? I only use a few social media clients without any games or soft like Photoshop. Also on the second screenshot it indicates that the apps are using just 1.13 GB and so I can't understand the situation. Also what is the "Other"? How to overcome Acrobat Reader error 131 with a pdfLaTeX doc? I'm generating a PDF document with pdflatex (more precisely, latexmk invoked from TexMaker 3.4. My LaTeX install is TeXLive 20120719). I can read the document fine in Okular and Gmail's attachment preview, but a colleague that runs Acrobat Reader on Windows reports: "There was a problem reading this document (131)".  EDIT: The error disappears when using \usepackage[demo]{graphicx}, which puts black boxes instead of the real images. They are all PDFs, PNGs and JPGs. Instead of using demo, I added the code for a \demoincludegraphics command, which makes the file work in Reader when using it to replace the logo in the headers, as shown below. But enabling it (using \includegraphics instead of \demoincludegraphics) and removing everything else also works in Reader! There is some strange global interaction that makes the file appear broken to Reader when enabling all the graphics...  My code compiles fine, reason why I only attach the preamble and a simple text for a Minimal Working Example:  \documentclass[12pt,lettersize,twoside]{article}  \usepackage[utf8]{inputenc} \usepackage[spanish,es-tabla]{babel}   \usepackage[fixlanguage]{babelbib} \usepackage{amssymb,amsmath} \usepackage{graphicx}    \usepackage{array}    \usepackage{caption} \usepackage{subfig} \usepackage{rotating} \usepackage{booktabs} \usepackage{multirow} \usepackage{calc} \usepackage{stfloats} \usepackage{fixltx2e} \usepackage{url} \usepackage[colorlinks=true,linkcolor=blue,citecolor=red]{hyperref} \usepackage[all]{hypcap}  % Margins \usepackage[top=2cm, bottom=2cm, left=3cm, right=2cm]{geometry}  % Make header show up in the first page \makeatletter \renewcommand\maketitle{\par \begingroup     \renewcommand\thefootnote{\@fnsymbol\c@footnote}%     \def\@makefnmark{\rlap{\@textsuperscript{\normalfont\@thefnmark}}}%     \long\def\@makefntext##1{\parindent 1em\noindent             \hb@xt@1.8em{%                 \hss\@textsuperscript{\normalfont\@thefnmark}}##1}%     \if@twocolumn     \ifnum \col@number=\@ne         \@maketitle     \else         \twocolumn[\@maketitle]%     \fi     \else     \newpage     \global\@topnum\z@   % Prevents figures from going at top of page.     \@maketitle     \fi     \thispagestyle{fancy}\@thanks % was {empty} \endgroup \setcounter{footnote}{0}% \global\let\thanks\relax \global\let\maketitle\relax \global\let\@maketitle\relax \global\let\@thanks\@empty \global\let\@author\@empty \global\let\@date\@empty \global\let\@title\@empty \global\let\title\relax \global\let\author\relax \global\let\date\relax \global\let\and\relax } \makeatother  % Remove date field in title; Remove preceding vertical space \makeatletter \renewcommand{\@maketitle}{ \vspace{-3cm} \begin{center}% {\LARGE \@title}\\ \vspace{2mm}% {\@author} \end{center}% \par} \makeatother \makeatletter  % Demo includegraphic boxes \newcommand{\demoincludegraphics}[2][]{%   \setlength{\fboxsep}{-\fboxrule}% Remove frame separator/gap   \framebox{\rule{0pt}{100pt}\rule{150pt}{0pt}}% Framed box of 150pt x 100pt }  % Headers Config  \usepackage{fancyhdr} \pagestyle{fancy} \fancyhead{} \renewcommand{\headrulewidth}{0.4pt} \setlength{\headheight}{2.2cm}  \addtolength{\textheight}{-1.62cm} \fancyhead[LO,RE]{\demoincludegraphics[height=1.6cm]{logo-slogan}} \fancyhead[LE,RO]{}  % Footers Config \fancyfoot{} \renewcommand{\footrulewidth}{0pt} \fancyfoot[C]{\thepage}  % Notas al pie etiquetadas por simbolos   \renewcommand{\thefootnote}{\fnsymbol{footnote}}  \title{Doc Title} \author{My name}  \begin{document}  \maketitle  This is a test. It includes graphics and other floats.   \end{document}   EDIT: I found a workaround: Processing the generated pdf file through Ghostscript, as follows:   gs -dSAFER -dBATCH -dNOPAUSE  -sDEVICE=pdfwrite -sOutputFile=output.pdf input.pdf   where input.pdf is the file generated by pdfLaTeX, and output.pdf the one that will be readable by Adobe Reader and the rest of the PDF readers. The standard titlepage is quite simple. I am looking for a very  and need help customizing my title page.  It should match the  classicthesis doctoral thesis template  Below a minimal working example:   \documentclass[12pt,a4paper,footinclude=true,twoside,headinclude=true]{scrbook} \XeTeXinputencoding iso-8859-1 \usepackage[marginparsep=8pt,left=3.5cm,right=3.5cm,top=3cm,bottom=3cm]{geometry}  \usepackage[parts,pdfspacing,dottedtoc]{classicthesis}    \frontmatter  \begin{document}     \pagestyle{scrheadings}        %this is the place to set up my cover sheet.       %*******************************************************      % Abstract     %*******************************************************    %\renewcommand{\abstractname}{Abstract}    \pdfbookmark[1]{Résumé}{Résumé}   \begingroup    \let\clearpage\relax    \let\cleardoublepage\relax    \let\cleardoublepage\relax       \chapter{Résumé}     Résumé de la thèse en français\dots      \vfill      \chapter*{Summary}   Résumé de la thèse en anglais\dots    \endgroup             \vfill     \end{document} Badge overflow on users with too many badges When a user has &quot;too much&quot; reputation and &quot;too many&quot; badges, the design of the user card breaks:    The badge counters overflowing to another line. Can anything be done about this? For example, expanding the rep and badges to their own lines:  NameRepBadges Why would Voldemort spare Lily? In one of Harry's dementor-induced flashbacks, he presumably hears his mother protesting against Voldemort:  Harry was falling again through thick white fog, and his mother’s voice was louder than ever, echoing inside his head – ‘Not Harry! Not Harry! Please – I’ll do anything –’ ‘Stand aside – stand aside, girl –’ ‘Harry!’  Voldemort tells her to stand aside - why did he not just kill her? Hole-and-nail paradox in special relativity Bounded measurable function and integral with charcteristic function Setup or "set up"? The word setup (as a noun), I feel, has become an acceptable part of the English language, at least insofar as technical manuals or computer textbooks are concerned. However I find that the word is used very often in verb form like so:     "You can setup the program quite easily ..."   when in fact my understanding is that the actual sentence should be:     "You can set up the program quite easily ..."   I would like to know if setup can be used as a verb. The situation seems be very similar to the word output (as a noun). Output is a purely technical word and its use is restricted solely to technical texts or at least in a technical context within non-technical books. However it is also used in verb form now like so:     "The system will output the signal ..."   I would be very grateful if this question is answered. Thank you I am trying to prove a result, for which I have got one part, but I am not able to get the converse part.  Theorem. Let $R$ be a commutative ring with $1$. Then $f(X)=a_{0}+a_{1}X+a_{2}X^{2} + \cdots + a_{n}X^{n}$ is a unit in $R[X]$ if and only if $a_{0}$ is a unit in $R$ and $a_{1},a_{2},\dots,a_{n}$ are all nilpotent in $R$.  Proof. Suppose $f(X)=a_{0}+a_{1}X+\cdots +a_{n}X^{n}$ is such that $a_{0}$ is a unit in $R$ and $a_{1},a_{2}, \dots,a_{r}$ are all nilpotent in $R$. Since $R$ is commutative, we get that $a_{1}X,a_{2}X^{2},\cdots,a_{n}X^{n}$ are all nilpotent and hence also their sum is nilpotent. Let $z = \sum a_{i}X^{i}$ then $a_{0}^{-1}z$ is nilpotent and so $1+a_{0}^{-1}z$ is a unit. Thus $f(X)=a_{0}+z=a_{0} \cdot (1+a_{0}^{-1}z)$ is a unit since product of two units in $R[X]$ is a unit.   I have not been able to get the converse part and would like to see the proof for the converse part. The ability to undo reviews