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A Julia set for a complex function 𝑓 is the set of all points in the complex plane where the iterates of 𝑓 do not form a normal family. A picture of the Julia set for a function can be generated with a computer by coloring pixels (which we consider to be small squares) based on the behavior of the point at the center of each pixel. We consider the accuracy of computer generated pictures of Julia sets. Such a picture is said to be accurate if each colored pixel actually contains some point in the Julia set. We extend previous work to show that the pictures generated by an algorithm for the family λe² are accurate, for appropriate choices of parameters in the algorithm. We observe that the Julia set for meromorphic functions with polynomial Schwarzian derivative is the closure of those points which go to infinity under iteration, and use this as a basis for an algorithm to generate pictures for such functions. A pixel in our algorithm will be colored if the center point becomes larger than some specified bound upon iteration. We show that using our algorithm, the pictures of Julia sets generated for the family λtan(z) for positive real λ are also accurate. We conclude with a cautionary example of a Julia set whose picture will be inaccurate for some apparently reasonable choices of parameters, demonstrating that some care must be exercised in using such algorithms. In general, more information about the nature of the function may be needed. / Ph. D.

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.

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

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.

No / Market structure affects the informational and real frictions faced by traders in equity markets. Using bid-ask spreads, we present evidence which suggests that while real frictions associated with the costs of supplying immediacy are less in order-driven systems, informational frictions resulting from increased adverse selection risk are considerably higher in these markets. Firm value, transaction size and order location are all major determinants of the trading costs borne by investors. Consistent with the stealth trading hypothesis of Barclay and Warner (1993), we report that informational frictions are at their highest for medium size trades that go through the order book. Finally, while there is no doubt that the total costs of trading on order-driven systems are lower for very liquid securities, the inherent informational inefficiencies of the trading format should not be ignored. This is particularly true for the vast majority of small to mid-size stocks that experience infrequent trading and low transaction volume.