Let $R$ be commutative ring with $1$. Let $f \in R$. Suppose $f$ is not nilpotent in $R$. We need to show $R_f \cong R[x]/(fx-1)$, with $D = \{f^n: n \ge 0\}$ and where $R_f = D^{-1}R$. I am given the following hint: [Show that the map $\phi : R[x] \to R_f$ gives a surjective ring homomorphism and the universal property in   Theorem 36 gives an inverse.]           Theorem 36: Let $R$ be a commutative ring with 1 and let $D$ be a multiplicatively closed subset of R containing 1. Then there is a commutative ring $D^{-1}R$ and a ring     homomorphism $\pi: R \to D^{-1}R$ satisfying the following universal property: for any     homomorphism $\psi : R \to S$ of commutative rings that sends 1 to 1 such that $\psi(d)$ is a     unit in $S$ for every $d \in D$, there is a unique homomorphism $\Psi: D^{-1}R \to S$ such that $\Psi \circ\pi = \psi$.      I'm not sure where to go after I define $\phi$. How does the theorem give me an inverse, and how does that show that the two rings are isomorphic? Help would highly, highly be appreciated.
In Dummit and Foote, Exercise 15.18 (p. 727 3rd edition) one is asked to prove that for a commutative ring with unity $R$, and $f$ a non-nilpotent element of $R$, that the localization $R_f$ is isomorphic to $R[x]/(xf - 1)$.  I have done this, using the strategy outlined in the problem, using the universal property of $R_f$ proved in Theorem 36. The problem I have is this:  If $R_f \cong R[x]/(xf - 1)$ and $\phi:R[x] \to R_f$ is a surjective homomorphism, it stands to reason that $\text{ker}(\phi)$ should be $(xf - 1)$. I cannot show this directly. The surjective homomorphism I have in mind is, for:  $p(x) = a_0 + a_1x + \cdots + a_mx^m$  $\phi(p) = \dfrac{a_0}{1} + \dfrac{a_1}{f} + \cdots + \dfrac{a_m}{f^m}$  It is clear to me that $\phi$ annihilates $(xf - 1)$, so that ideal is contained in the kernel. What I cannot show is the other containment, or equivalently:  If $\phi(g) = 0_{R_f}$ then $g(x) \in (xf - 1)$. The trouble I am running into is that $R$ might be a "bad" ring, containing a lot of zero-divisors, $f$ might even be a zero-divisor (being non-nilpotent is a considerably weaker property), so the usual polynomial tricks of the division algorithm or using the irreducibility of $xf - 1$ aren't much use, here.  I feel I am missing something obvious....what is it?
Though tangentially related to another post on MathOverflow (), the questions below are mainly out of curiosity. They may be very-well known ones with very well-known answers, but...    Suppose $\Sigma$ is a sigma-algebra over a set, $X$. For any given topology, $\tau$, on $X$ denote by $\mathfrak{B}_X(\tau)$ the Borel algebra over $X$ generated by $\tau$.     Question 1. Does there exist a topology, $\tau$, on $X$ such that $\Sigma = \mathfrak{B}_X(\tau)$?   If the answer to the previous question is affirmative, it makes sense to ask for the following too:     Question 2. Denote by ${\frak{T}}_X(\Sigma)$ the family of all topologies $\tau$ on $X$ such that $\Sigma = \mathfrak{B}_X(\tau)$ and let $\tau_X(\Sigma) := \bigcap_{\tau \in {\frak{T}}_X(\Sigma)} \tau$. Is $\Sigma = \mathfrak{B}_X({\frak{T}}_X(\Sigma))$?   Updates. Q2 was answered in the negative by Mike (). Why is the Star Trek 2009 reboot called a reboot? Doctor Who (2005) isn't called a reboot. It is called a continuation of the old series. This was made possible by that regeneration ju-ju and stuff.  On the other hand, Star Trek 2009 reincarnation is often called a reboot. Nobody dared call it continuation. I don't think I need to do citation because it's common.  Now, if we actually go into the details:   The new universe contains two Spock. One is from the original timeline. The Spock Prime even gives data from the original timeline like trans-warp equation, Khan etc. We see other past data from before the timeline diverged. For example, Scott told that he transwarped Admiral Archer's dog. "Archer's dog" existed in Star Trek: Enterprise TV series.   So, why didn't the legacy survive in-universe? Why isn't it considered a continuation of the old Star Trek TV shows and movies?  Or, is the reboot word used by mistaken fans and media only?  Update: After the release of Star Trek: Beyond (2016), this is no longer a confusion as the movie showed a photo from original timeline. Let be the next equivalence relation on $B= \{R \subseteq \mathbb{N}\times\mathbb{N} \ | \ R \ is \ a \ well-order \ on \  \mathbb{N}\}$ : $$"R \equiv R' \leftrightarrow \ (\mathbb{N},R) \cong (\mathbb{N},R') \ are \ well-ordering \ sets \  isomorphic" $$  Which is the cardinality of $B/\equiv$?  