$A$ has only finitely many minimal prime ideals $\implies\ (0)$ is decomposable?
Irreducible Components of the Prime Spectrum of a Quotient Ring and Primary Decomposition
Axiom of Choice: Family of non-empty sets mutually disjoint or not? Property of Entire Functions How can I get the size of an array from a pointer in C? Analysing Bounce Rate and Time on Page Is it possible to generate transcendental numbers of arbitrary irrationality measure ? I am thinking of something like an infinite series which depends on a parameter fixing the irrationality measure.   All I have found so far is bounds on the irrationality measure of famous numbers, I have not been able to find something going the other way around. Settings disappeared in Ubuntu 19.10 I usually go to the right top corner and at the drop down menu I select the tool icon to go to the settings.  However, 19.10 does not have a tool icon.  How can I go to settings now? I'm trying to prove that if $\alpha(t)=(x(t),y(t))$ is a $C^2$ regular curve $(\alpha'\neq0)$ with constant and positive curvature, then $\alpha$ is on the circunference and if $\alpha$ is the circunference, then its curvature is constant.  I solved the second part since $\alpha$ is the circunference, then $\alpha(t)=(r\cos\theta,r\sin\theta)$, so $|\alpha'(t)|=r$, I need help in the first part.  Thanks a lot. What would you call someone who imposes on other people's generosity? What would you call someone who isn't afraid to ask for money or any kind of favor or who misinterprets someone's generosity for a consistent resource for what they need? Why didn't the machine affect Bond in Spectre? Proving the Cauchy-Schwarz inequality by induction Is it ok to answer based someone's older answer on SE? Possible Duplicate:             This question will probably be closed in a few minutes. It is a duplicate but I couldn't find the question using with meta search.  Anyway, , . In this case, is vote to close the new question as a possible dublicate not the best choice? Looks to me a little bit not ethical. How to remove film from stainless steel pan I have an unsightly film around my stainless pan as shown here:    What is causing this to happen?  And how can I remove the film? I have soaked it in soapy water, scraped it with a plastic scraper, and run it in the dishwasher several times, but I can't seem to get rid of the film.   UPDATE: the pan is an All-Clad stainless, fully-clad frypan: I've recently been learning about differential equations, and my teacher has been giving some particularly difficult examples out for those of us who finish early. He gave us the following differential equation: $$x=y\frac{dy}{dx}-\left(\frac{dy}{dx}\right)^{2}$$ With the hint "Differentiate with respect to $y$, then let $\frac{dy}{dx}=p$". I've solved it after the lesson and would like someone to check whether or not I've done so correctly.    Following the hint, $$\frac{dx}{dy}=\frac{1}{p}=p+y\frac{dp}{dy}-2p\frac{dp}{dy}$$ Which rearranges to $$p-\frac{1}{p}+y\frac{dp}{dy}=2p\frac{dp}{dy}$$ Multiplying through by $\frac{dy}{dp}$, $$\left(p-\frac{1}{p}\right)\frac{dy}{dp}+y=2p$$ Rearranging into the form $\frac{dy}{dp}+f(p)y=g(p)$, and writing $p-\frac{1}{p}=\frac{p^{2}-1}{p}$ $$\frac{dy}{dp}+\frac{p}{p^{2}-1}y=\frac{2p^{2}}{p^{2}-1}$$ Our integrating factor is $$e^{\int\frac{p}{p^{2}-1}dp}=e^{\frac{1}{2}\ln(p^{2}-1)}=\sqrt{p^{2}-1}$$ So we have $$\sqrt{p^{2}-1}\frac{dy}{dp}+\frac{p}{\sqrt{p^{2}-1}}y=\frac{2p^{2}}{\sqrt{p^{2}-1}}$$ The left hand side is, by design, $\frac{d}{dp}(y\sqrt{p^{2}-1})$, and the right hand side can be integrated as follows: $$\int \frac{2p^{2}}{\sqrt{p^{2}-1}}dp=\int\frac{2(p^{2}-1)+2}{\sqrt{p^{2}-1}}dp=\int 2\sqrt{p^{2}-1}dp+\int \frac{2}{\sqrt{p^{2}-1}} dp$$ These can both be solved using the substitution $p=\cosh(u)$, and gives $$y\sqrt{p^{2}-1}=p\sqrt{p^{2}+1}-\cosh^{-1}(p)+\alpha$$ Where $\alpha$ is an arbitrary constant. Dividing, we finally have $$y=p+\frac{\cosh^{-1}(p)+\alpha}{\sqrt{p^{2}-1}}$$  Here I was stuck for a long time, until I realised that we also have the equation we began with: $x=py-p^{2}$, which we can solve as a quadratic in $p$ to also get $$p=\frac{y\pm \sqrt{y^{2}-4x}}{2} \implies p^{2}=\frac{y^{2}+y^{2}-4x \pm 2y\sqrt{y^{2}-4x}}{4}$$  Thus, $$y=\frac{y\pm \sqrt{y^{2}-4x}}{2}+\frac{\cosh^{-1}\left(\frac{y\pm \sqrt{y^{2}-4x}}{2}\right)+\alpha}{\sqrt{\frac{y^{2}-2x \pm y\sqrt{y^{2}-4x}}{2}-1}}$$  This is as good as I can do - we have a relation between $y$ and $x$ - am I done? Is this the correct answer? Is there an easier way?     