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Finding the limit $\lim \limits_{n \to \infty}\ (\cos \frac x 2 \cdot\cos \frac x 4\cdot \cos \frac x 8\cdots \cos \frac x {2^n}) $ Let $R=\frac{\mathbb{C}[x,y]}{\langle x^2-y^2-1 \rangle}$. Prove that (1) $R$ is an integral domain. (2) Any nonzero prime ideal of $R$ is maximal.  My idea: (1) The first part is easy. Since $x^2-y^2-1\in \mathbb{C}[y][x]$ is Eisenstein with respect to the prime $y+i$, it is irreducible. Hence the ideal generated by it is a prime ideal. (2) I am struggling with this part. I have one approach (not sure if correct). Approach : Let $I=\langle x^2-y^2-1 \rangle$. Then $R=\mathbb{C}[x,y]/I$.  Any ideal of $R$ is of the form $J/I$ where $J\subseteq \mathbb{C}[x,y]$ is an ideal containing $I$. If we can show that any prime ideal $J$ containing $I$ is maximal, then we are done. How to prove this? Is there any other way to prove the second part? Edit: I am trying to avoid Hilbert Nullstellensatz and the notion of dimension of a ring, as these concepts are not there in the syllabus of the exam from which I found this question. How can I measure the actual memory usage of an application or process? This question is covered  in great detail. How do you measure the memory usage of an application or process in Linux? From the blog article of , ps is not an accurate tool to use for this intent.  Why ps is &quot;wrong&quot; Depending on how you look at it, ps is not reporting the real memory usage of processes. What it is really doing is showing how much real memory each process would take up if it were the only process running. Of course, a typical Linux machine has several dozen processes running at any given time, which means that the VSZ and RSS numbers reported by ps are almost definitely wrong. Getting error message when trying to start a virtual machine I have been using VMWare on Windows for a long time. But after a long wait, I moved to VirtualBox on Ubuntu 11.10. I installed Ubuntu, 32 Bit, installed all available updates and installed Virtual Box. When I try to create a new Windows installation inside VirtualBox, I got the following error messages.  VirtualBox - Error  Failed to open a session for the virtual machine Windows XP. The virtual machine '**Windows XP**' has terminated unexpectedly during startup with exit code 1. Details  Result Code: NS_ERROR_FAILURE (0x80004005) Component: Machine Interface: IMachine {5eaa9319-62fc-4b0a-843c-0cb1940f8a91}   2nd error dialogue  Virtualbox - Error in suplibOsinit Kernal driver not installed (rc--1908) Please install the virtualbox-dkmbs package and execute 'modprobe vboxdrv' as root.    I have already tried reinstalling VirtualBox. Google result seem to indicate the the problem happens due to Kernel updates.   Is there anyway I can get this working? I need this for malware analysis and if VirtualBox is going to crash on me all the time, then I won't be able to use Ubuntu for work.  Output to dpkg -l | grep virtual   server rc  virtualbox                             4.1.2-dfsg-1ubuntu1                     x86 virtualization solution - base binaries rc  virtualbox-qt                          4.1.2-dfsg-1ubuntu1                     x86 virtualization solution - Qt based user interface  cute 'modprobe vboxdrv' as root.&lt;p&gt; Rollup Summary Trigger Help I am trying to create an Apex Trigger that performs a Rollup from a custom object to Contacts.   Here is the error that I am receiving:      Didn't understand relationship 'Assignments__r' in FROM part of query   call. If you are attempting to use a custom relationship, be sure to   append the '__r' after the custom relationship name. Please reference   your WSDL or the describe call for the appropriate names.   The child relationship name is Assignments.  Below is the code:      trigger RollupActiveAssigmentLevelsToContact on pse__Assignment__c (after insert, after update, after delete) {      set&lt;Id&gt; set_id = new set&lt;Id&gt;();      List&lt;Contact&gt; con_list = new List&lt;Contact&gt;();      if(trigger.isInsert || trigger.isUpdate){         for(pse__Assignment__c myAssignment : trigger.New) {             set_id.add(myAssignment.pse__Resource__c);         }     }     else if(trigger.isDelete){         for(pse__Assignment__c myAssignment : trigger.Old){             set_id.add(myAssignment.pse__Resource__c);         }     }      if(trigger.isAfter &amp;&amp; (trigger.isInsert || trigger.isUpdate || trigger.isDelete)){         con_list = [SELECT id,Total_Levels__c, (SELECT id,name FROM Assignments__r WHERE Active_Assignment__c = TRUE) FROM Contact WHERE id IN :set_id];          for(Contact con : con_list){             if(con.Assignments__r.size()&gt;0)                 con.Total_Levels__c = con.Assignments__r.size();             else                 con.Total_Levels__c = 0;         }         if(!con_list.isEmpty())         update con_list;     } }   Below is a snap of the Child Relationship name: is the group of rational numbers the fundamental group of some space? Centos mysql version is 5.5 however PHPmyadmin still says version 5.1 how to *completely* turn off automatic updates in Windows 10 I have disabled the services &quot;Windows Update&quot; and &quot;Windows Update Medic Service&quot; in using services.mcs. Furthermore, I made changes to the registry, creating a key HKEY_LOCAL_MACHINE\SOFTWARE\Policies\Microsoft\Windows\WindowsUpdate\AU with a DWORD (32-bit) Value &quot;AUOptions&quot; with value set to 2. I still see frequent queries to ctldl.windowsupdate.com sent to my DNS server. Which services need to be disables to completely shut this down? Lewis Carroll's Pillow Problem via Bayes' Theorem In Pillow Problems and a Tangled Tale Lewis Carroll presents the following problem:     A bag contains a counter, known to be either white or black. A white counter is put in, the bag is shaken, and a counter is drawn out, which proves to be white. What is now the chance of drawing a white counter?   