Computer Algebra Methods Research Articles

Computer Algebra Methods for Searching the Stationary Motions of the Connected Bodies System Moving in Gravitational Field

Computer Algebra Methods for Searching the Stationary Motions of the Connected Bodies System Moving in Gravitational Field

A Mathematical Model for a Transmissible Disease with a Variant

The outbreak of the Coronavirus (COVID-19) pandemic around the world has caused many health and socioeconomic problems, and the identification of variants like Delta and Omicron with similar and often even more transmissible modes of transmission has motivated us to do this study. In this article, we have proposed and analyzed a mathematical model in order to study the effect of health precautions and treatment for a disease transmitted by contact in a constant population. We determined the four equilibria of the system of ordinary differential equations representing the model and characterized their existence using exact methods of algebraic geometry and computer algebra. The model is studied using the stability theory for systems of differential equations and the basic reproduction number R 0 . The stability of the equilibria is analyzed using the Lienard-Chipart criterion and Lyapunov functions. The asymptotic or global stability of endemic equilibria is established, and the disease-free equilibrium is globally asymptotically stable if R 0 < 1 . Model simulation is done with Python software to study the effects of health precautions and treatment, and the results are analyzed. It is observed that if the rate of treatment and compliance with health precautions are high, the number of infections decreases in the classes of infectious and is canceled out over time. It is concluded that the high treatment rate accompanied by a suitable rate of compliance with health precautions allows for the control the disease.

Positive dimensional parametric polynomial systems, connectivity queries and applications in robotics

In this paper we introduce methods and algorithms that will help us solve connectivity queries of parameterized semi-algebraic sets. Answering these connectivity queries is applied in the design of robotic structures having similar kinematic properties (e.g. topology of the kinematic-singularity-free space). From these algorithms one also obtain solutions to connectivity queries of a specific parameter which is in turn related to kinematic-singularity free path-planning of a specific manipulator belonging to the family of robots with these properties; i.e. we obtain paths joining two given singularity free configurations lying in the same connected component of the singularity-free space.We prove in the paper how one reduces the problems related to connectivity queries of parameterized semi-algebraic sets to closed and bounded semi-algebraic sets. We then design an algorithm using computer-algebra methods for “solving” positive dimensional polynomial system depending on parameters. The meaning of solving here means partitioning the parameter space into semi-algebraic components over which the number of connected components of the semi-algebraic set defined by the input system is invariant. The complexity of this algorithm is singly exponential in the dimension of the ambient space. The algorithm scales enough to analyze automatically the family of UR-series robots.Finally we provide manual analysis of the family of UR-series robots, proving that the number of connected components of the complementary of kinematic singularity set of a generic UR-robot is eight.

Computer algebra methods for polynomial system solving at the service of image-based visual servoing

Visual servoing refers to different methods used for robot motion control based on data from computer vision, in general the projection of a number of characteristics from the scene on a camera mounted on the robot. When the control is carried out in the space of parameters extracted from the image, we refer to Image-Based Visual Servoing (IBVS).

Accuracy of the Laplace transform method for linear neutral delay differential equations

In this paper, we study and solve linear neutral delay differential equations. We investigate the reliability and accuracy of applying the Laplace transform to obtain the solutions of linear neutral delay differential equations. These types of equations are more difficult to solve because the time delay appears in the derivative of the state variable. We rely on computer algebra and numerical methods to determine the poles which are required for computing the inverse Laplace transform. The form of the resulting solution is a non-harmonic Fourier series. A sufficient degree of accuracy can often be achieved by using a relatively small number of terms in the associated truncated series. We compare these solutions with the solutions obtained by the classical method of steps and numerical solutions obtained by the discretization of the linear delay differential equations. It is shown that the Laplace transform method provides very reliable and accurate solutions.

