For any triangle $ABC$ what is the minimum value $\sum \frac{\sqrt{\sin A}}{\sqrt {\sin B}+\sqrt{\sin C}-\sqrt{\sin A}}$?
Find minimum value of $\sum \frac{\sqrt{a}}{\sqrt{b}+\sqrt{c}-\sqrt{a}}$
What is the distinction between a professional and amateur photographer? In the context of metric (and in general first-countable) topologies, it's reasonably clear what a closed set is: a set $F$ is closed if and only if every convergent sequence of points in $F$ converges to a point also in $F$. This naturally generalises to the definition of a closed set in an arbitrary topological space using the concept of limit points... but limit points are defined in terms of open sets, which are, to me, somewhat more mysterious than closed sets.  I was once told that ultrafilters axiomatise the concept of (sets of) big sets. I'm hoping here to find a similar conceptual picture of topologies defined by systems of open sets, preferably without reference to closed sets. One such explanation I've seen is that open set axiomatise the concept of nearness, which, I suppose, is fair enough at least for metric topologies. Indeed, if something is "happening" at points near $x \in X$, then it's usually the case that every open neighbourhood $U \subseteq X$ of $x$ contains an open subset $V \subseteq U$, $x \in V$, such that the something "happens" at all points in $V$. But what about non-metrisable topologies, particularly the coarse ones where there are no "small" open sets?  Consider, for example, the Zariski topology on affine $n$-space $\mathbb{A}^n$. $\mathbb{A}^n$ is irreducible, so every non-empty open set is dense. It seems reasonable to interpret this to mean that every non-empty open set is "large". Indeed, if we work over the complex numbers, then in the usual metric geometry, Zariski-open sets are unbounded and have full measure, so are very large indeed, so I don't think it's fair to say that open sets are capturing the notion of nearness here.  I'm also curious about the history of point set topology. When were the axioms first written down? What were the first "non-geometric" examples of topological spaces — "non-geometric" here referring to either non-metrisable topologies or topologies on sets other than sets of points of some intuitively geometric object — and were they part of the motivations for creating point set topology? I have read , but it was asked in the context of servers not desktops or notebooks.  If a good amount of RAM is installed on a notebook, say 8 GB, do we still need a swap space? Entire function with vanishing derivatives? Let $f:\mathbb{C}\rightarrow\mathbb{C}$ be an entire function. And assume that at each point, one of it's derivatives vanishes.  What can you say about $f$?  A hint suggests that $f$ must be a polynomial. Uninstall GRUB and use Windows bootloader I have Windows 8 pre-installed, and then I installed GRUB with Ubuntu. Ubuntu is not my thing, so now I want to remove it along with GRUB. From what I have learned, with UEFI, GRUB does not overwrite the windows bootloader in the EFI partition and is stored elsewhere. How would I remove GRUB and make my PC use the Windows bootloader instead? It should be noted that I created a separate /boot partition when installing Ubuntu. suppress page numbers in document's front matter I have a problem and I want that someone tell me how I can resolve that.  I want that the number of page in my report start in the page of introduction but in reality it starts in the didecas and liste of contents. I want that these pages have not number. This is my code  \documentclass[12pt,a4paper]{report} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage[french]{babel} \usepackage{color} \usepackage{xcolor} %les couleurs \definecolor{myblue}{rgb}{0.2,0.4,1} \usepackage{layout} \usepackage[top=2 cm, bottom=2 cm, left=2 cm, right=2 cm]{geometry} \usepackage{setspace} \usepackage[pdftex]{graphicx} \usepackage{array} %\documentclass[landscape]{scrartcl} \usepackage{geometry} \usepackage{multirow}  \usepackage[justification=centering]{caption}  \usepackage{pgfplots}   \usepackage{fancyhdr} \fancyhf{} % clear all header and footers   \begin{document} \pagestyle{empty} \include{title-content} \color{black} \doublespacing \include{dedicaces-content} %\include{remerciement}   \tableofcontents  \listoffigures \listoftables     %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \pagestyle{plain}  \pagenumbering{arabic} \chapter{Introduction générale} Excuse my lack of knowledge and expertise in math,but to me it would came naturally that the cubic root of $-8$ would be $-2$ since $(-2)^3 = -8$.  