卷 59, 编号 8 (2019)

Available methods for nonnegative matrix factorization make use of all elements of the original \(m \times n\) matrix, and their complexity is at least \(O(mn),\) which makes them extremely resource-intensive in the case of large amounts of data. Accordingly, the following natural question arises: given the nonnegative rank of a matrix, can a nonnegative matrix factorization be constructed using some of its rows and columns? Methods for solving this problem are proposed for certain classes of matrices, namely, for nonnegative separable matrices (for which there exists a cone spanned by several columns of the original matrix that contains all its columns), for nonnegative separable matrices with perturbations, and for nonnegative matrices of rank 2. In practice, the number of operations and the amount of storage used by the proposed algorithms depend linearly on \(m + n\).

Bifurcations of periodic solutions of the well-known Mackey–Glass equation from its unique equilibrium state under varying equation parameters are considered. The equation is used as a mathematical model of variations in the density of white blood cells (neutrophils). Written in dimensionless variables, the equation contains a small parameter multiplying the derivative, which makes this equation singular. It is shown that the behavior of solutions to the equation with initial data from a fixed neighborhood of the equilibrium state in the equation phase space is described by a countable system of nonlinear ordinary differential equations. This system has a minimal structure and is called the normal form of the equation in the neighborhood of the equilibrium state. One fast variable and a countable number of slow variables can be extracted from this system of equations. As a result, the averaging method can be applied to the obtained system. It is shown that the equilibrium states of the averaged system of equations in slow variables are associated with periodic solutions of the same stability type in the original equation. The possibility of simultaneous bifurcation of a large number of periodic solutions (multistability bifurcation) is shown. It is also shown that, with a further increase in the bifurcation parameter, each of the periodic solutions exhibits the transition to a chaotic attractor through a series of period-doubling bifurcations. Thus, the behavior of the solutions of the Mackey–Glass equation is characterized by chaotic multistability.

An implicit scheme with splitting with respect to physical processes is proposed for a stiff system of two-dimensional Euler gas dynamics equations with chemical source terms. For the first time, convection is computed using a bicompact scheme that is fourth-order accurate in space and third-order accurate in time. This high-order bicompact scheme is L-stable in time. It employs a conservative limiting method and Cartesian meshes with solution-based adaptive mesh refinement. The chemical reactions are computed using an L-stable second-order Runge–Kutta scheme. The developed scheme is successfully tested as applied to several problems concerning detonation wave propagation in a two-species ideal gas with a single combustion reaction. The advantages of bicompact schemes over the popular MUSCL and WENO5 schemes as applied to shock-capturing computations of detonation waves are discussed.

The slow longitudinal flow of a rarefied gas between two infinitely long coaxial cylinders driven by a prescribed temperature gradient is studied. The problem is formulated for a linearized kinetic model with specular-diffuse boundary conditions. The reduced heat and mass fluxes through the channel are obtained as functions of the tangential momentum accommodation coefficient and the Knudsen number for various ratios of the inner to outer radius of the cylinders. The values of these fluxes are found using Chebyshev polynomials. The resulting expressions are analyzed in the free-molecular and hydrodynamic limits.

The Extended-Rydberg potential has wide applicability in determining the properties of diatomic molecules. In this paper, we estimate the Extended-Rydberg potential using a novel approach based on the objective least square functions of differential, integral-differential and integral approaches for the estimation of the potential. Interesting research results are obtained as the numerical differentiation (differential approach), integration (integral-differential and integral approach) are in agreement with the experimental data sets of gold atoms. It is a well-known fact that the more parameters a semiempirical interatomic potential has, the more flexible and accurate it is for experimental curve fitting but it takes longer computational time. We establish via CPU time the efficiency and novelty of our approach for the five-parameter Extended-Rydberg potential.

A theory of integral equations for radial currents in the axisymmetric problem of scattering by a disk is constructed. The theory relies on the extraction of the principal part of a continuously invertible operator and on the proof of its positive definiteness. Existences and uniqueness theorems are obtained for the problem. An orthonormal basis is constructed for the energy space of the positive definite operator. Each element of the basis on the boundary behaves in the same manner as the unknown function. The structure of the matrix of the integral operator in this basis is studied. It is found that the principal part has an identity matrix, while the matrix of the next operator is tridiagonal.

The classical Germeier’s attack–defense game is generalized for the case of defense in depth (multilayered defense) that has a network structure. The generalization is based on the work by Hohzaki and Tanaka. In distinction from this work, the defense in each possible direction of motion between the network nodes given by directed arcs may have multiple layers, which leads in the general case to convex minimax problems that can be solved using the subgradient descent method. In particular, the proposed model generalizes the classical attack–defense model for the multilayered defense without the simplifying assumption that the effectiveness of defense is independent of the defense layer.