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A new numerical method is developed for the solution of the diffusion problem in a system of several materials. In such a system, the diffusion coefficients are piecewise continuous and jumps in their values can occur across the complex...
In financial markets under uncertainty, the classical Black-Scholes model cannot explain the empirical facts such as fat tails observed in the probability density. To overcome this drawback, during the last decade, Lévy process and...
The connections between algebra, geometry, and analysis have led the way for numerous results in many areas of mathematics, especially complex analysis. Considerable effort has been made to develop higher dimensional analogues of the...
Artificial neural networks (ANNs) are very popular nowadays and offer reliable solutions to many classification problems. Recent research indicates that these neural networks might be overparameterized and different solutions have been...
This dissertation presents studies of effective methods in two main areas of algebraic geometry: intersection theory and characteristic classes, and combinatorial algebraic geometry. We begin in chapter 2 by giving an effective algorithm...
In this dissertation, we evaluate existing Monte Carlo estimators and develop new Monte Carlo estimators for pricing financial options with the goal of improving precision. In Chapter 2, we discuss the conditional expectation Monte Carlo...
We develop partial differential equation methods with well-posed boundary conditions to price average strike options and swing options in the energy market. We use the energy method to develop boundary conditions that make a two space...
Quantifying uncertainty in modeling and simulation is crucial since the parameters of the physical system are inherently non-deterministic and knowledge of the system embodied in the model is incomplete or inadequate. The most well...
Crew costs are the second largest direct operating cost of airlines next to fuel costs. Therefore much research has been devoted to the planning and scheduling of crews over the last thirty years. The planning and scheduling of crews is...
The objective of this work is to quantify uncertainty and perform global sensitivity analysis for nonlinear models with a moderate or large number of stochastic parameters. We implement non-intrusive methods that do not require...
A unified, asymptotically-preserving method for simulating multiphase flows using an exactly mass, momentum, and energy conserving Cell-Integrated Semi-Lagrangian advection algorithm is presented. The new algorithm uses a semi-implicit...
Belief function theory, also known as Dempster-Shafer theory or evidence theory, gives a general framework for quantifying, representing, and managing uncertainty, and it is widely used in several applications from artificial...
Global sensitivity analysis studies how uncertainty in a model's output can be attributed to uncertainties in the model's inputs. This analysis has applications in various fields such as natural sciences, engineering, social sciences, ...
This dissertation develops and evaluates a computationally efficient and high-order numerical method to compute wave reflection and transmission from moving material boundaries. We use a discontinuous Galerkin spectral element...
New numerical algorithms are devised for high-order, efficient quadrature in domains arising from the intersection of a hyperrectangle and a manifold implicitly defined by level sets. By casting the manifold locally as the graph of a...
The objective of this dissertation is to study the queuing and point process models that try to capture as many features as possible of the high-frequency data of a limit order book. First, we use a generalized birth-death stochastic...
In this dissertation, we introduce a genetic algorithm approach to estimate the star discrepancy of a point set. This algorithm allows for the estimation of the star discrepancy in dimensions larger than seven, something that could not...
Most of the data encountered is bounded nonlinear data. The Universe is bounded, planets are sphere like shaped objects, and life growing on Earth comes in various shapes and colors that can hardly be represented as points on a linear...
We develop an adaptive spectral element method to price American options, whose solutions contain a moving singularity, automatically and to within prescribed errors. The adaptive algorithm uses an error estimator to determine where...
Our work builds on that of Barbot and Fenley to generalize Bonatti and Langevin's famous construction of a graph manifold with pseudo-Anosov flow in which all Seiftert fibered pieces of the torus decomposition are periodic. We provide...
Shape represents a complex and rich source of biological information that is fundamentally linked to underlying mechanisms and functions. Many fields of biology employ mathematical tools for the statistical analysis of shape variation....
In this thesis we present an algorithm that finds closed form solutions for homogeneous linear recurrence equations. The key idea is transforming an input operator Linp to an operator Lg with known solutions. The main problem of this...
The default threshold framework for credit risk modeling developed by Garreau and Kercheval [SIAM Journal on Financial Mathematics, 7:642-673, 2016] enjoys the advantages of both the structural form models and the reduced form models, ...
The neuroendocrine system presents a diversity of patterns in hormone secretion across a large range of temporal scales. These patterns of secretion are crucial to many aspects of animal survival and behavior. This offers a wealth of...
This dissertation investigates the practical expressive power of machine learning models as general function approximators and their practical applications in optimal filtering problems. The practical expressive power of these models...
Symmetric positive definite (SPD) matrices have become fundamental computational objects in many areas. It is often of interest to average a collection of symmetric positive definite matrices. This dissertation investigates different...
