Computer Algebra Software Research Articles

An experiment in the use of computer algebra in the classroom

This paper describes an expeirment to determine the effect of the use of a computer algebra system in the school classroom. A standard computer algebra package, muMath, was enhanced for use in the classroom and a comparison between it and conventional teaching methods was performed. The conclusion from the experiment indicates that such a system is a useful teaching aid in the classroom.

Computer algebra in nonlinear analyses of the straightline stability of combination vehicles

Recent developments in the theories of the dynamics of nonlinear systems and bifurcations provide a means for studying nonlinear oscillations and stability problems. The analysis methods require that equations of motion for the system of interest be derived in a certain form. Such derivation can involve substantial algebraic manipulation. However, by using recently developed computer algebra software, the entire derivation can be performed automatically by a computer. The level of automation associated with the analysis makes it practical for general use by engineers. An application from vehicle system dynamics is given as an example.

Interval estimation of a measure of diversity

Interval estimation of a well-known measure of heterogeneity or diversity is approached by fitting a distribution by the method of moments. Exact expressions for the moments of the distribution are derived and evaluated using a computer algebra package.

APPLICATIONS OF SYMBOLIC MANIPULATION FOR INVESTIGATING THE NONLINEAR DYNAMICS OF GROUND VEHICLES

SUMMARY Symbol manipulation tools can aid in the analysis of the nonlinear dynamics of ground vehicles in several ways. The equations of motion, derived automatically in symbolic form, can be numerically integrated to obtain a single simulated test. Alternatively, the nature of the overall nonlinear stability can be determined if the equations are derived in a certain form. Formulating equations for the stability analysis involves considerable algebraic manipulation, such that the analysis is practical only with computer aid. Recent improvements of computer hardware and recently developed computer algebra software now allow a level of automation that makes these analyses practical for general use by engineers. Applications from both rail and road vehicles are given in this paper as examples.

数式処理によるロボットの逆運動学問題の解法 （第１報） 幾何学的な不変量による解法

It is well known that the solution of the inverse kinematics for robotic manipulator requires a lot of labor and time. In this paper, they propose a new method to automatically generate the closed form inverse kinematics solutions for most industrial robots efficiently. This method is based on the subdivision of rigid displacement into elementary subspaces which have invariant geometric properties. Using the invariant geometric properties of the elementary substructures, kinematic invariant relations are obtained, which are independent of the joint variables of the substructures. These relations generate the symbolic inverse kinematic solutions by computer algebra software Mathematica. Their program has successfully solved the inverse kinematics of more than ten typical industrial manipulators.

On the Mathematical Modeling of ROV'S

Remotely operated underwater vehicles (ROV’s) are important tools for inspection and maintenance tasks in off-shore engineering and for submarine structures as well as for exploration and exploitation in deep sea mining. The communication, power, and control connection to the operation basis is usually performed by means of a tether cable (umbilical) which has a significant influence on the overall behavior of the system. In order to optimize the vehicle construction and to design an appropriate control system, knowledge about general characteristics of the dynamics of the system is required. The aim of this paper is to concentrate on the dynamic behavior of the cable and to outline a strategy to divide the cable into a quasistatic and dynamic part in order to reduce calculation time for on-line control purposes. For the numerical simulations hydrodynamic effects must properly be taken into account. In this fundamental investigation the cable is modeled as a planar multibody system with twenty elements; the generation of the symbolic equations of motion is performed by using computer algebra packages. The simulation of a typical robot manoeuver indicates the strong influence of the cable even for moderate currents.

Construction of singular integral equations for interacting straight cracks by using reduce

The method of singular integral equations has been applied to the solution of crack problems in plane and antiplane elasticity many times during the last 20 years. Here we revisit the case of an arbitrary number of interacting straight cracks in plane elasticity and we illustrate the possibility of constructing (algebraically) the corresponding system of singular integral equations by using computer algebra software. We present a procedure (computer program) using REDUCE as well as several examples of application of the present approach, extensions and generalizations of which follow rather trivially.

