Physical Sciences and Mathematics Commons^™

Asymptotic Expansion Of The L^2 Norms Of The Solutions To The Heat And Dissipative Wave Equations On The Heisenberg Group, Preston Walker

Theses and Dissertations

Motivated by the recent work on asymptotic expansions of heat and dissipative wave equations on the Euclidean space, and the resurgent interests in Heisenberg groups, this dissertation is devoted to the asymptotic expansions of heat and dissipative wave equations on Heisenberg groups. The Heisenberg group, $\mathbb{H}^{n}$, is the $\mathbb{R}^{2n+1}$ manifold endowed with the law $$(x,y,s)\cdot (x',y',s') = (x+x', y+y', s+ s' + \frac{1}{2} (xy' - x'y)),$$ where $x,y\in \mathbb{R}^{n}$ and $t\in \mathbb{R}$. Let $v(t,z)$ and $u(t,z)$ be solutions of the heat equation, $v_{t} - \mathcal{L} v=0$, and dissipative wave equation, $u_{tt}+u_{t} - \mathcal{L}u =0$, over the Heisenberg group respectively, where …

Optimization Framework For Reconstructing Biomedical Images By Efficient Sample-Based Parameterization, Paul Richard Arbic Ii

Theses and Dissertations

An efficient computational approach for optimal reconstruction of binary-type images suitable for models in various biomedical applications is developed and validated. The methodology includes derivative-free optimization supported by a set of sample solutions with customized geometry generated synthetically. The entire framework has an easy to follow design due to a nominal number of tuning parameters which makes the approach simple for practical implementation in various settings, adjusting it to new models, and enhancing the performance. High efficiency in computational time is achieved through applying the coordinate descent method to work with individual controls in the predefined custom order. This technique …

On Coupled Reaction Diffusion Equations And Their Applications, Juan J. Huerta

Theses and Dissertations

Reaction-diffusion equations are nonlinear partial differential equations that have been used extensively in mathematical modeling. An interesting case in this type of equation is the Fisher-Kolmogorov system, which has been used to study a low-grade glioma, a group of primary brain tumors. In the first part of this thesis, a stochastic version of the Fisher-Kolmogorov system will be studied, and exact and numerical solutions will be presented.

The second part of this thesis will show how the speed of information propagation affects disease spread and vaccination uptake through networks in epidemics. In this model, the information reaches different people at …

Bivariate Markov Chain Model Of Irritable Bowel Syndrome (Ibs) Subtypes And Abdominal Pain, Ricardo Reyna Jr.

Theses and Dissertations

Researchers use stochastic models like continuous-time Markov chains (CTMC) to model progression of morbidities of public health impact, like HIV and Hepatitis C. Most of the research in that area is done for a single disease. In this research, we use a bivariate continuous-time Markov chain (CTMC) to model progression of co-morbidities. In particular, we use a bivariate CTMC to model the joint progression of Irritable Bowel Syndrome (IBS) and abdominal pain. Symptoms of IBS are known to change throughout the duration of the disorder. Hence, patients are normally asked to make a journal of the stool type, symptoms, and …

Variable-Order Fractional Partial Differential Equations: Analysis, Approximation And Inverse Problem, Xiangcheng Zheng

Theses and Dissertations

Variable-order fractional partial differential equations provide a competitive means in modeling challenging phenomena such as the anomalous diffusion and the memory effects and thus attract widely attentions. However, variable-order fractional models exhibit salient features compared with their constant-order counterparts and introduce mathematical and numerical difficulties that are not common in the context of integer-order and constant-order fractional partial differential equations.

This dissertation intends to carry out a comprehensive investigation on the mathematical analysis and numerical approximations to variable-order fractional derivative problems, including variable-order time-fractional, space-fractional, and space-time fractional partial differential equations, as well as the corresponding inverse problems. Novel techniques …

The Second Law Of Thermodynamics And The Accumulation Theorem, Austin Maule

Theses and Dissertations

In Serrin's proof of the Accumulation Theorem, the presence of an ideal gas G is assumed.

In 1979 at the University of Naples, Serrin (allegedly) proved that the ideal system G can be replaced by a more general ideal system and still have the Accumulation Theorem hold.

In this paper, we attempt to reconstruct Serrin's proof and supply a proof for a more general theorem stated in a paper of Coleman, Owen and Serrin.

Asymptotic Probability Of Incidence Relations Over Finite Fields, Adam Buck

Theses and Dissertations

Given four generic lines in FP3, we ask, "How many lines meet the four?" The answer depends on the field. When F = C, the answer is two. When F = R, the answer is either zero or two.

If we work over a finite field there are only finitely many projective lines. We compute the probability four lines are met by two. The main result is that as q approaches infinity, this probability approaches 1/2. Asymptotically, the other half of the time zero lines will meet the four.

