javascript Object.prototype VS Anything.prototype
How does JavaScript .prototype work?
How to compute rational or integer points on elliptic curves This is an attempt to get someone to write a canonical answer, as discussed in this . We often have people come to us asking for solutions to a diophantine equation which, after some clever manipulation, can be turned into finding rational or integer points on an elliptic curve. See , , , , . (This list is biased towards questions I have answered because I remember them best; other people have answered such questions as well.) I would like an answer to which, after one of us has explained the cleverness, we could direct the OP.  An ideal answer would address    How to find both rational and integer solutions Good software solutions. Ideally, it would nice to have a walkthrough for doing these things with , so that people could find solutions without even installing anything. References for how to transform some standard presentations of elliptic curves into Weierstrass form, so that we don't have to write out the algebra every time. I'm thinking of  a cubic in $\mathbb{P}^2$ with a rational point that is not a flex,  $y^2 = \mbox{degree 4 polynomial}$, a $(1,1)$ curve on $\mathbb{P}^1 \times \mathbb{P}^1$, or an intersection of two quadrics in $\mathbb{P}^3$. I am familiar with the set-theoretical definition of functions.   A function $f : A \rightarrow B$ from a set $A$ to a set $B$ is a subset $f \subseteq A \times B$ of the Cartesian product $A \times B$ such that the following conditions hold:   $\forall a \in A : \exists b \in B : (a,b) \in f$ $\forall a \in A : \forall b, b' \in B : (a,b), (a,b') \in f \rightarrow b=b'$   Is there a purely set-theoretical definition of the function evaluation $f(a)$ for every $a \in A$? I get as far as defining  $f(a) = \{ b \in B : (a,b) \in f \}$ but that defines rather the set of values of a multi-valued function. Beamer: How to Disable `\pause` Temporarily I just refreshed the [iphone]-tag-site, and I'm seeing all kinds of questions with what appears to be a grey overlay (see image). What is this for/why am I seeing this? I have never seen anything like this before.. LaTeX doesn't recognize "et al." in the Bibliography I'm using Jabref to manage my Bibliography.  Sometimes, I put directly:     First Author et al.   when adding the article author's name in Jabref.  When compiling my .tex file, LaTeX doesn't recognize the et. al as more names. It takes it as a name/surname.  This is an example of a Bibliography entry in Jabref.  @ARTICLE{Wolf2003,     author = {Wolf, K.et al.},     title = {Compensation mechanism in tumor cell migration: mesenchymal–amoeboid     transition after blocking of pericellular proteolysis},     journal = {J. Cell Biol.},     year = {2003},     volume = {160},     pages = {267–277},    }   This is a .texMWE with the bib entry mentioned:  \documentclass[11pt]{book}   \usepackage{cite}   \bibliographystyle{unsrt} \begin{document}   Here is the bib entry mentioned\cite{Wolf2003}    \bibliography{bibfile}  \end{document}   Here's the output:   How can I solve this problem? Irfanview nearest equivalent for linux Using $\epsilon-\delta$ to prove that a limit is not a specific number I want to use $\epsilon-\delta$ formulation to prove that $\lim_{x\to 1 } 2x+3 \neq 6$.  I know that I need to show that there exists an $\epsilon&gt;0$ such that for every $\delta&gt;0$ there exists $x\in \mathbb{R}$ such that $0&lt;\left| x-1 \right| &lt;\delta$ but $|2x-3|\geq \varepsilon$.  Will someone please guide me through the process?  Thanks a lot in advance What is $\int_{-\pi}^\pi \cos(nx)\cos(mx)\,dx$? Does new Star Wars continuity only affect post-Jedi EU? I have read  a couple of times.     In order to give maximum creative freedom to the filmmakers and also preserve an element of surprise and discovery for the audience, Star Wars Episodes VII-IX will not tell the same story told in the post-Return of the Jedi Expanded Universe.   Does this describe material set after Jedi, or published after?  Is it also saying that only things published as "Legends" are now non-continuity?  I have read that it is only films + Clone Wars + Rebels that are now in the main continuity but can't see this sourced by Lucas Film Story Group. Does anyone have a reference for this? $f$ holomorphic on $D\setminus \{0\}$ and takes no values in $(-\infty,0],$ then $0$ removable If $f$ is holomorphic on $D\setminus \{0\}$ and takes no values in $(-\infty,0]$ then $0$ is a removable singularity.   I thought to prove this by elimination, but I can't really tell anything about the behavior of $f$ around $0$. How can one translate the information about the definition of $f$ in semi-open interval $(-\infty,0]$. My WiFi adapter is not working at all, how to troubleshoot? How to show $d(x,y)= \sqrt{d_1(x_1,y_1)^2+d_2(x_2,y_2)^2}$ is a metric? $d_1(x_1,y_1)$ and  $d_2(x_2,y_2)$ are metric on $X$ and $d(x,y)$ is defined as: $$d(x,y)= \sqrt{d_1(x_1,y_1)^2+d_2(x_2,y_2)^2}.