System Maple Research Articles

Reducing Effect on Amount of Interpolation Data for Curve Contour Using Optimum Circular Interpolation.

We compared the results of the optimum circular interpolation with those of the clothoid interpolation and the automatical circular interpolation function in the Maple NC automatic programming system, for three types of contour: an ellipse, an involute and a parallel index cam profile. The amount of interpolation data obtained by the optimum circular interpolation is less than that obtained by the Maple system. In particular, the reducing effect becomes obvious for a contour curve with a steep change in radius of curvature. Although the optimum circular interpolation is slightly more disadvantageous than the clothoid interpolation in terms of the amount of interpolation data, it is very practical and useful because the interpolation data can be directly applied in a general NC machine tool and the interpolation data for any offset curve can be direct obtained from those for the original curve.

The Maple Handbook: Maple V Release 3

The Maple Handbook: Maple V Release 3 is an essential reference tool for all users of the Maple system. It provides a complete listing of every command in the Maple language, categorized into logical categories (e.g., calculus, algebra, programming, etc.) and explained in the context of those categories. If a Maple command has different purposes in different categories, it is included more than once as appropriate. A short introductory tutorial starts the Handbook, and each category begins with a brief introduction to the related subject area. The Handbook is well referenced, with an alphabetical index of commands, and pointers to appropriate sections of the official Maple documentation (Springer-Verlag). This new approach to reference materials for Maple enhances the material found in Maple's on-line help files and provides a much more organized, intuitive resource for all users of the Maple system. The Handbook improves the efficiency of Maple users by supplying them with the information they need - at their fingertips. This new edition is comprised of the Maple V Release 3 symbolic computation language.

EXACT (1+1)-DIMENSIONAL SOLUTIONS TO A DISCRETE VELOCITY MODEL FOR COAGULATION-FRAGMENTATION

A family of discrete velocity models is proposed for the study of particle coagulation and fragmentation in gases. Patterned after the Broadwell model, the models allow inelastic collisions to occur, as well as elastic collisions. The models generate systems of highly coupled semilinear hyperbolic equations which approximate the Boltzmann equation. With the aid of the powerful computer algebra systems Mathematica and Maple, we find an exact four-parameter solution to the simplest model.

Irreducible decomposition of algebraic varieties via characteristics sets and Gröbner bases

This paper presents a method for decomposing algebraic varieties into irreducible components by using characteristic sets and Gröbner bases. A set of 16 examples is provided to explain the geometric meaning of the irreducible decomposition and as a test bed for our experiments with the method. A function that implements the presented method has been included in the author's characteristic sets package in the Maple system and used to decompose a number of non-trivial algebraic varieties. A few strategies adopted in our implementation and the timing statistics for the 16 test examples are given.

A heuristic irreducibility test for univariate polynomials

This paper describes a heuristic irreducibility test for univariate polynomials over the integers. The irreducibility test is based on finding a prime evaluation of a polynomial which, under appropriate conditions, is a witness to the irreducibility of the polynomial. The irreducibility test has been implemented as part of the polynomial factorization package available in the Maple symbolic algebra system. Several implementation details are important for the practicality of this approach, in particular, a probabilistic primality test is used. Therefore, the irreducibility test does not constitute a proof of irreducibility. Timing results are presented for a range of sample problems. The empirical results obtained compare well with the Berlekamp-Hensel algorithm for factoring polynomials, as it is implemented in the MAPLE, MACSYMA and REDUCE symbolic algebra systems.

Conformal mapping using Bergman's method and the MAPLE system

In this paper we use Bergman's Kernel Method (BKM) for solving conformal mapping problems. In contrast to most conformal mapping techniques, the approximation of the solution is an analytic function. The main drawback of the method, however, is that it involves an orthonormalization process and thus is numerically unstable. This disadvantage can be overcome by using the MAPLE system which makes no use of Gaussian quadrature, so that approximation errors are almost entirely avoided. We tested how some typical domains can be treated.Using theorems of Walsh (1929) and Simonenko (1978) we determine a number N such that the error of the N -th approximation of the mapping function remains under a desired value.We compare the obtained results to the sharp asymptotic estimates of Gaier (1988) and to numerical results of Levin, Papamichael and Sideridis (1978) and Papamichael and Warby (1986).

