It is known that finite gap potentials of Hill's equation y" + q(τ)y = Ey can be obtained as solutions of an integrable dynamical system: uncoupled harmonic oscillators constrained to move on the unit sphere in configuration space--The Neumann System. This Dissertation systematizes and generalizes this result. First, the theory of Baker-Akhiezer functions is placed on a solid mathematical foundation. Guided by the theory of Baker-Akhiezer functions and Riemann surfaces, trace formulas, particle systems, constraints, integrals and Lax pairs are systematically constructed for the particle system of the ℓ x ℓ matrix differential operator of order n.

Various patterns of the organization of Western tonal music exhibit hierarchical structure, among them the harmonic progressions underlying melodies and the metre underlying rhythmic patterns. Recognizing these structures is an important part of unconscious human cognitive processing of music. Since the prosody and syntax of natural languages are commonly analysed with similar hierarchical structures, it is reasonable to expect that the techniques used to identify these structures automatically in natural language might also be applied to the automatic interpretation of music. In natural language processing (NLP), analysing the syntactic structure of a sentence is prerequisite to semantic interpretation. The analysis is made difficult by the high degree of ambiguity in even moderately long sentences. In music, a similar sort of structural analysis, with a similar degree of ambiguity, is fundamental to tasks such as key identification and score transcription. These and other tasks depend on harmonic and rhythmic analyses. There is a long history of applying linguistic analysis techniques to musical analysis. In recent years, statistical modelling, in particular in the form of probabilistic models, has become ubiquitous in NLP for large-scale practical analysis of language. The focus of the present work is the application of statistical parsing to automatic harmonic analysis of music. This thesis demonstrates that statistical parsing techniques, adapted from NLP with little modification, can be successfully applied to recovering the harmonic structure underlying music. It shows first how a type of formal grammar based on one used for linguistic syntactic processing, Combinatory Categorial Grammar (CCG), can be used to analyse the hierarchical structure of chord sequences. I introduce a formal language similar to first-order predicate logical to express the hierarchical tonal harmonic relationships between chords. The syntactic grammar formalism then serves as a mechanism to map an unstructured chord sequence onto its structured analysis. In NLP, the high degree of ambiguity of the analysis means that a parser must consider a huge number of possible structures. Chart parsing provides an efficient mechanism to explore them. Statistical models allow the parser to use information about structures seen before in a training corpus to eliminate improbable interpretations early on in the process and to rank the final analyses by plausibility. To apply the same techniques to harmonic analysis of chord sequences, a corpus of tonal jazz chord sequences annotated by hand with harmonic analyses is constructed. Two statistical parsing techniques are adapted to the present task and evaluated on their success at recovering the annotated structures. The experiments show that parsing using a statistical model of syntactic derivations is more successful than a Markovian baseline model at recovering harmonic structure. In addition, the practical technique of statistical supertagging serves to speed up parsing without any loss in accuracy. This approach to recovering harmonic structure can be extended to the analysis of performance data symbolically represented as notes. Experiments using some simple proof-of-concept extensions of the above parsing models demonstrate one probabilistic approach to this. The results reported provide a baseline for future work on the task of harmonic analysis of performances.

We provide an L² theory for the local double Hilbert transform along an analytic surface (s, t ,φ(s, t )) in the Heisenberg group H¹, that is operator f ↦ Hφ f (x) := p.v.∫∣s∣,∣t∣≤1 f (x ∙ (s, t ,φ(s, t ))-¹) ds/s dt/t, where ∙ denotes the group operation in H1. This operator combines several features: it is amulti-parameter singular integral, its kernel is supported along a submanifold, and convolution is with respect to a homogeneous group structure. We reprove Hφ is always L²(H¹)→L²(H¹) bounded (a result first obtained in [Str12]) to illustrate the method and then refine it to characterize the largest class of polynomials P of degree less than d such that the operator HP is uniformly bounded when P ranges in the class. Finally, we provide examples of surfaces that can be treated by our method but not by the theory of [Str12].

This thesis is about computational theory in the setting of the mathematics of information. The first goal is to demonstrate that many commonly considered problems in optimisation theory cannot be solved with an algorithm if the input data is only known up to an arbitrarily small error (modelling the fact that most real numbers are not expressible to infinite precision with a floating point based computational device). This includes computing the minimisers to basis pursuit, linear programming, lasso and image deblurring as well as finding an optimal neural network given training data. These results are somewhat paradoxical given the success that existing algorithms exhibit when tackling these problems with real world datasets and a substantial portion of this thesis is dedicated to explaining the apparent disparity, particularly in the context of compressed sensing. To do so requires the introduction of a variety of new concepts, including that of a breakdown epsilon, which may have broader applicability to computational problems outside of the ones central to this thesis. We conclude with a discussion on future research directions opened up by this work.