$$ I am trying to show this is a metric. Can you give me some clue about proving the triangle inequality for $d$ ? Thank you for your help. Multivariable Delta Epsilon Proof $\lim_{(x,y)\to(0,0)}\frac{x^3y^2}{x^4+y^4}$ --- looking for a hint How does Earth's interior dynamo work? I'm interested in getting a basic physical understanding of how Earth's magnetic field is generated. I understand that it's a "dynamo" type of effect, driven by convection currents in the molten outer core. These currents cause charges to move, and this generates the field.  However, what I can't find a good explanation of is why there is a separation of charges in the first place. Presumably, moving neutrally charged molten iron would have the same effect as moving any other neutrally charged thing, i.e. it wouldn't create a field.  And presumably, if the fluid wasn't moving then it would become neutral pretty quickly, since molten iron is a good electrical conductor.  So am I right in thinking that the charge separation is the result of positive feedback, in that an intial deviation from neutrality would generate a field, and this would (somehow) cause a greater separation of charges, resulting in a kind of self-maintaining charge separation?  Or is there another explanation?  In either case, does anyone know of a good resource that explains the basic principles in physical terms?  I know that the interior dynamo is a very complex phenomenon, but I'd like something that gives a good physical picture of how the electromagnetic and fluid dynamical phenomena interact, rather than diving straight into partial differential equations. Using Illustrator CS6 to create shape out of multiple paths I am very new to Illustrator, in fact I am only experimenting with it in an attempt to help my friend develop a logo for a new business venture. I managed to draw out the shell of the logo using the line tool and modifying the lines with the anchor tool to get the shape I wanted.  However, I would like to somehow take all of these paths and create one object, or symbol which I can then apply a drop shadow, fill and various other effects too. Currently when I try to do this I get unwanted effects...  I'd appreciate any help on the subject. ^@ spam in tty (but seems to be system-wide) I recently noticed some weird behavior on my GNU/Linux machine. I've been trying to narrow it down but I am unsure where to go next. My setup uses the following:   i3 gdm arch systemd grub2 pulseaudio 4.13.3-1-ARCH   Here is how it manifests iself:   when trying to resize panes in i3 (alt+right click and hold) the action will be randomly canceled (happens on key presses) alt+dragging windows meets a similar fate booting to tty periodically adds ^@s to the login prompt, I'd estimate them to be ~0.5s intervals in tty I can see those being passed to vim when I open the command bar (:) using a pager like less in tty mode and searching with / will result in  (?) being spammed htop's search (/) is instantly canceled by the periodic keypress  holding a key and trying to have it be repeated (such as holding space) will stop the other occurrences from happening (e.g. only one or two happen) any tooltip (such as chrome tooltips) get instantly closed because of key presses   Here is what I have tried:   tried switching DE temporarily (gnome, cinnamon, xfce) tried killing programs one by one until I was left with a tty session and essentially systemd services + pulseaudio tried booting on an arch livecd - problem does not manifest  overriding grub2 with init=/bin/bash drops me into a shell where the problem does not manifest I have tried pressing all keys to verify it wasn't a hardware problem, however previous steps point to it not being hardware I have made a diff of dmesg, from the init=/bin/bash and the regular boot - the main difference seems to be systemd being started in the regular boot Made a diff of lsmod from init=/bin/bash and regular boot, tried to rmmod modules that seemed related   My system is up to date, I'm not quite sure what other steps I can do to track this down. Any other debugging tips? Prove if $n$ has a primitive root, then it has exactly $\phi(\phi(n))$ of them Prove if $n$ has a primitive root, then it has exactly $\phi(\phi(n))$ of them.   Let $a$ be the primitive root then I know other primitive roots will be among $\{a,a^2,a^3 \cdots\cdots a^{\phi(n)} \}$ because any other number will be congruent modulo $n$ to one of these. Then I figured as the answer is $\phi(\phi(n))$ so for $a^k$ to be primitive root, $k$ must be coprime with $\phi(n)$.But I don't know the reason.Am I missing any silly point? I have recently submitted a paper to arXiv, but the arXiv moderators put my paper in status "on hold", because they have to make a decision on the right classification for the paper.   To those who have experienced a similar situation, how long did it take before they made a decision and published the paper on the arXiv? I just want to have an idea of how long I will have to wait. update line based on previous line At the start of A New Hope, when the corvette is being attacked, C3PO says something to the effect of:     There'll be no escape for the princess this time.   But when Luke stumbles onto Leia's message in R2D2, C3PO seems to not know who the figure in the message is:     I'm afraid I'm not quite sure, sir. I think she was a passenger on our last voyage. A person of some importance, sir - I believe.   Is C3PO feigning ignorance to protect her, or does he really not recognize her?