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
S.A. Greenblatt

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 statistical test.

Shotgun Wounding Characteristics
G. Russell

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
D. Harper

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
B. Lisser

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 n=20 circles are given along with the minimal polynomials for the optimal solutions which were computed by Maple.

Application of Maple to General Relativity
J. Carminati

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 Problems
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:

• Brachistochrone Problem (Classical Mechanics)
• Perihelion Precession Rate for Mercury (General Relativity)
• Alpha-Particle Decay (Quantum mechanics)

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.

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
R. Lopez

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 Turtle Graphics. 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.