Find the orders of $M_3(\mathbb Z_2)$ and $GL_3(\mathbb Z_2)$ $|M_3(\mathbb Z_2)| = 2^{3^2} = 64$  $|GL_3(\mathbb Z_2)| = 64$ minus all singular matrices in $M_3(\mathbb Z_2)$ or all matrices with linearly independent columns.  My question is how do I find all singular matrices or all matrices with linearly independent columns. Please advise.
Order of general- and special linear groups over finite fields. Let $\mathbb{F}_3$ be the field with three elements. Let $n\geq 1$. How many elements do the following groups have?   $\text{GL}_n(\mathbb{F}_3)$ $\text{SL}_n(\mathbb{F}_3)$   Here GL is the , the group of invertible n×n matrices, and SL is the , the group of n×n matrices with determinant 1.
What is the difference between a symbolic link and a hard link? Recently I was asked this during a job interview. I was honest and said I knew how a symbolic link behaves and how to create one, but do not understand the use of a hard link and how it differs from a symbolic one. "I'm right, aren't I?" My dictionary reports that aren't is also an abbreviation for am not, when used in questions.  Is this use of aren't common? If it is common, are there any differences between the different English dialects?     Why aren't I being given a pay raise? The effect of gravitational & Coulomb forces on free electrons in a conductor In a metal, why don’t the free electrons fall to the bottom of the metal due to gravity?  Also, charges in a conductor are supposed to reside on the surface so why don’t the free electrons all go to the surface?   Regarding the first question, my hunch is that it is due to the electric force which is greater than the gravitational force and electric force act along the length of the conductor.  I don't have any intuition for the second. Are functions of independent variables also independent? Cube of an integer $\frac{x}{y}+\frac{y}{z}+\frac{z}{x}=k$ and $x, y, z, k$ are integers.  Prove that $xyz$ is cube of some integer number.    I was wondering about giving a parametrization for the rational points on the elliptic curve in $\mathbb{P}^2(\mathbb{R})$ $$ x^2 y + y^2 z + z^2 x = k xyz$$ but I am not too confident in the topic, so I am asking you to shed some light.  Thank you for your help. Batch normalize audio volume on .ogg-files Are there any simple tools to normalize an .ogg-library of 10'000+ songs so that the volume is the same throughout all songs?  Terminal or GUI doesn't matter, it only need to be simple.  One caveat though, I don't want soft interludes/intermissions and ballads blown out of proportion.  Preferably the process should find the overall gain of the album (I have all my CD's ripped into separate folders) and normalize the level thereafter. How to Check If A Plugin Is Enabled Through API? Difference between '>' and '-gt' What's the difference between these:  SEC=$(mysql -hxxx -Pxxx -uxxx -pxxx -se "SELECT Seconds FROM tablename")  SEC=100   Both return the same result (100). But when I test it with:  if [ ${SEC} &gt; 1 ] then     echo "SEC GT 1" else     echo "SEC LT 1" fi   The first one returns:  "SEC GT 1"   The second one returns:  "SEC LT 1"   If however, I change it to:  if [ ${SEC} -gt 1 ]   then the result is reversed.  I'm assuming it's got something to do with the datatype, and that I presumbaly need to cast one or the other to get the same result.  It's just that at present it's messing up my testing, as I have to remember to change the IF section every time. Deleting missing data with CountryData Let $f$ and $g$ be continuous at $ x=a$, with $f(a)<g(a)$. Prove that there exists a neighborhood $I$ of a such that $f(x)<g(x)$ for all $x \in I$. Inadvertent use of = instead of == It seems that   if (x=y) { .... }   instead of  if (x==y) { ... }    is a root of many evils.  Why don't all compilers mark it as error instead of a configurable warning?  I'm interested in finding out cases where the construct if (x=y) is useful. MongoDB could not connect local server I had mongdb installed and evrything was working fine. I wanted to delete the mongodb lock. By mistake the directory /var/lib/mongodb/ got deleted.  