No description available.

The main purpose of this thesis is to find the number of convex sets on a product of two totally ordered spaces. We will give formulas to find this number for specific cases and describe a process to obtain this number for all such spaces. In the first chapter we briefly discuss the motivation behind the work presented in this thesis. Also, the definitions and notation used throughout the paper are introduced here The second chapter starts with examining the product spaces of the form {1; 2; : : : ;n} × {1; 2}. That is, we begin by analyzing a two-row by n-column space for n > N. Three separate approaches are discussed, and verified, to find the total number of convex sets on the space. A general formula is presented to obtain this total for all n. In the third chapter we take the same {1; 2; : : : ;n} × {1; 2} spaces from Chapter 2 and consider all the scenarios for adding a second disjoint convex set to the space. Adding a second convex set gives a collection of two mutually disjoint sets. Again, a general formula is presented to obtain this total number of such collections for all n. The fourth chapter takes the idea from Chapter 2 and expands it to product spaces {1; 2; : : : ;n} × {1; 2; : : : ;m} consisting of more than two rows. Here the creation of convex sets having z rows from those having z − 1 rows is exploited to obtain a model that will give the total number of z-row convex sets on any n × m space, provided the set occupies z adjacent rows. Finally, the fifth chapter describes all possible scenarios for convex sets to be placed in the {1; 2; : : : ;n}×{1; 2; : : : ;m} space. This chapter then explains the process needed to acquire a count of all convex sets on any such space as well. Chapter 5 ends by walking through this process with a concrete example, breaking it down into each scenario. We conclude by briefly summarizing the results and specifying future work we would like to further investigate, in Chapter 6.

The first of NASA's high-level strategic goals is to extend and sustain human activities across the solar system. As the United States moves into the post-Shuttle era, meeting this goal is more challenging than ever. There are several desired outcomes for this goal, including development of an integrated architecture and capabilities for safe crewed and cargo missions beyond low Earth orbit. NASA's Flexible Path for the future human exploration of space provides the guidelines to achieve this outcome. Designing space system architectures to satisfy the Flexible Path starts early in design, when a downselection process works to reduce the broad spectrum of feasible system architectures into a more refined set that contains a handful of alternatives that are to be considered and studied further in the detailed design phases. This downselection process is supported by what is referred to as architecture space exploration (ASE). ASE is a systems engineering process which generates the design knowledge necessary to enable informed decision-making. The broad spectrum of potential system architectures can be impractical to evaluate. As the system architecture becomes more complex in its structure and decomposition, its space encounters a factorial growth in the number of alternatives to be considered. This effect is known in the literature as combinatorial explosion. For the Flexible Path, the development of new space system architectures can occur over the period of a decade or more. During this time, a variety of changes can occur which lead to new requirements that necessitate the development of new technologies, or changes in budget and schedule. Developing comprehensive and quantitative design knowledge early during design helps to address these challenges. Current methods focus on a small number of system architecture alternatives. From these alternatives, a series of 'one off' -type of trade studies are performed to refine and generate more design knowledge. These small-scale studies are unable to adequately capture the broad spectrum of possible architectures and typically use qualitative knowledge. The focus of this research is to develop a systems engineering method for system-level ASE during pre-phase A design that is rapid, exhaustive, flexible, traceable, and quantitative. Review of literature found a gap in currents methods that were able to achieve this research objective. This led to the development of the Set Theory-Influenced Architecture Space Exploration (STASE) methodology. The downselection process is modeled as a decision-making process with STASE serving as a supporting systems engineering method. STASE is comprised of two main phases: system decomposition and system synthesis. During system decomposition, the problem is broken down into three system spaces. The architecture space consists of the categorical parameters and decisions that uniquely define an architecture, such as the physical and functional aspects. The design space contains the design parameters that uniquely define individual point designs for a given architecture. The objective space holds the objectives that are used in comparing alternatives. The application of set theory across the system spaces enables an alternative form of representing system alternatives. This novel application of set theory allows the STASE method to mitigate the problem of combinatorial explosion. The fundamental definitions and theorems of set theory are used to form the mathematical basis for the STASE method. A series of hypotheses were formed to develop STASE in a scientific way. These hypotheses are confirmed by experiments using a proof of concept over a subset of the Flexible Path. The STASE method results are compared against baseline results found using the traditional process of representing individual architectures as the system alternatives. The comparisons highlight many advantages of the STASE method. The greatest advantage is that STASE comprehensively explores the architecture space more rapidly than the baseline. This is because the set theory-influenced representation of alternatives has a summation growth with system complexity in the architecture space. The resultant option subsets provide additional design knowledge that enables new ways of visualizing results and comparing alternatives during early design. The option subsets can also account for changes in some requirements and constraints so that new analysis of system alternatives is not required. An example decision-making process was performed for the proof of concept. This notional example starts from the entire architecture space with the goal of minimizing the total cost and the number of launches. Several decisions are made for different architecture parameters using the developed data visualization and manipulation techniques until a complete architecture was determined. The example serves as a use-case example that walks through the implementation of the STASE method, the techniques for analyzing the results, and the steps towards making meaningful architecture decisions.

