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If $a,b$ are positive integers such that $\gcd(a,b)=1$, then show that $\gcd(a+b, a-b)=1$ or $2$. and $\gcd(a^2+b^2, a^2-b^2)=1$ or $2 $ Implications and Ordinary language Already the early papers on supergravity were written using computer algebra software to do some calculations. What modern packages do people normally use for doing such calculations? Of course Mathematica and presumable a lot of other programs can be made to do such calculations, I am wondering if there are special packages or tutorials how to do such calculations in them. How can I disable Ctrl+Q for all applications (system wide)? A lot of applications like Firefox, Chrome etc, get closed when I accidentally press Ctrl+Q instead of Ctrl+W because of the proximity of the Q and W keys on the keyboard. Is there a way this shortcut can be removed or reset on a system wide basis? Seeking appropriate statistic to measure spatial autocorrelation of points with binary values I am trying to determine the level of spatial autocorrelation in a point dataset.  The attribute I'm interested in is binary (presence/absence of a species), for which Moran's I is not appropriate.  On the other hand, Joint Count statistics, which are typically recommended for binary or categorical data, apparently are not appropriate for point data. What is an appropriate statistic to measure global and/or local spatial autocorrelation of points when the attribute of interest is binary? Examples of provably${}^n$ unprovable statements Given any statement $A$ and a classical theory $T$ which we assume is at least as strong as Peano Arithmetic ($\sf PA$), we have that $T\vdash A$ implies $T\vdash T\vdash A$ (that is, if a statement is provable, then it is provably provable). Let $T_1=T$ and $T_{n+1}=T_n+{\sf Con}(T_n)$. (We need stronger theories if we want to prove unprovability.) Given this, simply by considering the truth values of the relevant statements, we get the following classification of statements by unprovability:   $T\vdash A$ ($A$ is provable) $T_2\vdash T\not\vdash A$ ($A$ is provably unprovable) $T_3\vdash T_2\not\vdash T\not\vdash A$ ($A$ is provably unprovably unprovable)   &nbsp;&nbsp;&nbsp;&nbsp;...  &nbsp;&nbsp;&nbsp;&nbsp;$\omega$. For all $n$, $T_n\not\vdash\dots \not\vdash T_2\not\vdash T\not\vdash A$ (we can't say anything about $A$'s provability)  We can refine this hierarchy by also considering the status of $\lnot A$. Let's call a class $m,n$ statement one such that $A$ is at level $m$ above and $\lnot A$ is level $n$ in the list. Since $T\vdash\lnot A$ implies $T_2\vdash T\not\vdash A$, the only nonempty classes at level $1$ are the class $1,2$ statements (provable statements) and the class $2,1$ statements (refutable statements). Independent statements are in class $2,2$. The class $1,1$ is nonempty iff the theory is inconsistent.  My question is: Are there any (preferably "natural") statements that fall further into this hierarchy? I know many examples of independent statements, like the continuum hypothesis or the axiom of choice relative to $\sf ZF$, or the Paris–Harrington theorem relative to $\sf PA$. But I know no examples of even any class $2,3$ statements. I expect there may also be natural examples of $\omega,\omega$ statements. My guess is that all the classes $m,n$ with $m,n\ne1$ are nonempty, but examples elude me. How do I defeat the Kayran in the first chapter of the Witcher 2? In the first chapter you are supposed to slay the Kayran, this enormous beast:    I'm having a hard time with it. I drank the Mongoose potion for poison resistance and I have the Kayran trap, but it just hits me too often with its tentacles. Especially when the Kayran swings a tentacle sideways, not from above, I have a very hard time evading it.  Is there any trick besides not getting hit by the tentacles?  With a lot of luck I'm getting as far as the point where it's throwing stones. Any idea what I'm supposed to do then? When we eat something, after munching and swallowing a bite, we are strongly stimulated to have one more bite. What is the taste that causes this effect? Color change of string delimiter in listings with macro What is "umask" and how does it work? I believe that umask is something that controls , but do not fully understand it.  After running umask 0644 in a , I cannot read the files I create with the command-line text editor nano. I noticed that the permissions of that file are set to 0022 instead of the default 0755.  How does umask work? I thought I could just remove the each digit in the umask from 0777, 7 - 6 = 1 and 7 - 4 = 3, so I expect the permissions to be 0133, but apparently, this is not the case.   What is umask exactly? Explain it to me like I was a "Linux noob" How do I calculate with umask? What are use cases for umask? Why does Yoda speak the way he does? Do "Best by: XX-XX-XX" foods have an expiration date? How long will foods that have "Best by: XX-XX-XX" on the product last past that date?  I have tried looking online, and all I can find is anecdotal evidence, which isn't very assuring for me since there seems to be no specific "x food will expire y weeks after its best by date".  Is there a rule of thumb or some guide out there that can tell me how long a "best by" food will last? Or is it safe to assume that they never expire, since it's not actually labelled on the product? How to make the space under footnotes disappear? How do I make the space under footnotes disappear? And to space over the footnote's bar?    