Showing papers in "Lobachevskii Journal of Mathematics in 2019"

AMGCL: An Efficient, Flexible, and Extensible Algebraic Multigrid Implementation

Abstract: The paper presents AMGCL—an opensource C++ library implementing the algebraic multigrid method (AMG) for solution of large sparse linear systems of equations, usually arising from discretization of partial differential equations on an unstructured grid. The library supports both shared and distributed memory computation, allows to utilize modern massively parallel processors via OpenMP, OpenCL, or CUDA technologies, has minimal dependencies, and is easily extensible. The design principles behind AMGCL are discussed and it is shown that the code performance is on par with alternative implementations.

On Inverse Boundary Value Problem for a Fredholm Integro-Differential Equation with Degenerate Kernel and Spectral Parameter

Abstract: In this paper are considered the questions of unique solvability and redefinitions of a nonlocal inverse problem for the Fredholm integro-differential equation of the second order with degenerate kernel, integral condition, and spectral parameter. Calculations of the value of the spectral parameter are reduced to the solve of trigonometric equations. Systems of algebraic equations are obtained. The singularities that arose in determining arbitrary constants are studied. A criterion for unique solvability of the problem is established and the corresponding theorem is proved.

Spectral Features of the Solving of a Fredholm Homogeneous Integro-Differential Equation with Integral Conditions and Reflecting Deviation

Abstract: The problems of solvability and construction of solutions of a nonlocal boundary value problem for the second-order Fredholm integro-differential equation with degenerate kernel, integral conditions, spectral parameters and reflecting deviation are considered. Using the method of the degenerate kernel, the boundary value problem is integrated as an ordinary differential equation. When we define the arbitrary integration constants there are possible five cases with respect to the first spectral parameter. Calculated values of the spectral parameter for each case. Further, the problem is reduced to solving systems of linear algebraic equations. Irregular values of the second spectral parameter are determined. At irregular values of the second spectral parameter the Fredholm determinant is degenerate. Other values of the second spectral parameter, for which the Fredholm determinant does not degenerate, are called regular values. Taking the values of the first spectral parameter into account for regular values of the second spectral parameter the corresponding solutions were constructed for each of five cases. The stability of the solution of the boundary value problem for given values in integral conditions is proved. The conditions under which the solution of the boundary value problem will be small are studied. For the irregular values of the second spectral parameter each of the five cases is checked separately. The orthogonality conditions are used. Cases are determined in which the problem has an infinite number of solutions and these solutions are constructed. For other cases, the absence of nontrivial solutions of the problem is proved.

Optimization of Thermal Protection Panels Subjected to Intense Heating and Mechanical Loading

Abstract: In this work we solve a higly-nonlinear structural optimization problem for the sandwich panel with external thermal protection layer that can be used in the spacecraft systems. Objective function of the problem is the mass per unit area of the panel. Constraints are formulated based on the simplified analytical solutions of structural mechanics and heat transfer problems, which are suitable for the preliminary design considerations. The set of design variables includes the geometric parameters of the panel and additional microstructural parameter—porosity of the heat protection material. Direct random search and simulated annealing method are applied to solve considered problem. Change of limit states and optimal configurations of the panel are studied for different levels of the mechanical loading.

Joint Supercomputer Center of the Russian Academy of Sciences: Present and Future

Abstract: Joint Supercomputer Center of the Russian Academy of Sciences (JSCC RAS) is the leading supercomputer center for the Russian Academy of Sciences. JSCC RAS uses new technology which is particularly based on native solutions that provide ultra-high dense layout of nodes in the computational field and energy efficiency. JSCC RAS offers users the latest architecture of computing nodes and communications infrastructure. The center has advanced energy-efficient “hot” and “cold” water-cooling systems and a wide range of engineering equipment, a system for monitoring and managing computational resources serving a distributed network of scientific supercomputer centers, a domestic system for scheduling and managing jobs, software development and maintenance tools, application packages for high-performance computing. The paper presents the analysis of the current state of JSCC RAS, and a review of its development plans in the main scientific and practical directions.

Non-Markovian Evolution of Multi-level System Interacting with Several Reservoirs. Exact and Approximate

Abstract: An exactly solvable model for the multi-level system interacting with several reservoirs at zero temperatures is presented. Population decay rates and decoherence rates predicted by exact solution and several approximate master equations, which are widespread in physical literature, are compared. The space of parameters is classified with respect to different inequalities between the exact and approximate rates.

