Complex Shepard Operators and Their Summability

In this paper, we construct the complex Shepard operators to approximate continuous and complex-valued functions on the unit square. We also examine the effects of regular summability methods on the approximation by these operators. Some applications verifying our results are provided. To illustrate the approximation theorems graphically we consider the real or imaginary part of the complex function being approximated and also use the contour lines of the modulus of the function.

Novel Construction of Copulas Based on ( α , β ) Transformation for Fuzzy Random Variables

The paper introduces a method for the construction of bivariate copulas with the usage of specific values of the parameters α and β ( α , β transformation) and the parameters κ and λ in their domain. The produced bivariate copulas are defined in four subrectangles of the unit square. The bounds of the produced copulas are investigated, while a novel construction method for fuzzy copulas is introduced, with the usage of the produced copulas via α , β transformation in four subrectangles of the unit square. Following this construction procedure, the production of an infinite number of copulas and fuzzy copulas could be possibly achieved. Some applications of the proposed methods are presented.

Intersection points of planar curves can be computed

Consider two paths ϕ , ψ : [ 0 ; 1 ] → [ 0 ; 1 ] 2 in the unit square such that ϕ ( 0 ) = ( 0 , 0 ), ϕ ( 1 ) = ( 1 , 1 ), ψ ( 0 ) = ( 0 , 1 ) and ψ ( 1 ) = ( 1 , 0 ). By continuity of ϕ and ψ there is a point of intersection. We prove that from ϕ and ψ we can compute closed intervals S ϕ , S ψ ⊆ [ 0 ; 1 ] such that ϕ ( S ϕ ) = ψ ( S ψ ).

A non-Borel special alpha-limit set in the square

We consider the complexity of special $\alpha $ -limit sets, a kind of backward limit set for non-invertible dynamical systems. We show that these sets are always analytic, but not necessarily Borel, even in the case of a surjective map on the unit square. This answers a question posed by Kolyada, Misiurewicz, and Snoha.

Linear spaces of games on the unit square with pure equilibrium points

The set of all linear spaces of continuous two-person zero-sum games on the unit square with pure equilibrium points is considered. It is shown that the set contains maximal linearspaces of any finite dimension greater than three.

NOTES ON THE K-RATIONAL DISTANCE PROBLEM

Let K be an algebraic number field. We investigate the K-rational distance problem and prove that there are infinitely many nonisomorphic cubic number fields and a number field of degree n for every $n\geq 2$ in which there is a point in the plane of a unit square at K-rational distances from the four vertices of the square.

Breaking bivariate records

We establish a fundamental property of bivariate Pareto records for independent observations uniformly distributed in the unit square. We prove that the asymptotic conditional distribution of the number of records broken by an observation given that the observation sets a record is Geometric with parameter 1/2.