John J. Wavrik

Computer simplification of formulas in linear systems theory

Currently, the three most popular commercial computer algebra systems are Mathematica, Maple, and MACSYMA. These systems provide a wide variety of symbolic computation facilities for commutative algebra and contain implementations of powerful algorithms in that domain. The Grobner basis algorithm, for example, is an important tool used in computation with commutative algebras and in solving systems of polynomial equations. On the other hand, most of the computation involved in linear control theory is performed on matrices, which do not commute, and Mathematica, Maple, and MACSYMA are weak in the area of noncommutative operations. The paper reports on applications of a powerful tool, a noncommutative version of the Grobner basis algorithm. The commutative version of this algorithm is implemented in most major computer algebra packages. The noncommutative version is relatively new.

Mathematics Education For The Gifted Elementary School Student

*The ideas discussed in this article were originally developed for a math club at Skyline Elementary School in Solana Beach, California. The club was a joint venture by myself, my wife Mary Wavrik, and Stephen Ludwiczak, the principal of the school. Dr. Charlotte Malone was instrumental in the establishment of the Math Course through the University of California Extension Program, allowing a much more extensive program to be developed. Any form of instruction rests, either explicitly or implicitly, on a philosophy or attitude toward the subject on the part of those designing curricula and teaching it. Standard school mathematics instruction evolved at a time when mathematics played a less vital role in society than at present. This article attempts to make as sharp a contrast as possible between two sets of philosophies and attitudes.

Commutativity theorems: examples in search of algorithms

Commutativity theorems are part of the study of polynomial identities in non-commutative rings. They are theorems which assert that, under certain conditions, the ring at hand must be commutative. The proofs of theorems of this sort in their general form require the structure theory for non-commutative rings. Instances of these theorems have a strongly computational flavor. They provide interesting test examples for algorithms which use rewrite rules and reduction theory for polynomial rings in noncommuting variables. This paper presents several examples of commutativity theorems with solutions. The solutions were obtained using a reduction process for non-commutative polynomials with integer coefficients. The reduction process blends a treatment of integer coefficients due to Buchberger with handling of non-commutative polynomials due to Mora. Some comparisons are made between automated solutions and solutions “by hand”.

Computers and the Multiplicity of Polynomial Roots.

Introduction. This article will not describe the assorted twists and turns of fate that lead a worker in a pure mathematics area like algebraic geometry to become involved with computers. It is quite likely, however, that as personal computers become more common an algebraist who acquires one will at some stage make a stab at using it for research work. This article, written by an algebraist who has become so involved, will try to prepare others for the shock, amazement, problems, pitfalls, and pleasures which lie in the wonderful world of machine computation. Algebraic geometry is an example of a field that has developed in a way which underemphasizes computation. Recent generations of students have been trained to accept an outlook on the subject in which even the geometry aspects have been all but hidden from view. Andre Weil comments upon this in the introduction to Foundations of Algebraic Geometry:

Using Groups32 in instruction

This article discusses the process of integrating the use a software system into a conventional course in Abstract Algebra. It is a description of how and why a software system was used. It uses a particular system, Groups32, as an example.

Use of Forth in a course in computer algebra

ABSTWCT: A course in Computer Algebra was developed at UCSD to teach mathematics majors about symbolic and seminumeric computation. Forth has been found to be an excellent language to acquaint these students with the ideas involved in representing and manipulating mathematics by computer. Forth, as a teaching language, has the distinct advantage of making visible some important things which most languages try to conceal. It does this without sacrificing high level language features which facilitate programming