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theses

This thesis deals with applications of experimental mathematics to a variety of fields. The first is partition identities. These identities, such as the Rogers-Ramanujan iden- tities, are typically (in generating function form) of the form “product side” equals “sum side,” where the product side enumerates partitions obeying certain congruence conditions, and the sum side counts partitions following certain initial conditions and difference conditions (along with possibly other restrictions). We use symbolic compu- tation to generate various such sum sides, and then use Euler’s algorithm to see which of them actually do produce elegant product sides, rediscovering many known identities and discovering new ones as conjectures. Furthermore, we examine how the judicious use of computers can help provide new proofs of old identities, with the experimenta- tion behind a “motivated proof” of the Andrews-Bressoud partition identities for even moduli. We also examine the usage of computers to study the Laurent phenomenon, an outgrowth of the Somos sequences, first studied by Michael Somos. Originally, these are recurrence relations that surprisingly produce only integers. The integrality of these sequences turns out to be a special case of the Laurent phenomenon. We will discuss methods for searching for new sequences with the Laurent phenomenon — with the conjecturing and proving both automated. Finally, we will exhibit a family of sequences of noncommutative variables, recursively defined using monic palindromic polynomials in Q [x], and show that each possesses the Laurent phenomenon, and will examine some two-dimensional analogues. Once again, computer experimentation was key to these discoveries.

ETD_7195

Ph.D.

Includes bibliographical references

by Matthew Christopher Russell

doi:10.7282/T3MC926D

ETD doctoral

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