Higher-order Corrections to the Redfield Equation with Respect to the System-bath Coupling Based on the Hierarchical Equations of Motion

Abstract: The Redfield equation describes the dynamics of a quantum system weakly coupled to one or more reservoirs and is widely used in theory of open quantum system. However, the assumption of weak system-reservoir coupling is often not fully adequate and higher-order corrections to the Redfield equation with respect to the system-bath coupling is required. Here we propose a general method of derivation of higher-order corrections to the Redfield quantum master equation based on the hierarchical equations of motion (HEOM). Also we derive conditions of validity of the Redfield equation as well as the additional secular approximation for it.

Maximization of the Overlap between Density Matrices for a Two-Level Open Quantum System Driven by Coherent and Incoherent Controls

Abstract: The article considers a two-level open quantum system evolving under the action of coherent and incoherent controls using the general control method proposed in Phys. Rev. A. 73, 062102 (2006). Coherent control determines the Hamiltonian aspects of the dynamics whereas incoherent control determines the dissipative aspects. The goal is to find controls which steer the initial density matrix into a state which maximizes overlap with a predefined target density matrix. The controlled dynamics is represented as evolution in the Bloch ball and is analyzed analytically using Pontryagin maximum principle and numerically using optimization either in the functional space of controls (with the conditional and projected gradient methods) or through reduction to a finite-dimensional control space.

Analytical Solution of Fractional Burgers-Huxley Equations via Residual Power Series Method

Abstract: This paper is aimed at constructing fractional power series (FPS) solutions of fractional Burgers-Huxley equations using residual power series method (RPSM). RPSM is combining Taylor’s formula series with residual error function. The solutions of our equation are computed in the form of rapidly convergent series with easily calculable components using Mathematica software package. Numerical simulations of the results are depicted through different graphical representations and tables showing that present scheme are reliable and powerful in finding the numerical solutions of fractional Burgers-Huxley equations. The numerical results reveal that the RPSM is very effective, convenient and quite accurate to time dependence kind of nonlinear equations. It is predicted that the RPSM can be found widely applicable in engineering.

Application of Hypergeometric Functions of Two Variables in Wireless Communication Theory

Abstract: Hypergeometric functions of several variables are widely used in applied problems, in particular, in the mathematical theory of communication in the problems of calculating the error probabilities of receiving signal constructions. Error probabilities can often be expressed through various types of special functions. One of the most important functions of this kind is the Marcum function and its generalization ℋ-function.

Consistent Equations of Nonlinear Multilayer Shells Theory in the Quadratic Approximation

Abstract: For the laminated shells on the basis of the Timoshenko model, taking into account the transversal compression used for each layer, two versions of two-dimensional equations describing geometrically nonlinear deformation for arbitrary displacements and small strains are constructed. They are based on previously proposed consistent relationships of the non-linear elasticity theory, usage of which does not lead to the appearance of “false” bifurcation solutions. The first version corresponds to the contact problem statement, in accordance with which the contact stresses into the contact points of the layers as unknowns are introduced, and the second version corresponds to the preliminary satisfaction of the kinematic coupling relations for the layers along the displacements. An example is given of the application of the derived equations for solving the linear problem of a plane stress-strain state of a straight beam under the action of normal surface forces applied to the front boundary surfaces is given.

Identification of Non-stationary Load Upon Timoshenko Beam

Abstract: This paper investigates an inverse non-stationary problem of the restoration of the spatial law of a homogeneous isotropic Timoshenko beam of finite length. Hinge support conditions are used as boundary conditions. Initial conditions are assumed to be zero. It is assumed that of the beam’s ends is fitted with sensors which in the course of corresponding experiment register the amount of deflection of the beam at the sensor points. The method of the solution of a direct problem is based on the principle of superposition where the deflection of the beam is associated with the space load the beam is exposed to, by means of an integral operator by the spatial coordinate and time. The kernel of such operator is so called influence function. This function is a fundamental solution of a system of differential equations of motion of the study beam. The construction of such solution represents a separate problem. The influence function is found by means of Laplace time transformation and expansion into Fourier series in a system of the problem’s eigenfunctions. The solution of the inverse problem at the first stage reduces to a system of algebraic equations for vector operator whose components are time convolutions of the coefficients of expansion series for an influence function with the desired coefficients of expansion of the load in a Fourier series. At the same time, the components of the vector of the rights parts are time dependencies registered by the sensors. The resulting system is ill-conditioned [1]. The second stage serves to resolve independent Volterra integral equations of the first kind for the desired coefficients of Fourier serials for the load.

