Computer Algebra System Research Articles

The Skies Are Falling: Mathematics for Non-Mathematicians

Mathematical education, both mass education, and university education of non-mathematicians, are in an abominable state, and rapidly degrading. We argue that the instruction of non-mathematicians should be dramatically reformed both as substance and style. With traditional approach, such a transformation would take decades, with unclear results. But we do not have this time. The advent of Computer Algebra Systems gives the mathematics community a chance to reverse the trend. We should make a serious attempt to seize this opportunity. In the present paper we describe one such project of reform implemented at the St Petersburg State University.

COMPUTER ALGEBRA TOOLS FOR GEOMETRIZATION OF MAXWELL'S EUQATIONS

Calculations of optical devices in the geometrized Maxwell’s theory use well-known formalisms of general theory of relativity and differential geometry. In particular, for such calculations it is required to know the analytical form of the geodesic equations, which leads to the need to calculate a large number of monotonous mathematical expressions. One of the purposes of computer algebra is to facilitate the researcher’s work by automating cumbersome symbolic computations. Thus, the use of computer algebra systems seems to be quite an obvious way. Several free implementations of symbolic computations for the apparatus of general relativity are considered. A practical example of symbolic computations for the geometrized Maxwell’s theory is given.

Implementation of the Adams method for solving ordinary differential equations in the Sage computer algebra system

This work is devoted to the implementation and testing of the Adams method for solving ordinary differential equations in the Sage computer algebra system. The Sage computer algebra system has, to some extent, trivial means for numerical integration of ordinary differential equations, but at the same time, it is worth noting that this environment is convenient and practical for conducting computer experiments related to symbolic numerical calculations in it. The article presents the FDM package developed on the basis of the RUDN, which contains the developments of recent years, performed by M. D. Malykh and his students, for numerical integration of differential equations. In this package, attention is paid to the visualization of the calculation results, including the construction of various kinds of auxiliary diagrams, such as Richardson diagrams, as well as graphs of dependence, for example, the value of a function or step from a moment in time. The implementation of the Adams method will be considered from this package. In this article, this implementation of the Adams method will be tested on various examples of input data, and the method will also be compared with the Jacobi system. Exact and approximate values will be found and compared, and an estimate for the error will be obtained.

Изгиб секторальной пластины: использование систем компьютерной алгебры

Purpose: To investigate the stress-strain state of a thin homogeneous isotropic plate in the form of a sector by numerical-analytical method. To consider the possibility of using computer algebra systems (CAS) to calculate sectoral plates operating under bending due to a transverse load. To demonstrate the effectiveness of applying one of these CASs through the example of the Maple system for calculations using the Ritz method — performing analytical transformations when calculating the integral that determines the total potential energy functional, forming and solving the main resolving system of linear algebraic equations with respect to unknown numerical coefficients in the formula approximating the deflection of the plate, visualization of the obtained solution. Methods: A direct method is used to solve the variational problem of minimizing the functional of the total potential energy of deformation of a thin homogeneous isotropic plate in the form of a sector — the Ritz method. The solution is constructed in the form of a series in terms of basis functions. As basis functions, polynomial functions are chosen that exactly satisfy all boundary conditions. Results: An approximate numerical-analytical solution has been obtained for the problem of bending a sectoral plate in the form of a quarter of a circle, clamped along the contour and loaded with a uniformly distributed load. The effectiveness of using the Maple analytical computing system for solving the problem of bending a sectoral plate by the variational Ritz method is demonstrated. It is shown that the resulting solution quickly converges both for deflection and for bending moments and stresses. Practical significance: The algorithm proposed in the article for solving the problems of bending sectoral plates can be recommended for practical use in determining the stress-strain state of such plates.

On application of the new mapping method to magneto-optic waveguides having Kudryashov’s law of refractive index

This paper obtains solitons and other exact traveling wave solutions to magneto-optic waveguides that maintains Kudryashov’s law of refractive index using the new mapping method and computer algebraic systems (CAS) such as Maple or Mathematica. These solutions are important for understanding the behavior of waveguides in telecommunications, optical computing and sensing. Dark solitons, bright solitons, bright-dark solitons, combo solitons, singular solitons and periodic solutions are obtained. Further, some 2D and 3D graphs of the obtained exact traveling wave solutions are shown. Finally, comparison between our new results and the well-known results is made.

