A dissertation submitted in fulfillment of the requirements for the degree of Master of Science. August 2016. / The six scalar fields in N = 4 super Yang-Mills theory enjoy a global SO(6) symmetry, and large N but non-planar limits of this theory are well-described by adopting a group representation approach. Studies have shown that the one-loop dilatation operator is highly determined by the action of the su(2)=su(3) subalgebras on restricted Schur polynomials. These actions involve the traces of products of projection operators. In this dissertation, exact analytical formulae for these traces are found which in turn are used to find the exact action of these algebras on restricted Schur polynomials. The potential of the su(2) algebra to determine the one-loop dilatation operator is also explored. This is done by exploiting necessary symmetry conditions and moving to a continuum limit in order to derive a number of partial differential equations which determine the dilatation operator. The ultimate goal of this work is to provide tools to find the exact one-loop dilatation operator in the non-planar limit. / LG2017

A Dissertation submitted to the Faculty of Science, University of the Witwatersrand, in fulfilment of the requirements for the degree of Master of Science. 14 May 2013 / We study the action of the dilatation operator on restricted Schur polynomials labeled by Young diagrams with p long columns or p long rows. A new version of Schur-Weyl duality provides a powerful approach to the computation and manipulation of the symmetric group operators appearing in the restricted Schur polynomials. Using this new technology, we are able to evaluate the action of the one loop dilatation operator. The result has a direct and natural connection to the Gauss Law constraint for branes with a compact world volume. Generalzing previous results, we find considerable evidence that the dilatation operator reduces to a decoupled set of harmonic oscillators. This strongly suggests that integrability in N = 4 super Yang-Mills theory is not just a feature of the planar limit, but extends to other large N but non-planar limits.

Benson spaces of higher order are introduced extending the idea of N. Krugljak and M. Milman, A distance between orbits that controls commutator estimates and invertibilty of operators, Advances in Mathematics 182 (2004), 78-123. The concept of Benson shift operators is introduced and a class of spaces equipped with these operators is considered. Commutator theorems of higher order on orbit spaces generated by a single element are proved for this class. It is shown that these results apply to the complex method of interpolation and to the real method of interpolation for the case q=1. Two new characterizations are presented of the domain space of the "derivation" operator in the context of orbital methods. Comparisons to the work of others are made, especially the unifying paper of M. Cwikel, N. Kalton, M. Milman and R. Rochberg, A United Theory of Commutator Estimates for a Class of Interpolation Methods, Advances in Mathematics 169 2002, 241-312. / by Eva, Kasprikova / Thesis (Ph.D.)--Florida Atlantic University, 2009. / Includes bibliography. / Electronic reproduction. Boca Raton, Fla., 2009. Mode of access: World Wide Web.

by Chi-keung Ng. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1991. / Includes bibliographical references. / Introduction --- p.1 / Chapter Chapter 0. --- Preliminary --- p.4 / Chapter 0.1 --- Topological vector spaces / Chapter 0.2 --- Ordered vector spaces / Chapter 0.3 --- Ordered normed spaces / Chapter 0.4 --- Ordered topological vector spaces / Chapter 0.5 --- Ordered bornological vector spaces / Chapter Chapter 1. --- Results on Ordered Normed Spaces --- p.23 / Chapter 1.1 --- Results on e∞-spaces and e1-spaces / Chapter 1.2 --- Complemented subspaces of ordered normed spaces / Chapter 1.3 --- Half injections and Half surjections / Chapter 1.4 --- Strict quotients and strict subspaces / Chapter Chapter 2. --- Helley's Selection Theorem and Local Reflexivity Theorem of order type --- p.55 / Chapter 2.1 --- Helley's selection theorem of order type / Chapter 2.2 --- Local reflexivity theorem of order type / Chapter Chapter 3. --- Operator Modules and Ideal Cones --- p.68 / Chapter 3.1 --- Operator modules and ideal cones / Chapter 3.2 --- Space cones and space modules / Chapter 3.3 --- Injectivity and surjectivity / Chapter 3. 4 --- Dual and pre-dual / Chapter Chapter 4. --- Topologies and Bornologies Defined by Operator Modules and Ideal Cones --- p.95 / Chapter 4.1 --- Generalized polars / Chapter 4.2 --- Topologies and bornologies defined by β and ε / Chapter 4. 3 --- Injectivity and generating topologies / Chapter 4.4 --- Surjectivity and generating bornologies / Chapter 4.5 --- The solid property and the generating topologies / Chapter 4.6 --- The solid property and the generating bornologies / Chapter Chapter 5. --- Semi-norms and Bounded disks defined by Operator Modules and Ideal Cones --- p.129 / Chapter 5.1 --- Results on semi-norms / Chapter 5.2 --- Results on bounded disks / References --- p.146 / Notations --- p.149

