Fractional Fourier theory has provided a generalization of the classical Fourier transform, and as a result has become a rich area of new concepts and applications. For instance, the implicit relationship that exists between the fractional Fourier transform (FrFT) and time-frequency representations has revealed a continuum of time-frequency (T-F) rotated domains of which the well-known frequency domain is simply a special case. Consequently, the existence of such domains allows for the generalization of Fourier filtering in ways that make it possible to easily realize various time-varying operators. This can in turn lead to more effective signal processing approaches for a range of practical applications. The main focus of this thesis is on the novel concept of fractional Fourier-based filtering. Particularly the work looks into the design of single, as well as multi-stage, systems for the restoration of both simulated and real-world signals. The thesis starts by first examining some of the essential properties of the fractional Fourier transform which relate to filtering. Precisely, the concept of rotated domains in the joint time-frequency plane is elaborated and further exploited for filtering. Results and improvements achieved are demonstrated and discussed through different application examples over the chapters of this thesis.

We consider the problem of asymptotic reconstruction of state and parameter values in dynamical systems that cannot be transformed into a canonical adaptive observer form. A solution to this problem is proposed for certain classes of systems for which parameters enter the model nonlinearly. In addition to asymptotic Lyapunov stability, we provide for these classes, a reconstruction technique based on the notions of weakly attracting sets and nonuniform convergence.

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The Ultra Weak Variational Formulation (UWVF) is a powerful numerical method for the simulation of acoustic, elastic, and electromagnetic waves. Key to its strength is the superior approximation properties of the Trefftz basis of local solutions of the homogeneous form of the equation to be solved. In this thesis we consider time harmonic acoustic wave propagation in two dimensions, as modelled by the Helmholtz equation. We investigate enrichment of the UWVF basis for wave scattering and propagation problems, with applications in geophysics. A new Hankel basis is implemented in the UWVF, allowing greater flexibility than the traditional plane wave basis. We use ray tracing techniques to provide a good a priori choice of direction of propagation for the UWVF basis. A reduction in the number of degrees of freedom required for a given level of accuracy is achieved for the case of scattering by a smooth convex obstacle. The use of the UWVF for forward seismic modelling is considered, simulating wave propagation through a synthetic sound speed profile of the subsurface of the Earth. The practicalities of implementation in a domain of highly varying sound speed are discussed, and a ray enhanced basis is trialled. Wave propagation from a source on the interior of the domain is simulated, representative of an explosive sound source positioned at depth. The UWVF typically has difficulties representing the inhomogenous Helmholtz equation. An augmentation to the UWVF called the Source Extraction UWVF is presented which allows the superior approximation properties of the Trefftz basis to be maintained.

Spaces of real valued functions form many examples in the theory of ordered vector spaces and vector lattices. For example the space of real -valued polynomials on a bounded subset of the reals is not in general a vector lattice but does have the Riesz interpolation property (RIP). Another example of an ordered vector space which is in general not a lattice is the space of differentiable functions, again this space has the Riesz decomposition property (RDP) which is equivalent to the RIP for ordered vector spaces. In this thesis we replace the real -valued function versions of these spaces with Banach lattice-valued functions with appropriate definitions and investigate when they have the RIP/RDP. We also consider when the Banach lattice-valued polynomials form a vector lattice. Spaces of continuous real-valued functions on a (locally) compact Hausdorff space are very important in Banach lattice theory. Generalisations to Banach lattice-valued functions have already been made and many analogous results to the real case have been proved. Including some extension and separation results. In the thesis this space is further generalised by considering functions which are continuous with respect to the weak topology on the Banach lattice instead of the norm topology. These spaces of functions may not be Banach lattices in general in contrast to the norm continuous versions which always are. We provide conditions under which it will be a Banach lattice and prove several other desirable properties for a Banach lattice under varying conditions. We also give some extension and separtion results under fairly restrictive conditions.

