Digital Commons Network^™

Multiscale Modelling Of Brain Networks And The Analysis Of Dynamic Processes In Neurodegenerative Disorders, Hina Shaheen

Theses and Dissertations (Comprehensive)

The complex nature of the human brain, with its intricate organic structure and multiscale spatio-temporal characteristics ranging from synapses to the entire brain, presents a major obstacle in brain modelling. Capturing this complexity poses a significant challenge for researchers. The complex interplay of coupled multiphysics and biochemical activities within this intricate system shapes the brain's capacity, functioning within a structure-function relationship that necessitates a specific mathematical framework. Advanced mathematical modelling approaches that incorporate the coupling of brain networks and the analysis of dynamic processes are essential for advancing therapeutic strategies aimed at treating neurodegenerative diseases (NDDs), which afflict millions of …

Dynamical Aspects In (4+1)-Body Problems, Ryan Gauthier

Theses and Dissertations (Comprehensive)

The n-body problem models a system of n-point masses that attract each other via some binary interaction. The (n + 1)-body problem assumes that one of the masses is located at the origin of the coordinate system. For example, an (n+1)-body problem is an ideal model for Saturn, seen as the central mass, and one of its outer rings. A relative equilibrium (RE) is a special solution of the (n+1)-body problem where the non-central bodies rotate rigidly about the centre of mass. In rotating coordinates, these solutions become equilibria.

In this thesis we study dynamical aspects of planar (4 + …

Opposite Trees, Theo Goossens

Theses and Dissertations (Comprehensive)

A spanning tree of a graph G is a connected acyclic subgraph of G that includes all of the vertices in G. The degree of a vertex is the number of edges incident to that vertex. Given a spanning tree T of a graph G, an opposite tree of T is a spanning tree of G where the degree of each of its vertices is different from its degree in T. For complete, complete bipartite, and complete multipartite graphs, we give the conditions spanning trees of these graphs must satisfy in order to have an opposite tree.

The Kepler Problem On Complex And Pseudo-Riemannian Manifolds, Michael R. Astwood

Theses and Dissertations (Comprehensive)

The motion of objects in the sky has captured the attention of scientists and mathematicians since classical times. The problem of determining their motion has been dubbed the Kepler problem, and has since been generalized into an abstract problem of dynamical systems. In particular, the question of whether a classical system produces closed and bounded orbits is of importance even to modern mathematical physics, since these systems can often be analysed by hand. The aforementioned question was originally studied by Bertrand in the context of celestial mechanics, and is therefore referred to as the Bertrand problem. We investigate the qualitative …

Rationality In Bargaining By Finite Automata, Jim Bell

Theses and Dissertations (Comprehensive)

Aspects of behavioral decision-making can be integrated into game-theoretic models of two-player bargaining using finite automata which can represent bargaining strategies in combination with various behavioral traits. The automata are used as bargaining agents who must jointly agree upon a fixed allocation of transferable utility in an infinite-horizon Rubinstein bargaining game. At each turn, the automata are given the opportunity to accept a proposed portion of the transferable utility, or to reject the proposal and make a counter-offer of their own. A round-robin tournament and ecological simulations were run to explore strategic dominance under different conditions. Principles of bargaining strategy …

Nonlinear Coupled Effects In Nanomaterials, Sia Bhowmick

Theses and Dissertations (Comprehensive)

Materials at the nanoscale have different chemical, structural, and optoelectrical properties compared to their bulk counterparts. As a result, such materials, called nanomaterials, exhibit observable differences in certain physical phenomena. One such resulting phenomenon called the piezoelectric effect has played a crucial role in miniature self-powering electronic devices called nanogenerators which are fabricated by using nanostructures, such as nanowires, nanorods, and nanofilms. These devices are capable of harvesting electrical energy by inducing mechanical strain on the individual nanostructures. Electrical energy created in this manner does not have environmental limitations. In this thesis, important coupled effects, such as the nonlinear piezoelectric …

Abelian Subalgebras Of Maximal Dimension In Euclidean Lie Algebras, Mark Curro

Theses and Dissertations (Comprehensive)

In this paper we define, discuss and prove the uniqueness of the abelian subalgebra of maximal dimension of the Euclidean Lie algebra. We also construct a family of maximal abelian subalgebras and prove that they are maximal.

