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Applications Of Statistical Physics To Ecology: Ising Models And Two-Cycle Coupled Oscillators, Vahini Reddy Nareddy

Doctoral Dissertations

Many ecological systems exhibit noisy period-2 oscillations and, when they are spatially extended, they undergo phase transition from synchrony to incoherence in the Ising universality class. Period-2 cycles have two possible phases of oscillations and can be represented as two states in the bistable systems. Understanding the dynamics of ecological systems by representing their oscillations as bistable states and developing dynamical models using the tools from statistical physics to predict their future states is the focus of this thesis. As the ecological oscillators with two-cycle behavior undergo phase transitions in the Ising universality class, many features of synchrony and equilibrium …

A Representation For Cmc 1 Surfaces In H^3 Using Two Pairs Of Spinors, Tetsuya Nakamura

Doctoral Dissertations

For Bryant's representation $\Phi\colon \widetilde{M} \rightarrow \SL_2(\C)$ of a constant mean curvature (CMC) $1$ surface $f\colon M\rightarrow \Hyp^3$ in the $3$-dimensional hyperbolic space $\Hyp^3$, we will give a formula expressed only by the global $\tbinom{P}{Q}$ and local $\tbinom{p}{q}$ spinors and their derivatives. We will see that this formula is derived from the Klein correspondence, understanding $\Phi$ as a null curve immersion into a $3$-dimensional quadric. We will show that, if $f$ is a CMC $1$ surface with smooth ends modeled on a compact Riemann surface, the linear change of $\tbinom{P}{Q}\oplus \tbinom{p}{-q}$ by some $\Sp(\C^4)$ matrices gives rise to a transformtion …

Combinatorial Algorithms For Graph Discovery And Experimental Design, Raghavendra K. Addanki

Doctoral Dissertations

In this thesis, we study the design and analysis of algorithms for discovering the structure and properties of an unknown graph, with applications in two different domains: causal inference and sublinear graph algorithms. In both these domains, graph discovery is possible using restricted forms of experiments, and our objective is to design low-cost experiments. First, we describe efficient experimental approaches to the causal discovery problem, which in its simplest form, asks us to identify the causal relations (edges of the unknown graph) between variables (vertices of the unknown graph) of a given system. For causal discovery, we study algorithms …

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 …

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 …

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 …

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 …

On The Iwasawa Invariants Of Nonordinary Modular Forms, Rylan J. Gajek-Leonard

Doctoral Dissertations

We extend known results on the behavior of Iwasawa invariants attached to Mazur-Tate elements for p-nonordinary modular forms of weight k=2 to higher weight modular forms with a_p=0. This is done by using a decomposition of the p-adic L-function due to R. Pollack in order to construct explicit lifts of Mazur-Tate elements to the full Iwasawa algebra. We then study the behavior of Iwasawa invariants upon projection to finite layers, allowing us to express the invariants of Mazur-Tate elements in terms of those coming from plus/minus p-adic L-functions. Our results combine with work of Pollack and Weston to relate the …

General Covariance With Stacks And The Batalin-Vilkovisky Formalism, Filip Dul

Doctoral Dissertations

In this thesis we develop a formulation of general covariance, an essential property for many field theories on curved spacetimes, using the language of stacks and the Batalin-Vilkovisky formalism. We survey the theory of stacks, both from a global and formal perspective, and consider the key example in our work: the moduli stack of metrics modulo diffeomorphism. This is then coupled to the Batalin-Vilkovisky formalism–a formulation of field theory motivated by developments in derived geometry–to describe the associated equivariant observables of a theory and to recover and generalize results regarding current conservation.

An Optimal Transportation Theory For Interacting Paths, Rene Cabrera

Doctoral Dissertations

In this work we study a modification of the Monge-Kantorovich problem taking into account path dependence and interaction effects between particles. We prove existence of solutions under mild conditions on the data, and after imposing stronger conditions, we characterize the minimizers by relating them to an auxiliary Monge-Kantorovich problem of the more standard kind. With this notion of how particles interact and travel along paths, we produce a dual problem. The main novelty here is to incorporate an interaction effect to the optimal path transport problem. This covers for instance, N-body dynamics when the underlying measures are discrete. Lastly, …

Extensions And Bijections Of Skew-Shaped Tableaux And Factorizations Of Singer Cycles, Ga Yee Park

Doctoral Dissertations

This dissertation is in the field of Algebraic and Enumerative Combinatorics. In the first part of the thesis, we study the generalization of Naruse hook-length formula to mobile posets. Families of posets like Young diagrams of straight shapes and d-complete posets have hook-length product formulas to count linear extensions, whereas families like Young diagrams of skew shapes have determinant or positive sum formulas like the Naruse hook-length formula (NHLF). In 2020, Garver et. al. gave determinant formulas to count linear extensions of a family of posets called mobile posets that refine d-complete posets and border strip skew shapes. We give …

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 …

Anticanonical Models Of Smoothings Of Cyclic Quotient Singularities, Arie A. Stern Gonzalez

Doctoral Dissertations

In this thesis we study anticanonical models of smoothings of cyclic quotient singularities. Given a surface cyclic quotient singularity $Q\in Y$, it is an open problem to determine all smoothings of $Y$ that admit an anticanonical model and to compute it. In \cite{HTU}, Hacking, Tevelev and Urz\'ua studied certain irreducible components of the versal deformation space of $Y$, and within these components, they found one parameter smoothings $\Y \to \A^1$ that admit an anticanonical model and proved that they have canonical singularities. Moreover, they compute explicitly the anticanonical models that have terminal singularities using Mori's division algorithm \cite{M02}. We study …

Moving Polygon Methods For Incompressible Fluid Dynamics, Chris Chartrand

Doctoral Dissertations

Hybrid particle-mesh numerical approaches are proposed to solve incompressible fluid flows. The methods discussed in this work consist of a collection of particles each wrapped in their own polygon mesh cell, which then move through the domain as the flow evolves. Variables such as pressure, velocity, mass, and momentum are located either on the mesh or on the particles themselves, depending on the specific algorithm described, and each will be shown to have its own advantages and disadvantages. This work explores what is required to obtain local conservation of mass, momentum, and convergence for the velocity and pressure in a …