Thank you for your time.    Following the suggestion by @Valentin $$\frac{dy}{dp}=1+\frac{\frac{1}{\sqrt{p^{2}-1}}\sqrt{p^{2}-1}-p\frac{\cosh^{-1}(p)+\alpha}{\sqrt{p^{2}-1}}}{p^{2}-1}=1+p\frac{1-\frac{\cosh^{-1}(p)+\alpha}{\sqrt{p^{2}-1}}}{p^{2}-1}$$ Hence, $$dx=\frac{dy}{p}=dp\left(\frac{1}{p}+\frac{1-\frac{\cosh^{-1}(p)+\alpha}{\sqrt{p^{2}-1}}}{p^{2}-1}\right)$$ Which manga chapters correspond to the 2nd season (or "Season 3" on Netflix) of the anime, "Revival of the Commandments"? Prove that $\sqrt{n} \le \sum_{k=1}^n \frac{1}{\sqrt{k}} \le 2 \sqrt{n} - 1$ is true for $n \in \mathbb{N}^{\ge 1}$ I can understand this requirement for the old PPC RISC systems and even for x86-64, but for the old tried-and-true x86? In this case, the stack needs to be aligned on 4 byte boundaries only. Yes, some of the MMX/SSE instructions require 16byte alignments, but if that is a requirement of the callee, then it should ensure the alignments are correct. Why burden every caller with this extra requirement? This can actually cause some drops in performance because every call-site must manage this requirement. Am I missing something? Update: After some more investigation into this and some consultation with some internal colleagues, I have some theories about this:  Consistency between the PPC, x86, and x64 version of the OS It seems that the GCC codegen now consistently does a sub esp,xxx and then &quot;mov&quot;s the data onto the stack rather than simply doing a &quot;push&quot; instruction. This could actually be faster on some hardware. While this does complicate the call sites a little, there is very little extra overhead when using the default &quot;cdecl&quot; convention where the caller cleans up the stack.  The issue I have with the last item, is that for calling conventions that rely on the callee cleaning the stack, the above requirements really &quot;uglifies&quot; the codegen. For instance, what some compiler decided to implement a faster register-based calling style for its own internal use (ie any code that isn't intended to be called from other languages or sources)? This stack-alignment thing could negate some of the performance gains achieved by passing some parameters in registers. Update: So far the only real answers have been consistency, but to me that's a bit too easy of an answer. I have well over 20 years experience with the x86 architecture and if consistency, not performance, or something else concrete, is really the reason then I respectfully suggest that is a bit naive for the developers to require it. They're ignoring nearly three decades of tools and support. Especially if they're expecting tools vendors to quickly and easily adapt their tools for their platform (maybe not... it is Apple...) without having to jump through several seemingly unnecessary hoops. I'll give this topic another day or so then close it... Related I've heard that special relativity makes the concept of magnetic fields irrelevant, replacing them with relativistic effects between charges moving in different velocity frames. Is this true? If so, how does this work? Squeezing all the SEO out of a URL as possible I am working on an ecommerce site, I told our SEO consultant that I plan to make the URL scheme: /products/&lt;id&gt;/&lt;name&gt;. This is similar to Stackoverflow's URLs which are /questions/&lt;id&gt;/&lt;title&gt;.  He asked me if I could change the URL scheme to /p/&lt;id&gt;/&lt;name&gt; instead. I know why he wants this change, the word "products" isn't needed to find the correct product, and it doesn't offer any SEO, so shortening it to just p would make the relevant keywords in the &lt;name&gt; weigh more.  