An accurate, intuitive and tractable transform as been identified and developed from which the spatial harmonics of reticle patterns defined in polar coordinates can be obtained. A description of reticles and generic methods for describing them mathematically are presented along with some background on general harmonic analysis. Focus then turns to candidate transforms for analysis of reticle patterns and the most promising are investigated in more detail. A fast linear algorithm is devised to overcome some problems with implementation of a fast transform and this is followed by analysis of the transform basis functions to assist with interpretation of the transform in azimuth and radius. A sampling guideline is presented so that aliasing can be avoided and, finally, the transforms of some representative reticle patterns are shown along with some insight into their interpretation. The transformations presented provide a first step toward raising the resolution and harmonic content required in simulation image scenes that will ultimately result in optimal use of computing resources for the simulation of reticle seekers. / thesis (MSc(AppliedPhysics))--University of South Australia, 2005.

A large part of the structure of the objects in the theory of Dooley and Wildberger [Funktsional. Anal. I Prilozhen. 27 (1993), no. 1, 25-32] and that of Rouviere [Compositio Math. 73 (1990), no. 3, 241-270] can be described by considering a connected, finite-dimentional symmetric space G/H (as defined by Rouviere), with ???exponential map???, Exp, from L G/L H to G/H, an action, ???: K ??? Aut??(G) (where Aut?? (G) is the projection onto G/H of all the automorphisms of G which leave H invariant), of a Lie group, K, on G/H and the corresponding action, ???# , of K on L G/L H defined by g ??? L (???g), along with a quadruple (s, E, j, E#), where s is a ???# - invariant, open neighbourhood of 0 in L G/L H, E is a test-function subspace of C??? (Exp s), j ?? C??? (s), and E# is a test-function subspace of C??? (s) which contains { j.f Exp: f ?? E }. Of interest is the question: Is the function ???: ?? ??? ????, where ??: f ??? j.f Exp, a local associative algebra homomorphism from F# with multiplication defined via convolution with respect to a function e: s x s ??? C, to F, with the usual convolution for its multiplication (where F is the space of all ??? - invariant distributions of E and F# is the space of all ???# - invariant distributions of E#)? For this system of objects, we can show that, to some extent, the choice of the function j is not critical, for it can be ???absorbed??? into the function e. Also, when K is compact, we can show that ??? ker ?? = { f ?? E : ???k f (???g) dg = 0}. These results turn out to be very useful for calculations on s2 ??? G/H, where G = SO(3) and H??? SO(3) with H ??? SO(2) with ??? : h ??? Lh, as we can use these results to show that there is no quadruple (s, E, j, E#) for SO(3)/H with j analytic in some neighbourhood of 0 such that ??? is a local homomorphism from F# to F. Moreover, we can show that there is more than one solution for the case where s, E and E# are as chosen by Rouviere, if e is does not have to satisfy e(??,??) = e(??,??).

In this thesis we study ideals in the Fourier algebra, A(G), of a locally compact group G. For a locally compact abelian group G, necessary conditions for a closed ideal in A(G) to be weakly complemented are given, and a complete characterization of the complemented ideals in A(G) is given when G is a discrete abelian group. The closed ideals in A(G) with bounded approximate identities are also characterized for any locally compact abelian group G. When G is an arbitrary locally compact group, we exploit the natural operator space structure that A(G) inherits as the predual of the group von Neumann algebra, VN(G), to study ideals in A(G). Using operator space techniques, necessary conditions for an ideal in A(G) to be weakly complemented by a completely bounded projection are given for amenable G, and the ideals in A(G) possessing bounded approximate identities are completely characterized for amenable G. Ideas from homological algebra are then used to study the biprojectivity of A(G) in the category of operator spaces. It is shown that A(G) is operator biprojective if and only if G is a discrete group. This result is then used to show that every completely complemented ideal in A(G) is invariantly completely complemented when G is discrete. We conclude by proving that for certain discrete groups G, there are complemented ideals in A(G) which fail to be complemented or weakly complemented by completely bounded projections.

In this thesis we study ideals in the Fourier algebra, A(G), of a locally compact group G. For a locally compact abelian group G, necessary conditions for a closed ideal in A(G) to be weakly complemented are given, and a complete characterization of the complemented ideals in A(G) is given when G is a discrete abelian group. The closed ideals in A(G) with bounded approximate identities are also characterized for any locally compact abelian group G. When G is an arbitrary locally compact group, we exploit the natural operator space structure that A(G) inherits as the predual of the group von Neumann algebra, VN(G), to study ideals in A(G). Using operator space techniques, necessary conditions for an ideal in A(G) to be weakly complemented by a completely bounded projection are given for amenable G, and the ideals in A(G) possessing bounded approximate identities are completely characterized for amenable G. Ideas from homological algebra are then used to study the biprojectivity of A(G) in the category of operator spaces. It is shown that A(G) is operator biprojective if and only if G is a discrete group. This result is then used to show that every completely complemented ideal in A(G) is invariantly completely complemented when G is discrete. We conclude by proving that for certain discrete groups G, there are complemented ideals in A(G) which fail to be complemented or weakly complemented by completely bounded projections.