Energy-Сonstrained Diamond Norms and Quantum Dynamical Semigroups

Abstract: In the developing theory of infinite-dimensional quantum channels the relevance of the energy-constrained diamond norms was recently corroborated both from physical and information-theoretic points of view. In this paper we study necessary and sufficient conditions for differentiability with respect to these norms of the strongly continuous semigroups of quantum channels (quantum dynamical semigroups). We show that these conditions can be expressed in terms of the generator of the semigroup. We also analyze conditions for representation of a strongly continuous semigroup of quantum channels as an exponential series converging w.r.t. the energy-constrained diamond norm. Examples of semigroups having such a representation are presented.

Mathematical Modeling of the Coronary Circulation During Cardiac Pacing and Tachycardia

Abstract: In this work, we consider blood flow in the coronary arteries during the abnormal heart rhythm conditions due to the cardiac pacing and pathological type of the tachycardia. We adapt the ID model of haemodynamics to the coronary circulation conditions, which are characterized by the periodic vessels contraction. The range of the convergence of the Newton’s method is studied for the boundary conditions at the outlets of the terminal vessels. Simulation of the cardiac pacing conditions supports the hypothesis, that right ventricle pacing produce the most pronounced effect to the coronary flow. We also observe, that 0.03 s delay during right ventricular pacing produce substantial increase of the systolic blood flow in the left coronary artery. Simulations of the tachycardia conditions show, that the ratio between systolic and diastolic phase is a critical parameter for the possible ischemia development. We also observe, that the supraventricular tachycardia in paediatric patients results in the distinctive decrease of the coronary flow at some heart rates.

Assessments of the Economic Sectors Needs in Digital Technologies

Abstract: An assessment of the economic sectors needs for the solutions based on end-to-end technologies (E2ET) using the Technology readiness level (TRL) and Manufacture readiness level (MRL) metrics has been carried out. The created “E2ET-needs map” matrix shows the needs of nine sectors of the economy in nine blocks of E2ET. It reflects the priorities of the required state support for the development of E2ET in the different economy sectors. It was considered that each block of E2ET includes dozens of sub-technologies. A 3-level tree of about 30 needs assessment criteria were built. Given the presence of many non-quantitative factors that characterize the E2ET-needs the author’s convergent approach including the cognitive modelling and network expertise (e-Expertise) technology was applied. The approach is based on the combination of methods of inverse problems solving on topological spaces, controllable thermodynamics, and category theory. For synthesis and verify cognitive models and the reliability of the experts’ work the Big Data analysis was exploited. Domestic databases of media publications were used as verification arrays. The approach ensures the creation of the necessary conditions for achieving the strategic goals of the development of the digital economy in Russia using the E2ET. The rules of regulation of the annual monitoring of the E2ET needs under consideration were proposed.

Vectorization of High-performance Scientific Calculations Using AVX-512 Intruction Set

Abstract: Modern calculation codes used in supercomputing are very demanding of computing resources. For their effective appliance requires the use of parallelization at all levels, starting with the use of multiprocess and multi-threaded programming, and ending with vectorization. The AVX-512 instruction set, first introduced in Intel Xeon Phi Knights Landing and Intel Xeon Skylake microprocessors, opens up broad possibilities for vectorizing code and allows to speed up the execution of applications in several times. This article discusses some aspects of the application of vectorization in the program code of some kinds, which is found in high-performance scientific computing.

Exact Solutions of Three Nonclassical Equations, and Their Construction with Maple System

Abstract: Since the second half of the twentieth century, wide studies of Sobolev-type equations are undertaken. These equations contain items that are derivatives with respect to time of the second order derivatives of the unknown function with respect to space variables. They can describe nonstationary processes in semiconductors, in plasm, phenomenons in hydrodynamics and other ones. It is important to notice that wide studies of qualitative properties of solutions of Sobolev-type equations exist. Namely, results about existense and uniqueness of solutions, their asymptotics and blow-up are known. But there are few results about exact solutions of Sobolev-type equations. There are books and papers about exact solutions of partial equations, but they are devoted mainly to classical equations, where the first or second order derivative with respect to time or the derivative with respect to time of the first order derivative of the unknown function with respect to the space variable is equal to a stationary expression. Therefore it is interesting to study exact solutions of Sobolev-type equations. In the present paper, three nonclassical nonlinear partial equations are studied. Some results about existense, uniqueness and blow-up of their solutions are already known. Here, we construct some classes of their exact solutions with the help of Maple System. We use the method of travelling waves, the method of short-cut decompositions and construction of solutions of some special forms. Also, we discuss the way of realizing of these investigations with the help of Maple system of computer mathematics.

On Continuous Multifunctions in Ideal Topological Spaces

Abstract: The purpose of the present paper is to introduce the concepts of upper and lower *-continuous multifunctions. Several characterizations of upper and lower *-continuous multifunctions are investigated. The relationships between upper and lower *-continuous multifunctions and the other types of continuity are discussed.

