Vol 98, No 2 (2018)
- Year: 2018
- Articles: 31
- URL: https://journals.rcsi.science/1064-5624/issue/view/13879
Mathematics
On Pronormal Subgroups in Finite Simple Groups
Abstract
A subgroup H of a group G is called pronormal if, for any element g of G, the subgroups H and Hg are conjugate in the subgroup they generate. Some problems in the theory of permutation groups and combinatorics have been solved in terms of pronormality, and the characterization of pronormal subgroups in finite groups is a problem of importance for applications of group theory. A task of special interest is the study of pronormal subgroups in finite simple groups and direct products of such groups. In 2012 E.P. Vdovin and D.O. Revin conjectured that the subgroups of odd index in all finite simple groups are pronormal. We disproved this conjecture in 2016. Accordingly, a natural task is to classify finite simple groups in which the subgroups of odd index are pronormal. This paper completes the description of finite simple groups whose Sylow 2-subgroups contain their centralizers in the group and the subgroups of odd index in which are pronormal.
The Basis Property of the System of Root Functions of the Oblique Derivative Problem
Abstract
The oblique-derivative spectral problem for the Laplacian in a disk D is studied. The asymptotic properties of its eigenvalues are established, and the system of root functions of this problem is proved to be a basis with brackets in L2 (D).
On Correctness Conditions for Algebra of Recognition Algorithms with μ-Operators over Pattern Problems with Binary Data
Abstract
The concept of an Ω-weakly regular problem is introduced. On the basis of the Zhuravlev operator approach combined with the neural network paradigm, it is shown that, for each such problem, a correct algorithm and a six-level spatial neural network reproducing the computations executed by this algorithm can be constructed. Moreover, the set of Ω-weakly regular problems includes the set of Ω-regular problems. It turns out that a three-level spatial network (μ-block) is a forward propagation network whose inner loop under estimation of the class membership for each test object consists of a single iteration. As a result, the amount of computations required for the six-level network is reduced noticeably.
Frequency Tests for the Existence and Stability of Bounded Solutions to Differential Equations of Higher Order
Abstract
To study a vector-matrix differential equation of order n, the method of integral equations is used. When the Lipschitz condition holds, an existence and uniqueness theorem for a bounded solution and its estimates are obtained. This solution is almost periodic if the nonlinearity is almost periodic, and it is asymptotically Lyapunov stable if the matrix characteristic polynomial is a Hurwitz polynomial. Under a Lipschitztype condition, a theorem on the existence of at least one bounded solution is proved; among the bounded solutions, there is at least one recurrent solution if the nonlinearity is almost periodic. The equation is S-dissipative if the matrix characteristic polynomial is a Hurwitz polynomial.
On the Periodicity of Continued Fractions in Hyperelliptic Fields over Quadratic Constant Field
Abstract
We give a description of the cubic polynomials f(x) with coefficients in the quadratic number fields \(\mathbb{Q}(\sqrt{5})\) and \(\mathbb{Q}(\sqrt{-15})\) for which the continued fraction expansion of the irrationality \(\sqrt {f\left( x \right)} \) is periodic.
A Mesh Free Stochastic Algorithm for Solving Diffusion–Convection–Reaction Equations on Complicated Domains
Abstract
A mesh free stochastic algorithm for solving transient diffusion–convection–reaction problems on domains with complicated structure is suggested. For the solutions of this kind of equations exact representations of the survival probabilities, the probability densities of the first passage time and position on a sphere are obtained. Based on these representations we construct a stochastic algorithm which is simple in implementaion for solving one- and three-dimensional diffusion–convection–reaction equations. The method is continuous both in space and time, and its advantages are particularly well manifested in solving problems on complicated domains, calculating fluxes to parts of the boundary, and other integral functionals, for instance, the total concentration of the particles which have been reacted to a time instant t.
Numerical Characteristics of the Genetic Code for the Possibility of Overlapping Genes
Abstract
Random changes in the standard genetic code in which its structure is preserved are considered. The hypothesis that the best possibility of overlapping genes is provided by the standard genetic code of given structure is justified. This code is determined by the condition that the total number of admissible substitutions opening closed reading frames is maximal, while the number of closed reading frames for which there is no such substitution is minimal. In terms of this criterion, the estimated probability of a random choice of the standard genetic code is less than 10–6.
Superfast Iterative Solvers for Linear Matrix Equations
Abstract
Superfast algorithms for solving large systems of linear equations are developed on the basis of an original method for multistep decomposition of a linear multidimensional dynamical system. Examples of analytical synthesis of iterative solvers for matrices of the general form and for large numerical systems of linear algebraic equations are given. For the analytical case, it is shown that convergence occurs at the second iteration.
