Localisation isomorphic to a quotient of polynomial ring Let $R$ be a commutative ring and $A=\{1,a,a^2,\dots\}$ for some $a\in R$. Prove that $A^{-1}R$ is isomorphic to $R[T]/(aT-1)$.    I guess I'm meant to find a surjective homomorphism between $A^{-1}R$ and $R[T]$ and then use first isomorphism theorem. What homomorphism should I use?
Localisation is isomorphic to a quotient of polynomial ring I am having trouble with the following problem.      Let $R$ be an integral domain, and let $a \in R$ be a non-zero element. Let $D = \{1, a, a^2, ...\}$. I need to show that $R_D \cong R[x]/(ax-1)$.    I just want a hint.  Basically, I've been looking for a surjective homomorphism from $R[x]$ to $R_D$, but everything I've tried has failed. I think the fact that $f(a)$ is a unit, where $f$ is our mapping, is relevant, but I'm not sure. Thanks
How to run firefox and firefox ESR simultaneously The main reason I want to use firefox ESR is to use a proprietary database tool for work that only allows oracle-java based web logins for linux (or a windows only thin client). I also want to use the full version of the outlook web application which only works in firefox. I don't want to run the NPAPI plugins for normal web browsing and email.   I have the repo version of firefox installed and firefox ESR installed in /opt/firefox and linked to /usr/local/firefox-esr  When I run firefox any attempt to run firefox-esr either brings up a window without the NPAPI plugins.   If I try running with the --new-instance or --no-remote options I receive the following error:  Firefox is already running, but is not responding. To open a new window, you must first close the existing Firefox process, or restart your system.   Regardless which one I start first, I can only open that one, and get the above error if I attempt to sidestep.  What makes this an especially annoying problem is that I often have to respond to emails that I need to access the database to deal with. So either I close and open many windows. How to effectively close a page? Finding all solutions for a system of equations with constraint on the sum of absolute values I am looking for a fast algorithm to find all integer solutions for the following system of linear equations ($c_1,\ldots,c_n \in \mathbb{Z}$ and $r\in \mathbb{N}$ ):  $x_1+x_2=c_1$  $x_3+x_4=c_2$  ...  $x_{2n-1}+x_{2n}=c_n$  $|x_1|+|x_2|+\ldots+|x_{2n}|=r$  $x_1,\ldots,x_{2n} \in \mathbb{Z}$   Also can we find the number of solutions before solving the system? Does Earth really have two high-tide bulges on opposite sides? The bit that makes sense – tidal forces My physics teacher explained that most tidal effect is caused by the Moon rotating around the Earth, and some also by the Sun. They said that in the Earth and Moon system, the bodies are in freefall about each other. But that points on the surface of Earth, not being at Earth's centre of gravity, experience a slightly different pulls towards the Moon. The pull is a little greater if they are on the Moon's side, and slightly lesser if they are on the side away from the Moon. Once freefall is removed, on the Moon side this feels like a pull towards the Moon and on the the opposite side it feels like a repulsion from the Moon. This makes sense to me, and is backed up by other questions and answers here, like  and also  Phys.SE question. The bit that doesn't make sense – tidal bulges They also said that there are &quot;tidal bulges&quot; on opposite sides of the Earth caused by these forces. The bulges stay still relative to the Moon and Earth rotating through the bulges explains why we get two tides a day. They drew a picture like this one…  An image search for  finds hundreds of similar examples, and  from a scientist on Twitter. …But, if there is a tidal bulge on both sides of Earth, like a big wave with two peaks going round and around, how can an island, like Great Britain where I live, simultaneously have a high tide on one side and a low tide on the other? For example:   on the West coast  on the East  Two ports with tides 6 hours, or 180º apart. It's high tide at one while low tide at the other. But they are only  by road. Great Britain is much smaller than Earth. It's probably not even as big as the letter &quot;A&quot; in the word &quot;TIDAL&quot; in that picture.  To prove this isn't just Britain being a crazy anomaly, here is another example from New Zealand:     Two ports that are 180º (6 hours) apart, but separated by  through a national park. New Zealand, unlike the UK, is in fairly open ocean. How to split the output and store it in an array? This is the output:  3,aac-lc, 93.8, aaclc, 77.3, h.264, 1024.6, h.264, 1029.1, 31, 31, 0,0,0.000000,31,31,0,0,0.000000,7,0,0,0.000000,30,1280 720,10,0,0,0.000000,30,1280 720   I tried with 2 scenario:   Storing in an array    @arr=split(',',$stats);   echo "statistics: $stats"  Storing in a variable   echo $stats | cut -d ',' -f | read s1  echo $s1    But neither scenario is working. Advantages of parallel over serial connections As far as my understanding goes, parallel data connections are faster than serial at the same frequency. However, when you increase the frequency in parallel you start getting bits 'cross-talking' and getting transferred onto the next bus. Serial however, since it's only a single bus line, can have the frequency increased to make it faster than parallel without a problem. Also, surely parallel is more expensive as it uses more lines.  So my question is, what actually is the benefit of using a parallel data connection over a serial one? Are there any? If not, why do we still use parallel connections? One of my favorite quotes is by Alvin Toffler which says:     The illiterate of the 21st century will not be those who cannot read   and write, but those who cannot learn, unlearn, and relearn.   The way I understand "relearn" in the quote is to learn something new that is different from what was originally learned.  