Vol 100, No 3 (2019)
- Year: 2019
- Articles: 23
- URL: https://journals.rcsi.science/1064-5624/issue/view/13886
Mathematics
Generalized Localization for Spherical Partial Sums of Multiple Fourier Series
Abstract
Abstract—In this paper the generalized localization principle for the spherical partial sums of the multiple Fourier series in the L2 class is proved, that is, if f ∈ L2(TN) and f = 0 on an open set Ω ⊂ TN, then it is shown that the spherical partial sums of this function converge to zero almost-everywhere on Ω. It has been previously known that the generalized localization is not valid in Lp(TN) when \(1 \leqslant p < 2\). Thus the problem of generalized localization for the spherical partial sums is completely solved in Lp(TN), p ≥ 1: if p ≥ 2 then we have the generalized localization and if p < 2, then the generalized localization fails.
Optimal Control and Maximum Principle in (B)-Spaces. Examples for PDE in (H)-Spaces and ODE in\({{\mathbb{R}}^{n}}\)
Abstract
Observation and control problems in Banach (B)-spaces are investigated. A criterion for controllability and optimal controllability is formulated on the basis of the BUME method and the monotone mapping method. An inverse controllability problem is introduced, and an abstract maximum principle is formulated in (B)-spaces. For PDE in Hilbert (H)-spaces and for ODE in \({{\mathbb{R}}^{n}}\), an integral maximum principle is proved and an optimality system is presented.
Factor Model for the Study of Complex Processes
Abstract
An original concept of building a factor model based on frames has been proposed. A method has been developed for calculating the values of factors by searching for the eigenvalues of matrices of pairwise comparisons of the degree of influence of slots. Possible applications of the factor model in the study of complex processes and systems are discussed.
Combined DG Scheme That Maintains Increased Accuracy in Shock Wave Areas
Abstract
A combined scheme for the discontinuous Galerkin (DG) method is proposed. This scheme monotonically localizes the fronts of shock waves and simultaneously maintains increased accuracy in the regions of smoothness of the computed weak solutions. In this scheme, a nonmonotone version of the third-order DG method is used as a baseline scheme and a monotone version of this method is used as an internal scheme, in which a nonlinear correction of numerical fluxes is used. Tests demonstrate the advantages of the new scheme as compared to standard monotonized variants of the DG method.
Complete Radon–Kipriyanov Transform: Some Properties
Abstract
The even Radon–Kipriyanov transform (Kγ-transform) is suitable for studying problems with the Bessel singular differential operator \({{B}_{{{{\gamma }_{i}}}}} = \frac{{{{\partial }^{2}}}}{{\partial x_{i}^{2}}} + \frac{{{{\gamma }_{i}}}}{{{{x}_{i}}~}}\frac{\partial }{{\partial {{x}_{i}}}},{{\gamma }_{i}} > 0\). In this work, the odd Radon–Kipriyanov transform and the complete Radon–Kipriyanov transform are introduced to study more general equations containing odd B-derivatives \(\frac{\partial }{{\partial {{x}_{i}}}}~B_{{{{\gamma }_{i}}}}^{k},~~k = 0, 1, 2,~ \ldots \) (in particular, gradients of functions). Formulas of the Kγ-transforms of singular differential operators are given. Based on the Bessel transforms introduced by B.M. Levitan and the odd Bessel transform introduced by I.A. Kipriyanov and V.V. Katrakhov, a relationship of the complete Radon–Kipriyanov transform with the Fourier transform and the mixed Fourier–Levitan–Kipriyanov–Katrakhov transform is deduced. An analogue of Helgason’s support theorem and an analog of the Paley–Wiener theorem are given.
Constructing a Numerically Statistical Model of a Homogeneous Random Field with a Given Distribution of the Integral over One of the Phase Coordinates
Abstract
A numerically implementable model of a three-dimensional homogeneous random field in a “horizontal” layer 0 < z < H is constructed assuming that the integral of the field with respect to the “vertical” coordinate z has a given infinitely divisible one-dimensional distribution and a given correlation function. An aggregate of n independent elementary horizontal layers of thickness h = H/n shifted vertically by a random variable uniformly distributed in the interval (0, h) is considered as a basic model.
Stability Defect Estimation for Sets in a Game Approach Problem at a Fixed Moment of Time
Abstract
On a Characterization Theorem ona-adic Solenoids
Abstract
According to the Heyde theorem the Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form in independent random variables given another. We prove an analogue of this theorem for linear forms in two independent random variables with values in an a-adic solenoind without elements of order 2, assuming that the characteristic functions of the random variables do not vanish, and coefficients of the linear forms are topological automorphisms of the a-adic solenoid.
On the Estimation of Coefficients of Irreducible Factors of Polynomials over a Field of Formal Power Series in Nonzero Characteristic
Abstract
We discuss some results and problems related to the Newton–Puiseux algorithm and its generalization for nonzero characteristic obtained by the author earlier. A new method is suggested for obtaining effective estimates of the roots of a polynomial in the field of fractional power series in the case of arbitrary characteristic.
NP-Hardness of Quadratic Euclidean 1-Mean and 1-Median 2-Clustering Problem with Constraints on the Cluster Sizes
Abstract
We consider the problem of clustering a finite set of N points in d-dimensional Euclidean space into two clusters minimizing the sum (over both clusters) of the intracluster sums of the squared distances between the cluster elements and their centers. The center of one cluster is defined as a centroid (geometric center). The center of the other cluster is determined as an optimized point in the input set. We analyze the variant of the problem with given cluster sizes such that their sum is equal to the size of the input set. The strong NP-hardness of this problem is proved.
