Computers and Mathematics

Advances in computer technology have had a tremendous impact on mathematics in the last two decades. In June of 1989, an international conference was held at MIT, bringing together mathematicians and computer scientists, to survey the work that has been done in computational mathematics, to report recent results in this field, and to discuss research directions as well as educational issues. This book presents a fascinating collection of contributions on topics ranging from computational algebra, and parallel computing, to mathematics education. Mathematicians interested in the computational aspects of their discipline as well as computer scientists interested in mathematical applications will enjoy the integrative view provided by this book. Front Matter....Pages i-xi A Completion Procedure for Computing a Canonical Basis for a k -Subalgebra....Pages 1-11 Summation of Harmonic Numbers....Pages 12-20 Algorithm and Implementation for Computation of Jordan Form over A [ x 1 ,..., x m ]....Pages 21-26 Fast Group Membership Using a Strong Generating Test for Permutation Groups....Pages 27-36 Finite-Basis Theorems and a Computation-Integrated Approach to Obstruction Set Isolation....Pages 37-45 Practical Determination of the Dimension of an Algebraic Variety....Pages 46-52 A Computer Generated Census of Cusped Hyperbolic 3-Manifolds....Pages 53-59 Classicality of Trigonal Curves of Genus Five....Pages 60-65 Symmetric Matrices with Alternating Blocks....Pages 66-72 Cohomology to Compute....Pages 73-81 Use of symbolic methods in analyzing an integral operator....Pages 82-90 Computer Algebraic Methods for Investigating Plane Differential Systems of Center and Focus Type....Pages 91-99 An Example of Computer Enhanced Analysis....Pages 100-108 An Algorithm for Symbolic Computation of Hopf Bifurcation....Pages 109-118 Application of the REDUCE Computer Algebra System to Stability Analysis of Difference Schemes....Pages 119-129 Signs of Algebraic Numbers....Pages 130-134 Efficient Reduction of Quadratic Forms....Pages 135-139 A Story About Computing with Roots of Unity....Pages 140-144 Exact Algorithms for the Matrix-Triangularization Subresultant PRS Method....Pages 145-155 Computation of Fourier Transforms on the Symmetric Group....Pages 156-165 Integration in Finite Terms and Simplification with Dilogarithms: A Progress Report....Pages 166-171 Why Integration is Hard....Pages 172-181 Liouvillian Solutions of Linear Differential Equations with Liouvillian Coefficients....Pages 182-191 Recipes for Classes of Definite Integrals Involving Exponentials and Logarithms....Pages 192-201 Logic and Computation in MATHPERT: An Expert System for Learning Mathematics....Pages 202-214 Representation of Inference in Computer Algebra Systems with Applications to Intelligent Tutoring....Pages 215-227 Bunny Numerics....Pages 228-239 Advanced Mathematics from an Elementary Viewpoint: Chaos, Fractal Geometry, and Nonlinear Systems....Pages 240-249 Iterated Function Systems and the Inverse Problem of Fractal Construction Using Moments....Pages 250-259 Working with ruled surfaces in solid modeling....Pages 260-268 Using MACSyma to Calculate the Extrinsic Geometry of a Tube in a Riemannian Manifold....Pages 269-278 Computer Algebra in the Theory of Ordinary Differential Equations of Halphen Type....Pages 279-288 Symbolic Derivation of Equations for Mixed Formulation in Finite Element Analysis....Pages 289-298 Semantics in Algebraic Computation....Pages 299-307 Symbolic Computation with Symmetric Polynomials an Extension to MACSYMA....Pages 308-320 Simultaneous Computations in Fields of Different Characteristics....Pages 321-326