In this research, I utilize the theoretical perspective Knowledge In Pieces to identify the knowledge resources students utilize while in the process of completing various differential equations tasks. In addition I explore how this utilization changes over the course of a semester, and how resources related to the concepts of function and rate of change supported the students in completing the tasks. I do so using data collected from a series of task-based individual interviews with two students enrolled in separate differential equations courses. The results provide a fine-grained description of the knowledge students consider to be productive with regard to completing various differential equations tasks. Further the analysis resulted in the identification of five ways students interpret differential equations tasks and how these interpretations are related to the knowledge resources students utilize while completing the various tasks. Lastly, this research makes a contribution to mathematics education by illuminating the knowledge concerning function and rate of change students utilize and how this knowledge comes together to support students in drawing connections between differential equations and their solutions, structuring those solutions, and reasoning with relationships present in the differential equations. / Ph. D.

Yes / We introduce a novel approach to face models, which exploits the use of Partial Differential Equations (PDE) to generate the 3D face. This addresses some common problems of existing face models. The PDE face benefits from seamless merging of surface patches by using only a relatively small number of parameters based on boundary curves. The PDE face also provides users with a great degree of freedom to individualise the 3D face by adjusting a set of facial boundary curves. Furthermore, we introduce a uv-mesh texture mapping method. By associating the texels of the texture map with the vertices of the uv mesh in the PDE face, the new texture mapping method eliminates the 3D-to-2D association routine in texture mapping. Any specific PDE face can be textured without the need for the facial expression in the texture map to match exactly that of the 3D face model.

yes / The spine of an object is an entity that can characterise the object¿s topology and describes the object by a lower dimension. It has an intuitive appeal for supporting geometric modelling operations. The aim of this paper is to show how a spine for a PDE surface can be generated. For the purpose of the work presented here an analytic solution form for the chosen PDE is utilised. It is shown that the spine of the PDE surface is then computed as a by-product of this analytic solution. This paper also discusses how the of a PDE surface can be used to manipulate the shape. The solution technique adopted here caters for periodic surfaces with general boundary conditions allowing the possibility of the spine based shape manipulation for a wide variety of free-form PDE surface shapes.

The problem of determining the stability of a set of linear differential equations has been of interest to mathematicians and engineers for a considerable length of time. The problem is attacked by obtaining the characteristic equation of the original set of equations and determining the stability of this equation. The stability of the characteristic equation is first considered in terms of a continued fraction expansion. Necessary and sufficient conditions are given for the characteristic equation to be stable. The stability of the equation is then determined by means of a determinant sequence, which was the manner originally presented by A. Hurwitz in 1895. The Nyquist criterion, which is a graphical method for determining whether the equation is stable, is then presented. An example is given for each of the above methods to illustrate the procedure used in determining whether the equation is stable or unstable. Also included is a brief analysis of stability for non-linear equations. / Master of Science

Reduced-order models have long been used to understand the behavior of nonlinear partial differential equations. Naturally, reduced-order modeling techniques come at the price of either computational accuracy or computation time. Optimization techniques are studied to improve either or both of these objectives and decrease the total computational cost of the problem. This thesis focuses on the dynamic mode decomposition (DMD) applied to nonlinear PDEs with periodic boundary conditions. It provides one study of an existing optimization framework for the DMD method known as the Optimized DMD and provides another study of a newly proposed optimization framework for the DMD method called the Split DMD. / Master of Science / The Navier-Stokes (NS) equations are the primary mathematical model for understanding the behavior of fluids. The existence and smoothness of the NS equations is considered to be one of the most important open problems in mathematics, and challenges in their numerical simulation is a barrier to understanding the physical phenomenon of turbulence. Due to the difficulty of studying this problem directly, simpler problems in the form of nonlinear partial differential equations (PDEs) that exhibit similar properties to the NS equations are studied as preliminary steps towards building a wider understanding of the field. Reduced-order models have long been used to understand the behavior of nonlinear partial differential equations. Naturally, reduced-order modeling techniques come at the price of either computational accuracy or computation time. Optimization techniques are studied to improve either or both of these objectives and decrease the total computational cost of the problem. This thesis focuses on the dynamic mode decomposition (DMD) applied to nonlinear PDEs with periodic boundary conditions. It provides one study of an existing optimization framework for the DMD method known as the Optimized DMD and provides another study of a newly proposed optimization framework for the DMD method called the Split DMD.

