How to allow another user in a group to read a file
Understanding Group Permissions
A geometry question: Let $ABCD$ be a convex quadrilateral. Prove that AD is parallel to BC. Let $ABCD$ be a convex quadrilateral such that the length of the segment connecting midpoints of the two opposite sides $AB$ and $CD$ equals $\frac{AD+BC}{2}$ . Prove that AD is parallel to BC. I assume that AD is not parallel to BC but I can't find any contradiction. Note: I am posting a question from a former student of mine unable to post on his own for technical reasons.  Given an iid sample $x_1,\ldots,x_n$ from a Weibull distribution with pdf $$ f_k(x) = k x^{k-1} e^{-x^k} \quad x&gt;0 $$ is there a useful missing variable representation  $$f_k(x) = \int_\mathcal{Z} g_k(x,z)\,\text{d}z$$and hence an associated EM (expectation-maximisation) algorithm that could be used to find the MLE of $k$, instead of using straightforward numerical optimisation? What is the purpose of a serial number? "Distribution worker" vs. "distributed worker" I cannot differentiate the meanings of the two words. Could you please clarify?   In general, when should I use a noun (distribution) or an adjective (distributed) for verbs similar to distribute, like organize, create? Is there a way to forget node names between TikZ pictures? I have a document with several TikZ pictures in it.  I generally use simple names for the nodes such as:  \node (A) at .....  \node (B) at .....   and so forth.  Because of this I have name clashes between the various pictures.  That's generally not a problem.  However I find myself in a position where I may or may not have defined a certain node within a given picture.  So I use the following snippet to test whether the node has been defined:  \makeatletter \long\def\ifnodedefined#1#2#3{%%   \@ifundefined{pgf@sh@ns@#1}{#3}{#2}} \makeatother   Unfortunately this fails if I've already created a node by the same name in another picture.  Is there a way to scope the definitions of nodes or is there a way to undefine nodes once you no longer need them?  P.S.  I'm aware that I could embed each picture in a standalone file and import it using \includegraphics.  Unfortunately, this approach will not work for the document I'm creating.  MWE  \documentclass{article} \usepackage{tikz} \usetikzlibrary{calc}  \makeatletter \long\def\ifnodedefined#1#2#3{%%   \@ifundefined{pgf@sh@ns@#1}{#3}{#2}} \makeatother  \setlength{\parindent}{0pt} \pagestyle{empty} \begin{document}   There's no node \textbf{D} here, so I draw a triangle:\par \begin{tikzpicture}    \coordinate (A) at (0,0);   \coordinate (B) at (1,0);   \coordinate (C) at (1,1); %  \coordinate (D) at (0,1);    \ifnodedefined{D}     {  \draw (A) foreach \myn in {B,C,D} {  -- (\myn) } --cycle;  }     {  \draw (A) foreach \myn in {B,C}   {  -- (\myn) } --cycle;  }  \end{tikzpicture}     There is a node \textbf{D} here, so I draw a square:\par \begin{tikzpicture}    \coordinate (A) at (0,0);   \coordinate (B) at (1,0);   \coordinate (C) at (1,1);   \coordinate (D) at (0,1);    \ifnodedefined{D}     {  \draw (A) foreach \myn in {B,C,D} {  -- (\myn) } --cycle;  }     {  \draw (A) foreach \myn in {B,C}   {  -- (\myn) } --cycle;  }  \end{tikzpicture}      There isn't supposed to be a node \textbf{D} here.  I'm expecting a triangle, but get a square.\par \begin{tikzpicture}    \coordinate (A) at (0,0);   \coordinate (B) at (1,0);   \coordinate (C) at (1,1); %  \coordinate (D) at (0,1);    \ifnodedefined{D}     {  \draw (A) foreach \myn in {B,C,D} {  -- (\myn) } --cycle;  }     {  \draw (A) foreach \myn in {B,C}   {  -- (\myn) } --cycle;  }  \end{tikzpicture}  \end{document}   produces: FileReader.js nothing happens in IE9 Limit $c^n n!/n^n$ as $n$ goes to infinity Let $c&gt;0$ be a real number. I would like to study the convergence of $a_n := c^n n!/n^n$, when $n \to \infty$, in function of $c$.   I know (from  question) that $n!&gt;(n/e)^n$, so that $c^n n!/n^n&gt;1$ for $c ≥e$. But this doesn't imply that the sequence goes to infinity. And I'm not sure what to do for $c&lt;e$. I tried usual tests (D'Alembert...), without any success. I would like to avoid using Stirling approximation.  Thank you for your help! My passport was detached from the front cover but all the pages are intact. what shall i do? How can I publish my thesis? I have a written thesis, as my Final year project. I have submitted in university.   but now i want to make it professionally mine.   I want to publish it some where but i don't have such kind of experience before. I am total Rookie in this field.  Any Expert Opinion?  Note: Thesis in About "Handwriting Recognition " Given the length ot the sides $a , b$ and $c$  of $ \triangle ABC$.  What is the length of the radius of the circumcribed circle?  