A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of requirements for the degree of Master of Science. Johannesburg, 09 February 2016. / The main goal of this dissertation is to construct a better understanding of the subtleties that arise in the holographic computation of extremal correlators. It is well known that these computations, in the gravitational description, su er from divergences, but the interpretation and origin of these divergences is unclear. The study starts with detailed computations of two- and three-point functions of a scalar eld minimally coupled to gravity on Euclidean AdSd space, three-point functions of two giant gravitons and one light graviton, and three-point functions of the Kaluza-Klein gravitons, using supergravity theory. Further, we also give the computation of these same correlators in the dual CFT. These involve novel techniques in the matrix model, including methods that employ Schur polynomials in the dual gauge theory analysis. By employing the usual AdS=CFT dictionary, we argue that extremal correlators are naturally related to collinear particles. There are divergences that arise in collinear amplitudes as a consequence of the fact that the particles momenta are parallel. We therefore reach the suggestive idea that the divergences in extremal correlator computations are linked to collinear divergences. Much remains to be done to really establish this connection. / LG2018

Thesis (Ph.D.)--Indiana University, Dept. of Mathematics, 2004. / Title from PDF t.p. (viewed Dec. 2, 2008). Source: Dissertation Abstracts International, Volume: 66-01, Section: B, page: 0373. Chair: Lawrence S. Moss.

Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2006. / Source: Dissertation Abstracts International, Volume: 67-07, Section: B, page: 3915. Adviser: Eric de Sturler. Includes bibliographical references (leaves 117-121) Available on microfilm from Pro Quest Information and Learning.

In the first part of the thesis, we characterize exactly the expressiveness of a family of typed database transaction languages based on a traversal (or recursion) scheme devised by Fegaras, Sheard and Stemple (in (FSS92)), and we capture the complexity of reflection in a first order calculus framework. Reflection in a programming language refers to the ability to generate a program and execute it in the same environment. Using a first-order interpretation from PSPACE to the quantified Boolean formula value problem, we are able to derive an easy proof of the fact that reflection when added to first-order algebra captures exactly the problems in PSPACE. Further, we give a partial answer toward resolving the complexity of higher order reflection. In the second part, eschewing traditional (static) complexity classes for measuring the complexity of database query languages, we propose a complexity theoretic framework for studying dynamic complexity classes. In this thesis we define the requisite dynamic complexity classes. In particular, we introduce and investigate a natural logic for a parallel dynamic complexity class, Dynamic First-Order Logic (Dyn-FO). We show that many interesting graph problems are in Dyn-FO, including, among others, graph connectivity and the computation of minimum spanning trees. We prove that certain standard complete problems for static complexity classes, such as AGAP for P remain complete via these new reductions. On the other hand, we prove that other such problems including GAP for NL and 1GAP for L are no longer complete via bounded expansion reductions. Our results shed light on some of the interesting differences between static and dynamic complexity. We examine the dynamic complexity of maintaining approximate solutions to NP optimization problems in the descriptive framework. We introduce a new approximation class called BMAXSNP that is a subset of MAX SNP and includes interesting problems such as MAX CUT, MAX SAT and MAX 3SAT. We define reductions that honor dynamic complexity while preserving approximations. We then show the completeness, via such reductions, of a number of NP optimization problems for BMAXSNP, and we further show that this class has constant approximable solutions which can be maintained in Dyn-FO.

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In his paper B.K.Harrison concludes with the observation that his "solutions ... are presented as raw material for further research in General Relativity". In the same spirit, the present work started out as an attempt to process that raw material in a production-line powered by a computer. Harrison's solutions uould be fed in at one end, and the finished product, as yet undecided, would appear at the other. In the event, however, the project became more like an exercise in quality control, to continue the analogy. A search was made for algebraic criteria which would distinguish between those solutions which were acceptable for further analysis with particular regard to Gravitational radiation, and those which were not. Regrettably, no criteria could be found which characterised radiative solutions unequivocally, and, at the same time, lent themselves to a computer approach. The result is that the discussion of radiative solutions has had to be relegated to an appendix (Appendix 1), while the main body of the work is concerned with the determination of those quantities (the Newman-Penrose scalars) which would seem to be the foundation of any future computer-based analysis of gravitational radiation. Chapter 1 is an account of the underlying mathematical formulation, defining the terms, concepts and processes involved. In Chapter 2 the transformation of some of the ideas of Chapter 1 into computer software is presented. Chapter 3 is concerned with the specific metrics (the Harrison metrics) and the extent to which they have heen processed. The project has leaned heavily on papers by Harrison for the "raw material", by D' Inverno and Russell Clark, who pioneered the techniques and classified the Harrison metrics, and by Sachs for the treatment of gravitational radiation. However, the analysis of diagonal metrics, the special tetrad of Chapter 2 and the results in Appendix 2 are new.

This dissertation advances a novel approach to formal specification and verification of systems-level computation. The approach is based on a purely finite state model of computation, and makes use of algebraic and syntactic techniques which have not been previously applied to the problem. The approach makes use of a modal extension of the primitive recursive functions to specify system behavior, and uses an algebraic feedback product of automata to provide semantic content for specifications. The modal functions are shown to provide highly compact and expressive notation for defining, composing, and verifying the properties of large-scale finite state machines. The feedback product allows for a very general model of composition, multi-level dynamic behavior, and concurrency. Techniques are developed for specifying both detailed operation of algorithms, and abstract system properties such as liveness and safety. Several significant examples are provided to illustrate application of the method to complex algorithms and designs.

Higher bandwidths in computer networks have made application with real-time constraints, such as control, command, and interactive voice and video communication feasible. We describe two congestion control mechanisms that utilize properties of real-time applications. First, many real-time applications, such as voice and video, can tolerate some loss due to signal redundancy. We propose and analyze a congestion control algorithm that aims to discard packets if they stand little chance of reaching their destination in time as early on their path as possible. Dropping late and almost-late packets improves the likelihood that other packets will make their deadline. Secondly, in real-time systems with fixed deadlines, no improvement in performance is gained by arriving before the deadline. Thus, packets that are late and have many hops to travel are given priority over those with time to spare and close to their destination by introducing a hop-laxity priority measure. Simulation results show marked improvements in loss performance. The implementation of the algorithm within a router kernel for the DARTnet test network is described in detail. Because of its unforgiving real-time requirements, packet audio was used as one evaluation tool; thus, we developed an application for audio conferencing. Measurements with that tool show that traditional source models are seriously flawed. Real-time services are one example of traffic whose perceived quality of service depends not only on the loss rate but also on the correlation of losses. We investigate the correlation of losses due to buffer overflow and deadline violations in both continuous and discrete-time queueing systems. We show that loss correlation does not depend on value of the deadline for M/G/1 systems and is generally only weakly influenced by buffer sizes. Per-stream loss correlation in systems with periodic and Bernoulli on/off sources are evaluated analytically. Numerical examples indicate that loss correlation is of limited influence as long as each stream contributes less than about one-tenth of the overall network load.

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