Normal forms of double Hopf oscillators with radial nonlinearities

We are concerned with the simplest normal forms of non-resonant double Hopf oscillators with radial nonlinearities. Due to the required Lie algebraic structure for preserving the radial structure in the normal form process, we appeal to the Lie algebra of radial and rotational transformation generators. The structure constants of the Lie algebra are presented, where we notice a Lie ideal structure for the rotational vector fields. This gives rise to an effective reduction technique through the use of its corresponding quotient Lie algebra. Yet, computations of kernel spaces of the generalized homological maps seem a formidable project and thus, we employ the multiple Lie brackets method to skip explicit kernel space derivations. The simplest normal forms and orbital normal forms of the nonlinear family are presented. We also address their parametric normal forms as multiple parametric perturbations of the corresponding vector fields. Transformation symmetry groups associated with truncated normal forms are also derived. An efficient algorithm is provided for the normal form computations using computer algebra systems. Finally, we derive state-feedback controllers for a two-arm vibrator using normal forms of radial double Hopf singularity. This is to illustrate how our normal forms can be applied on classical control engineering problems as a new alternative nonlinear controller technique to the existing methods in nonlinear control theory.

The method of manipulability control of a robot

The paper presents the results of a numerical experiment that demonstrates the effectiveness of a control algo-rithm that maintains a sufficiently high numerical value of the robot's manipulability function , which ensures «remoteness» from a singular configuration. The problem of approaching the end effector of the space manipula-tor Canadarm2, which has seven degrees of freedom (one redundant generalized coordinate), with an object mov-ing along a known trajectory, is considered. To the six parameters that determine the position and orientation of the gripper in absolute space, a «kinematic» function , depending on the relative rotation angles of the links, has been added. In this case, the kinematics equations became closed with respect to the rotation angles. The control law includes a condition for maintaining a certain numerical value that exceeds the minimum allowable value. Illustrations in the form of three-dimensional animation obtained on the basis of numerical integration of the equa-tions of controlled motion of the manipulator and the object were built using the Mathematica computer algebra system.

Some Reflections About the Success and Bibliographic Impact of the Dynamic Geometry System GeoGebra

The authors were surprised by the number of articles that used or cited the computer algebra system DERIVE more than 10 years after it was discontinued and developed a small bibliographic study about it, published in 2019. Now they address in a similar way the very successful dynamic geometry system GeoGebra that, although created 20 years ago, later than the other great dynamic geometry systems (Cabri Geometry II, The Geometer’s Sketchpad and Cinderella), has now dozens of millions of users around the world. Not surprisingly, the cites to GeoGebra in the well known bibliographic databases Scopus, Web of Science and Google Scholar show an impressive growth.

The stability of mathematics students’ beliefs about working with CAS

In Victoria, Australia, senior secondary mathematics students are expected to use technology and thus need to make decisions about using pen-and-paper (P&P) or technology when solving mathematics problems. The predominant technology is a Computer Algebra System (CAS). This study investigated the beliefs about CAS held by twelve Year 11 students as they learnt to use CAS and whether these beliefs were stable over time. These students held a range of beliefs related to the usefulness of CAS, speed of CAS compared to P&P, whether CAS is proper mathematics, choice of CAS or P&P, ease of use, the correctness of answers and solving problems in Mathematical Methods (i.e. the mathematics subject studied). Beliefs are often described as being stable (e.g. McLeod, 1992), but some researchers stress stability needs to be determined empirically rather than being seen as a characteristic of beliefs (e.g. Liljedahl et al., 2012). For this sample of students, stability (rather than instability) is a feature of students’ beliefs about CAS.

Uncertainty propagation, entropy, and relative entropy in the homogenization of some particulate composites

AbstractThe main idea of this work is an application of probabilistic entropy and also relative entropy in the numerical analysis of uncertainty propagation in the homogenization of some composite materials. The homogenization method is based on the determination of deformation energy for the representative volume elements computed with the use of some specific finite element method experiments. Uncertainty propagation concerns material and geometrical design parameters of particulate composites and is performed thanks to an application of polynomial responses; probabilistic moments of the effective tensor are computed via the iterative generalized stochastic perturbation technique and the semi‐analytical probabilistic method. Probabilistic entropy is determined according to Shannon theory, while relative entropies equations employed here follow mathematical models created by Kullback & Leibler, Jeffreys, Hellinger, and Bhattacharyya. Deterministic analyses have been performed with the use of the system ABAQUS, while all the remaining procedures have been programmed in the computer algebra system MAPLE.

