Textbook on group theory
Comprehensive book on group theory for physicists?
Can you read my proof and tell me if it's correct? Thanks.  Let $V$ be a vector space over a complete topological field say $\mathbb R$ (or $\mathbb C$) with $\dim(V) = n$, base $e_i$ and norm $\|\cdot\|$. Let $v_k$ be a Cauchy sequence w.r.t. $\|\cdot\|$. Since any two norms on a finite dimensional space are equivalent, $\|\cdot\|$ is equivalent to the $l^1$-norm $\|\cdot\|_1$ which means that for some constant $C$, $\varepsilon &gt; 0$, $k,j$ large enough, $$ \varepsilon &gt; \|v_j - v_k\| \geq C \|v_j - v_k\|_1 e_i= C \sum_{i=1}^n |v_{ji} - v_{ki}| \geq |v_{ji} - v_{ki}|$$ for each $1 \leq i \leq n$. Hence $v_{ki}$ is a Cauchy sequence in $\mathbb R$ (or $\mathbb C$) for each $i$. $\mathbb R$ (or $\mathbb C$) is complete hence $v_i = \lim_{k \to \infty} v_{ki} $ is in $\mathbb R$ (or $\Bbb C$) for each $i$. Let $v = (v_1, \dots , v_n) = \sum_i v_i e_i$. Then $v$ is in $V$ and $\|v_k - v\| \to 0$:  Let $\varepsilon &gt; 0$. Then $$ \|v_k - v\| \leq C \|v_k - v\|_1 = C \sum_{i=1}^n |v_{ki} - v_i| \leq C^{'}n \varepsilon$$  for $k$ large enough. Is the below brief explanation correct and complete (at least in context)? Can it be improved/simplified without getting technical? I am especially concerned about the last sentence: Is that reasoning absolutely clear to a layman?            The problem can be viewed as drawing the above graph without lifting your hand and without retracing a line. Observe that each dot is connected by three lines. Also observe that you have to draw a line to arrive at a dot, and you have to draw a line to leave that dot. Because each dot is connected by three lines, each must be visited twice. But it is impossible to visit each dot twice (even number of times) and end up with each dot connected with three lines (odd number of times). The category Set has the property that any morphism f can be decomposed into an epimorphism $e$ and a monomorphism $m$ with $f$ = $m\circ e$. (The intermediary set object is the image of $f$.)  Is there a name for this construction?  Is there a name for this property of categories?  In Set, the intermediary object is unique up to isomorphism. Is this the case for all categories with the property? Where $\phi(n)$ is the Euler phi function, how do you find all $n$ such that $\phi(n)|n$? I have a typical scientific manuscript in a LaTeX .tex file, and I need to convert it to MS Word .doc file.  The reason for having to convert to MS Word is I'm submitting the manuscript to an academic journal and they only accept MS Word (I know...)  The manuscript includes title page, figures, tables, equations (inline and in their own align environment), footnotes, bibliography, and an annex.  The tables are in their own separate tables.tex file, which I include using the \include{tables} command.  Most tables take up a whole landscape page, and were generated sing the package pdflscape.  I am using Windows 7 Professional.    My plan is to use pandoc to go from .tex to .odt, open the latter in Libre Office, and convert to .doc.  I have read a related  but it is too general.  Similarly the examples in the Pandoc  are too simple.  I have played around but I am unable to accomplish what I want.  This is surprising since converting a scientific manuscript is probably the most common use case for Pandoc. Here are some sample failures:  Example 1  I open a command line in the project folder, and execute the following:  pandoc -s document.tex -o document.odt   I get this error message:  pandoc: figure1: openFile: does not exist &lt;no such file or directory&gt;   where figure1 is the name of a figure file (e.g. figure1.png) in the project folder referenced in a line as \includegraphics[width=5.8in]{figure1}.  I suspect pandoc expects a .png extension but not sure how to provide it.  Example 2  Next I try .html, and excute the following:  pandoc -s document.tex -o document.html   The program executes fine.  I open HTML file.  Footnotes are there but figures are missing, tables are displayed as LaTeX, bibliography is missing, in-line math displays well, but math in align environment does not, section labels are displayed, and some other minor issues.  So given that mine is probably a typical use case scenario, my question is this: What commands should I use to get the .odt file I want?  I could not find a fully worked out example on the web.    Here is a specific list of errors.  