Thesis (MSc)--University of Stellenbosch, 2007. / ENGLISH ABSTRACT: Buying and selling in financial markets are driven by demand. The demand can be quantified by the imbalance in the number of shares QB and QS transacted by buyers and sellers respectively over a given time interval t. The demand in an interval t is given by (t) = QB − QS. The local noise intensity is given by = h|aiqi − haiqii|i where i = 1, . . . ,N labels the transactions in t, qi is the number of shares traded in transaction i, ai = ±1 denotes buyer- initiated and seller- initiated trades respectively and h· · · i is the local expectation value computed from all the transactions during the interval t. In a paper [1] based on data from the New York Stock Exchange Trade and Quote database during the period 1995-1996, Plerou, Gopikrishnan and Stanley [1] reported that the analysis of the probability distribution P( | ) of demand conditioned on the local noise intensity revealed the surprising existence of a critical threshold c. For < c, the most probable value of demand is roughly zero; they interpreted this as an equilibrium phase in which neither buying nor selling predominates. For > c two most probable values emerge that are symmetrical around zero demand, corresponding to excess demand and excess supply; they interpreted this as an out-of-equilibrium phase in which the market behaviour is buying for half of the time, and selling for the other half. It was suggested [1] that the two-phase behaviour indicates a link between the dynamics of a financial market with many interacting participants and the phenomenon of phase transitions that occurs in physical systems with many interacting units. This thesis reproduces the two-phase behaviour by means of experiments using sequences of random variables. We reproduce the two-phase behaviour based on correlated and uncorrelatd data. We use a Markov modulated Bernoulli process to model the transactions and investigate a simple interpretation of the two-phase behaviour. We sample data from heavy-tailed distributions and reproduce the two-phase behaviour. Our experiments show that the results presented in [1] do not provide evidence for the presence of complex phenomena in a trading market; the results are a consequence of the sampling method employed. / AFRIKAANSE OPSOMMING: Aankope en verkope in finansi¨ele markte word deur aanvraag gedryf. Aanvraag kan gekwantifiseer word in terme van die ongebalanseerdheid in die getal aandele QB en QB soos onderskeidelik verhandel deur kopers en verkopers in ’n gegewe tyd-interval t. Die aanvraag in ’n interval t word gegee deur (t) = QB −QS. Die lokale geraasintensiteit word gegee deur = h|aiqi − haiqii|i waar i = 1, . . . ,N die transaksies in t benoem, qi die getal aandele verhandel in transaksies verwys, en h· · · i op die lokale verwagte waarde dui, bereken van al die tansaksies tydens die interval t. In ’n referaat [1] wat op data van die New York Effektebeurs se Trade and Quote databasis in die periode tussen 1995 en 1996 geskoei was, het Plerou, Gopikrishnan en Stanley [1] gerapporteer dat ’n analise van die waarskynlikheidsverspreiding P( | ) van aanvraag gekondisioneer op die lokale geraasintensiteit , die verrassende bestaan van ’n kritieke drempelwaarde c na vore bring. Vir < c is die mees waarskynlike aanvraagwaarde nagenoeg nul; hulle het dit ge¨ınterpreteer as ’n ekwilibriumfase waartydens n`og aankope n`og verkope die oormag het. Vir > c is die twee mees waarskynlike aanvraagwaardes wat te voorskyn kom simmetries rondom nul aanvraag, wat oorenstem met ’n oormaat aanvraag en ’n oormaat aanbod; hulle het dit geinterpreteer as ’n buite-ewewigfase waartydens die markgedrag die helfte van die tyd koop en die anderhelfte verkoop. Daar is voorgestel [1] dat die tweefase gedrag op ’n verband tussen die dinamiek van ’n finansiele mark met baie deelnemende partye, en die verskynsel van fase-oorgange wat in fisieke sisteme met baie wisselwerkende eenhede voorkom, dui. Hierdie tesis reproduseer die tweefase gedrag deur middel van eksperimente wat gebruik maak van reekse van lukrake veranderlikes. Ons reproduseer die tweefase gedrag gebaseer op gekorreleerde en ongekorreleerde data. Ons gebruik ’n Markov-gemoduleerde Bernoulli proses om die transaksies te moduleer en ondersoek ’n eenvoudige interpretasie van die tweefase gedrag. Ons seem steekproefdata van “heavy-tailed” verspreidings en reproduseer die tweefase gedrag. Ons ekperimente wys dat die resultate in [1] voorgested is nie bewys lewer vir die teenwoordigheid van komplekse verskynsel in’n handelsmark nie; die resultate is as gevolg van die metode wat gebruik is vir die generering van die steekproefdata.