Firstly, we can notice that each well-ordering set is isomorphic to an ordinal $\alpha \ $such that $\alpha &lt; \aleph_{1}$, and so it is enough to count how many ordinals are isomorphic to a subset of $\mathbb{N}$. For example, if we consider $A$ the set of the even numbers and $B$ the set of odd numbers with the usual order relation, $A \oplus B$ is isomorphic to $\omega + \omega$. Another example could be: if we consider the order of $\mathbb{N}$ that we use in Sharkowski Theorem, that order is isomorphic to $ \omega^{2} + \omega$. Working with prime numbers we could build also $\omega^{n} \ \forall n \in \mathbb{N}$, and so $\omega^{\omega}$ and maybe something bigger. However, we know that each countable ordinal is isomorphic to a subset of $\mathbb{Q}$, but I bet that the same result is not true for $\mathbb{N}$ because it is "too small".  What can I do? Changing parameter values according to other parameter choice in Processing script using QGIS 3? I couldn't find anything about this problem.  Suppose to have the first parameters in a Processing script defined as an Enum:     self.addParameter(         QgsProcessingParameterEnum(             self.CHOICE,             self.tr('Choose the parameter'),             ['param1', 'param2']         ) )   While the second one as a number:  self.addParameter(     QgsProcessingParameterNumber(         name=self.VALUE,         description=self.tr('My value'),         type=QgsProcessingParameterNumber.Double,         defaultValue=40,         optional=False,         minValue=0,         maxValue=100     ) )   Is it possible to change the minValue, maxValue, description, ecc.. of the second parameter depending on the first or second choice of the first parameter already in the GUI?  e.g. if the user choose param1 then the second parameter should become:  self.addParameter(     QgsProcessingParameterNumber(         name=self.VALUE,         description=self.tr('My value'),         type=QgsProcessingParameterNumber.Double,         defaultValue=40,         optional=False,         minValue=0,         maxValue=100     ) )   while if he choose param2 it should became:  self.addParameter(     QgsProcessingParameterNumber(         name=self.VALUE,         description=self.tr('My AWESOME value'),         type=QgsProcessingParameterNumber.Double,         defaultValue=0,         optional=False,         minValue=50,         maxValue=500     ) ) Prove that if $p$ and $p^2+2$ are prime then $p^3+2$ is prime too Optimizing a "black box" function: Linear Regression or Bayesian Optimization... what's the difference? Goal: I have a function $f(x,y)=z$ (two variables for illustration only) which I know almost nothing about--it has a compact domain which I can determine, it is non-negative, and bounded above.  My goal is to find the maximum value $f(x,y)$ takes on its domain.  I can query the function and I know the values at a few points: $f(x_1, y_1)=z_1, f(x_2, y_2)=z_2,..., f(x_n, y_n)=z_n$.  Optimization via Linear Regression:  I have used linear regression (and related methods like the lasso) to model functions like this in the past; this is the method that I am very comfortable with.  Typically, I might try to find a polynomial (of degree $k$) which relatively accurately models the data. The end result would be to say that:   $$f(x,y)\approx \beta_1 + \beta_2 x + \beta_3 y + \beta_4 x^2 + \beta_5 xy + \beta_6 y^2 + ... + \beta_{m}y^{k}.$$  Determining the coefficients is done by solving the .  The "best degree" for the polynomial might be determined using $k$-fold cross validation.  Once the "best" polynomial has been determined, one can choose any of a number of methods to optimize it (Newton's method, gradient descent, hill climbing, etc).  Then, if all goes well, the optimal value on the polynomial corresponds to an optimal value of $f(x,y)$.  Bayesian Optimization:  Recently, I was describing this technique (linear regression) to a colleague who asked: "What is the difference between that and Bayesian Optimization?"  I've been attempting to answer this question for the last couple days, and so far I don't have anything really conclusive.  After reading , trying to read  and getting stuck reading  I think I understand the following about BO:   Start with a prior (a guess as to what family of functions the black box belongs to... or at least some family of functions which you believe might be able to describe $f$ well).  