Local Connectedness Of Bowditch Boundary Of Relatively Hyperbolic Groups, Ashani Dasgupta

Theses and Dissertations

If the Bowditch boundary of a finitely generated relatively hyperbolic group is connected, then, we show that it is locally connected. Bowditch showed that this is true provided the peripheral subgroups obey certain tameness condition. In this paper, we show that these tameness conditions are not necessary.

Earth Mover's Distance Between Grade Distribution Data With Fixed Mean, Jan Kretschmann

Theses and Dissertations

The Earth Mover's Distance (EMD) is examined on all theoretically possible grade distributions with the same grade point average (GPA). The numbers of distributions with the same EMD and GPA are encoded in the coefficients of a generating function. The theoretical mean EMD for grade distributions, that are sampled uniformly and independently at random, is computed from this function, and compared to real world grade data taken from several years. The data is further examined regarding the appearance of clusters that change when varying the distance threshold.

Algebraic Relations Via A Monte Carlo Simulation, Alison Elaine Becker

Theses and Dissertations

The conjugation action of the complex orthogonal group on the polynomial functions on $n \times n$ matrices gives rise to a graded algebra of invariants, $\mathcal{P}(M_n)^{O_n}$. A spanning set of this algebra is in bijective correspondence to a set of unlabeled, cyclic graphs with directed edges equivalent under dihedral symmetries. When the degree of the invariants is $n+1$, we show that the dimension of the space of relations between the invariants grows linearly in $n$. Furthermore, we present two methods to obtain a basis of the space of relations; we construct a basis using an idempotent of the group algebra …

A Dynamic Programming Approach To Impulse Control Of Brownian Motions, Robin Braun

Theses and Dissertations

This thesis considers an impulse control problem of a standard Brownian motion under a discounted criterion, in which every intervention incurs a strictly positive cost. The value function and an optimal $(\tau_{*}, Y_{*})$ policy are found using the dynamic programming principle together with the smooth pasting technique. The thesis also performs a sensitivity analysis by analyzing the limiting behaviors of the value function and the $(\tau_{*}, Y_{*})$ policy when the fixed intervention cost converges to zero. It is demonstrated that the limits agree with the classic fuel follower problem.

The thesis next formulates and analyzes an $N$-player stochastic game of …

Dictionary-Based Data Generation For Fine-Tuning Bert For Adverbial Paraphrasing Tasks, Mark Anthony Carthon

Theses and Dissertations

Recent advances in natural language processing technology have led to the emergence of

large and deep pre-trained neural networks. The use and focus of these networks are on transfer

learning. More specifically, retraining or fine-tuning such pre-trained networks to achieve state

of the art performance in a variety of challenging natural language processing/understanding

(NLP/NLU) tasks. In this thesis, we focus on identifying paraphrases at the sentence level using

the network Bidirectional Encoder Representations from Transformers (BERT). It is well

understood that in deep learning the volume and quality of training data is a determining factor

of performance. The objective of …

Analysis On Some Basic Ion Channel Modeling Problems, Zhen Chao

Theses and Dissertations

The modeling and simulation of ion channel proteins are essential to the study of many vital physiological processes within a biological cell because most ion channel properties are very difficult to address experimentally in biochemistry. They also generate a lot of new numerical issues to be addressed in applied and computational mathematics. In this dissertation, we mainly deal with some numerical issues that are arisen from the numerical solution of one important ion channel dielectric continuum model, Poisson-Nernst-Planck (PNP) ion channel model, based on the finite element approximation approach under different boundary conditions and unstructured tetrahedral meshes. In particular, we …

Gross's Proof Of Local Existence For The Coupled Maxwell-Dirac Equations, Kimberly Jane Harry

Theses and Dissertations

The Maxwell-Dirac equations are a model for the interaction of a relativistic electron with an electromagnetic field. It is to be expected that the initial value problem will have an unique solution which exists for all time t >0, for all appropriate initial conditions. This is not yet known, but in 1966, Leonard Gross proved a local existence theorem. In this thesis, we will present an overview of Leonard Gross's proof.

Analysis Of The Continuity Of The Value Function Of An Optimal Stopping Problem, Samuel Morris Nehls

Theses and Dissertations

In order to study model uncertainty of an optimal stopping problem of a stochastic process with a given state dependent drift rate and volatility, we analyze the effects of perturbing the parameters of the problem. This is accomplished by translating the original problem into a semi-infinite linear program and its dual. We then approximate this dual linear program by a countably constrained sub-linear program as well as an infinite sequence of finitely constrained linear programs. We find that in this framework the value function will be lower semi-continuous with respect to the parameters. If in addition we restrict ourselves to …

Complete Integrability And Discretization Of Euler Top And Manakov Top, Austin Marstaller

Theses and Dissertations

The Euler top is a completely integrable system with physical system implications and the Manakov top is its four-dimensional extension. We are concerned about their complete integrability and the preservation of this property under a specific discretization known as the Hirota-Kimura Discretization. Surprisingly, it is not guaranteed that under any discretization the conserved quantities are preserved and therefore they must be discovered. In this work we construct the Poisson bracket and Lax pair for each system and provide the Lie algebra background needed to do such such constructions.