Applications that manipulate sparse data structures contain memory reference patterns that are unknown at compile time due to indirect accesses such as A[B[i]]. To exploit parallelism and improve locality in such applications, prior work has developed a number of Run-Time Reordering Transformations (RTRTs). This paper presents the Sparse Polyhedral Framework (SPF) for specifying RTRTs and compositions thereof and algorithms for automatically generating efficient inspector and executor code to implement such transformations. Experimental results indicate that the performance of automatically generated inspectors and executors competes with the performance of hand-written ones when further optimization is done.

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We apply anisotropic Banach space methods together with operator renewal theory to obtain polynomial rates of decay of correlations for a class of generalized baker's transformations. The polynomial rates were proved for a smaller class of observables in a 2013 paper of Bose and Murray by fundamentally different methods. Our approach provides a direct analysis of the Frobenius-Perron operator associated to a generalized baker's transformation in contrast to the paper of Bose and Murray where decay rates are obtained for a factor map and lifted to the full map. / Graduate

Thesis (MSc (Physics))--University of Stellenbosch, 2005. / The goal of this thesis is the development and implementation of a non-perturbative solution method for Wegner’s flow equations. We show that a parameterization of the flowing Hamiltonian in terms of a scalar function allows the flow equation to be rewritten as a nonlinear partial differential equation. The implementation is non-perturbative in that the derivation of the PDE is based on an expansion controlled by the size of the system rather than the coupling constant. We apply this method to the Lipkin model and obtain very accurate results for the spectrum, expectation values and eigenstates for all values of the coupling and in the thermodynamic limit. New aspects of the phase structure, made apparent by this non-perturbative treatment, are also investigated. The Dicke model is treated using a two-step diagonalization procedure which illustrates how an effective Hamiltonian may be constructed and subsequently solved within this framework.

Thesis (MSc)--Stellenbosch University, 2003. / ENGLISH ABSTRACT: This thesis presents an overview of the flow equations recently introduced by Wegner. The little known mathematical framework is established in the initial chapter and used as a background for the entire presentation. The application of flow equations to the Foldy-Wouthuysen transformation and to the elimination of the electron-phonon coupling in a solid is reviewed. Recent flow equations approaches to the Lipkin model are examined thoroughly, paying special attention to their utility near the phase change boundary. We present more robust schemes by requiring that expectation values be flow dependent; either through a variational or self-consistent calculation. The similarity renormalization group equations recently developed by Glazek and Wilson are also reviewed. Their relationship to Wegner's flow equations is investigated through the aid of an instructive model. / AFRIKAANSE OPSOMMING: Hierdie tesis bied 'n oorsig van die vloeivergelykings soos dit onlangs deur Wegner voorgestel is. Die betreklik onbekende wiskundige raamwerk word in die eerste hoofstuk geskets en deurgans as agtergrond gebruik. 'n Oorsig word gegee van die aanwending van die vloeivergelyking vir die Foldy-Wouthuysen transformasie en die eliminering van die elektron-fonon wisselwerking in 'n vastestof. Onlangse benaderings tot die Lipkin model, deur middel van vloeivergelykings, word ook deeglik ondersoek. Besondere aandag word gegee aan hul aanwending naby fasegrense. 'n Meer stewige skema word voorgestel deur te vereis dat verwagtingswaardes vloei-afhanklik is; óf deur gevarieerde óf self-konsistente berekenings. 'n Inleiding tot die gelyksoortigheids renormerings groep vergelykings, soos onlangs ontwikkel deur Glazek en Wilson, word ook aangebied. Hulle verwantskap met die Wegner vloeivergelykings word bespreek aan die hand van 'n instruktiewe voorbeeld.

Quantum systems with finite Hilbert space are considered. Position and mo- mentum states and their relation through a Fourier transform, displacement in the position-momentum phase-space, and symplectic transformations are introduced and their properties are studied. Symplectic Sp(2l;Zp) trans- formations in l-partite finite system are explicit constructed. The general method is applied to bi-partite and tri-partite systems. The effect of these transformations on the correlations is discussed. Entanglement calculations between the subsystems in a bi-partite system and a tri-partite system are presented. The effect of measurements is also studied.

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