It looks using a  as a prior is popular. Specify whatever is needed to build your prior (for GP this is a covariance function). As you go along, you maintain two evolving functions, a "mean" function $m(x,y)$ and a "confidence interval" function $c(x,y)$.  Basically, you can predict the value at any point by evaluating $m(x,y)$ and you are confident that $f(x,y)\in [m(x,y)-c(x,y), m(x,y)+c(x,y)]$. Bayesian optimization provides a systematic way of updating/exploiting the model by alternatively selecting points where the confidence interval is wide (you are very uncertain about the value) or at optimal points on the model $m(x,y)$.   My Question: Is it fair/correct to say that the major difference between Bayesian Optimization and simply optimizing the linear regression model is that Bayesian Optimization provides the systematic way of sampling with the goal of optimizing the model and reducing uncertainty whereas the linear regression model ignores the idea of uncertainty and doesn't explicitly provide a means of updating the model?  Said another way, is the linear regression model an example of what might be a "mean" function $m(x,y)$ in Bayesian Optimization?  If I stick with linear regression, specify how to compute the uncertainty, and provide a means of updating/improving the model (exploit/explore) am I just doing Bayesian Optimization (without calling it that)?  Background: My background is in pure mathematics (graph theory) but my new job has me working with a lot of data (so I'm self learning statistics).  With that in mind, my statistics vocabulary is somewhat limited (so I appreciate answers that either avoid technical terminology or briefly explain any technical terminology used)--for example, Monday was the first time I heard/read the word "prior" used in a stats sense and I'm still not 100% sure I understand that correctly. Changing background color to transparent in GeoServer? How to set the background color to transparent. I want use style which describe in *.sld file for all my photo with tif format. How to do that? Suppose $\sum_{n=1}^\infty \frac{1}{n} = S$ where $S$ is finite.  Then  $$S =\sum_{n=1}^\infty \frac{1}{n}= \sum_{n=1}^\infty \frac{1}{2n-1} + \frac{1}{2n} &gt; \sum_{n=1}^\infty \frac{1}{2n} + \frac{1}{2n} = S$$  which is a contradiction.  Is this valid? $\lim\limits_{n\to \infty} \int_0^1 |f(x)-a_nx-b_n| dx=0$ implies $(a_n)_n,(b_n)_n$ are convergent Let $f:[0,1] \to \mathbb R$ be a continuous function and the sequences $(a_n)_n,(b_n)_n$ s.t. $$\lim_{n\to \infty}  \int_0^1 |f(x)-a_nx-b_n| dx=0.$$ Prove that $(a_n)_n,(b_n)_n$ are convergent.   I know that $$\left|\int_0^1f(x)dx\right| \le \int_0^1|f(x)|dx.$$ Can somebody help me, please? Can we theoretically balance a perfectly symmetrical pencil on its one-atom tip? I was asked by an undergrad student about this question. I think if we were to take away air molecules around the pencil and cool it to absolute zero, that pencil would theoretically balance.  Am I correct?  . I have a text file named "Hex_data.txt". I want to load content of Hex_data.txt into a variable name RAM in verilog. When I try this, I get an error that the text file can't be found. Where is this file loaded from?  The content of "hex_data.txt" :            21            a4          2a          26          7a          ff           6f   I want to create a variable name "RAM" defined like:   reg [6:0] RAM; in which the above content save like:  RAM[0]=21     RAM[1]=a4           RAM[2]=2a             RAM[3]=26            RAM[4]=7a             RAM[5]=ff             RAM[6]=6f   I wrote a code:  module file_read;      reg [7:0] RAM [0:6];          initial $readmemh("Hex_data.txt", RAM);   integer i;               initial begin                    $display("data:");                for (i=0; i &lt; 7; i=i+1)                $display("%d:%h",i,RAM[i]);              end                 endmodule        But the above code not working. How can i get desired output? Can file uploads time out in PHP? Hi im quite new to PHP, i have created a form for very large csv files to be uploaded to my server. Some one mentioned to me that the browser can time out due to the uploading file being to big, is this true? and if so, can it be prevented?  Thanks for your help! Replacing fonts in EPS/PDF files What would be the quickest way to construct a Python binding to a C or C++ library?  (I am using Windows if this matters.) Android cancel Toast when exiting the app and when toast is being shown Mrs Reed safe combination Mrs Reed forgets the number combination to her safe. The combination has $4$ numbers each is a $1$ digit number. The $4$ digits add up to $10$ and there are no zeros. How many times does Mrs Reed have to try to open her safe?  My attempt: ${9 \choose 3} = 84$ but the answer is $80$. Delete all files except in a certain subdirectory with find I want to recursively delete all files not accessed in a while in folder a, except all files in the subfolder b.  find a \( -name b -prune \) -o -type f -delete   However, I get an error message:     find: The -delete action automatically turns on -depth, but -prune does   nothing when -depth is in effect.  If you want to carry on anyway,   just explicitly use the -depth option.   Adding -depth causes all files in b to be included, which must not happen.  Anyone know a safe way to make this work? How do I prepare best for my first wedding photography event? I am going to shoot a wedding this weekend, and I want to do my best for my friend's big day.    What can I do to prepare for the day in order to minimize failures and mistakes? How do I avoid missing important shots? What backup plan should I have?   I have been shooting for 6 years or so, and I know wedding is one of the hardest thing to do. I have told my friend what I end up producing will be of no match with the work of dedicated wedding photographers, not even the ones that are about to fall out of business.  They are on a tight budget, they said some photos are still better than none, they really must minimize the cost as much as they can. I exaggerated again that my photos will be worse off than the cheapest wedding photographer you can find.  She has been a close friends for years, I want to do everything I can to help her.  Their wedding is at 12:30pm coming Sunday, I have visited the venue last Sunday at 12:30pm to check traffic condition and also the lighting.  The setup is mainly outdoor (no church), the indoor area will not have a good ceiling for bounce. Being at the middle of the day, this can mean some tricky fill flash. I have jot down a list of must-have shots. I am also asking for the seating plan and the event run-down.  Gear:   Canon EOS 60D , EF-S 15-85mm f/3.5-5.6 IS USM Canon EOS 450D (or I can borrow a 550D), EF 50mm f/1.4 Speedlite 580 EXII Speedlite 270 EXII Yongnuo 460 (a cheap but powerful manual flash as backup) Samsung EX-1 f/1.8 point and shoot (backup of a backup...?)   Other lenses that I have but I don't think will be useful:   50mm f/1.8 EF-S 50-250mm f/4.0-5.6 IS II EF-S 18-135mm f/3.5-5.6 IS (I have the better 15-85 on 60D, should I put this on the 450D as a backup allrounder?) EF-S 60mm f/2.8 Macro (should I replace 50mm f/1.4 with this for added macro? You can hardly use f/1.4 under direct sunlight if you want to have fill-flash)   I have two batteries for all cameras (60D 450D and EX1). I have like in total 64GB SD card (I am really inclined to get more, or bring a laptop for on-the-fly backup, or do both). I recently repaired my 580EXII, it has a brand new flash head so the recycle time is as good as it can be, I just need a tonne of batteries. No I don't have external power pack or battery grip for my cameras.  I have a great monopod, but a cheap tripod.  I am happy to bring everything, just putting them in the boot of my car. The carpark is on-site, walking from venue to my car is under 1 minute.  I will have my girlfriend carry the 450D and shoot details / non-portraits.  One concern that I have is fill flash, I really do not have a large surface to bounce off  from. I have shot events before but they are not as dynamic/critical as a wedding.  How do I prepare for differing weather conditions?  If its sunny, fill flash is probably going to be in every single shot. If its cloudy, I probably will turn off the flash. If it rains, my 60D should be fine I have used it in slight rain before, but I really don't know their plan and if they have extendable shelters etc.  What else should I prepare for? Having no experience I want to be as prepared as I can be. Maybe I have overlooked something, maybe there is something else I can do to ensure the shooting will go smoothly. Your help is greatly appreciated! Explain reflection laws at the atomic level The "equal angles" law of refection on a flat mirror is a macroscopic phenomenon. To put it in anthropomorphic terms, how do individual photons know the orientation of the mirror so as to bounce off in the correct direction? Working with objects created with Activator.CreateInstance