I'm wondering in general if "relearn" convey learning the same thing again, or a different thing, or it depends on the context of the sentence. Is the same thing true for "learn again"? Are $\mathbb{Q}_p$ and $\mathbb{Q}_q$ homeomorphic? Evaluating: $\lim\limits_{x\to0}\left(\frac{\sin x}{x}\right)^{{6}/{x^{2}}}$ I am trying to evaluate the following but without result.  $$\lim_{x\to0}\left(\frac{\sin x}{x}\right)^{{6}/{x^{2}}}$$  Can you please give me some hints? I have tried to put log to both sides but it hasn't lead me somewhere... Thanks a lot Evaluating $\int_{0}^{1}\int_{0}^{1}\sqrt{x^{2}+y^{2}}dxdy$ using polar coordinates. I want to evaluate the integral $\int_{0}^{1}\int_{0}^{1}\sqrt{x^{2}+y^{2}}dxdy$ using the substitution $x=r\cos\theta$ and $y=r\sin\theta$ where the Jacobian is given as $|J(r, \theta)|=r$ and thus $dxdy=rdrd\theta$.  I need to set the range for $r$ and $\theta$. For $\theta$, it seems obvious that it ranges from $0$ to $\frac{\pi}{2}$ to cover all parts of the rectangle. But I am confused with the range for $r$. Would it be OK to set minimum $r$ as $0$ and maximum $r$ as $\sqrt{2}$, which is the diagonal of the square domain? I think I am confused with the fundamentals of integration. Chat arrangement question: Left or right? I'm building a small chat. Now I see but in other chats (whatsapp, facebook mobile chat, sms ...) that the arrangement is always the same:  Left: the chat partner. right: myself      Why is that? How can I make my IP address appear to be from another country? How do I make it appear that my IP address is coming from one country while I'm located in another? I live in Germany and some websites (like Hulu or Youtube) don't work because my IP isn't in the US. How do I get around this? Do I have to use a proxy or something?  Moderator note Super User does not endorse nor defend any activity which may be used to circumvent local/state/national laws. How many physical degrees of freedom does the $\mathrm{SU(N)}$ Yang-Mills theory have? The $\mathrm{U(1)}$ QED case has two physical degrees of freedom, which is easy to understand because the free electromagnetic field must be transverse to the direction of propagation. But what are the physical degrees of freedom the $\mathrm{SU(N)}$ Yang-Mills theory?   I believe that the $\mathrm{SU(N)}$ Y.M. theory has four degrees of freedom, and it is easy to see that by gauge fixing (such as the Lorentz condition or the Coulomb condition) that we can always remove one redundant degree of freedom. But then we still haven't completely fixed the redundant degrees of freedom. Thus my question is:     How many physical degrees and how many are redundant degrees of freedom does the Y.M. theory have?    I would be interested to understand how we can determine this mathematically, and also to understand what the physical intuition behind this is? Firefox - Websites with custom fonts not displaying How can I change the local origin of an object, without changing the object location in the world space.  This is the object located at position (0,0,0) in the world space    This is the same object located at position (0,1,0) in the world space, but with its local origin offset so that the object is in the same place    Is there a way to do this in a single step? USB flash drive not working or is appearing as an empty disk drive, Disk Management reports "No Media" with 0 bytes size I have a USB flash drive which is no longer recognized by my computer. Windows Disk Management and DiskPart report No Media with no storage space (0 bytes) on the drive and I cannot partition or format the drive:     If the drive appears in Windows Explorer, trying to access it returns an error message indicating that there is no disk inserted, such as the following:     Please insert a disk into drive X:.   Various disk partitioning and data recovery utilities don't recognize the drive or only give a generic name for the drive and cannot access the contents of the drive.  What can I do? How do I recover data from the drive?     This question comes up often and the answers are usually the same. This post is meant to provide a definitive, canonical answer for this problem. Feel free to edit the answer to add additional details. Integral vanishes on all intervals implies the function is a.e. zero I am having trouble with the following problem:      $f:\mathbb{R}\to \mathbb{R}$ is a measurable function such that for all $a$:   $$\int_{[0,a]}f\,dm=0.$$   Prove that $f=0$ for $m$ almost every $x$ (here $m$ is the Lebesgue measure).    I have no problem proving this for $f$ non-negative, or under the assumption that $f$ is integrable. But the question only assumes that $f$ is measurable and no more.  My idea was the usual thing; we look at the set of points where $f$ is positive and negative and assume one of these has measure greater than zero. Then I wanted to estimate one of these by an open set, look at the integral on the open set and show that it had to be greater than zero, a contradiction. But a key part of this attack is the assumption of the absolute continuity of the integral, which only holds in the case where $f$ is integrable.  Alternatively, if it were integrable one could simply estimate $f$ by a continuous function, where the result is quite obvious.  Ultimately we are going to show that $f$ is integrable, but it is not clear to me how to show this before showing it is zero a.e. So there must be a simpler way. Does anyone have suggestions? Different PDF output with same source files (NeurIPS 2021 package) What is "it" in the following sentence: It is clear that Bob likes doughnuts I am very confused. Unless I am mistaken, I know "it" has to be a noun of some sort, but I am unable to figure out what noun "it" is referring to.   What is "it" in the following sentence:      It is clear that Bob likes doughnuts.   Heres another couple of examples:     It is impossible to fly. Change Windows 7/8/10 system language - welcome screen, login, etc