MTN Special Issue, 1994
Maple in Mathematics and the Sciences
- The Status of Fermat's Last Theorem - mid 1994
A. Granville, M.B. Monagan
This is a news article on Fermat's last theorem which outlines
the direction of Wiles' important attack on
Fermat's last theorem (as well as the far more general
Taniyama conjecture), other recent developments in
this field, and what rôle Maple has played in
research on this subject.
- Wavelets in Econometrics
This article presents a brief introduction to wavelets. It then shows
how Maple may act as an exploratory tool allowing researchers to
detect outliers in a data series with the help of a new wavelet-based
- Shotgun Wounding Characteristics
This article demonstrates that
high rates of wounding are a necessary consequence of using shotguns to
shoot flying ducks. It shows that high wounding rates are a
statistical consequence of the shotgun pellet and aim
error distributions. Shooters in the skill range establised by
published studies will wound (and not retrieve) 50 to 150 ducks for
every 100 they "bag".
- Disturbing Function Expansions
This article describes the use of Maple to obtain an expansion
of the planetary disturbing function (the mutual gravitational potential of two
bodies orbiting a third, much more massive, body) as a Poisson
series to eighth order in the small parameters which characterise the system.
Such an expansion is the starting point of much of analytical celestial
mechanics. This work represents the first time that such an expansion
has been generated to such a high order, and the first time that two
different computer algebra systems have been used to obtain expansions
independently as a means of verification.
- Solving Kostant's Conjecture Using Maple
This article concerns an important conjecture of Kostant about relations
between simple Lie groups and finite simple groups; both are
classes of well defined mathematical structures that have been
classified into families, much like the chemical elements are
classified in the periodic table. The simple Lie groups are big
(in fact infinite) structures, and it is sometimes possible to
indicate a finite subset that has the structure of a finite simple
group; due to the severe restrictions imposed by the mathematical
structure, this is however a rare event. In fact, for most
combinations of a simple Lie group and a finite simple group,
the possibility of such an embedding has been ruled out.
The conjecture is that for the remaining list of combinations
an embedding actually exists.
One cannot be certain however about this until an embedding is
actually constructed, which means that specific group elements
have to be indicated and certain properties that are
required for a proper embedding have to be tested.
- The History of Packing Circles in a Square
D. Würtz, M. Monagan, R. Peikert
This article reviews the history of the
circle packing problem, that is, finding the
densest packing of
n circles into a square.
The optimal packings for up to
are given along with the minimal polynomials
for the optimal solutions which were computed by Maple.
- Application of Maple to General Relativity
General relativity is both conceptually and computationally
very complex. In recent times considerable progress has been made in the
application and understanding of this theory through the use of powerful
symbolic manipulators such as Maple. This article describes how Maple
was used to help solve certain problems that arose in determining exact
solutions of the Einstein field equations and scalar invariants of the
Riemann tensor in a 4-dimensional Lorentzian manifold.
- Applications of Maple to Mathematical, Scientific and Engineering
T. Scott, G. Fee, R. Corless, M. Monagan
This is a review article which reports various Maple "victories" from
Quantum Mechanics, to Audio Engineering, to General Relativity to Asbestos
Fiber Analysis to Fluid Mechanics. Although brief, this review articles
gives some idea of the potential and versatility of the Maple system as a
problem-solving tool in research. This article has been previously shown
in issue no. 6 of the MTN but is updated with new material.
Maple in Education
- Maple in Science Education
T. Scott, B. Madore, R.M. Corless
The first article of this group presents a set of three very
definitive problems in physics and applied mathematics
as prepared on worksheets using Maple V Release 3.
Each problem was taken from a major area in Physics (and Applied
Mathematics) i.e. classical mechanics, general relativity and
quantum mechanics, namely:
The authors hope these examples will encourage Maple users to make
similar types of worksheets
and will show users the type of accessible applications we wish to
include within the Maple Share Library.
This article is a continuation to "Maple in Education" as
presented in issue no. 7 and its sequel in volume 1, no. 1 of the MTN.
- Brachistochrone Problem (Classical Mechanics)
- Perihelion Precession Rate for Mercury (General Relativity)
- Alpha-Particle Decay (Quantum mechanics)
- Mathematics, Biology, and Maple
J. Herod, E. Yeargers
This is a parallel the previous "Maple in Education" article
but devoted to biology and medicine. It is also a reflection of the
conviction by the authors
that Maple enables biologists with a bent toward modeling to
analyze mathematical models even though they might have had
limited experience in upper-division mathematics. It is the
authors' belief that Maple opens to students of biology an array
of ideas and tools not possible only a few years before.
- Saw-Toothed Oscillator: Exact and Approximate Solutions
This is a Maple worksheet which addresses the following pedagogical problem.
A damped linear oscillator is driven with a saw-toothed forcing function. The
resulting system is solved analytically by Laplace transforms, and then solved
approximately, by Fourier series. The exact solution is plotted, and this
graph is used as a basis of comparison for the convergence of the
eigenfunction expansion based on the Fourier series of the driving term. The
two parts of this exploration illustrate a pair of lessons studied in a Maple
based course in ordinary differential equations.
- An Implementation of "Turtle Graphics" in Maple
E. Roanes Lozano, E. Roanes Macias
The Turtle Geometry was introduced
in the seventies and is the basis behind what is often called
The novelty of the approach is to avoid the use of Cartesian
coordinates, relating the orders to a moving cursor on the graphics
screen that behaves as a little animal (which is the origin of the name).
This authors of this article as users of both Turtle Graphics
and Maple, have tried to
join the graphics power of the former with the advantages of the exact
arithmetic, programming facilities and portability of the later. Therefore,
the article represents more than just another implementation of
the Turtle Geometry, it also reveils
new possibilities for the Turtle users.
HTML originally written by Reid M. Pinchback
Copyright 1996, Massachusetts Institute of Technology
Last modified: 96/09/25