We establish necessary conditions, in the form of the positivity of Pick-matrices, for the existence of a solution to the spectral Nevanlinna-Pick problem. We approach this problem from an operator theoretic perspective. We restate the problem as an interpolation problem on the symmetrized polydisc Γ(κ). We establish necessary conditions for a κ-tuple of commuting operators to have Γ(κ) as a complete spectral set. We then derive necessary conditions for the existence of a solution of the spectral Nevanlinna- Pick problem. The final chapter of this thesis gives an application of our results to complex geometry. We establish an upper bound for the Caratheodory distance on int Γ(κ).

The use of a stochastic model to predict the likelihood of future outcomes forms an integral part of decision optimization under uncertainty. In classical stochastic modeling uncertain parameters are often assumed to be driven by a particular form of probability distribution. In practice however, the distributional form is often difficult to infer from the observed data, and the incorrect choice of distribution can lead to significant quality deterioration of resultant decisions and unexpected losses. In this thesis, we present new approaches for evaluating expected future performance that do not rely on an exact distributional specification and can be robust against the errors related to committing to a particular specification. The notion of comprehensive robustness is promoted, where various degrees of model misspecification are studied. This includes fundamental one such as unknown distributional form and more involved ones such as stochastic moments and moment outliers. The approaches are developed based on the techniques of moment-based optimization, where bounds on the expected performance are sought based solely on partial moment information. They can be integrated into decision optimization and generate decisions that are robust against model misspecification in a comprehensive manner. In the first part of the thesis, we extend the applicability of moment-based optimization to incorporate new objective functions such as convex risk measures and richer moment information such as higher-order multivariate moments. In the second part, new tractable optimization frameworks are developed that account for various forms of moment uncertainty in the context of decision analysis and optimization. Financial applications such as portfolio selection and option pricing are studied.

The use of a stochastic model to predict the likelihood of future outcomes forms an integral part of decision optimization under uncertainty. In classical stochastic modeling uncertain parameters are often assumed to be driven by a particular form of probability distribution. In practice however, the distributional form is often difficult to infer from the observed data, and the incorrect choice of distribution can lead to significant quality deterioration of resultant decisions and unexpected losses. In this thesis, we present new approaches for evaluating expected future performance that do not rely on an exact distributional specification and can be robust against the errors related to committing to a particular specification. The notion of comprehensive robustness is promoted, where various degrees of model misspecification are studied. This includes fundamental one such as unknown distributional form and more involved ones such as stochastic moments and moment outliers. The approaches are developed based on the techniques of moment-based optimization, where bounds on the expected performance are sought based solely on partial moment information. They can be integrated into decision optimization and generate decisions that are robust against model misspecification in a comprehensive manner. In the first part of the thesis, we extend the applicability of moment-based optimization to incorporate new objective functions such as convex risk measures and richer moment information such as higher-order multivariate moments. In the second part, new tractable optimization frameworks are developed that account for various forms of moment uncertainty in the context of decision analysis and optimization. Financial applications such as portfolio selection and option pricing are studied.

Many geometry problems can be solved by transformation to graph problems. Often, both the geometry version and graph version of the problem are NP-hard - and therefore not likely to be solved in polynomial time. One approach to solving these hard problems is to use fixed parameter tractable (FPT) algorithms. We present a framework for developing FPT algorithms for graph problems using dynamic programming, monadic second order logic of graphs, tree-width, and bidimensionality. We use this framework to obtain FPT results for covering tour problems on grid-graphs with turn costs. The results for these problems are not practical, but they demonstrate how the basic framework can be used to quickly obtain FPT results. We provide suggestions on further research to improve our FPT results and to apply our framework to obtain new FPT results.

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This work deals with the application of supermathematics to supersymmetrical problems arising in physics. Some recent developments are presented in detail. A reduction scheme for general supermanifolds to vector bundles is presented, which significantly simplifies their mathematical treatment in a physical context. Moreover, some applications of this new approach are worked out, such as the Fermi oscillator.

Thesis (Ph. D.)--Ohio State University, 2007. / Title from first page of PDF file. Includes bibliographical references (p. 124-130).

Thesis (Ph.D.)--Bowling Green State University, 2007. / Document formatted into pages; contains xiii, 183 p. Includes bibliographical references.

Thesis (Ph.D.)--Pennsylvania State University, 2007. / Mode of access: World Wide Web.