Mathematical Physics Branches: Identifying Mixed Type Equations

Abstract: The article focuses on the problem of defining paradigmatic relations between definitions of certain fields in mathematical physics. The ultimate goal is to outline the hierarchical relations between the terms that can be used when searching on the mathematical resources along with additional classification parameters set in secondary documents. A thesaurus entry is selected as an information model. The thesaurus was formed by analyzing the original works of classics of mathematical analysis and differential calculus, and a representative list of articles was organized for that purpose. Following the example of thesaurus on the ‘problem of mixed type equations’ domain, a way of employing formulas in a mathematical article search is proposed. The paper covers a work script of a user, who is familiar with the subject domain and deals with papers done with the help of TeX-notation. A natural document indexing mechanism is set by key words in cited secondary documents. Such an approach helps to specify the search query with mathematical notation regardless the source language. The semantic links effect is based on usage of terms from the mathematical subject domain thesaurus stored with formulas that serve as a background for a mathematical search. It results in lower level of information noise and reduced search time.

Levy Laplacian on Manifold and Yang—Mills Heat Flow

Abstract: A covariant definition of the Levy Laplacian on an infinite dimensional manifold is introduced. It is shown that a time-depended connection in a finite dimensional vector bundle is a solution of the Yang—Mills heat equations if and only if the associated flow of the parallel transports is a solution of the heat equation for the covariant Levy Laplacian on the infinite dimensional manifold.

Embedding Semigroup C*-algebras into Inductive Limits

Abstract: The note is concerned with inductive systems of Toeplitz algebras and their *-homomorphisms over arbitrary partially ordered sets. The Toeplitz algebra is the reduced semigroup C*-algebra for the additive semigroup of non-negative integers. It is known that every partially ordered set can be represented as the union of the family of its maximal upward directed subsets indexed by elements of some set. In our previous work we have studied a topology on this set of indexes. For every maximal upward directed subset we consider the inductive system of Toeplitz algebras that is defined by a given inductive system over an arbitrary partially ordered set and its inductive limit. Then for a base neighbourhood Ua of the topology on the set of indexes we construct the C*-algebra $$\mathfrak{B}$$ a which is the direct product of those inductive limits. In this note we continue studying the connection between the properties of the topology on the set of indexes and properties of inductive limits for systems consisting of C*-algebras $$\mathfrak{B}$$ a and their *-homorphisms. It is proved that there exists an embedding of the reduced semigroup C*-algebra for a semigroup in the additive group of all rational numbers into the inductive limit for the system of C*-algebras $$\mathfrak{B}$$ a.

On the Eigen Frequencies of Rectangular Resonator with a Hole in the Wall

Abstract: The rectangular waveguide is attached to a hole in a wall of rectangular resonator and the corresponding to the hole part of the waveguide boundary is cross-section of the waveguide. The problem of excitation of the resonator by an eigen wave of the waveguide is investigated. The condition on a hole is obtained from the condition defining the wave in the waveguide, outgoing from cross-section. The initial diffraction problem is reduced to some infinite set of linear algebraic equations. Numerical experiment shows that the dependence of expansion coefficients of the field in the resonator on frequency of exciting wave has resonant nature. We propose to use the real values of the resonance frequencies of the running on the hole wave as the initial approximations for eigen frequencies of the resonator with a hole in the wall.

On Linear Structure of Non-commutative Operator Graphs

Abstract: We continue the study of non-commutative operator graphs generated by resolutions of identity covariant with respect to unitary actions of the circle group and the Heisenber-Weyl group as well. It is shown that the graphs generated by the circle group has the system of unitary generators fulfilling permutations of basis vectors. For the graph generated by the Heisenberg-Weyl group the explicit formula for a dimension is given. Thus, we found a new description of the linear structure for the operator graphs introduced in our previous works.

Acoustic Wave Incidence on a Medium Containing Multifractional Bubbly Liquid

Abstract: The mathematical model that determines reflection and transmission of acoustic wave through a medium containing multifractional bubbly liquid is presented. For the water-water with bubbles-water-waterwith bubbles-water model the wave reflection and transmission coefficients are calculated. It is revealed that the special dispersion and dissipative properties of the layer of bubbly liquid can significantly influence on the reflection and transmission of acoustic waves in multilayer medium.

Numerical Method for Solving the Inverse Problem of Nonisothermal Filtration

Abstract: In this work a mathematical model of thermohydrodynamic processes occurring in the oil reservoir and the horizontal well bore is developed. On the basis of the proposed model and regularization methods, a computational algorithm for solving the inverse coefficient problem is proposed. Data of temperature changes, registered simultaneously by several depth gauges installed in the different locations of the horizontal part of the well bore, is taken as the initial information.

