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Extensions Of Commutative Rings With Linearly Ordered Intermediate Rings, Michael Scott Gilbert

Doctoral Dissertations

This work concerns λ-extensions of (commutative) rings, which are defined as ring extensions R ⊆ T whose set of intermediate rings is linearly ordered by inclusion. λ-extensions from a subclass of the ∆-extensions of Gilmer-Huckaba and generalize the adjacent extensions of Ferrand-Oliver, Modica, and Dechene.

In Chapter I, we characterize λ-domains (i.e., integral domains R with quotient field K such that R ⊆ K is a λ-extension), showing that they form a subclass of quasilocal i-domains. These results parallel results of Gilmer-Huckaba for ∆-domains. We relate λ-domains to divided domains and pseudovaluation domains.

Chapter II concerns λ-extensions R ⊆ T …

Applications Of Optimal Control, Katherine Renee Fister

Doctoral Dissertations

In this dissertation, we investigate optimal control of partial and ordinary differential equations. We prove the existence of an optimal control for which the objective functional is maximized. The goal is to characterize the optimal control in terms of the solution of the optimality system. The optimality system consists of the state equations coupled with the adjoint equations. To obtain the optimality system we differentiate the objective functional with respect to the control. This process is applied to harvesting in a predator-prey parabolic system, to analyzing surface runoff in a parabolic problem, and to controlling the effect of the HIV …