# MTN Volume 2 Number 2, Fall 1995

## Maple Functionality

The Maple V Release 4 Math Engine
M.B. Monagan
This article shows some of the new features of Maple V Release 4.
Tips for Maple Users and Programmers
M.B. Monagan
The first part of this issue's column tackles six questions about how to work with various features of Maple. The second part shows some important "little" improvements in the Maple V Release 4 for manipulating expressions. Some of these improvements will be useful in programming, some will be useful when using Maple interactively.
Maple Package on Formal Power Series
D. Gruntz, W. Koepf
This article describes the Maple implementation of a method to compute the power series of a given function in closed form. The described method is available through the Maple share library.
A Maple 5.3 Package for the Analytical Solving of Partial Differential Equations
E.S.Cheb-Terrab, K. von Bülow
A package of commands for working with and analytically solving partial differential equations, implemented using Maple V R.3, is presented. It includes a PDE-solver, a command for changing variables and some other related tool-commands. Though the package was prepared to work without specific user indications, the user can optionally participate in the solving process by giving the solver an extra argument as to the functional form of the indeterminate function, and by changing variables in PDE's as desired. Also, there is a special routine for testing the PDE-solver's results.

## Maple Applications

Implementation of an Umbral Calculus Package
A. Bottreau, A. Di Bucchianico, D. Loeb
Rota's umbral calculus uses sequences of Sheffer polynomials to count certain combinatorial objects. The main tool in computing these polynomials is the use of linear operators on the vector space of polynomials. The authors review this theory and show how we implemented it in a Maple package. In particular, they explain how linear operators can be specified in several ways in our package. Finally, they show how some generalizations of the umbral calculus are implemented.
Manipulation of Real Roots of Polynomials: Isolating Intervals or Thom's Codes
L. Gonzalez-Vega, L. Ceballos
This article shows how Maple can be used to deal with the exact computation of real roots for univariate polynomials. The capabilities of Maple to solve this problem are presented: these include Sturm sequences, isolating intervals, the use of Maple's `fsolve`, etc. It is also shown how Maple can be used to compute the real roots of some bad polynomials (those with two roots very close to each other) by using Thom's codes.
Formal Power Series Solution of Functional Equations
K. Briggs
Functional equations are equations in which the unknown is a function. They have many applications in dynamical systems theory. The author discusses several classical examples and shows how to find series solutions of them. The resulting algorithms are interesting examples of formal power series manipulations.
A Form Factor Evaluation in Heavy Meson Decay
A.H. Fariborz
This qualifies as MapleTech's first article oriented towards high-energy physics. This work concerns the evaluation of a double integrals from quantum field theoretical study of a decay of the heavy mesons (bound states of quark-antiquark pair.) Maple is used to evaluate this double integrals in three different limits where the mass of the light-quark can be ignored.
Maple and Exact Solutions of Einstein's Equations
S.R. Czapor
In this article, the author describes the derivation of a particular class of analytic (exact) solutions of Einstein's field equations from general relativity theory - one of the "classic" areas of application for computer algebra systems. The computations involved range from calculation of standard (but somewhat unwieldy) tensorial quantities such as the Ricci and Einstein tensors in a four-dimensional space (which yield a system of coupled, nonlinear PDE's), to solution of the systems of nonlinear algebraic equations which arise in the final analysis of our exact solutions.

## Maple in Education

Maple and the Putnam Competition
P. Dumas, B. Salvy
The authors apply Maple to the solution of problems taken from the Putnam Competition, a distinguished yearly mathematical contest at the advanced undergraduate level. Out of the twelve problems of the 1993 contest, Maple is shown to help the resolution of half of them.
Generating Problems in Linear Algebra II
J. Hausen
In vol. 1, no. 2 of MapleTech, Jürgen Hausen presented an article where exercises in basic topics of linear algebra were generated by making use of Maple and its random number generator. In this sequel the author generates exercises belonging to more advanced topics of linear algebra, such as eigenvalues and the normal form problem. He also present examples of applications to calculus and differential equations.
Investigating Eigensystems in an Introductory Linear Algebra Course
W.C. Bauldry, H.P. Hirst
The authors discuss using Maple in an introductory linear algebra course, and demonstrate an approach to calculating eigenvalues and eigenvectors designed to reinforce the basic theory. The eigenvectors are constructed using the theorem: Given an eigenvalue L of A, x is an eigenvector if and only if (LI-A)x=0. The eigenvectors are found by performing symbolic Gaussian elimination on this equation, during which the characteristic polynomial appears in the reduced matrix and must be set equal to zero "by hand". This exercise shows an aspect of why eigensystem estimation can be a computationally difficult calculation.

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