Example of integral domain with infinitely ascending chain of ideals. I am looking for an integral domain in which we have an infinitely ascending chain of ideals.   Clearly, this can't be a PID. Also, I am looking for examples other than infinite dimensional fields, integral domains like $k[x_1,x_2,\dots]$.
Example of a ring with an infinite inclusion chain of ideals I'm trying to track down an example of a ring in which there exists an infinite chain of ideals under inclusion. (i.e. $I_1 \subsetneq I_2 \subsetneq I_3 \subsetneq...$)
Declare IDisposable for the class or interface? How can I pad a value with leading zeros? How to sort an array of objects containing null elements? optexp environment error when compiling with XeLaTeX I have troubles with optexp environment. There is a code  \documentclass[pstricks,margin=12pt]{standalone}  \usepackage{pst-optexp}  \begin{document} \begin{pspicture}(3.2,5) \pnodes(0,3){A}(1,3){BS}(3,3){M1}(1,5){M2}(1,1){PD} \psset{mirrortype=extended, mirrordepth=0.2}  \begin{optexp} \beamsplitter[bsstyle=plate, compname=BS](A)(BS)(PD) \mirror[compname=M1](BS)(M1)(BS) \mirror[compname=M2, variable](BS)(M2)(BS) \lens[compname=L](BS)(PD) \optdetector[compname=Det, dettype=diode](BS)(PD) \addtopsstyle{Beam}{beamwidth=0.2, fillstyle=solid,fillcolor=green!20!white} \drawwidebeam(A){BS}{M1}{BS}{M2}{BS}{L}{Det} \end{optexp} \end{pspicture}  \end{document}   and the result is this    and I get "MiKTeX GPL Ghostscript 9.19: Unrecoverable error, exit code 1" error. When I remove optexp environment, so  \documentclass[pstricks,margin=12pt]{standalone}  \usepackage{pst-optexp}  \begin{document} \begin{pspicture}(3.2,5) \pnodes(0,3){A}(1,3){BS}(3,3){M1}(1,5){M2}(1,1){PD} \psset{mirrortype=extended, mirrordepth=0.2}   \beamsplitter[bsstyle=plate, compname=BS](A)(BS)(PD) \mirror[compname=M1](BS)(M1)(BS) \mirror[compname=M2, variable](BS)(M2)(BS) \lens[compname=L](BS)(PD) \optdetector[compname=Det, dettype=diode](BS)(PD) \addtopsstyle{Beam}{beamwidth=0.2, fillstyle=solid,fillcolor=green!20!white} \drawwidebeam(A){BS}{M1}{BS}{M2}{BS}{L}{Det}  \end{pspicture}  \end{document}   the result is     which is correct but optical components are behind beam. I use XeLatex. What am I doing wrong? Which packages are missing? Thank you (example is from manual "Drawing optical experimental setups" where the code works correctly). For a presentation with Beamer, I would like to have a slide where a long list of formulas (maybe mixed with text) scrolls automatically within a certain part of the slide. This should look like e.g. closing credits in movies.   I have tried the idea from  based on animateinline from the , but whenever I use environments like \begin{equation} ... \end{equation} or commands like \textcolor{red}{...} I get an error:  TeX capacity exceeded, sorry [main memory size=3000000]. \end{frame}   Ideally, the formulas/the text should be given within the document itself, but if a solution with an external PDF works fine, I'll take it!  Any ideas how that might work? Solving $\sqrt{x+5} = x - 1$ I'm currently learning about radicals and simplifying them, and I came across this problem on the internet and tried to solve it:  $$\sqrt{x+5} = x - 1$$  So I used this logic:  $$ \begin{align} \sqrt{x+5} &amp;= x - 1 \\ x + 5 &amp;= (x-1)^2 \\ x + 5 &amp;= (x-1)(x-1) \\ x + 5 &amp;= x^2 - 2x + 1 \\ 0 &amp;= x^2 - 3x - 4 \\ 0 &amp;= (x-4)(x+1) \\ \end{align} $$  Therefore, $x = -1$ and $x = 4$ satisfy the equation $0 = (x-4)(x+1)$.  But then I tried to plug them in the original problem $\sqrt{x+5}=x-1$:  $$ \begin{align} \sqrt{4 + 5} &amp;= 4 - 1 \\ \sqrt{9} &amp;= 3 \\ 3 &amp;= 3 \end{align} $$  So using 4 works as expected, but when using $-1$:  $$ \begin{align} \sqrt{-1 + 5} &amp;= -1 - 1 \\ \sqrt{4} &amp;= -2 \\ 2 &amp;\ne -2 \end{align} $$  At what stage am I going wrong?  And according the , the solution is $x = 4$. Finding what's using all the space in *nix Which sequence of commands will tell me which files are the largest starting from a particular directory, including all sub directories? I want to know where all the space went.  Preferably just with regular unix'y commands.  If possible, try to keep it compatible with Sun OS 5.10 (perhaps in addition to regular Linux answers, not as a replacement, to keep it as general as possible.) How can I prevent false score reports to global highscore tables? Browser and mobile games commonly have global highscore tables. It's also common for those tables to contain scores of 2,147,483,647 &mdash; where people have figured out the webservice call that reports scores and used it to record a fictitious score.  For simple puzzle games, we can defend against this by including a record of every move the player made (and any random seeds used to generate the level) with the score-reporting call. The entire game can then be reproduced and verified on the server.  However, this quickly becomes infeasible for anything larger than Pac-man.  