Document Type

Dissertation

Date of Degree

Summer 2013

Degree Name

PhD (Doctor of Philosophy)

Degree In

Mathematics

First Advisor

Palle Jorgensen

First Committee Member

Isabel Darcy

Second Committee Member

Victor Camillo

Third Committee Member

Colleen Mitchell

Fourth Committee Member

Ionut Chifan

Abstract

By first looking at the orthonormal basis: Γ = {∑i 4 ibi ∈{0, 1}, finite sums} and the related orthonormal basis 5Γ = {5∑i 4i bi : bi ∈ {0, 1}, finite sums} we find several interesting relationships with the unitary matrix Uα,β arising from the operator U: Γ → 5Γ. Further, we investigate the relationships between U and the operators So : Γ → 4Γ defined by Soe where eγ = e2ΠiΓ and S1: Γ → 4Γ+1 defined by S1eγ = e4γ+1.

Most intriguing, we found that when taking powers of the aforementioned Uα,β matrix that although there are infinitely many 1's occurring in the entries of Uα,β only one such 1 occurs in the subsequent higher powers Ukα,β. This means that there are infinitely many γ ∈ Γ ∩ 5Γ, but only one such γ in the intersection Γ and 5kΓ, for k ≥ 2.

Keywords

infinite matrices

Pages

vi, 58 pages

Bibliography

Includes bibliographical references (pages 57-58).

Copyright

Copyright 2013 Corissa Marie Goertzen

Included in

Mathematics Commons

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