The Algorithm for Decision-Making Supporting on the Selection of Processing Means for Big Arrays of Natural Language Data

Abstract: In this paper decision support algorithm for choosing the processing means of natural language big data arrays is proposed. In the process the algorithm uses the program for evaluating the effectiveness of text analyzers. This program is based on the operation of a fuzzy choice system, which serves to calculate the integral indicator of the text analyzer effectiveness. The quality and efficiency of getting answers to test questions are taken into account when evaluating the effectiveness of text analyzers.

Discovery of Time Series Motifs on Intel Many-Core Systems

Abstract: A motif is a pair of subsequences of a longer time series, which are very similar to each other. Motif discovery is applied in a wide range of subject areas involving time series: medicine, biology, entertainment, weather prediction, and others. In this paper, we propose a novel parallel algorithm for motif discovery using Intel MIC (Many Integrated Core) accelerators in the case when time series fit in the main memory. We perform parallelization through thread-level parallelism and OpenMP technology. The algorithm employs a set of matrix data structures to store and index the subsequences of a time series and to provide an efficient vectorization of computations on the Intel MIC platform. The experimental evaluation shows the high scalability of the proposed algorithm.

On Extensions of Semigroups and Their Applications to Toeplitz Algebras

Abstract: The paper deals with the normal extensions of cancellative commutative semigroups and the Toeplitz algebras for those semigroups. By the Toeplitz algebra for a semigroup S one means the reduced semigroup C*-algebra C*(S). We study the normal extensions of cancellative commutative semigroups by the additive group ℤn of integers modulo n. Moreover, we assume that such an extension is generated by one element. We present a general method for constructing normal extensions of semigroups which contain no non-trivial subgroups. The Grothendieck group for a given semigroup and the group of all integers are involved in this construction. Examples of such extensions for the additive semigroup of non-negative integers are given. A criterion for a normal extension generated by an element to be isomorphic to a numerical semigroup is given in number-theoretic terms. The results concerning the Toeplitz algebras are the following. For a cancellative commutative semigroup S and its normal extension L generated by one element, there exists a natural embedding the semigroup C*-algebra C*(S) into C*(L). The semigroup C*-algebra C*(L) is topologically ℤn-graded. The results in the paper are announced without proofs.

Inductive Limits for Systems of Toeplitz Algebras

Abstract: By the Toeplitz algebra we mean the reduced semigroup C*-algebra for the additive semigroup of non-negative integers. This article deals with inductive systems of Toeplitz algebras over arbitrary directed sets. For such a system the family of its connecting injective *-homomorphisms is defined by a set of natural numbers satisfying a factorization property. The motivation for the study of those inductive systems comes from our previous work on the inductive sequences of Toeplitz algebras defined by sequences of numbers and the limit automorphisms for the inductive limits of such sequences. We show that there exists an isomorphism in the category of unital C*-algebras and unital *-homomorphisms between the inductive limit of an inductive system of Toeplitz algebras over a directed set defined by a set of natural numbers and a reduced semigroup C*-algebra for a semigroup in the group of all rational numbers. The inductive systems of Toeplitz algebras over arbitrary partially ordered sets defined by sets of natural numbers are also studied.

Projections and Traces on von Neumann Algebras

Abstract: Let P, Q be projections on a Hilbert space. We prove the equivalence of the following conditions: (i) PQ + QP ≤ 2(QPQ)p for some number 0 < p ≤ 1; (ii) PQ is paranormal; (iii) PQ is M*-paranormal; (iv) PQ = QP. This allows us to obtain the commutativity criterion for a von Neumann algebra. For a positive normal functional φ on von Neumann algebra $$\mathcal{M}$$ it is proved the equivalence of the following conditions: (i) φ is tracial; (ii) φ(PQ + QP) ≤ 2φ((QPQ)p) for all projections P,Q ∈ $$\mathcal{M}$$ and for some p = p(P, Q) ∈ (0,1]; (iii) φ(PQP) ≤ φ(P)1/pφ(Q)1/q for all projections P, Q ∈ $$\mathcal{M}$$ and some positive numbers p = p(P, Q), q = q(P, Q) with 1/p+ 1/q = 1, p ≠ 2. Corollary: for a positive normal functional φ on $$\mathcal{M}$$ the following conditions are equivalent: (i) φ is tracial; (ii) φ(A + A*) ≤ 2φ(∣A*∣) for all A ∈ $$\mathcal{M}$$ .