Let $A = \mathbb Q [x, y] / (x ^ 2 + y ^ 2 - 1)$ and note that $A$ is a domain. How to show that $\operatorname{Quot} (A)$ (or $\operatorname{Frac} (A)$, i.e. the "field of fractions") is isomorphic to $\mathbb Q (t)$?   I know that the parametrization of the classic circle is $$t\mapsto \left(\frac{1 - t ^ 2}{ 1 + t ^ 2}, \frac{2t}{1 + t ^ 2}\right)$$
How can I prove that $\mathbb{Q}[x,y]/\langle x^2+y^2-1 \rangle$ is an integral domain?  Also, I need to prove that its field of fractions is isomorphic to the field of rational functions $\mathbb{Q}(t)$?    (The question is taken from )
Is "mens" a valid word? I've been living in Ireland for almost a year now and I start noticing they use the word "mens" a lot. I can see it used in:   Shops, to denote the area where you can find men's clothes In sport, when they talk about "mens team".    My guess is just that they are lazy about the use of quotes, so that mens should actually be men's. However, there may be some rule I'm not aware of. So, is "mens" only limited to Irish English? When I'm allowed to use it? If 20 independent Bernoulli trials are carried out each with a different probability of success and therefore failure. What is the probability that exactly n of the 20 trials was successful?  Is there a better way of calculating these probabilities rather than simply summing together the combinations of success and failure probabilities? How can i render the reflections i see in the Look Dev viewport in Blender 2.8 i'm finding blender 2.8 confusing and i'm relatively new to blender in general anyway. While looking at my model in the Look Dev view port i can see reflections.        However, when i render my scene in eevee (and cycles), the reflections aren't rendered     Is there something i'm doing wrong or an option i haven't enabled? Prove recursively defined sequence converges I would like some advice on how to solve problems like the following: Let $(x_n)$ be a sequence defined by $x_1= 3$ and $x_{n+1} = \frac{1}{4-x_n}$. Prove that the sequence converges.  My strategy is to use the Monotone Convergence Theorem, but I am having trouble showing that the sequence is decreasing and bounded below. Here's my work so far:  Decreasing: The first 4 values are $3,1,1/3,3/11$, so let's assume $x_n \leq x_{n-1} \leq \ldots \leq x_1$. Want to show $x_{n+1} \leq x_n$. We have $x_{n+1} = \frac{1}{4-x_n}$. I want to have an upper bound for the RHS, but can't find one and don't really know where to go from here.  Bounded below: I wanted to show that all values are positive, but if we assume $x_n &gt; 0$, that doesn't rule out $x_{n+1}$ from being negative. Code examples, like in the TikZ/PGF manual I like very much the 'side-by-side' code examples in the TikZ/PGF manual...    ... and I would like to use the same thing in a manual I'm writing. I have found the source code for the codeexample environment that the TikZ/PGF manual uses for this purpose (specifically, it's located at /usr/local/texlive/2012/texmf-dist/doc/generic/pgf/macros/pgfmanual-en-macros.tex).  My (probably rather basic) question is: how can I use the same codeexample environment in my own documents? I ask because I tried compiling the example in the related question below, but the file couldn't be found. How did Anakin Skywalker survive Order Sixty Six? . When activated, troopers would immediately perceive Jedi as a threat and kill them.  Since Anakin was a Jedi until around the time of the Order, how did the troopers know not to kill Anakin? How can I prevent Google mistakenly offering to translate a page? Several of my site's pages are appearing in search results with [Translate this page] next to it. When I click that it takes me to Google Translate and translates my page "from Catalan to English".  The pages are in English but have a couple of foreign words (actually Japanese romanisations, not Catalan) that appear to be tripping Google up.  A few weeks ago I set the html tag to &lt;html lang="en"&gt; which from research appears to be the best method to specify the language of a document. Google has cached the pages with this attribute but it is still offering to translate.  More research led me to a "notranslate" attribute which prevents translation entirely: &lt;html lang="en" class="notranslate"&gt;. The problem now is users cannot translate from English to their desired language!  Are there any other solutions that force Google to parse my site as English only? Question mark or bold citation key instead of citation number I've browsed the forums and found a number of posts that have addressed this issue, but none of the solutions seem to work for me. I have the following script that I just copied from the bibtex home page to get familiar with it. Instead of the citation number I get a question mark. I compile using Latex+Bibtex+Latex+Latex+PDFLatex+ViewPDF just as has been previously suggested and the problem persists.   \documentclass[11pt]{article} \usepackage{cite}  \begin{document}  \title{My Article} \author{Nobody Jr.} \date{Today} \maketitle  Blablabla said Nobody ~\cite{Nobody06}.  \bibliography{mybib} \bibliographystyle{plain} \end{document}   My bibliography (Bib.bbl)  @misc{ Nobody06, author = "Nobody Jr", title = "My Article", year = "2006" }    Looking at previous posts one thing that is concerning is that my .bbl looks empty as shown below. Further, I don't have a .blg  \begin{thebibliography}{}  \end{thebibliography} Is there a converter that will convert LaTeX to HTML? I have tried several services but the LaTeX diagrams and some of the maths came out weird.  