If $H \leq G$ such that $[G:H]=2$ then $a^2 \in H $ for every $a\in G $   If $a \in H $ then $a^2 \in H $ so I took $a\in G-H $ and wrote $G=Hx \cup Hy $ for some $x,y \in G $ but wasn't able to arrive anywhere. Any hint?
I'm having some trouble understanding cosets.  If I understand it correctly, cosets form a partition. So, if I have $|G:H| = 2$, then $G = H \cup xH$. Right?  In an exercise, I'm asked the following:  "For $G$ group and $H \le G$ such that $|G:H| = 2$, prove that $x^2 \in H$ for any $x \in G$"  If $x \in H$, then $x^2 \in H$ because $H$ is a group. But, if $x \notin H$, then $x \in gH$ for any $g \in G$, right? Then $x^2$ = $(gh)^2$ but I can't see why this is in $H$.  Can you give me some hint? Thanks for stopping by.
I am used to Turbo C++ and I am new to Ubuntu.  I really need Turbo C++ to get started.  Can somebody please help me to install Turbo C++ on Ubuntu. Why do mathematicians use single-letter variables? We always say that tree levels are classical but loop diagrams are quantum.  Let's talk about a concrete example： $$\mathcal{L}=\partial_a \phi\partial^a \phi-\frac{g}{4}\phi^4+\phi J$$ where $J$ is source.   The equation of motion is $$\Box \phi=-g \phi^3+J$$  Let's do perturbation, $\phi=\sum \phi_{n}$ and $\phi_n \sim \mathcal{O}(g^n) $. And define Green function $G(x)$ as $$\Box G(x) =\delta^4(x)$$  Then   Zero order:  $\Box \phi_0 = J$   $\phi_0(x)=\int d^4y G(x-y) J(y) $  This solution corresponds to the following diagram:  First order:  $\Box \phi_1 = -g \phi_0^3   $  $\phi_1(x)=-g \int d^4x_1 d^4x_2 d^4x_3 d^4x_4 G(x-x_1)G(x_1-x_2)G(x_1-x_3)G(x_1-x_4)J(x_2)J(x_3)J(x_4) $  This solution corresponds to the following diagram:  Second order:  $\Box \phi_2 = -3g \phi_0^2\phi_1  $  $\phi_2(x)= 3g^2 \int d^4x_1 d^4x_2 d^4x_3 d^4x_4 d^4x_5 d^4yd^4z  G(x-y)G(y-x_1)G(y-x_2)G(y-z)G(z-x_3)G(z-x_4)G(z-x_5) J(x_1)J(x_2)J(x_3)J(x_4)J(x_5) $ This solution corresponds to the following diagram:  Therefore, we've proved in brute force that up to 2nd order, only tree level diagram make contribution.  However in principle the first order can have the loop diagram, such as  but it really does not occur in above classical calculation.  My question is:   What's the crucial point in classical calculation, which forbids the loop diagram to occur? Because the classical calculation seems similiar with quantum calculation. How to prove the general claim rigorously that loop diagram will not occur in above classical perturbative calculation. I am using XeTeX and have the following short document:  \documentclass[a4paper,12pt]{article} \usepackage{amsmath,amsthm} \usepackage[MnSymbol]{mathspec} \usepackage[no-sscript]{xltxtra} \defaultfontfeatures{Mapping=tex-text} \setallmainfonts{Minion Pro}  \begin{document}  $\phi, \varphi, \delta \ldots A \vee B$  \end{document}    Now no matter what font I am using the punctuation marks in math mode (commas, dots, \ldots etc.) come out in Computer Modern. Is there a way to change this? Why does HTML require that multiple spaces show up as a single space in the browser? I have long recognized that any set of whitespace in an HTML file will only be displayed as a single space.  For instance, this:  &lt;p&gt;Hello.        Hello. Hello. Hello.                       Hello.&lt;/p&gt;   displays as:  Hello.        Hello. Hello. Hello.                       Hello.  This is perfectly fine, as if you need multiple spaces of pre-formatted text you can just use the &lt;pre&gt; tag.  But what is the reason?  More precisely, why is this in the specification for HTML? In the new header could we please move the chat link back to where it was? (One click away)  As a daily frequent to the JavaScript room that's the one link I'd love to have visible 100% of the time.  Aside from that, I actually love the new header. (Maybe take it a step further an add a customizable header with (X) link spots for your most used links?)  Just my thoughts How do you apply colormanagement settings to EXR files? How do you apply colormanagement settings to EXR files?  When i do some color adjustment by using colormanagement and try to save exr file and open it in photoshop it looks like colormanagement settings was not applied to it. All  I just install a fresh new Ubuntu 16.04 on my new desktop which has a 8th generation Intel CPU i5 8600K. When OS installed, I found there is only one option in display setting: 1024 x 768 (4:3),  My monitor is a HDMI monitor with 1920x1080 resolution  Do anyone have any idea how to make the resolution setting right? Thanks.  BTW. I do not install any graphic card, I use the internal GPU of Intel Raise bounty on question I have this question here:  I chose 500 then before accepting decided that was too much and halfed it, but now I just want the answer so I'd like to raise it to 500 but I can't find how to it. Also it's just such an irritating problem that I'd give all my rep away if I could to the clever person who has the answer. It would be nice to have a few minutes after creating the bounty to be able to raise it. But for now could a mod or something please raise it for me to 500. Knowing the interface of ubuntu and editing them Where are the files for the desktop or the interface that appears on the screen stored? And how can we edit them. How are they programmed?  