Let $F$ be a field. Apparently we know that $\langle x,y \rangle \neq \langle g(x,y) \rangle$ for any $g \in F[x,y]$. Why is this the case?
The ideal $I= \langle x,y \rangle\subset k[x,y]$ is not a principal ideal.   I don't know how to consider it. Any suggestions?
FoxTrot Bill Amend Problems So I found this on the Wolfram website today:     So I was wondering about how one might be able to (if possible) solve those four problems by hand. Here are the problems, $\LaTeX$ed:    $ \lim_{x \to +\infty} \dfrac {\sqrt{x^3-x^2+3x}}{\sqrt{x^3}-\sqrt{x^2}+\sqrt{3x}} $ $ \displaystyle\sum_{k=1}^{\infty} \dfrac {(-1)^{k+1} k^2}{k^3+1} $  $ \dfrac {\mathrm{d}}{\mathrm{d}u} \left[ \dfrac {u^{n+1}}{(n+1)^2} \cdot \left[ (n+1) \ln u - 1 \right] \right] $  $ \displaystyle\int_0^{2\pi}\displaystyle\int_0^{\frac{\pi}{4}}\displaystyle\int_0^4 \left( \rho \cos \phi \right) \rho^2 \sin \phi \, \mathrm{d}\rho \mathrm{d}\phi \mathrm{d}\theta$     Ideas   The degree of the numerator is $\frac{3}{2}$. The degree of the denominator is $\frac{3}{2}$. The expression is of an indeterminate form, namely $\frac{\infty}{\infty}$, so we use l'Hoptial's Rule.  No ideas, really.  Derivatives are always pretty easy, although this one is a bit bashy. Basically bash Product and Chain Rules, etc.  Integration by Parts bash? The $\mathrm{d}\theta$ part is very trivial. I mean for the $\rho$ and $\phi$. How to redirect output of an entire shell script within the script itself? Is it possible to redirect all of the output of a Bourne shell script to somewhere, but with shell commands inside the script itself? Redirecting the output of a single command is easy, but I want something more like this: #!/bin/sh if [ ! -t 0 ]; then     # redirect all of my output to a file here fi  # rest of script...  Meaning: if the script is run non-interactively (for example, cron), save off the output of everything to a file.  If run interactively from a shell, let the output go to stdout as usual. I want to do this for a script normally run by the FreeBSD periodic utility.  It's part of the daily run, which I don't normally care to see every day in email, so I don't have it sent.  However, if something inside this one particular script fails, that's important to me and I'd like to be able to capture and email the output of this one part of the daily jobs. Update: Joshua's answer is spot-on, but I also wanted to save and restore stdout and stderr around the entire script, which is done like this: # save stdout and stderr to file  # descriptors 3 and 4,  # then redirect them to &quot;foo&quot; exec 3&gt;&amp;1 4&gt;&amp;2 &gt;foo 2&gt;&amp;1  # ...  # restore stdout and stderr exec 1&gt;&amp;3 2&gt;&amp;4 Is there any advantage in marketing my product for £99.99 instead of £100? I used to deploy metadata using ant migration tool successfully but from last night I am getting the below error:  John@John:~/mm_workspace/TEST_Project/deploy$ ant deployUnpackaged  Buildfile: /home/john/mm_workspace/TEST_Project/deploy/build.xml  deployUnpackaged:  BUILD FAILED /home/john/mm_workspace/TEST_Project/deploy/build.xml:72: Failed to submit request to https://cs19.salesforce.com/services/Soap/m/35.0/00D293456000Qaf for user john@test.com.dev. Error: {http://schemas.xmlsoap.org/soap/envelope/}Client - Maximum size of request reached. Maximum size of request is 52428800 bytes.  Total time: 2 minutes 10 seconds   I think this is because of too much metadata that I am trying to deploy. For now, I am deploying classes and then VF pages and so on but I am looking for an option to deploy everything in one go without running into the above error.  I am sure that people are working on more complex projects than mine with lots of metadata and I am wondering how are they deploying with out hitting this error?  Thanks Can I run an HTTP GET directly in SQL under MySQL? I'd love to do this:  UPDATE table SET blobCol = HTTPGET(urlCol) WHERE whatever LIMIT n;   Is there code available to do this? I known this should be possible as the  include an example of adding a function that does a DNS lookup.  MySQL / windows / Preferably without having to compile stuff, but I can.  (If you haven't heard of anything like this but you would expect that you would have if it did exist, A "proly not" would be nice.)  