Let $S/R$ be a ring extension and $\beta \in S^*$. Show that for all $\alpha \in R[\beta]\cap R[\beta^{-1}]$ is integral over R.
Prove that $B[x] \cap B[x^{-1}]$ is integral over $B$
A term for words that change pronunciation with part of speech I'm talking about words like:   construct: CON-struct(n.), cun-STRUCT(v.) present: PRE-sent(n.), pre-SENT(v.) record: RE-cord(n.), ri-CORD(v.)   They are pronounced differently based on whether they are a noun or a verb.  Here are the terms I DON'T think would apply:   Homograph: same spelling, different meanings: for example, bark (the sound of a dog) and bark (the skin of a tree). Homophone: same pronunciation, different meaning: for example, to and two Homonym: same pronunciation and spelling, different meanings: for example: bank (river bank or savings bank) Heteronym: same spelling,  different pronunciations and meanings: for example bow (the front of a ship) and bow (a ranged weapon)   Bow is particularly interesting. It is a heteronym based on the two nouns I described. But in the sense of the body movement, the noun and the verb are pronounced alike (rhyming with wow), thus, not belonging to the category I've explained.  In fact none of the terms explained  satisfy my requirement.  I'm looking for the term for words that have same (or related at best, NOT different) meanings, same spelling and different pronunciation.  Specifically, different pronunciation depending on the part of speech. As against the words which have the same pronunciation in different parts of speech like bill, turn, case (or bigger ones like manoeuvre)  TL;DR:  Fill in the blank:     The noun and verb forms of construct are pronounced differently   because the word construct is a _____   or      The noun and verb forms of construct are pronounced differently   because they are _____s. How does CoffeeScript's existential operator work? Anti-virus comparison for Mac OS X This is not a "I use OS X, do I need Anti-Virus?" question.    I am looking at comparison and user opinion of existing AV products. I know I can avoid most (if not all) threats by being careful on the sites I visit, and the emails I download. But I need an anti-virus, for a variety of reasons - mostly to prevent my Mac as a carrier for viruses. (Many friends' USB drives turn out to be infected).  I was looking for the "Best" of the anti virus software out there.  I am looking prominently at anti-virus. "Internet Security" and the like are not important, and would try to avoid them if it would save money or not make my mac slower.  By "Best" I mean the best combination of detect-rate (or whatever that is called) and performance.  I don't intend to run it always in the background. Maybe I would do a onetime scan of my system and hard drives, and then it would be mostly on a need basis, especially on USB drives/portable hard disks of friends. So primary objective: High Detect rate  I have used  and  (got this with one of those software bundles sometime before, not the latest version) and had problems with both. Both sometimes hog my CPU, and somehow "lock" my USB drives preventing me from Ejecting it (I had to force eject them or just pull them off from the port) (thought off late Norton seemed a little better)  Ok, thats a long one. But my question is simple: What anti-virus do you recommend for the Mac?  Thanks. latexmk: jobname option and clean up incompatibility? I'm using  in a Makefile to compile my document. Since I want my output filename to be different from the input filename, I use latexmk's option jobname.  The problem is: it seems to be incompatible with latexmk -c to clean temporary files.  Here's the specific entry in my Makefile:  these.pdf: header.tex $(OBJS)     latexmk                 \         -pdf                \         -pdflatex="$(TEX)"  \         -jobname="these"    \         -use-make $&lt;   Input files are header.tex containing the latex instruction header and other latex content files. The output files are these.{aux,bbl,bcf,...}. Then latexmk -c does not remove temporary files: .{aux,bbl,bcf,...