by Wai-Shing Luk. / Thesis (Ph.D.)--Chinese University of Hong Kong, 1996. / Includes bibliographical references (leaves 104-113). / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Functional Extension of Iterative Methods --- p.2 / Chapter 1.2 --- Applications in Circuit Simulation --- p.2 / Chapter 1.3 --- Multigrid Acceleration --- p.3 / Chapter 1.4 --- Why Hilbert Space? --- p.4 / Chapter 1.5 --- Parallel Implementation --- p.5 / Chapter 1.6 --- Domain Decomposition --- p.5 / Chapter 1.7 --- Contributions of This Thesis --- p.6 / Chapter 1.8 --- Outlines of the Thesis --- p.7 / Chapter 2 --- Waveform Relaxation Methods --- p.9 / Chapter 2.1 --- Basic Idea --- p.10 / Chapter 2.2 --- Linear Operators between Banach Spaces --- p.14 / Chapter 2.3 --- Waveform Relaxation Operators for ODE's --- p.16 / Chapter 2.4 --- Convergence Analysis --- p.19 / Chapter 2.4.1 --- Continuous-time Convergence Analysis --- p.20 / Chapter 2.4.2 --- Discrete-time Convergence Analysis --- p.21 / Chapter 2.5 --- Further references --- p.24 / Chapter 3 --- Waveform Krylov Subspace Methods --- p.25 / Chapter 3.1 --- Overview of Krylov Subspace Methods --- p.26 / Chapter 3.2 --- Krylov Subspace methods in Hilbert Space --- p.30 / Chapter 3.3 --- Waveform Krylov Subspace Methods --- p.31 / Chapter 3.4 --- Adjoint Operator for WBiCG and WQMR --- p.33 / Chapter 3.5 --- Numerical Experiments --- p.35 / Chapter 3.5.1 --- Test Circuits --- p.36 / Chapter 3.5.2 --- Unstructured Grid Problem --- p.39 / Chapter 4 --- Parallel Implementation Issues --- p.50 / Chapter 4.1 --- DECmpp 12000/Sx Computer and HPF --- p.50 / Chapter 4.2 --- Data Mapping Strategy --- p.55 / Chapter 4.3 --- Sparse Matrix Format --- p.55 / Chapter 4.4 --- Graph Coloring for Unstructured Grid Problems --- p.57 / Chapter 5 --- The Use of Inexact ODE Solver in Waveform Methods --- p.61 / Chapter 5.1 --- Inexact ODE Solver for Waveform Relaxation --- p.62 / Chapter 5.1.1 --- Convergence Analysis --- p.63 / Chapter 5.2 --- Inexact ODE Solver for Waveform Krylov Subspace Methods --- p.65 / Chapter 5.3 --- Experimental Results --- p.68 / Chapter 5.4 --- Concluding Remarks --- p.72 / Chapter 6 --- Domain Decomposition Technique --- p.80 / Chapter 6.1 --- Introduction --- p.80 / Chapter 6.2 --- Overlapped Schwarz Methods --- p.81 / Chapter 6.3 --- Numerical Experiments --- p.83 / Chapter 6.3.1 --- Delay Circuit --- p.83 / Chapter 6.3.2 --- Unstructured Grid Problem --- p.86 / Chapter 7 --- Conclusions --- p.90 / Chapter 7.1 --- Summary --- p.90 / Chapter 7.2 --- Future Works --- p.92 / Chapter A --- Pseudo Codes for Waveform Krylov Subspace Methods --- p.94 / Chapter B --- Overview of Recursive Spectral Bisection Method --- p.101 / Bibliography --- p.104

by Ye Yuanling. / "August 2000." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2000. / Includes bibliographical references (p. 82-85). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.

Chifan, Kida, and myself introduced a new class of non-amenable groups denoted by ${\bf NC} \cap {\bf Quot}(\mathcal C_{rss})$ which gives rise to \emph{prime} von Neumann algebras. This means that for every $\G\in {\bf NC} \cap {\bf Quot}(\mathcal C_{rss})$ its group von Neumann algebra $L(\G)$ cannot be decomposed as a tensor product of diffuse von Neumann algebras. The class ${\bf NC} \cap {\bf Quot}(\mathcal C_{rss})$ is fairly large as it contains many natural examples of groups, some intensively studied in various areas of mathematics: all infinite central quotients of pure surface braid groups; all mapping class groups of (punctured) surfaces of genus $0,1,2$; most Torelli groups and Johnson kernels of (punctured) surfaces of genus $0,1,2$; and, all groups hyperbolic relative to finite families of residually finite, exact, infinite, proper subgroups. In a separate investigation, de Santiago and myself were able to extend the previous techniques that allowed us to eliminate the usage of the {\bf NC} condition and ultimately classify all the possible tensor factorization of the von Neumann algebras of groups that belong solely to ${\bf Quot}(\mathcal C_{rss})$. This provides a far-reaching generalization of the aforementioned primeness results; for instance, we were able to show that if $\Gamma$ is a poly-hyperbolic group, then whenever we have a tensor decomposition $L(\G)\cong P_1\bar\otimes P_2 \bar \otimes \cdots \bar\otimes P_n$ then there exists a product decomposition $\G\cong \G_1\times \G_2 \times \cdots \times \G_n$ with $\G_i \in {\bf Quot}(\mathcal C_{rss})$ and, up to amplifications, we have $L(\G_i)\cong P_i$ for all $i=1,n$.

Thesis (M.S.)--Florida State University, 2004. / Advisor: Dr. Xiuwen Liu, Florida State University, College of Arts and Sciences, Dept. of Computer Science. Title and description from dissertation home page (viewed June 17, 2004). Includes bibliographical references.

published_or_final_version / Physics / Master / Master of Philosophy

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