We discuss the dependence of the measure with respect to which a sequence of polynomials defined by a three-term recurrence relation is orthogonal on the initial data.
We consider an ordinary 2m-th order differential operator with purely integral conditions. In this case, the domain of definition of the corresponding operator is not dense in L2(0,1). Under certain conditions for the weight functions included in the integral conditions, the Abel summability of the system of root functions of the corresponding differential operator is proved. \( \)
We study a partial integral operator with a weak singularity in Rn in the anisotropic space of continuous functions with values in the space Lp, p ∈ (1,∞). We prove theorems on the uniform boundedness and equicontinuity of the operator under study. These theorems imply a theorem on the mapping of any bounded and equicontinuous set into a precompact set by a partial integral operator with a weak singularity acting from the space of functions continuous in some variables with values in Lp with respect to the other part of the variables to the space of continuous functions.
This study examines the small oscillations of an ideal stratified fluid within bounded domains. In classical formulations of these problems, the lateral boundaries of the container are typically assumed to align with the gravity vector g and the direction of density stratification. This research investigates non-classical cases where the container walls and the stratification direction form a specific angle. This geometric discrepancy results in a qualitative transformation of the internal wave spectrum. Analysis of a tilted rectangular vessel demonstrates that the angle between the domain boundary and the vector g significantly affects the spectral formation. Specifically, the spectrum of the small-oscillation operator is no longer purely discrete but includes regions of a continuous spectrum. Identifying the boundaries of the continuous spectrum is essential for the accurate resolution of non-homogeneous evolution problems. The tilt angle and the geometric parameters of the vessel determine both these boundaries and the transition points between the discrete and continuous spectra. The results indicate that the orientation of the cavity relative to the gravitational field is a primary factor determining the properties of internal waves in a bounded volume of stratified fluid.
The oscillatory properties of the spectrum of a problem on the natural vibrations of a cross-shaped rod system are investigated. The model is reduced to a fourth-order boundary value problem on a graph with rigid joint conditions for the rods. A method is proposed for reducing the original problem to a multipoint boundary value problem on a selected route, allowing the system to be interpreted as an elastically supported rod. A justification for this method is provided, and a condition for the oscillatory nature of the spectrum is formulated.
This work is a continuation of the research conducted in [2,3]. When forming the capital structure of enterprises (determining the shares of borrowed and equity funds), the balance of interests of owners and managers plays an important role. In this case, the owner is interested in obtaining the maximum return on his capital (maximizing its rentability ROE), and the manager — in increasing the financial stability of the company by reducing the weighted average cost of capital WACC. Resolving the conflict situation is possible, as shown by the authors, within the framework of nonantagonistic game theory, namely in the context of a bimatrix game and a cooperative game of two people. The owners of enterprises Πi were chosen as player A, and the ROE values of enterprises from a representative sample, observed in a certain period T, were chosen as the components of his game matrix. The role of player B was played by the managers of enterprises Πi, and the components of his game matrix were values of the form 1/(1+ WACC) determined in the same period T. The algorithms for finding the Nash equilibrium capital structure for both models were tested using real data for five enterprises in the metallurgical segment observed over a period of 5 years. As a result, it was possible not only to obtain new methods for determining the balanced capital structure, but also to analyze the stability of the obtained solutions with respect to perturbations of the initial data (reduced game matrices), which made it possible to calculate the corridors of possible changes in ROE and WACC for the reference set of enterprises Πα in the selected (shortened) observation period, as well as to outline paths for further development of this topic.
In a Bergman Hilbert space, we study the problem of estimating the best approximation of an analytic function in the unit disk by algebraic polynomials via averaging its modulus of smoothness. A general condition on the weight function is found that allows us to obtain an exact estimate. This condition is analogous to the Shabozov-Yusupov condition but additionally takes into account the specific features of the Bergman space. The resulting upper bounds are applied to calculating the diameters of certain classes of functions in the Kolmogorov, Gelfand, and Bernstein classes, as well as linear and projection diameters, in the Bergman space.