Symbolic computation of flow in a rotating pipe

AbstractA perturbation solution of the fully developed flow through a pipe of circular cross‐section, which rotates uniformly around an axis oriented perpendicularly to its own, is considered. The perturbation parameter is given by R = 2Ωa2/ν in terms of the angular velocity Ω, the pipe radius a and the kinematic viscosity ν of the fluid. The two coupled non‐linear equations for the axial velocity ω and the streamfunction ϕ of the transverse (secondary) flow lead to an infinite system of linear equations. This system allows first the computation of a given order ϕn, n ϕ 1, of the perturbation expansion ϕ = ∑ Rnϕn in terms of ωn‐1, the (n‐1)‐th order of the expansion ω = ∑ Rnωn, and of the lower orders ϕ1,…,ϕn − 1. Then it permits the computation of ωn from ω0,…,ωn − 1 and ϕ1,…,ϕ;n. The computation starts from the Hagen–Poiseuille flow ω0, i.e. the perturbation is around this flow.The computations are performed analytically by computer, with the REDUCE and MAPLE systems. The essential elements for this are the appropriate co‐ordinates: in the complex co‐ordinates chosen the two‐dimensional harmonic (Laplace, Δ) and biharmonic (Δ2) operators are ideally suited for (symbolic) quadratures. Symmetry considerations as well as analysis of the equations for ωn, ϕn and of the boundary conditions lead to general (polynomial) formulae for these functions, with coeffcients to be determined. Their determination, order by order, implies, in complex co‐ordinates, only (symbolic) differentiation and quadratures. The coefficients themselves are polynomials in the Reynolds number c of the (unperturbed) Hagen–Poiseuille flow. They are tabulated in the paper for the orders n ⩽ 6 of the perturbation expansion.

Speed and data structures in computer algebra systems

Comparing the speed of computation in algebra systems is a perennial occupation of system designers, algorithm implementors, and, more recently, marketing personnel. At least some people have observed that for many problems, the choice of a system makes much less difference than the approach used to represent the problem. The mapping from mathematics to a data representation and the choice of algorithms can make significant, and separate, contributions to efficiency. Systems which have the flexibility to provide several data structures and algorithms can provide an advantage in this respect. Macsyma [1] is probably the system with the largest selection, currently. On the other hand, Macsyma has not taken advantage of recent advances such as the extensive use of hash-coding incorporated in the University of Waterloo's Maple [2] system. For the one somewhat artificial benchmark we discuss in this paper, it appears that the Maple system does considerably better than any representation in Macsyma by precisely this mechanism.

Examples of Maple applied to problems from the American Mathematical Monthly

The following pages show the output of the Maple system applied to some selected problems from the Problem Section of the American Mathematical Monthly. The purpose of assembling these examples is to show some of the facilities and commands of the Maple system. These examples were run on our Vax 11/785 computer, under the Unix operating system. This is a rather random sample, and does not represent a systematic search of problems in the AMM.

A tutorial introduction to Maple

The Maple computer algebra system is described. Brief sample sessions show the user syntax and the mathematical power of the system for performing arithmetic, factoring, simplification, differentiation, integration, summation, solving algebraic equations, solving differential equations, series expansions, and matrix manipulations. Time and space statistics for each sample session show that the Maple system is very efficient in memory space utilisation, as well as in time. The Maple programming language is presented by describing the most commonly used features, using some non-trivial computations to illustrate the language features.

The maple symbolic computation system

The Maple system for algebraic computation is introduced. Samples of user/system interaction and of Maple programming are presented.

EVALUATION OF INTEGRALS OF FUNCTIONS OF COMPLEX VARIABLE BY USING THE RESIDUES IN COMPUTER MATHEMATICS SYSTEMS

The article describes the methodology of teaching one of the major topics of the course of complex analysis using the system Maple. It also demonstrates how the use of computer mathematics systems facilitates the evaluation of students by the teacher during organization of independent work.