Computing Closed Forms For The Convergent Series $\Displaystyle\Sum_{N \In \Mathbb{Z}}\Frac{1}{(N^3+Bn^2+Cn+D)^K}$, Gagandeep K. Virk

Theses and Dissertations (Comprehensive)

In this thesis we discuss the various approaches that will be taken to evaluate and find a finite closed form for the sum $$\sum_{n \in \mathbb{Z}} \frac{1}{(n^3+Bn^2+Cn+D)^k}$$ where $B, C, D \in \mathbb{C}$ and $k$ is a positive integer. We begin this thesis by studying the cubic equations and discussing briefly various methods of finding their roots. Cardano's method (1545) for finding the roots of cubic polynomials is explored in detail as this method is used in later parts of the thesis to make calculations while evaluating the sums. Various tools and techniques from Fourier analysis are reviewed for these …

Community Detection Detailed For Online Social Networks, Christopher J. Hogan

Theses and Dissertations (Comprehensive)

Ever since the internet became publicly available it has allowed users to interact with each other across virtual networks. With this large amounts of data being collected the clustering of this information has become an even more powerful tool for recognize patterns and trends in a network. In this research we look build a model for Community Detection in these online social networks. We combine the ideas from both discrete mathematics and sociology, to build an algorithm with the specific intent on discovering communities that exist in an online social network. We present many of the sociology theories behind the …

Relative Equilibria Of Isosceles Triatomic Molecules In Classical Approximation, Damaris Miriam Mckinley

Theses and Dissertations (Comprehensive)

In this thesis we study relative equilibria of di-atomic and isosceles tri-atomic molecules in classical approximations with repulsive-attractive interaction. For di-atomic systems we retrieve well-known results. The main contribution consists of the study of the existence and stability of relative equilibria in a three-atom system formed by two identical atoms of mass $m$ and a third of mass $m_3$, constrained in an isosceles configuration at all times.

Given the shape of the binary potential only, we discuss the existence of equilibria and relative equilibria. We represent the results in the form of energy-momentum diagrams. We find that fixing the masses …

Mathematical Modeling And Control Of Nonlinear Oscillators With Shape Memory Alloys, Mohamed Bendame

Theses and Dissertations (Comprehensive)

Shape memory alloys (SMAs) belong to an interesting type of materials that have attracted the attention of scientists and engineers over the last few decades. They have some interesting properties that made them the subject of extensive research to find the best ways to utilize them in different engineering, biomedical, and scientific applications. In this thesis, we develop a mathematical model and analyze the behavior of SMAs by considering a one degree of freedom nonlinear oscillator consisting of a mass connected to a fixed frame through a viscous damping and a shape memory alloy device. Due to the nonlinear and …

Computational And Theoretical Aspects Of N-E.C. Graphs, Alexandru Costea

Theses and Dissertations (Comprehensive)

We consider graphs with the n-existentially closed adjacency property. For a positive integer n, a graph is n-existentially closed (or n-e.c.) if for all disjoint sets of vertices A and B with \A∪ B\ = n (one of A or B can be empty), there is a vertex 2 not in A∪B joined to each vertex of A and no vertex of B. Although the n-e.c. property is straightforward to define, it is not obvious from the definition that graphs with the property exist. In 1963, Erdos and Rényi gave …

Strict-Dominance Solvability Of Games On Continuous Strategy Spaces, Andrew Campbell Elkington

Theses and Dissertations (Comprehensive)

The concept of strict dominance provides a technique that can be used normatively to predict the play of games based only on the assumption of individual rationality. Such predictions, unlike those based on Nash equilibria, do not depend on players’ beliefs about the behaviour of others. One strategy strictly dominates another if and only if the payoff from the first strategy is strictly greater than the payoff from the second, no matter how the opponent(s) plays. It is possible for iterated elimination of strictly dominated strategies to remove all but a single choice for each player, in which case we …

Simulation Studies On Estimation Of Variance Components For Multilevel Models, Sara Vakilian

Theses and Dissertations (Comprehensive)

With the presence of unequal sampling in a multilevel model, the weight inflated estimators for variance components can be biased even though the use of survey weights results in design consistent estimators of the parameters. In this thesis I will carry out a simulation study to examine the performance of current existing methods and I will examine the resampling method for correcting bias of estimators of variance components of a multilevel model with covariates. This study will be based on these three papers: “Weighting for Unequal Selection Probabilities in Multilevel Models” by D. Pfeffermann , C. J. Skinner, D.J. Holmes, …

