Let $\mathcal{A}$ be a Lie algebra, and $$  \operatorname{Aut}(\mathcal{A})=\{a \in \operatorname{End}(\mathcal{A}) \mid[a(x), a(y)]=a([x, y]), \forall x, y \in \mathcal{A}\}, $$ where $\operatorname{End}(\mathcal{A})$ denotes the set of linear maps $\mathcal{A}\to\mathcal{A}$. I have to show that $\operatorname{Aut}(\mathcal{A})$ is a closed Lie subgroup of $GL(\mathcal{A})$, and I've been provided the hint that I should first show that $\mathrm{Aut}(\mathcal{A})$ is the isotropy group (a.k.a. the little group or the stabilizer) of some point in some representation of $GL(\mathcal{A})$. Could someone give me a sketch or clue of how to prove the latter?
Let $G$ be a lie group with lie algebra $\frak{g}$. Let $Aut(\frak g)$ be the automorphism group of $\frak{g}$.   Its clear to me that $Aut(\frak{g})$ $\subset GL(\frak{g})$ since any automorphism of $\frak{g}$ is also a linear transformation. Now i have several questions:   How do i prove that $Aut(\frak{g})$ is actually a lie subgroup of $GL(\frak{g})$? I've tried playing with actions to realize $Aut(\frak g)$ as an orbit or an isotropy subgroup but nothing worked out... Say $H \subset G$ is an abstract subgroup (not necessarily lie). Are the following statements equivalent?   $$\{T_e H \text{  is closed under the lie bracket of } \frak{g} \}$$  $$ \{H \text{is a lie subgroup of } G\}$$   How does the lie algebra of $Aut(\frak g)$ look like? When does $Aut(\frak g)$ $= G$? what does $Aut(\frak g)$ tell me about $G$?
I have made a script that will sort files according to their extension and place them in the proper folder. For example, place abc.jpg in the directory jpg.  #!/bin/bash #this script sorts files according to their extensions oldIFS=$IFS IFS=$'\n' (find . -type f) &gt; /tmp/temp for var in `cat /tmp/temp` do name=`basename "$var"` ext=`echo $name | cut -d'.' -f2- | cut -d'.' -f2- | cut -d'.' -f2- | cut -d'.' -f2- | cut -d'.' -f2- | cut -d'.' -f2- | cut -d'.' -f2-` mkdir -p $ext mv "$var" $ext/ 2&gt; /dev/null done IFS=$oldIFS   problem with this script:    it involves use of IFS, it is said to avoid use of IFS, as much as possible it does not sorts file without file extensions  it will sort files like abc.tar.bz in folder named bz, but however such a file should go in tar.bz folder see line 9 of my script; if any file contain more no. of dots(in its name) than no. of cut -d'.' -f2- in the script than if will result in file name taken in extension part. for example, a file named i.am.live.in.india.and.i.study.computer.science.txt will be placed in folder named study.computer.science.txt   you may also suggest any tweaks to make this script more smaller and neat. We have duplicates in our all contacts in Marketing Cloud. To solve this, we want to delete all contacts (audience builder).   We use the Marketing Cloud connector, and send emails to synchronized data extensions. Because of this connector the all contacts will be filled with fresh data after 15 minutes right?  Is there any reason not to do this? What can go wrong? Setting "checked" for a checkbox with jQuery I'd like to do something like this to tick a checkbox using jQuery:  $(".myCheckBox").checked(true);   or  $(".myCheckBox").selected(true);   Does such a thing exist? Nginx rewite rules 403 error I am having trouble with converting .htaccess files to nginx. I have 3 .htaccess files. The first .htaccess file is located in the document root and is as follows:  Options +FollowSymLinks RewriteEngine On RewriteRule ^img-(.*)\.html img.php?id=$1 [L] RewriteRule ^slide-(.*)\.html slider.php?id=$1 [L] RewriteRule ^page-(.*)\.html page.php?name=$1 [L] RewriteRule ^contact\.html$ contact.php [QSA,L,NC]   The second .htaccess file is located in a folder called upload:  RewriteEngine On RewriteCond %{HTTP_REFERER} !^http://(.+\.)?foo\.com/ [NC] RewriteCond %{HTTP_REFERER} !^$ RewriteRule .*\.(jpe?g|gif|bmp|png)$ nohotlink.gif [L] &lt;Files ~ "\.(php|sql|php3|php4|phtml|pl|py|jsp|asp|htm|shtml|sh|cgi)$"&gt;   order allow,deny   deny from all &lt;/Files&gt;   And the third and final .htaccess file is found in a subdirectory of the upload folder named "small":  RewriteEngine Off   Now I have created a folder in /etc/nginx called includes and made 3 seperate .access files with these rewrite rules:  For the first .htaccess file located in the document root directory I have created an file in /etc/nginx/includes called root.access. In this file I have:  # nginx configuration   location /img {  rewrite ^/img-(.