A unitary operator acting in the Krein space and possessing an invariant subspace that is maximal nonnegative and decomposes into a direct sum of a uniformly positive (i.e., equivalent to a Hilbert space with respect to the inner pseudoscalar product) and a finite-dimensional neutral subspace is considered. The existence of a spectral function with a finite number of spectral singularities and a difference expression for this operator transforming the infinite-in-both-sides sequence of momenta generated by this operator into a sequence representable as the difference of positive sequences of momenta is proven. In the special case of a cyclic unitary operator in a Pontryagin space, a function space is constructed in which the operator under study is modelled as the operator of multiplication by an exponential with an imaginary argument.
Asymptotic approximations to stationary solutions are constructed for two classes of diffusion–reaction systems describing the interactions of biological species (competition and predator– prey) in a heterogeneous habitat. The initial models are supplemented with periodicity conditions on a one-dimensional area; the diffusion coefficients are assumed to be small and, generally speaking, multi-scale. Under the assumption that the degenerate system admits a solution in the form of an ideal free distribution (IFD), methods of singular perturbation theory are used to obtain explicit analytical formulas for the leading terms of the asymptotics. It is established that diffusion corrections to the IFD are proportional to the local curvature of the resource profile \( p''(x)/p(x) \) and are determined by both the parameters of interspecies interaction and the ratio of the diffusion coefficients. For a predator–prey system, it is shown that resource heterogeneity significantly influences the predator distribution, while the prey distribution remains close to the IFD even at relatively high diffusion values. A quantitative criterion for the applicability of the asymptotic model is formulated, allowing for an a priori assessment of its accuracy for an arbitrary resource profile. The reliability of the analytical results is confirmed by numerical calculations.
In this paper, we consider a model of an emulsion of two liquids in a three-dimensional domain filled with one liquid and having small inclusions of another liquid. For analysis in such combined media, it is assumed that the structure of the domain is periodic with rapidly alternating parameters, while the characteristic size of the alternation is taken as a small parameter ε 0. The question of the existence of a sound propagation front for this model is investigated for the corresponding effective boundary value problem with integro-differential equations with slowly varying coefficients obtained by the method of asymptotic averaging at ε → 0. It is shown that with a certain smoothness of the density change between phases, there is a leading edge in the effective model. Thus, it is proved that with a finite initial perturbation of the combined medium under consideration, the propagation of vibrations will have a finite velocity.
We study the initial-boundary value problem for a second-order nonhomogeneous hyperbolic equation in a plane half-strip with constant coefficients, containing a mixed derivative, and with zero and nonzero potentials. This equation is the equation for the transverse oscillations of a moving finite string. We consider the case of Dirichlet-Neumann boundary conditions: the left end is fixed, and the right end is free. It is assumed that the roots of the characteristic equation are simple and lie on the real axis on opposite sides of the origin. We seek a classical solution (or a solution almost everywhere, sometimes called a “strong solution”) to this problem. A spectral problem associated with the original initial-boundary value problem, generated by an ordinary differential operator-valued function (pencil) of second order, is investigated. The asymptotic behavior of the eigenvalues and resolvent is determined, the operator-valued function is linearized in the corresponding space of vector functions, and a theorem on the expansion of the first component of the vector function in root functions of the spectral problem is proved. A theorem on the uniqueness of the classical solution is formulated and proved, and a formula for the classical solution is derived in the form of a series of contour integrals. Then, using these formulas in the case of zero potential, theorems on finite formulas for the classical solution in special cases are proved, and based on these, a finite formula for the classical solution in the general case is obtained.
We consider a non-self-adjoint ordinary differential operator defined on a finite interval by an \( n \)th-order linear differential expression with a nonzero coefficient of the \( (n-1) \)th derivative and two-point Birkhoff regular boundary conditions. We study the uniform equiconvergence of expansions of a given function in a biorthogonal series in eigenfunctions and associated functions (or, briefly, root functions) of this operator and in an ordinary trigonometric Fourier series, as well as an estimate of the difference of the corresponding partial sums (or, briefly, the rate of equiconvergence) under the most general conditions on the expanded function and the coefficient of the \( (n-1) \)th derivative. We obtain estimates for the difference of the expansions in terms of general (integral) moduli of continuity of the expanded function and the coefficient of the \( (n-1) \)th derivative uniform inside the fundamental interval. From these estimates, corresponding estimates are derived in the case where moduli of continuity are bounded from above by slowly varying functions and, in particular, by logarithmic functions. Based on this, sufficient conditions for equiconvergence in the indicated cases are formulated. These results are obtained using the author's previously obtained estimate for the difference between the partial sums of expansions of a given function in a biorthogonal series in eigenfunctions and associated functions of the differential operator under consideration and in a modified trigonometric Fourier series, as well as analogues of the Steinhaus theorem. The modification of the trigonometric Fourier series consisted in applying a very specific bounded operator to the ordinary trigonometric Fourier series expressed through the coefficient of the \( (n-1) \)th derivative and its inverse operator to the expanded function.
