On a conjecture of Ma

In this paper, we prove a result concerning a conjecture of Ma from diophantine equations, which is connected to an open problem on abelian difference sets of multiplier ?1.     

On the convexity of images of nonlinear integral operators

We study continuous nonlinear Urysohn-type integral operators acting from the spaces of vector functions with integrable components to the space of continuous functions. We obtain conditions under which the images of sets defined by pointwise constraints have a convex closure under the action of these operators. The result is used to justify a method of constructive approximation of these images and to derive a necessary solvability condition for Urysohn-type integral equations. A numerical method for finding the residual of equations of this type on the sets under consideration is justified.     

Prime paucity for sums of two squares

Almost all solutions of the diophantine equations , with all x_j prime numbers, are diagonal ones.This is established here in quantitative form.     

Positive integer solutions of certain diophantine equations

In this study, the diophantine equations \(x^2 -32B_nxy-32y^2 =\pm 32^{r}\), \(x^4 -32B_nxy-32y^2 =\pm 32^{r}\) and \(x^2 -32B_nxy-32y^4 =\pm 32^{r}\) are considered and determined when these equations have positive integer solutions. Moreover, all positive integer solutions of these diophantine equations in terms of balancing and Lucas-balancing numbers are also found out.     

A separation theorem for nonlinear inverse images of convex sets

This paper offers first- and higher-order necessary conditions for the local disjointness of a finite system of sets that are nonlinear inverse images of convex sets. The proof is based on the characterizations of α-admissible and α-tangent variations to nonlinear inverse images of convex sets and a necessary condition for the local disjointness in terms of these variations. As an application, the results are used to obtain first- and higher-order necessary conditions of optimality in constrained optimization problems.     

On a Diophantine Equation Concerning the Number of Integer Points in Special Domains

In this paper a family of diophantine equations concerning the number of integer points in special domains is investigated. Some of the equations are completely solved.     

Estimation of fractal dimension and fractal curvatures from digital images

Most of the known methods for estimating the fractal dimension of fractal sets are based on the evaluation of a single geometric characteristic, e.g. the volume of its parallel sets. We propose a method involving the evaluation of several geometric characteristics, namely all the intrinsic volumes (i.e. volume, surface area, Euler characteristic, etc.) of the parallel sets of a fractal. Motivated by recent results on their limiting behavior, we use these functionals to estimate the fractal dimension of sets from digital images. Simultaneously, we also obtain estimates of the fractal curvatures of these sets, some fractal counterpart of intrinsic volumes, allowing a finer classification of fractal sets than by means of fractal dimension only. We show the consistency of our estimators and test them on some digital images of self-similar sets.     

Approximation of quadrilaterals by rational quadrilaterals in the plane

Many questions about triangles and quadrilaterals with rational sides, diagonals and areas can be reduced to solving certain diophantine equations. We look at a number of such questions including the question of approximating arbitrary triangles and quadrilaterals by those with rational sides, diagonals and areas. We transform these problems into questions on the existence of infinitely many rational solutions on a two parameter family of quartic curves. This is further transformed to a two parameter family of elliptic curves to deduce our main result concerning density of points on a line which are at a rational distance from three collinear points (Theorem 4). We deduce from this a new proof of density of rational quadrilaterals in the space of all quadrilaterals (Theorem 39). The other main result (Theorem 3) of this article is on the density of rational triangles which is related to analyzing rational points on the unit circle. Interestingly, this enables us to deduce that parallelograms with rational sides and area are dense in the class of all parallelograms. We also give a criterion for density of certain sets in topological spaces using local product structure and prove the density Theorem 6 in the appendix section. An application of this proves the density of rational points as stated in Theorem 31.     

On the algebraic points of a definable set

This paper studies diophantine properties of sets definable in an o-minimal structure over the real field. The main theorem of the author’s recent paper with A. J. Wilkie is refined, and used to deduce a strong result on the density of algebraic points of such sets.      

Global Classification of Two-Component Approximately Integrable Evolution Equations

We globally classify two-component evolution equations, with homogeneous diagonal linear part, admitting infinitely many approximate symmetries. Important ingredients are the symbolic calculus of Gel’fand and Dikiĭ, the Skolem–Mahler–Lech theorem, an algorithm of Smyth, and results on diophantine equations in roots of unity obtained by Beukers.      

Which sets are images of disk-algebra functions?

We give a characterization of those compact sets in the plane with finitely many holes that are images of disk-algebra functions. We also show that the image of the closed unit disk via a polynomial is, in general, not polynomially convex.     

On the global distance between two algebraic points on a curve

We prove diophantine inequalities involving various distances between two distinct algebraic points of an algebraic curve. These estimates may be viewed as extensions of classical Liouville's inequality. Our approach is based on a transcendental construction using algebraic functions. Next we apply our results to Hilbert's irreducibility Theorem and to some classes of diophantine equations in the circle of Runge's method.     

广义Lebesgue-Ramanujan-Nagell方程研究的新进展

广义Lebesgue-Ramanujan-Nagell方程是数论中一类重要的Diophantine方程.本文介绍了此类方程的近期结果和尚未解决的问题.     

Arithmetic of algebraic curves from diophantus to Poincaré

This article contains an analysis of the methods for solving diophantine equations of second and third degrees in two unknowns, from the most ancient times to Poincaré. Special attention is paid to the works of Fermat, Euler, and Jacobi.     

On the Frobenius problem for Beatty sequences

We study the solvability of linear diophantine equations in two variables within the context of Beatty sequences.     

整图的构造

In this paper, some new classes of integral graphs are given in two new ways. It is proved that the problem of finding such integral graphs is equivalent to the problem of solving diophantine equations. Some classes are infinite. The discovery of these classes is a new contribution to the search of such integral graphs.     

A complete diophantine characterization of the rational torsion of an elliptic curve

We give a complete characterization for the rational torsion of an elliptic curve in terms of the (non-)existence of integral solutions of a system of diophantine equations.     

Classification of critical sets and their images for quadratic maps of the plane

We provide a complete classification of the critical sets and their images for quadratic maps of the real plane. Critical sets are always conic sections, which provides a starting point for the classification. The generic cases, maps whose critical sets are either ellipses or hyperbolas, was published by Delgado et al. in 2013. This work completes the classification by including all the nongeneric cases: the empty set, a single point, a single line, a parabola, two parallel lines, two intersecting lines, or the whole plane. We describe all possible images for each critical set case and illustrate the geometry of representative maps for each case.     

On the rank of certain matrices

This note is an appendix to the paper “Osculating spaces and diophantine equations (with an Appendix by Pietro Corvaja and Umberto Zannier)” by M. Bolognesi and G. Pirola.     

Zero-divisors among finite structures of a fixed type

Conditions are established which guarantee that a finite structure be a zero-divisor relative to the class of all finite structures of its type. These conditions are concerned with reducts, disjointness conditions on ranges and/or projections of operations and/or relations, and diophantine equations induced by a relation. Presented by R. McKenzie.