This dissertation is concerned with the study of how various properties such as malnormality and maximality of surface groups embedded in a 3-manifold group give us information about the topology of a 3-manifold. In this direction we...
This dissertation considers optimization problems on a Riemannian matrix manifold ℳ⊆ℝ[superscript m x n] with an additional rank inequality constraint. A novel technique for building new rank-related geometric objects from known...
Diseases such as tuberculosis, chronic pneumonia, and inner ear infections are caused by bacterial biofilms. Biofilms can form on any surface such as teeth, floors, or drains. Many studies show that it is much more difficult to kill the...
We study the asymptotics of various short-range interactions, namely, minimal Riesz $s$-energy, maximal Riesz $s$-polarization, best packing, and best covering, on sets satisfying minimal smoothness assumptions. In particular, we present...
In the field of data analysis, it is important to reduce the dimensionality of data, because it will help to understand the data, extract new knowledge from the data, and decrease the computational cost. Principal Component Analysis (PCA...
The focus of this work is pulsatile prolactin (PRL) secretion, which includes in vivo experiments on PRL release in female rats and mathematical modeling at the system and single-cell level . First we investigate the generation of the...
This dissertation examines the study of rigidity of collections of objects in various geometric spaces, and the correspondences shared between geometries. In particular, we take a look at vectors and lines in Lorentz $(n+1)$-space, ...
Microfiltration is one of the most efficient filtration techniques used to remove pathogenic protozoa, bacteria, and other colloidal matter from feed water. During the filtration process these pollutants accumulate on the filter surface...
In this dissertation, we create three theoretical models to answer questions raised by recent experiments that lie beyond the scope of current theory. In the landmark-effect model, we determine size, shape and location for a territory...
The present work is about partially hyperbolic dynamics on closed 3-manifolds and closed periodic curves. Partially hyperbolic systems are dynamical systems characterized by a dominated splitting of their tangent bundle into directions...
We present three algorithms to reduce homogeneous linear differential equations to their simplest form. Factoring a differential operator reduces a differential equation L(y)=0 to equations of minimal order, but this is not the only...
In this dissertation, we describe the biophysical mechanisms underlying the relationship between the structure and function of the KcsA K+ channel. Because of the conciseness of electro-diffusion theory and the computational advantages...
The use of time-inhomogeneous additive models in option pricing has gained attention in recent years due to their potential to adequately price options across both strike and maturity with relatively few parameters. In this thesis two...
In Seminaire de Geometrie Algebrique 4 (SGA4), Expose XVIII, Pierre Deligne proves that to any Picard stack one can associate a complex of abelian sheaves of length 2. He also studies the morphisms between such stacks and shows that such...
Biofilms are thin layers of microorganisms in which cells adhere to each other and stick to a surface. They are resistant to antibiotics and disinfectants due to the protection from extracellular polymeric substance (EPS), which is a gel...
The cerebral cortex is a thin folded sheet of neural tissue forming the outmost layer of the cerebrum (brain). Several biological hypotheses have suggested dierent mechanisms involved the development of its folding pattern into sulci ...
The standard Lucas asset pricing model makes two common assumptions of homogeneous agents and rational expectations equilibrium. However, these assumptions are unrealistic for real financial markets. In this work, we relax these...
Like nerve and many other endocrine cells, pancreatic beta-cells are electrically excitable and produce electrical impulses in response to elevations in glucose. These electrical impulses typically come in the form of bursting. One type...
Of late, path-breaking advances are taking place and flourishing in the field of solid-state Nuclear Magnetic Resonance (ssNMR)spectroscopy. One of the major applications of ssNMR techniques is to high resolution three-dimensional...
In this contribution, we focus on the study of multiphase flow using the phase field approach. Multiphase flow phenomena are ubiquitous. Common examples include coupled atmosphere and ocean system (air and water), oil reservoir (water, ...
Methods from shape analysis are used in morphometry, which is the quantitative analysis of macroscopic anatomical features. We assume that anatomy is flexible, and this brings us to the first problem of resolving how ``shape'' should be...
Iterative methods are widely used to solve algebraic equations and matrix equations. In fact, they can also be used to solve differential equations. Unlike discretization methods which generate point solutions, iterative methods generate...
The biaxiality of the steady state solutions and their stability to in-plane disturbances in shearing flows of nematic liquid crystalline polymers are studied by using simplified Wang (2002) model. We obtain all the steady states of Wang...
Canine distemper virus (CDV), similar to the human measles virus, is an extremely contagious virus that can cause outbreaks in dense canine populations. Previous models showed an outbreak of CDV in animal shelters can prevent or reduce...
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