Solving a second-order algebro-differential equation with respect to the derivative

We consider a second-order algebro-differential equation. Equations and systems of second-order differential equations describe the operation of an electronic triode circuit with feedback, rotation of a rigid body with a cavity, reading information from a disk, etc. The highest derivative is preceded by an irreversible operator. This is a Fredholm operator with index zero, kernel of arbitrary dimension, and Jordan chains of arbitrary lengths. Equations with irreversible operators at the highest derivative are called algebro-differential. In this case, the solution to the problem exists under certain conditions on the components of the desired function. To solve the equation with respect to the derivative, the method of cascade splitting of the equation is used, which consists in the stepwise splitting of the equation into equations in subspaces of decreasing dimensions. Cases of one-step and two-step splitting are considered. The splitting uses the result on the solution of a linear equation with Fredholm operator. In each case, the corresponding result is formulated as a theorem. To illustrate the result obtained in the case of one-step splitting, an illustrative example of the Cauchy problem is given.

Symbols in Berezin quantization for representation operators

The basic notion of the Berezin quantization on a manifold M is a correspondence which to an operator A from a class assigns the pair of functions F and F^♮ defined on M. These functions are called covariant and contravariant symbols of A. We are interested in homogeneous space M=G/H and classes of operators related to the representation theory. The most algebraic version of quantization — we call it the polynomial quantization — is obtained when operators belong to the algebra of operators corresponding in a representation T of G to elements X of the universal enveloping algebra Env g of the Lie algebra g of G. In this case symbols turn out to be polynomials on the Lie algebra g. In this paper we offer a new theme in the Berezin quantization on G/H: as an initial class of operators we take operators corresponding to elements of the group G itself in a representation T of this group. In the paper we consider two examples, here homogeneous spaces are para-Hermitian spaces of rank 1 and 2: a) G=SL(2;R), H — the subgroup of diagonal matrices, G/H — a hyperboloid of one sheet in R^3; b) G — the pseudoorthogonal group SO_0 (p; q), the subgroup H covers with finite multiplicity the group SO_0 (p-1,q -1)×SO_0 (1;1); the space G/H (a pseudo-Grassmann manifold) is an orbit in the Lie algebra g of the group G.

On ring solutions of neural field equations

The article is devoted to investigation of integro-differential equation with the Hammerstein integral operator of the following form: ∂_t u(t,x)=-τu(t,x,x_f )+∫_(R^2)▒〖ω(x-y)f(u(t,y) )dy, t≥0, x∈R^2 〗. The equation describes the dynamics of electrical potentials u(t,x) in a planar neural medium and has the name of neural field equation.We study ring solutions that are represented by stationary radially symmetric solutions corresponding to the active state of the neural medium in between two concentric circles and the rest state elsewhere in the neural field. We suggest conditions of existence of ring solutions as well as a method of their numerical approximation. The approach used relies on the replacement of the probabilistic neuronal activation function f that has sigmoidal shape by a Heaviside-type function. The theory is accompanied by an example illustrating the procedure of investigation of ring solutions of a neural field equation containing a typically used in the neuroscience community neuronal connectivity function that allows taking into account both excitatory and inhibitory interneuronal interactions. Similar to the case of bump solutions (i. e. stationary solutions of neural field equations, which correspond to the activated area in the neural field represented by the interior of some circle) at a high values of the neuronal activation threshold there coexist a broad ring and a narrow ring solutions that merge together at the critical value of the activation threshold, above which there are no ring solutions.

Stability of a weak solution for a hyperbolic system with distributed parameters on a graph

In the work, the stability conditions for a solution of an evolutionary hyperbolic system with distributed parameters on a graph describing the oscillating process of continuous medium in a spatial network are indicated. The hyperbolic system is considered in the weak formulation: a weak solution of the system is a summable function that satisfies the integral identity which determines the variational formulation for the initial-boundary value problem. The basic idea, that has determined the content of this work, is to present a weak solution in the form of a generalized Fourier series and continue with an analysis of the convergence of this series and the series obtained by its single termwise differentiation. The used approach is based on a priori estimates of a weak solution and the construction (by the Fayedo–Galerkin method with a special basis, the system of eigenfunctions of the elliptic operator of a hyperbolic equation) of a weakly compact family of approximate solutions in the selected state space. The obtained results underlie the analysis of optimal control problems of oscillations of netset-like industrial constructions which have interesting analogies with multi-phase problems of multidimensional hydrodynamics.

Maximal linked systems on families of measurable rectangles

Linked and maximal linked systems (MLS) on π -systems of measurable (in the wide sense) rectangles are considered (π-system is a family of sets closed with respect to finite intersections). Structures in the form of measurable rectangles are used in measure theory and probability theory and usually lead to semi-algebra of subsets of cartesian product. In the present article, sets-factors are supposed to be equipped with π-systems with “zero” and “unit”. This, in particular, can correspond to a standard measurable structure in the form of semialgebra, algebra, or σ-algebra of sets. In the general case, the family of measurable rectangles itself forms a π -system of set-product (the measurability is identified with belonging to a π - system) which allows to consider MLS on a given π -system (of measurable rectangles). The following principal property is established: for all considered variants of π -system of measurable rectangles, MLS on a product are exhausted by products of MLS on sets-factors. In addition, in the case of infinity product, along with traditional, the “box” variant allowing a natural analogy with the base of box topology is considered. For the case of product of two widely understood measurable spaces, one homeomorphism property concerning equipments by the Stone type topologies is established.

