Vol 98, No 1 (2018)
- Year: 2018
- Articles: 27
- URL: https://journals.rcsi.science/1064-5624/issue/view/13878
Mathematics
On the Parameters of the Singular Part of the Horn—Sergeichuk Regularizing Decomposition
Abstract
A simple method is given for a priori determination of the number of Jordan blocks and their orders in the regularizing decomposition of a square matrix. This decomposition was proposed by R. Horn and V. Sergeichuk. It is attained by congruence transformations and makes it possible to separate the regular and singular parts of the matrix.
Clustering Coefficient of a Spatial Preferential Attachment Model
Abstract
The clustering structure of a graph in a spatial preferential attachment model whose similarity to real-world networks has been shown in many aspects is considered. The behavior of the local clustering coefficient is studied. Namely, the asymptotic behavior of its average value over all graph vertices of a certain degree as the graph size tends to infinity is examined. This characteristic has not been previously analyzed in the SPA model, and it reflects the typical dependence of the clustering structure near some vertex on its degree in the graph. Additionally, it is shown that, with a high probability, there is a vertex for which the value of the clustering coefficient differs from its average.
Programmed Iteration Method in Differential Games with a Functional Target Set
Abstract
A variant of the programmed iteration method is developed for solving game problems of realizing trajectories of a nonlinear conflict controlled system in a given set of functions. Solving these open-loop control problems yields an iterative process in a space of sets whose elements are functional positions, and the corresponding limit determines the set of successful solvability of the original problem in the class of set-valued quasi-strategies (nonanticipating responses to disturbances). For the conflict controlled system, the conditions of generalized uniqueness and uniform boundedness of programmed motions are assumed to hold. The use of an infinite-dimensional space of functional positions is essential, since the arising differential game generally does not satisfy the alternative solvability conditions in classes of feedback strategies.
Optimal Disturbances of Bistable Time-Delay Systems Modeling Virus Infections
Abstract
For bistable time-delay dynamical systems modeling the dynamics of viral infections and the virusinduced immune response, an efficient approach is proposed for constructing optimal disturbances of steady states with a high viral load that transfer the system to a state with a low viral load. Functions approximating the behavior of drugs within the framework of well-known pharmacokinetic models are used as basis functions. Optimal disturbances are sought in the W21 norm. It is shown that optimal disturbances found in this norm are superior to those found in the L2 norm as applied to the development of adequate therapeutic strategies.
Spectral Decomposition Formulas for Zeta Functions of Imaginary Quadratic Fields of Class Number One
Abstract
The squared absolute value of the Dedekind zeta functions of imaginary quadratic fields of class number one on the critical line is expressed in terms of averaged values associated with the spectrum of the automorphic Laplacian with respect to the full modular group.
Strict Embeddings of Rearrangement Invariant Spaces
Abstract
A Banach space E of measurable functions on [0,1] is called rearrangement invariant if E is a Banach lattice and equimeasurable functions have identical norms. The canonical inclusion E ⊂ F of two rearrangement invariant spaces is said to be strict if functions from the unit ball of E have absolutely equicontinuous norms in F. Necessary and sufficient conditions for the strictness of canonical inclusion for Orlicz, Lorentz, and Marcinkiewicz spaces are obtained, and the relations of this concept to the disjoint strict singularity are studied.
On the Uniqueness of the Solution of the Inverse Sturm–Liouville Problem with Nonseparated Boundary Conditions on a Geometric Graph
Abstract
For the first time, the inverse Sturm–Liouville problem with nonseparated boundary conditions is studied on a star-shaped geometric graph with three edges. It is shown that the Sturm–Liouville problem with general boundary conditions cannot be uniquely reconstructed from four spectra. Nonseparated boundary conditions are found for which a uniqueness theorem for the solution of the inverse Sturm–Liouville problem is proved. The spectrum of the boundary value problem itself and the spectra of three auxiliary problems are used as reconstruction data. It is also shown that the Sturm–Liouville problem with these nonseparated boundary conditions can be uniquely recovered if three spectra of auxiliary problems are used as reconstruction data and only five of its eigenvalues are used instead of the entire spectrum of the problem.
Polynomial Computability of Fields of Algebraic Numbers
Abstract
We prove that the field of complex algebraic numbers and the ordered field of real algebraic numbers have isomorphic presentations computable in polynomial time. For these presentations, new algorithms are found for evaluation of polynomials and solving equations of one unknown. It is proved that all best known presentations for these fields produce polynomially computable structures or quotient-structures such that there exists an isomorphism between them polynomially computable in both directions.
Fejér Sums for Periodic Measures and the von Neumann Ergodic Theorem
Abstract
The Fejér sums of periodic measures and the norms of the deviations from the limit in the von Neumann ergodic theorem are calculated, in fact, using the same formulas (by integrating the Fejér kernels), so this ergodic theorem is, in fact, a statement about the asymptotics of the growth of the Fejér sums at zero for the spectral measure of the corresponding dynamical system. As a result, well-known estimates for the rates of convergence in the von Neumann ergodic theorem can be restated as estimates of the Fejér sums at the point for periodic measures. For example, natural criteria for the polynomial growth and polynomial decrease in these sums can be obtained. On the contrary, available in the literature, numerous estimates for the deviations of Fejér sums at a point can be used to obtain new estimates for the rate of convergence in this ergodic theorem.
