As a generalization of the famous four square theorem of Lagrange, Ramanujan and Willerding found all positive definite integral quaternary quadratic forms that represent all positive integers. In this paper, we find all positive definite integral quinary quadratic forms that represent all positive definite integral binary quadratic forms. We also discuss recent results on positive definite int...

In this work, we consider rational ordinary differential equations dy/dx = Q(x, y)/P (x, y), with Q(x, y) and P (x, y) coprime polynomials with real coefficients. We give a method to construct equations of this type for which a first integral can be expressed from two independent solutions of a second–order homogeneous linear differential equation. This first integral is, in general, given by a...

In computer aided geometric design and computer graphics, high quality fair surfaces with G2 smoothness are sometimes required. In this paper we derive a general sixth order geometric partial differential equation from minimizing a curvature integral functional. The obtained equation is used to solve several surface modeling problems such as free-form surface design, surface blending and N-side...

planar d-bar integral equation is one of the inverse scattering solution methods for complex problems including inverse conductivity considered in applications such as electrical impedance tomography (eit). recently two different methodologies are considered for the numerical solution of d-bar integrals equation, namely product integrals and multigrid. the first one involves high computational ...

In this paper, we present a method for solving the rst kind Abel integral equation. In thismethod, the rst kind Abel integral equation is transformed to the second kind Volterraintegral equation with a continuous kernel and a smooth deriving term expressed by weaklysingular integrals. By using Sidi's sinm - transformation and modied Navot-Simpson'sintegration rule, an algorithm for solving this...

in this paper, a numerical efficient method based on two-dimensional block-pulse functions (bpfs) is proposed to approximate a solution of the two-dimensional linear stochastic volterra-fredholm integral equation. finally the accuracy of this method will be shown by an example.