Show p(x,y) is a metric on R
Show that $d_2$ defined by $d_2(x,y)=\frac{|x-y|}{1+|x-y|}$ is a metric
Contour graphing algorithm I am not sure if this belongs on Mathematics Stack Exchange, but it is somewhat relevant here.       The Problem If you've installed any graphing/plotting apps on your smartphone, you will notice that the app will only take the equation in form of a single variable for 2D (e.g  $y=f(x)$) or in 2 variables for a 3D plot (e.g $z=f(x,y)$). This is problematic as many equations are tough to reduce to the form of $y=f(x)$ or $z=f(x,y)$. The plotters like this because this is the only way they can plot it in form of lines connecting multiple points. These plotters compute the $y$ value at fixed intervals of $x$ and join the point to the previous one. The brute force approach to counter this is to check if the function satisfies it at each point on the image. The problem is that this not only takes more time, but gives discontinuous graph for the function as the image cannot have infinite resolution.         The Question The only way to fix this is to identify which point comes after the next in the plot and join them together by a line. That is where I am stuck. How do you identify which point is sequentially the next one for a given function? Keep in mind that for a human, this might be very easy to do. But, for a computer, this requires a general algorithm. Any ideas? How can I do a line break (line continuation)? What do the E and R prefixes stand for in the names of Intel 32-bit and 64-bit registers? Simple Linq query has duplicated join against same table? Does deleting iCloud photos affect them on devices? If I delete a photo from my iCloud will it also be removed from my devices that are connected to my account? All Stack Exchange data dumps Connecting multiple grounds As I'm relatively new to the electronics world, I was wondering if you can connect multiple  ground terminals (corresponding to several different voltage outputs) to the same ground. Or do they each need their own?  In my case I have 5 lines, one 12V, one 3.3V, two 5V and one variable from a 12V in using a  voltage regulator that already has its own ground. So I'm guessing that each would need its own ground, but it is never bad to ask. Is there a way to set my current location manually on Windows 8? Is there a way to set my current location manually on Windows 8? I don't have a GPS module, but the device is always located at the same place. The automatic detection via network sets a place 200 kilometers from here.  I searched for a setting that allows me to set the location by hand, but I didn't find one. Iphone 4 won't update past a 7.1.2 How can I find posts related to lost reputation due to a "user was removed" message? Today I noticed that my reputation got truncated by 64 points, and when I looked at my reputation history, I saw  notifying me of points removal along to a "user was removed" message. Each has a slightly different timestamp, ranging from 12:39:04 to 12:39:43.    I can't find the questions/answers concerned by this (and I don't remember having won as much as 65 points once, making this even stranger). I would like to find out how I could have lost 65 points while I never won as much in a single question/answer.  Is there a way I can check which of my posts were impacted? Launchers for installed snap applications (including Ubuntu Software) disappeared after installing GNOME Software What's the purpose of cycling gloves? I haven't cycled for many years already and long ago when I was cycling I didn't pay much attention to equipment. Now I see that many cyclists wear a certain kind of gloves - that only cover the palm and leave the fingers uncovered.  What's the purpose of wearing such gloves (and not usual gloves for example)? In what non-cycling activities can such gloves be useful? Limit $\lim_{n\to\infty}\left(1-\frac{1}{3}\right)^2\left(1-\frac{1}{6}\right)^2\cdots\left(1-\frac{1}{\frac{n(n+1)}{2}}\right)^2$ How can I find the following limit :$$\lim_{n\to\infty}\left(1-\frac{1}{3}\right)^2\left(1-\frac{1}{6}\right)^2\left(1-\frac{1}{10}\right)^2\cdots\left(1-\frac{1}{\frac{n(n+1)}{2}}\right)^2$$There are four options $(a)\frac1{3}(b)\frac1{9}(c)\frac1{81}(d)0$ Thanks! How do I pass a function as a parameter to in elisp? I'm trying to pass one method to another in elisp, and then have that method execute it.  Here is an example:  (defun t1 ()   "t1")  (defun t2 ()   "t1")  (defun call-t (t)   ; how do I execute "t"?   (t))  ; How do I pass in method reference? (call-t 't1) I have keypair (u,r), and two pieces of data, X and Y.  I reveal X publicly but keep Y secret. Y has high entropy.  I then concatenate X and Y, into Z.  I encrypt Z, into "encrypt(Z,u) = Q". r is needed to decrypt Q back into Z.  Later, (when it doesn't matter), r will be revealed and everyone will decrypt Q into Z. At that point, everyone will know that, all along, Q "contained" X. Can I prove that Q contains X before r is revealed (including interactive zero-knowledge proofs)? Another point of view that $\mathbb{Z}/m\mathbb{Z} \times\mathbb{Z}/n\mathbb{Z}$ is cyclic. I was thinking that the product of groups $\mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/2\mathbb{Z}$ is not cyclic, but $\mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/p\mathbb{Z}$ is cyclic if p is an odd prime. Then I found logic that $\mathbb{Z}/m\mathbb{Z} \times\mathbb{Z}/n\mathbb{Z}$ is certainly cyclic if $m$ and $n$ are coprimes.  I searched on google a proof of that, to compare with mine and the answer is that $\mathbb{Z}/m\mathbb{Z} \times\mathbb{Z}/n\mathbb{Z}$ is isomorphic to $\mathbb{Z}/mn\mathbb{Z}$ (which is clearly cyclic) $\iff$ $m$ and $n$ are coprimes.  But I got another proof and I want to know what you think about it :     I claim that $([1]_m,[1]_n)$ generates $\mathbb{Z}/m\mathbb{Z}$   $\times\mathbb{Z}/n\mathbb{Z}$, where $[a]_d$ is the canonical   projection of $a\in \mathbb{Z}$ into $\mathbb{Z}/d\mathbb{Z}$.       Let's take multiples of $([1]_m,[1]_n)$ until we get $m \cdot$   $([1]_m,[1]_n) = ([0]_m,[m]_n)$ (where I suppose $m=\min(m,n)$...   otherwise, consider $\mathbb{Z}/n\mathbb{Z}$   $\times\mathbb{Z}/m\mathbb{Z}$ and $\min(m,n)=n$).       The next step is $([1]_m,[m+1]_n)$. This pair is not equal to   $([1]_m,[1]_n)$, since $m+1 \not\equiv1$ mod $n$ ($m$ and $n$ are   supposed to be coprimes). We can now continue adding $([1]_m,[1]_n)$   to the results.      The last step is precisely reached when you got   $([mn]_m,[mn]_n)=([0]_m,[0]_n)$. At this point, the fact that $m$ and $n$ are coprimes is absolutely necessary because if they are not coprimes, $lcm(m,n)\ne m\cdot n$ and   $([lcm(m,n)]_m,[lcm(m,n)]_n)=([0]_m,[0]_n)$ but all the $m \cdot n$ possible pairs are not reached, only the $lcm(m,n)$ are ! High Memory Usage on Linux Server Which Distribution Does the Data Point Belong to? I have two distributions which are derived from 2 separate sets of data. These distributions are not normal, and it is not clear at this point if they belong to any family of known pdfs (they are not symmetric either). Given a data point, I need to decide which distribution it is most likely to belong to. If these were normal distributions, I could do usual parametric tests and go from there, but in this case I am not sure how to proceed. I searched around a bit, but I couldn't find anything, probably because I am not using the right keywords. Any help is much appreciated.  Edit to clarify: I should have mentioned that it was univariate. I might as well explain the actual problem here as well. We have time-spent data for users on websites. We also have information on whether a user liked or disliked some of those sites (about 4% of all the sites). So we know the distribution of time spent for like and dislike. An obvious question, then, is for a random user who spent x amount of time, are they more likely to have liked the page or dislike the page. Our time spent information is based on seconds, so the distributions are really discreet, but in real life, they are continuous. Can anyone show me how to find invertible elements in $\mathbb{Z}/m\mathbb{Z}$, $m$ is say $28$?  Also I'm not very clear about what it mean by 'invertible element in $\mathbb{Z}/m\mathbb{Z}$' Error propagation with asymmetric uncertainties I red the Wikipedia  on error propagation. I get into some trouble when I want to calculate the error propagation in a specific case. I have real numbers $A$ and $B$ with related upper ($\Delta A_u$, $\Delta B_u$) and lower ($\Delta A_l$, $\Delta B_l$) uncertainties. I have two functions calculated as follows: $$F = \sqrt{A^2+B^2}$$ and  $$G = \tan^{-1}\frac{B}{A}$$ What are the related upper and lower errors of $F$ and $G$?  I found general formulae for Gaussian error distributions ($\Delta A_u = \Delta A_l$ and $\Delta B_u = \Delta B_l$), for instance the first formula of the first answer on . Is there a way to generalize that formula for arbitrary distributions with asymmetric errors as in my example?