} files. Start new chapter on same page How does a calculator calculate the sine, cosine, tangent using just a number? Sine $\theta$ = opposite/hypotenuse   Cosine $\theta$  = adjacent/hypotenuse   Tangent $\theta$  = opposite/adjacent      In order to calculate the sine or the cosine or the tangent I need to know $3$ sides of a right triangle. $2$ for each corresponding trigonometric function. How does a calculator calculate the sine, cosine, tangent of a number (that is actually an angle ?) without knowing any sides? Does driving an microcontroller's GPIO when it is powered off weaken the chip? We are using the  in a project. The way our design is now, the IC's GPIO pins will sometimes be driven high (to 3.3V) or low when it is powered-off. Will this weaken the chip? The absolute maximum ratings state the voltage on any digital I/O pin is -0.3V to VSupply+0.3V. If the answer is no, could the IC be weakened during power up?  The LTC5800-IPM does not have the most detailed datasheet. If it is not clear for that device in particular, I'd be interested to know for CMOS microcontroller GPIOs in general. Let $a=p_1^{x_1}p_2^{x_2}\cdots\cdot p_q^{x_q}$ and $b=p_1^{y_1}p_2^{y_2}\cdots p_r^{x_r}$ ,   say — Since gcd(a,b) is the largest common divisor of a and b and is divisible by any other common divisor of the two,   GCD$(a,b)=p_1^{\min(x_1,y_1)}p_2^{\min (x_2,y_2)} \cdots p_q^{\min (x_q,y_q)}$  LCM$(a,b)= p_1^{\max(x_1,y_1)}p_2^{\max(x_2,y_2)}\cdots p_r^{\max (x_r,y_r)}$  I understand GCD(a,b) has to divide both $a,b$. Therefore the exponent of any prime factor $p_i$ in GCD has to be in both $a,b$  — therefore $p_i^{\min(x_i, y_i)}$. But I'm muddled and anxious. Why does $\min$ appear in GREATEST common divisor? Why does $\max$ appear in LOWEST common multiple? In the new header could we please move the chat link back to where it was? (One click away)  As a daily frequent to the JavaScript room that's the one link I'd love to have visible 100% of the time.  Aside from that, I actually love the new header. (Maybe take it a step further an add a customizable header with (X) link spots for your most used links?)  Just my thoughts This question is the flip-side of the commonly asked food temperature safety concerns. I'd like to know, if a meat-based soup stock (chicken/pork/beef/etc) were to be held constantly at a safe hot temperature of 165F/73C-180F/82C, how long could it stay at this temperature and still be considered safe to eat?  Obviously, it would reduce over time, but suppose it's reconstituted with water regularly so that it could continue being held... Is there ANY sort of time limit (even completely unreasonable spans like, months, years, etc)? I have one bash source run.sh as follows,  #!/bin/bash if [ $# -ne 1 ]; then     exit fi ...   when I execute it in two ways, there are different behaviors. The first way is,  source run.sh   It will close the terminal after execution. The second way is,  ./run.sh   this will simply finish running the script, and stay on the terminal. I am asking if there is a command for exiting a bash scripts for both source run.sh and ./run.sh execution. I have tried return too, which does not work well under ./run.sh execution.  More generally, I am interested in why this is happening, and what's difference between using "source" and "." for script execution? How do I keep both the background and foreground in the image in focus at the same time? How do you have both the background and foreground in the image in focus at the same time?  Last week I was attempting a shot where I had the back of a hat in focus and in the distance you could see a mountain. I wanted both in focus, but could either focus on the mountain or the hat. I ended up taking the in-focus shots of both and merging them in Photoshop. While that isn't difficult I would prefer to shoot the shot as accurate as possible. From what I have read aperture is important in the depth of field but what are the techniques for keeping both the background and foreground in focus?  I am using a Nikon D5100 with a 18-55mm f/3.5-5.6 (standard kit lens) and a 55-200mm  f/4.5-5.6.    almost answers my question, but I am a bit confused. Can someone explain this a bit differently? Thanks! First off, sorry for my english. I am currently developing a new website that may be deployed in early 2017. This site may contain user login but that may be shipped later in that year.   Now, I googled about https and I hear mixed opinions, google wants the web in https and may punish those who don't use https in the future. Mozilla will also favor https over http for its Firefox. Let's Encrypt make it for free. So far so good.  What is worrying me is I read that many sites that switched to https suffered much lower revenues of adsense (I read also some terrible things about seo and ranking suffering after switching) and many returned again to http.  