I'll update how I corrected them based on community suggestions:   Figures not rendering.  Solved by adding .png extension to .tex file in \includegraphics command.  Now figures are included but they are huge, with half of each figure outside the page. No bibliography.  Solved. First, I have one huge consolidated Latex .bib file where I keep all my citations.  I manage it using .  This was giving me problems as I do not keep the cleanest .bib file in town.  So I reduced the problem by using a neat trick in JabRef that allows you to subset your master .bib file using the .aux file generated by Latex when compiling your manuscript.  In JabRef click on Tools > New Subdatabase based on AUX file.  This way I generated a much smaller biblio.bib file with only the articles referenced in my manuscript.  Running pandoc -s document.tex -o document.odt --bibliography=biblio.bib did the trick. Display math. Math in \begin{align} environment displayed in verbatim \latex;  (A partial solution is to use the  Libre Office extension.  Copy and paste the latex math code in the .odt file created by Pandoc into the equation editor, and so on.  Surely this could be built into a macro that can post-process all remaining math.)  UPDATE: Display math works very well using --mathjax extension. Inline math. Inline equation do not always render properly.  Bold math is a problem.  E.g. $\Sigma=\sigma^2\bm{I}$ displays as $\Sigma=\sigma^2\bm{I}$; Labels are displayed (e.g. section labels show as [sec:empirical] blah blah]; All tables display as raw latex. I am attempting to write a filter using something like sed or awk to do the following:   If a given pattern does not exist in the input, copy the entire input to the output If the pattern exists in the input, copy only the lines after the first occurrence to the output   This happens to be for a "git clean" filter, but that's probably not important. The important aspect is this needs to be implemented as a filter, because the input is provided on stdin.  I know how to use sed to delete lines up to a pattern, eg. 1,/pattern/d but that deletes the entire input if /pattern/ is not matched anywhere.  I can imagine writing a whole shell script that creates a temporary file, does a grep -q or something, and then decides how to process the input. I'd prefer to do this without messing around creating a temporary file, if possible. This needs to be efficient because git might call it frequently. Why is the sum of two random variables a convolution? For long time I did not understand why the "sum" of two random variables is , whereas a mixture density function sum of $f(x)$ and $g(x)$ is $p\,f(x)+(1-p)g(x)$; the arithmetic sum and not their convolution. The exact phrase "the sum of two random variables" appears in google 146,000 times, and is elliptical as follows. If one considers an RV to yield a single value, then that single value can be added to another RV single value, which has nothing to do with convolution, at least not directly, all that is is a sum of two numbers. An RV outcome in statistics is however a collection of values and thus a more exact phrase would be something like "the set of coordinated sums of pairs of associated individual values from two RV's is their discrete convolution"...and can be approximated by the convolution of the density functions corresponding to those RV's. Even simpler language: 2 RV's of $n$-samples are in effect two n-dimensional vectors that add as their vector sum.           Please show the details of how the sum of two random variables are a convolution and a sum. How does the EMH hold onto a phaser while having bullets go through him? In the Voyager episode: "Future's End, Part II" the doctor rescues B'elanna and Chakotay.     PORTER: They've got lasers! A black man and some bald guy.  (The EMH   enters. Porter and Butch's bullets go straight through him.)       PORTER: God in heaven help us.       EMH: Divine intervention is unlikely.  (The EMH phaser stuns them,   then unties Torres and Chakotay.)   In an earlier episode of Voyager, "Phage":     PARIS: But a hologram is just a projection of light held in a magnetic   containment field. There's no real matter involved. (The EMH slaps   Paris.)       EMH: Now, you hit me.  (Paris's hand passes through the EMH's head.)       EMH: The magnetic containment field that creates the illusion of my   body can be modulated to allow matter to pass through it or be   stopped. I might be able to modulate the holographic lungs in the same   way, allowing oxygen and carbon dioxide to pass from the lungs to the   blood stream.   