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Equilibrium Analysis For An Epidemic Model With A Reservoir For Infection, Istvan Lauko, Gabriella Pinter, Rachel Elizabeth Tewinkel

Mathematical Sciences Student Articles

We consider a system of non-linear differential equations describing the spread of an epidemic in two interacting populations. The model assumes that the epidemic spreads within the first population, which in turn acts as a reservoir of infection for the second population. Weexplore the conditions under which the epidemic is endemic in both populations and discuss the global asymptotic stability of the endemic equilibrium using a Lyapunov function and results established for asymptotically autonomous systems. We discuss monkeypox as an example of an emerging disease that can be modelled in this way and present some numerical results representing the model …

The Strong Law Of Large Numbers For U-Statistics Under Random Censorship, Jan Höft

Theses and Dissertations

We introduce a semi-parametric U-statistics estimator for randomly right censored data. We will study the strong law of large numbers for this estimator under proper assumptions about the conditional expectation of the censoring indicator with re- spect to the observed life times. Moreover we will conduct simulation studies, where the semi-parametric estimator is compared to a U-statistic based on the Kaplan- Meier product limit estimator in terms of bias, variance and mean squared error, under different censoring models.

Dynamic Pricing With Variable Order Sizes For A Model With Constant Demand Elasticity, Nyles Kirk Breecher

Theses and Dissertations

We investigate a dynamic pricing model under constant demand elasticity which accounts for customers ordering multiple items at once. A closed form expression for the optimal expected revenue and pricing strategy is found. Models with the same demand are shown to have asymptotically similar expected revenue and pricing strategies, even if the order size distributions of the customers are different. Surprisingly, the relative difference between comparable models is shown to be independent of time and the magnitude of demand. Variations of the model are considered, including different low inventory behavior as well as the effect of advertising. Some numerical simulations …

Triebel-Lizorkin Spaces Estimates For Evolution Equations With Structure Dissipation, Jingchun Chen

Theses and Dissertations

This work is concerned with the long time decay estimates of the generalized heat equations and the generalized wave equations in the homogeneous Triebel-Lizorkin spaces. We first extend the known results for the generalized heat equations in the real Hardy spaces. We also extend the known results for the generalized wave equations with structure dissipation in the real Hardy spaces.

The main tools employed are the decomposition of the unit, duality property in Triebel-Lizorkin spaces and the multiplier theorems in different function spaces such as Lebesgue spaces, real Hardy spaces and Triebel-Lizorkin spaces.

Internal And External Harmonic Functions In Flat-Ring Coordinates, Lijuan Bi

Theses and Dissertations

The goal of this dissertation is to derive expansions for a fundamental solution of Laplace's equation in flat-ring coordinates in three-dimensional Euclidean space. These expansions are in terms of harmonic functions in the interior and the exterior of two different types of regions, "flat rings" and "peanuts" according to their shapes. We solve Laplace's equation in the interior and the exterior of these regions using the method of separation of variables. The internal and external "flat-ring" and "peanut" harmonic functions are expressed in terms of Lamé functions.

Compactifications Of Manifolds With Boundary, Shijie Gu

Theses and Dissertations

This dissertation is concerned with compactifications of high-dimensional manifolds.

Siebenmann's iconic 1965 dissertation \cite{Sie65} provided necessary and

sufficient conditions for an open manifold $M^{m}$ ($m\geq6$) to be

compactifiable by addition of a manifold boundary. His theorem extends easily

to cases where $M^{m}$ is noncompact with compact boundary; however when

$\partial M^{m}$ is noncompact, the situation is more complicated. The goal

becomes a \textquotedblleft completion\textquotedblright\ of $M^{m}$, ie, a

compact manifold $\widehat{M}^{m}$ containing a compactum $A\subseteq\partial

M^{m}$ such that $\widehat{M}^{m}\backslash A\approx M^{m}$. Siebenmann did

some initial work on this topic, and O'Brien \cite{O'B83} extended that work

to an important special case. …

Z-Structures And Semidirect Products With An Infinite Cyclic Group, Brian Walter Pietsch

Theses and Dissertations

Z-structures were originally formulated by Bestvina in order to axiomatize the properties that an ideal group boundary should have. In this dissertation, we prove that if a given group admits a Z-structure, then any semidirect product of that group with an infinite cyclic group will also admit a Z-structure. We then show how this can be applied to 3-manifold groups and strongly polycyclic groups.

