Sovremennaâ matematika. Fundamentalʹnye napravleniâ

In a separable Hilbert space, for an abstract linear parabolic equation with a weighted integral condition of a special type in time on the solution, the existence and uniqueness of a weak solution are proved. For this, the problem is solved approximately by the semidiscrete Galerkin method. A priori estimates are established for a sequence of approximate solutions, after which it is proved that the weak limit of this sequence is the exact solution of the original problem.

The problem of perturbation of the spectrum of a linear operator by a linear operator is solved thanks to the introduced concepts of holomorphic families of operators of type (A) and in the sense of Kato. The Rayleigh-Schr¨odinger series constructed in this case already converged in the usual sense, and not asymptotically. In this paper, conditions for holomorphy with respect to a small parameter of eigenpairs are found in the situation when a linear operator is perturbed by a nonlinear operator generated by a product in a Banach algebra.

Singular pseudodifferential operators created on the base of the mixed Fourier–Bessel transform are usually called Kipriyanov singular pseudodifferential operators (SPDO). The paper provides an overview of three types of such operators. The Kipriyanov SPDOs are adapted to work with singular Bessel operators \(B_{\gamma_i}=\dfrac{\partial^2}{\partial x_i^2}+\dfrac{\gamma_i}{x_i}~\dfrac{\partial}{\partial x_i},\) \(\gamma_i>-1.\) The main attention in our work is paid to two modifications that arose on the base of the “even \(\mathbb{J}\)-Bessel transforms” (i.e., for \(\gamma\in(-1,0)\)) and the “even-odd \(\mathbb{J}\)-Bessel–Kipriyanov–Katrakhov transforms”. The latter are introduced to study differential equations with singular differential operators \(\dfrac{\partial}{\partial x_i}B_{\gamma_i}\) with a negative parameter of the Bessel operator \(\gamma_i\in(-1,0).\)

We investigate a model that associates random walks in a finite-dimensional Euclidean coordinate space of a classical system with random quantum walks, i.e. random transformations of the set of states of a quantum system arising from quantization of a classical system. As is known, the convolution semigroup of Gaussian measures on a coordinate space admits a representation by a semigroup of self-adjoint contractions in the space of square-integrable functions described by the heat equation. We obtain a representation of the convolution semigroup of Gaussian measures on a coordinate space by a quantum dynamic semigroup in the space of nuclear operators. We give a description of the quantum dynamic semigroup by solutions of the Cauchy problem for a degenerate diffusion equation. We establish the generalized convergence in distribution of a sequence of quantum random walks to an operator-valued random process with values in the Abelian algebra of shift operators by a vector with a normal distribution.

The converter of continual systems of relays (also known as the Preisach converter) is a wellknown model applicable to describe the hysteresis relationships of a wide range. This article provides a review of works devoted to the study of systems from various subject areas (physics, economics, biology), where the continual system of relays plays a key role in the formalization of hysteresis dependencies. The first section of the work is devoted to a description of the input-output correspondences of the classical converter of continual systems of relays, its main properties are established, methods for constructing the output using the formalism of the demagnetization function are described, and a generalization of the classical converter of continual system of relays to the case of vector input-output correspondences is given. Applications of the Preisach model, classified according to various natural science fields, are given in the second section. Various generalizations of the model as applied to systems containing ferromagnetic and ferroelectric materials are described there. The main attention was paid to experimental works, where the model of continual system of relays was used to analytically describe the dependences observed in experiments. Special attention in the review is paid to technical applications of the model such as energy storage devices, systems using the piezoelectric effect, models of systems with long-term memory. The review also presents the results of using the converter of continual systems of relays in biology and medicine, as well as in economics. The third part of the review describes the properties of the converter of continual system of relays in terms of its response to stochastic external influences and provides a generalization of the converter model to the case of stochasticity of the threshold numbers of its elementary components. In addition, the review contains fresh results in the field of dynamics of systems with a converter of continual system of relays: a method for identifying dynamic modes is given, based on a modification of Benettin’s algorithm for calculating Lyapunov exponents in systems with nonsmooth multivalued characteristics.