MTN Volume 2 Number 2, Fall 1995
- The Maple V Release 4 Math Engine
This article shows some of the new features of Maple V Release 4.
- Tips for Maple Users and Programmers
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.
- Implementation of an Umbral Calculus Package
A. Di Bucchianico,
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
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
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
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
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
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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Copyright 1996, Massachusetts Institute of Technology
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