A new method based on the Laplace transform and Fourier series for solving linear neutral delay differential equations

In this paper, we present a new method for solving linear neutral delay differential equations. We derive and illustrate the main features of this novel approach, that combines the Laplace transform method with (harmonic) Fourier series theory. Linear neutral delay differential equations are generally more difficult to solve because the time delay appears in the derivative of the state variable. We rely on computer algebra and numerical methods to implement the method. In addition, we derive an approximate formula for the location of the complex poles, which are required for computing the inverse Laplace transform. The form of the resulting solution, when only the Laplace method is used is a non-harmonic Fourier series. The accuracy of this solution can be improved by including more terms in the associated truncated series, but the convergence to the correct solution is slow. The main goal of this paper is to present a modified method which enables us to account for the terms which are excluded from these truncated Laplace series. That is, the terms in the tail of the infinite series. We include several examples where we compare the solutions generated by the standard Laplace method and the proposed Laplace-Fourier approach. Both solutions require using Cauchy’s residue theorem and finding the real and complex poles. It is shown that the Laplace-Fourier solution provides more accurate solutions than the conventional Laplace transform solution. Finally, since the Laplace-Fourier method generates a solution which is valid for all times, it allows us to accurately approximate the solution at any point with a single calculation.

Evolution equations of the multi-planetary problem with variable masses

AbstractWe investigated the influence of the variability of the masses of planets and the parent star on the dynamic evolution of n planetary systems, considering that the masses of bodies change isotropically with different rates. The methods of canonical perturbation theory, which developed on the basis of aperiodic motion over a quasi-conical cross section and methods of computer algebra were used. 4n evolutionary equations were obtained in analogues of Poincare elements. As an example, the evolutionary equations of the three-planet exosystem K2 − 3 were obtained explicitly, which is a system of 12 linear non-autonomous differential equations. Further, the evolutionary equations will be investigated numerically.

Complex associated to some systems of PDE

In this paper, we construct the complex associated to an overdetermined system of PDE of order one with constant coefficients using the theory of exterior differential system of Cartan (see [1]). We show that if the tableau associated to the system of PDE is in involution then the complex contains only operators of order one given by the successive torsions of the system (see section 2 for the details). Moreover in section 3, we give a simple direct proof of “the involution of tableau” by prolongation process in this particular case (theorem of Cartan-Kuranishi in this context). Finally in the last section, we construct the complex associated to the Cauchy-Fueter operator (in fact, more general system of PDE containing Cauchy-Fueter equations) using the previous results. We can note that the system of PDE associated to the Cauchy-Fueter operator is not in involution contrarily of the De Rham or Dolbeault operators (this complex was first constructed by [5] by using computer algebra methods, and by [6] and [7] with the help of theory of Leray's spectral sequence). We can add that the key point to obtain many of the above results is a useful characterization of the involution of a tableau given in section 2.

On Algorithm of Integrability Classification of the Nonlinear Dynamical Systems via Computer Algebra Methods

There is developed an algorithm to classify integrable nonlinear dynamical systems via Wolfram Mathematica. The hierarchy of conservation laws for the nonlinear dynamical system can be cal-culated by this algorithm. There are demonstrated some modifications of nonlinear Korteweg-de Vries equations integrated by inverse scatering method.

Dynamic Simulation of Quantum Entanglement in Finite Quantum Mechanics: A Computer Algebra Approach

In this paper, we investigate the behavior of quantum entanglement in the process of unitary evolution in constructive models of multicomponent quantum systems. Symmetry groups of quantum systems that admit the occurrence of geometric structures associated with quantum entanglement are described. Algorithms for dynamic simulation of quantum entanglement are based on methods of computer algebra and computational group theory. Some examples of practical computations are presented.

Invariant Coordinate Subspaces of Normal Form of a System of Ordinary Differential Equations

For systems of ordinary differential equations (ODEs) with a nondegenerate linear part in the general and Hamiltonian cases, the problem of finding invariant coordinate subspaces in the coordinates of the normal form calculated in the vicinity of the equilibrium is stated. Conditions for the existence of such invariant subspaces in terms of the resonant relations between the eigenvalues of the linear part of the system are obtained. An algorithm for finding the resonant relations between the eigenvalues without their explicit calculation is described; this algorithm substantially uses computer algebra methods and the q-analog of the polynomial subresultants. The implementation of this algorithm in three popular computer algebra systems—Mathematica, Maple, and SymPy—is discussed. Interesting model examples are provided.