The Computation of Generalized Invariants

In this paper, an algorithm is presented that calculates the module of generalized invariants. Generalized invariants are useful in the study of modular invariant theory, where the characteristic of the base field divides the order of the group. The structure theorems of generalized invariants were given by the author in 2016, and computational aspects of these results are studied in this note. Implementation codes are also provided for MAGMA, computational algebra system.

Skew Cyclic Codes over $\mathbb{Z}_{8}+u\mathbb{Z}_{8}+v\mathbb{Z}_{8}$

In this paper, we study the skew cyclic codes over the ring $S=\mathbb{Z}_{8}+u\mathbb{Z}_{8}+v\mathbb{Z}_{8}$, where $u^{2}=u$, $v^{2}=v$, $uv=vu=0$. We consider these codes as left $S[x,\theta]$-submodules and use the Gray map on $S$ to obtain the $\mathbb{Z}_{8}$-images. The generator and parity-check matrices of a free $\theta$-cyclic
 code of even length over $S$ are determined. Also, these codes are generalized to double skew-cyclic codes. We give some examples using Magma computational algebra system.

On the Computation of the Codimension of Map Germs Using the Lie Algebra Associated with a Restricted Left–Right Group

The codimension is an important invariant, which measures the complexity of map germs and play an important role in classification and recognition problems. The restricted A-equivalence was introduced to obtain a classification of reducible curves. The aim was to classify simple parameterized curves with two components, one of them being smooth with respect to the A-equivalence in characteristic p. In characteristic 0, the corresponding classification was given by Kolgushkin and Sadykov. The aim of this article is to present an algorithm to compute the codimension of germs of singularities under a restricted left–right equivalence (A-symmetry). We also give the implementation of this algorithm in the computer algebra system singular.

COMPUTER EXPERIMENT IN TEACHING MATHEMATICS

Mathematical experiment has always been a key source for mathematical discovery. Over the past 50 years, thanks to digital technologies, its role in mathematical research has grown significantly. Digital technologies have opened up fundamentally new opportunities for experimentation in mathematics education, bringing the majority of students of mathematical education closer to mathematical research. Such an approximation is especially desirable in the modern world, where it becomes possible thanks to digital technologies. The article discusses the results of the work of the authors over the past decades on the application of a computer mathematical experiment at different levels of school and university education. Particular attention is paid to dynamic geometry environments. The possibilities of using computer algebra systems are also considered. The project of schoolchildren’s work on generalizations of Napoleon’s theorem is considered in detail.

THE SKIES ARE FALLING: MATHEMATICS FOR NON-MATHEMATICIANS

Mathematical education, both mass education, and university education of non-mathematicians, are in an abominable state, and rapidly degrading. We argue that the instruction of non-mathematicians should be dramatically reformed both as substance and style. With traditional approach, such a transformation would take decades, with unclear results. But we do not have this time. The advent of Computer Algebra Systems gives the mathematics community a chance to reverse the trend. We should make a serious attempt to seize this opportunity. In the present paper we describe one such project of reform implemented at the St Petersburg State University.

Динамика плоского движения космического крана-манипулятора типа руки с учетом изгиба звеньев

The paper considers the plane motion dynamics of a three-link crane-manipulator consisting of two rods elastic in bending and a massive solid body attached to them at their end (load capture). All links are interconnected by hinge joints with the given (controlled) relative rotation angles. The first link is also connected to the mobile base. The system mathematical model was developed to calculate its non-stationary oscillations and dynamic loading of links at arbitrary in time and large relative angles of the links rotation as the solid bodies. The rods bending deformations (elastic transverse displacements and rotation angles of the axes) were considered to be insignificant (linear). Each rod ending was represented by the Ritz method using two given shape functions with the unknown coefficients, which were taken as the generalized coordinates. In this case, the “cantilever” bending forms of each rod with the given transverse displacement and angle of rotation at their ends were used. A simplified version was considered: each rod bending was represented by only one given form of its bending by a single moment applied at the end. System oscillation equations were obtained based on the principle of probable displacements in the generalized coordinates (four equations or two in a simplified form). Since rotation angles at the ends of the elastic rods were added to the given rotation angles of the non-deformable rods, total angles were left under the signs of sines and cosines without linearization for convenience of calculations and increasing accuracy. As a result, such equations of non-stationary oscillations remained non-linear. They were integrated by numerical methods using standard programs in one of the computer algebra systems. Examples are provided with the comparative results estimates obtained in the two-term and one-term approximations of the function in regard to the bending form of the system bending rod joints.

