This paper will examine analogues of Cantor sets, called fractal squares, and some of the geometric ways in which fractal squares raise issues not raised by Cantor sets. Also discussed will be a technique using directed graphs to prove bilipschitz equivalence of two fractal squares. / Ph. D.

The classification problem in discriminant analysis involves identifying a function that accurately classifies observations as originating from one of two or more mutually exclusive groups. Because no single classification technique works best for all problems, many different techniques have been developed. For business applications, neural networks have become the most commonly used classification technique and though they often outperform traditional statistical classification methods, their performance may be hindered because of failings in the use of training data. This problem can be exacerbated because of small data set size. In this dissertation, we identify and discuss a number of potential problems with typical random partitioning of neural network training data for the classification problem and introduce deterministic methods to partitioning that overcome these obstacles and improve classification accuracy on new validation data. A traditional statistical distance measure enables this deterministic partitioning. Heuristics for both the two-group classification problem and k-group classification problem are presented. We show that these heuristics result in generalizable neural network models that produce more accurate classification results, on average, than several commonly used classification techniques. In addition, we compare several two-group simulated and real-world data sets with respect to the interior and boundary positions of observations within their groups' convex polyhedrons. We show by example that projecting the interior points of simulated data to the boundary of their group polyhedrons generates convex shapes similar to real-world data group convex polyhedrons. Our two-group deterministic partitioning heuristic is then applied to the repositioned simulated data, producing results superior to several commonly used classification techniques. / Ph. D.

Nearly Euclidean Thurston maps are simple generalizations of rational Lattes maps. A Thurston map is called nearly Euclidean if its local degree at each critical point is 2 and it has exactly four postcritical points. We investigate when such a map has the property that the associated pullback map on Teichmuller space is constant. We also show that no Thurston map of degree 2 has constant pullback map. / Ph. D.

This dissertation examines characteristics of a class of semi-infinite linear programming problems designated as C/C semi-infinite linear programming problems. Semi-infinite programming problems which belong to this class are problems of the form [See document] where S is a compact, convex subset of Euclidean m space and u_i : S→R, i=1,...,n are strictly concave functions while u _n+1 : S→R is convex. Certain properties of the C/C semi-infinite linear programming problems give rise to efficient solution techniques. The solution techniques are given as well as examples of their use. Of significant importance is the intimate relationship between the class of C/C semi-infinite linear programming problems and certain convex fuzzy set programming problems. The fuzzy set programming problem is defined as [See document] The convex fuzzy set programming problem is transformed to an equivalent semi-infinite linear programming problem. Characterizations of the membership functions are given which cause the equivalent semi-infinite linear programming problems to fall within the realm of C/C semi-infinite linear programming problems. Some extensions of the set inclusive programming problem are also given. / Ph. D.

This paper is concerned with certain properties of derivatives and some characterizations of linear point sets with derivatives. In 1946, Zygmunt Zahorski published a letter on this topic listing a number of theorems without proof, and no proof of these assertions has been published. Some of the theorems presented here are paraphrases of Zahorski's statements, developed in a slightly different order.

This thesis considers algorithmic and structural aspects of high-dimensional convex sets with respect to the standard Gaussian measure. Among our contributions, (i) we introduce a notion of "influence" for convex sets that yields the first quantitative strengthening of Royen's celebrated Gaussian correlation inequality; (ii) we investigate the approximability of general convex sets by intersections of halfspaces, where the approximation quality is measured with respect to the standard Gaussian distribution; and (iii) we give the first lower bounds for testing convexity and estimating the distance to convexity of an unknown set in the black-box query model. Our results and techniques are inspired by a number of fundamental ingredients and results---such as the influence of variables, noise sensitivity, and various extremal constructions---from the analysis of Boolean functions in complexity theory.

The problem of computing the minimum distance between two convex hulls has applications to many areas including robotics, computer graphics and path planning. Moreover, determining the minimum distance between two convex hulls plays a significant role in support vector machines (SVM). In this study, a new algorithm for finding the minimum distance between two convex hulls is proposed and investigated. A convergence of the algorithm is proved and applicability of the algorithm to support vector machines is demostrated. The performance of the new algorithm is compared with the performance of one of the most popular algorithms, the sequential minimal optimization (SMO) method. The new algorithm is simple to understand, easy to implement, and can be more efficient than the SMO method for many SVM problems.