*)\.html /img.php?id=$1 break;  }  location /slide {      rewrite ^/slide-(.*)\.html /slider.php?id=$1 break;  }  location /page {      rewrite ^/page-(.*)\.html /page.php?name=$1 break;  }  location /contact {      rewrite ^/contact\.html$ /contact.php break;  }   For the second file located in the "upload" folder I have crated a file in /etc/nginx/includes called upload.access. This file contains the following:  # nginx configuration   location /upload {  if ($http_referer !~ "^http://(.+\.)?foo\.com/"){  rewrite .*\.(jpe?g|gif|bmp|png)$ /nohotlink.gif break;  }  }  location ~ \.(php|sql|php3|php4|phtml|pl|py|jsp|asp|htm|shtml|sh|cgi)$ {  deny all;  }   And for the third and final file I have created a file in /etc/nginx/includes names small.access. The contents of this file are as follows:  # nginx configuration   location /upload/small {  }   In the server block configuration file I have:   location / {               try_files $uri $uri/ /index.php;               include /etc/nginx/includes/root.access;      }   location /upload {               include /etc/nginx/includes/upload.access;      }   location /upload/small {               include /etc/nginx/includes/small.access;      }   Now with this configuration when trying the site I receive 403 errors. Nginx error log reports:  [error] 18156#0: *7 access forbidden by rule, client: 111.**.**.**, server: foo.com, request: "POST /upload.php HTTP/1.1", host: "foo.com", referrer: "http://foo.com/"   Now under apache everything works no problems. But I can not see why I am getting 403 errors. I am also worried that the rewrite rules as I have stated them, beyond the 403 error will not function properly at all. Can anyone help me with this? I can not see what is wrong here. MacBook Pro locked with Find My Mac and won't let me boot Trouble enabling Cube surfer addon Every time I have tried to enable the Cube Surfer addon but this error message comes up. FYI I have the correct versions of everything.  I have had this problem with some other add ons as well. Sometimes it's when I have tried to click a button in an addon (after 'successfully' installing) How to use sed to replace only the first occurrence in a file? Which folder is recycle bin in on Win7? How to check files there? I was using another account before. Let me call it account A. And I deleted a lot of files to the Recycle Bin at that time. But before I removed account A I forgot to empty the Recycle Bin.   My question is, are my files still there? I've searched and found that the Recycle Bin may live in $Recycle.Bin, but I can't open files there so I can't know what they are. Any help how to open deleted files or how to delete them permanently? Decreasing font size in verbatim How can I modify the following command to make it work?  \newcommand{\code}[1]{\begingroup \fontsize{10pt}{10pt}\selectfont \begin{verbatim} #1 \end{verbatim} \endgroup}   I currently get the following error on the close of the following verbatim block:  ! Undefined control sequence. &lt;argument&gt; ...endgroup ^^M^^M$\eta_{xi} = r_{xi} -                                                    q_i \cdot p_x$ \\^^M$q_i ... l.227     \end{verbatim} Do you capitalize both parts of a hyphenated word in a title? Do you capitalize both parts of a hyphenated word in a title?  "My Ex-Wife Hates Me"  or  "My Ex-wife Hates Me" Seeking detailed description of qgs file structure? Is there a detailed description on the .qgs file structure somewhere?   I checked at  and got     The requested URL /qgis.dtd was not found on this server.   and at  which wasn't very detailed. At what moment will matter falling into a black hole affect its size? Does the mass of matter falling into a black hole affect the size of an event horizon the moment it passes through it, or when it has been incorporated into the singularity? Image looks embossed when converted to SVG What are the chances of getting $K$ identical outcomes in a row in a series of $Y$ independent events? Each event has two possible outcomes: a and b.  Each event is independent.  Outcome a has $X\%$ chance of happening.  If we run $Y$ number of such events, what are the chances of getting $K$ outcomes a in a row, at least once?  