# Mathematics

## Substantiation of Fourier Method in Mixed Problem with Involution

In this paper the mixed problem for the first order differential equation with involution is investigated. Using the received specified asymptotic formulas for eigenvalues and eigenfunctions of the corresponding spectral problem, the application of the Fourier method is substantiated. We used techniques, which allow to transform a series representing the formal solution on Fourier method, and to prove the possibility of its term by term differentiation. At the same time on the initial problem data minimum requirements are imposed.

## Applicathion the Pontryagin‘s Maximum Principle to Optimal Economics Models

In this paper three models of firm are considerd as the discrete optimal control problems. The algorithm for solution is based on Pontryagin‘s Maximum Principle. The paper contains numerical examples.

## On Congruences of Partial n-ary Groupoids

Ri-congruence is defined for partial n-ary groupoids as a generalization of right congruence of a full binary groupoid. It is proved that for any i the Ri-congruences of a partial n-ary groupoid G form a lattice, where the congruence lattice of G is not necessary a sublattice. An example is given, demonstrating that the congruence lattice of a partial n-ary groupoid is not always a sublattice of the equivalence relations lattice of G. The partial n-ary groupoids G are characterized such that for some i, all the equivalence relations on G are its Ri-congruences.

## Cramer’s Formulas for Systems of Linear Equations and Inequalities Over Boolean Algebra

There obtained analogies of classical Cramer’s formulas for systems of linear equations and inequalities with square matrix of coefficients from Boolean algebra.

## Polynomials, Orthogonal on Non-Uniform Grids

Asymptotic properties of polynomials pˆn(t), orthogonal with weight ∆tj on any finite set of N points from segment [−1, 1] are investigated. Namely an asymptotic formula is proved in which asymptotic behaviour of these polynomials as n tends to infinity together with N is closely related to asymptotic behaviour of the Lasiandra polynomials. Furthermore are investigated the approximating properties of the sums by Fourier on these polynomials..

## Generalization of Method A. A. Dorodnicyn Close Calculation of Eigenvalues and Eigenvectors of Symmetric Matrices on Case of Self-Conjugate Discrete Operators

Let the discrete self-conjugate operator A operates in separable Hilbert space H and has the kernel resolvent with simple spectrum. Self-conjugate and limited operator B operates also in H. Then it is possible to find such number ε > 0, that eigenvalues and eigenfunctions of the perturbation operatorA+εB will be calculated on a method of Dorodnicyn.

## tolerant space, space of tolerant loops, homotopy groups of tolerant space.

In the article is proved the theorem about isomorphism between homotopy groups of initial tolerant space and homotopy groups with descremented by one dimension of space of tolerant loops.

## The Classification of Complexes of Lines in Zeroth Order Frame in F ̄2 3 Space

Complexes of lines in hyperbolic type of biflag space introduced by the author are studied by the method of external Cartan forms. We prove that 5 non-special variants of complexes exist in mentioned space in zero order neighborhood. For every complex a first-order moving flag was drawn.

## Approximating Properties of Solutions of the Differential Equation with Integral Boundary Condition

With the use of the solution of the first-order differential equation the approximations to the continuous functions with integral boundary conditions are constructed.

## Hilbert Generalizations b-Bessel Systems

The notion of b-Bessel systems that generalizes the known classic notion of Bessel systems is introduced, the criteria of Bessel property of the systems are established. Some properties of the space of coefficients corresponding to the b-basis generalizing the classic notion of Schauder basis are studied.