Construction of the equation of caustics in dynamic plane elasticity problems with the help of reduce

The method of caustics has become a very efficient tool in crack, hole and many additional plane elasticity problems. Unfortunately, the fundamental equation of the caustics frequently requires complicated algebraic computations including that of a Jacobian determinant. Here we show that computer algebra software can prove very efficient in these computations, using as a vehicle for this illustration the already known fundamental equation of caustics in dynamic plane elasticity (both for crack problems in fracture mechanics as well as for hole and additional problems). We have used the programming capabilities of REDUCE, a very popular computer algebra system, for our algebraic computations. Moreover, we illustrate the ‘learning’ abilities of REDUCE especially for the derivation of the complex form of this equation. The case of static plane elasticity results simply as a special case of dynamic plane elasticity. Additional possibilities are suggested in brief.

The crack tip elastic stress field using computer algebra software

In various cases, we are interested in the equations for the elastic stress field components near a crack tip, expressed in series forms. Frequently, the dominant terms in these series are not sufficient and additional terms are taken into account. Here we use the computer algebra system DERIVE in order to derive these series symbolically. (Additional popular computer algebra systems can also be successfully used.) The case of a finite straight crack under normal loading at infinity was selected for the illustration of the approach. The obtained symbolic results for the Westergaard complex potential, its derivative and the stress components are displayed in detail. The present results show clearly that the automatic symbolic series expansion of the components of the elastic field of interest near a crack tip is just a routine process if computer algebra software facilities are used.

Displacements in a clamped annular plate transversely loaded at an arbitrary point by a concentrated force

An annular plate clamped at both inner and outer edges and loaded at an arbitrary point by a transverse concentrated force is mechanically analysed via a purely flexural model. A series solution in terms of plate deflections is obtained, whose coefficients are analytically evaluated by resorting to a computer algebra package. A series acceleration technique is exploited for computational purposes. Several practically significant isometric projections and diagrams are presented which reproduce the plate deflections for a variety of plate geometries and load locations.

Using a computer algebra system to teach double integration

Evaluating double integrals is an important topic in calculus. The mechanics of evaluating interated integrals can be assisted with a computer algebra system. Methods using the computer algebra software MAPLE are presented.

A Proof of the Macdonald–Morris Root System Conjecture for $F_4 $

This paper gives a proof of the $q = 1$ case of the Macdonald–Morris root system conjecture for $F_4 $ that draws on ideas from Zeilberger’s recent proof of the $G_2^ \vee $ case and Kadell’s proof of the $q - BC_n $ case. The present proof depends on much computer computation. As in Zeilberger’s proof, the problem is reduced to solving a system of linear equations. A FORTRAN program generated the equations, which were solved using the computer algebra package MAPLE.

Some problems in computational representation theory

Modern computers and computer algebra systems yield powerful tools for mathematicalresearch. In this article, some applications of computer algebra packages and scientific software in the theory of modular representations of finite groups are described. Ten mathematical problems are mentioned, the solutions of which require new algorithms and most likely extensions of the present mathematical software. It is hoped that the corresponding computer experiments stimulate further mathematical research. Furthermore, they will show the importance of the development of computer algebra for theoretical mathematics.

Principal internal resonances in 3-DOF systems subjected to wide-band random excitation

The autoparametric interaction of two normal modes in a three-degree-of-freedom (3-DOF) structural model subjected to a wide-band random excitation is investigated. The analysis deals primarily with the structure response in the neighborhood of three different internal resonance conditions. The Fokker-Planck equation approach, together with a non-Gaussian closure scheme, is used. The analysis is carried out with the aid of the computer algebra software MACSYMA, and leads to 69 differential equations in the first through fourth order moments of the response co-ordinates. Contrary to the Gaussian closure solution, the non-Gaussian closure scheme yields a strictly stationary response. The Gaussian closure scheme fails to predict the system response when two non-adjacent modes are internally tuned. According to the non-Gaussian solution, the autoparametric interaction is found to be sensitive to a relatively high level of excitation spectral density only if the first and second or the first and third normal modes are internally tuned. However, for the case of second and third normal mode interaction, the non-linear response is critical to lower excitation levels. The random responses of the three cases are characterized by energy exchange between the interacted modes. The sensitivity of autoparametric interaction to a certain excitation level is mainly dependent upon the system dynamic properties.