EDIT: I known this would open a whole can-o-worms re security, however in my cases, the only access to the DB is via the mysql console app. Its is not a world accessible system. It is not a web back end. It is only a local data logging system I exported a map and everything looks fine when viewing the PDF. But when printing it on my plotter, all the text disappears. The scale bar would bring but the text associated to the scale bar wouldn't. The title block also disappears. I've tried messing around with the settings. I've checked/unchecked &quot;Embed All Document Fonts&quot; and &quot;Convert Marker Symbols to Polygons&quot;. I'm not sure what I'm doing wrong here. What's weird is I exported this map a few weeks ago and it printed just fine. Today, I needed to make a change to the map and exported it out, and now I am experiencing this issue. I must have changed one of the export settings in the past few weeks but I'm not sure what it is. Has anyone experienced this? Variables losing their values How to remove Windows 10 from Windows Boot Manager after uninstalling it? I installed Windows 7 on one partition and formatted the other where I had Windows 10 installed leaving only that one Windows 7 OS on my drive, but I still have two OSs offered in Windows Boot Manager. What is even weirder is that their names are reversed, when I choose Windows 10, it's Windows 7 that gets loaded. What is the purpose of a self executing function in javascript? In javascript, when would you want to use this:  (function(){     //Bunch of code... })();   over this:  //Bunch of code... Can you allocate an array with something equivalent to make_shared? Gravitational field intensity inside a hollow sphere In Java, why do people prepend fields with `this`? When referencing class variables, why do people prepend it with this? I'm not talking about the case when this is used to disambiguate from method parameters, but rather when it seems unnecessary.  Example:  public class Person {             private String name;      public String toString() {         return this.name;     } }   In toString, why not just reference name as name?  return name;   What does this.name buy?   a stackoverflow question whose code has this pre-pending. I've noticed that the majority of Ubuntu help websites present apt commands in the following format:  sudo add-apt-repository ppa:[ppa-address] sudo apt-get update sudo apt-get install [application-name]   But I know that apt commands like this one work too:  sudo add-apt-repository ppa:[ppa-address] &amp;&amp; sudo apt-get update &amp;&amp; sudo apt-get install [application-name]   Is there a particular reason that most sites present in the first format? Are there known issues with chaining commands with &amp;&amp;, or to put it another way are there any benefits/drawbacks to using either method?  Further, other then loss of internet, are there any circumstances in which an apt command might fail? Calculate the sum of the infinite series $\sum_{n=0}^{\infty} \frac{n}{4^n}$ A previous problem had us solving $\sum_{n=0}^{\infty} \frac{1}{4^n}$ which I calculated to be $\frac{4}{3}$ using a bit of mathematical manipulation.   Wonderful. Thank you for all the prompt responses.  Could anyone suggest an alternate technique that does not involve differentiation? Parent chat user still links to discuss.area51 When viewing a user who has their parent user on chat set to area51, it still links to discuss.area51, instead of the new URL, area51.meta. See:      So can we update these links? Who should I be damage buffing as Mercy I just started playing Overwatch (played a few games during open beta, but wasn't able to play much). I'm enjoying Mercy, but who (in a group situation) should be getting the damage buff? Additionally, when my team splits up, who should I follow to support? Allowing line break at ',' in inline math mode? In the inline math mode ($...$), if the formula is too long, LaTeX will try to break it on operators, e.g.  very long text followed by a very long equation like $a+b+c+d+e+f+g+h+i+j+k+l$ etc   may be rendered as  very long text followed by a very long equation like a+b+c+d+e+f+g+h+i+ j+k+l etc   However, the break won't happen if they are separated by commas, e.g.  very long text followed by a very long equation like $a,b,c,d,e,f,g,h,i,j,k,l$ etc   will overflow the page like  very long text followed by a very long equation like a,b,c,d,e,f,g,h,i,j,k,l etc   How to make LaTeX able to insert line breaks after a comma too? What is the status icon that looks like a house? Can you switch to the other Unarmored Defense by multiclassing? PHB p164 states:      If you already have the Unarmored Defense feature, you can't gain it again from another class.   So if I start as a Monk and gain the UD(Wis) feature, and then at next level up, take a level in Barbarian, can I switch to the alternate UD(Con)? Or am I stuck with UD(Wis), even if I take 19 levels in Barbarian?  I'm unclear if the rule is stating this or stating that you can't use both at the same time, which I know. Make Vim show ALL white spaces as a character