When is a holomorphy ring a PID? I will use the notation and language of Stichtenoth, Algebraic Function Fields and Codes.  Let $F$ be a function field over a finite field $\mathbb F_q$, $S$ a non empty set of places (possibly infinite) and $O_S$ the holomorphy ring of $S$, i.e., $O_S:=\bigcap_{P\in S} O_P$ (being $O_P$ the valuation ring of the place $P$).  Give necessary and sufficient conditions for $O_S$ to be a Principal Ideal Domain.  Many thanks in advance!  G.
When a holomorphy ring is a PID? I will use the notation and language of Stichtenoth, Algebraic Function Fields and Codes.  Let $F$ be a function field over a finite field $\mathbb F_q$, $S$ a non empty set of places (possibly infinite) and $O_S$ the holomorphy ring of $S$, i.e., $O_S:=\bigcap_{P\in S} O_P$ (being $O_P$ the valuation ring of the place $P$).  Give necessary and sufficient conditions for $O_S$ to be a Principal Ideal Domain.  Many thanks in advance!  G.
Writing a string array to text file separated by new line character Faster language/keyboard swapping I like making a habit to use the different languages I study, to keep that knowledge fresh. So I like typing in those languages when I get the chance. But swapping keyboards under OSX is just a little too cumbersome.  For example, I was just typing the name of one of my co-workers in its original 汉子 form instead of as an English transliteration. It's nice. It reminds me to pronounce her name properly, with tone, etc., which otherwise doesn't come naturally. The transliteration of her name is especially misleading. Everyone at the company is pronouncing it wrong.   But to swap into writing 汉子 I have to mouse up to the top menu bar, select a new keyboard, type, mouse back up to the top bar, and then select back into English. I know there's a keyboard shortcut, but with 5 different keyboards active in my menu, cycling through 5 to get to the one I want feels just as cumbersome.  Is there a way to attach a hotkey to a specific keyboard, or some other way to make swapping character sets lighter? Are the known exoplanets representative of the population of exoplanets? An  states that most of the 715 newly announced exoplanets orbit very close to their host stars.   Do the theories predict that this is a representative sample and that most planets are very close to their host stars? Or is it just that our current detection techniques are predisposed to finding only those planets close to their stars (because the sensors look for dimming of the host star for example)? Migrating to full Ubuntu after dual installation on WinXP I previously ran 12.04 alongside WinXP to see how it performed on my mom's older laptop.  i also migrated her email to Thunderbird. If I now convert from dual WinXP/Ubuntu to only run Ubuntu will things like her Thunderbird emails be transferred as part of the Ubuntu full install? Or must I do something else? I am fairly tech aware, but not a full IT boffin! Adding materials to more than one object Is it possible to add a texture or material to more than one object at a time? For any prime $p > 3$, why is $p^2-1$ always divisible by 24? Let $p&gt;3$ be a prime. Prove that $24 \mid p^2-1$.   I know this is very basic and old hat to many, but I love this question and I am interested in seeing whether there are any proofs beyond the two I already know. Find all functions $ f : \Bbb R \to \Bbb R$ which satisfy $f(x)f(y) = f(x + y) + xy$ for all $x, y ∈ \Bbb R$. Find all functions $f : \Bbb R \to \Bbb R$ which satisfy $f(x)f(y) = f(x + y) + xy$ for all  $x, y \in \Bbb R$. The text of the  spell reads:      A flickering flame appears in your hand. The flame remains there for the duration and harms neither you nor your equipment. The flame sheds bright light in a 10-foot radius and dim light for an additional 10 feet. The spell ends if you dismiss it as an action or if you cast it again.   Emphasis mine. The druid in our party asked our DM if he is able to cast produce flame in a dark cave, then opt not to attack with it, and wield a shield and sword. The goal is to continue emitting light, since he doesn't have the light spell. He claims that since the flame does not harm his equipment, he is able to hold a shield or sword with the fiery hand with no problem.  