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Dvr-Matroids Of Algebraic Extensions, Anna L. Lawson

Doctoral Dissertations

A matroid is a finite set E along with a collection of subsets of E, called independent sets, that satisfy certain conditions. The most well-known matroids are linear matroids, which come from a finite subset of a vector space over a field K. In this case the independent sets are the subsets that are linearly independent over K. Algebraic matroids come from a finite set of elements in an extension of a field K. The independent sets are the subsets that are algebraically independent over K. Any linear matroid has a representation as an algebraic matroid, but the converse is …

Survivor Bond Models For Securitizing Longevity Risk, Priscilla Mansah Codjoe

Doctoral Dissertations

"Longevity risk is the risk that a reference population’s mortality rates deviate from what is projected from prior life tables. This is due to discoveries in biological sciences, improved public health measures, and nutrition, which have dramatically increased life expectancy. Longevity risk raises life insurers’ liability, increasing product costs and reserves. Securitization through longevity derivatives is a way of dealing with this risk.

To enhance the pricing of life contingent products, we present an additive type mortality model in the style of the Lee-Carter. This model incorporates policyholder covariates. By using counting processes and martingale machinery, we obtain close form …

Numerical Studies Of Correlated Topological Systems, Rahul Soni

Doctoral Dissertations

In this thesis, we study the interplay of Hubbard U correlation and topological effects in two different bipartite lattices: the dice and the Lieb lattices. Both these lattices are unique as they contain a flat energy band at E = 0, even in the absence of Coulombic interaction. When interactions are introduced both these lattices display an unexpected multitude of topological phases in our U -λ phase diagram, where λ is the spin-orbit coupling strength. We also study ribbons of the dice lattice and observed that they qualitative display all properties of their two-dimensional counterpart. This includes flat bands near …

Characteristic Sets Of Matroids, Dony Varghese

Doctoral Dissertations

Matroids are combinatorial structures that generalize the properties of linear independence. But not all matroids have linear representations. Furthermore, the existence of linear representations depends on the characteristic of the fields, and the linear characteristic set is the set of characteristics of fields over which a matroid has a linear representation. The algebraic independence in a field extension also defines a matroid, and also depends on the characteristic of the fields. The algebraic characteristic set is defined in the similar way as the linear characteristic set.

The linear representations and characteristic sets are well studied. But the algebraic representations and …

Sequential Deformations Of Hadamard Matrices And Commuting Squares, Shuler G. Hopkins

Doctoral Dissertations

In this dissertation, we study analytic and sequential deformations of commuting squares of finite dimensional von Neumann algebras, with applications to the theory of complex Hadamard matrices. The main goal is to shed some light on the structure of the algebraic manifold of spin model commuting squares (i.e., commuting squares based on complex Hadamard matrices), in the neighborhood of the standard commuting square (i.e., the commuting square corresponding to the Fourier matrix). We prove two types of results: Non-existence results for deformations in certain directions in the tangent space to the algebraic manifold of commuting squares (chapters 3 and 4), …

A Weak Fractional Calculus Theory And Numerical Methods For Fractional Differential Equations, Mitchell D. Sutton

Doctoral Dissertations

This dissertation is comprised of four integral parts. The first part comprises a self-contained new theory of weak fractional differential calculus in one-dimension. The crux of this new theory is the introduction of a weak fractional derivative notion which is a natural generalization of integer order weak derivatives; it also helps to unify multiple existing fractional derivative definitions.

The second part of this work presents three new families of fractional Sobolev spaces and their accompanying theory in one-dimension. The new construction and theory are based on a newly developed notion of weak fractional derivatives, which are natural generalizations of the …

A Coarse Approach To The Freudenthal Compactification And Ends Of Groups, Hussain S. Rashed

Doctoral Dissertations

The main purpose of this work is to present a coarse counterpart to the Freudenthal compactification and its corona (the space of ends) that generalizes the Freudenthal compactification of a Freudenthal topological space X (connected, locally connected, locally compact and σ-compact) and its corona; then applying it to groups as coarse space to obtain generalizations to many well-known results in the theory of ends of groups. To this end, we present two constructions:

1. The Coarse Freudenthal compactification of a proper metric space which is a coarse compactification that coincides with the Freudenthal compactification when the metric space is geodesic. …

Some Results About Reproducing Kernel Hilbert Spaces Of Certain Structure, Jesse Gabriel Sautel

Doctoral Dissertations

The theory of reproducing kernel Hilbert spaces has been crucial to the development of many of the most significant modern ideas behind functional analysis. In particular, there are two classes of reproducing kernel Hilbert spaces that have seen plenty of interest: that of complete Nevanlinna-Pick spaces and de Branges-Rovnyak spaces.

In this dissertation, we prove some results involving each type of space separately as well as one result regarding their potential overlap. It turns out that a de Branges-Rovnyak space is also of complete Nevanlinna-Pick type as long as there exists a multiplier satisfying a certain identity.

Further, we extend …