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No description available.

This thesis discusses a theorem on the convergence of a continued fraction.

While the traditional form of continued fractions is well-documented, a new form, designed to approximate real numbers between 1 and 2, is less well-studied. This report first describes prior research into the new form, describing the form and giving an algorithm for generating approximations for a given real number. It then describes a rational function giving the rational number represented by the continued fraction made from a given tuple of integers and shows that no real number has a unique continued fraction. Next, it describes the set of real numbers that are hardest to approximate; that is, given a positive integer $n$, it describes the real number $\alpha$ that maximizes the value $|\alpha - T_n|$, where $T_n$ is the closest continued fraction to $\alpha$ generated from a tuple of length $n$. Finally, it lays out plans for future work.

This dissertation consists of three parts. In the first part we consider the equidistribution of roots of quadratic congruences. The roots of quadratic congruences are known to be equidistributed. However,we establish a bound for the discrepancy of this sequence using a spectral method involvingautomorphic forms, especially Kuznetsov's formula, together with an Erdős-Turán inequality. Then we discuss the implications of our discrepancy estimate for the reducibility problem of arctangents of integers. In the second and third part of this dissertation we consider some aspects of Farey fractions. The set of Farey fractions of order at most [mathematical formula] is, of course, a classical object in Analytic Number Theory. Our interest here is in certain sumsets of Farey fractions. Also, in this dissertation we study Farey fractions by working in the quotient group Q/Z, which is the modern point of view. We first derive an identity which involves the structure of Farey fractions in the group ring of Q/Z. Then we use these identities to estimate the asymptotic magnitude of the size of the sumset [mathematical formula]. Our method uses results about divisors in short intervals due to K. Ford. We also prove a new form of the Erdős-Turán inequality in which the usual complex exponential functions are replaced by a special family of functions which are orthogonal in L²(R/Z). / text

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A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of requirements for the degree of Master of Science. 2015. / We consider continued fractions with coe cients that are in K, the quaternions. In particular we consider coe cients in the Hurwitz integers H in K. These continued fractions are expressed as compositions of M¨obius maps in M R4 1 that act, by Poincar´e extension, as isometries on H5. This dissertation explores groups of 2 2 matrices over K and two particular determinant type functions acting on these groups. On the one hand we find M R4 1 , the group of orientation preserving M¨obius transformations acting on R4 1 in terms of a determinant D [19],[38]. On the other hand K may be considered as a Cli ord algebra C3 based on two generators i and j, or more generally i1 and i2, where i j = k or i1i2 = k. It is shown this group of matrices over C4 defined in terms of a pseudo-determinant [1],[37] can also be used to establish M R4 1 . Through this relationship we are able to connect the determinant D to the pseudo-determinant when acting on the matrices that generate M R4 1 . We explore and build on the results of Schmidt [30] on the subdivision of a Farey simplex into 31 Farey simplices. These results are reinterpreted in H5 with boundary K1 using the group of M¨obius transformations on R4 1 [19], [38]. We investigate the unimodular group G = PS DL(2;K) with its generators and derive a fundamental domain for this group in H5. We relate this domain to the 24-cells PU and r that tessellate K. We define the concepts of Farey neighbours, Farey geodesics and Farey simplices in the Farey tessellation of H5. This tessellation of H5 by a Farey pentacross under a discrete subgroup G of M R4 1 is analogous to the Farey tessellation by Farey triangles of H2 under the modular group [31]. The result in Schmidt [30], that for each quaternion there is a chain of Farey simplices that converge to , is reinterpreted as a continued fraction, with entries from H, that converges to . We conclude with a review of Pringsheim’s theorem on convergence of continued fractions in higher dimensions [5].

This body of work examines the plausibility of applying partial di erential equations and time-fractional partial di erential equations to images. The standard di usion equation is coupled with a nonlinear cubic source term of the Fitzhugh-Nagumo type to obtain a model with di usive properties and a binarizing e ect due to the source term. We examine the e ects of applying this model to a class of images known as document images; images that largely comprise text. The e ects of this model result in a binarization process that is competitive with the state-of-the-art techniques. Further to this application, we provide a stability analysis of the method as well as high-performance implementation on general purpose graphical processing units. The model is extended to include time derivatives to a fractional order which a ords us another degree of control over this process and the nature of the fractionality is discussed indicating the change in dynamics brought about by this generalization. We apply a semi-discrete method derived by hybridizing the Laplace transform and two discretization methods: nite-di erences and Chebyshev collocation. These hybrid techniques are coupled with a quasi-linearization process to allow for the application of the Laplace transform, a linear operator, to a nonlinear equation of fractional order in the temporal domain. A thorough analysis of these methods is provided giving rise to conditions for solvability. The merits and demerits of the methods are discussed indicating the appropriateness of each method.

Thesis (Ph.D.). / Written for the Division of Experimental Medicine. Title from title page of PDF (viewed 2008/05/09). Includes bibliographical references.

Thesis (Ph. D.)--Georgia State University, 2007. / Title from file title page. Christine Thomas, committee chair; Deborah Najee-Ullah, Barbara Kawulich, Clara Nosegbe-Okoka , Pier Junor-Clarke, committee members. Electronic text (286 p. : ill.) : digital, PDF file. Description based on contents viewed June 9, 2008. Includes bibliographical references (p. 233-256).