Applications Of New Diffusion Models To Barrier Option Pricing And First Hitting Time In Finance, Keang Ly

Theses and Dissertations (Comprehensive)

The main focus of this thesis is in the application of a new family of analytical solvable diffusion models to arbitrage-free pricing exotic financial derivatives, such as barrier options. The family of diffusions is the so-called “Drifted Bessel family” having nonlinear (smile-like) local volatility with multiple adjustable parameters. In particular, the drifted Bessel-K diffusion is used to model asset (stock) price processes under a risk-neutral measure whereby discounted asset price are martingales.

Closed-form spectral expansions for barrier option values are derived within the Bessel-K family of models. This follow from the closed-form spectral expansions for the transition probability …

First-Passage Time Models With A Stochastic Time Change In Credit Risk, Hui Li

Theses and Dissertations (Comprehensive)

Many authors have used a time-changed Brownian motion as a model of log-stock returns. Using a Levy process as a stochastic time change, one obtains well known asset price models such as the variance gamma (VG) and normal inverse Gaussian (NIG) models. Following on the heels of these asset price models, it is natural to extend structural credit models by using a time-changed geometric Brownian motion and other jump-diffusion processes to model the value of a firm. To avoid the difficulties that arise in computing the associated first passage time distribution and in analogy to the time-changed Markov chain models, …

Mathematical Modeling Of Quantum Dots With Generalized Envelope Functions Approximations And Coupled Partial Differential Equations, Dmytro Sytnyk

Theses and Dissertations (Comprehensive)

No abstract provided.

Models For On-Line Social Networks, Noor Hadi

Theses and Dissertations (Comprehensive)

On-line social networks such as Facebook or Myspace are of increasing interest to computer scientists, mathematicians, and social scientists alike. In such real-world networks, nodes represent people and edges represent friendships between them. Mathematical models have been proposed for a variety of complex real-world networks such as the web graph, but relatively few models exist for on-line social networks.

We present two new models for on-line social networks: a deterministic model we call Iterated Local Transitivity (ILT), and a random ILT model. We study various properties in the deterministic ILT model such as average degree, average distance, and diameter. We …

Modelling Asset Prices Under Regime Switching Diffusions Via First Passage Time, Xiaojing Xi

Theses and Dissertations (Comprehensive)

In this thesis we focus on the development of a new class of stochastic models for asset price processes and their application to option pricing and hedging. The asset price process involves analytical treatments for calculating first-hitting (or first-passage) times for a regular diffusion with killing in combination with Markov state-switching. The dynamics is naturally dictated by the underlying diffusion process itself rather than arising from some addition exogenous process. To date, this class of asset pricing models appears to be novel in the literature and, moreover, offers a significant to the standard geometric Brownian motion commonly used in the …

Portfolio Selection In Gaussian And Non-Gaussian Worlds, Jing Wang

Theses and Dissertations (Comprehensive)

In the case of minimizing risk with a given level of expected return, we discuss the portfolio selection problem with the asset returns are characterized by a Gaussian distribution and heavy tailed distribution.

More specifically, under the Gaussian assupmtion, we give the explicit solutions to the problems of minimizing risk variance, CaR and EaR respectively. When a compound Poisson process is assumed, we derive explicit solutions to the variance, CaR and EaR. Furthermore, we give the explicit soultion for the CaR when a Lévy distribution is considered.

For the more realistic process-normal inverse process, we are able to obtain the …

Partial Separability And Partial Additivity For Orderings Of Binary Alternatives, Md. Abul Bashar

Theses and Dissertations (Comprehensive)

In Multiple-Criteria Decision Analysis (MCDA), a good way to find the best alternative is to construct a value function that represents a Decision Maker’s (DM) preferences. For multidimensional alternatives, an additive value function is easiest to work with because it assesses the alternatives in a simple and transparent manner. A DM’s preferences over consequences on a subset of the set of criteria may or may not depend on consequences on the rest of the criteria. Preferences that are free from all such interdependence are said to be separable. The existence of an additive value function implies separability and, when consequences …