Hilbert's ε-operator is a powerful logical instrument which can be used to simplify-the formulation of logical calculi and mathematical theories and to facilitate metamathematical investigations of the predicate calculus. The purpose of this thesis is to investigate the nature of the c-operator and to demonstrate the useful role it can play in metamathematica and in the formulation of mathematical theories. In chapter I a semantic interpretation is given to first-order languages containing the ε-symbol (ε-languages). By defining the notion of a logical closure, we establish an abstract model-theoretic property of E-languages which provides new proofs of the compactness theorems for e-languages and c-free languages. These proofs are model-theoretic in the sense that they do not depend on a particular choice of axioms and rules of inference for the languages concerned. Furthermore, this abstract property is used to establish the completeness of the ε-calculus which is defined in this chapter. In chapter II a new system of natural deduction is defined, and with the help of the ε-symbol the equivalence of this system and the predicate calculus is established. This result, which is analogous to Gentzen's Hauptsatz, provides new proofs of Hilbert's E-theorems and Herbrand's theorem. Also the ε-symbol is used to explain the conditions which must be imposed on the rule of existential instantiation in systems of natural deduction. In chapter IQ a proof is given of the eliminability of the c-symbol for E-calculi which contain the axiom schema Vx(A<->B) -> εxA = εxB. This result, which constitutes a strengthened form of Hilbert's second E-theorem, is then used to clarify the role which the ε-symbol plays in axiomatic set theory, particularly in connection with the axiom of choice.

In this thesis we explore (1,0) supersymmetric theories in six dimensions. The first part of the thesis focuses on the investigation of supersymmetric solutions of (1,0) six dimensional supergravity theory coupled to any number of tensor, vector and scalar multiplets. The methodology used to solve the Killing spinor equations will be based on the spinorial geometry technique. Therefore, we begin by giving details of the spinorial geometry approach in the first chapter. In the chapter that follows six dimensional supergravity coupled to tensor, vector and scalar multiplets is described. Once we have given details of the theory under consideration the solutions to the Killing spinor equations are discussed in some detail. In particular, we find that there are backgrounds preserving 1, 2, 3, 4 and 8 supersymmetries broadly falling into two cases; those with Killing spinors that have compact isotropy groups and those with non-compact isotropy groups. We then discuss the integrability conditions of the Killing spinor equations. In the fourth chapter we analyse the supersymmetric near horizon geometries of (1,0) six dimensional supergravity coupled to arbitrary number of tensor and scalar multiplets. In order to do this we make use of Gaussian null coordinates as well as the solutions of the Killing spinor equations. We find that there are two classes of near horizon geometries. One class is isometric to R1;1 S, where S is a suitable 4-manifold, and the other class is isometric to AdS3 3, where 3 is a homology 3-sphere. In the final chapter we investigate a more recent development, namely (1,0) superconformal theories in six dimensions. In particular we find the BPS solutions of (1,0) superconformal theory in all cases. In addition, we analyse the half supersymmetric solutions to some specic models in detail and give examples of string and 3-brane solutions.

This thesis studies statistical properties of intermittent maps. We obtain three new results. First we use an Ulam-type discretization scheme to provide {\em{pointwise}} approximations for invariant densities of interval maps with a neutral fixed point. We prove that the approximate invariant density converges pointwise to the true density at a rate $C^*\cdot\frac{\ln m}{m}$, where $C^*$ is a computable fixed constant and $\frac{1}{m}$ is the mesh size of the discretization. We then study intermittent maps in a random setting. In particular, we study a random map $T$ which consists of intermittent maps $\{\tau_{k}\}_{k=1}^{K}$ and a position dependent probability distribution $\{p_{k,\varepsilon}(x)\}_{k=1}^{K}$. We prove existence of a unique absolutely continuous invariant measure (ACIM) for the random map $T$. Moreover, we show that, as $\varepsilon$ goes to zero, the invariant density of the random system $T$ converges in the $L^{1}$-norm to the invariant density of the deterministic intermittent map $\tau_{1}$. The outcome of Chapter \ref{chapACIM} contains a first result on stochastic stability, in the strong sense, of intermittent maps. Finally, we study the problem of correlation decay of random map built from finitely many intermittent maps with a common neutral fixed point. Using a Young-tower technique, we show that the map with the fastest relaxation rate dominates the asymptotics. In particular, we prove that the rate of correlation decay for the annealed dynamics of the random map is the same as the {\em sharp rate} of correlation decay for the map with the fastest relaxation rate.

