Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1981. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Bibliography: leaves 100-102. / by Robert Alan Proctor. / Ph.D.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1981. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Bibliography: p. 135-138. / by James William Walker. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1982. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE / Bibliography: leaves 106-109. / by Rae Michael Andrew Shortt. / Ph.D.

by Zang Rui. / "Nov 2004." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (p. 78-82) / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.

The manufacturer claims that there is only one solution to the puzzle Instant Insanity II. However, a recent paper shows that there are two solutions. Our goal is to find ways in which we only have one solution. We examine the permutation groups of the puzzle and use modern algebra to attempt to fix the puzzle. First, we find the permutation group for the case when there is only one empty slot at the top. We then examine the scenario when we add an extra column or an extra row to make the game a 4 × 5 puzzle or a 5 x 4 puzzle, respectively. We consider the possibilities when we delete a color to make the game a 3 × 3 puzzle and when we add a color, making the game a 5 × 5 puzzle. Finally, we determine if solution two is a permutation of solution one.

This project investigates the development of four different proofs of the law of quadratic reciprocity, in order to study the critical reasoning process that drives discovery in mathematics. We begin with an examination of the first proof of this law given by Gauss. We then describe Gauss’ fourth proof of this law based on Gauss sums, followed by a look at Eisenstein’s geometric simplification of Gauss’ third proof. Finally, we finish with an examination of one of the modern proofs of this theorem published in 1991 by Rousseau. Through this investigation we aim to analyze the different strategies used in the development of each of these proofs, and in the process gain a better understanding of this theorem.

This thesis discusses the topic of Boolean algebras. In order to build intuitive understanding of the topic, research began with the investigation of Boolean algebras in the area of Abstract Algebra. The content of this initial research used a particular notation. The ideas of partially ordered sets, lattices, least upper bounds, and greatest lower bounds were used to define the structure of a Boolean algebra. From this fundamental understanding, we were able to study atoms, Boolean algebra isomorphisms, and Stone’s Representation Theorem for finite Boolean algebras. We also verified and proved many properties involving Boolean algebras and related structures. We then expanded our study to more thoroughly developed theory. This comprehensive theory was more abstract and required the use of a different, more universal, notation. We continued examining least upper and greatest lower bounds but extended our knowledge to subalgebras and families of subsets. The notions of cardinality, cellularity, and pairwise disjoint families were investigated, defined, and then used to understand the Erdös-Tarski Theorem. Lastly, this study concluded with the investigation of denseness and incomparability as well as normal forms and the completion of Boolean algebras.

Stastical pattern recognition is the most successful approach to automatic speech and speaker recognition (ASASR). Of all the statistical pattern recognition techniques, the hidden Markov model (HMM) is the most important. The Gaussian mixture model (GMM) and vector quantisation (VQ) are also effective techniques, especially for speaker recognition and in conjunction with HMMs. for speech recognition. However, the performance of these techniques degrades rapidly in the context of insufficient training data and in the presence of noise or distortion. Fuzzy approaches with their adjustable parameters can reduce such degradation. Fuzzy set theory is one of the most, successful approaches in pattern recognition, where, based on the idea of a fuzzy membership function, fuzzy C'-means (FCM) clustering and noise clustering (NC) are the most, important techniques. To establish fuzzy approaches to ASASR, the following basic problems are solved. First, a time-dependent fuzzy membership function is defined for the HMM. Second, a general distance is proposed to obtain a relationship between modelling and clustering techniques. Third, fuzzy entropy (FE) clustering is proposed to relate fuzzy models to statistical models. Finally, fuzzy membership functions are proposed as discriminant functions in decison making. The following models are proposed: 1) the FE-HMM. NC-FE-HMM. FE-GMM. NC-FEGMM. FE-VQ and NC-FE-VQ in the FE approach. 2) the FCM-HMM. NC-FCM-HMM. FCM-GMM and NC-FCM-GMM in the FCM approach, and 3) the hard HMM and GMM as the special models of both FE and FCM approaches. Finally, a fuzzy approach to speaker verification and a further extension using possibility theory are also proposed. The evaluation experiments performed on the TI46, ANDOSL and YOHO corpora showbetter results for all of the proposed techniques in comparison with the non-fuzzy baseline techniques.

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.

This thesis presents a partition of the class of homomorphisms between groupoids of n-tuples in a system g = (G,&,@), where G = { a,b,c,d,e }is a set of five elements such that: 1) a is the &-identity and annihilates all elements under @; 2) b is the @-identity; 3) d absorbs all all elements except e under & and all elements except a and e under @; 4) e absorbs all elements under & and all elements except a under @; 5) & is a binary operation on G and is commutative in G; 6) @ is a binary operation on G and is left-distributive over & in G.Matrices over g were examined for characteristics which would determine different atomic properties of homomorphisms. A matrix operation @ was defined, which allowed the homomorphisms of groupoids of the form, (G(n) , &), to be modeled by a matrix equation. Using the atomic proper ties, a partition of the class of homomorphisms between groupoids was developed, and an example of an element in each of its disjoint subsets was presented. A listing of theorems was also derived.Ball State UniversityMuncie, IN 47306