Optimization of discounted income for a structured population exposed to harvesting

A structured population the individuals of which are divided into n age or typical groups x_1,…,x_n. is considered. We assume that at any time moment k, k = 0,1,2… the size of the population x(k) is determined by the normal autonomous system of difference equations x(k+1)=F(x(k)), where F(x)=col(f_1 (x),…,〖 f〗_n (x) ) are given vector functions with real non-negative components f_i (x), i=1,…n. We investigate the case when it is possible to influence the population size by means of harvesting. The model of the exploited population under discussion has the form x(k+1)=F((1-u(k) )x(k) ), where u(k)= (u_1 (k),…,u_n (k))∈〖[0; 1]〗^n is a control vector, which can be varied to achieve the best result of harvesting the resource. We assume that the cost of a conventional unit of each of n classes is constant and equals to C_i≥0, i=1,…,n. To determine the cost of the resource obtained as the result of harvesting, the discounted income function is introduced into consideration. It has the form H_α (u ̅,x(0))=∑_(j=0)^∞▒〖∑_(i=1)^n▒〖C_i x_i (j) u_i (j) e^(-αj) 〗,〗 where α>0 is the discount coefficient. The problem of constructing controls on finite and infinite time intervals at which the discounted income from the extraction of a renewable resource reaches the maximal value is solved. As a corollary, the results on the construction of the optimal harvesting mode for a homogeneous population are obtained (that is, for n = 1).

On application of the i-smooth analysis methodology to elaboration of numerical methods for solving functional differential equations

The article discusses a number of aspects of the application of i -smooth analysis in the development of numerical methods for solving functional differential equations (FDE). The principle of separating finite- and infinite-dimensional components in the structure of numerical schemes for FDE is demonstrated with concrete examples, as well as the usage of different types of prehistory interpolation, those by Lagrange and Hermite. A general approach to constructing Runge–Kutta-like numerical methods for nonlinear neutral differential equations is presented. Convergence conditions are obtained and the order of convergence of such methods is established.

On an ill-posed boundary value problem for the Laplace equation in a circular cylinder

In this paper, we consider a mixed problem for the Laplace equation in a region in a circular cylinder. On the lateral surface of a cylidrical region, the homogeneous boundary conditions of the first kind are given. The cylindrical area is bounded on one side by an arbitrary surface on which the Cauchy conditions are set, i.e. a function and its normal derivative are given. The other border of the cylindrical area is free. This problem is ill-posed, and to construct its approximate solution in the case of Cauchy data known with some error it is necessary to use regularizing algorithms. In this paper, the problem is reduced to a Fredholm integral equation of the first kind. Based on the solution of the integral equation, an explicit representation of the exact solution of the problem is obtained in the form of a Fourier series with the eigenfunctions of the first boundary value problem for the Laplace equation in a circle. A stable solution of the integral equation is obtained by the Tikhonov regularization method. The extremal of the Tikhonov functional is considered as an approximate solution. Based on this solution, an approximate solution of the problem in the whole is constructed. The theorem on convergence of the approximate solution of the problem to the exact one as the error in the Cauchy data tends to zero and the regularization parameter is matched with the error in the data is given. The results can be used for mathematical processing of thermal imaging data in medical diagnostics.

A counterexample to the stochastic version of the Brouwer fixed point theorem

It is shown that the stochastic counterpart of the classical fixed point theorem for continuous maps in a finite dimensional Euclidean space (“Brouwer’s theorem”) is not, in general, true. This result implies, in particular, that a careful choice of invariant sets in the stochastic version of Brouwer’s theorem is necessary in the theory of stochastic nonlinear operators.

Lagrange principle and its regularization as a theoretical basis of stable solving optimal control and inverse problems

The paper is devoted to the regularization of the classical optimality conditions (COC) — the Lagrange principle and the Pontryagin maximum principle in a convex optimal control problem for a parabolic equation with an operator (pointwise state) equality-constraint at the final time. The problem contains distributed, initial and boundary controls, and the set of its admissible controls is not assumed to be bounded. In the case of a specific form of the quadratic quality functional, it is natural to interpret the problem as the inverse problem of the final observation to find the perturbing effect that caused this observation. The main purpose of regularized COCs is stable generation of minimizing approximate solutions (MAS) in the sense of J. Warga. Regularized COCs are: 1) formulated as existence theorems of the MASs in the original problem with a simultaneous constructive representation of specific MASs; 2) expressed in terms of regular classical Lagrange and Hamilton–Pontryagin functions; 3) are sequential generalizations of the COCs and retain the general structure of the latter; 4) “overcome” the ill-posedness of the COCs, are regularizing algorithms for solving optimization problems, and form the theoretical basis for the stable solving modern meaningful ill-posed optimization and inverse problems.