But when I checked Wolfram Alpha for , real it tells me it doesn't exist.  I came to trust Wolfram Alpha so I thought I'd ask you guys, to explain the sense of that to me. If I have bought X tickets, do I have X hand baggage allowances? If I have bought 3 tickets for the same flight, can I bring three times the cabin allowance allowed per ticket?  The way I see it I am doing the airline a favor  Related: Where is the password stored for ssh authentication? It's known that when you're connecting to a wrong server via ssh and send your password  to it, your password comes in a plain text and a server saves it by default. Probably. I wonder, where I can retrieve it? I don't see it in /var/log/auth.log, although I do see all the attempts to authenticate. Will two-letter font style commands (\bf , \it , …) ever be resurrected in LaTeX? I understand that \bf and \it are now obsolete in LaTeX and that \textbf and \textit are proper, as they produce more sophisticated (in particular, cumulative and properly kerned) changes to font style. I have read the  and I understand the rules and their reasons, as well as the several other commands that are affected.  However, I find it convenient to use LaTeX for notetaking during lectures, and in that rushed environment, shortening a command by any number of keystrokes helps keep me from falling behind. \bf and its two-letter kin are still very useful to me for that reason, and once a presentation is finished I can go through and replace all appearances of \bf et al. with \textbf et al.  My question is this: is there a plan eventually to replace the short font style commands like \bf with the implementations of \textbf etc. some day, or should I expect \bf always to remain in existence but obsolete, for reasons of backward compatibility with original TeX? Original TeX has been greatly improved on in countless ways, but in the heat of transcription I sometimes miss its conciseness. Cauchy condensation test The following is Cauchy condensation test:  If $b_n \ge 0$ and $b_n \ge b_{n+1}$ then $\sum b_n$ converges if and only if $\sum 2^n b_{2^n}$ converges.  Is the following modified Cauchy condensation test possible:  Let $N \in \mathbb N$. Then if $b_n \ge 0$ and $b_n \ge b_{n+1}$ then $\sum b_n$ converges if and only if $\sum N^n b_{N^n}$ converges? I was looking at  and I didn't realize that Minecraft had some Easter eggs in it. Besides the Christmas chest Easter egg, and the sheep Easter egg, what else is there? Should you declare methods using overloads or optional parameters in C# 4.0? I was watching , and it got me thinking about when optional parameters are available in C# what is going to be the recommended way to declare methods that do not need all parameters specified?  For example something like the FileStream class has about fifteen different constructors which can be divided into logical 'families' e.g. the ones below from a string, the ones from an IntPtr and the ones from a SafeFileHandle.  FileStream(string,FileMode); FileStream(string,FileMode,FileAccess); FileStream(string,FileMode,FileAccess,FileShare); FileStream(string,FileMode,FileAccess,FileShare,int); FileStream(string,FileMode,FileAccess,FileShare,int,bool);   It seems to me that this type of pattern could be simplified by having three constructors instead, and using optional parameters for the ones that can be defaulted, which would make the different families of constructors more distinct [note: I know this change will not be made in the BCL, I'm talking hypothetically for this type of situation].  What do you think? From C# 4.0 will it make more sense to make closely related groups of constructors and methods a single method with optional parameters, or is there a good reason to stick with the traditional many-overload mechanism? How can I copy coverart from flac to opus with ffmpeg I am trying to convert some flac files which all have coverart embedded to opus with ffmpeg. All but the coverart gets copied over just fine, but I find myself unable to move the cover over as well. Tried all sorts of -mapmetadata stuff I could find so far, but no luck, the result is always missing the coverart.  While the original .flac reports to have a video stream with the coverart  Duration: 00:02:53.51, start: 0.000000, bitrate: 910 kb/s    Stream #0:0: Audio: flac, 44100 Hz, stereo, s16    Stream #0:1: Video: mjpeg, yuvj420p(pc, bt470bg/unknown/unknown), 500x500 [SAR 1:1 DAR 1:1], 90k tbr, 90k tbn, 90k tbc   the created opus files does not have any graphics:  size=    2815kB time=00:02:53.51 bitrate= 132.9kbits/s speed= 104x video:0kB audio:2793kB subtitle:0kB other streams:0kB global headers:0kB muxing overhead: 0.811556%   I tried -vcodec copy, -mapmetadata 0, -mapmetadata 0:s:0 so far, with no luck. Doesnt seem to have anything todo with opus, as I have the same problem converting to aac.  