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This work develops theoretical and applied results for variational convex analysis. First we present the basic tools of analysis necessary to develop the core theory and applications. New results concerning duality principles for systems originally modeled by non-linear differential equations are shown in chapters 9 to 17. A key aspect of this work is that although the original problems are non-linear with corresponding non-convex variational formulations, the dual formulations obtained are almost always concave and amenable to numerical computations. When the primal problem has no solution in the classical sense, the solution of dual problem is a weak limit of minimizing sequences, and the evaluation of such average behavior is important in many practical applications. Among the results we highlight the dual formulations for micro-magnetism, phase transition models, composites in elasticity and conductivity and others. To summarize, in the present work we introduce convex analysis as an interesting alternative approach for the understanding and computation of some important problems in the modern calculus of variations. / Ph. D.

If (X,Ƭ) is a topological space, then a quasi-uniformity U on X is compatible with Ƭ if the quasi-uniform topology, Ƭ_u = Ƭ. This paper is concerned with local properties of quasi-uniformities on a set X that are compatible with a given topology on X. Chapter II is devoted to the construction of Hausdorff completions of transitive quasi-uniform spaces that are members of the Pervin quasi-proximity class. Chapter III discusses locally complete, locally precompact, locally symmetric and locally transitive quasi-uniform spaces. Chapter IV is devoted to function spaces of quasi-uniform spaces. Chapter V and the Appendix are concerned with the topological homeomorphism groups of quasi-uniform spaces. / Ph. D.

This thesis deals with the relationships between covering properties and properties of compatible quasi-uniformities of a topological space. The covering properties considered in this work are orthocompactness, metacompactness and paracompactness; some generalizations of orthocompactness are also defined and studied. / Ph. D.

In this thesis we answer a number of unsolved problems in generalized paracompact topological spaces. Examples satisfying the T₄ separation axiom are constructed showing the relationship between the properties B(D, ω₀)-refinability, B(D, λ)-refinability, and weak θ̅-refinability. The properties B(D, λ)-refinability and weak θ̅-refinability are shown to be strictly weaker than B(D, ω₀)-refinability. Sum theorems, mapping theorems, and o—product theorems are obtained for B(D, ω₀)-refinability, weak θ̅-refinability, and several other properties. The σ—product theorem for B(D,ω₀)-refinability, weak θ̅-refinability, and other properties are shown to follow from a new special B(D,ω₀) sum theorem. / Ph. D.

Open land has historically been considered an expendable resource. This concept has abetted the sprawl which occurs in all urbanizing areas of our country. With the United States urbanizing at a rate of more than a million acres a year, this concept can no longer be tolerated. In the last fifteen years, urban development has consumed approximately two-thirds as much new land as it did in all the previous years in the history of our country. Virginia has not been an innocent bystander in this process. This thesis points up the necessity for immediate action to preserve open space in the State of Virginia. It examines the human as well as the economic values that can be derived from open space through an analysis of both the active and passive uses to which open space or low-density use lands can be put. This is accomplished through the establishment and critical analysis of the goals and objectives of an open space program for Virginia. The law on open space in Virginia is a mass of detail buried in traditional legal categories developed for other purposes. This thesis has examined the existing constitutional and statutory powers available to local governmental subdivisions for developing an open space program. In addition, it examines and appraises significant proposals for acquiring and controlling open space. From this analysis, alternative programs for open space acquisition and control are presented including the acquisition of development rights, the land bank, fresh concepts of zoning and subdivision regulations, and expanded concepts in the use of the taxing power. / Master of Science

This thesis is a study of public spaces with a focus on personal safety and human behavior in public spaces. It establishes literature in the realm of public spaces, safety standards, and behavioral research and aims to identify common ground or conflicts between people behavior in public spaces and safety standards of public spaces. Research is supported through detailed on-site analysis and various techniques of behavioral research of two plazas in a campus setting. The first part of the research focuses on literature to understand origin of public space, importance of safety, and evolution of safety standards in the context of Crime Prevention Through Environmental Design (CPTED). The later part of this thesis is based on preliminary site investigations, which help in identifying two public plazas on the Virginia Tech campus to provide a platform to conduct research and help identify common grounds or conflicts between safety standards and human behavior. The research also aims to help revise techniques of safety evaluation of public spaces, based on human needs and behavior. The research is primarily qualitative in nature supported with a concise quantitative data analysis to ascertain participant demographics and social needs. / Master of Landscape Architecture

Those who eat share a relationship with design. The Museum of Gastronomy, a museum located at the Washington D.C. National Mall, would dedicate a space to a full-sense experience pursuing a relationship between food, culture, and design. The National Mall is a symbolic setting dedicated to U.S. heritage via museums, memorials, and the honoring of legacies accessible to the public. As the U.S. is a melting pot of different nationalities, this museum would fill an open niche devoted to celebrating the cultural heritage of cooking throughout the cycle of food consumption. As consumers, regardless of notice or intention, we have an intimate relationship with food as means of survival. Cooking, the method for preparing this vital element, is also an expression of culture, style, and use of available resources. In essence, cooking is an everyday design: a crucial aspect of our very survival. Therefore: food → cooking → design → survival → food → cooking Similarly, it is the architect's mission to reach an effective design with available "ingredients," forces, or kit-of-parts within a project. In the Museum of Gastronomy, the architect would become the unifying source between a relationship literally built between food, people, and design in an interactive experience. This would result in a space dedicated to showcasing and combining the concept of "cooking is designing." Incorporating these components into a museum located at the National Mall would personify an experience of the cooking processes as a building block of every-day observation of design. / Master of Architecture