How much truth is in the suffering of revenues due to switching to https? My website is still in development so I have actually no links needed to be reindexed. My site isn't a blog type. It mainly provides profiles on local artists in my country.  The problem is I need every penny to cover the server and cdn hosting costs. I can't be sure how much using https will hurt the revenues. Are the suggested edit audits too easy? Yes, this might be another post complaining about audits. But, in this case, I want the audits to be more difficult.  Specifically, I'm concerned that the suggested edits audits are just too darn easy to spot. They don't behave anything like actual edits, and so after a tiny bit of experience (and electric shocks from audit failures) I think they wouldn't slow down a roboreviewer at all.  Unlike the other queues, which use real data (and are legitimately, in my opinion, difficult), the suggested edits audits look like this:    In about 500ms I can tell that is a bogus edit (less time than it takes for the buttons to unfreeze). After just a few of these, I'm about 99% confident I could tell you which edits were audit edits and which ones were not, and I don't think I actually have to spend extra time on them. On the other queues, picking out the audits is very hard because they use real data, and so I wouldn't bother to do so.  Were I a roboreviewer, I could get on with my Accept-happy life, only pausing to reconsider when someone actually changes the text. (Seriously -- how many real edits have you see which change the text in such a drastic manner? 95% of edits are grammar, spelling, formatting or some combination of those).  Are the suggested edits audits too easy? Is the "Markov-chain random-edit-word-salad" too easily detectable? Or does it legitimately slow down and/or deter robo-reviewers? Should we (and can we) be using real data for the suggested audits queue too? I am aware of what hyperfocal distance is and the general principles behind it. However, most of the tutorials I have seen and read regarding it make use of lenses with focus distance indicators. Unfortunately, neither of my lenses (Nikkor 18-55mm &amp; 35mm f1.8) have indicators.  I have read that you can 'cheat' and simply focus a third of the way into the shot (which I have done previously simply by spot-focusing on the lowest focus point in my VF). I also assume you could pace out the distance and note an object to focus on when you get back to the camera (though that might be easier said than done).  Does anyone have any suggestions on how to achieve maximum sharpness from front to back without focus distance markers. A new lens is not an option; global economic crisis and all that! How can I undo git reset --hard HEAD~1? What is 'Currying'? I've seen references to curried functions in several articles and blogs but I can't find a good explanation (or at least one that makes sense!) Consider the following:   $1 = 1^2$ $2 + 2 = 2^2$ $3 + 3 + 3 = 3^2$   Therefore,   $\underbrace{x + x + x + \ldots + x}_{x \textrm{ times}}= x^2$   Take the derivative of lhs and rhs and we get:   $\underbrace{1 + 1 + 1 + \ldots + 1}_{x \textrm{ times}} = 2x$   Which simplifies to:   $x = 2x$   and hence    $1 = 2$.     Clearly something is wrong but I am unable pinpoint my mistake. Wrapper for package managers There are a lot of language-specific package managers like cpan for perl, cabal for haskell etc. When we want to install some software, it can sometimes be installed from a corresponding repo, with installation instruction just like (say, ) npm install yaxy or cpan something or cabal install something.   These package managers have slightly different syntaxes, they all need root privileges and/or custom setup, or (in case of Yaxy) just a newer version of interpreter (nodejs packaged by Ubuntu was too old to run it).  Is there a wrapper for different package managers that unifies/automates setup for different package managers?  Edit: what I want is a way to execute installation command like that npm install yaxy and make it working without manual installing system npm+nodejs, getting non-functional executable, finding the simplest way to install fresher nodejs on my system, reinstalling... There are two 'Library' directories in my OS X 10.11 installation, one being /Library/, the other being /System/Library/. What is the difference between different Library folders?