No description available.

Research on multisensor data fusion aims at providing the enabling technology to combine information from several sources in order to form a unifi ed picture. The literature work on fusion of conventional data provided by non-human (hard) sensors is vast and well-established. In comparison to conventional fusion systems where input data are generated by calibrated electronic sensor systems with well-defi ned characteristics, research on soft data fusion considers combining human-based data expressed preferably in unconstrained natural language form. Fusion of soft and hard data is even more challenging, yet necessary in some applications, and has received little attention in the past. Due to being a rather new area of research, soft/hard data fusion is still in a edging stage with even its challenging problems yet to be adequately de fined and explored. This dissertation develops a framework to enable fusion of both soft and hard data with the Random Set (RS) theory as the underlying mathematical foundation. Random set theory is an emerging theory within the data fusion community that, due to its powerful representational and computational capabilities, is gaining more and more attention among the data fusion researchers. Motivated by the unique characteristics of the random set theory and the main challenge of soft/hard data fusion systems, i.e. the need for a unifying framework capable of processing both unconventional soft data and conventional hard data, this dissertation argues in favor of a random set theoretic approach as the first step towards realizing a soft/hard data fusion framework. Several challenging problems related to soft/hard fusion systems are addressed in the proposed framework. First, an extension of the well-known Kalman lter within random set theory, called Kalman evidential filter (KEF), is adopted as a common data processing framework for both soft and hard data. Second, a novel ontology (syntax+semantics) is developed to allow for modeling soft (human-generated) data assuming target tracking as the application. Third, as soft/hard data fusion is mostly aimed at large networks of information processing, a new approach is proposed to enable distributed estimation of soft, as well as hard data, addressing the scalability requirement of such fusion systems. Fourth, a method for modeling trust in the human agents is developed, which enables the fusion system to protect itself from erroneous/misleading soft data through discounting such data on-the-fly. Fifth, leveraging the recent developments in the RS theoretic data fusion literature a novel soft data association algorithm is developed and deployed to extend the proposed target tracking framework into multi-target tracking case. Finally, the multi-target tracking framework is complemented by introducing a distributed classi fication approach applicable to target classes described with soft human-generated data. In addition, this dissertation presents a novel data-centric taxonomy of data fusion methodologies. In particular, several categories of fusion algorithms have been identifi ed and discussed based on the data-related challenging aspect(s) addressed. It is intended to provide the reader with a generic and comprehensive view of the contemporary data fusion literature, which could also serve as a reference for data fusion practitioners by providing them with conducive design guidelines, in terms of algorithm choice, regarding the specifi c data-related challenges expected in a given application.

In order to store and process natural phenomena in Geographic Information Systems (GIS) it is necessary to model the real world to form computational representation. Since classical set theory is used in conventional GIS software systems to model uncertain real world, the natural variability in the environmental phenomena can not be modeled appropriately. Because, pervasive imprecision of the real world is unavoidably reduced to artificially precise spatial entities when the conventional crisp logic is used for modeling. An alternative approach is the fuzzy set theory, which provides a formal framework to represent and reason with uncertain information. In addition, linguistic variable concept in a fuzzy logic system is useful for communicating concepts and knowledge with human beings. In this thesis, a system to enhance commercial GIS software, namely ArcGIS, with fuzzy set theory is designed and implemented. The proposed system allows users to (a) incorporate human knowledge and experience in the form of linguistically defined variables into GIS-based spatial analyses, (b) handle impreiii cision in the decision-making processes, and (c) approximate complex ill-defined problems in decision-making processes and classification. The operation of the proposed system is presented through case studies, which demonstrate its application for classification and decision-making processes. This thesis shows how fuzzy logic approach may contribute to a better representation and reasoning with imprecise concepts, which are inherent characteristics of geographic data stored and processed in GIS.

In "Badiou, Political Nihilism, and a Small-Scale Solution", I argue that Badious presentation of politics, exclusively on a large scale that of the nation-state betrays his underlying set-theoretic ontology. The consequence of presenting politics on this scale is that political events, opportunities for genuine political engagement, are extremely rare. This leaves potential political actors with little reason to believe they will have the opportunity to engage in politics. The absence of meaningful engagement, along with Badious unique conception of truth, gives rise to the problem of political nihilism. But, just as sets are both composed of sets and couched within others, situations too should be viewed as scalable. Re-presenting politics on a multiplicity of scales overcomes the worry about nihilism, while better capturing the real complexity and texture of political commitments.

No description available.

Thesis(M.S.)--Auburn University, 2005. / Abstract. Vita. Includes bibliographic references.

Thesis (Ph. D.)--University of New South Wales, 1998. / Also available online.