Physical Sciences and Mathematics Commons^™

Magnetic Phases Of Large-Spin Ultracold Bosons: Quantum Dimer Models And Spin Liquid Phases, Todd C. Rutkowski

Graduate Dissertations and Theses

This thesis investigates the plausibility of producing a quantum spin liquid (QSL) with ultracold bosonic atoms optically confined to the Mott insulating state. QSLs have received a great deal of attention for being an antiferromagnetic groundstate with many exotic properties, including the absence of local order, long-range entanglement, and fractionalized excitations. However, the identification and characterization of these phases in solid state systems remains a great challenge. Here we outline an alternate route to uncovering the QSL phase, which from the nature of spin angular momentum for ultracold atoms encounters many properties unique to these systems along the way. This …

On A Generalization Of The Hanoi Towers Group, Rachel Skipper

Graduate Dissertations and Theses

In 2012, Bartholdi, Siegenthaler, and Zalesskii computed the rigid kernel for the only known group for which it is non-trivial, theHanoi towers group. There they determined the kernel was the Klein 4 group. We present a simpler proof of this theorem. In thecourse of the proof, we also compute the rigid stabilizers and present proofs that this group is a self-similar, self-replicating, regular branch group.

We then construct a family of groups which generalize the Hanoi towers group and study the congruence subgroup problem for the groups in this family. We show that unlike the Hanoi towers group, the groups …

On A Pseudodifferential Calculus With Modest Boundary Decay Condition, Binbin Huang

Graduate Dissertations and Theses

A boundary decay condition, called vanishing to infinite logarithmic order is introduced. A pseudodifferential calculus, extending the b-calculus of Melrose, is proposed based on this modest decay condition. The mapping properties, composition rule, and normal operators are studied. Instead of functional analytic methods, a geometric approach is invoked in pursuing the Fredholm criterion. As an application, a detailed proof of the Atiyah-Patodi-Singer index theorem, including a review of Dirac operators of product type and construction of the heat kernel, is presented.

A Thermally Stabilized Fluorescent Organic Chromophore, Brendan P. Hughes

Graduate Dissertations and Theses

A fluorescent chromophore, hereby designated as cyanine dye, has been bound to generated zinc oxide nanoparticles. It has been shown that the material not only retains the characteristic absorption peak of the native chromophore, but it also continues to fluoresce with comparable intensity. The binding occurs through a Zn – O₂C bridge, which allows for vibrational relaxation as the dye heats up, yet also preserves the optical properties of the dye. This bond linkage can be observed from the comparison of FT-IR spectra of the nanoparticles and the native dye, designated by a shift in one of the …

Interaction Graphs Derived From Activation Functions And Their Application To Gene Regulation, Simon Joyce

Graduate Dissertations and Theses

Interaction graphs are graphic representations of complex networks of mutually interacting components. Their main application is in the field of gene regulatory networks, where they are used to visualize how the expression levels of genes activate or inhibit the expression levels of other genes.

First we develop a natural transformation of activation functions and their derived interaction graphs, called conjugation, that is related to a natural transformation of signed digraphs called switching isomorphism. This is a useful tool for the analysis of interaction graphs used throughout the rest of the dissertation.

We then discuss the question of what restrictions, if …