The purpose of this study was to increase the student's understanding of the measures of area and perimeter of polygons. The goal of the project was to create a supplemental geometry unit to develop the concept of the area and perimeter of a polygon without the use of formulas and numbers and to measure the effectiveness of this unit on student understanding. Two high school geometry classes with under 28 students each participated in this study.

The specification of enterprise information systems using formal specification languages enables the formal verification of these systems. Reasoning about the properties of a formal specification is a tedious task that can be facilitated much through the use of an automated reasoner. However, set theory is a corner stone of many formal specification languages and poses demanding challenges to automated reasoners. To this end a number of heuristics has been developed to aid the Otter theorem prover in finding short proofs for set-theoretic problems. This dissertation investigates the applicability of these heuristics to next generation theorem provers. / Computing / M.Sc. (Computer Science)

Esta tese de doutorado divide-se em duas partes: a primeira trata de uma Teoria dos Conjuntos Rítmicos, e a segunda traz o Portfolio de Composições desenvolvidas no período do doutorado. A Teoria dos Conjuntos Ritmicos apresenta um sistema de organização rítmica paralelo ao sistema de organização de alturas, tendo como ponto de partida a Teoria dos Conjuntos Musicais (TCM), tal como organizada por FORTE (1973), além de uma adaptação do time-point-system (BABBITT, 1962). A partir da sistematização da TCM, e também de noções da Teoria dos Conjuntos Diatônicos (TCD), essa abordagem sintética permite estabelecer uma conexão entre aspectos básicos da harmonia e da cifragem de acordes com a organização rítmica. A um só tempo, em um catálogo completo, são relacionadas as famílias de conjuntos de alturas e cifras cordais, com suas respectivas contrapartes rítmicas. A motivação musical para esta investigação acerca dos ritmos surgiu pelo interesse nos ritmos dançantes e repetitivos, denominados timelines (TOUSSAINT, 2013), comumente utilizados na chamada música popular As timelines dançantes refletem propriedades similares às dos conjuntos diatônicos e, por essa razão, esta tese apresenta algumas propriedades dos conjuntos diatônicos de alturas, traçando um paralelo com suas contrapartes rítmicas. Essas relações também aparecem no portfolio de composições, caracterizando alguns procedimentos utilizados. O portfolio de composições, que inclui uma composição para orquestra sinfônica, é tematizado a partir da dualidade transparência/opacidade, abordando algumas diferenças essenciais, do ponto de vista da audibilidade, entre os resultados oriundos de técnicas variadas de composição. Este estudo sobre Teoria dos Conjuntos Rítmicos ajudará na abordagem analítica da minha produção composicional na música popular, trazendo uma maneira sistemática de entender e extrapolar alguns aspectos já utilizados na minha prática como compositor e improvisador. / This doctoral dissertation is divided into two parts: the first deals a rhythmic set theory, and the second contains the portfolio of compositions developed during this period of studies. This dissertation presents a system of rhythmic organization parallel to the musical set theory pitch class organization FORTE (1973), as well as an adaptation of the time-point-system (BABBITT, 1962). From the standpoint of the traditional set theory, and also from the diatonic set theory, this unified approach allows to estabilish a connecting tissue of basic aspects: from the harmony and chords symbols to the rhythmic organization. At one time, in a complete catalog, the families of pitch class sets and chord symbols are related to their respective rhythmic counterparts. The musical motivation for this research came from my interest in the swinging and groovy repetitive rhythms called timelines (TOUSSAINT, 2013), commonly used in popular music. These dancing timelines have properties similar to those of the diatonic sets, and for this reason, this dissertation presents some properties of the diatonic pitch class sets, drawing a parallel with their rhythmic counterparts. These relationships also appear in the portfolio of compositions, characterizing some procedures used. The portfolio of compositions, which includes a composition for symphony orchestra, is presented form the standpoint of a duality between transparency and opacity. This duality address the essential differences in the audibility of the results from various composition techniques. This study of Rhythmic Set Theory will serve as an analytical approach of my compositional output in popular music, with a systematic way to understant and to extrapolate some aspects already used in my practice as composer and improviser.

