I am not that well acquainted with commutative algebra, so that I am kind of struggling to understand the concept of "localizing a ring at a prime ideal."  For one thing I can say is that localizing alludes to a geometric meaning.   Can somebody clarify this concept from both perspectives: algebraic and geometric ? Since a ring ( for example $\mathbb{Z}$ ) can contain many prime ideals, I presume that the introduced localization is not unique. Is this correct ? What are the consequences ? How then do you make a choice on picking up the most suitable prime ideal for the purpose of the localization ?  How do you procede in constructing such "localization" ?   Many thanks.
Let $R$ be a commutative ring with unit, $I$ an ideal of $R$ and consider the following three constructions.   The  $R_I$ of $R$ at $I$ (i.e. the localization of $R$ at the multiplicative system $R\setminus I$) gives a morphism $$ f_1:\operatorname{Spec}(R_I)\to \operatorname{Spec}(R)=:X $$ The  $\widehat{R}_I$ of $R$ at $I$ gives a morphism $$ f_2:\operatorname{Spec}(\widehat{R}_I)\to X $$ For the special case $I=(a)$, the localization $R_a$ (i.e. the localization of $R$ at the multiplicative system $\{1,a,a^2,\ldots\}$) gives a morphism $$ f_3:\operatorname{Spec}(R_a)\to X $$   My question is:     What is the geometric meaning of all three constructions and how are they related?   This is what I ''know'' already or describes at least the style of answer I would appreciate. As remarked in the comments, $I$ has to be a prime ideal.   $R\to R_I$ is injective iff $R\setminus I$ contains no zero divisors which is the case if $I$ is a prime ideal. The scheme $\operatorname{Spec}(R_I)$ is the intersection of all the neightbourhoods of $I$ in $X$. This is a little contra-intuitive for me since the last statement sounds like ''$f_1$ is injective'' but the $\operatorname{Spec}$-operator should turn 'injective'' and ''surjective'' around somehow (I know this is literally not true but I only want to get a feeling like ''is contained'', ''is bigger'', ''is smaller'', etc.). $\operatorname{Spec}(R_a)$ is somehow the opposite of $\operatorname{Spec}(R_{(a)})$ (= the intersection of all the neightbourhoods of $(a)$ in $X$) because $\operatorname{Spec}(R_a)$ seems to be something like the union of all the open sets of $X$ not containing the point $(a)$. $R\to\widehat{R}_I$ is injective iff $\cap I^n=(0)$ and this holds very often (e.g. for $R$ noetherian and either an integral domain or a local ring). Hence (this is probably false intuition as remarked before), $f_2$ should be something like a ''projection'' (from something ''big'' into something ''small''). But what is geometrically the difference between localization and completion. I don't have a geometric idea of completion at all.   As remarked by Qiaochu Yuan in the comments below, one should not think of $\operatorname{Spec}$ as injective-surjective switching.
Best way to transfer files from Windows to Windows via Internet I have a set of Photoshop files which consist of a base image with a series of layers with colored overlays. The overlays are labeled using the layer names.  My task is to get these overlays as SVG paths, somehow preserving the labels. (If I can preserve the fill colors as well, that would be cool, but we can define arbitrary fill colors if needed.)  So far we have tried exporting the layers as Illustrator paths, and then exporting to SVG from Illustrator. This gets us reasonable SVG paths, but (a) we lose the labels, and (b) if a layer has more than one discrete section in its overlay, it is separated into several paths.  Is there a way to get these layers into SVG while retaining the labels? Or should we do our Export > Illustrator > SVG  route on a layer-by-layer basis?  We're working with CS5, if that's important.  ETA to answerers: We've long since finished this project and delivered the job, so I can't accept an answer, but feel free to pitch in your potential solutions for others. Q1. Find the general solution to the difference equation  $$ a_{n} - 4a_{n-1} + 3a_{n-2} = 6 $$    Q2. Solve the difference equation   $$ a_{n} - a_{n-1} - 2a_{n-2} = 0, a_0 = 2, a_1 = 4 $$    I am completely lost in solving recurrence relation questions. Can anyone guide me in steps to solve the following 2 questions? Why KL divergence is non-negative? Why is KL divergence non-negative?  From the perspective of information theory, I have such an intuitive understanding:  Say there are two ensembles $A$ and $B$ which are composed of the same set of elements labeled by $x$. $p(x)$ and $q(x)$ are different probability distributions over ensemble $A$ and $B$ respectively.   From the perspective of information theory, $\log_{2}(P(x))$ is the least amount of bits that required for recording an element $x$ for ensemble $A$. So that the expectation $$\sum_{x \in  ensemble}-p(x)\ln(p(x))$$ can be interpreted as at least how many bits that we need for recording an element in $A$ on average.   Since this formula puts a lower bound on the bits that we need on average, so that for a different ensemble $B$ which brings about a different probability distribution $q(x)$, the bound that it gives for each element $x$ will surely not bit that is given by $p(x)$, which means taking the expectation, $$\sum_{x\in ensemble}-p(x)\ln(q(x))$$ this average length will surely be greater than the former one, which leads to $$\sum_{x\in ensemble }p(x)\frac{\ln(p(x))}{\ln(q(x))} &gt; 0$$  I don't put $\ge$ here since $p(x)$ and $q(x)$ are different.  This is my intuitive understanding, is there a purely mathematical way of proving KL divergence is non-negative? The problem can be stated as:  Given $p(x)$ and $q(x)$ are both positive over real line, and $\int_{-\infty}^{+\infty}p(x)dx = 1$, $\int_{-\infty}^{+\infty}q(x)dx = 1$.  