$\sigma$-algebra $\mathcal{G} \subset \mathcal{F}$ is countably generated iff there is a random variable $X$ such that $\mathcal{G} = \sigma(X)$.
Every countably generated sigma algebra is generated by a random variable
Background: I have the opportunity to pursue a Computer Science graduate degree at a well respected program (around top 20). I spoke with the professors at the university and I feel like I am a good fit with their research program. My ultimate pursuits are to complete a PhD in CS. I've done research as an undergraduate and it has only confirmed my notion that I want to pursue a research track career.  More background: I applied to jobs (in case I was rejected to all the programs I applied for) prior to hearing back from my respective graduate programs. I accepted a position for an industry job (this was to secure I wouldn't be both unemployed &amp; not in school).  Dilemma: Turns out my top graduate school is very interested in me, and I'm very interested in them. I would rather go to graduate school than work in industry forever. The industry job pays very well. I am split between A) working for a maximum of 1 year(It would only be 1 year, seriously I do not care about the money enough to work past a year) and B) going straight to graduate school.   My concerns are as follows, if I choose A):    Could I potentially defer my admissions? If I am not allowed to defer my admissions, would I have a good chance of re-applying and being accepted a year later? Would my potential advisers look down on me for deciding to work a year?   I already made my mind up it would be a 1 year gig if I decide A). I know some people say once you make money, you may not be able to readjust to the graduate salary pay. But I don't think that will apply to because I'm going to live very frugally with or without industry pay. The example bellow comes from the  (p610). It does not work my computer. Note that it works when I comment the option [prefix=figures/]. Any idea of what could go wrong here?  \documentclass{article} % main document, called main.tex \usepackage{tikz} \usetikzlibrary{external} \tikzexternalize[prefix=figures/] % activate  \begin{document} \tikzsetnextfilename{trees} \begin{tikzpicture} % will be written to ’figures/trees.pdf’ \node {root} child {node {left}} child {node {right} child {node {child}} child {node {child}} }; \end{tikzpicture} \tikzsetnextfilename{simple} A simple image is \tikz \fill (0,0) circle(5pt);. % will be written to ’figures/simple.pdf’ \begin{tikzpicture} % will be written to ’figures/main-figure0.pdf’ \draw[help lines] (0,0) grid (5,5); \end{tikzpicture} \end{document} Review overview shows two suggested edits for a long time, but the review queue is empty When I went to the review overview on Meta on 08.01.2015, 19:16, it listed two open items in the suggested-edits queue, however when I clicked on the queue, I was told that the queue had been cleared. When I went back to the review overview, however, the edits were there again.  Since the icons for recent reviews had not moved, I suppose that these were still the same two items waiting for review, and since phenomenon lasted for half an hour now, I would rule out  or  to be the reason for this.  After a while, the phenomenon ceased, with one new reviewer popping up in the icon list of recent reviewers.  I had not performed or skipped any suggested-edit review from the queue at that point.    A similar phenomenon occurred for several hours up to 11.04.2015, 11:05: The review overview showed two suggested edits and when I went to the suggested-edits queue, it was cleared. Sometimes, edits that I could review appeared, but two phantom edits remained. I have a homework question that asks     Find necessary and sufficient conditions on $1 \leq i &lt; j \leq n$ so that $(i \, j)$ and $(1 \, 2 \, \dotsc \, n)$ generate $S_n$.   Here is what I have done so far.  Call $c = (1 \, 2 \, \dotsc \, n)$.  I made the observation that $$ \overbrace{c c \dotsb c}^{n - j + 1} (i \, j) \overbrace{c^{-1} c^{-1} \dotsb c^{-1}}^{n - j + 1} = (1 + i - j + n \, 1). $$ Thus, $\langle (i \, j), c \rangle = \langle (1 \, i - j + n + 1), c \rangle$, so it suffices to look at $\langle (1 \, k), c \rangle$ for positive integers $2 \leq k \leq n$.  