What programming language is used to built the interfaces? Automatic field should take the required entry Wikipedia claims this but only cites an offline proof:  $$\lim_{n\to\infty} 2^n \sqrt{2-\sqrt{2+\cdots+ \sqrt 2}} = \pi$$  for $n$ square roots and one minus sign. The formula is not the "usual" one, like Taylor series or something like that, so I can't easily prove it. I wrote a little script to calculate it and it's clearly visible, but that's not a proof. "one of the cables that runs" or "one of the cables that run"? Recently at work I was writing the following, and I have not been able to get a firm answer on which version is the most grammatically correct. The sentence is:     We would like to use one of the cables that _________ between rooms.   Is the correct conjugation run or runs?  I think 'run' is correct, because I think that the verb in the prepositional phrase is referring to 'cables' not 'one.' However, a number of colleagues have said that 'runs' is correct because one is singular. footnotes at the end of the book Is it possible to have all footnotes placed at one specific place in my document instead of bottom of each page? I have multiple footnotes scattered in the article and it would be clearer if all of them are at the very end of the article  Edit (moved from an ``answer'')  is it possible in endnotes to have links in each number where I placed footnote to the actual footnote at the end? Gaming with Virtualbox, has it worked for you? Currently I am using Windows 8. I have a virtual box with Ubuntu set up for development. Aside from that I sometimes just want to relax and enjoy a game (Battlefield 3, Skyrim, Saints Row: The Third, etc) after hours of research and development.   I want to have my main machine with linux, and install windows on a virtual box.   What I want to know, from those who have tried or seen it:   How is the performance? I've never been clear on this but, even if I install the correct drivers on the windows box, will it matter? At least on this part, I'm just curious how it offload. Which drivers will matter the most.  Mouse and Keyboard integration, I'll be honest, sometimes the mouse integration in Virtual Box is a bit frustrating, mouse cursor disappears, etc.    I'm not sure if this is the right place to ask. I'm merely trying to come back to linux, yet I'm a PC gamer so I can't throw away Steam. I'm aware of the linux client, but most of those games won't work.   If there's a right place to ask this whether StackExchange sites or any other site let me know.   Cheers and I hope those in the same position find a middle ground. Proof of $\int_{[0,\infty)}pt^{p-1}\mu(\{x:|f(x)|\geq t\})d\mu(t)=\int_{[0,\infty)}\mu(\{x:|f(x)|^p\geq s\})d\mu(s)$ Money from my parents from abroad to my bank account in UK I am from Japan. I have moved to UK recently on a spouse visa.   I still have my Japanese bank account which I have been using even before I came to UK. I have saving in that account and also my parents transfer me some money per month in my bank account in Japan. I have an International Debit Card on this account.  I am withdrawing this Money in UK via ATM / Debit Card.  In this case, do I have to pay taxes in UK or declare this as income in some form. I would like to have a similar biblatex behaviour as described , i.e. separate bibliographies for different publication types, each bibliography should 1) be prefixed with a corresponding label (C for conference publications, J for journals etc), 2) sorted so that the newest entry is at the top (defined by the publication year) and 3) with reference numbers sorted in descending order per bibliography, i.e. the newest entry should have the highest number.  Journal articles ---------------- [J3]    reference from 2017 [J2]    reference from 2016 [J1]    reference from 2015  Conference publications ----------------------- [C4]    reference from 2017 [C3]    reference from 2017 [C2]    reference from 2016 [C1]    reference from 2015   I currently have the following:  \usepackage[backend=biber,         isbn=true,         giveninits=true,         style=numeric,         maxnames=99,         sorting=ydnt,         defernumbers=true,         autocite=superscript]{biblatex}  \subsection*{Journal articles} \begin{refcontext}[labelprefix=J] \printbibliography[type=article] \end{refcontext}  \subsection*{Conference publications} \begin{refcontext}[labelprefix=C] \printbibliography[type=inproceedings]   Separate bibliographies works fine, labelprefix also works as desired, sorting by year in descending order as well. However, if I try the solution mentioned in the above question I get very high reference numbers (in the 500s).  Is the above behaviour possible at all and, if yes, what is the most elegant solution using current biblatex etc.?  Thanks a lot in advance! How to mount read-write an ext4 partition on Windows? If every polynomial of degree 2 is reducible then $K$ is algebraically closed?