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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 + …

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 …

Asset Pricing Under Randomized Solvable Diffusions, Hiromichi Kato

Theses and Dissertations (Comprehensive)

By employing a randomization procedure on the geometric Brownian motion (GBM) model, we construct our new pricing models with stochastic volatility exhibiting symmetric smiles in the log-forward moneyness, and admitting simple closed-form analytical expressions for European-style option prices. We assume that there are no infinitesimal correlations between the underlying asset prices and their volatility, and the integrated squared volatility processes are random variables with well-known probability density functions. Under some regularity conditions, closed-form expressions are obtained by taking the expectation of option prices under diffusion models over the integrated squared volatility process, which relate to the Bayesian framework in the …

Analysis Of Clmr Trees For European And Asian Option Pricing Under Regime-Switching Jump-Diffusion Models, Yaode Sui

Theses and Dissertations (Comprehensive)

In this paper, we study the convergence rates of the multinomial trees constructed by [Costabile, Leccadito, Massabo and Russo, Journal of Computational and Applied Mathematics, 256 (2014), 152 - 167] for European option pricing under the regime-switching jump-diffusion model, which is named as CLMR tree. We also extend the CLMR tree to the pricing of Asian options under the models. Numerical examples are carried out to confirm the theoretical results and the accuracy of computation.

Multiscale Mathematical Modelling Of Nonlinear Nanowire Resonators For Biological Applications, Rosa Fallahpourghadikolaei

Theses and Dissertations (Comprehensive)

Nanoscale systems fabricated with low-dimensional nanostructures such as carbon nanotubes, nanowires, quantum dots, and more recently graphene sheets, have fascinated researchers from different fields due to their extraordinary and unique physical properties. For example, the remarkable mechanical properties of nanoresonators empower them to have a very high resonant frequency up to the order of giga to terahertz. The ultra-high frequency of these systems attracted the attention of researchers in the area of bio-sensing with the idea to implement them for detection of tiny bio-objects. In this thesis, we originally propose and analyze a mathematical model for nonlinear vibrations of nanowire …

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 …

Disease Models With Immigration, Reem Almarashi

Theses and Dissertations (Comprehensive)

In this thesis we focus first on studying the susceptible, exposed, and infected ($SEI$) disease model without immigration. We determine the basic reproduction number $\mathcal{R}_0$, which can be interpreted as the expected number of new cases that can be produced by a single infection in a completely susceptible population. Further, by using the Jacobian matrix, we determine the local stability of the disease model. Then we have the result that when $\mathcal{R}_0<1$ the DFE point is locally asymptotically stable(L.A.S). In contrast, when $\mathcal{R}_0>1$ we find that the endemic equilibrium is L.A.S. After that, we analyze the $SEI$ model with immigration of infected individuals.

Furthermore, we investigate the direction that 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 …

Financial Securities Under Nonlinear Diffusion Asset Pricing Model, Andrey Vasilyev

Theses and Dissertations (Comprehensive)

In this thesis we investigate two pricing models for valuing financial derivatives. Both models are diffusion processes with a linear drift and nonlinear diffusion coefficient. The forward price process of these models is a martingale under an assumed risk-neutral measure and the transition probability densities are given in analytically closed form. Specifically, we study and calibrate two different families of models that are constructed based on a so-called diffusion canonical transformation. One family follows from the Ornstein-Uhlenbeck diffusion (the UOU family) and the other—from the Cox-Ingersoll-Ross process (the Confluent-U family).

The first part of the thesis considers single-asset and multi-asset …

First Passage Time Problem For Multivariate Jump-Diffusion Processes: Models, Computation, And Applications In Finance, Di Zhang

Theses and Dissertations (Comprehensive)

The first passage time (FPT) problems are ubiquitous in many applications, from physics to finance. Mathematically, such problems are often reduced to the evaluation of the probability density of the time for a process to cross a certain level, a boundary, or to enter a certain region. While in other areas of applications the FPT problems can often be solved analytically, in finance we usually have to resort to the application of numerical procedures, in particular when we deal with jump-diffusion stochastic processes (JDP). The application of the conventional Monte-Carlo procedure is possible for the solution of the resulting model, …

Markov Switching And Jump Diffusion Models With Applications In Mathematical Finance, Shengkun Xie

Theses and Dissertations (Comprehensive)

In this thesis, we study some jump diffusion models with Markov switching and transition densities for Markov switching diffusion processes with and without an absorbing barrier. We work out some analytical results, which have useful applications in mathematical finance and other related fields. The first-passage time problem for a Markov switching model is also studied and European type options and lookback options are computed in closed-form as examples to show that these models can be applied in practice. We apply optimization methods and kernel smoothing techniques to produce some important numerical results that show that jump diffusion with Markov switching …