A discrete time Markov Chain is used to model enrollment at a regional university. A preliminary analysis is conducted on the data set in order to determine the classes for the Markov chain model. The semester, yearly, and long term results of the model are examined thoroughly. A sensitivity analysis of the probability matrix entries is then conducted to determine the overall greatest influence on graduation rates.

The spread of diseases and opinions has profoundly affected the development of human societies. The structure of the underlying social network may change as a result of individuals changing their social connections in response to an ongoing epidemic or opinion spreading, either for self protection or as an expression of personal values. The interaction of spreading processes and the underlying network structure has been a focus of many recent studies. In this dissertation, we construct models to better incorporate heterogeneous responses to disease spread and attempted opinion spread.;We first model the simultaneous spread of an epidemic and awareness about the epidemic on an adaptive social network. A previous Susceptible-Infectious-Susceptible (SIS) model with avoidance rewiring is extended. Susceptible and infectious nodes are each divided into aware and unaware types. Aware nodes affect the network structure by rewiring their connections to reduce disease exposure. Public media information is considered as an external source of node awareness. The effects of awareness on disease spread and network structure are explored using stochastic simulations and mean field equations. Network adaptation can generate steady state behavior or periodic oscillations. The epidemic threshold is predicted using two methods that improve upon mean field predictions, and a critical media rate controlling the existence of an epidemic threshold under fast rewiring is given.;Node-to-node communication is then introduced as another source of node awareness, and its influence on disease levels and epidemic thresholds is compared with public media information. A relationship of the thresholds under different awareness sources is derived. Our results in both models indicate that node awareness can play a significant role in minimizing disease spread, and in some cases media information is more effective at controlling disease than communication.;We also model the competition of two opinions on a social network. A small fraction of committed supporters of a new opinion is randomly distributed in the network among supporters of a previous opinion. We introduce a new process, exacerbation, in which committed supporters of an opinion may drive their contacts away from that opinion and toward strong commitment to the opposing viewpoint. In addition to network simulations, a mass action model of the process is studied. We find that exacerbation can change the final outcome of opinion competition. The influence of the initial fraction committed to the new opinion is also explored.

In this work, we propose a framework to develop testable hypotheses for the effects of changes in the experimental conditions on the dynamics of a biological system using mathematical models. We discuss the uncertainties present in this process and show how information from different experiment regimes can be used to identify a region in the parameter space over which subsequent mathematical analysis can be conducted. To determine the significance of variation in the parameters due to varying experimental conditions, we propose using sensitivity analysis. Using our framework, we hypothesize that the experimentally observed decrease in the survivability of bacterial populations of Xylella fastidiosa (causal agent of Pierce’s Disease) upon addition of zinc, might be because of starvation of the bacteria in the biofilm due to an inhibition of the diffusion of the nutrients through the extracellular matrix of the biofilm. We also show how sensitivity is related to uncertainty and identifiability; and how it can be used to drive analysis of dynamical systems, illustrating it by analyzing a model which simulates bursting oscillations in pancreatic β-cells. For sensitivity analysis, we use Sobol’ indices for which we provide algorithmic improvements towards computational efficiency. We also provide insights into the interpretation of Sobol’ indices, and consequently, define a notion of the importance of parameters in the context of inherently flexible biological systems. / A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Spring Semester 2019. / April 16, 2019. / Bacterial growth, Dynamical systems, Mathematical modeling, Sensitivity analysis, Sobol Indices, Xylella fastidiosa / Includes bibliographical references. / Nick Cogan, Professor Co-Directing Thesis; M.Y. Hussaini, Professor Co-Directing Thesis; Eric Chicken, University Representative; Harsh Jain, Committee Member; Richard Bertram, Committee Member; Washington Mio, Committee Member.

As one of the emerging optical molecular imaging modalities, bioluminescence tomography is for reconstruction of the light source distribution inside a small animal body from the measured photon data on its surface. Such a light source distribution can be either induced through the excitation of an external light source, or generated by an internal bioluminescent source. The propagation of light within a biological medium is accurately described by the radiative transfer equation. In this thesis, a bioluminescent source recovery problem is considered for the steady state radiative transfer equation. The recovery model is based on the minimization of combined effects of equation residual, boundary condition residual, and boundary measurement residual. The total variation of the light source is taken as one of the regularization terms so that the minimization of surface area of a light source is achieved. An alternating direction multiplier method is applied to decouple the system governing the light distribution and source data. Three different formulations and algorithms are introduced for the total variation regularization term. Convergence analysis is provided for each algorithm and numerical experiments are presented to show the performance of the algorithms.

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"October 13, 2003" Bibliography: leaves 102-105. This work describes methods associated with general random effects models. Part one describes a technique for investigating mean-variance relationships in random effects models. Part two derives and approximation to the likelihood function using a Laplace expansion to the fourth order.

<p>This study investigates the effect of using the different benchmarks stated above to calculate the beta of some Swedish stocks and to form a high risk stock vis-a-vis a low risk stock. The stocks will be combined in different forms (scenarios) i.e. High beta stocks, low beta stocks and a mixture of both high and low beta stocks to form a portfolio of stocks and tested to see the performance level of the individual scenarios.<strong></strong></p>

<p>This Licentiate Thesis is devoted to the presentation and discussion of some new contributions in applied mathematics directed towards scientific computing in sports engineering. It considers inverse problems of biomechanical simulations with rigid body musculoskeletal systems especially in cross-country skiing. This is a contrast to the main research on cross-country skiing biomechanics, which is based mainly on experimental testing alone. The thesis consists of an introduction and five papers. The introduction motivates the context of the papers and puts them into a more general framework. Two papers (D and E) consider studies of real questions in cross-country skiing, which are modelled and simulated. The results give some interesting indications, concerning these challenging questions, which can be used as a basis for further research. However, the measurements are not accurate enough to give the final answers. Paper C is a simulation study which is more extensive than paper D and E, and is compared to electromyography measurements in the literature. Validation in biomechanical simulations is difficult and reducing mathematical errors is one way of reaching closer to more realistic results. Paper A examines well-posedness for forward dynamics with full muscle dynamics. Moreover, paper B is a technical report which describes the problem formulation and mathematical models and simulation from paper A in more detail. Our new modelling together with the simulations enable new possibilities. This is similar to simulations of applications in other engineering fields, and need in the same way be handled with care in order to achieve reliable results. The results in this thesis indicate that it can be very useful to use mathematical modelling and numerical simulations when describing cross-country skiing biomechanics. Hence, this thesis contributes to the possibility of beginning to use and develop such modelling and simulation techniques also in this context.</p>

<p>Compact Riemann surfaces of genus greater than 1 can be realized as quotient spaces of the hyperbolic plane by the action of Fuchsian groups. The Teichmüller space is the set of all complex structures of Riemann surfaces and the moduli space the set of conformal equivalence classes of Riemann surfaces. For genus greater than two the branch locus of the covering of the moduli space by the Teichmüller space can be identified wi the set of Riemann surfaces admitting non-trivial automorphisms. Here we give the orbifold structure of the branch locus of surfaces of genus 5 by studying the equisymmetric stratification of the branch locus. This gives the orbifold structure of the moduli space.</p><p>We also show that the strata corresponding to surfaces with automorphisms of order 2 and 3 belong to the same connected component for every genus. Further we show that the branch locus is connected with the exception of one isolated point for genera 5 and 6, it is connected for genus 7 and it is connected with the exception of two isolated points for genus 8.</p>