Vol 225 (2023)
Статьи
Initial-value problem for an integro-differential equation with difference kernels and an inhomogeneity in the linear part
Abstract
A global theorem on the existence and uniqueness of a nonnegative solution of the initial-value problem for an integro-differential equation with difference kernels, power nonlinearity, and inhomogeneity in the linear part is proved by the method of weight metrics in the cone of the space of continuous functions. It is shown that the solution can be found by the method of successive approximations of the Picard type. An estimate of the rate of their convergence is obtained.
3-13
Application of the method of similar operators to some classes of difference operators
Abstract
Abstract. Two second-order difference operators defined by their infinite matrices are considered: an operator with an ordinary potential and an operator with an involutive potential. The spectral properties of these operators under various conditions were performed by the method of similar operators. Results concerning the asymptotics of the eigenvalues in the case of a potential with involution are obtained.
14-27
Stability criteria for systems of ordinary differential equations
Abstract
Abstract. In this paper, we present criteria of stability in the Lyapunov sense for systems of ordinary differential equations based on transformations of difference schemes. The purpose of the transformations is to obtain the dependence of the magnitude of the perturbation of the solution at an arbitrary point in time on the perturbation of the initial data.
28-37
Kac–Siegert formula for oscillatory random processes
Abstract
Abstract. A general scheme for calculating the characteristic functions of random variables represented by quadratic functionals of the trajectories of elementary Gaussian processes based on the Feynman—Kac method is described. This scheme is applied to the oscillatory random process , . The characteristic function of the random variable of its random trajectories is calculated.
38-58
Quasi-nonmonodromic systems of first-order differential equations with a parameter
Abstract
Введено понятие квазибезмонодромной особой точки системы дифференциальных уравнений первого порядка с параметром и аналитическими на комплексной плоскости коэффициентами, как такой особой точки, некоторая степень матрицы монодромии М которой при всех допустимых значениях параметра пропорциональна единичной матрице. При этом коэффициент пропорциональности может как зависеть, так и не зависеть от параметра. Для системы двух уравнений сформулированы условия на матрицу М, её след и определитель, необходимые и достаточные для того, чтобы особая точка системы была квазибезмонодромной. Приведены примеры систем двух уравнений с такими особыми точками, включая точки ветвления одного из коэффициентов системы.
59-68
Extreme paths on graphs with simultaneously varying arc durations
Abstract
Abstract. In this paper, we propose an algorithm for finding the fastest path on a graph with two weights on each arc, namely, the times required to pass the arc before the beginning of rush hour and during rush hours, if the time of the beginning of rush hours is also indicated. The algorithm proposed can be considered a modification of the classical E. Dijkstra algorithm.
69-72
On main equation for inverse Sturm–Liouville operator with discontinuous coefficient
Abstract
In this work, a boundary-value problem for the Sturm–Liouville operator with discontinuous coefficient is examined. The main equation for the inverse problem for the boundary-value problem is obtained and the uniqueness of its solution is proved.
73-86
On the local extension of the group of parallel translations of four-dimensional space
Abstract
The problem of the search for all locally boundedly exactly doubly transitive extensions of the group of parallel translations of a four-dimensional space is reduced to the calculation of the Lie algebras of locally boundedly exactly doubly transitive extensions of the group of parallel translations. Some locally boundedly exactly doubly transitive transformation Lie groups with decomposable Lie algebras are found.
87-107
On the search for a time-optimal boundary control using the method of moments for systems governed by the diffusion-wave equation
Abstract
For a system described by a one-dimensional, inhomogeneous diffusion-wave equation on a segment, two types of optimal boundary control problems are considered: the problem of finding a control with a minimum norm for a given control time and the problem of finding a control that brings the system to a given state in a minimum time under a given constraint on the norm of the control. Various ways of specifying conditions on the final state are considered. The finite-dimensional l-problem of moments is analyzed, to which the optimal control problem can be reduced. We show that under the conditions of well-posedness and solvability of this problem, the problem of finding a control with a minimum norm always has a solution, while the problem of finding a control with a minimum transition time may not have a solution.
108-114
On the decay rate of solutions to the stationary Schrödinger equation with a potential depending on one variable
Abstract
In 1982, E. M. Landis posed the problem of exact estimates for the exponential decay rate of solutions to the stationary Schrödinger equation. A few years later, this problem in its original formulation was solved by the Voronezh mathematician V. Z. Meshkov. He constructed an example of a solution that decreases superlinearly at infinity, which gives a negative answer to the original question in Landis’ problem. In this paper, we prove that for some potentials of a special form, nevertheless, the answer to the question in Landis’ problem may be positive. Some generalizations and modern results in this direction are also presented.
115-122
On an exact estimate of the number of real invariant lines of polynomial vector fields of degree n
Abstract
In this paper, we prove that a polynomial vector field of degree n that has two invariant sets, each of which consists of n-1 pairwise real invariant straight lines, has at most 2n+4 invariant straight lines, where is odd and n≥3.
123-133
Step scaling functions and the Chrestenson system
Abstract
A review of methods for constructing step scaling functions on the positive half-line associated with the Chrestenson functions is presented. The conditions under which such step functions generate orthogonal wavelets and tight frames are discussed. A detailed bibliography is provided.
134-149
Completeness criteria of an exponential system in geometric terms of breadth in the direction
Abstract
In this paper, we establish criterions for the completeness of an exponential system in spaces of functions that are continuous on a convex compact set and holomorphic in the interior of this compact set and in spaces of holomorphic functions in a convex domain in terms of the directional width of a compact set or a domain. The main results are formulated exclusively in terms of the relationship between the breadth in the direction or the diameter of a compact set or a domain, on the one hand, and the so-called logarithmic submeasures or logarithmic block densities of the distribution of exponential system indicators, on the other hand.
150-159