Vol 25, No 129 (2020)
Articles
On connection between continuous and discontinuous neural field models with microstructure: II. Radially symmetric stationary solutions in 2D (“bumps”)
Abstract
We suggest a method allowing to investigate existence and the measure of proximity between the stationary solutions to continuous and discontinuous neural fields with microstructure. The present part involves results on proximity of the stationary solutions to specific homogenized neural field equations with continuous and discontinuous activation functions. The results of numerical investigation of radially symmetric stationary solutions (bumps) to the neural field with a discontinuous activation function and a given microstructure are presented.
Russian Universities Reports. Mathematics. 2020;25(129):6-17
6-17
On coincidence points of mappings in generalized metric spaces
Abstract
Let X be a space with ∞ -metric ρ (a metric with possibly infinite value) and Y a space with ∞ -distance d satisfying the identity axiom. We consider the problem of coincidence point for mappings F,G:X→Y , i.e. the problem of existence of a solution for the equation F(x)=G(x) . We provide conditions of the existence of coincidence points in terms of a covering set for the mapping F and a Lipschitz set for the mapping G in the space X×Y . An α -covering set (α>0) of the mapping F is a set of (x,y) such that ∃u ∈X F(u)=y, ρ(x,u)≤ α -1 d(F(x),y), ρ(x,u)<∞ , and a β - Lipschitz set (β≥0) for the mapping G is a set of (x,y) such that ∀u ∈X G(u)=y ⇒d(y,G(x))≤βρ(u,x) . The new results are compared with the known theorems about coincidence points.
Russian Universities Reports. Mathematics. 2020;25(129):18-24
18-24
On the study of the spectral properties of differential operators with a smooth weight function
Abstract
In this paper we study the spectral properties of a third-order differential operator with a summable potential with a smooth weight function. The boundary conditions are separated. The method of studying differential operators with summable potential is a development of the method of studying operators with piecewise smooth coefficients. Boundary value problems of this kind arise in the study of vibrations of rods, beams and bridges composed of materials of different densities. The differential equation defining the differential operator is reduced to the solution of the Volterra integral equation by means of the method of variation of constants. The solution of the integral equation is found by the method of successive Picard approximations. Using the study of an integral equation, we obtained asymptotic formulas and estimates for the solutions of a differential equation defining a differential operator. For large values of the spectral parameter, the asymptotics of solutions of the differential equation that defines the differential operator is derived. Asymptotic estimates of solutions of a differential equation are obtained in the same way as asymptotic estimates of solutions of a differential operator with smooth coefficients. The study of boundary conditions leads to the study of the roots of the function, presented in the form of a third-order determinant. To get the roots of this function, the indicator diagram wasstudied. The roots of this equation are in three sectors of an infinitely small size, given by the indicator diagram. The article studies the behavior of the roots of this equation in each of the sectors of the indicator diagram. The asymptotics of the eigenvalues of the differential operator under study is calculated. The formulas found for the asymptotics of eigenvalues allow us to study the spectral properties of the eigenfunctions of the differential operator under study.
Russian Universities Reports. Mathematics. 2020;25(129):25-47
25-47
Asymptotic solution of the Cauchy problem for the first-order equation with perturbed Fredholm operator
Abstract
We consider the Cauchy problem for a first-order differentialequation in a Banach space. The equation contains a small parameter in the highest derivative and a Fredholm operator perturbed by an operator addition on the right-hand side. Systems with small parameter in the highest derivative describe the motion of a viscous flow, the behavior of thin and flexible plates and shells, the process of a supersonic viscous gas flow around a blunt body, etc. The presence of a boundary layer phenomenon is revealed; in this case, even a small additive has a strong influence on the behavior of the solution. Asymptotic expansion of the solution in powers of small parameter is constructed by means of the Vasil’yeva- Vishik-Lyusternik method. Asymptotic property of the expansion is proved. To construct the regular part of the expansion, the equation decomposition method is used. It is consisted in a step-by-step transition to similar problems of decreasing dimensions.
Russian Universities Reports. Mathematics. 2020;25(129):48-56
48-56
About unbounded complex operators
Abstract
The concept of an unbounded complex operator as an operator acting in the pull-back of a Banach space is introduced. It is proved that each such operator is linear. Linear operations of addition and multiplication by a number and also the operation of multiplication are determined on the set of unbounded complex operators. The conditions for commutability of operators from this set are indicated. The product of complex conjugate operators and the properties of the conjugation operation are considered. Invertibility questions are studied: two contractions of an unbounded complex operator that have an inverse operator are proposed, and an explicit form of the inverse operator is found for one of these restrictions. It is noted that unbounded complex operators can find application in the study of a linear homogeneous differential equation with constant unbounded operator coefficients in a Banach space.
Russian Universities Reports. Mathematics. 2020;25(129):57-67
57-67
Maximal linked systems and ultrafilters: main representations and topological properties
Abstract
Questions connected with representation of the ultrafilter (UF) set for widely understood measurable space are investigated; this set is considered as a subspace of bitopological space of maximal linked systems (MLS) under equipment with topologies of Wallman and Stone types (measurable structure is defined as a π -system with “zero” and “unit”). Analogous representations connected with generalized variant of cohesion is considered also; in this variant, for corresponding set family, it is postulated the nonemptyness of intersection for finite subfamilies with power not exceeding given. Conditions of identification of UF and MLS (in the above-mentioned generalized sense) are investigated. Constructions reducing to bitopological spaces with points in the form of MLS and n -supercompactness property generalizing the “usual” supercompactness are considered. Finally, some characteristic properties of MLS and their corollaries connected with the MLS contraction to a smaller π -system are being studied. The case of algebras of sets is selected separately.
Russian Universities Reports. Mathematics. 2020;25(129):68-84
68-84
On the uniqueness of solution to the inverse problem of the atmospheric electricity
Abstract
We investigate the inverse problem of determination of two unknown numerical parameters occuring linearly and nonlinearly in the higher coefficient of a linear second order elliptic equation of the diffusion-reaction type in a domain Ω diffeomorphic to a ball layer under special boundary conditions by observation in neighborhoods of the correspondent amount of points. For an analogous inverse problem under Dirichlet boundary conditions, sufficient conditions of solution uniqueness was obtained by the author formerly, but they had an abstract character and so were inconvenient for practical usage. In the paper, these conditions are extended to the case of different boundary conditions and rendered concrete for the case of the exponential type higher coefficient. The inverse problem investigated in the paper refers to research of electric processes in the Earth atmosphere in the frame of global electric circuit in the stationary approximation and arises from needs of recovering the unknown higher coefficient of the equation on the base of observation data obtained from two local transmitters.
Russian Universities Reports. Mathematics. 2020;25(129):85-99
85-99
In memory of Professor Alexander Ivanovich Bulgakov
Russian Universities Reports. Mathematics. 2020;25(129):100-102
100-102