To prove that every element of $G$ has finite order. Let $G$ be a group such that the intersection of all its subgroups which are different from $e$ is a subgroup different from $e$. Prove that every element of $G$ has finite order.   Can i get some hints to get started?  Thanks.  EDIT   As per suggestion of user Slade, i have proceeded according to this   If $g$ has infinite order, then the cyclic group that $g$ generates will have elements of the form $\{g,g^{2},g^{3},......)$.   The claim is that no two elements are same, i.e. there are no repetitions. To show this let us assume that there are repetitions, say $a^{i}=a^{j}$, for some $i,j$. Then $a^{i-j}=e$ and this contradiction of $a$ having infinite order.   What should i do next?
Prove every element of $G$ has finite order. Let $G$ be a group such that the intersection of all its subgroups which are different from $\{e\}$, is a subgroup different from $\{e\}$. Prove that every element of  $G$  has finite order.  Assume that $a$ has infinite order in $G$. Then let $y\neq e$ belong to the intersection of all subgroups of $G$ different from $\{e\}$. Then $y=a^n$ and $y$ will also belong to subgroup generted by $a^2$ hence  $y=a^{2m}$ for some integers $m$. This contradicts that $a$ has infinte order.  Is this correct?
How to solve 'I found no database files--while read file something.aux' Hi I am trying to include references into my paper but it keeps showing the following when I compile. I am using texstudio. Message: I'm skipping whatever remains of this command The style file: plain.bst I found no database files---while reading file term paper.aux Warning--I didn't find a database entry for "Astrov"  My code is like this:   \documentclass[letter,scriptaddress,twocolumn, prl,showkeys]{revtex4}  \usepackage{amssymb, amsmath, amstext, amscd, verbatim, mathtools, graphicx, float, caption, subcaption} \usepackage[utf8]{inputenc} \usepackage{natbib} \begin{document} This is my reference\cite[1]{Astrov}. \bibliography{term paper} \bibliographystyle{plain} \end{document}   They are all in the same directory. I am really panicing about this at the moment and apologize if similar questions have been asked before. Thanks in advance for your help! $\sqrt{c+\sqrt{c+\sqrt{c+\cdots}}}$, or the limit of the sequence $x_{n+1} = \sqrt{c+x_n}$ (Fitzpatrick Advanced Calculus 2e, Sec. 2.4 #12)  For $c \gt 0$, consider the quadratic equation $x^2 - x - c = 0, x &gt; 0$.  Define the sequence $\{x_n\}$ recursively by fixing $|x_1| \lt c$ and then, if $n$ is an index for which $x_n$ has been defined, defining  $$x_{n+1} = \sqrt{c+x_n}$$  Prove that the sequence $\{x_n\}$ converges monotonically to the solution of the above equation.  Note: The answers below might assume $x_1 \gt 0$, but they still work, as we have $x_3 \gt 0$.    This is being repurposed in an effort to cut down on duplicates, see here: .  and here: . I'm working on a VAR model and am doing a log transformation of the raw data.   x = log(x)   After differencing, running various tests and running a VAR(2) model, I add the predicted differences cumulatively to the original time series (logged) like this:  prd.diffs &lt;- as.ts(prd$mean$d.x1) U &lt;- tail(undiff,2) cumu &lt;- cumsum(prd.diffs) + U[1]   However, while the output now looks as it should, it is still in logged values.  My question is as follows:  How does one return the logged output to original values? I'm under the impression that using exp() will not give the correct results, and having tried some variations it hasn't worked for me.   Does the fact that it is a VAR($p$) model (and possibly what lag it has) influence the method, and how should that be implemented to get the correct output?  P.S. I realise that the above code does not constitute a replicable example as I didn't deem that necessary for the question, but should that be desired I can add some more code. tput: No value for $TERM and no -T specified I am trying to run a bash script I have via cron, and I am getting the following error at the beginning of the execution:  tput: No value for $TERM and no -T specified   Here is what is in my crontab:  0 8 * * 1-5 cd /var/www/inv/ &amp;&amp; /var/www/inv/unitTest run all 2&gt;&amp;1| mail -r "user@example.com" -s "Daily Inventory Unit Test Results" user@example.com I just visited  and noticed that my profile picture is not the same as on all other sites.  According to  the image will be synchronized when synchronizing the complete profile. I did that but the image is still the same.  