Pipe Flow Of Newtonian And Non-Newtonian Fluids, Erick Sanchez

Theses and Dissertations

We consider an incompressible, viscous fluid in a cylindrical pipe. We obtain velocity profile for both Newtonian fluid and non-Newtonian fluids such as shear-thinning, shear- thickening and Bingham plastic fluids. The flow is governed by the equation of continuity (conservation of mass) and the momentum equation. After presenting the governing system in the cylindrical coordinate system and assuming that the flow is due to the pressure drop and wall shear stress, we derive the expressions for the velocity component in the axial direction for these cases. Some computational results of the velocity profiles for various cases are presented. We will …

The Period Of The Coefficients Of The Gaussian Polynomial[N+33], Arturo J. Martinez

Theses and Dissertations

Definition 1. For any N, the central coefficient(s) of [N+33] is denoted by C0(N) and the coefficient that is x ''away" from the central coefficient(s) of [N+33] is denoted by Cx(N).

In [1] the following result is proved:

Theorem 2. The central coefficient(s) of the Gaussian polynomial [N+33] are described by the generating function

[Special characters omitted]

This generating function has period 4.

The main goal of this thesis is to generalize Theorem 0.2 by way of proving the following conjecture:

Conjecture 3. For any x the …

A Study Of The Efficacy Of Machine Learning For Diagnosing Obstructive Coronary Artery Disease In Non-Diabetic Patients, Demond Larae Handley

Theses and Dissertations

According to the Centers for Disease Control and Prevention, about 18.2 million adults age 20 and older have Coronary Artery Disease in the United States. Early diagnosis is therefore of crucial importance to help prevent debilitating consequences, and principally death for many patients. In this study we use data containing gene expression values from peripheral blood samples in 198 non-diabetic patients, with the goal of developing an age and sex gene expression model for diagnosis of Coronary Artery Disease. We employ machine learning methods to obtain a classification based on genetic information, age and sex. Our implementation uses feed forward …

Evaluation Of Text Document Clustering Using K-Means, Lisa Beumer

Theses and Dissertations

The fundamentals of human communication are language and written texts. Social media is an essential source of data on the Internet, but email and text messages are also considered to be one of the main sources of textual data. The processing and analysis of text data is conducted using text mining methods. Text Mining is the extension of Data Mining to text files to extract relevant information from large amounts of text data and to recognize patterns. Cluster analysis is one of the most important text mining methods. Its goal is the automatic partitioning of a number of objects into …

Numerical Solution Of A Class Of Stochastic Functional Differential Equations With Financial Applications, Laszlo Nicolai Fertig

Theses and Dissertations

After a brief review of the Euler and Milstein numerical schemes and their convergence results

for stochastic differential equations (SDEs) and stochastic functional differential equations

(SFDEs), the thesis next proposes two specific SFDEs. The classical Euler and Milstein

schemes are developed to find the numerical solutions of these SFDEs, which are then compared

with the Ornstein-Uhlenbeck and a modified Ornstein-Uhlenbeck processes. These

results are further used to build four different but related stochastic models for stock prices.

The fitness of these models is analyzed by comparing real market data. The thesis concludes

with a numerical study for option pricing for …

Smoothed Quantiles For Claim Frequency Models, With Applications To Risk Measurement, Ponmalar Suruliraj Ratnam

Theses and Dissertations

Statistical models for the claim severity and claim frequency variables are routinely constructed and utilized by actuaries. Typical applications of such models include identification of optimal deductibles for selected loss elimination ratios, pricing of contract layers, determining credibility factors, risk and economic capital measures, and evaluation of effects of inflation, market trends and other quantities arising in insurance. While the actuarial literature on the severity models is extensive and rapidly growing, that for the claim frequency models lags behind. One of the reasons for such a gap is that various actuarial metrics do not possess ``nice'' statistical properties for the …

Analysis Of Inventory Models With Random Supply Using A Long-Term Average Criterion, Lars Moestue

Theses and Dissertations

In this thesis we will use different numerical algorithms for inventory models, where the inventory level is described by a stochastic differential equation and therefore random. Furthermore we assume that order supply is randomly distributed. The goal is to find the optimal order strategy to minimize the long-term average costs.\\