I've checked that, for the value $k=2$, $\langle (1 \, 2), c \rangle = S_n$.  This immediately implies $\langle (1 \, n), c \rangle = S_n$.  I suspect that no other values of $k$ will work, but am not sure how to prove it. I am looking for a book that goes through the mathematical aspects of neural networks, from simple forward passage of multilayer perceptron in matrix form or differentiation of activation functions, to back propagation in CNN or RNN (to mention some of the topics).  Do you know any book that goes in depth into this theory? I've had a look at a couple (such as Pattern Recognition and Machine Learning by Bishop or Deep Learning by Goodfellow, Bengio and Courville) but still have not found a rigorous one (with exercises would be a plus). Do you have any suggestions? How to access folder named "x y" (x space y) using Ubuntu Bash script in Windows? How do I send HTML email using linux 'mail' command? latex table border dont appear / to big / to small still working on that table. Here's my new question reguarding the table borders:  It appears, that i do something wrong setting boarders. Here is my code, and a screenshot:  \documentclass{article} \usepackage{multirow} % Used for the vertical cells \usepackage{rotating} % Used for the vertical cells  \usepackage{xcolor} % Used for cell-colors \usepackage{colortbl} % Used for cell-colors  \usepackage[a4paper]{geometry}% Used to set the pdf margins manually \newgeometry{left=2cm,bottom=0.1cm}  \definecolor{grey}{HTML}{F2F2F2} % Define grey for the cells \definecolor{red}{HTML}{FF0000} % Define red for testing  \begin{document} \scriptsize % Use a small font size  \begin{tabular}{|p{0.6cm}|p{0.6cm}|p{0.6cm}|p{0.6cm}|p{0.6cm}|p{0.6cm}|p{0.6cm}|p{0.6cm}|p{0.6cm}|p{0.6cm}|p{0.6cm}|p{0.6cm}|p{0.6cm}|p{0.6cm}|p{0.6cm}|p{0.6cm}|p{0,6cm}|}  \cline{10-17}  \multicolumn{9}{|c|}{} &amp; \multicolumn{2}{p{\dimexpr (0.6cm)*2+2\arrayrulewidth+2\tabcolsep\relax}|}{\cellcolor{grey}Work to retired ratio} &amp; \multicolumn{4}{p{\dimexpr (0.6cm)*4+1\arrayrulewidth+2\tabcolsep\relax}|}{Health care sizing (per thousand)} &amp; \multicolumn{2}{p{\dimexpr (0.6cm)*2+1\arrayrulewidth+2\tabcolsep\relax}|}{\cellcolor{grey}Inverse dependency ratio}\\\cline{3-9}  \multicolumn{2}{|c|}{} &amp; \multicolumn{4}{c|}{\cellcolor{grey}Population (mio.)} &amp; \multicolumn{3}{c}{Workforce (1000)}  &amp; \multicolumn{2}{|c|}{\cellcolor{grey}} &amp; \multicolumn{2}{l|}{Physicians by} &amp; \multicolumn{2}{l}{Nurses by} &amp; \multicolumn{2}{|l|}{\cellcolor{grey}Old agers by}\\\hline  \multicolumn{2}{|c|}{} &amp; \begin{sideways}\cellcolor{grey}\parbox{4cm}{Overall}\end{sideways} &amp; \begin{sideways}\cellcolor{grey}\parbox{4cm}{Young age}\end{sideways} &amp; \begin{sideways}\cellcolor{grey}\parbox{4cm}{Working age}\end{sideways} &amp; \begin{sideways}\cellcolor{grey}\parbox{4cm}{Old age}\end{sideways} &amp; \begin{sideways}\parbox{4cm}{Overall}\end{sideways} &amp; \begin{sideways}\parbox{4cm}{Physicians (Proj: 2025)}\end{sideways} &amp; \begin{sideways}\parbox{4cm}{Nurses (Proj: 2030)}\end{sideways} &amp; \begin{sideways}\parbox{4cm}{\cellcolor{grey}Working age by Old age}\end{sideways} &amp; \begin{sideways}\parbox{4cm}{\cellcolor{grey}Work force by Old age}\end{sideways} &amp; \begin{sideways}\parbox{4cm}{Workforce}\end{sideways} &amp; \begin{sideways}\parbox{4cm}{Working age}\end{sideways} &amp; \begin{sideways}\parbox{4cm}{Workforce}\end{sideways} &amp; \begin{sideways}\parbox{4cm}{Working age}\end{sideways} &amp; \begin{sideways}\parbox{4cm}{\cellcolor{grey}1 physicians}\end{sideways} &amp; \begin{sideways}\parbox{4cm}{\cellcolor{grey}1 nurse}\end{sideways}\\\hline  \multicolumn{2}{|l}{USA} &amp; \multicolumn{4}{l}{\cellcolor{grey}} &amp; \multicolumn{3}{l}{} &amp; \multicolumn{2}{l}{\cellcolor{grey}} &amp; \multicolumn{4}{l}{} &amp; \multicolumn{2}{l|}{\cellcolor{grey}}\\\hline  1990/1 &amp; &amp; 248,6 &amp; 71,3 &amp; 146,1 &amp; 31,2 &amp; 127.500 &amp; 615,4 &amp; 1491,6 &amp; 4,68 &amp; 4,09 &amp; 4,83 &amp; 4,21 &amp; 11,70 &amp; 10,21 &amp; 50,7 &amp; 20,9\\\hline  1 &amp; 2 &amp; 3 \\\hline  \end{tabular}  \end{document}     On 1 the Top border of Population is not visible although i did used \\\cline{3-9}\cline{12-17} on that line.  In 2 both borders are pushed a bit to the right for no reason i can see.  