The problem of the form of the 'arctic' curve of the six-vertex model with domain wall boundary conditions in its disordered regime is addressed. It is well-known that in the scaling limit the model exhibits phase-separation, with regions of order and disorder sharply separated by a smooth curve, called the arctic curve. To find this curve, we study a multiple integral representation for the em...

Let (M, g) be a time-oriented Lorentzian manifold and d the Lorentzian distance on M . The function τ(q) := supp<q d(p, q) is the cosmological time function of M , where as usual p < q means that p is in the causal past of q. This function is called regular iff τ(q) <∞ for all q and also τ → 0 along every past inextendible causal curve. If the cosmological time function τ of a spacetime (M, g) ...

An amicable pair for an elliptic curve E/Q is a pair of primes (p, q) of good reduction for E satisfying #Ẽp(Fp) = q and #Ẽq(Fq) = p. In this paper we study elliptic amicable pairs and analogously defined longer elliptic aliquot cycles. We show that there exist elliptic curves with arbitrarily long aliqout cycles, but that CM elliptic curves (with j 6= 0) have no aliqout cycles of length greate...

Everybody knows from school how to solve a quadratic equation of the form $x^2-px+q=0$ graphically. But this method can become tedious if several equations ought be solved, as for each pair $(p,q)$ new parabola has drawn. Stunningly, there is one single curve that used every via drawing tangent lines through given point curve. In article we derive in an elementary way and generalize it $x^n-p...

We consider the quotient of the Hermitian curve defined by the equation yq + y = xm over Fq2 where m > 2 is a divisor of q + 1. For 2 ≤ r ≤ q + 1, we determine the Weierstrass semigroup of any r-tuple of Fq2 rational points (P∞, P0b2 , . . . , P0br ) on this curve. Using these semigroups, we construct algebraic geometry codes with minimum distance exceeding the designed distance. In addition, w...

The paper formulates a precise relationship between the Tate-Shafarevich group of an elliptic curve $E$ over ${\mathbb Q}$ with quotient classgroup Q}(E[p])$ on which $Gal({\mathbb Q}(E[p]/{\mathbb Q}) = GL_2({\mathbb Z}/p)$ operates by its standard 2 dimensional representation Z}/p$. We establish such in most cases.