Scaling Behavior Of Drug Transport And Absorption In In Silico Cerebral Capillary Networks, William Langhoff, Alexander Riggs, Peter Hinow

Mathematical Sciences Faculty Articles

Drug delivery to the brain is challenging due to the presence of the blood-brain barrier. Mathematical modeling and simulation are essential tools for the deeper understanding of transport processes in the blood, across the blood-brain barrier and within the tissue. Here we present a mathematical model for drug delivery through capillary networks with increasingly complex topologies with the goal to understand the scaling behavior of model predictions on a coarse-to-fine sequence of grids. We apply our model to the delivery of L-Dopa, the primary drug used in the therapy of Parkinson's Disease. Our model replicates observed blood flow rates and …

Robust Estimation Of Parametric Models For Insurance Loss Data, Chudamani Poudyal

Theses and Dissertations

Parametric statistical models for insurance claims severity are continuous, right-skewed, and frequently heavy-tailed. The data sets that such models are usually fitted to contain outliers that

are difficult to identify and separate from genuine data. Moreover, due to commonly used actuarial “loss control strategies,” the random variables we observe and wish to model are affected by truncation (due to deductibles), censoring (due to policy limits), scaling

(due to coinsurance proportions) and other transformations. In the current practice, statistical inference for loss models is almost exclusively likelihood (MLE) based, which typically results in non-robust parameter estimators, pricing models, and risk measures. …

Fitting A Complex Markov Chain Model For Firm And Market Productivity, Julia Ruth Valder

Theses and Dissertations

This thesis develops a methodology of estimating parameters for a complex Markov chain model for firm productivity. The model consists of two Markov chains, one describing firm-level productivity and the other modeling the productivity of the whole market. If applicable, the model can be used to help with optimal decision making problems for labor demand. The need for such a model is motivated and the economical background of this research is shown. A brief introduction to the concept of Markov chains and their application in this context is given. The simulated data that is being used for the estimation is …

Numerical Solution Of Stochastic Control Problems Using The Finite Element Method, Maritn Gerhard Vieten

Theses and Dissertations

Based on linear programming formulations for infinite horizon stochastic control problems, a numerical technique in fashion of the finite element method is developed. The convergence of the approximate scheme is shown and its performance is illustrated on multiple examples. This thesis begins with an introduction of stochastic optimal control and a review of the

theory of the linear programming approach. The analysis of existence and uniqueness of solutions to the linear programming formulation for fixed controls represents the first contribution of this work. Then, an approximate scheme for the linear programming formulations is established. To this end, a novel discretization …

Optimal Insurance With Background Risk: An Analysis In The Presence Of Moderate Negative Dependence, Julian Johannes Dursch

Theses and Dissertations

As an individual or a corporation, there are various types of risks one faces. For many of these risks, there are insurance policies available for purchase that provide some protection against potential losses. However, there are also risks that are not insurable. These risks remain present as a background factor and affect the insured's final wealth. Consequentially, they have an impact on the optimal insurance for the insurable risk through the dependence structure between the insurable and uninsurable risk.

In this thesis, we take a look at the optimal insurance problem given an insurable risk Xand a background risk Y …

Optimal Deductibles: A Theoretical Analysis From An Insured's Perspective, Alexander Kreienbring

Theses and Dissertations

A stop-loss policy as a tool for protection against a large loss is one of the most common insurance forms. For fixed premiums and therefore a uniquely determined insurance deductible, it has been well-established that the stop-loss form is superior to all other common

insurance forms (Arrow, 1963). Using the expected premium principal, one can relax the assumption of a fixed premium and allow the insured to choose an arbitrary deductible that fits their needs.

This thesis presents a stop-loss insurance policy model from an insured's perspective for a flexible premium. It shows the existence and uniqueness of an optimal …

Exact Sampling And Prefix Distributions, Sebastian Oberhoff

Theses and Dissertations

This thesis explores some new means to generate random numbers without incurring any numerical

inaccuracies along the way. In the context of continuous distributions this leads to the discussion of

prex distributions { discrete distributions that fully capture a continuous distribution by describing

their initial digits. These are rst studied graphically, then analytically, which also leads to a general

examination of the behavior of the distribution of trailing digits of continuous distributions. Finally,

some slightly novel, related results from the theory of computation are presented.

Orthogonal Abelian Cartan Subalgebra Decompositions Of Classical Lie Algebras Over Finite Commutative Rings, Songpon Sriwongsa

Theses and Dissertations

Orthogonal decompositions of classical Lie algebras over the complex numbers of types A, B, C and D were studied in the early 1980s and attracted further attention in the past decade, especially in the type A case, due to its application in quantum information theory. In this dissertation, we consider the orthogonal decomposition problem of Lie algebras of type A, B, C and D over a finite commutative ring with identity. We first establish the appropriate definition of orthogonal decomposition under our setting, and then derive some general properties that rely on the finite commutative rings theory. Our goal is …

Numerical Solutions Of Fractional Nonlinear Advection-Reaction-Diffusion Equations, Sophia Vorderwuelbecke

Theses and Dissertations

In this thesis nonlinear differential equations containing advection, reaction and diffusion terms are solved numerically, where the diffusion term is modelled by a fractional derivative. One of the methods employed is a finite difference method for temporal as well as spatial discretization. Furthermore, exponential time differencing schemes under consideration of different matrix exponential approximations are exploited for the temporal discretization, whereas finite differences are used for the spatial approximation. The schemes are applied to the homogeneous Burgers, Burgers-Fisher and Burgers-Huxley equation and compared with respect to convergence and efficiency in a numerical investigation.