So is there a way to get the cover out of the flac file and move it over to the new file in on go using ffmpeg, or do I have to extract it first and then glue it back again later to the new file with extra commands? Marginal probability derived from joint probability distribution of a poisson process and binomial process I thought I figured out this problem, but my answer is different than the solutions manual. Hoping someone here can stear me in the correct direction:     Assume the number of automobile collisions that occur on a given stretch of highway per year is a Poisson random variable $X$ with $\mu = 20$. The probability is $p=0.05$ that there will be one or more fatalities in each accident; occurrences of fatalities are independent from one collision to the next. If $Y$ is the number of collisions with one or more fatalities on this stretch of road in one year, find the probability law for $Y$.   So as said, X is poisson, so:  $$p_X(x) = \frac{20^x}{x!}e^{-20}$$  The conditional probability of $Y$ given $X=x$ is a Binomial random variable:  $$p_{Y\mid X}(y\mid x) = \binom{x}{y}p^yq^{x-y}$$  where $Y =$ number of fatal traffic accidents and $x =$ given number of accidents (from poisson distribution)  Therefore the joint probability for $X$ and $Y$ is:  $$p_{X,Y}(x,y)= p_{Y\mid X}\cdot p_X(x) = \binom{x}{y}p^yq^{x-y}\frac{20^x}{x!} e^{-20}$$  to find the marginal probability of $Y$, I sum over $P_{X,Y}$ for all $x$:  \begin{align} p_Y(y) &amp; = \sum_{x=y}^\infty{\binom{x}{y}p^yq^{x-y}\frac{20^x}{x!}e^{-20}} = \frac{(20p)^ye^{-20}}{y!} \sum_{x=y}^\infty {20^{x-y}\frac{q^{x-y}}{(x-y)!}} \\[10pt] &amp; = \frac{(20p)^ye^{-20}}{y!} \sum_{x-y=0}^\infty {20^{x-y}\frac{q^{x-y}}{(x-y)!}} = \frac{(20p)^ye^{-20}}{y!}e^{20q} \end{align}  So:   $$p_Y(y) = \frac{(20p)^ye^{-20p}}{y!}$$  however, the correct answer is apparently $p_Y(y)=\dfrac{e^{-1}}{y!}$ Wrong place for hyperref link in scaled tikz picture I have a TikZ picture with a node containing a minipage with an itemize environment; each item is a hyperref to other sections of the document.   When scaling the image (for example with \begin{tikzpicture}[scale=.7, transform shape] I've a strange behavior: links in itemize are not in the right place: they present a somewhat vertical offset, as if they don't scale as the rest of image.   Here the MWE:  \documentclass[a4paper]{article} \usepackage{tikz} \usepackage[pdftex]{hyperref} \begin{document}  Here something interesting\label{text:interesting-one} \newpage  Here something interesting, too\label{text:interesting-two} \newpage  and here there are very very interesting stuff\label{text:interesting-three} \newpage  \newpage Now the problem: links rectangles have some offset, making links unusable in case of long itemize  \begin{figure}[htb] \begin{tikzpicture}[scale=.7, transform shape] \node at (0,0) (mynode) { \begin{minipage}{4cm} \begin{itemize} \item \hyperref[text:interesting-one]{First link} \item \hyperref[text:interesting-two]{Second link} \item \hyperref[text:interesting-three]{Third link} \item \hyperref[text:interesting-one]{First link again} \item \hyperref[text:interesting-two]{Second link again} \item \hyperref[text:interesting-three]{Third link again} \end{itemize} \end{minipage} }; \end{tikzpicture} \end{figure} \end{document}   And here a snapshot of the problem    Note: This is evident using the default settings of hyperref package, which draws the red rectangle around links, but it took me some time to found it when only the text is coloured, i.e. when using \usepackage[linkcolor=blue,colorlinks=true]{hyperref} Protect .NET code from reverse engineering? How to delete duplicate LiDAR points? How can I delete duplicate LiDAR points? I incorrectly read the same las twice and saved as a one file resulting in duplicate XY points.  How should I delete these duplicate points? Using "on" vs. "in"? Consider the following sentence:     I am developing an application to be installed on Android.   And this:     This has been a major flaw in Android...   To be clear I am unsure of the usage of "in" vs. "on". When should I use "in" and when to use "on"? I really enjoyed that site. I've only posted on it once, but I enjoyed seeing other people's posts and appreciated reading about entrepreneurial dilemmas.