His main priority is maximizing SEO, but the part that I don't think he is considering is how this effects the semantics of the site. Also having the word "products" looks like it has meaning and a reason for being there, just having a p looks chaotic and ugly to me. I also don't think it makes that much of a difference does it? Stackoverflow doesn't use /q/&lt;id&gt;/&lt;title&gt; and they do just fine, I do realize that theres many factors at play here though, not just the URL.  So I want some outside opinions on which is the better way and why? Extra "lines" are being added when creating/editing questions/answers in Edge Lately, I've noticed that extra lines are being added when I create questions/answers. For example, if I write:   Your problem is X, Y, Z. Fix it by doing:      Code  Then maybe also do some other thing.  Oh, and some other thing as well.   When I post it, then go back into the editor, it's been changed to:   Your problem is X, Y, Z. Fix it by doing:       Code   Then maybe also do some other thing.   Oh, and some other thing as well.   Which is mildly annoying since that causes less to be displayed in the box at once, but whatever.  The weird thing is though, the added line isn't properly treated as a newline, even though it does appear to be caused by a newline. A lot of lines that look like they should be empty don't even get rendered. If I delete the original lines that I entered leaving only the "new" newlines, I get weird behavior:    I tried selecting the characters involved and inspecting them in Python, and they just appear to be newlines, so I have no idea what the problem is.    The only browser I've been able to reproduce it in is Edge (version 44.18362.329.0). It does not happen in Chrome. This has been happening for ~3 days now.    There is also a problem with link embedding and code formatting. If while editing I highlight some text and either try to embed a link (interlinked circle button) or format it as code ({} button), it often puts the wrong text between the […]s when embedding, or formats code that I didn't highlight (and doesn't format the code that I did highlight). How do I open PDF in Adobe Reader DC without locking the file from being updated by pdflatex? I don't know if this is possible on Adobe Reader DC on Windows, but asking in case someone has managed to do it.   When I am working on a large document in LaTeX (I use TexStudio) I sometimes like to keep a second copy of my PDF on a second screen so that I can easily refer to what I wrote before, while on my main screen I would have TexStudio with its preview window, which is typically automatically in sync with my current editing location.  On Ubuntu (which uses evince as its default PDF viewer) I used to do it quite nicely. It also seems to watch the PDF file for updates, and keeps in sync of any new versions of the PDF file and reloads it (staying on the same page) when I re-run pdflatex from TexStudio.  At the moment I am working on a Windows machine, and I am trying to recreate the same workflow. I would really like to open a second viewer on my second screen so that I can navigate easily to previous chapters without losing my current position in TexStudio's preview window (which will anyway go back to the current editing location when I run pdflatex).  The only problem however is that when I open the PDF in Adobe Reader DC to put it on the second screen and then I try to recompile the PDF I get the error "I can't write to file". I presume that for some stupid reason Adobe Reader DC maintains some write lock on the file, probably keeping it open while the PDF is visible on screen.  Is there a way to disable this stupid file-locking behaviour from Adobe Reader DC?  UPDATE  After some valuable comments, a workaround is possible using a different viewer. One could either use  or  (the default PDF viewer on Ubuntu, but also available for Windows). They are not so feature rich, and maybe do not look so nice, but in this case do the job.  A similar question was posted  for reference.