We consider the discretization in time of a parabolic equation, using a representation of the solution as an integral along a smooth curve in the complex left half plane. The integral is then evaluated to high accuracy by a quadrature rule. This reduces the problem to a finite set of elliptic equations, which may be solved in parallel. The procedure is combined with finite element discretizatio...

We show that the abc-conjecture implies that few quadratic twists of a given hyperelliptic curve have any non-trivial rational or integral points; and indicate how these considerations dovetail with other predictions.

We study the integral expression of a knot invariant obtained as the second coefficient in the perturbative expansion of Witten's Chern-Simons path integral associated with a knot. One of the integrals involved turns out to be a generalization of the classical Crofton integral on convex plane curves, and it is related with the invariants of generic plane curves recently defined by Arnold, with ...

A new proof is given for the explicit formulae for the non archime dean canonical height on an elliptic curve This arises as a direct calculation of the Haar integral in the elliptic Jensen formula

It is frequently important to approximate a rational Bézier curve by an integral, i.e., polynomial one. This need will arise when a rational Bézier curve is produced in one CAD system and is to be imported into another system, which can only handle polynomial curves. The objective of this paper is to present an algorithm to approximate rational Bézier curves with polynomial curves of higher deg...

Fix integers d, g, r such that r ≥ 3, g > 0 and d ≥ g + r. Here we prove the existence of an integral non-special curve C in an r-dimensional projective space such that deg(C) = d, pa(C) = g, C has exactly one node and C has maximal rank (i.e. it has the expected postulation), i.e., the general non-special embedding of a general curve with a single node has maximal rank. M.S.C. 2010: 14H51, 4N05.

In this paper, we study Riemann boundary value problems on the Curve of Parabola. We characterized functions which are intergrable also asymptotic behaviors Cauchy-type integral and Cauchy principal Parabola at infinity. At end, discuss for sectionally holomorphic with as their jump curve obtain explicit form.

We implement in Matlab a Gauss-like cubature formula on bivariate domains whose boundary is a piecewise smooth Jordan curve (curvilinear polygons). The key tools are Green’s integral formula, together with the recent software package chebfun to approximate the boundary curve close to machine precision by piecewise Chebyshev interpolation. Several tests are presented, including some comparisons ...

We study the A-hypergeometric system associated with a monomial curve XA. We explicitly construct all rational solutions and show that XA is arithmetically Cohen-Macaulay if for all values of the exponents, the holonomic rank of the system equals the degree d of the curve. In the case of integral exponents we construct d independent solutions in terms of the roots of the generic univariate poly...

There is a beautiful theory of integral closure of ideals in regular local rings of dimension two, due to Zariski, several aspects of which were later extended to modules. Our goal is to study integral closures of modules over normal domains by attaching divisors/determinantal ideals to them. They will be of two kinds: the ordinary Fitting ideal and its divisor, and another ‘determinantal’ idea...