Esta tese de doutorado divide-se em duas partes: a primeira trata de uma Teoria dos Conjuntos Rítmicos, e a segunda traz o Portfolio de Composições desenvolvidas no período do doutorado. A Teoria dos Conjuntos Ritmicos apresenta um sistema de organização rítmica paralelo ao sistema de organização de alturas, tendo como ponto de partida a Teoria dos Conjuntos Musicais (TCM), tal como organizada por FORTE (1973), além de uma adaptação do time-point-system (BABBITT, 1962). A partir da sistematização da TCM, e também de noções da Teoria dos Conjuntos Diatônicos (TCD), essa abordagem sintética permite estabelecer uma conexão entre aspectos básicos da harmonia e da cifragem de acordes com a organização rítmica. A um só tempo, em um catálogo completo, são relacionadas as famílias de conjuntos de alturas e cifras cordais, com suas respectivas contrapartes rítmicas. A motivação musical para esta investigação acerca dos ritmos surgiu pelo interesse nos ritmos dançantes e repetitivos, denominados timelines (TOUSSAINT, 2013), comumente utilizados na chamada música popular As timelines dançantes refletem propriedades similares às dos conjuntos diatônicos e, por essa razão, esta tese apresenta algumas propriedades dos conjuntos diatônicos de alturas, traçando um paralelo com suas contrapartes rítmicas. Essas relações também aparecem no portfolio de composições, caracterizando alguns procedimentos utilizados. O portfolio de composições, que inclui uma composição para orquestra sinfônica, é tematizado a partir da dualidade transparência/opacidade, abordando algumas diferenças essenciais, do ponto de vista da audibilidade, entre os resultados oriundos de técnicas variadas de composição. Este estudo sobre Teoria dos Conjuntos Rítmicos ajudará na abordagem analítica da minha produção composicional na música popular, trazendo uma maneira sistemática de entender e extrapolar alguns aspectos já utilizados na minha prática como compositor e improvisador. / This doctoral dissertation is divided into two parts: the first deals a rhythmic set theory, and the second contains the portfolio of compositions developed during this period of studies. This dissertation presents a system of rhythmic organization parallel to the musical set theory pitch class organization FORTE (1973), as well as an adaptation of the time-point-system (BABBITT, 1962). From the standpoint of the traditional set theory, and also from the diatonic set theory, this unified approach allows to estabilish a connecting tissue of basic aspects: from the harmony and chords symbols to the rhythmic organization. At one time, in a complete catalog, the families of pitch class sets and chord symbols are related to their respective rhythmic counterparts. The musical motivation for this research came from my interest in the swinging and groovy repetitive rhythms called timelines (TOUSSAINT, 2013), commonly used in popular music. These dancing timelines have properties similar to those of the diatonic sets, and for this reason, this dissertation presents some properties of the diatonic pitch class sets, drawing a parallel with their rhythmic counterparts. These relationships also appear in the portfolio of compositions, characterizing some procedures used. The portfolio of compositions, which includes a composition for symphony orchestra, is presented form the standpoint of a duality between transparency and opacity. This duality address the essential differences in the audibility of the results from various composition techniques. This study of Rhythmic Set Theory will serve as an analytical approach of my compositional output in popular music, with a systematic way to understant and to extrapolate some aspects already used in my practice as composer and improviser.

In this work, we concern ourselves with particular topics in Polish space theory. We first consider the space A(U) of complex-analytic functions on an open set U endowed with the usual topology of uniform convergence on compact subsets. With the operations of point-wise addition and point-wise multiplication, A(U) is a Polish ring. Inspired by L. Bers' algebraic characterization of the relation of conformality, we show that the topology on A(U) is the only Polish topology for which A(U) is a Polish ring for a large class of U. This class of U includes simply connected regions, simply connected regions excluding a relatively discrete set of points, and other domains of usual interest. One thing that we deduce from this is that, even though C has many different Polish field topologies, as long as it sits inside another Polish ring with enough complex-analytic functions, it must have its usual topology. In a different direction, we show that the bounded complex-analytic functions on the unit disk admits no Polish topology for which it is a Polish ring. We also study the Lie ring structure on A(U) which turns out to be a Polish Lie ring with the usual topology. In this case, we restrict our attention to those domains U that are connected. We extend a result of I. Amemiya to see that the Lie ring structure is determined by the conformal structure of U. In a similar vein to our ring considerations, we see that, again for certain domains U of usual interest, the Lie ring A(U) has a unique Polish topology for which it is a Polish Lie ring. Again, the Lie ring A(U) imposes topological restrictions on C. That is, C must have its usual topology when sitting inside any Polish Lie ring isomorphic to A(U). In the last chapter, we introduce a new ideal of subsets of Polish spaces consisting of what we call residually null sets. From this ideal, we introduce an algebra consisting of what we call R-sets which is consistently a strict extension of the algebra of Baire property sets. We show that the algebra of R-sets is closed under the Alexandrov-Suslin operation and generalize Pettis' Theorem. From this, we provide new automatic continuity results and give a generalization of a result of D. Montgomery which shows that minimal assumptions on the continuity of group operations of an abstract group G with a Polish topology imply that G is actually a Polish group. We also see that many results pertaining to the algebra of Baire property sets generalize to the context of R-sets.