To put it in practical terms imagine a series of coin tosses. Each coin toss has two possible outcomes: a = heads and b = tails. Outcome a = heads has a chance of $50\%$. If we run $Y = 100$ number of coin tosses, what are the chances of getting $K = 10$ outcomes a=heads in a row, at least once?   I'm using the practical example only so I can express the idea more clearly. I'm interested in finding out the general formulas and computations so that I can run the calculations for any $X$, $Y$, or $K$.  I would also be very interested in knowing how the probabilities would change if the events would not be independent. What if the more outcomes a we have in a row the bigger the chance of getting an outcome a in the next event? What if it's the other way around, and the more outcomes a in a row we have the lower the chance of getting another a? Where does a spinning figure skater's energy go when she slows down? \end{abstract} conflicts with arabtext package I am using elsarticle  Everything was find before I used \usepackage{arabtext}.   When I compiled to PDF, I got the following error message.    And if I type "s", everything became fine again but I don't want to type an "s" every time I compile PDF preview.   Please anyone advise.  Thank you very much in advance for your help.  Here is the code  \documentclass[12pt, authoryear]{elsarticle}    % use "amsart" instead of "article" for AMSLaTeX format \usepackage[colorlinks=true,linkcolor=black, citecolor=blue, urlcolor=blue]{hyperref} \usepackage{geometry}                       % See geometry.pdf to learn the layout options. There are lots. \geometry{a4paper}                          % ... or a4paper or a5paper or ...  %\geometry{landscape}                       % Activate for rotated page geometry %\usepackage[parfill]{parskip}          % Activate to begin paragraphs with an empty line rather than an indent \usepackage{graphicx}               % Use pdf, png, jpg, or epsÂ§ with pdflatex; use eps in DVI mode                             % TeX will automatically convert eps --&gt; pdf in pdflatex         \usepackage{amssymb}  \usepackage{amsmath} \usepackage{tipa} \usepackage{arabtex} %SetFonts  %SetFonts   \title{Translating from English to Bahasa Indonesia: Which procedures should be taught to EFL students preparing to be a translator?} \author[mkl]{Mohammad Kholid}  \author[fai]{Faisal Mustafa\corref{cor1}} \ead{faisal.mustafa@unsyiah.ac.id}  \cortext[cor1]{Corresponding author}  \address[mkl]{Syiah Kuala University, Banda Aceh, Indonesia} \address[fai]{Syiah Kuala University, Banda Aceh, Indonesia} \date{}                         % Activate to display a given date or no date  \begin{document} %\section{} %\subsection{}  \begin{frontmatter}  \begin{abstract}   Abstract content is here.  \end{abstract}  \begin{keyword}  translation procedures \sep English to Bahasa Indonesia \sep teaching translation \end{keyword} \end{frontmatter} The quotient norm on $\ell^{\infty}(\mathbb{N}) / c_0 (\mathbb{N})$ is given by $\limsup_{n \in \mathbb{N}} |x_n|$. I try to show that the norm on the quotient space $\ell^{\infty}(\mathbb{N}) / c_0 (\mathbb{N})$ is given by $\limsup_{n \in \mathbb{N}} |x_n|$, where $x = (x_n)_{n \in \mathbb{N}} \in \ell^{\infty} (\mathbb{N})$. My attempt is the following:     By definition, the quotient norm $\| \cdot \|_{\ell^{\infty} / c_0}$ is defined as   $$ \| x + c_0 (\mathbb{N}) \|_{\ell^{\infty} / c_0} = d( x + c_0 (\mathbb{N}), 0 + c_0 (\mathbb{N}) ), $$   which can be re-written as    $$ \| x + c_0 (\mathbb{N}) \|_{\ell^{\infty} / c_0} = \inf \{ \| x + y \|_{\infty} \; | \; y \in c_0 (\mathbb{N}) \}. $$   Since $0 = (0,0,...) \in c_0 (\mathbb{N})$, it follows that   $$ \| x + c_0 (\mathbb{N}) \|_{\ell^{\infty} / c_0} \leq \| x \|_{\infty} = \sup_{n \in \mathbb{N}} |x_n|. $$   I thought about using Bolzano-Weiserstrass' Theorem at a certain point, but I am not sure how to obtain the wished equality from here.    Any help is appreciated. Thanks in advance. Can you allocate an array with something equivalent to make_shared? I was wondering why I couldn't delete my own question. Well actually, I have accepted the question itself to be wrong and even I feel that I have accepted a wrong answer, but when I try to delete my question, I get a message saying that,"This question can't be deleted, flag it for moderator attention". Why is this happening and how can I solve this? Unable to see anything after importing .obj file