Python: Is it possible to setup a folder trigger?
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$\sqrt{1-x^2}$ is not a rational function If $k$ is a field in which $1 + 1 \neq 0$, prove that $\sqrt{1-x^2}$ is not a rational function. Hint. Mimic the classical proof that $\sqrt{2}$ is irrational   It seems rather odd to discuss the square root function in the context of an arbitrary field, but here goes nothing!  Proof:   Suppose that $\sqrt{1-x^2}$ is a rational function in $k(x)$. Then there exist polynomials $p(x)$ and $q(x) \neq 0$ that are relatively prime with $\sqrt{1-x^2} = \frac{p(x)}{q(x)}$; clearly we may take $p(x) \neq 0$. Then $1-x^2 = \frac{p(x)^2}{q(x)^2}$ or $q(x)^2(1-x^2) = p(x)^2$, which says $q^2 | p^2$, and since $q | q^2$, we can infer $q | p^2$. Moreover, since $p|p^2$ and $(p,q)=1$, then $pq|p^2$ or $q|p$, which contradicts the fact that the polynomials are relatively prime. Hence $\sqrt{1-x^2}$ cannot be a rational function.    How does this sound? Aside from the problem of discussing $\sqrt{~~}$ in the context of an arbitrary field, I am worried about not using the fact that $1+1 \neq 0$, at least not explicitly. Where exactly is this assumed used, if at all?  EDIT:  Suppose that $q(x) =1$, and let $f(x) = a_n x^n + ... a_1 x + a_0$. Then $1-x^2 = f(x)^2$. If $x=0$, $1 = a^2_0$ and therefore $a_0$ must be a unit. Letting $x=1$ we get $0 = (a_n + ... + a_1 + a_0)^2$; and letting $x=-1$ we get $0 = (-a_n - ... - a_1  + a_0)^2$. Since we are working in a field, there can be no nonzero nilpotent which means that $a_n + ... + a_1 + a_0$ and $-a_n - ... - a_1  + a_0$ are both zero. Adding the two equations together yields $2a_0 = 0$, and since $1+1 \neq 0$, $a_0 = 0$ which contradicts the fact that $a_0$ is a unit.   How does this sound?  Another attemtpt:  If $f(x)^2 = 1-x^2$, then $2 = \deg(f^2) = 2 \deg (f)$ implies $\deg(f) = 1$. Yet if $x=-1$, we get $f(-1)^2 = 1 - 1 = 0$ and therefore $f(-1)=0$ since there are no nonzero nilpotent elements in a field; similarly, $f(1)=0$. Since $1+1 \neq 0$, $f$ has two distinct roots in $k$ yet is only a $1$-st degree polynomial, a contradiction. Calculate $\sum_{k=1}^n (-1)^{k+1} \binom{n}{k}\frac{1}{k}$ How to prevent freehub body damage? I feel like this is poor design. I have two wheelsets that are less than a year old which have damaged freehub bodys due to cogs digging in to the splines on the freehub. It's so bad that I can't remove the cassette spacer without filing down the splines. I had to use  pliers and chain whip to remove the cogs that were embedded in the freehub when trying to remove the cassette.  Most of the damage is on the center of the cassette as the largest 3 cogs are held together by carrier. I'm using a shimano 105 cassette and  Novatec disc hub and Velocity ATB convertible disc hubs (which were somewhat damaged in the same fashion, but not as bad as the novatec).  I'm using this bike for cyclocross, offroad exploring, light singletrack use, Rail trails and some road riding.    How can I prevent this damage to the freehub from happening? Drawing/Sketching Tool for elaborating questions More often than not I need to elaborate my question put up on Stack Overflow with a custom sketch, UML class diagram, flow chart.    It is in these times I have to open up a custom paint software, factor out some diagrams, export it as an image, attach it to Stack Overflow. I wouldn't know how complicated it is to invest in having such a sketching or drawing tool integrated, It could even be a basic pen stripped off all the advanced features leaving it a bare minimum.  Such a tool if incorporated into the Stack Overflow "ask question" UI would be certainly helpful.  As of Today this Question has been visited around 170 times and 7 answers, This was meant to be the voice of Stack Overflow users. Would you need a Drawing Tool for you on the Stack Overflow site, would it help you to express you questions and answers better? Sadly, only 2, or 3 answers are in the context of the question. The bounty was meant to be a bait and even that ends tomorrow. Hope you'll get the will and the zest to comment on this question today and in the future . Missing return statement in a non-void method compiles What is an intuitive way of understanding Ostrogradsky instability? Is there a way of explaining why no differential equations in physics exceed order two without delving into Lagrangian and Hamiltonian mechanics - i.e. from Newtonian mechanics? Moreover, is there such an alternative proof, which focuses more on physical reasoning than abstract mathematics? Topologist's sine curve is connected I just came across the example of the topologist's sine curve that is connected but not path-connected. The rigorous proof of the non-path-connectedness can be found .   But how can I prove that the curve is connected? To be honest, even intuitively I am not being able to see that the curve is connected. I am thinking if it is proved that the limit point of $\sin(1/x)$ as $x \to 0=0$, then it would be proved. But, why is this true? IMO, this limit doesn't exist. Intuitively also, it seems that the graph would behave crazily and not approach a particular value as a tends to $0.$  EDIT (Brett Frankel): There are a few different working definitions of the topologist's since curve. For the sake of clarity/consistency, I have copied below the definition used in the linked post: $$ y(x) = \begin{cases} \sin\left(\frac{1}{x}\right) &amp; \mbox{if $0\lt x \lt 1$,}\\\ 0 &amp; \mbox{if $x=0$,}\end{cases}$$ Is there a reason why people can't just take the helicopter to mount Everest? Why is it that helicopters can't reach that high? This question led to a new package:      I wish to typeset some relatively long URLs in a piece of text, and when I use \url{..}, the resulting text does not respect the margin boundaries that govern the main text body, instead going all the way to the edge of the paper before wrapping around.   