Modeling critical phenomena in a generalized van der Pol model

The van der Pol model has a long history of being used in both the physical and biological sciences. This paper considers modeling critical phenomena in one of three-time scales fast-slow autonomous dynamical systems. We investigate a model that is a generalization of the classical van der Pol model. In the framework of this paper, we construct an invariant manifold with variable stability for the generalized van der Pol model. Expansions of the manifold and corresponding gluing function were constructed using the flow curvature method. We show the existence of a sufficiently smooth invariant surface (a black swan surface) for the model. Computer algebra methods are used for the quantitative analysis of the model.

Solving the Cauchy Problem for a Two-Dimensional Difference Equation at a Point Using Computer Algebra Methods

An algorithm for finding the solution to the Cauchy problem for a two-dimensional difference equation with constant coefficients at a point using computer algebra is described. In the one-dimensional case, solving the Cauchy problem is easy; however, already in the two-dimensional case the number of unknowns rapidly increases at each step. To automate the process of computing the solution to the Cauchy problem for a two-dimensional difference equation with constant coefficients at a given point, an algorithm in MATLAB is developed in which the input data are the matrix of coefficients obtained on the basis of the structure of the two-dimensional polynomial difference equation, coordinates of the points that specify the structure and the size of the matrix of initial data, and the matrix of the initial data. The algorithm produces the solution to the Cauchy problem for the given two-dimensional difference equation at the given point.

International Conference «Marchuk Scientific Readings 2020» (MSR-2020), dedicated to the 95th anniversary of the birthday of Academician Guri I. Marchuk

PrefaceThese Conference Proceedings contain the selected papers of the International Conference «Marchuk Scientific Readings 2020» (MSR-2020) held on October 19 - 23, 2020, in Akademgorodok, Novosibirsk, Russia.The conference is dedicated to the 95th anniversary of the birthday of Academician Guri I. Marchuk. He was the founder and the first Director of the Computer Centre of the Siberian Branch of the USSR Academy of Sciences (currently, the Institute of Computational Mathematics and Mathematical Geophysics SB RAS). In different periods, he was the Chairman of the Siberian Branch of the USSR Academy of Sciences, Chairman of the USSR State Committee for Science and Technology, the last President of the USSR Academy of Sciences, Chairman of the Council of the USSR Academy of Sciences for Space. Hero of Socialist Labour and Lenin Prize winner, Academician Guri I. Marchuk was an outstanding scientist, recognized specialist in computational mathematics and mathematical modelling, nuclear reactor theory, atmospheric physics and geophysics, space, and environmental sciences, medicine, and immunology. He authored more than 350 scientific papers and monographs.Academician Marchuk was a distinguished organizer of science. He created a huge scientific school and was actively working on the introduction of scientific achievements to industry. Guri I. Marchuk made a significant contribution to the development of nuclear reactor technology. He was one of the pioneers of space exploration, the organizer of the international project “World ocean”, international systems for remote sensing of the Earth from space and space station flights to Venus.The purpose of the «Marchuk Scientific Readings» conference series is to gather specialists in numerical analysis, applied mathematics, mathematical modelling, computational technologies, information systems, artificial intelligence, and machine learning to discuss, identify and systematize current results, applications, and promising directions in computational mathematics and computer science.The conference program included 31 plenary, 298 oral, and 46 poster presentations. They were presented in 11 sections and two mini-symposia.The conference sections were:• Numerical solution of differential and integral equations• Computational algebra and approximation methods• Numerical statistical modeling and Monte-Carlo methods• Mathematical geophysics• Mathematical models of atmospheric physics, ocean and environment• Programming• Supercomputer computations• Inverse problems• Informational and computational systems• Machine learning and artificial intelligence methods• Computer biology, medicine and biotechnologyTwo mini-symposia were held:• Computational geosciences across the scales• Particle-based methods: numerical aspects and modern applicationsAll contributions selected for publication in this volume have been peer-reviewed according to rigorous international standards. We are grateful to the reviewers for maintaining the manuscripts’ standards and quality throughout the reviewing process. However, the authors take full responsibility for the content of their papers.The event was organized by the Institute of Computational Mathematics and Mathematical Geophysics SB RAS and Novosibirsk State University. It was supported by the Russian Foundation for Basic Research (under project 20-01-20025) and the Siberian Branch of the Russian Academy of Sciences. We gratefully acknowledge all our sponsors.EditorsCorresponding Member of the RAS Gennady A. MikhailovCorresponding Member of the RAS Sergey I. KabanikhinProfessor Yurii M. LaevskiiProfessor Vladimir V. PenenkoProfessor Boris M. GlinskyPh.D. Alexey V. Penenko

