Mathematics Commons^™

Limit Theorems For L-Functions In Analytic Number Theory, Asher Roberts

Dissertations, Theses, and Capstone Projects

We use the method of Radziwill and Soundararajan to prove Selberg’s central limit theorem for the real part of the logarithm of the Riemann zeta function on the critical line in the multivariate case. This gives an alternate proof of a result of Bourgade. An upshot of the method is to determine a rate of convergence in the sense of the Dudley distance. This is the same rate Selberg claims using the Kolmogorov distance. We also achieve the same rate of convergence in the case of Dirichlet L-functions. Assuming the Riemann hypothesis, we improve the rate of convergence by using …

A Causal Inference Approach For Spike Train Interactions, Zach Saccomano

Dissertations, Theses, and Capstone Projects

Since the 1960s, neuroscientists have worked on the problem of estimating synaptic properties, such as connectivity and strength, from simultaneously recorded spike trains. Recent years have seen renewed interest in the problem coinciding with rapid advances in experimental technologies, including an approximate exponential increase in the number of neurons that can be recorded in parallel and perturbation techniques such as optogenetics that can be used to calibrate and validate causal hypotheses about functional connectivity. This thesis presents a mathematical examination of synaptic inference from two perspectives: (1) using in vivo data and biophysical models, we ask in what cases the …

Role Of Influence In Complex Networks, Nur Dean

Dissertations, Theses, and Capstone Projects

Game theory is a wide ranging research area; that has attracted researchers from various fields. Scientists have been using game theory to understand the evolution of cooperation in complex networks. However, there is limited research that considers the structure and connectivity patterns in networks, which create heterogeneity among nodes. For example, due to the complex ways most networks are formed, it is common to have some highly “social” nodes, while others are highly isolated. This heterogeneity is measured through metrics referred to as “centrality” of nodes. Thus, the more “social” nodes tend to also have higher centrality.

In this thesis, …

At The Interface Of Algebra And Statistics, Tai-Danae Bradley

Dissertations, Theses, and Capstone Projects

This thesis takes inspiration from quantum physics to investigate mathematical structure that lies at the interface of algebra and statistics. The starting point is a passage from classical probability theory to quantum probability theory. The quantum version of a probability distribution is a density operator, the quantum version of marginalizing is an operation called the partial trace, and the quantum version of a marginal probability distribution is a reduced density operator. Every joint probability distribution on a finite set can be modeled as a rank one density operator. By applying the partial trace, we obtain reduced density operators whose diagonals …

Tropical Cyclone Hazards In Relation To Propagation Speed, Jiehao Huang

Dissertations and Theses

As the population and infrastructure along the US East Coast increase, it becomes increasingly important to study the characteristics of tropical cyclones that can impact the coast. A recent study shows that the propagation speed of tropical cyclones has slowed over the past 60 years, which can lead to greater accumulation of precipitation and greater storm surge impacts. The study presented herein is meant to examine and analyze the relationships that exist between the propagation speed of tropical cyclones, their surface wind strength, displacement angles, and cyclone averaged winds. This analysis is focused on tropical cyclones spanning from 1950-2015 in …

Physical Applications Of The Geometric Minimum Action Method, George L. Poppe Jr.

Dissertations, Theses, and Capstone Projects

This thesis extends the landscape of rare events problems solved on stochastic systems by means of the \textit{geometric minimum action method} (gMAM). These include partial differential equations (PDEs) such as the real Ginzburg-Landau equation (RGLE), the linear Schroedinger equation, along with various forms of the nonlinear Schroedinger equation (NLSE) including an application towards an ultra-short pulse mode-locked laser system (MLL).

Additionally we develop analytical tools that can be used alongside numerics to validate those solutions. This includes the use of instanton methods in deriving state transitions for the linear Schroedinger equation and the cubic diffusive NLSE.

These analytical solutions are …

Stochastic Processes And Their Applications To Change Point Detection Problems, Heng Yang

Dissertations, Theses, and Capstone Projects

This dissertation addresses the change point detection problem when either the post-change distribution has uncertainty or the post-change distribution is time inhomogeneous. In the case of post-change distribution uncertainty, attention is drawn to the construction of a family of composite stopping times. It is shown that the proposed composite stopping time has third order optimality in the detection problem with Wiener observations and also provides information to distinguish the different values of post-change drift. In the case of post-change distribution uncertainty, a computationally efficient decision rule with low-complexity based on Cumulative Sum (CUSUM) algorithm is also introduced. In the time …