Symmetries of Differential Equations: From Sophus Lie to Computer Algebra

The topic of this article is the symmetry analysis of differential equations and the applications of computer algebra to the extensive analytical calculations which are usually involved in it. The whole area naturally decomposes into two parts depending on whether ordinary or partial differential equations are considered. We show how a symmetry may be applied to lower the order of an ordinary differential equation or to obtain similarity solutions of partial differential equations. The computer algebra packages SODS and SPDE, respectively, which have been developed to perform almost all algebraic manipulations necessary to determine the symmetry group of a given differential equation, are presented. Furthermore it is argued that the application of computer algebra systems has qualitatively changed this area of applied mathematics.

Structural modal interaction with combination internal resonance under wide-band random excitation

In this paper the non-linear interaction of a three degree of freedom structural model subjected to a wide-band random excitation is examined. The non-linearity of the system results in different critical regions of internal resonance and this has a significant effect on the response statistics. With reference to the combination internal resonance of the summed type the system response is analyzed by using the Fokker-Planck equation approach together with a non-Gaussian closure scheme. The non-Gaussian closure is based on the cumulant properties of order greater than three. As a first order approximation the scheme yields 209 first order differential equations in first through fourth order joint moments of the response co-ordinates. The analysis is carried out with the aid of the computer algebra software MACSYMA. The response statistics are determined, numerically in the time and frequency (internal detuning) domains. Contrary to the Gaussian closure scheme, the non-Gaussian closure solution yields a strictly stationary response in addition to a number of complex response characteristics not previously reported in the literature of the area of non-linear random vibration. These include multiple solutions and jump phenomena (jump and collapse in the response mean squares) at internal detuning slightly shifted from the exact internal resonance condition. At exact internal resonance the system response possesses a unique limit cycle in a stochastic sense. The regions of multiple solutions are defined in terms of system parameters (damping ratios and non-linear coupling parameter) and excitation spectral density level.

Public key encryption

The RSA method is used for the interchange of secret messages via insecure channels. It is elegant in theory and fast and reliable in practice. Applications are in the field of communication networks.The method is initialized by choosing some suitable large prime numbers. Encrypting and decrypting of a message are done by modular arithmetic. The modulus is a large integer (e.g. 200 decimal digits). Any attempt to break the system amounts to factoring the modulus.The basic operations that are needed for an implementation are fundamental for any computer algebra package, e.g. SAC-2 offers them in an suitable way. This project therefore requires only little programming, but some insight into theory and the ability to find and put together existing components. An optional part of the project deals with the security of the method.

Application of the MuMATH computer algebra system to sets of first order kinetic equations of significance in chemistry

AbstractAnalytical solutions to systems of first order differential equations have long been available; however, they become so involved that they are seldom used. Numerical simulations are preferred in practice, although they produce bulky output that is difficult to interpret. Even worse, they are occasionally seriously in error owing to “stiffness” in the equations, intrinsic instability, or other pathological behavior. Computer algebra systems can handle the complicated manipulations that make the analytical solutions of first order kinetic systems inconvenient and can serve as a check on numerical simulations. We show applications of the small MuMATH(R) computer algebra package to a variety of first‐order process, including relaxation kinetics and the analysis of stability of steady states in nonlinear kinetic systems.

The first step in an experiment to fully integrate computers into a university physics course

Describes the thinking behind the installation of a network of IBM PCs in the Department of Physics at the University of Liverpool for the use of undergraduate physics students. Students were canvassed as to their attitudes to computers and computing and the results of this survey are discussed. Their expectations of the system are also included. One of the important outcomes of this work is that the use of various aspects of the system are to be made compulsory for the students on the undergraduate physics programme. Computer algebra, word processing and teaching packages, related and unrelated to practical work, will be included.

Amplitudes for scattering of electrons by hydrogenic and alkali-like atomic systems

Amplitudes for scattering of electrons by hydrogenic and alkali-like atomic systems