Is this okay to do RAW, or are we missing something?  This is specifically not asking about casting spells with a sword in hand. He wants to draw his weapon after casting and holding the flames.  Related, but not the same question: , , . Having recently switched to Google Chrome, I was wondering what files and directories I should be including in my backups to make sure that bookmarks and history are included.  Along the same lines, is there a process to restore these settings on a new install?  This is on Windows Vista. List creation/manipulation Suddenly my Ubuntu stopped playing video. When I open a video, the player suggests that I install  gstreamer1.0-libav. However, it fails to install.  I tried installing it manually, here is the error message:  $ sudo apt-get install gstreamer1.0-libav Reading package lists... Done Building dependency tree        Reading state information... Done Some packages could not be installed. This may mean that you have requested an impossible situation or if you are using the unstable distribution that some required packages have not yet been created or been moved out of Incoming. The following information may help to resolve the situation:  The following packages have unmet dependencies:  gstreamer1.0-libav : Depends: libavcodec54 (&gt;= 6:9.1-1) but it is not going to be installed or                                libavcodec-extra-54 (&gt;= 6:9.13) but it is not going to be installed                       Depends: libavformat54 (&gt;= 6:9.1-1) but it is not going to be installed E: Unable to correct problems, you have held broken packages. (p → q) ∧ ¬p ⇒ ¬q  I have tried to use the rules of inference in many ways to prove this argument to be valid. However, I cannot seem to find a way to validate the argument, but I am afraid I have missed something.  Using a truth table I find that when p and q is 0, then p → q is 1, ¬p is 1 and ¬q is 1. Would that not make the argument valid?  I apologize in advance if I have misunderstood the whole concept. Any answer would be much appreciated. Thank you How do you performance test JavaScript code? Adding newmode with Xrandr - "800x480_60.00" I have 5-inch a 800x480 tft connected to the LVDS connector on a Portwell WADE-8020 mini-ITX embedded system board. I would like to build a XMBC/Kodi box for streming my music from my freenas PC with the tft showing what is been played, I got as fare as this...   sudo xrandr --newmode "800x480_60.00"  29.50  800 824 896 992  480 483 493 500 -hsync +vsync`enter code here   then,  sudo xrandr --addmode LVDS1 800x480_60.00   now that I get the correct resolution, I now have a black band on the top of the screen.  sudo xrandr --output LVDS1 --set "scaling mode"  "Full aspect"   However this did not solve the issue, also I know I need to amend the /bin/sh file with this from googling so I don't need to keep doing this at every boot. But I don't know how to as I'm very new to Linux.  #!/bin/sh xrandr --newmode "800x480_60.00"  29.50  800 824 896 992  480 483 493 500 -hsync +vsync xrandr --addmode LVDS1 800x480_60.00 I would like to place a lower limit under two consecutive \sums, somewhere in between. So I have \sum_{j=1}^{\infty}\sum_{k=1}^{\infty} and would like to place j&gt;k in the lower limit between the two sums. I believe the multiple integral operator has this kind of functionality. Possible Duplicate:             When I listen to major news programs, often I notice that they seem to intentionally omit "the" before the name of the government organizations. For example:     We contacted E.P.A. for comment but they refused our requests.   or     Others consider the actions of treasury to be detrimental to the economy.   These just sound wrong to me. I think it should be "the E.P.A" or "the treasury". However I most often hear this on very credible news programs (The PBS News Hour and Frontline come to mind) so I am sure they know what they're doing...  