Traditional methods for evaluating investments in state-of-the-art technology are sometimes found lacking in providing equitable recommendations for project selection. The major cause for this is the inability of these methods to handle adequately uncertainty and imprecision, and account for every aspect of the project, economic and non-economic, tangible and intangible. Fuzzy set theory provides an alternative to probability theory for handling uncertainty, while at the same time being able to handle imprecision. It also provides a means of closing the gap between the human thought process and the computer, by enabling the establishment of linguistic quantifiers to describe intangible attributes. Fuzzy set theory has been used successfully in other fields for aiding the decision-making process. The intention of this research has been the application of fuzzy set theory to aid investment decision making. The research has led to the development of a structured model, based on theoretical algorithms developed by Buckley and others. The model looks at a project from three different standpoints- economic, operational, and strategic. It provides recommendations by means of five different values for the project desirability, and results of two sensitivity analyses. The model is tested on a hypothetical case study. The end result is a model that can be used as a basis for promising future development of investment analysis models. / Master of Science / incomplete_metadata

A structured and disciplined system engineering process is essential for the efficient and effective development of products and systems which are both responsive to customer needs and globally competitive. Rigor and discipline during the later life-cycle phases of design and development (preliminary and detailed) cannot compensate for an ill-conceived system concept and for premature commitments made during the conceptual design phase. This significance notwithstanding, the nascent stage of system design has been largely ignored by the research and development community. This research is unique. It focuses on conceptual system design and formalizes analysis and evaluation activities during this important life-cycle phase. The primary goal of developing a conceptual design analysis and evaluation methodology has been achieved, including complete integration with the system engineering process. Rather than being a constraint, this integration led to a better definition of conceptual design activity and the coordinated progression of synthesis, analysis, and evaluation. Concepts from fuzzy set theory and the calculus of fuzzy arithmetic were adapted to address and manipulate imprecision and subjectivity. A number of design decision aids were developed to reduce the gap between commitment and project specific knowledge, to facilitate design convergence, and to help realize a preferred system design concept. / Ph. D.

Available cost-variance investigation models are reviewed and evaluated in Chapter Three of this study. As shown in the chapter, some models suffer from ignoring the costs and benefits of the investigation. Other models, although meeting the cost-benefit test, fail to capture the essence of the real-world problem. For example, they fail to handle the imprecision (fuzziness) surrounding the investigation decision. They are also based on the unrealistic assumptions of (1) a two-state system, and (2) constant level of accuracy and precision. In addition, the models suffer from the lack of applicability. They require precise numerical inputs to the analysis that are difficult, if not impossible, to attain. This dissertation provides a new cost-variance investigation model that may overcome some of these problems. The new model utilizes the calculus of fuzzy set theory which was introduced by Zadeh in 1965 as a means for dealing with fuzziness. The theory is also intended to reduce the need for precise measures that are difficult to obtain. Consequently, the theory seems to be well suited for handling the investigation problem. (Chapter Two provides a summary of the theory and its applications in the decision making area.) The new model is presented in Chapter Four and extended in Chapter Five. The performance is assumed to be described by·a transformation function, S_t+1 = f(S_t,D_t), where S_t, D_t, and S_t+1 represent the sets of the input states, available decisions, and output states, respectively. The transformation function can be deterministic, stochastic, or fuzzy. Methods are suggested to obtain the optimal decision for the three cases of transformation functions. These methods are based on formulating a fuzzy optimal decision set D_O = {u_{D_O}(d_j)d_j}, where u_{D_O}(d_j) represents the compatibility (i.e., relative merit) of decision d_j with the optimal decision set. The optimal decision is the decision having the highest compatibility with the fuzzy optimal decision set. In addition to allowing for different transformation functions, the new model allows for varying degrees of out-of-controllness. The model also provides for the fuzziness (imprecision) surrounding (1) the states of performance, (2) the net benefits from the investigation, and (3) the probabilities. This is done by employing the basic concept in fuzzy set theory, namely, the membership function concept. The new model was examined (in Chapter Six) for feasibility. First, the model was computerized. Then, it was applied to an actual investigation problem encountered by a manufacturing company. As the application may indicate, the new model can be applied to real-world situations. / Ph. D.