One can easily reason that in the end, the bag must be in one of three states:   Black remains, new white taken New white remains, original white taken Original white remains, new white taken   In two of those states, a white counter remains to be drawn. Therefore, the chance is 2/3.  How can one arrive to this answer using Bayes' theorem? Find the determinant of $I + A$ I would like to capture the contents of AJAX requests using Greasemonkey.  Does anybody know how to do this? What's the proof that the Euler totient function is multiplicative? That is,     why is $\varphi(AB)=\varphi(A)\varphi(B)$, if $A$ and $B$ are two coprime positive integers?   It's not just a technical trouble—I can't see why this should be, intuitively: I bellyfeel that its multiplicativity should be an approximation at most. And why the minimum value for $\varphi (n)$ should be $ \frac{4n}{15} $ completely passes me by. Double integral of an function odd with respect to $y$ over a domain symmetric with respect to $x$ axis Let $D$ be a region given as the set of $(x, y)$ with $$a \leq x \leq b\quad\text{and}\quad-\Phi(x) \leq y \leq \Phi(x)$$ where $Φ$ is a nonnegative continuous function on the interval $[a, b]$.  Let $f(x, y)$ be a function on $D$ such that $$f(x, y) = - f(x, - y)$$ for all $(x , y) \in D$.  Argue that $$\displaystyle \iint_D f(x, y) dA = 0.$$ Math contest integer triplet problem: $x + y + z + xy + yz + xz = xyz − 1$ The label toolbar uses fields in the attribute table of vector layers to fine tune labeling (Data defined labeling). That's the main reason i'm looking for that, but it may have more applications.  I'd like to automatically add some format columns instead of adding them manually every time.   Do you know a plugin or a way to do this manipulation automatically on multiples layers ?  I tried to get the QGIS tool "Add a field to attribute table" into a quick model which works properly, but it creates a new memory layer. And I'd like just new fields.  ##Add_ETIKT_Field=name ##myvlayer=vector ##myvlayer=output vector outputs_QGISADDFIELDTOATTRIBUTESTABLE_2=processing.runalg('qgis:addfieldtoattributestable', myvlayer,'ETIKT_Y',1,15.0,4.0,None) outputs_QGISADDFIELDTOATTRIBUTESTABLE_1=processing.runalg('qgis:addfieldtoattributestable', outputs_QGISADDFIELDTOATTRIBUTESTABLE_2['OUTPUT_LAYER'],'ETIKT_X',1,15.0,5.0,myvlayer)   What I'm looking to automate is :   Open the attribute table Edit button Field Calculator button Add Column + Type + length + Precision   It seems there's some ideas in , but I'd love to avoid coding anything, or at least at minimum go through the modeler.     Adding and Removing Fields       To add fields (attributes), you need to   specify a list of field definitions. For deletion of fields just   provide a list of field indexes.   if caps &amp; QgsVectorDataProvider.AddAttributes:     res = layer.dataProvider().addAttributes([QgsField("mytext", QVariant.String), QgsField("myint", QVariant.Int)])  if caps &amp; QgsVectorDataProvider.DeleteAttributes:     res = layer.dataProvider().deleteAttributes([0]) I've always managed my iPhone/iPad/iPod Manually, I recently updated to iTunes 11 and now I can see no way to get my music onto my device.  I used to be able to just drag songs from my library to my device, now there is seemingly no way to do this.  I can view the music on my device by clicking on it on the top right corner of iTunes, but I can't seem to get to a state where I can drag music from my computer onto my device.  I'm using iTunes 11.0.1.12 for Windows if that makes a difference. How do R and Python complement each other in data science? In many tutorials or manuals the narrative seems to imply that R and python coexist as complementary components of the analysis process. To my untrained eye, however, it seems that both languages sort of do the same thing.  So my question is if there are really specialized niches for the two languages or if it's just a personal preference whether to use one or the other? Unable to lock the administration directory (/var/lib/dpkg/) is another process using it? I get this error when trying to use apt-get:  E: Could not get lock /var/lib/dpkg/lock - open (11 Resource temporarily unavailable) E: Unable to lock the administration directory (/var/lib/dpkg/) is another process using it?     How can I fix this? Creating triangle commutative diagram with tikz-cd I've just started using TikZ and need to create a simple triangle commutative diagram like this one.    The closest I've been able to get is this.   How do I get the h arrow to go from A2 to A3? Everything I've tried just causes errors.  Here's my current code:  \begin{tikzcd}[column sep=small] A \arrow{r}{i}  \arrow{rd}{g}    &amp; A3 \arrow{d}{h} \\     &amp; A2 \end{tikzcd} PHP mail function doesn't complete sending of e-mail