For reference, my preamble looks like this:  \documentclass[12pt]{article} \usepackage{mathptmx} \usepackage{fullpage} \usepackage[pdftex]{hyperref}   and example of the url I'm including is:  \begin{itemize} \item Some text  \url{very-long-url} ... \end{itemize}   I'm compiling using pdflatex, although I doubt this makes a difference. I tried logging in with a Google account, but I got the following error: Do I need a visa for a connecting flight to Japan? I'm going to the Philippines this October and would like to layover in Japan. I found a reasonably priced ticket with a 15-hour layover to Haneda, however, I have to transfer to Narita. Do I need a visa since I'm going to have to stay at a hotel and ride a bus to travel to Narita airport? Thank you! If $a,b,c$ are positive , show that $$\dfrac{b^2+c^2}{b+c}+\dfrac{c^2+a^2}{c+a}+\dfrac{a^2+b^2}{a+b} \ge a+b+c$$  Trial: Here  I proceed in this way $$\dfrac{b^2+c^2}{b+c}+\dfrac{c^2+a^2}{c+a}+\dfrac{a^2+b^2}{a+b} \ge  \dfrac{2bc}{b+c}+\dfrac{2ca}{c+a}+\dfrac{2ab}{a+b}$$ then how I proceed. Please help. Javascript "Variable Variables": how to assign variable based on another variable? I have a set of global counter variables in Javascript:  var counter_0 = 0; var counter_1 = 0; var counter_2 = 0;   etc  I then have a Javascript function that accepts an 'index' number that maps to those global counters. Inside this function, I need to read and write to those global counters using the 'index' value passed to the function.  Example of how I'd like it to work, but of course doesn't work at all:  function process(index) {     // do some processing      // if 'index' == 0, then this would be incrementing the counter_0 global variable     ++counter_+index;       if (counter_+index == 13)     {         // do other stuff     } }   I hope what I'm trying to accomplish is clear. If not I'll try to clarify. Thanks.  EDIT Clarification:  I'm not trying to increment the name of the counter, but rather the value the counter contains. How to add startup applications in lubuntu? Is there a 'Startup applications' app for lubuntu? Or how do I add apps to startup? I attempted a tutorial to add entries to ~/.config/autostart, but it did not work. Isn't there an easy way? Any ideas please? I have a matrix which is kind of symmetrical with the other diagonal, i.e., something like  $$A = \left[ \begin{array}{c c c c}    a &amp; b &amp; c &amp; d \\    e &amp; f &amp; g &amp; c \\    h &amp; i &amp; f &amp; b \\    j &amp; h &amp; e &amp; a  \end{array} \right]$$  Does this matrix have a special name in literature? What are it's properties?  And a matrix that is symmetrical by both diagonals   $$A = \left[ \begin{array}{c c c c}    a &amp; b &amp; c &amp; d \\    b &amp; e &amp; f &amp; c \\    c &amp; f &amp; e &amp; b \\    d &amp; c &amp; b &amp; a  \end{array} \right]$$  What's the name of it? Any interesting properties? A number of times I have asked a certain kind of ‘attractive nuisance’ question – that is, one in which I anticipate a certain kind of answer would be given that I already know I am going to find insufficient or entirely inadequate. Some examples:      A more abstract example would be asking ‘is it possible to do this 100% correctly’, where people itch to come up with hacks, workarounds and half-baked solutions of various degrees of robustness – while the point of the question is obtaining a fully correct and reliable solution.0 I try to be very explicit in the question body about what kinds of answers I am not interested in. And yet, every time I ask such a question, someone gives such an answer anyway, or an outright XY-type, i.e. frame-challenge answer – where in fact the framing of the question is exactly how I want it, and I don’t want it challenged. Since I have already said in the question body that such an answer would be inaqeduate, when I get one, I downvote it. I believe it’s only appropriate, but it tends not to be received very well – I get insulting comments and downvotes in retaliation. I realise that a proper answer would be much harder to come up with than an ‘obvious’ inadequate answer. I suspect that often a real answer should be negative: ‘there is no way to do that’ or ‘there is no way to answer that for sure’. Nevertheless, if that is the right answer, this is the one I want posted – I don’t want the Internet polluted with more tumbleweed of what is already well-known (and may even be subtly wrong), and I don’t want to give ‘A for effort’ upvotes. What is the most appropriate way to act in such a situation?  0 I am also reminded of when Gary Bernhardt  while trying to avoid reflexive ‘don’t use eval’ replies. Of course, expecting nuance from Twitter of all places is silly, but that’s beside the point. I am flying from Orange County, California to San Francisco, California and then on to Seoul, South Korea. If I check a bag will I have to pick it up in San Francisco and re-check it and go through security or customs for the international flight?  Everything I am reading leads me to believe that it's only necessary to re-check baggage and go through security and/or customs when you are first arriving into a country. However, since my first leg is staying within the United States I assume the airline will transfer my bag to the next flight and I can just go straight to the gate for my international flight?  Both legs of the flight are through the same airline (United Airlines) and there are only the two legs: OC -> SF, then SF -> Seoul. How to achieve function overloading in C? Is there any way to achieve function overloading in C? I am looking at simple functions to be overloaded like    foo (int a)   foo (char b)   foo (float c , int d)   I think there is no straight forward way; I'm looking for workarounds if any exist. How to make git mark a deleted and a new file as a file move? My competitor is using Alexa to see what I am doing. I need to get stop Alexa from including my sites in their rankings. How can I do so?   Is it possible to deny access to any visitor with the Alexa toolbar and deny access to Alexa bot requests?