Then I wanted to change the image in the Area51 profile directly - but I can only change to a gravatar image and can't use the new and fancy Stack Exchange image.  Is that a bug? What does the "yield" keyword do? How do I automate a key press in AppleScript? I am curious if it is possible to have a VPS that has data on it that is not readable by the hosting provider, but is still usable on the VPS.  Obviously there are some things that you could do to prevent them from reading anything...   You could change all the passwords, including root. But then, they could still use some alternate boot to reset the password, or they could just mount the disk another way. So, you could encrypt the disk or at least some of the contents on the disk. But then it seems that if you decrypted the content, they could still "peer in" to see what you were doing at the console, because after all, the virtualization platform should allow this. And even if you could stop that, it seems they could just read the RAM of the VPS directly.   Of course, the VPS can store data on it and as long as the key is not on the VPS and the data is never decrypted there, then the host cannot get the data.  But it seems to me that if any point the data on the VPS is decrypted...for use on the VPS...then the hosting provider can get the data.  So, my two questions are:   Is this correct? Is it true that there is no way to 100% secure data on a VPS from a host from seeing it, while keeping it accessable by the VPS? If it is possible to make it 100% secure, then how? If it is not possible, then what is the closest you can get to hiding data from the web host? Sum of Cosine/Exponential According to the , 3rd edition:        I wish to understand why this decision has been made. What's wrong with generic exceptions?  (As far as I know, generics are simply compile-time syntactic sugar, and they will be translated to Object anyway in the .class files, so effectively declaring a generic class is as if everything in it was an Object. Please correct me if I'm wrong.) AES Mix Column Transformation Section 3.4.3 "The MixColumns Step"(page 39) of , it states     The columns of the state are considered as polynomials over   $\operatorname{GF}(2^8)$ and multiplied modulo $x^4 + 1$ with a fixed polynomial $c(x)$.      The polynomial $c(x)$ is given by   $c(x) = \mathtt{03} \cdot x^3 + \mathtt{01} \cdot x^2 + \mathtt{01} \cdot x + \mathtt{02}$      Let $b(x) \equiv c(x) \cdot a(x) \pmod{x^4 + 1}$. Then         How this polynomial multiplication modulo $x^4+1$ is converted to Matrix Multiplication? I know several questions were asked about the difference between "going to" and "will". Based on several answers (see, for instance, ,  and ), I understood that "will" is more spontaneous and "going to" is used with more planned actions.  So, it seems that everything is pretty fine. However, in , Kosmonaut has an answer in which he states: "Let's say that tomorrow you will walk your dog from 7 - 8 AM".  On the one hand, you probably planned to walk your dog long before and thus I should use "Let's say that tomorrow you're going to walk your dog from 7 - 8 am".   On the other hand, since I'm saying "let's say...", I'm deciding right now (thus, unplanned) that you will walk your dog. So, even though in this hypothetical situation you made a plan, I'm in a more spontaneous mood deciding right now that that's what you will do tomorrow, and, thus, I should use "will".  Which one (if any) of the above explanations is right? It would be nice to see what users have recently been active. This is somewhat standard with many forum-style websites, showing how active the site has been in the last n-minutes or so. As a functional advantage, this also serves to let you know when is a good time to get a question in. Change \headheight for a \pagestyle I have a document class that auto generates header from some user input. The problem is the first page should have a different page header than the others and the headline can be multi line. So \headheight is too small. Here is a demonstration of the problem:  \documentclass[twoside]{scrartcl} \newcommand{\longfirstpagehead}{% \begin{minipage}{10cm} some\\long\\text\\text\\text\\text\\text\\text \end{minipage}} \newcommand{\normalhead}{normal} \usepackage{scrpage2}  \newpagestyle{firstpage}{%     {\longfirstpagehead\hfill} %twoside links     {\longfirstpagehead\hfill} %twoside rechts     {\longfirstpagehead\hfill} %oneside     (\textwidth,1pt) }{%     {\pagemark\hfill} %twoside links     {\hfill\pagemark} %twoside rechts     {\hfill\pagemark} %oneside } %  \newpagestyle{normal}{%     {\normalhead\hfill} %twoside links     {\normalhead\hfill} %twoside rechts     {\normalhead\hfill} %oneside     (\textwidth,1pt) }{%     {\pagemark\hfill} %twoside links     {\hfill\pagemark} %twoside rechts     {\hfill\pagemark} %oneside } %    \pagestyle{normal} \AtBeginDocument{     \thispagestyle{firstpage}}  \usepackage{blindtext} \begin{document}     %\vspace*{1cm}     \blindtext      \blindtext     \pagebreak \end{document}   I tried to read the height with \settototalheight and:   add a \vspace at beginning of document (unnice hack):  still the bad box warning (of cause) first section has different alignment  manually change \headheight in pagestyle  text height and page foot is wrong (can't change this because some scoping?)  use Komascript in pagestyle  \recalctypearea warning  geometry in pagestyle  pretty much the same problem like the manual attemp    So I would like to change the \headheight for the pagestyle firstpage, but the page should still have the same page margins, etc. The problem is that this has to work with \thispagestyle. (Better Ideas are also welcome) Something like \thispagegeometry in geometry would be great. ;) I'm having some problems with listings and UTF-8 in my document. Maybe someone can help me? Some characters work, like é and ó, but á and others appear at the beginning of words...  \documentclass[12pt,a4paper]{scrbook} \KOMAoptions{twoside=false,open=any,chapterprefix=on,parskip=full,fontsize=14pt}  \usepackage[portuguese]{babel} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage{listingsutf8}  \usepackage{inconsolata} \lstset{     language=bash, %% Troque para PHP, C, Java, etc... bash é o padrão     basicstyle=\ttfamily\small,     numberstyle=\footnotesize,     numbers=left,     backgroundcolor=\color{gray!10},     frame=single,     tabsize=2,     rulecolor=\color{black!30},     title=\lstname,     escapeinside={\%*}{*)},     breaklines=true,     breakatwhitespace=true,     framextopmargin=2pt,     framexbottommargin=2pt,     extendedchars=false,     inputencoding=utf8 }  \begin{document} \begin{lstlisting} &lt;?php  echo 'Olá mundo!'; print 'Olá mundo!'; \end{lstlisting}  \end{document} \end{lstlisting} Why can't energy be created or destroyed? My physics instructor told the class, when lecturing about energy, that it can't be created or destroyed.  Why is that?  Is there a theory or scientific evidence that proves his statement true or false?  I apologize for the elementary question, but I sometimes tend to over-think things, and this is one of those times. :) Short story with competitors complaining to government about a new very popular synthetic food Synthetic foods are all the rage, and while some are new and novel, many are attempts to recreate the taste and texture of a natural foods, with varying degrees of success. One company introduces a new synthetic food that becomes wildly popular. The competition analyzes it, and based on discovering what natural creature it is based on, they petition the government to ban it.      They discover that the product is synthetic human flesh, so consuming it is not technically cannibalism, but it is similar, as they explain to the panel of government officials. After migration to MacOSX 10.11 "El Capitan", how to run simple utilities such as "texdoc" and "kpsewhich"? Last night, I migrated my MacBook's operating system from 10.10 "Yosemite" to 10.11 "El Capitan". Everything seemed to go smoothly. Happily, regarding my TeX-related work, I am able to use the GUI front ends "TeXworks" and "TeXshop" just fine since completing the OS upgrade. I use MacTeX2015, with all the latest updates installed.  Oddly, though, since completing the upgrade I can no longer run any MacTeX command-line utilities such as texdoc and kpsewhich. (Actually, as I just discovered, I can no longer run pdflatex either from a command line. However, I never do run pdflatex from a command line prompt, so that's not an urgent problem for me -- at least not for now.)  The suggestions given in the posting  do not seem to be applicable to this case, as they all seem to revolve around setting up /usr/texbin correctly -- /usr/texbin is no longer allowed under MacOSX 10.11. The document  by Herb Schulz deals in depth with adjusting settings in the GUI front ends, but it appears not to touch on what to do about /usr/texbin, which appears to be the now-missing link to TeX's command line utilities.   Pointers and solutions most welcome! How to make a bootable ISO image from a bootable DVD? When do I use a dot, arrow, or double colon to refer to members of a class in C++?