Prove $$\int_{-\infty}^{+\infty}p(x)\ln\frac{p(x)}{q(x)}$$ is non-negative.  How can this be proved? Or can this be proved without extra conditions? How is it the Voyagers are a few seconds closer to Earth than earlier? The  of March 01, 2013 put it's distance at 14 hrs 04 mins 23 secs of light-travel time from Earth. A  (earlier today) tweet says it is 14 hrs 04 mins 22 secs of light-travel time from Earth. An  tweet put it still further out at 14 hrs 04 mins 25 secs of light-travel time.  The Voyager 2 tweets on Voyager 1's proximity to Earth too has reduced similarly to  earlier today from  reported on Feb 28 2013  It could quite simply be a typo ... If not a typo then my curiosity is piqued; I was under the impression the Voyagers are outbound from the Solar System now without any major in-system activities now.   How is it the Voyagers are closer to Earth than before? Reactjs convert html string to jsx Journals selection - alternative criteria to impact factor While trying to select the journals I can submit my present work and future work, I realize the time between the date the authors send the article to the journal and when the revised article is submitted. Usually this is something between 6 months and more than 12 months.  Althought I am focusing on the Impact Factor (IF) to select journals, I think that 12 months to have a feedback that might include "add more variables analysis","this article will be more valuable if you test also model x,y and z", can have a huge impact on you work. It will make you stop what you are currently doing and possible spend up to 2 months making the changes. This might make a big mess on your work if you have deadlines to respect, however this happens to everyone in academia.  My question is, besides the IF journal based selection, how can we select a journal with enought quality with lower IF? What alternative criteria should one use to select the journal? Should we choose newly created Journals? How to find a global minimum for $\frac{1}{2}x^tQx-b^tx$ In my class design, I use abstract classes and virtual functions extensively. I had a  feeling that virtual functions affects the performance. Is this true? But I think this performance difference is not noticeable and looks like I am doing premature optimization. Right? What is the correct form in the following sentences, "both" and "both of them"?      a) "Both of them are unique."      b) "Both are unique."   If "both" by itself already contains "both of them", so I can conclude that the first option is not correct. Isn't it? Let $P(X)$ denote the power set of $X$ (the set of all subsets of $X$) with a partial order given by inclusion.  If $F: P(X) \to  P(X)$ is monotone (order preserving), then $F$ has a fixed point.  How are we going to prove that without using the term (complete lattice)?! Install Alongside Windows or Something Else - Which should I use? I have old notebook with BIOS (not EFI) installed with Windows 8 RTM, want to install Ubuntu 12.10 by booting to a LiveCD.   Doing some research, I found method either to choose Something Else or Install Alongside... Which method should I use? Identical creatures entering the battlefield If I have two cards called  in hand and I play my first, Sin Collector (A). I use its triggered ability.   "When Sin Collector enters the battlefield, target opponent reveals his or her hand. You choose an instant or sorcery card from it and exile that card."  Now, I play Sin Collector (B), I use B's triggered ability.  But now my question. Does Sin Collector (A)'s ability trigger again as well? 'Sin Collector' entered the battlefield again, right? Structure of ideals in the product of two rings $R$ and $S$ are two rings. Let $J$ be an ideal in $R\times S$. Then there are $I_{1}$, ideal of $R$, and $I_{2}$, ideal of $S$ such that $J=I_{1}\times I_{2}$.  For me it's obvious why $\left\{ r\in R\mid \left(r,s\right)\in J\text{ for some } s\in S\right\}$ is an ideal of $R$ (and the same for $S$) so I can prove $J$ is a subset of the product of two ideals and also that the product of these two ideals is an ideal. But I can't see how to prove equality without assuming existence of unity or commutativity. Trying basically to show that if $\left(r,s\right)\in J$ then also $\left(r,0\right),\left(0,s\right)\in J$. Thank you for your help! Unable to lock the administration directory (/var/lib/dpkg/) is another process using it? How to move boot and root partitions to another drive So in the first 5 or 6 books, it was very clear that the wand chooses the wizard and the power/level of the spells the wizard produces is based on the knowledge and skill of the wizard.  This is referenced by people constantly being telling Harry that he wasn't skilled enough to perform certain spells, or Bellatrix telling Harry that he needed to mean the Cruciatus curse for it to be effective.  However, in the 7th book, this entire idea is changed to the wand being the thing that's powerful and regardless of the wizard's/witch's skill or knowledge, he/she could still perform a very advanced spell if it's something the wand could handle.  Goyle accidentally creating a fire serpent is something that instantly comes to mind from the movies(I don't remember if that happened in the book).  So, my question is, what actually holds the power, the wizard/witch, or the wand? jsp:useBean scope Variable importance from GLMNET I have a SNP table looks like this  A N N N N N N N N N N N N C N N N C N N N N N N N N N N N N N N N N N N N C N N N C N N N N N N N N N N N N N N N N N N C C N N N N N N N N N N C C N N N C C N N N N N N N N N N N N N N N N N   I want to count total number of N in each row? any idea how to do that