In an era where use and dependence of information systems is significantly high, the threat of incidents related to information security that could jeopardize the information held by organizations is more and more serious. Alarming facts within the literature point to inadequacies in information security practices, particularly the evaluation of information security controls in organizations. Research efforts have resulted in various methodologies developed to deal with the information security controls assessment problem. A closer look at these traditional methodologies highlights various weaknesses that can prevent an effective information security controls assessment in organizations. This dissertation develops a methodology that addresses such weaknesses when evaluating information security controls in organizations. The methodology, created using the Fuzzy Logic Toolbox of MATLAB based on fuzzy theory and fuzzy logic, uses fuzzy set theory which allows for a more accurate assessment of imprecise criteria than traditional methodologies. It is argued and evidenced that evaluating information security controls using fuzzy set theory addresses existing weaknesses found in the literature for traditional evaluation methodologies and, thus, leads to a more thorough and precise assessment. This, in turn, results in a more effective selection of information security controls and enhanced information security in organizations. The main contribution of this research to the information security literature is the development of a fuzzy set theory-based assessment methodology that provides for a thorough evaluation of ISC in organizations. The methodology just created addresses the weaknesses or limitations identified in existing information security control assessment methodologies, resulting in an enhanced information security in organizations. The methodology can also be implemented in a spreadsheet or software tool, and promote usage in practical scenarios where highly complex methodologies for ISC selection are impractical. Moreover, the methodology fuses multiple evaluation criteria to provide a holistic view of the overall quality of information security controls, and it is easily extended to include additional evaluation criteria factor not considered within this dissertation. This is one of the most meaningful contributions from this dissertation. Finally, the methodology provides a mechanism to evaluate the quality of information security controls in various domains. Overall, the methodology presented in this dissertation proved to be a feasible technique for evaluating information security controls in organizations.

Multitarget tracking is the process of jointly determining the number of targets present and their states from noisy sets of measurements. The difficulty of the multitarget tracking problem is that the number of targets present can change as targets appear and disappear while the sets of measurements may contain false alarms and measurements of true targets may be missed. The theory of random finite sets was proposed as a systematic, Bayesian approach to solving the multitarget tracking problem. The conceptual solution is given by Bayes filtering for the probability distribution of the set of target states, conditioned on the sets of measurements received, known as the multitarget Bayes filter. A first-moment approximation to this filter, the probability hypothesis density (PHD) filter, provides a more computationally practical, but theoretically sound, solution. The central thesis of this work is that the random finite set framework is theoretically sound, compatible with the Bayesian methodology and amenable to immediate implementation in a wide range of contexts. In advancing this thesis, new links between the PHD filter and existing Bayesian approaches for manoeuvre handling and incorporation of target amplitude information are presented. A new multitarget metric which permits incorporation of target confidence information is derived and new algorithms are developed which facilitate sequential Monte Carlo implementations of the PHD filter. Several applications of the PHD filter are presented, with a focus on applications for tracking in sonar data. Good results are presented for implementations on real active and passive sonar data. The PHD filter is also deployed in order to extract bacterial trajectories from microscopic visual data in order to aid ongoing work in understanding bacterial chemotaxis. A performance comparison between the PHD filter and conventional multitarget tracking methods using simulated data is also presented, showing favourable results for the PHD filter.

This thesis presents an analysis of Copland's first major serial work, the Quartet for Piano and Strings (1950), using pitch-class set theory and tonal analytical techniques.

In this paper several equivalences of the axiom of choice are examined. In particular, the axiom of choice, Zorn's lemma, Tukey's lemma, the Hausdorff maximal principle, and the well-ordering theorem are shown to be equivalent. Cardinal and ordinal number theory is also studied. The Schroder-Bernstein theorem is proven and used in establishing order results for cardinal numbers. It is also demonstrated that the first uncountable ordinal space is unique up to order isomorphism. We conclude by encountering several applications of the axiom of choice. In particular, we show that every vector space must have a Hamel basis and that any two Hamel bases for the same space must have the same cardinality. We establish that the Tychonoff product theorem implies the axiom of choice and see the use of the axiom of choice in the proof of the Hahn- Banach theorem.

This paper is a study of the relation between smoothness of the norm on a normed linear space and the property that every Chebyshev subset is convex. Every normed linear space of finite dimension, having a smooth norm, has the property that every Chebyshev subset is convex. In the second chapter two properties of the norm, uniform Gateaux differentiability and uniform Frechet differentiability where the latter implies the former, are given and are shown to be equivalent to smoothness of the norm in spaces of finite dimension. In the third chapter it is shown that every reflexive normed linear space having a uniformly Gateaux differentiable norm has the property that every weakly closed Chebyshev subset, with non-empty weak interior that is norm-wise dense in the subset, is convex.