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Planar Algebras: A Category Theoretic Point Of View, Shamindra Kumar Ghosh

Doctoral Dissertations

We define Jones's planar algebra as a map of multicategories and construct a planar algebra starting from a 1-cell in a pivotal strict 2-category. We introduce the concept of affine representations of a planar algebra and prove some finiteness results for the affine representations of finite depth planar algebras. We also show that the radius of convergence of the dimension of an affine representation of the planar algebra associated to a finite depth subfactor is at least as big as the inverse-square of the modulus.

An Investigation Of Pre-Service Teachers' And Professional Mathematicians' Perceptions Of Mathematical Proof At The Secondary School Level, David M. Gray

Doctoral Dissertations

The National Council of Teachers of Mathematics [NCTM] states that by the time students graduate high school, they should learn to present "arguments consisting of logically rigorous deductions of conclusions from hypotheses" (NCTM, 2000, p. 56) in "written forms that would be acceptable to professional mathematicians" (NCTM, 2000, p. 58). Research studies indicate, however, that students and teachers have difficulty with many aspects of mathematical proof, including its nature and meaning. In addition, there appears to be a disconnect between school teachers' and university mathematicians' expectations for their respective students regarding mathematical proof. This study examined the perceptions of thirteen …

Cupolets: Chaotic Unstable Periodic Orbits Theory And Applications, Kourosh Zarringhalam

Doctoral Dissertations

Recent theoretical work suggests that periodic orbits of chaotic systems are a rich source of qualitative information about the dynamical system. The presence of unstable periodic orbits located densely on the attractor is a typical characteristic of chaotic systems. This abundance of unstable periodic orbits can be utilized in a wide variety of theoretical and practical applications [19]. In particular, chaotic communication techniques and methods of controlling chaos depend on this property of chaotic attractors [12, 13].

In the first part of this thesis, a control scheme for stabilizing the unstable periodic orbits of chaotic systems is presented and the …

Exploring Lesson Study As A Form Of Professional Development For Enriching Teacher Knowledge And Classroom Practices, Melissa K. Mitcheltree

Doctoral Dissertations

This study tracked the development of teacher knowledge through a professional development experience called lesson study. Lesson study is a Japanese professional development process in which a group of teachers develop a series of lessons using the following stages: planning lessons, teaching/observing the lessons, reflecting on lessons taught as well as sharing and discussing the lessons with their colleagues (Lewis, 2002). The aim of this study was to explore how the lesson study process influenced teacher knowledge of mathematics content and pedagogical content.

Study participants were four secondary mathematics teachers from a rural high school in the Northeast. All participants …