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by Lam Shek Ki. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 59-60). / Abstract also in Chinese. / Acknowledgment --- p.i / Abstract --- p.ii / Introduction --- p.2 / Chapter Chapter 1 --- Prelimnaries / Chapter §1.1 --- A Summary on C*-algebras and their representation theory --- p.4 / Chapter §1.2 --- Primitive ideals --- p.7 / Chapter §1.3 --- Canonical form of a matrix --- p.12 / Chapter Chapter 2 --- Uniqueness and basic properties of OA / Chapter §2.1 --- Uniqueness of Cuntz-Krieger Algebra --- p.13 / Chapter §2.2 --- Basic properties of OA --- p.25 / Chapter Chapter 3 --- A generalization of OA / Chapter §3.1 --- Universal Cuntz-Krieger Algebras --- p.33 / Chapter §3.2 --- Gauge action on AO A --- p.36 / Chapter Chapter 4 --- The ideal structure of OA and AO A / Chapter §4.1 --- Gauge-invariant ideals of AO A --- p.39 / Chapter §4.2 --- The primitive ideal space of AO A --- p.49 / Bibliography --- p.59

In the 1960s French mathematician George de Rham found a relationship between two invariants of knots. He found that there exist representations of the fundamental group of a knot into a group G of upper right triangular matrices in C with determinant one that is described exactly by the roots of the Alexander polynomial. I extended this result to find that the representations of the fundamental group of a link into G are described by the multivariable Alexander polynomial of the link.

<p>In this master thesis I have looked on two different kinds of representations of the Lie algebras <em>su</em>(2) and <em>sl</em>(2), and the tensor products of the representations.</p><p>In the first case I looked at a tensor product involving a representation similar to one that appears in an article by A. van Tonder. This representation and tensor product was investigated mainly to get a good comprehension in the subject and to understand some of the problems that can arise.</p><p>In the other case, which is the main problem in this thesis, I looked at a tensor product and representations that appears in an article by M. R. Gaberdiel. Here we deal with a tensor product of representations of <em>su</em>(2) with a specific value for the level at k = -4/3 and a specific eigenvalue of the Casimir operator at -2/9. This was done in the frame of finite dimensional Lie algebra and affine Lie algebra and not in the case of fusion rules as in the article by M. R. Gaberdiel.</p><p>In both cases some of the calculations where done from <em>in situ</em> and the investigation of the representations behaviour due to the step operators, theirs eigenvalue and theirs weight system.</p><p>Results and conclusions of the investigations are discussed in the last part of this thesis.</p><p> </p>

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In this master thesis I have looked on two different kinds of representations of the Lie algebras su(2) and sl(2), and the tensor products of the representations. In the first case I looked at a tensor product involving a representation similar to one that appears in an article by A. van Tonder. This representation and tensor product was investigated mainly to get a good comprehension in the subject and to understand some of the problems that can arise. In the other case, which is the main problem in this thesis, I looked at a tensor product and representations that appears in an article by M. R. Gaberdiel. Here we deal with a tensor product of representations of su(2) with a specific value for the level at k = -4/3 and a specific eigenvalue of the Casimir operator at -2/9. This was done in the frame of finite dimensional Lie algebra and affine Lie algebra and not in the case of fusion rules as in the article by M. R. Gaberdiel. In both cases some of the calculations where done from in situ and the investigation of the representations behaviour due to the step operators, theirs eigenvalue and theirs weight system. Results and conclusions of the investigations are discussed in the last part of this thesis.

Using both quantitative and qualitative data collection and analyses techniques, this study examined representations used by sixteen (n = 16) teachers while teaching the concepts of converting among fractions, decimals, and percents. The classroom videos used for this study were recorded as part of the Middle School Mathematics Project (MSMP). The study also compared teacher-selected and textbook representations and examined how teachers‘ use of idiosyncratic representations influenced representational choices on the number test by the teachers‘ five hundred eighty-one (N = 581) students. In addition to using geometric figures and manipulatives, a majority of the teachers used natural language such as the words nanny, north, neighbor, dog, cowboy, and house to characterize fractions and mathematical procedures or algorithms. Coding of teacher-selected representations showed that verbal representations deviated from textbook representations the most. Some teachers used the words or phrases bigger, smaller, doubling, tripling, breaking-down, and building-up in the context of equivalent fractions. There was widespread use of idiosyncratic representations by teachers, such as equations with missing or double equal signs, numbers and operators written as superscripts, and numbers written above and below the equal sign. Although use of idiosyncratic representations by teachers influenced representational choices by students on the number test, no evidence of a relationship between representational forms and degree of correctness of solutions was found. The study did reveal though that teachers‘ use of idiosyncratic representations can lead to student misconceptions such as thinking that multiplying by a whole number not equal to 1 gives an equivalent fraction. Statistical tests were done to determine if frequency of representation usage by teachers was related to the textbook, highest degree obtained by teacher, certification, number of years spent teaching mathematics, number of years teaching mathematics at grade level, number of hours completed on professional development related to their textbook, and total number of days spent on the Interagency Education Research Initiative (IERI) professional development. The results showed representation usage was related to all the above variables, except the highest degree obtained and the total number of days spent on the IERI professional development.

Thesis (Ph. D.)--Ohio State University, 2003. / Title from first page of PDF file. Document formatted into pages; contains vi, 93 p. Includes abstract and vita. Advisor: Stephen Rallis, Dept. of Mathematics. Includes bibliographical references (p. 91-93).

Thesis (Ph. D.)--University of Oregon, 2003. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 69-71). Also available for download via the World Wide Web; free to University of Oregon users.