In order to make the above adjustments the doctor had to touch several buttons on a control panel. I realize the time I refer to above he was using the holo-emitter instead.    How does it work exactly that the doctor can have bullets pass straight through him but he's able to hold on to a phaser without it falling out of his hand? Can he simultaneously have part of his body be transparent and the rest be more like matter at the same time? What if the bullets hit the hand holding the phaser? I think one had a shotgun as I recall so there probably would have been pellets going all over the place. Solve $x^2+5x+6 \equiv 187 \mod 187$ Solution $$x^2+5x+6 \equiv 187 \mod 187$$ $$ (x+\frac{5}{2})^2 \equiv \frac{1}{4}$$ $$ 4(x+\frac{5}{2})^2 \equiv 1$$ $$ y:= x+\frac{5}{2} $$ $$ 4y^2 \equiv 1 \mod 11 \wedge 4y^2 \equiv 1 \mod 17  $$ $$ ( 2y \equiv 1 \mod 11 \vee 2y \equiv 10 \mod 11 ) \wedge ( 2y \equiv 1 \mod 17  \vee 2y \equiv 13 \mod 17)  $$ $$ ( y \equiv 6 \mod 11 \vee y \equiv 5 \mod 11 ) \wedge ( y \equiv 9 \mod 17  \vee y \equiv 15 \mod 17)  $$ Combining that from CRT I got: $$ y \in \left\{49, 60,83,94 \right\} $$ and for example: $$ x+\frac{5}{2}  \equiv 94 \mod 187$$ $$ 2x  \equiv 183 \mod 187$$ some calculus and get... $$x \equiv 185 $$ And the same thing for each other case. Question Is there any faster (or smarter) way to solve equations like that? How can I install Steam? How can I install and run  on Ubuntu? There is a classic problem:  Suppose that $X_1,\ldots,X_n$ form an i.i.d. sample from a uniform distribution on the interval $(0,\theta)$, where $\theta&gt;0$ is unknown. I would like to find the MLE of $\theta$.  The pdf of each observation will have the form: $$ f(x\mid\theta) =  \begin{cases} 1/\theta\quad&amp;\text{for }\, 0\leq x\leq \theta\\ 0 &amp;\text{otherwise}. \end{cases} $$ The likelihood function therefore has the form: $$ L(\theta) = \begin{cases} 1/\theta^n \quad&amp;\text{for }\; 0\leq x_i \leq \theta\;\; \text{for all }i,\\ 0 &amp;\text{otherwise}. \end{cases} $$ The general solution is usually that the MLE of theta must be a value of $\theta$ for which $\theta \geq x_i$ and which maximizes $1/\theta^n$ among all such values.   The reasoning is that since $1/\theta^n$ is a decreasing function of $\theta$, the estimate will be the smallest possible value of $\theta$ such that $\theta\geq x_i$.  Therefore, the mle of $\theta$, $\hat{\theta}$, is $\max(X_1,\ldots,X_n)$.  Here, I do not understand why we cannot just differentiate the likelihood function with respect to theta and then set it equal to $0$?  Thanks! Allow an edit to notify downvoters: "I think I've fixed the issue now - please check" How can I get very vivid colors in night cityscape pictures? I am asked to determine if a series converges or not:  $$\displaystyle\sum\limits_{n=1}^{\infty} \frac{(2^n)n!}{(n^n)}$$  So I'm using the $n$th root test and came up with $\lim_{n \to {\infty}}\frac{2}{n}\times(\sqrt[n]{n!})$ I know that the limit of $\frac{2}{n}$ goes to $0$ when $n$ goes to infinity but what about the $(\sqrt[n]{n!})$? Getting the ID of the element that fired an event Mount ext4 partition with >4KiB block size I'm attempting to mount a partition from a NAS and the block size is 65536 bytes, this is above my distribution's precompiled kernel memory paging size of 4KiB, so I cannot mount it normally.  Which kernel compilation parameters would I need to change to get a memory paging size large enough?  Would it be possible under the x86-64 architecture?  EDIT: There doesn't seem to be an option for it in make xconfig unless I'm just missing it.  From page_types.h under arch\x86\include\asm  /* PAGE_SHIFT determines the page size */ #define PAGE_SHIFT  12 #define PAGE_SIZE   (_AC(1,UL) &lt;&lt; PAGE_SHIFT) #define PAGE_MASK   (~(PAGE_SIZE-1))`   It doesn't look to be configurable.  I could change this number directly to 16 and it would probably make pretty explosions and destroy Tokyo.  I shall probably try it later this evening. Recommended way to initialize srand? multiple prints on the same line in Python I've been working with Corel Draw previously and now i use Illustrator more.Still getting used to some of their features though. I want to insert an image into a circle I've drawn but i don't know how to go about it.It is called power clipping in Corel. Any answers would be appreciated. DataTable merge without primary key column