The overarching goal of this project is to develop intelligent, self-adapting numerical algorithms for the time discretization of complex real-world problems with Q-Learning methodologies. The specific application is ordinary differential equations which can resolve problems in mathematics, social and natural sciences, but which usually require approximations to solve because direct analytical solutions are rare. Using the traditional Brusellator and Lorenz differential equations as test beds, this research develops models to determine reward functions and dynamically tunes controller parameters that minimize both the error and number of steps required for approximate mathematical solutions. Our best reward function is based on an error that does not overly punish rejected states. The Alpha-Beta Adjustment and Safety Factor Adjustment Model is the most efficient and accurate method for solving these mathematical problems. Allowing the model to change the alpha/beta value and safety factor by small amounts provides better results than if the model chose values from discrete lists. This method shows potential for training dynamic controllers with Reinforcement Learning. / Master of Science / This research applies Q-Learning, a subset of Reinforcement Learning and Machine Learning, to solve complex mathematical problems that are unable to be solved analytically and therefore require approximate solutions. Specifically, this research applies mathematical modeling of ordinary differential equations which are used in many fields, from theoretical sciences such and physics and chemistry, to applied technical fields such as medicine and engineering, to social and consumer-oriented fields such as finance and consumer purchasing habits, and to the realms of national and international security and communications. Q-Learning develops mathematical models that make decisions, and depending on the outcome, learns if the decision is good or bad, and uses this information to make the next decision. The research develops approaches to determine reward functions and controller parameters that minimize the error and number of steps associated with approximate mathematical solutions to ordinary differential equations. Error is how far the model's answer is from the true answer, and the number of steps is related to how long it takes and how much computational time and cost is associated with the solution. The Alpha-Beta Adjustment and Safety Factor Adjustment Model is the most efficient and accurate method for solving these mathematical problems and has potential for solving complex mathematical and societal problems.

This work investigates modeling, approximation, estimation, and control for classes of nonlinear systems whose state evolves in space $mathbb{R}^n times H$, where $mathbb{R}^n$ is a n-dimensional Euclidean space and $H$ is a infinite dimensional Hilbert space. Specifically, two classes of nonlinear systems are studied in this dissertation. The first topic develops a novel framework for adaptive estimation of nonlinear systems using reproducing kernel Hilbert spaces. A nonlinear adaptive estimation problem is cast as a time-varying estimation problem in $mathbb{R}^d times H$. In contrast to most conventional strategies for ODEs, the approach here embeds the estimate of the unknown nonlinear function appearing in the plant in a reproducing kernel Hilbert space (RKHS), $H$. Furthermore, the well-posedness of the framework in the new formulation is established. We derive the sufficient conditions for existence, uniqueness, and stability of an infinite dimensional adaptive estimation problem. A condition for persistence of excitation in a RKHS in terms of an evaluation functional is introduced to establish the convergence of finite dimensional approximations of the unknown function in RKHS. Lastly, a numerical validation of this framework is presented, which could have potential applications in terrain mapping algorithms. The second topic delves into estimation and control of history dependent differential equations. This study is motivated by the increasing interest in estimation and control techniques for robotic systems whose governing equations include history dependent nonlinearities. The governing dynamics are modeled using a specific form of functional differential equations. The class of history dependent differential equations in this work is constructed using integral operators that depend on distributed parameters. Consequently, the resulting estimation and control equations define a distributed parameter system whose state, and distributed parameters evolve in finite and infinite dimensional spaces, respectively. The well-posedness of the governing equations is established by deriving sufficient conditions for existence, uniqueness and stability for the class of functional differential equations. The error estimates for multiwavelet approximation of such history dependent operators are derived. These estimates help determine the rate of convergence of finite dimensional approximations of the online estimation equations to the infinite dimensional solution of distributed parameter system. At last, we present the adaptive sliding mode control strategy developed for the history dependent functional differential equations and numerically validate the results on a simplified pitch-plunge wing model. / Ph. D. / This dissertation aims to contribute towards our understanding of certain classes of estimation and control problems that arise in applications where the governing dynamics are modeled using nonlinear ordinary differential equations and certain functional differential equations. A common theme throughout this dissertation is to leverage ideas from approximation theory to extend the conventional adaptive estimation and control frameworks. The first topic develops a novel framework for adaptive estimation of nonlinear systems using reproducing kernel Hilbert spaces. The numerical validation of the framework presented has potential applications in terrain mapping algorithms. The second topic delves into estimation and control of history dependent differential equations. This study is motivated by the increasing interest in estimation and control techniques for robotic systems whose governing equations include history dependent nonlinearities.

