Csaba Vincze

Reconstruction of hv-Convex Sets by Their Coordinate X-Ray Functions

Gardner and Kiderlen (Adv. Math. 214:323–343, 2007) presented an algorithm for reconstructing convex bodies from noisy X-ray measurements with a full proof of convergence in 2007. We would like to present some new steps into the direction of reconstructing not necessarily convex bodies by the help of the continuity properties of so-called generalized conic functions. Such a function measures the average taxicab distance of the points from a given compact set

A Robbins–Monro-type algorithm for computing global minimizer of generalized conic functions

K\subset \mathbb {R}^{N}

On a new geometrical derivation of two-dimensional Finsler manifolds with constant main scalar

by integration. The basic result (Vincze and Nagy in J. Approx. Theory 164:371–390, 2012) is that the generalized conic function associated to a compact planar set determines the coordinate X-rays and vice versa. Vincze and Nagy (Submitted to Aequationes Math., 2014) proved continuity properties of the mapping which sends connected compact hv-convex sets having the same axis parallel bounding box to the associated generalized conic functions. We use these results to present an algorithm for the reconstruction of compact connected hv-convex planar bodies given by their coordinate X-rays. The basic method is varied with the quota system scheme. Greedy and anti-greedy versions are also presented with examples.

An observation on Asanov's Unicorn metrics

We generalize the notion and some properties of the conic function introduced by Vincze and Nagy in 2012. We provide a stochastic algorithm for computing the global minimizer of generalized conic functions, we prove almost sure and -convergence of this algorithm.

Algebraic dependency of roots of multivariate polynomials and its applications to linear functional equations

In this paper we investigate the problem (it what kind of (two-dimensional) Finsler manifolds have a conformal change leaving the mixed curvature of the Berwald connection invariant?) We establish a differential equation for such Finslerian energy functions and present the solutions under some simplification. As we shall see they are essentially the same as the singular Finsler metrics with constant main scalar determined by L. Berwald.

On the characteristic polynomials of linear functional equations

Finsleroid-Finsler metrics form an important class of singular (y-local) Finslerian metrics. They were introduced by G. S. Asanov in 2006. As a special case Asanov produced examples of Landsberg spaces of dimension at least three that are not of Berwald type. These are called Unicorns [5]. The existence of regular (y - global) Landsberg metrics that are not of Berwald type is an open problem up to this day. In this paper we prove that Asanovs Unicorns belong to the class of generalized Berwald manifolds. More precisely we prove the following theorems: a Finsleroid-Finsler space is a generalized Berwald space if and only if the Finsleroid charge is constant. Especially a Finsleroid-Finsler space is a Landsberg space if and only if it is a generalized Berwald manifold with a semi-symmetric compatible linear connection.

On the curvature of the indicatrix surface in three-dimensional Minkowski spaces

In this paper we prove that a multivariate polynomial has algebraically dependent roots if and only if the coefficients are algebraic numbers up to a common proportional term; for the problem see section 4.4 in Varga-Vincze (On the characteristic polynomials of linear functional equations, Period Math Hungar 71(2):250–260, 2015). The case of univariate polynomials belongs to basic algebra. As far as we know the case of multivariate polynomials is not discussed in the literature. As an application we formulate a sufficient and necessary condition for the existence of non-trivial solutions of special types of linear functional equations. The criteria is based only on the algebraic properties of the parameters in the functional equation.

On functional equations characterizing derivations: methods and examples

The solutions of a linear functional equation are typically generalized polynomials. The existence of their non-trivial monomial terms strongly depends on the algebraic properties of some related families of parameters. In extremal cases (the parameters are algebraic numbers or the parameters form an algebraically independent system) we have elegant methods to decide the existence of non-trivial solutions. In this paper we are going to extend and unify the treatment of the existence problem by introducing the characteristic polynomials of a linear functional equation such that the algebraic properties of the roots allows us to conclude the existence of non-trivial solutions.

On spectral synthesis in varieties containing the solutions of inhomogeneous linear functional equations

As it is well-known a Minkowski space is a finite dimensional real vector space equipped with a Minkowski functional F. If the square of the Minkowski functional is quadratic then we have an Euclidean space and the indicatrix hypersurface S:= F-1 (1) has constant 1 curvature. In his classical paper [1] F. Brickell proved that the converse is also true provided that the indicatrix is symmetric with respect to the origin. M. Ji and Z. Shen investigated the (sectional) curvature of Randers indicatrices and it always turned out greater than zero and less or equal than 1; see [3]. In this note we give a general lower and upper bound for the curvature in terms of the norm of the Cartan tensor.

On a special class of generalized conics with infinitely many focal points

Functional equations satisfied by additive functions have a special interest not only in the theory of functional equations, but also in the theory of (commutative) algebra because the fundamental notions such as derivations and automorphisms are additive functions satisfying some further functional equations as well. It is an important question that how these morphisms can be characterized among additive mappings in general. The paper contains some multivariate characterizations of higher order derivations. The univariate characterizations are given as consequences by the diagonalization of the multivariate formulas. This method allows us to refine the process of computing the solutions of univariate functional equations of the form