Sorin G. Gal

Almost periodic fuzzy-number-valued functions

In this paper we develop a theory of almost periodic fuzzy functions, i.e. of the almost periodic functions of real variable and with values fuzzy real numbers, Although the class of fuzzy real numbers does not form a linear normed space, the majority of main properties of almost periodic functions with values in Banach spaces are extended to this case. Applications to fuzzy differential equations and to (fuzzy) dynamical systems are given.

Quadrature rules for integrals of fuzzy-number-valued functions

In this paper, we introduce some quadrature rules for the Henstock integral of fuzzy-number-valued mappings by giving error bounds for mappings of bounded variation and of Lipschitz type. We also consider generalizations of classical quadrature rules, such as midpoint-type, trapezoidal and three-point-type quadrature. Finally, we study δ-fine quadrature rules and we present some numerical applications.

Approximation by complex genuine Durrmeyer type polynomials in compact disks

Abstract In this paper, the order of simultaneous approximation and Voronovskaja kind results with quantitative estimate for the complex genuine Durrmeyer polynomials attached to analytic functions on compact disks are obtained. In this way, we put in evidence the overconvergence phenomenon for the genuine Durrmeyer polynomials, namely the extensions of the approximation properties (with quantitative estimates) from real intervals to compact disks in the complex plane.

Two-dimensional geometries, topologies, trigonometries and physics generated by complex-type numbers

In the recent monograph [8], G.L. Naber provides an interesting introduction to the special theory of relativity that is mathematically rigorous and yet spells out in considerable detail the physical significance of the mathematics.His mathematical model is based on a special indefinite inner product of index one and its associated group of orthogonal transformations (the Lorentz group). Also, in the same monograph, the Hawking, King and McCarthy’s topology [6] is presented. This topology is physically well motivated and has the remarkable property that its homeomorphism group is essentially just the Lorentz group.Starting from the remark that the inner product and the topology above can be generated by the so-called hyperbolic complex numbers, in this paper we introduce and study two-dimensional geometries and physics generated in a similar manner, by the more general so-called complex-type numbers, i.e. of the typez=x+qy,q ∉ ℝ, whereq2=A+q(2B), A,A, B ∉ ℝ fixed.

Approximation and Shape Preserving Properties of the Bernstein Operator of Max-Product Kind

Starting from the study of the Shepard nonlinear operator of max-prod type by Bede et al. (2006, 2008), in the book by Gal (2008), Open Problem 5.5.4, pages 324–326, the Bernstein max-prod-type operator is introduced and the question of the approximation order by this operator is raised. In recent paper, Bede and Gal by using a very complicated method to this open question an answer is given by obtaining an upper estimate of the approximation error of the form 𝐶𝜔1√(𝑓;1/𝑛) (with an unexplicit absolute constant 𝐶g0) and the question of improving the order of approximation 𝜔1√(𝑓;1/𝑛) is raised. The first aim of this note is to obtain this order of approximation but by a simpler method, which in addition presents, at least, two advantages: it produces an explicit constant in front of 𝜔1√(𝑓;1/𝑛) and it can easily be extended to other max-prod operators of Bernstein type. However, for subclasses of functions 𝑓 including, for example, that of concave functions, we find the order of approximation 𝜔1(𝑓;1/𝑛), which for many functions 𝑓 is essentially better than the order of approximation obtained by the linear Bernstein operators. Finally, some shape-preserving properties are obtained.

n-Dimensional hyperbolic complex numbers

Abstract and IntroductionDirect product rings have received relatively little attention, perhaps because they are sometimes labeled “trivial” [8, p.6]. Nevertheless, the 2-dimensional direct product ring of the reals, when expressed in the “hyperbolic basis”, is analogous in many ways to the system of complex numbers and also has a physical interpretation. This prompted an exploratory foray into the world ofn-dimensional direct product rings of the reals to see how much can be extended from the 2-dimensional case (see, e.g. [3,4,5]). Section 1 provides algebraic notation, up to the point of defining polar coordinates. Section 2 uses analysis to explore differentiability and conformality.

Approximation and geometric properties of complex Favard-Szász-Mirakjan operators in compact disks

In this paper, first we obtain quantitative estimates of the convergence and of the Voronovskajas theorem in compact disks, for complex Favard-Szasz-Mirakjan operators attached to analytic functions satisfying some suitable exponential-type growth condition. Then, we prove that beginning with an index, these operators preserve the starlikeness, convexity and spirallikeness in the unit disk.

On the defect of additivity of fuzzy measures

It is well-known that fuzzy measures are non-additive. In this paper we introduce the concept of defect which gives us the degree of non-additivity of fuzzy measures. For large classes of fuzzy measures this defect is calculated and estimated. Three kinds of applications are given: to the approximative calculation of some fuzzy integrals, to the best approximation of a fuzzy measure by classical measures and to the introduction of an invariant at translations distance on the set of fuzzy measures.

Approximation and Shape Preserving Properties of the Nonlinear Meyer–König and Zeller Operator of Max-Product Kind

Starting from the study of the Shepard nonlinear operator of max-prod type in [2, 3; 6, Open Problem 5.5.4], the Meyer–König and Zeller max-product type operator is introduced and the question of the approximation order by this operator is raised. The first aim of this article is to obtain the order of pointwise approximation for these operators. Also, we prove by a counterexample that in some sense, in general this type of order of approximation with respect to ω1(f; ·) cannot be improved. However, for some subclasses of functions, including for example the continuous nondecreasing concave functions, the essentially better order (of uniform approximation) ω1(f; 1/n) is obtained. Several shape preserving properties are obtained including the preservation of quasi-convexity.

Saturation and inverse results for the Bernstein max-product operator

In this paper we obtain the saturation order and a local inverse result in approximation by the Bernstein max-product operator.