First Advisor

John Caughman

Term of Graduation

Spring 2024

Date of Publication

4-1-2024

Document Type

Dissertation

Degree Name

Doctor of Philosophy (Ph.D.) in Mathematical Sciences

Department

Mathematics and Statistics

Language

English

DOI

10.15760/etd.3743

Physical Description

1 online resource (viii, 96 pages)

Abstract

In Linear algebra, the concept of Leonard pair (LP) was motivated by the theory of Q-polynomial distance-regular graphs. In this dissertation, we will first give a brief introduction to LPs and to two closely-related classes of objects: (i) bipartite Leonard pairs (BLPs) and (ii) almost bipartite Leonard pairs (ABLPs). Taking these as departure points, we will introduce a new class of object - doubly almost bipartite Leonard pairs (DABLPs). The primary aim of our work is to fully classify (up to isomorphism) this new family. In addition, since there is known to be a natural correspondence between Leonard pairs and families of orthogonal polynomials, we reveal which families of orthogonal polynomials correspond to the DABLPs. Several related objects, such as Leonard triples, modular Leonard triples, spin Leonard pairs, and near-bipartite Leonard pairs have corresponding notions for the doubly almost bipartite case. These analogous objects are also defined and briefly explored.

Rights

© 2024 Shuichi Masuda

In Copyright. URI: http://rightsstatements.org/vocab/InC/1.0/ This Item is protected by copyright and/or related rights. You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s).

Persistent Identifier

https://archives.pdx.edu/ds/psu/42218

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