Problem Formulations, Solving Strategies, Implementation Methods & Applications of Selective Harmonic Elimination for Multilevel Converters

Multilevel converters are commonly recommended for converting DC power to AC as per load requirements owing to several advantages such as common mode voltage, less distorted input current, low switching frequency and reduced THD. However, the reliability, performance and efficiency of MLIs depend in particular on the switching technique used. The Selective Harmonic Elimination PWM approach is more efficient among existing modulation techniques to remove undesired harmonics from the output of MLI, since it operates at low switching frequency, particularly at fundamental frequency. The fundamental switching frequency operation of SHEPWM reduce the switching losses and improves inverter efficiency. The principal problem, however, is the empirical approach to non-linear equations. Numerous algorithms have been developed and implemented over the last decades including computational approaches, analytical methods, algebraic methods and optimization algorithms to solve the SHE equations and to eliminate the unintended harmonics. This article presented various aspects of SHE problem formulations, the comprehensive philosophy of operation of various SHE problem-solving approaches and the application of multilevel inverters. PWM waveforms, with single and multiple transitions, are examined and shown at each voltage level for solution frameworks that suit multiple solutions for pulse width modulation to allow researchers gain a deeper understanding and solution of SHE problems.

Computer Algebra Methods in Modeling the Dynamics of Particle Beams in Accelerators

This article is devoted to the methods of numerical and symbolical methods of calculations, which are based on perturbation theory and are implemented in terms of matrix formalism. These methods were used to solve various problems of modeling and optimization in accelerator physics. The basic theoretical principles are verified using methods to neural network technologies, machine learning and others.

An Algorithm for Constructing Irreducible Decompositions of Permutation Representations of Wreath Products of Finite Groups

We describe an algorithm for decomposing permutation representations of wreath products of finite groups into irreducible components. The algorithm is based on the construction of a complete set of mutually orthogonal projections to irreducible invariant subspaces of the Hilbert space of the representation under consideration. In constructive models of quantum mechanics, the invariant subspaces of representations of wreath products describe the states of multicomponent quantum systems. The suggested algorithm uses methods of computer algebra and computational group theory. The C implementation of the algorithm is capable of constructing irreducible decompositions of representations of wreath products of high dimensions and ranks. Examples of calculations are given. Bibliography: 15 titles.

Exact solution of Boussinesq equations for propagation of nonlinear waves

In this paper, we consider two Boussinesq models that describe propagation of small-amplitude long water waves. Exact solutions of the classical Boussinesq equation that represent the interaction of wave packets and waves on solitons are found. We use the Hirota representation and computer algebra methods. Moreover, we find various solutions for one of the variants of the Boussinesq system. In particular, these solutions can be interpreted as the fusion and decay of solitary waves, as well as the interaction of more complex structures.

Computational aspects of Burnside rings part II: important maps

The Burnside ring $$\mathcal {B}(G)$$ of a finite group G, a classical tool in the representation theory of finite groups, is studied from the point of view of computational algebra. In the first part (cf. Kreuzer and Patil, Beitr Algebra Geom 58:427–452, 2017) we examined the ring theoretic properties of $$\mathcal {B}(G)$$ using the methods of computer algebra. In this part we shift our focus to important maps between two Burnside rings and make several well-known maps of representation theory explicitly computable. The inputs of all our algorithms are the tables of marks of the two groups, and the outputs are matrices of integers representing the maps via their image in the ghost ring. All algorithms have been implemented in the computer algebra system ApCoCoA and are illustrated by applying them to explicit examples. Especially, we study the restriction and induction maps, the projection and inflation maps, the conjugation isomorphism, and the Frobenius-Wielandt homomorphism.

Distributive laws between the operads Lie and Com

Using methods of computer algebra, especially, Gröbner bases for submodules of free modules over polynomial rings, we solve a classification problem in theory of algebraic operads: we show that the only nontrivial (possibly inhomogeneous) distributive law between the operad of Lie algebras and the operad of commutative associative algebras is given by the Livernet–Loday formula deforming the Poisson operad into the associative operad.