Monitored network data allows network managers and operators to gain valuable insight into the health and status of a network. Whilst such data is useful for real-time analysis, there is often a need to post-process historical network performance data. Storage of the monitored data then becomes a serious issue as network monitoring activities generate significant quantities of data. This thesis is part of the EPSRC sponsored MASTS (Measurements in All Scales in Time and Space) project. MASTS is a joint project between Loughborough, Cambridge and UCL and focuses on measuring, analyzing, compressing and storing network characteristics of JANET (UK's research/academic network). The work in this thesis is motivated by the need of measuring the performance of high-speed networks and particularly of UKLight. UKLight connects JANET to U.S.A and the rest of Europe. Such networks produce large amounts of data over a long period of time, making the storage of this information practically inefficient. A possible solution to this problem is to use lossy compression on an on-line system that intelligently compresses computer network measurements while preserving the quality in important characteristics of the signal and various statistical properties. This thesis contributes to the knowledge by examining two threshold estimation techniques, two threshold application techniques, the impact of window size on the lossy compression performance. In addition eight different wavelets were examined in terms of compression performance, energy preservation, scaling be- haviour, quality attributes (mean, standard deviation, visual quality and PSNR) and Long Range Dependence. Finally, this thesis contributes by presenting a technique for precise quality control of the reconstructed signal and an additional use of wavelets for detecting sudden changes. The results of the thesis show that the proposed Gupta-Kaur (GK) based algorithm compresses on average delay signals 17 times and data rate signals 11.2 times while accurately preserving their statistical properties.

Within this thesis we consider locally stationary wavelet process models for one and two dimensional data. Unlike the traditional stationary Fourier models this locally stationary wavelet framework allows the second order structure to vary as a function of time (one dimension) or location (two dimensions). We consider three distinct problems within this broader setting. First of all we consider the impact upon one dimensional spectral estimates of the choice of analysing wavelet. As a result of the definition of a locally stationary wavelet process it is required that the same wavelet is used for analysis as generated the process. We show the impact that using an alternative wavelet has on the spectral estimates, showing that, depending on whether the analysing wavelet is smoother than the generating wavelet or not, the spectrum will be under or over estimated. The second problem relates to replicated data. In circumstances where we have data from a number of subjects relating to the same process we may be more interested in the population effects rather than individual subjects. We consider the extension of a mixed effects model to model the spectral structure of replicated locally stationary wavelet processes in order to estimate the population second order structure. Finally we consider the impact of aliasing on locally stationary images. We consider how subsampling an image can lead to artefacts in the spectral structure of the image. By taking advantage of the nature of the contamination in locally stationary two dimensional wavelet processes we show how the artefact may be estimated and a test for the absence of aliasing may be developed.

The thesis concerns linear systems and scattering theory. In particular, it presents lineal' systems for some integrable systems and finds discrete analogues for many well known results for continuous variables. It introduces some new tools from linear systems and applies them to standard integrable systems. We begin by expressing the first Painleve equation as the compatibility condition of a certain Lax pair and introduce the Korteweg-de Vries partial differential equation. We introduce the spectral curve for algebraic families and the Toda lattice. The Fredholm determinant of a trace class Hankel integral operator gives rise to a tau function. Dyson used the tau function to solve an inverse spectral problem for Schrodinger operators. When a plane wave is subject to Schrodinger's equation and scattered by a potential u, the output is described at great distances by a scattering function. The spectral problem is to find the spectrum of Schrodinger's operator in L2 and hence the scattering function. The inverse spectral problem is to find the potential given the scattering function. The scattering and inverse scattering problems are linked by the Gelfand- Levitan equation. In this thesis, for a discrete linear system, we introduce a scattering function and Hankel matrix and a version of the Gelfand-Levitan equation for discrete linear systems. We introduce the discrete operator ∑∞/k=n AkBCAk and use it to solve the Gelfand-Levitan equation and compute Fredholm determinants of Hankel operators. We produce a discrete analogue of a calculation of Poppe giving a solution to the Korteweg-de Vries equation and via the methods of linear systems find an analogous solution in terms of Hankel matrices. We then produce a discrete analogue of the Miura transform. Thus the main new contributions of this thesis are the discrete analogues of the R operator, the Gelfand- Levitan equation, the Lyapunov equation and the Miura transform.

No description available.

The integrable KdV equation is a model for shallow water. However, another integrable equation, the Camassa Holm equation is also claimed to be a model for long wave shallow water. The Camassa Holm equation was derived from the Green Naghdi equations which itself is a model for water. Robin lohnson found that the derivation of the Camassa Holm equation from the Green Naghdi model was inconsistent in the reduction to right moving waves. Dullin, Gottwald and Holm then derived the Camassa Holm equation from the shallow water wave equations. Since the publication of this paper Camassa Holm equation was ubiquitously used as model for water waves. In chapter 2 it is shown that the Camassa Holm equation is inconsistent with the long wave asymptotic expansion. Symbolic representation is used to find the approximate symmetries of perturbed integrable equations. The use of symbolic representation in integrable systems is well developed. In this thesis it is used to find near identity transformations of perturbed integrable equations to integrable equations. The question arises whether the conditions necessary for there to exist approximate symmetries are sufficient for there to exist such a transformation. In chapter 5 we study reciprocal coordinate transformations of perturbed integrable equations and asymptotic expansions of discrete Lax operators.