Code:  \documentclass[12pt,french]{report} \usepackage{mathptmx} \usepackage{helvet} \usepackage{courier} \usepackage[T1]{fontenc} \usepackage[cp1250]{inputenc} \usepackage[a4paper]{geometry} \geometry{verbose,margin=1in,headheight=12pt,headsep=25pt} \usepackage{fancyhdr} \pagestyle{fancy} \setlength{\parskip}{\medskipamount} \setlength{\parindent}{0pt} \usepackage{babel} \makeatletter \addto\extrasfrench{%    \providecommand{\og}{\leavevmode\flqq~}%    \providecommand{\fg}{\ifdim\lastskip&gt;\z@\unskip\fi~\frqq}% } \makeatother \usepackage{latexsym} \usepackage{pifont} \usepackage{color} \usepackage{float} \usepackage{textcomp} \usepackage{graphicx} \usepackage{setspace} \usepackage[unicode=true,  bookmarks=true,bookmarksnumbered=true,bookmarksopen=false,  breaklinks=true,pdfborder={0 0 0},backref=false,colorlinks=false]  {hyperref} \hypersetup{pdftitle={tito},  pdfauthor={abc},  pdfkeywords={kiko}} \usepackage{lipsum}  \makeatletter \usepackage{multicol} \usepackage{fancyhdr} \usepackage{multirow} \usepackage{longtable} \setcounter{secnumdepth}{4} \setcounter{tocdepth}{4} \usepackage{array} \newcommand{\ts}{\textsuperscript} \renewcommand\theparagraph{\alph{paragraph}.} \renewcommand\paragraph{\@startsection{paragraph}{4}{\z@}%                                     {3.25ex \@plus1ex \@minus.2ex}%                                     {0.5ex \@plus .2ex}%                                     {\normalfont\normalsize\bfseries}} \renewcommand*\l@paragraph{\@dottedtocline{4}{10em}{2em}}  \def\thickhrulefill{\leavevmode \leaders \hrule height 1ex \hfill \kern \z@}  \def\@makechapterhead#1{% \vspace*{10\p@}% {\parindent \z@ {\reset@font \usefont{OT1}{phv}{m}{n} \LARGE Chapitre \thechapter\par\nobreak}% \par\nobreak \vspace*{30\p@} \interlinepenalty\@M \usefont{OT1}{ptm}{b}{n} {\raggedright \Huge #1}% \par\nobreak \vskip 20\p@ \hrule height 1pt \par\nobreak \vskip 45\p@ }}  \def\@makeschapterhead#1{% \vspace*{10\p@}% {\parindent \z@ {\raggedleft \reset@font \scshape \vphantom{\@chapapp{} \thechapter} \par\nobreak}% \par\nobreak \vspace*{30\p@} \interlinepenalty\@M \usefont{OT1}{ptm}{b}{n} {\raggedright \Huge #1}% \par\nobreak \par\nobreak \vskip 45\p@ }}  \lhead{\nouppercase\leftmark} \chead{} \rhead{} \lfoot{} \cfoot{\thepage} \rfoot{} \renewcommand{\footrulewidth}{0.4pt}  \pretolerance=10000 \hyphenpenalty=10000 \exhyphenpenalty=10000  \AtBeginDocument{   \def\labelitemi{\Pisymbol{psy}{183}}   \def\labelitemii{\(\rhd\)}   \def\labelitemiii{\(\rhd\)}   \def\labelitemiv{\(\ast\)} } \makeatother  \begin{document} \chapter{Chap} \lipsum[1]\footnote{\lipsum[1]} \lipsum[1-8] \end{document} One of the double sockets in our garage needed replacing (damaged after a washing pole hit it) - a pretty trivial job that I'd done before. So I flipped the MCB for the ring main, used a non-contact voltage tester to check nothing was there, and set to work. I disconnected the old socket fine, and was lining the cables up to fit into the new socket when... ow.  My finger bridged the live and earth pins, and suffice to say the live wire was very much alive. Bar a few finger spasms I was fine fortunately, and believe me I realise I was very lucky in that respect (I'm in the UK, so that was 240V.) Before finishing the job I then shut off the entire property's electrics - I was taking no more chances.  Further inspection confirmed two things I immediately suspected at this point - the MCB in the garage (installed before we moved into the house) was shutting off power via the neutral, not the live, and the voltage tester was dodgy (I had tested it on the same socket earlier and it seemed fine, further testing showed it presumably had a loose connection internally and only worked sporadically.) Needless to say it's now in the bin, and an electrician has been called to sort the MCB and check the rest of the house for similar issues.  Unsurprisingly this has shaken me up a bit - I would say I'm a rather safety conscious person, and in this case thought I'd done all the checks properly (including taping over the relevant trip switch to ensure no-one flipped it back on while I was working.) In some ways I can't help feeling I was unlucky that the MCB was wired by a cowboy, in combination with the voltage tester failing in the way that it did, at the time that it did. However, I'm not one to solely blame tools - clearly some, if not most of the blame here lies with myself also.  So in short, what should I have done differently - and was I primarily being stupid, or really quite unlucky? Find the probability that a geometric random variable $X$ is an even number Proving $\frac{1}{n+1} + \frac{1}{n+2}+\cdots+\frac{1}{2n} &gt; \frac{13}{24}$ for $n&gt;1,n\in\Bbb N$  To solve it I used induction but it is leading me nowhere my attempt was as follows:  Lets assume the inequality is true for $n = k$ then we need to prove that it is true for $k+1$  so we need to prove $\frac1{k+2} + \frac1{k+3}+\cdots+\frac1{2(k+1)} &gt; 13/24$  I don't know where to go from here please help. I have a samba share on my LAN that requires a username and password to obtain write access (otherwise it defaults to read-only).  I can't figure out where in the Unity desktop I can connect to the share and be prompted for my login credentials.  I can mount the drive, of course, but then I'm browsing as a guest (read-only).  What gives? Search results showing questions without the searched for tag in results Recently, I was surfing on stackoverflow.com &amp; I have found one bug in Question filter.  I was looking under the java tag and yet I saw questions which were not tagged java    The incorrectly shown question was Gradient borders Gduated symbology not taking attribute into account using QGIS I m working on a project on QGIS. I have a point shapefile vector that represents my store. In the attribute table, i have bunch of data such as size, unit per transaction and most importantly store turnover. When I want to graduate the symbology, the turnover isn't present in the list I can choose from. Does someone know why, and how to fix it?