# Mathematics

## Tolerant Bundle of Ways and Gurevich’s Theorem for Tolerant Spaces

In the article the tolerant bundle of ways with tolerant collapsible bundle space is constructed and the tolerant analogue of the Gurevich’s the oremisproved by mean sofexact homotopic sequence and homology spectral sequence of tolerant bundles .

## Solution of Inverse Problem for the Diffusion Operator in a Symmetric Case

In the paper uniqueness of reconstruction of the diffusion operator by aspectrum is proved and sufﬁcient solvability conditions are provided.

## On Riesz Basises of Eigenfunctions of Integral Operators with Kernels Discontinuous on Broken Lines

For the integral operator, which kernel has jump discontinuities on the sides and diagonals of the four equal subsquares of the unit square 0 ≤ x, t ≤ 1, Riesz basisness of its eigen and associated functions is proved.

## On Spectral Property of Matrix Operators in Banach Space

The paper is covers to the investigation of spectral property of matrix operators in Banach space. One matrix operator obtained on linearization of a polynomial operator bundle is being searched resolution of identity for its spectral properties.

## On the Same Theorem on a Equiconvergence at the Whole Segment for the Functional Differential Operators

The equiconvergence of expansions in eigen- and adjoint functions of functional-differential operator with involution, containing the potentials, and simplest functional-differential operator at the whole segment of Fourier series is established.

## Absolute Convergence of Single and Double Fourier Series on Multiplicative Systems

Two-dimensional analogs of famous Zygmund and Szasz tests for absolute convergence of Fourier – Vilenkin series are established. Also it is proved that two-dimensional Szasz test is the best possible in the certain sense.

## Covariant and Contrvariant Homomorphisms of Games with Preference Relations

We consider Nash equilibrium and some its modiﬁcations as principles of optimality for games with preference relations. For description of the optimal solutions of games with preferencerelations covariantly and contrvariantly complete families of homomorphisms are introduced.

## Equilibrium in Games with Ordered Outcomes

We consider some conditions of existence of equilibrium points in mixed extensions of games with ordered outcomes.General methods for description of the set of equilibrium points and also for Nash equilibrium points are proposed.

## On Determinant Zeros of Boolean Matrices

The properties of exteriority and interiority of square matrices with elements from arbitrary Boolean algebra are studied in this paper. The exteriorandinte riorparts formade generatepart of a matrix with zerodeterminant. Itisshown, inparticular, that the setof exteriorparts is a normal setinthe Boolean algebraofall Booleans quarematrices and it is a lower semilattice. The set of interior parts is an upper semilattice. Moreover linear combinations and even polynomials of the interiorities also belong to it.

## Set-Theoretical Approach to the Structure of Concept Generators

Set-theoretical methods for the description of the structure of concept generators are developed.