After some formula substitution I came to the monster formula:  $$ \frac {a b c}{\sqrt{2a^2b^2+2a^2c^2+2b^2c^2-a^4-b^4-c^4}} $$  Can this formula be simplified? Do compound subjects like "everyone ... and everyone..." take singular or plural verb agreement? Here are the examples of compound subjects:      Everything on the bed and everything in the closet was organised   in under an hour.   Everybody who witnessed the shooting and everybody   in the room were interviewed.   Anyone on the soccer team and   anybody on the basketball team was eligible for the scholarship.   Should I also use was instead of were in the second example? Or the first and the third examples are incorrect that I should use were instead of was. How to identify multiple USB-serial adapters under Ubuntu 10.1 Can Hydrogen Fusion via CNO Cycle Occur in First Generation Stars CNO cycle requires the presence of carbon, nitrogen and oxygen to undergo hydrogen fusion. Does this mean that for first generation stars, no matter how big they are, can't undergo hydrogen fusion by CNO cycle because there is no carbon, nitrogen and oxygen present?   I can't seem to find any website which mentions this. All they say is if a star is large enough, it will undergo hydrogen fusion via CNO cycle, they don't specifically mention it having to be a second generation star. I posted an answer (pointing to another question already asked) but when I hit submit, I saw my answer appear as a comment to the question instead.   I tried it a couple of times and still the same result. The answer gets eaten up and is transformed into a comment.   Is this a bug? Is the 5-foot range of the Help action my range to the ally, the enemy, or both? Is there an easy way to get to www.stackexchange.com from any of the SE sites? One thing I've noticed about the relatively new sitewide layout is that I can't find a quick way to get back to www.stackexchange.com without finding StackExchange in the sites list. As a result, I am typing stackexchange in my toolbar a whole lot more often now. In particular, I tend to enjoy browsing the hot questions list and my own personal feed quite often so visit these pages on a daily basis. Is there an easy link to them that I'm missing?  (On the android app, there is a feed in the menu that I really appreciate, something like that on the desktop would be ideal!) Problem with Refsection Combinatorics - binary - very interesting question I tried to solve this problem without any success. I really hope that you will know the answer. $f(n,m)$ presents the number of binary strings (empty string included) that include at most n times '1', and at most m times '0'.      Prove: $$f(n,m) = \binom{n+m+2}{n+1}-1$$ I think my son was trying to guess the passcode! What do I do?  I can't wait 42 years! If $f:E\rightarrow S$ is an elliptic curve over a scheme $S$ (so $f$ is proper and smooth of relative dimension one with geometrically connected fibers of genus one, equipped with a section $0:S\rightarrow E$), then is the sheaf $\underline{\omega}_{E/S}:=f_*\Omega_{E/S}^1$ actually free of rank one? According to the statement of Grothendieck-Serre duality in Hida's book Geometric Modular Forms and Elliptic Curves, there should be a canonical isomorphism $\underline{\omega}_{E/S}\cong\mathcal{Hom}_{\mathscr{O}_S}(R^1f_*\mathscr{O}_E,\mathscr{O}_S)$, and the first result on elliptic curves in this book is that $R^1f_*\mathscr{O}_E\cong\mathscr{O}_S$. It's also not clear to me whether $S$ is being assumed (locally) Noetherian, but if that's necessary for Grothendieck-Serre duality to hold, then I'm fine with assuming it. I can't find another reference with a statement of Grothendieck-Serre duality in this generality which does not use the language of derived categories (which I unfortunately don't understand).  The reason I'm kind of skeptical about this is that in Hida's book, as well as in Katz-Mazur, it is said that $\underline{\omega}_{E/S}$ is invertible, so that an $\mathscr{O}_S$-basis $\omega$ for $\underline{\omega}_{E/S}$ can be found locally on $S$. If the invertible sheaf in question were really trivial then one would be able to choose a global $\mathscr{O}_S$-basis for $\underline{\omega}_{E/S}$, and there would be no reason to talk about doing so locally. Hida goes on to say that, choosing an $\mathscr{O}_S$-basis $\omega$ locally on $S$ allows one to regard $(\Omega_{E/S}^1,\omega)$ as a relative effective Cartier divisor in $E/S$, which also doesn't make complete sense to me because if we can only find $\omega$ locally, how are we getting a global section $\omega\in H^0(E,\Omega_{E/S}^1)=H^0(S,\underline{\omega}_{E/S})$ (unless $\underline{\omega}_{E/S}$ really is trivial)?