No

In the main hypothesis for this paper, H and K are Hilbert spaces, F:H->K is a function with continuour second Fréchet differential such that dF(x)dF(x)* is onto for all x in H.

Dengue viral infection is a mosquito-borne disease with four distinct strains, where the interactions between these strains have implications on the severity of the disease outcomes. The two competing hypotheses for the increased severity during secondary infections are antibody dependent enhancement and original antigenic sin. Antibody dependent enhancement suggests that long-lived antibodies from primary infection remain during secondary infection but do not neutralize the virus. Original antigenic sin proposes that T cells specific to primary infection dominate cellular immune responses during secondary infections, but are inefficient at clearing cells infected with non-specific strains. To analyze these hypotheses, we developed within-host mathematical models. In previous work, we predicted a decreased non-neutralizing antibody effect during secondary infection. Since this effect accounts for decreased viral clearance and the virus is in quasi-equilibrium with infected cells, we could be accounting for reduced cell killing and the original antigenic sin hypothesis. To further understand these interactions, we develop a model of T cell responses to primary and secondary dengue virus infections that considers the effect of T cell cross-reactivity in disease enhancement. We fit the models to published patient data and show that the overall infected cell killing is similar in dengue heterologous infections, resulting in dengue fever and dengue hemorrhagic fever. The contribution to overall killing, however, is dominated by non-specific T cell responses during the majority of secondary dengue hemorrhagic fever cases. By contrast, more than half of secondary dengue fever cases have predominant strain-specific T cell responses. These results support the hypothesis that cross-reactive T cell responses occur mainly during severe disease cases of heterologous dengue virus infections. Finally, using the results from our within-host models, we develop a multiscale model of dengue viral infection which couples the within-host virus dynamics to the population level dynamics through a system of partial differential equations. We analytically determine the relationship between the model parameters and the characteristics of the solutions, and find thresholds under which infections persist in the population. Furthermore, we develop and implement a full numerical scheme for our model. / Ph. D. / Dengue viral infection is a mosquito-borne disease with four distinct strains, where the interactions between these strains have implications on the severity of the disease outcomes. The two competing hypotheses for the increased severity during secondary infections are antibody dependent enhancement and original antigenic sin. Antibody dependent enhancement suggests that long-lived antibodies from primary infection remain during secondary infection but do not neutralize the virus. Original antigenic sin proposes that T cells specific to primary infection dominate cellular immune responses during secondary infections, but are inefficient at clearing cells infected with non-specific strains. To analyze these hypotheses, we developed within-host mathematical models. In previous work, we predicted a decreased non-neutralizing antibody effect during secondary infection. Since this effect accounts for decreased viral clearance and the virus is in quasi-equilibrium with infected cells, we could be accounting for reduced cell killing and the original antigenic sin hypothesis. To further understand these interactions, we develop a model of T cell responses to primary and secondary dengue virus infections that considers the effect of T cell cross-reactivity in disease enhancement. We fit the models to published patient data and show that the overall infected cell killing is similar in dengue heterologous infections, resulting in dengue fever and dengue hemorrhagic fever. The contribution to overall killing, however, is dominated by non-specific T cell responses during the majority of secondary dengue hemorrhagic fever cases. By contrast, more than half of secondary dengue fever cases have predominant strain-specific T cell responses. These results support the hypothesis that cross-reactive T cell responses occur mainly during severe disease cases of heterologous dengue virus infections. Finally, using the results from our within-host models, we develop a multiscale model of dengue viral infection which couples the within-host virus dynamics to the population level dynamics through a system of partial differential equations. We analytically determine the relationship between the model parameters and the characteristics of the solutions, and find thresholds under which infections persist in the population. Furthermore, we develop and implement a full numerical scheme for our model.