I am a math undergrad, I know some basics of algebraic geometry and hence, elliptic curves. I was attending a talk today which was about Poncelet's theorem. In the end of the talk, the speaker discussed complex elliptic curves and wrote     $$\Bbb{C}/\Lambda \simeq \text{torus}$$ i.e. complex elliptic curves   are tori. Where $\Lambda$ is a lattice in $\Bbb{C}$.   Is there an intuitive explanation to this? I know some basics of quotient topology and elliptic curves.
In a conference on elliptic curves (an introduction to the subject), the speaker said that an elliptic curve (I.e. an equation of the form $y^2=x^3+ax+b $ where the RHS has distinct roots) is, in the complex space, a torus/Riemann surface of genus 1.  What is meant by that? Are we talking about a 2-dimensional manifold the 4D space?
Labeling linguistic examples with language information Video showing up as white space On  question, there is a YouTube video in , on the Android app the video shows up as a white space. There is no indication that it is a video.  Would it be possible to show the video in the app or link to the video on YouTube instead? A little bird told me that if $2^n+1$ is prime, then $n$ is a power of $2$. I tend not to trust talking birds, so I'm trying to verify that statement independently.  Suppose $n$ is not a power of $2$. Then $n = a \cdot 2^m$ for some $a$ not a power of $2$ and some integer $m$. This gives $2^n+1 = 2^{a \cdot 2^m}+1$. Now I suspect there's a way to factor that, but I don't see how. Can someone give me a hint? Solving a implicit, recursive function Assuming $F(0,0)=0$, can I solve $F(F(x,y),y)=0$ for $y=f(x)$ near $(0,0).$ Now this might be a very stupid question for many readers, but I am getting really confused.   I was reading about electricity and current, and I read this:  Suppose we connect a wire to the ends of a battery, and a bulb in the circuit, current flows and lights up the bulb. The battery is responsible for doing work on the positive charges to move them from negative terminal to positive terminal, that is moving against the field. Now, these energised positive charges move through the external circuit, and within the circuit they give some of their energy to light the bulb. I understood this very well.  Now, reading further, it was written that "Electrons are actually responsible for current flow". It was written that there is concentration of energised negative charges on the negative terminal, and these electrons then move through the external circuit from negative to positive terminal. This is really confusing.     So my question: What actually flows? The negative charge or the   positive charge?   Further reading, I also read that in semi-conductors, fluorescent light bulbs etc., both charges move through the circuit.     Now, one more doubt: If both charges move through the circuit in this   case, shouldn't the charges stop in the middle of the wire, because   opposite charges will attract each other, hence stopping their   movement? Modifying list while iterating Xmonad with Unity / HUD I want to use xmonad and still use the HUD from Unity.  Is this possible? Could Xmonad be using while using Unity?  is there a way to use the HUD inside of Xmonad?  I'm in 12.04, soon to be 12.10. Permutation with prime order, fixed point Let $M$ be a finite set and $\varphi$ a permutation of this set with prime order $q$. How do I show that $\varphi$ has a fixed point if $q$ is not a divisor of $|M|$? Can you upgrade the RAM of an Android device? 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They represent 4 variables describing an identical shoreline. From these 4 overlapping polylines, I would like to create a resulting polyline that has segments for each unique set of the 4 input lines.    My approach was to use an intersect. This, however, breaks down when I try to run it on 4 huge feature classes.  My question is whether there is a more efficient way to combine polyline attributes stored in separate layers. I thought a union would be good, but it only works on polygons. Spatial Join seems inefficient, as I would end up with many to many joins for a single segment. Turning the segment end points into a point layer, then splitting the line at those points?  I have access to arcpy, PostGIS, and shapely.  Has anyone had experience doing this before on large datasets? Is there a more efficient way to store these variables? Duplicate objects transformation applies to both objects I'm facing this frustrating "feature" that I can't seem to remedy; when I duplicate an object (and now, even when I create two completely new objects), their transformations/scaling is synchronized. This is extremely annoying when trying to use any sort of transformation keyboard shortcut.  It should be noted by going to the properties menu (N) and manually setting the properties works just fine.  For instance, here is just the camera selected (trust me, I made sure):   And when I try to transform by clicking+dragging on the X axis OR by hitting G->X, this happens:   They are not grouped (verified by going to the Groups list up in the Scene panel) and they do not have parents (verified by going to Object->Relations).  I have tried the unlink command as well, to no avail.  I have seriously run out of options; what did I change to have my G->X shortcuts transform multiple objects? It seems completely at random, too; certain objects seemed to be "linked" to other random objects, and this "linking" is guaranteed when duplicating even when I do a non-linking duplicate (Shift+D).  What gives? In  Deathly Hallows, when Harry, Ron, and Hermione are at No. 12, Grimmauld Place, Harry summons Kreacher from the kitchens at Hogwarts where the elf has been working on Harry’s orders given a year previously.  When the trio’s heist in the Ministry of Magic ends with Hermione inadvertently bringing Yaxley inside the perimeter of the Fidelius charm that protects Grimmauld Place, the trio leg it and dare not return there for fear that it’s now crawling with Death Eaters.  Before they leave for the Ministry in the morning, Kreacher is talking about making a stew or pot roast or something for dinner. 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We know that house-elves can go against their masters’ wishes, but this usually entails them punishing themselves rather brutally, and Kreacher does not seem to have been constantly injuring himself for the better part of a year at Hogwarts.  So when and how does Kreacher leave his master’s house without orders and return to Hogwarts? How do I install VMware player on a Mac os X? How to write special set notation by hand? In League of Legends, there are a number of effects which revive a champion upon death.  The item  is the most well-known and widely used of these.  I was wondering, however, whether its effect would come before or after other revive effects?  I have two examples.  First, Zilean's ultimate  will provide a champion with a Guardian Angel -like buff for a limited time.  If a champion has this buff in addition to a Guardian Angel, which effect will happen first?  Second, Anivia's passive  allows her to come back to life if her egg is not destroyed after she is killed.  If she has a Guardian Angel, will her passive be triggered before or after it? How does this compare to  and ?  It seems like in one case it is more beneficial to have Guardian Angel's precedence be greater, and in the other it is more beneficial to have it be less.  In the first scenario, if the Guardian Angel was triggered first, then Zilean would have a good chance of simply wasting his ultimate.  On the other hand, if it is triggered after these revival effects, then Anivia could not get the full benefit of her passive.  Is there a set order of precedence, or does each revival effect have their own unique "priority level" that is explicitly detailed for each ability? Continuous function whose right hand derivative equals 0 is constant? Norm is sup of inner products (proof). Let $V$ be a vector space with an inner product $\langle.,. \rangle$ and associated norm $|| . ||$ Then:    Could I have a proof of this fact? Prove that in any GCD domain every irreducible element is prime