Using a predator-prey model in a heterogeneous environment, we have created a mathematical model that describes the interaction between populations with different evolutionary strategies. The model is based on partial differential equations and allows for the consideration of multifactor taxis. We propose modified functions of local predator-prey interactions, which provide a variety of evolutionary strategies for the system. Key parameters responsible for the formation of ideal free distribution strategies have been investigated, and conditions for the parameters of diffusion and migration have been given under which ideal free distribution-like strategies can be implemented. Results from computational experiments demonstrating stationary and oscillating modes have been presented.
An approach to describing the solution of the initial-boundary value problem for the wave equation on a finite and bounded geometrical graph \(\Gamma\) is implemented. The linear transmission conditions have a more general form than that considered in previous works. The approach is based on interpreting the behavior of the solution at the vertices of \(\Gamma\) as boundary regimes with respect to adjacent edges. The set of these boundary regimes turns out to be a solution to the initial value problem for a system of delays differential equations on \([0;+\infty)\) with the number of delaying arguments infinitely increasing with infinitely increasing of the argument.
In this paper, we study a boundary-value problem for a mathematical model describing the motion of aqueous polymer solutions. Based on the approximation-topological method, we investigate the existence of weak solutions of the problem under study. We consider the case of medium motion both in a bounded domain of two-dimensional or three-dimensional space and in an unbounded domain.
We consider the problem of estimating coverage intervals (both one-sided and two-sided) of the standard two-sided power distribution (STSP-distribution) based on sample data. We check the quality of the obtained estimates using the Monte Carlo method. We study the properties of the maximum likelihood estimates of the parameters of the original distribution and estimate the influence of their bias on the quality of estimating coverage intervals. We also give examples demonstrating that the obtained estimates can be used for continuous distributions that can be approximated by a family of STSP-distributions.
In this paper, an initial-boundary value problem is studied for one-dimensional equations of the dynamics of a compressible viscous mixture. A theorem is proved for the existence and uniqueness of a solution to the initial-boundary value problem without any restrictions on the structure of the viscosity matrix other than the standard physical requirements of symmetry and positive definiteness.
In this study, we analyze the heat transfer of single-chamber and double-chamber glass units installed in the outer and inner sashes of a composite window unit with the inter-glass space filled with dehumidified air and inert gases. We construct the mathematical model based on the solution of the heat conductivity equation with constant coefficients in a two-dimensional setting, taking into account the layered structure of the structure and using boundary conditions of the III and IV kind. Our numerical implementation of the problem uses the finite difference method on a uniform grid using the C++ programming language. To take into account convective heat exchange through glass units, we perform a series of numerical calculations in ANSYS Fluent software. We show that convective heat loss in glass units can be reduced by increasing the thickness of the spacer frame and using inert gases with low thermal conductivity. We identify the optimal thickness of the gas-filled chamber of a single-chamber glass unit (when filled with air, dry air, argon, krypton, xenon), ensuring maximum thermal resistance.
The present paper is devoted to the study of the abstract nonlocal boundary value problem with integral type Samarskii–Ionkin conditions for the differential equation of elliptic type \[\hspace{-6em} -u''(t)+Au(t)=f(t)\quad (0\leq t\leq T),\quad u\left( 0\right) =\varphi,\quad u'\left( 0\right) =u'\left( T\right) +\int\limits_{0}^{T}\alpha \left( s\right) u(s)ds+\psi.\quad\] in an arbitrary Banach space \(E\) with the positive operator \(A\). The well-posedness of this problem in various Banach spaces is established. In applications, theorems on the well-posedness of several nonlocal boundary value problems for elliptic equations with integral type Samarskii–Ionkin conditions are proved.
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.