And in 3 again the top borders are not visible despite using \cline{10-17} on that line. What did i do wrong?  Thanks! Can I levitate an object without using an electromagnet? If $\mathfrak p \subset R$ is a prime ideal, prove that for every nilpotent $r \in R$ it follows that $r \in \mathfrak p$.   The only hint that my tutor gave me was to use induction. Can someone explain what he means by this?  Thanks for the help! java.lang.NoClassDefFoundError: org/apache/xpath/XPathAPI How do I do simple user input in python? How many distinct functions can be defined from set A to B? The kernel and range of the powers of a self-adjoint operator Suppose $L: V \rightarrow V$ self-adjoint (on a finite-dimensional space $V$); $b$ is a real number. Show   The minimal polynomial has distinct roots; $\ker(L) = \ker(L^k)$ for $k\geq1$; $\operatorname{im}(L) = \operatorname{im}(L^k)$ for $k\ge 1$; $\ker(L-bI) = \ker((L-bI)^k)$ for $k\geq1$;     My thoughts:  1) I think I'm supposed to assume root is repeated and reach a contradiction but I'm not sure how to proceed mathematically with that.  2-4) pretty sure there's some huge theorem that I'm missing. Pipe to multiple files in the shell I have an application which will produce a large amount of data which I do not wish to store onto the disk. The application mostly outputs data which I do not wish to use, but a set of useful information that must be split into separate files. For example, given the following output:  JUNK JUNK JUNK JUNK A 1 JUNK B 5 C 1 JUNK   I could run the application three times like so:  ./app | grep A &gt; A.out ./app | grep B &gt; B.out ./app | grep C &gt; C.out   This would get me what I want, but it would take too long. I also don't want to dump all the outputs to a single file and parse through that.  Is there any way to combine the three operations shown above in such a way that I only need to run the application once and still get three separate output files? Improper integral $\sin(x)/x $ converges absolutely, conditionaly or diverges?  We have that $$\int_1^{\infty}\frac{\sin x}{x}dx$$ $$u=\frac{1}{x}$$ $$du=-\frac{1}{x^2}dx$$ $$dv=\sin xdx$$ $$v=-\cos x$$ $$\int_1^{\infty}\frac{\sin x}{x}dx=-\cos(1)-\int_1^{\infty}\frac{\cos x}{x^2}dx$$ $$\int_1^{\infty}\frac{\cos x}{x^2}dx&lt;\int_1^{\infty}\frac{1}{x^2}dx$$ So it converges. Now I need to find out if $\int_1^{\infty}|\frac{\sin x}{x}|dx$ converges or diverges. Can sum of a rational number and its reciprocal be an integer?   My brother asked me this question and I was unable to answer it.   The only trivial solutions which I could think of are $1$ and $-1$.  As to what I tried, I am afraid not much. I have never tried to solve such a question, and if someone could point me in the right direction, maybe I could complete it on my own.  Please don't misunderstand my question.   I am looking for a rational number $r$ where $r + \frac{1}{r}$ is an integer. "All our X" vs. "all of our X" Is the word "of" optional in this instance? Is either of these considered preferable to the other?        Taste all our delicious treats.   Taste all of our delicious treats. First, if you have a better suggestion for title I'm really opened to suggestions, but it is the best I came up with.  If I touch an electric line and the earth, I will get electrocuted.  However, if I touch the positive of a car battery, people told me the same thing won't happen...why is it?  It seems I need to touch the battery positive and the battery ground. I have just bought a new bike and had it about a month now. I don't really know much about bikes but I do know how to repair an inner tube.  I have had one puncture and just bought a new inner tube. It worked for a few days and then went down again so this time I got a puncture repair kit. I took the inner tube out to check for any punctures and found nothing. I put it in water and pumped it up and there were no bubbles. I went around the entire tube pushing it next to my ear and heard nothing. I checked the tire to see if anything was wrong - there was nothing that I could see.   So I put it all back together and took it for a ride and got about half a mile and the tire was flat. I pumped it up and got about another half a mile and it was flat again.  Do you have any suggestions for me?