So, I uninstalled mongo db and installed again. But now i always keep getting this error:  Error: couldn't connect to server 127.0.0.1:27017 at src/mongo/shell/mongo.js:145 exception: connect fail   The error remains same even if I try to login from terminal or rockmongo or my application. I am using ubuntu12.04 and have latest version of mongo. Error in .bib entry, "Too many commas" The code for reference is this  @article{matcontcl,  title= {MATCONT and CL MATCONT},  author = {A. Dhooge, W. Govaerts, Yu.A. Kuznetsov, W. Mestrom, A.M. Riet and  B.Sautois},  year = {2006},  journal = {Continuation toolboxes in matlab}, }   This is the error that follows  This is BibTeX, Version 0.99d (TeX Live 2015/W32TeX) The top-level auxiliary file: PropuestaInvestigacion.aux The style file: abbrv.bst Database file #1: biblio.bib Too many commas in name 1 of "A. Dhooge, W. Govaerts, Yu.A. Kuznetsov, W. Mestrom, A.M. Riet and B.Sautois" for entry matcontcl while executing---line 1049 of file abbrv.bst Too many commas in name 1 of "A. Dhooge, W. Govaerts, Yu.A. Kuznetsov, W. Mestrom, A.M. Riet and B.Sautois" for entry matcontcl while executing---line 1049 of file abbrv.bst Too many commas in name 1 of "A. Dhooge, W. Govaerts, Yu.A. Kuznetsov, W. Mestrom, A.M. Riet and B.Sautois" for entry matcontcl while executing---line 1090 of file abbrv.bst Too many commas in name 1 of "A. Dhooge, W. Govaerts, Yu.A. Kuznetsov, W. Mestrom, A.M. Riet and B.Sautois" for entry matcontcl while executing---line 1090 of file abbrv.bst (There were 4 error messages)   How could I fix it? Volumes of n-balls: what is so special about n=5? The volume of an $n$-dimensional ball of radius $1$ is given by the classical formula $$V_n=\frac{\pi^{n/2}}{\Gamma(n/2+1)}.$$  For small values of $n$, we have $$V_1=2\qquad$$ $$V_2\approx 3.14$$ $$V_3\approx 4.18$$ $$V_4\approx 4.93$$ $$V_5\approx 5.26$$  $$V_6\approx 5.16$$ $$V_7\approx 4.72$$ It is not difficult to prove that $V_n$ assumes its maximal value when $n=5$.   Question. Is there any non-analytic (i.e. geometric, probabilistic, combinatorial...) demonstration of this fact? What is so special about $n=5$? I also have a similar question concerning the $n$-dimensional volume $S_n$ ("surface area") of a unit $n$-sphere. Why is the maximum of $S_n$ attained at $n=7$ from a geometric point of view?   note: the question has also been  for those curious to other answers. override parent theme configuration in child functions.php Difference between Groovy Binary and Source release? I have a CSV with an address field (no lat/long), so I went to Google My Maps, imported the csv and geocoded of the records.  Now I export the layer as KML and added the KML to QGIS and Google Earth Pro, but in non of them I can see the markers. In fact in QGIS, the KML layer is added as datatable (no geometry).  What's happening? How can I export a geometry layer from google My Maps? Pretty formattings for warning and info blocks I'm looking for pretty formatting for warnings, and others for some info or tips.  For the moment, I use the following formatting but maybe someone has one better idea.    To obtain this, I use the following code.  \documentclass[10pt,a4paper]{article} \usepackage[utf8]{inputenc} \usepackage[x11names,svgnames]{xcolor} \usepackage{graphicx} \usepackage{fourier-orns} \usepackage[framemethod=tikz]{mdframed} \usetikzlibrary{calc}  \makeatletter % Default settings for warnings     \newcommand{\warningSymbol}{\raisebox{0.8\depth}{\danger}}      \definecolor{warningColorText}{named}{Red3}     \definecolor{warningColorLine}{named}{Red3}     \definecolor{warningColorBack}{named}{LemonChiffon1}     \definecolor{warningColorBackSymbol}{named}{white}  % Inline mode     \newcommand{\warning}[1]{%         \textcolor{warningColorText}{\warningSymbol{}\,#1}%     }  % Block mode     \tikzset{         warningsymbol/.style={             rectangle,             draw  = warningColorText,             fill  = warningColorBackSymbol,             scale = 1,             overlay         }     }      \mdfdefinestyle{warning}{%         hidealllines      = true,         leftline          = true,         skipabove         = 12,         skipbelow         = 12pt,         innertopmargin    = 0.4em,%         innerbottommargin = 0.4em,%         innerrightmargin  = 0.7em,%         rightmargin       = 0.85em,%         innerleftmargin   = 1.1em,%         leftmargin        = 0.85em,%         middlelinewidth   = .2em,%         linecolor         = warningColorLine,%         backgroundcolor   = warningColorBack,%         fontcolor         = warningColorText,%         firstextra        = {             \path let \p1=(P), \p2=(O) in ($(\x2,\y1-4)$)             node[warningsymbol] {\warningSymbol};         },%         secondextra       = {             \path let \p1=(P), \p2=(O) in ($(\x2,\y1-4)$)              node[warningsymbol] {\warningSymbol};         },%         middleextra       = {             \path let \p1=(P), \p2=(O) in ($(\x2,\y1-4)$)              node[warningsymbol] {\warningSymbol};         },%         singleextra       = {             \path let \p1=(P), \p2=(O) in ($(\x2,\y1-4)$)              node[warningsymbol] {\warningSymbol};         },%     }      \newmdenv[style=warning]{@Warning}     \newenvironment{Warning}{\let\warning\relax\begin{@Warning}}{\end{@Warning}} \makeatother  \usepackage{lipsum}  \begin{document}  \warning{IMPORTANT !} \lipsum[1]  \begin{Warning}      \lipsum[1] \end{Warning}  \end{document} linux: touch date precision Why was Severus Snape placed in Slytherin?