Optimization of Randomized Monte Carlo Algorithms for Solving Problems with Random Parameters
Abstract
Randomized Monte Carlo algorithms intended for statistical kernel estimation of the averaged solution to a problem with random baseline parameters are optimized. For this purpose, a criterion for the complexity of a functional Monte Carlo estimate is formulated. The algorithms involve a splitting method in which, for each realization of the parameters, a certain number of trajectories of the corresponding baseline process are constructed.
On Conditions for L2-Dissipativity of Linearized Explicit QGD Finite-Difference Schemes for One-Dimensional Gas Dynamics Equations
Abstract
An explicit two-level in time and spatially symmetric finite-difference scheme approximating the 1D quasi-gasdynamic system of equations is studied. The scheme is linearized about a constant solution, and new necessary and sufficient conditions for the L2-dissipativity of solutions to the Cauchy problem are derived, including, for the first time, the case of a nonzero background velocity and depending on the Mach number. It is shown that the condition on the Courant number can be made independent of the Mach number. The results provide a substantial development of the well-known stability analysis of the linearized Lax–Wendroff scheme.
Fejér Sums and Fourier Coefficients of Periodic Measures
Abstract
The Fejér sums of periodic measures and the norms of the deviations from the limit in the von Neumann ergodic theorem are calculating in terms of corresponding Fourier coefficients, in fact, using the same formulas. As a result, well-known estimates for the rates of convergence in the von Neumann ergodic theorem can be restated as estimates for the Fejér sums at a point for periodic measures. In this way, natural sufficient conditions for the polynomial growth and polynomial decay of these sums can be obtained in terms of Fourier coefficients. Besides, for example, it is shown that every continuous 2π-periodic function is uniquely determined by its sequence of Fejér sums at any two points whose difference is incommensurable with π.
Heterogeneous Computing in Resource-Intensive CFD Simulations
Abstract
A software package implementing a fully heterogeneous mode of computations on CPUs and GPU accelerators for efficient use of hybrid supercomputers has been developed at the Keldysh Institute of Applied Mathematics of the Russian Academy of Sciences. The package involves a distributed preprocessor ensuring work with fine unstructured meshes. Combined compression of the grid topology is used to reduce the amount of storage required for superlarge grid data. The study involves petascale computational resources.
On Non-Uniqueness of Probability Solutions to the Two-Dimensional Stationary Fokker–Planck–Kolmogorov Equation
Abstract
The problem of uniqueness of probability solutions to the two-dimensional stationary Fokker–Planck–Kolmogorov equation is considered. Under broad conditions, it is proved that the existence of two different probability solutions implies the existence of an infinite set of linearly independent probability solutions.
Green’s Function of Ordinary Differential Operators and an Integral Representation of Sums of Certain Power Series
Abstract
The eigenvalues and eigenfunctions of certain operators generated by symmetric differential expressions with constant coefficients and self-adjoint boundary conditions in the space of Lebesgue squareintegrable functions on an interval are explicitly calculated, while the resolvents of these operators are integral operators with kernels for which the theorem on an eigenfunction expansion holds. In addition, each of these kernels is the Green’s function of a self-adjoint boundary value problem, and the procedure for its construction is well known. Thus, the Green’s functions of these problems can be expanded in series in terms of eigenfunctions. In this study, identities obtained by this method are used to calculate the sums of convergent number series and to represent the sums of certain power series in an intergral form.
Nonasymptotic Estimates for the Closeness of Gaussian Measures on Balls
Abstract
Upper bounds for the closeness of two centered Gaussian measures in the class of balls in a separable Hilbert space are obtained. The bounds are optimal with respect to the dependence on the spectra of the covariance operators of the Gaussian measures. The inequalities cannot be improved in the general case.
Variance Reduction in Monte Carlo Estimators via Empirical Variance Minimization
Abstract
For Monte Carlo estimators, a variance reduction method based on empirical variance minimization in the class of functions with zero mean (control functions) is described. An upper bound for the efficiency of the method is obtained in terms of the properties of the functional class.
On Sobolev Classes Containing Solutions to Fokker–Planck–Kolmogorov Equations
Abstract
The main result of this paper answers negatively a long-standing question and shows that a density of a probability measure satisfying the Fokker–Planck–Kolmogorov equation with a drift integrable with respect to this density can fail to belong to the Sobolev class W1,1(ℝd). There is also a version of this result for densities with respect to Gaussian measures. On the other hand, we prove that the solution density belongs to certain fractional Sobolev classes.
Well-Posedness and Spectral Analysis of Volterra Integro-Differential Equations with Singular Kernels
Abstract
Integro-differential equations with unbounded operator coefficients in a Hilbert space are considered. Such equations arise in viscoelasticity theory, thermal physics, and homogenization problems in multiphase media. Initial–boundary value problems for the indicated equations are proved to be well posed, and their spectral analysis is performed.