Wavelet thresholding is an increasingly popular method of nonparametric smoothing. Both real-valued and complex-valued Daubechies wavelets exist. However, to date, complex-valued wavelets have attracted little attention in the statistical literature compared to real-valued wavelets. The broad aim of this thesis is to further the application of complex-valued Daubechies wavelets within the wavelet thresholding framework. Much of the previous work that applied complex-valued wavelets focused upon their application to real-valued data. However, complex-valued data exist and arise in multiple scientific areas. This thesis firstly examines how one method of applying complex-valued wavelets performs on complex-valued data before modifying the methodology to allow for native denoising of complex-valued data. A large number of smoothing regimes exist within the wavelet framework, known as 'thresholding rules'. The majority of these thresholding rules have been designed for use in conjunction with real-valued wavelets. One such thresholding rule is known as 'block thresholding' whereby the data is considered in blocks, rather than as individual data points, to allow for correlation between neighbouring data points. A further thresholding method is known as the 'fiducial thresholding' method which attempts to circumvent perceived problems within the Bayesian approach. Within this thesis these two thresholding rules are developed to allow for the application of complex-valued wavelets; the difference in their performance when using real- and complex-valued wavelets is investigated by simulation.

In this thesis the method of Zubov for obtaining domains of attraction for systems of autonomous ordinary differential equations is investigated. The necessary theorems of continuity, existence and uniqueness along with Zubov and Lyapunov theorems are listed in Chapter 1 as a starting point.

In 2 , the following extension of the higher order Rellich inequality / AV(x) 2>7(n,a,i) / /(x) 2- (1) JRn lxl JRn lxl was proven by W. Allegretto for all / G C&pound; (Rn {0}). The constant 7 is calculated explicitly by the author for all n > 2, a > 0 and j 6 N, giving the value of the constant in the previously unknown case n < a + 4j. Hence proving that 7 is equal to zero if and only if n < a + 4j and n a = 0 (mod 2). In this problematic case, the author finds that the higher order Rellich inequality (1) can be forced to be non-trivial if further restrictions are placed on the function in n_1. An alternative method to restricting the functional class is to look at the Rellich type inequality / AA/(x) 2 >*(n>a,4) / l/(x) 2 (2) JRn lxl JRn lxl found by W.D. Evans and R.T. Lewis in 15 for n = 2,3,4. The magnetic Laplacian is of the form Aa = (V zA)2 where in spherical coordinates H*(*i)ei ifn = 2, with e L (0,27r) and (0) = (2r). The potential A is of Aharonov- Bohm type and the constant $ is dependant upon the distance of the magnetic flux to the integers Z. By finding the discrete spectrum of the Friedrichs extension of A a in L2(Sn_1), the author is able to extend the Rellich type inequality (2) to all n > 2 and a > 0. Consequently, the higher order Rellich type inequality / Ai/(x) a>n(n,a,*,j)/ l/(x) 2 (4) JRn Ix x j can be constructed. The inequality (4) is shown to be non-trivial for all n < a + and n a = 0 (mod 2), the previously problematic case. The Rellich type inequality (4) enables an analysis of the spectral properties of perturbations of the magnetic operator AA to be undertaken in L2(IRn), n > 2. Furthermore, a CLR type bound for the number of negative eigenvalues of the operator AA can be found in L2(R8), a space in which there is no CLR bound for the operator A4.

An in-depth account of the wavelet regularization mechanisms acting in this method of continuous relaxation spectrum recovery is given. It is shown that that scaling parameter of the wavelets controls the resolution of the spectrum, whilst the number of basis functions controls the sparsity of the approximation.

In this thesis, we study semiclassical phase-space methods for quantum evolution in Hermitian and non-Hermitian systems. We first present the dynamics of Gaussian wave packets under non-Hermitian Hamiltonians and interpret them as a classical dynamics for complex Hamiltonians. We use these to derive exact dynamics for wave packets in the quadratic Swanson oscillator. We show that in the case of unbroken PT-symmetry there can be periodic divergences in this system and relate this to the fact that any operator mapping the system to a Hermitian counterpart is unbounded. We apply the semiclassical wave-packet dynamics to two further anharmonic example systems: a PT-symmetric wave guide, a version of which we propose as a filtering device for optical beams, and a non-Hermitian single-band tight-binding model, for which we use classical equations of motion to model both narrowly and widely distributed initial states. We further develop an exact quantum propagator for non-Hermitian dynamics using lattices of wave packets, whose evolution is governed by the semiclassical equations of motion. We demonstrate that this accurately reproduces quantum dynamics compared to the split operator method. Finally we study a Hermitian two-mode many-particle model for bosonic atom-molecule conversion, for which the classical phase-space structure is an orbifold. We show that standard semiclassical tools may be applied to recover features such as the dynamics, spectrum and density of states of the many-particle system.