Is it proper to use "the" before the name of a government organization, or is it optional? How to prevent system applications (like the Software Center) from asking for password? I was playing around with some integrals and noticed that some integrals of the form:  $$I(a,b,c)=\int_a^\infty \frac{\arctan(x+b)}{x^2+c}dx$$ Does have a closed form. I am trying to find for what constant $c$ will this work. In case you wonder why only $c$ is problematic, I will try to show by an example.  $$I=I(1,3,16)=\int_1^\infty \frac{\arctan(x+3)}{x^2+16}dx$$ Let's start by letting $x-1=t\,$ thus: $$I=\int_0^\infty \frac{{\arctan(\color{blue}{t+4})}}{t^2+2t+17}dt$$ With $\displaystyle{t=\frac{17}{y}\rightarrow dt=-\frac{17}{y^2}dy}$ $$I=\int_0^\infty \frac{{\arctan\left(\color{red}{\frac{17}{y}+4}\right)}}{\left(\frac{17}{y}\right)^2 +\frac{34}{y}+17 }\frac{17}{y^2}dy\overset{y=t}=\int_0^\infty \frac{{\arctan\left(\color{red}{\frac{17}{t}+4}\right)}}{t^2+2t+17}dt$$ $$2I=\int_0^\infty \frac{{\arctan(\color{blue}{t+4})+{\arctan\left(\color{red}{\frac{17}{t}+4}\right)}}}{t^2+2t+17}dt$$ $${\arctan(\color{blue}{t+4})+{\arctan\left(\color{red}{\frac{17}{t}+4}\right)}}=\arctan\left(\frac{\color{blue}{t+4}+\color{red}{\frac{17}{t}+4}}{1-(\color{blue}{t+4})\left(\color{red}{\frac{17}{t}+4}\right)}\right)$$ $$=\arctan\left(\frac{x^2+8x+17}{x}\frac{x}{-4(x^2+8x+17}\right)=\pi-\arctan\left(\frac14\right)$$ Above follows since the original integral is positive so we take $\arctan(-x)$ as $\pi-\arctan x $ and therefore getting a negative answer will not be an issue. $$I=\frac12 \left(\pi -\arctan\left(\frac14\right)\right)\int_0^\infty \frac{1}{t^2+2t+17}dt$$ Well, now the inner integral is not hard to compute and the final answer happens to be: $$I=\frac12 \left(\pi -\arctan\left(\frac14\right)\right)\frac14\arctan\left(\frac{t+1}{4}\right)\bigg|_0^\infty =\frac{\pi^2}{16}-\frac{3\pi}{16}\arctan\left(\frac14\right)+\frac18\arctan^2\left(\frac14\right) $$ There are more examples that I found by checking and try such as:  $$I(1,2,9)=\int_1^\infty \frac{\arctan(x+2)}{x^2+9}dx$$ $$I(2,1,6)=\int_2^\infty \frac{\arctan(x+1)}{x^2+6}dx$$ $$I(2,2,13)=\int_2^\infty \frac{\arctan(x+2)}{x^2+13}dx$$ And so on... All those can be solved by the same method: First substitute $x-a=t$, then let $t=\frac{\alpha}{y}$, where $\alpha$ is the "free of x" coefficient from the denominator.   The problem is that I tried more than $100$ combinations to get those integrals which is not that nice. How can we "smartly" find $c$ so that $I(a,b,c)$ is evaluable by symmetry? Or put in other word what should be $c$ if one wants to compute by symmetry $I(7,13,c)$? Citing multiple authors with alternate separator than "and" using BibTeX I'm using LyX on Linux Mint 14 ...  Using BibTeX with the apalike style and some modifications in the preamble, I've managed to get the bibliography as it should be except for one thing. When I cite a work with multiple authors it gets printed with the separator "and" which isn't correct for my language (Danish). Curiously it's displayed correctly in the user interface of Lyx - for example "(Wittgenstein og Waismann 2003 - which led me to believe everything was ok but in the pdf the separator "and" is used both in the reference and in the bibliography.  Is there any way to fix this using BibTeX, or can it only be done with biber 1.5, as suggested in the last comment here:   I really hope their is because getting biber to work using the following guide has thus far proven to be impossible:   Help much appreciated ... I really hope I won't have to relapse into setting my master thesis up in LibreOffice. I'm brand new to jQuery and have some experience using Prototype. In Prototype, there is a method to "flash" an element &mdash; ie. briefly highlight it in another color and have it fade back to normal so that the user's eye is drawn to it. Is there such a method in jQuery? I see fadeIn, fadeOut, and animate, but I don't see anything like "flash". Perhaps one of these three can be used with appropriate inputs?