Monotone Finite-Difference Scheme Preserving High Accuracy in Regions of Shock Influence
Abstract
An explicit combined shock-capturing finite-difference scheme is constructed that localizes shock fronts with high accuracy and simultaneously preserves the high order of convergence in all domains where the computed weak solutions are smooth. In this scheme, Rusanov’s explicit nonmonotone scheme of the third order is used as a basis one, while the internal scheme is based on the second-order monotone CABARET. The advantages of the new scheme as compared with the WENO scheme of the fifth order in space and third order in time are demonstrated in test computations.
Confidence Sets for Spectral Projectors of Covariance Matrices
Abstract
A sample X1,...,Xn consisting of independent identically distributed vectors in ℝp with zero mean and a covariance matrix Σ is considered. The recovery of spectral projectors of high-dimensional covariance matrices from a sample of observations is a key problem in statistics arising in numerous applications. In their 2015 work, V. Koltchinskii and K. Lounici obtained nonasymptotic bounds for the Frobenius norm \(\parallel {P_r} - {\hat P_r}{\parallel _2}\) of the distance between sample and true projectors and studied asymptotic behavior for large samples. More specifically, asymptotic confidence sets for the true projector Pr were constructed assuming that the moment characteristics of the observations are known. This paper describes a bootstrap procedure for constructing confidence sets for the spectral projector Pr of the covariance matrix Σ from given data. This approach does not use the asymptotical distribution of \(\parallel {P_r} - {\hat P_r}{\parallel _2}\) and does not require the computation of its moment characteristics. The performance of the bootstrap approximation procedure is analyzed.
Mathematical Physics
Efficient Parallel Algorithm for Calculating Electric Currents in a Toroidal Plasma
Abstract
In the six-dimensional (6D) phase space, a new statement of the problem of calculating the electric current due to the pressure gradient in a toroidal plasma is considered. A semi-Lagrangian approach is applied. A new efficient parallel algorithm is developed for the numerical solution of a 6D kinetic equation with the Coulomb collision operator. By the DiFF-PK code developed by the authors, a 5D problem is solved under the conditions of ITER-scale facilities. The bootstrap current of electrons is calculated. A good agreement with the previously known limit cases is demonstrated. The method proposed can be applied to the high-precision computation of a large class of problems whose solution within existing approaches is complicated. For these purposes, the mini-supercomputer at the Research Institute for Systems Analysis, Russian Academy of Sciences, or pentaflop class supercomputers can be used. The problems include, for example, the calculation of the bootstrap current of alpha particles–a product of thermonuclear fusion, the calculation of the electric current produced by neutrals injected into a plasma, and simulation of radial electric fields and instabilities of plasma.
Dynamics of a Delay Logistic Equation with Diffusion and Coefficients Rapidly Oscillating in Space Variable
Abstract
The applicability of the averaging principle in the study of the dynamics of a practically important delay logistic equation with diffusion and coefficients rapidly oscillating with respect to the space variable is analyzed. A task of special interest is to address equations with rapid oscillations of the delay coefficient or a quantity characterizing the deviation of the space variable. Bifurcation problems arising in critical cases for the averaged equation are studied. Results concerning the existence, stability, and asymptotic behavior of periodic solutions to the original equation are formulated.
Computer Science
Mathematical Models for Calculating the Development Dynamics in the Era of Digital Economy
Abstract
Mathematical models for practical calculations of technological progress (total productivity of production factors) and economic growth in the era of widespread digitalization and robotization of national economies, where the main factor of production is technological information, are developed and verified. For this purpose, models using different modes of information production are proposed for the first time. It is shown that the economic effect of the digitalization of an economy will not come immediately, but with a lag of about eight years. For the US economy, forecast calculations show that this will happen in 2022–2026 with total productivity increasing by 1.1 percentage points up to 2.5% per year.
Control Theory
Control of a Rigid Body Carrying an Oscillator under Incomplete Information
Abstract
A two-body system consisting of a rigid body with a linear oscillator attached to it is considered. The body moves along a horizontal line under the action of a control force and a small unknown disturbance. The phase state of the oscillator is assumed to be not available for measurement. A bounded feedback control is proposed which brings the body to a prescribed terminal state in a finite time.
Holistic Theory of Economic Equilibrium: Modified Cassel–Wald Model
Abstract
A modified Cassel–Wald model is used as an example of the holistic approach to creating a mathematical theory of market demand and economic equilibrium that is an alternative to methodological individualism (sociological reductionism). A theorem on the existence and uniqueness of equilibrium is proved.