Issued in English in 1937 as Thesis (Ph. D.)--Cincinnati. / From Annales scientifiques de l'Ecole normale supérieure. 3. sér., t. LVI, fasc. 1. Bibliography: p. 70.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1927. / Thesis note stamped on cover. Reprinted from the Transactions of the American mathematical society, vol. 31, no. 1. eContent provider-neutral record in process. Description based on print version record.

Inaug.-diss.--Kaiser Wilhelms-Universität zu Strassburg. / Lebenslauf.

No description available.

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

Thirty years ago, Effros and Shen classified the simple dimension groups with rank two. Every such group is parametrized by an irrational number, and can be constructed as an inductive limit using that number's continued fraction expansion. There is a natural generalization of continued fractions to higher dimensions, and this invites the following question: What dimension groups correspond to multidimensional continued fractions? We describe this class of groups and show how some properties of a continued fraction are reflected in the structure of its dimension group. We also consider a related issue: an Effros-Shen group has been shown to arise in a natural way from the tail equivalence relation on a certain sequence space. We describe a more general class of sequence spaces to which this construction can be applied to obtain other dimension groups, including dimension groups corresponding to multidimensional continued fractions.

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.

None

Thirty years ago, Effros and Shen classified the simple dimension groups with rank two. Every such group is parametrized by an irrational number, and can be constructed as an inductive limit using that number's continued fraction expansion. There is a natural generalization of continued fractions to higher dimensions, and this invites the following question: What dimension groups correspond to multidimensional continued fractions? We describe this class of groups and show how some properties of a continued fraction are reflected in the structure of its dimension group. We also consider a related issue: an Effros-Shen group has been shown to arise in a natural way from the tail equivalence relation on a certain sequence space. We describe a more general class of sequence spaces to which this construction can be applied to obtain other dimension groups, including dimension groups corresponding to multidimensional continued fractions.

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].