This thesis develops a new efficient and robust approximation framework, the Generalised Fourier B-spline (GFBS) method, for option pricing in the setting of continuous-time asset models from the family of exponen- tial semi martingale processes. The GFBS method introduces B-spline in- terpolation theory to derivative pricing to provide an accurate closed-form representation of the option pricing under an inverse generalised Fourier transform. The GFBS methodology is developed through a series of ap- plications that explore the interplays between financial mathematics and actuarial science. The first application considered is the pricing of non-life reinsurance con- tracts in the presence catastrophe bonds. Using the generalised Fourier transform, an appropriate financial pricing formula is derived, and the fast- and fractional fast Fourier transforms are used to evaluate prices. This methodology is extended to a wider range of reinsurance contracts, in- cluding individual excess of loss reinsurance with reinstatements. The GFBS method is then derived in the context of pricing European op- tions. This is a new efficient and robust framework for option pricing under continuous-time asset models from the family of exponential semimartin- gale processes. The GFBS method introduces B-spline interpolation the- ory to derivative pricing to provide an accurate closed-form representa- tion of the option price under an inverse generalised Fourier transform. The GFBS method is extremely fast and accurate and is demonstrated to be more efficient than existing numerical methods. This suggests a wide range of applications, including the use of more realistic asset mod- els in high frequency trading. Examples considered include pricing under a range of asset models, including stochastic volatility and jump diffusion, computation of the Greeks, and the inverse problem of cross-sectional cal- ibration. Next, the GFBS method is shown to be applicable in the space of exotic derivatives for pricing discrete lookback options for which, in general, no closed-form pricing formula exist. Using B-spline interpolation, an accu- rate closed-form representation of the lookback option price is obtained under an inverse generalised Fourier transform. This provides lookback option prices across the quantum of strike prices with greater efficiency than for a single strike price under existing methods. An explicit representation for the characteristic function of the maximum of a discretely observed stochastic process is derived, which provides a significant improvement in terms of numerical efficiency over the Spitzer- recurrence formula. This is of fundamental importance and could have a wide range of applications where the Spitzer formula is utilised. Several examples are considered covering a range of asset models that exhibit closed-form solutions for the price of a European option, which is required as an input to the GFBS pricing method as presented here. Finally, the GFBS method for discrete lookback options is extended to the case where the underlying asset model is less tractable and does not exhibit a closed-form solution for the price of European options. This is an important result that opens up the possibility of using more realistic asset models when pricing discrete lookback option.

This thesis consists of three chapters. In Chapter one, we introduce some notions and definitions for basic concepts of the theory of integrable bi-Hamiltonian systems. Brief statements of several open problems related to our main results are also mentioned in this part. In Chapter two, we applied the so-called Jordan-Kronecker decomposition theorem to study algebraic properties of the pencil P generated by two constant compatible Poisson structures on a vector space. In particular, we study the linear automorphism group GP that preserves P. In classical symplectic geometry, many fundamental results are based on the symplectic group, which preserves the symplectic structure. Therefore in the theory of bi-Hamiltonian structures, we hope GP also plays a fundamental role. In Chapter three, we study one of the famous Poisson pencils which is sometimes called 'argument shift pencil'. This pencil is defined on the dual space g * of an arbitrary Lie algebra g. This pencil is generated by the Lie-Poisson bracket { , } and constant bracket { , }a for a ε g * . Thus we may apply the Jordan-Kronecker decomposition theorem to introduce the so-called Jordan-Kronecker invariants of a finite-dimensional Lie algebra g. These invariants can be understood as the algebraic type of the canonical Jordan-Kronecker form for the 'argument shift pencil' at a generic point. Jordan-Kronecker invariants are found for all low-dimensional Lie algebras (dim g ≤ 5) and can be used to construct the families of polynomials in bi-involution. The results are found to be useful in the discussion of the existence of a complete family of polynomials in bi-involution w.r.t. these two brackets { , } and { , }a.