I actually converted the Latex to PDF without a problem. Then I used  to convert from PDF to HTML. The square root signs did not appear properly and some of the Cartesian coordinate plane diagrams had nothing on them except the grid.   \documentclass[12pt,a4paper]{report} \usepackage{amssymb} \usepackage{amsthm} \usepackage{amsmath} \usepackage{pstricks,pst-plot} \usepackage{pst-eucl} \addtolength{\hoffset}{0.2cm} \addtolength{\textwidth}{1.1cm} \addtolength{\textheight}{1.0cm} \addtolength{\voffset}{-0.7cm} \linespread{1.3} \begin{document}  If $x$ is non-positive then $x \leq -\sqrt{\frac{1}{2}}$     $\hspace{30 mm} $     (6)\\  (4) and (6) tell us that\\  (4)  $-1 \leq x \leq 0$  \\  (6)  If $x$ is non-positive then $x \leq -\sqrt{\frac{1}{2}}$\\  Therefore $-1 \leq x \leq -\sqrt{\frac{1}{2}}$\\  \vspace{50 mm}  \begin{align*} \psgrid[unit=2cm](0,0)(-2,-2)(2,2) \pnode(-1.8,.87){b}\uput[50](b){b} \pnode(-1.8,1.8){a}\uput[50](a){a} \psset{linecolor=green} \psline[linewidth=1pt](-1.8, 0)(-1.8,4) \psset{linecolor=green} \psline[linewidth=1pt](-1.8, 0)(-1.8,-4) \psset{linecolor=blue} \psellipticarc(0,0)(2,2){0}{180} \psset{linecolor=red} \psline[linewidth=1pt,linestyle=dashed,dash=2pt](-2,2)(-1.41,1.41) \psset{linecolor=red} \psline[linewidth=1pt](-1.41,1.41) \psset{linecolor=red} \psline[linewidth=1pt]{-&gt;}(4,-4) \psset{linecolor=red} \psline[linewidth=1pt]{-&gt;}(-2,2)(-4,4) \end{align*}  \end{document}   PDF output:    PDF->HTML output: Difference between "theoretical computer science" and (beta) "computer science" Possible Duplicate:          Title says it all. From what I can see, "theoretical computer science" questions seem to be slightly more isolated to discussing algorithms, whereas "computer science" seems to have questions that are slightly broader and higher-level. Is there any place on the Stack Exchange sites where they give a longer description?  I'm just curious as to what niche the beta "computer science" site is meant to fill, as I see many topics in both forums that overlap. Can Blender render animated gifs? After beating the final boss, can I return to the original castle? Blender 2.8 contains a number of HDRi's in it by default for the LookDev. I'd like to know how to use them in final renders (EEVEE or Cycles), because quite frankly they appear fantastic. How to automatically fetch missing dependencies when installing software from .deb? I'm trying to install  on my Ubuntu box (11.04). The website has a Ubuntu .deb available for download (for 10.10 and 10.04 so I chose 10.10).   However,   sudo dpkg -i mysql-workbench-gpl-5.2.34-1ubu1010-amd64.deb   yields:  (Reading database ... 194069 files and directories currently installed.) Preparing to replace mysql-workbench-gpl 5.2.34-1ubu1010 (using mysql-workbench-gpl-5.2.34-1ubu1010-amd64.deb) ... Unpacking replacement mysql-workbench-gpl ... dpkg: dependency problems prevent configuration of mysql-workbench-gpl:  mysql-workbench-gpl depends on libctemplate0; however:   Package libctemplate0 is not installed.  mysql-workbench-gpl depends on libpython2.6 (&gt;= 2.6); however:   Package libpython2.6 is not installed.  mysql-workbench-gpl depends on libzip1 (&gt;= 0.8); however:   Package libzip1 is not installed.  mysql-workbench-gpl depends on python-paramiko; however:   Package python-paramiko is not installed.  mysql-workbench-gpl depends on python-pysqlite2; however:   Package python-pysqlite2 is not installed. dpkg: error processing mysql-workbench-gpl (--install):  dependency problems - leaving unconfigured Processing triggers for bamfdaemon ... Rebuilding /usr/share/applications/bamf.index... Processing triggers for desktop-file-utils ... Processing triggers for python-gmenu ... Rebuilding /usr/share/applications/desktop.en_US.utf8.cache... Processing triggers for python-support ... Errors were encountered while processing:  mysql-workbench-gpl   My question is, is there a way to tell dpkg to automatically fetch missing dependencies, or do I need to manually apt-get install missing packages like libctemplate0 and  libpython2.6?   (Or alternatively, is there some other way to get MySQL Workbench easily up &amp; running?) What are the benefits of playing an alignment other than True Neutral? Plotting two variables as lines using ggplot2 on the same graph Installing .pkg with terminal ? Is it possible to install a .pkg using a terminal ? (I wanted to install an app through ssh). Why is polyploidy lethal for some organisms while for others is not? is the multiplication of number of chromosomal sets from 2n to 3n (triploidy), 4n (tetraploidy) and so on. It is quite common in plants, for example many crops like wheat or Brassica forms. It seems to be rarer in animals but still it is present among some amphibian species like Xenopus.  As I know in mammals polyploidy is lethal (I don't mean tissue - limited polyploidy). I understand that triploidy is harmful due to stronger influence of maternal or paternal epigenetic traits that cause abnormal development of placenta, but why there is no tetraploid mammals? swap x and y axis on data generated plot What is the "Templates" folder in the home directory for? For as long as I can remember, every time I install Ubuntu there is a folder in the home directory called "Templates." I've always just deleted it, but I'm curious: what's it supposed to be used for, and why is it in the default install?