Calculation of a Strong Resonance Condition in a Hamiltonian System

A method for symbolic computation of a condition of existence of a third- and fourth-order resonance for investigations of formal stability of an equilibrium state of a multiparameter Hamiltonian system with three degrees of freedom in the case of general position is proposed. This condition is formulated in the form of zeros of a quasi-homogeneous polynomial of the coefficients of the characteristic polynomial of the linear part of the Hamiltonian system. Computer algebra (Gröbner bases of elimination ideals) and power geometry (power transformations) are used to represent this condition for various resonance vectors in the form of rational algebraic curves. Given a linear approximation of the characteristic polynomial in the space of its coefficients, these curves are used to obtain a description of a partition of the stability domain into parts in which there are no strong resonances. An example of a description of resonance sets for a two-parameter pendulum-type system is given. All computations are carried out in the computer algebra system Maple.

Massively parallel computation of tropical varieties, their positive part, and tropical Grassmannians

We present a massively parallel framework for computing tropicalizations of algebraic varieties which can make use of symmetries using the workflow management system GPI-Space and the computer algebra system Singular. We determine the tropical Grassmannian TGr0(3,8). Our implementation works efficiently on up to 840 cores, computing the 14763 orbits of maximal cones under the canonical S8-action in about 20 minutes.Relying on our result, we show that the Gröbner structure of TGr0(3,8) refines the 16-dimensional skeleton of the coarsest fan structure of the Dressian Dr(3,8), except for 23 orbits of special cones, for which we construct explicit obstructions to the realizability of their tropical linear spaces. Moreover, we propose algorithms for identifying maximal-dimensional cones which belong to positive tropicalizations of algebraic varieties. We compute the positive Grassmannian TGr+(3,8) and compare it to the cluster complex of the classical Grassmannian Gr(3,8).

IMPROVING EFFICIENCY OF SSCA KIDYM DYNAMICS PROGRAM MODULE BY USING AUTOMATICALLY GENERATED DLL-PROCEDURE AND BY OTHER MEANS

The paper presents the results of research on the significant increase in the speed of solving the problems of the dynamics of discrete mechanical systems of an arbitrary form with arbitrary links by a special system of computer algebra (SSCA) KiDyM. In particular, an algorithm for determining the parameters of the additional rotation of the central coordinate systems (CS) of solid bodies in the presence of a non-diagonal inertia tensor to transform them into the main central SCs, in which Euler's equations have the simplest form, is implemented. The main central moments of inertia of the body are determined. A simpler description of SC transformations of robot links with open kinematic chains is proposed. It consists in a separate description of the positions of the SC of the links and the positions of the center of mass and the central SC in relation to the SC of the link. This approach leads to localized description of the kinematics of the mechanism and complete information about the inertial parameters of the link, such as mass, inertia tensor, position of the central SC. The possibility of errors in model descriptions is thereby reduced. On the example of the model of the lower limbs of an human like robot, the gain in the number of machine operations of this description is shown. The use of the software-generated DLL procedure for saving the mathematical model of the system is analyzed in detail. If necessary, it is loaded from an external medium (disk) into the computer's memory during numerical integration. For this purpose, a special procedure has been created in SSCA KiDyM, which transforms the internal representation of analytical expressions of the computer algebra system into program code in C++, calls the appropriate compiler, which creates and saves the DLL-procedure on the disk. In the numerical integration block, the system can load it and calculate the right-hand sides of the Cauchy form of the dynamic equations by the Runge-Kutta procedure. The time gain of this integration method compared to direct calculations of formula trees in PC memory for various tasks is shown. The time gain increases with the increase in the complexity of the mechanical system. A criterion for automatically switching such an algorithm depending on the amount of C++ code in the DLL procedure is proposed.

Graphical User Interface for Design, Analysis, Validation, and Reporting of Continuous-Time Systems Using Wolfram Language

A graphical user interface presented is intended for fast design, symbolic analysis, accurate simulation, exact verification, and test report preparation. It helps to skip the gap between theory and practice in electrical engineering because the numeric analysis is usually approximate, and the power of symbolic systems has insufficient speed even, for simple engineering problems. The software is written using a computer algebra system that is free on small computers. The mathematical representation of the system can be obtained automatically from the schematic description. Further automated symbolic manipulations are possible according to the user’s aspirations.