How else can cheating of this kind be prevented? What does it mean when multimeter accuracy is marked as: ±0,03%+10Digit? I have a digital multimeter and its accuracy for VDC is marked like this:      ±0,03%+10Digit   This multimeter has maximum display of 80000. So in the 80 V range it can show for example 79.999V.  0.03% of 80V is 0.024V - that is clear for me. But what does the +10Digit mean?  The device in question in . Currently, the only way to vote to delete (and undelete, since you can't see deleted posts) in the Android app is to open the post in a browser, and then vote to delete from there.  Can we get delete vote functionality within the app, as well as the ability to delete our own posts?  Worth to mention: users , so it's possible in the API.  Related: Why does the beta distribution become U shaped when $\alpha$ and $\beta$ <1? In the Beta distribution (used to model Bernoulli probabilities), the $\alpha$ and $\beta$ parameters can be interpreted as the number of heads$+1$ and the number of tails$+1$ seen. So, if they were both $2$, it would lean towards the coin being fair and have a maximum at $0.5$. If they are both $20$, the distribution would become even surer we're dealing with a fair coin and peak even more at $p=0.5$.  What I don't get is its behavior when $\alpha$ and $\beta$ both become $&lt;1$.   In that case, it becomes U-shaped and the density peaks at $p=0$ and $p=1$. Meaning the coin is likely to be two-sided. I know there is an intuition for this since I think I had an idea about it a long time ago. However, I've been trying to recollect all day and can't piece it together. Does anyone have an intuition? Can a magus with a high BAB cast two spells or attack and cast, with their full attack action? Suppose you are a level 10 Magus or something similar. (Something with a decent BAB progression and spellcasting abilities) When you hit +6/+1, can you cast a spell, then attack, vice-versa, or simply case two spells? Everywhere I look I can't get a clear answer. Calculate probability (area) under the overlapping area of two normal distributions Show that $\mathbb{Q}(\sqrt{2})$ is the smallest subfield of $\mathbb{C}$ that contains $\sqrt{2}$ Why is quantum entanglement considered to be an active link between particles? From everything I've read about quantum mechanics and quantum entanglement phenomena, it's not obvious to me why quantum entanglement is considered to be an active link. That is, it's stated every time that measurement of one particle affects the other.  In my head, there is a less magic explanation: the entangling measurement affects both particles in a way which makes their states identical, though unknown. In this case measuring one particle will reveal information about state of the other, but without a magical instant modification of remote entangled particle.  Obviously, I'm not the only one who had this idea. What are the problems associated with this view, and why is the magic view preferred? Ok, so at my school, everybody was supplied with a Macbook Pro. I set up the Messages app and it was all great until the county decided to block it. Now, instead of receiving text messages on my phone they go to my Mac. I have no idea how to fix this because I can't open the Messages app. Please help?! Is there a term for solving a sub-sub problem of a problem so far removed from the main problem the link is not clear? In some situations, trying to solve a problem uncovers another problem. Attempts to fix that uncover yet another. A few more rounds and one can be doing something seemingly completely unrelated to the main problem. Is there an expression for that? Evaluation of the limit $\lim\limits_{n \to \infty } \frac1{\sqrt n}\left(1 + \frac1{\sqrt 2 }+\frac1{\sqrt 3 }+\cdots+\frac1{\sqrt n } \right)$ Evaluate the limit : $$\lim_{n \to \infty } {1 \over {\sqrt n }}\left( {1 + {1 \over {\sqrt 2 }} + {1 \over {\sqrt 3 }} + \cdots + {1 \over {\sqrt n }}} \right)$$  I can use the sandwich principle, certain convergence criteria, Cesaro mean theorem, limit arithmetic.. things around this area.  Any help would be greatly appreciated, thanks!  Sorry for not elaborating more at the beginning, rookie first-post mistake I suppose. :) Is there anyway to get tuple operations in Python to work like this:  &gt;&gt;&gt; a = (1,2,3) &gt;&gt;&gt; b = (3,2,1) &gt;&gt;&gt; a + b (4,4,4)   instead of:  &gt;&gt;&gt; a = (1,2,3) &gt;&gt;&gt; b = (3,2,1) &gt;&gt;&gt; a + b (1,2,3,3,2,1)   I know it works like that because the __add__ and __mul__ methods are defined to work like that. So the only way would be to redefine them? I often end up issuing complex commands over ssh; these commands involve piping to awk or perl one-lines, and as a result contain single quotes and $'s.  I have neither been able to figure out a hard and fast rule to do the quoting properly, nor found a good reference for it.  For instance, consider the following:  # what I'd run locally: CMD='pgrep -fl java | grep -i datanode | awk '{print $1}' # this works with ssh $host "$CMD": CMD='pgrep -fl java | grep -i datanode | awk '"'"'{print $1}'"'"   (Note the extra quotes in the awk statement.)  But how do I get this to work with, e.g. ssh $host "sudo su user -c '$CMD'"?  Is there a general recipe for managing quotes in such scenarios?..