In this study, twenty-seven experiments of transport and hydrographic observations were conducted at 9 bridges along the Love River during the period between 2001/12/20 and 2003/9/10. In each experiment repeated measurements were made at each bridge every two hours for a total period of 12 hours. An additional experiment (the 28th experiment) was conducted at 2004/6/16 in the Chihsien Bridge which consists of continuous measurements of velocity, depth and turbidity with a total length of 27 hours. Our results indicate that the Love River is influenced by the incoming tides from the river mouth up to the Dingxin Bridge. The tidal range is approximately 1m during spring tides (10cm during neap tides) at river mouth. The velocity at the Chihsien Bridge has a high frequency variation with a period of approximately 1 hour and amplitude of 20 cm/s during the 28th experiment. Salinity variations are in phase with tides, while turbidity and tides are out of phase. Harmonic analysis of depth, velocity, salinity and turbidity data all indicate that K1 is the principal tidal constituent followed by M2. River transport in the lower estuary is mainly comprised of two parts: tide and river discharge. The tidal induced transport is estimated to be ¡Ó30 CMS and net river discharge is about 1~9 CMS. The upper estuary is affected by two transport mechanisms: agricultural runoff and rain precipitation. After analyzing the measurement results, the transport of the upper branches is estimated to be 0~0.5 CMS during dry seasons and 1~5 CMS during rainy seasons. The agricultural transport reaches its maximum value in January with an estimated rate of 1~2.8 CMS. For the upper branches of Love River, the ratio£\between the hydraulic depth D and hydraulic radius R is found to reach a constant value of 0.9~1.0 when the transport Q is less than 2CMS, and£\is 0.8~0.9 when Q is greater than 2 CMS. The relationship between Q and the section factor TaDb, where T is the channel width, is found to be TD5/3=7.171Q (Dingxin Bridge) and TD5/3=0.744Q (Hougang Bridge) based on Manning formula. Finally, the relationship between Q and D is found to be D=1.811Q0.2981 (Longxin bridge) and D=0.266Q0.256 (Hougang Bridge).

The studies of tidal current and sea level variation in the Taiwan Strait are popular topics in recent years. The sea level data, to be applied to data analysis or model forcing and validation, are mostly observed in the near shore region. It is relative not easier to obtain real tidal data in the offshore area. This study intended to obtain sea level data within Taiwan Strait, using in-situ water depth measurements collected by EK500 of research vessels OR1, OR2 and OR3 during 1989-2003. The basic assumption of this work is that the changes of sea level and topographical depth equal to observed water depth. By using a large set of field measurements, it is possible to get bottom topography such that tidal data can be extracted by harmonic analysis of long-term discrete time series of water depth data. A total of 1513 cruises of water depth data were collected, which account for nearly 6 million samples. These data were screened through a series of criteria for quality control. Firstly, data were plotted cruise by cruise ( longitude vs latitude , longitude vs depth , time vs depth), then reasonable range of time, depth and region were choosed manually. Second, outliers, defined as values greater than 3 standard deviations on 5 point moving mean along the cruise track (or time), were replaced by linear interpolation values. Finally, a 2-minute moving average was applied to the along track time series water depth data. This step was trying to remove the effect of surface waves. The original huge records were reduced to about 550,000 valuable samples for the 1513 cruises data. According to the density distribution of water depth samples in Taiwan Strait, 32 sub-region were selected for topography and harmonic analyses. In each sub-region, the bottom topography was mapped by an optimal interpolation method through a Gaussian weighting function. The radius of Gaussian weighting function applied is 3 time of the distance of grid. Water depth samples subtracted topographical depth of nearby grid to form a set of sea level data ready for harmonic analysis. The phase and amplitude of semi-diurnal tides (M2) and diurnal tides (K1¡BO1) in each sub-region were computed for the 32 regions in Taiwan Strait. The water depth measurements derived sea level variations were compatible with that of a global tidal model (OSU) and a set of moored long-term pressure records in the middle of the strait. Especially, the tidal phase among these results were quite close. However, the tidal amplitudes of water depth data derived were smaller. Sensitivity analysis showed that the errors, differences between OSU model and depth derived sea levels, were small with regions of high density of water depth measurements. Both harmonic derived sea level variations and OSU model predictions indicated a southward propagating tidal wave, which matched with the scenario of Kevin wave propagation in Taiwan Strait. Our analysis also showed that the sea level variations in the northern part of the strait were dominated by M2 and K1 components while the southern part of the strait were dominated by M2 and O1 components.