General Embedding Theorem
Abstract
The quotient space of an arbitrary Banach space B by any subspace B1 ⊂ B equipped with the norm \(||b|B/{B_1}|| = \mathop {\inf }\limits_{f \in B/{B_1}} ||f|b||\) is considered. In the case where the infimum is a minimum, i.e., it is attained at some element, a formula for this element is presented. The proof is based on restating the original problem in dual spaces with the help of corresponding Legendre transforms. Although the original problem is nonlinear, it is found that its formulation in dual spaces is always linear and solvable. The results are applied to the general theory of boundary value problems for differential equations of mathematical physics.
Generating Functions in the Knapsack Problem
Abstract
The knapsack problem with Boolean variables and a single constraint is considered. Combinatorial formulas for calculating and estimating the cardinality of the set of feasible solutions and the values of the functional in various cases depending on given parameters of the problem are derived. The coefficients of the objective function and of the constraint vector and the knapsack size are used as parameters. The baseline technique is the classical method of generating functions. The results obtained can be used to estimate the complexity of enumeration and decomposition methods for solving the problem and can also be used as auxiliary procedures in developing such methods.
On Spectral Asymptotics for a Family of Finite-Dimensional Perturbations of Operators of Trace Class
Abstract
Spectral asymptotics for a family of finite-dimensional perturbations of operators of trace class are found. The results are used to find exact asymptotics of small ball probabilities in the Hilbert norm for finitedimensional perturbations of Gaussian functions. As an example, Durbin processes appearing in the study of empirical processes with estimated parameters are considered.
On Two-Dimensional Polynomially Integrable Billiards on Surfaces of Constant Curvature
Abstract
The algebraic version of the Birkhoff conjecture is solved completely for billiards with a piecewise C2-smooth boundary on surfaces of constant curvature: Euclidean plane, sphere, and Lobachevsky plane. Namely, we obtain a complete classification of billiards for which the billiard geodesic flow has a nontrivial first integral depending polynomially on the velocity. According to this classification, every polynomially integrable convex bounded planar billiard with C2-smooth boundary is an ellipse. This is a joint result of M. Bialy, A.E. Mironov, and the author. The proof consists of two parts. The first part was given by Bialy and Mironov in their two joint papers, where the result was reduced to an algebraic-geometric problem, which was partially studied there. The second part is the complete solution of the algebraic-geometric problem presented below.
Convergence of Eigenfunction Expansions of a Differential Operator with Integral Boundary Conditions
Abstract
For a second-order ordinary differential operator on an interval of the real line with integral boundary conditions, conditions for the unconditional basis property and uniform convergence of the expansion of a function in terms of the eigen- and associated functions of this operator are established. The convergence and equiconvergence rates of this expansion and the equiconvergence rate of the trigonometric Fourier expansion of this function are estimated. The uniform convergence of its expansion in the adjoint system is studied.
On the Closeness of Solutions of Unperturbed and Hyperbolized Heat Equations with Discontinuous Initial Data
Abstract
The influence exerted by the second time derivative with a small parameter added to the heat equation in the case of discontinuous periodic initial data is investigated. It is shown that, except for the initial instants of time, the error of hyperbolization vanishes as the square root of the addition.
Modeling of Galactic Wind Formation from Supernovae Using High-Performance Computations
Abstract
Fundamental processes in the dynamics of the interstellar medium, namely, galactic wind, i.e., ejections of interstellar matter from central regions of galaxies, which are presumably caused by the formation of supernovae, are mathematically modeled in detail on high-performance parallel computer systems. The mathematical simulation is based on a kinetically consistent gasdynamic approach developed for such class of problems in astrophysics. A kinetically consistent algorithm is well adapted to the architecture of high-performance computer systems with massive parallelism, so that complex large-scale astrophysical phenomena can be efficiently studied with a high resolution. The approach, method, and algorithms are described, and numerical results are presented.
Mathematical Physics
Modeling of Ultrasonic Waves in Fractured Rails with an Explicit Approach
Abstract
Ultrasonic wave propagation in steel rails with explicit identification of flaws is numerically simulated. The problem is to detect a vertical crack in a railhead by applying ultrasonic nondestructive testing techniques. The propagation of elastic waves in the rail profile is simulated for various sizes and positions of the crack. It is shown that the finite-difference grid-characteristic method in the time domain and full-wave simulation can be used to analyze the effectiveness of rail flaw detection by applying ultrasonic nondestructive testing techniques. Full-wave simulation is also used to demonstrate the failure of the widely used echo-mirror method to detect flaws of certain types. It is shown that techniques for practical application of the ultrasonic delta method can be developed using full-wave supercomputer simulation. The study demonstrates a promising potential of geophysical methods as adapted to the analysis of ultrasonic nondestructive testing results.