This stochastic problem can be reformulated as non-linear optimization problem. However the problems are too complex to solve by hand, so we need to use numerical optimization algorithms and for some of the models even numerical integration methods. \\

These algorithms then can be used to analyze some properties and …

Discrete Moment Problems With Logconcave And Logconvex Distributions, Talal Alharbi

Theses and Dissertations

We introduce new shape constraints, logconcavity and logconvexity, to discrete moment problems for bounding the k-out-of-n type probabilities and expectations of higher order convex functions of discrete random variables with non-negative and finite support. The bounds are obtained as the optimum values of non-convex and convex nonlinear optimization problems, where the non-convex problem is reformulated as a bilinear optimization problem. We present numerical experiments to show the improvement in the tightness of the bounds when the shape of underlying unknown probability distribution is prescribed into discrete moment problems. We apply our optimization based bounding methodology in an insurance problem to …

Fitting Of Lotka-Volterra Model For Coupled Population Growth Data Through Least-Squares Estimation Of Parameters, Jessica Ann Harter

Theses and Dissertations

The population of two types of bacteria found in the Gulf Coast of Florida, V.chagasii and V. harveyi, can be described by the Lotka-Voltera competition model. Using data gathered in experiments conducted by Bury and Pickett (2015), we take a different approach to find parameter estimates using numerical methods in R. In particular, we find a numerical solution to the coupled set of ODEs and minimize the sum of squared errors in order to obtain the optimal parameter estimates that will fit the data best. In order to get a sense of accuracy of these parameter estimates, we use bootstrap …

The Fundamental System Of Units For Cubic Number Fields, Janik Huth

Theses and Dissertations

Let $K$ be a number field of degree $n$. An element $\alpha \in K$ is called integral, if the minimal polynomial of $\alpha$ has integer coefficients. The set of all integral elements of $K$ is denoted by $\mathcal{O}_K$. We will prove several properties of this set, e.g. that $\mathcal{O}_K$ is a ring and that it has an integral basis. By using a fundamental theorem from algebraic number theory, Dirichlet's Unit Theorem, we can study the unit group $\mathcal{O}_K^\times$, defined as the set of all invertible elements of $\mathcal{O}_K$. We will prove Dirichlet's Unit Theorem and look at unit groups for …

Modeling Of Cloud Droplet Formation: Software Development And Sampling Strategies, Niklas Selke

Theses and Dissertations

Updraft speeds are an important factor in the formation of cloud droplets which play an important role in an atmospheric simulation. The updraft speeds are varying very strongly in small areas of space. Current models do not account for this kind of variability. Support for a probability density function (PDF) based approach in representing the variability of the updraft speeds has been implemented in the Energy Exascale Earth System Model (E3SM). Specifics of the implementation process have been discussed.

Different sampling strategies were tested to analyze the convergence behavior of the new approach to the cloud droplet formation process. It …

A Statistical Model For The Prediction Of The Taxi Trip Time In The City Of Chicago, Frank Bitter

Theses and Dissertations

This thesis addresses the problem of prediction of taxi trip duration for any given

day, time, pickup point and dropo point. Data on taxi trips from the Chicago Data

Portal is used. The main idea of the model is to cluster similar trips together and use

the mean duration of all those clustered taxi trips to predict the duration of a new taxi

trip in that cluster. Furthermore, for a possible additional reduction of prediction error,

estimators from dierent days which are not signicantly dierent from each other are

pooled together. It is shown that this procedure improves prediction error.

Optimal Control Of The Second Order Elliptic Equations With Biomedical Applications, Saleheh Seif

Theses and Dissertations

Dissertation analyzes optimal control of systems with distributed parameters described by the general boundary value problems in a bounded Lipschitz domain for the linear second order uniformly elliptic partial differential equations (PDE) with bounded measurable coefficients. Broad class of elliptic optimal control problems under Dirichlet or Neumann boundary conditions are considered, where the control parameter is the density of sources, and the cost functional is the L2-norm difference of the weak solution of the elliptic problem from measurement along the boundary or subdomain. The optimal control problems are fully discretized using the method of finite differences. Two types of discretization …

A Mathematical Development Of Minimal Surface Theory: From Soap Films To Black Holes, Timothy Pitts

Theses and Dissertations

Minimal surfaces are a special subset of surfaces that have gone through a long and extensive development and have also led to many fruitful findings in mathematics. Several periods that are key to the progression of the theory are coined as Golden Ages for the field’s development. Here, a historical and mathematical development of minimal surface theory is presented that spans from its inception in the late 18th century to the present day. Along with the development, there is an emphasis on showing connections of minimal surfaces to various natural phenomena that occur such as soap films, black holes, biological …