On the Dirichlet Problem for an Elliptic Functional Differential Equation with Affine Transformations of the Argument
Abstract
The Dirichlet problem for an elliptic functional differential equation involving a shifted and contracted argument under the Laplacian sign is studied. Sufficient conditions for the unique solvability of the problem are established. It is shown that the problem may have an infinite-dimensional solution manifold.
Identities on Algebras and Combinatorial Properties of Binary Words
Abstract
Polynomial identities and codimension growth of nonassociative algebras over a field of characteristic zero are considered. A new approach is proposed for constructing nonassociative algebras starting from a given infinite binary word. The sequence of codimensions of such an algebra is closely connected with the combinatorial complexity of the defining word. These constructions give new examples of algebras with abnormal codimension growth. The first important achievement of the given approach is that the algebras under study are finitely generated. The second one is that the asymptotic behavior of codimension sequences is widely different from all previous examples.
On Mutually Inverse Transforms of Functions on a Half-Line
Abstract
Two transforms of functions on a half-line are considered. It is proved that their composition gives a concave majorant for every nonnegative function. In particular, this composition is an identity transform in the class of nonnegative concave functions. Applications of this result in Hilbert space operator theory and the theory of quantum systems are indicated. Several open problems are formulated.
Method for Estimating the Hurst Exponent of Fractional Brownian Motion
Abstract
Fractional Brownian motion is studied. Statistical estimators of the Hurst exponent are proposed, and their properties are examined. This stochastic process is widely used in model development, trend forecasting, and, in particular, as a special case of long-memory processes. The first model involving the Hurst exponent appeared in the British hydrologist Harold Hurst’s research published in 1951, where he analyzed the flow of the Nile River. Later, an improved model of fractional Brownian motion was widely used in different financial market studies. Since such stochastic processes are of great interest, the extrapolation of fractional Brownian motion and the point estimation of the Hurst exponent H have become important problems. A new approach to the point estimation of the Hurst exponent is proposed in this article.
On the Gardner Problem for Phase-Locked Loops
Abstract
This paper shows the possibilities of solving the Gardner problem of determining the lock-in range for multidimensional phase-locked loop models. Analytical estimates of the lock-in range for a third-order system are obtained for the first time by developing analogues of classical stability criteria for the cylindrical phase space.
Solidification of Binary Alloys and Nonequilibrium Phase Transitions
Abstract
A model was constructed for reconstructing the initial stage of solidification of binary alloys treated as a nonequilibrium phase transition with a diffusion stratification mechanism. Numerical experiments concerning the self-excitation of a homogeneous state by applying a melt cooling boundary control condition were performed.
Kinetic Algorithms for Modeling Conductive Fluids Flow on High-Performance Computing Systems
Abstract
Processes in the dynamics of electrically conducting fluid flows in complex heat transfer systems are mathematically modeled in detail on high-performance parallel computing systems. The study is based on the kinetically consistent magnetogasdynamic approach adjusted to this class of problems. The kinetically consistent algorithm is well adapted to the architecture of high-performance computing systems with massive parallelism, so that complex heat transfer systems can be effectively studied with a high resolution. The approach, method, and algorithms are described, and numerical results are presented.
Mathematical Physics
On One Method for Constructing Exact Solutions of Nonlinear Equations of Mathematical Physics
Abstract
A new method for constructing exact solutions of nonlinear equations of mathematical physics is proposed. The method is based on nonlinear integral transformations in combination with the splitting principle. The effectiveness of the method is illustrated by nonlinear reaction–diffusion equations that involve two or three arbitrary functions. New exact functional separable solutions and generalized traveling-wave solutions are presented.
Numerical Modeling of Wave Processes in Multilayered Media with Gas-Containing Layers: Comparison of 2D and 3D Models
Abstract
Today, the exploration of the Arctic region is one of the most important direction of research in our country, because large amounts of unexplored oil and gas deposits are located there. Large hydrocarbon deposits are situated within the Northern seas. Their development is complicated by gas explosions resulting from an accidental opening and further spread of the gas. Since the region with gas layers cannot be frequently monitored, an area with already detected gas deposits is numerically modeled instead. In this work, seismic waves propagating in multilayered geological models with gas-containing inclusions were numerically modeled over a four-year interval by applying the grid-characteristic method. Wave patterns of seismic reflections and seismograms for the described problem were obtained. The wave patterns and seismograms obtained in the two- and three-dimensional cases were compared. The results were found to be in good agreement.
Computer Science
Simulation of an Inhomogeneous Plasma Microfield
Abstract
The optical properties of plasma are determined by the presence of a fluctuating microscopic electric field. A simple ab initio model of a plasma microfield accounting for its inhomogeneity up to an octupole term has been constructed for the first time. A comparison with experiments has shown that only this model correctly describes the observed number of spectral lines.
Duality Principle and New Forms of the Inverse Laplace Transform for Signal Propagation Analysis in Inhomogeneous Media with Dispersion
Abstract
New equations for Laplace transform inversion are obtained. The equations satisfy the causality principle. The impulse response of a channel is determined in order to analyze dispersion distortions in inhomogeneous media. The impulse response excludes the possibility that the signal exceeds the speed of light in the medium. The transmission bandwidth, the angular spectrum, and the Doppler shift in the ionosphere are computed.