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In this thesis we study the modular representation theory of diagram algebras, in particular the Brauer and partition algebras, along with a brief consideration of the Temperley-Lieb algebra. The representation theory of these algebras in characteristic zero is well understood, and we show that it can be described through the action of a reflection group on the set of simple modules (a result previously known for the Temperley-Lieb and Brauer algebras). By considering the action of the corresponding affine reflection group, we give a characterisation of the (limiting) blocks of the Brauer and partition algebras in positive characteristic. In the case of the Brauer algebra, we then show that simple reflections give rise to non-zero decomposition numbers. We then restrict our attention to a particular family of Brauer and partition algebras, and use the block result to determine the entire decomposition matrix of the algebras therein.

This thesis addresses the synthesis problem of state deadbeat regulator using state space techniques. Deadbeat control is a linear control strategy in discrete time systems and consists of driving the system from any arbitrary initial state to a desired final state infinite number of time steps. Having described the framework for development of the thesis which is in the form of a lower linear-fractional transformation (LFT), the conditions for internal stability based on the notion of coprime factorization over the set of proper and stable transfer matrices, namely RH, is discussed. This leads to the derivation of the class of all stabilizing linear controllers, which are parameterized affinely in terms of a stable but otherwise free parameter Q, usually known as the Q-parameterization. In this work, the classical Q- parameterization is generalized to deliver a parameterization for the family of deadbeat regulators. Time response characteristics of the deadbeat system are investigated. In particular, the deadbeat regulator design problem in which the system must satisfy time domain specifications and minimize a quadratic (LQG-type) performance criterion is examined. It is shown that the attained parameterization for deadbeat controllers leads to the formulation of the synthesis problem in a quadratic programming framework with Q regarded as the design variable. The equivalent formulation of this objective as a quadratic integral in the frequency domain provides the means for shaping the frequency response characteristics of the system. Using the LMI characterization of the standard H problem, a new scheme for shaping the system frequency response characteristics by minimizing the infinity norm of an appropriate closed-loop transfer function is introduced. As shown, the derived parameterization of deadbeat compensators simplifies considerably the formulation and solution of this problem. The last part of the work described in this thesis is devoted to addressing the synthesis problem of deadbeat regulators in a robust way, when the plant is subject to structured norm-bounded parametric uncertainties. A novel approach which is expressed as an LMI feasibility condition has been proposed and analysed.

A hybrid model for simulating rogue waves in random seas on a large time and space scale is proposed in this thesis. It is formed by combining the derived fifth order Enhanced Nonlinear Schrödinger Equation based on Fourier transform (ENLSE-5F), the fully nonlinear Enhanced Spectral Boundary Integral (ESBI) method and its simplified version. The numerical techniques and algorithm for coupling three models on time scale are provided. Using them, and the switch between the three models during the computation is triggered automatically according to wave nonlinearities. Numerical tests are carried out and the results indicate that this hybrid model could simulate rogue waves both accurately and efficiently. In some cases showed, the hybrid model is more than 10 times faster than just using the ESBI method.

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An aortic dissection is a tear of the intima of the aortic wall that spreads into the media or between the media and adventitia. In addition to the original lumen for blood flow, the dissection creates a new flow channel, the `false' lumen that may cause the artery to narrow or even close over entirely. Aortic dissection is a medical emergency and can quickly lead to death. The mechanical property of the aorta has been described by the strain energy function given by Holzapfel et al. [2000]. The aorta is idealized as an elastic axisymmetric thickwalled tube with 3 layers. We focus on the dissection in media, which is considered as a composite reinforced by two families of fibres. We assume the dissection in the media is axisymmetric. The mathematical model for the dissection is presented. The 2D plane crack problem in linear elastic infinity plane and 2D strip, the axisymmetric crack problem in linear elastic compressible and incompressible tube, the axisymmetric crack problem in an incompressible axisymmetric aorta are applied to obtain solutions to three different problems. And the fluid flow inside the crack has been studied. The 2D plane crack problem in linear elastic infinity plane has been solved analytically. The 2D plane crack problem in linear elastic compressible and incompressible strip is modelled respectively and solved numerically. The models for axisymmetric crack problem in linear elastic compressible and incompressible tube are presented respectively. The numerical solutions for the crack problems are expressed, and the results are analyzed. The mathematical model of the incompressible aorta axisymmetric dissection is given, and the solutions are found numerically. The results change along with the different parameters in the strain energy function, which are analyzed and compared. The fluid flow inside the tear is assumed very thin which is expressed as the lubrication theory. We use the implicit method to model the Stokes equation numerically, and test the crack opening change along with time.

Collapsible channel flows have been attracting the interest of many researchers, because of the physiological applications in the cardiovascular system, the respiratory system and urinary system. The linear stability analysis of the collapsible channel flows in the Fluid-Beam Model can be finalized as a large sparse asymmetric generalized eigenvalue problem, where the stiffness matrix is sparse, asymmetric and nonsingular, and the mass matrix is sparse, asymmetric and singular. The dimensions of the both matrices can reach about ten thousand or more, and the traditional QZ Algorithm is so expensive for this size of eigenvalue problem, due to its large requirement of computational resources and the quite long elapsed time. Unlike the traditional direct methods, the projection methods are much more efficient for solving some specified eigenpairs of the large scale eigenvalue problems, because normally a small subspace is made use of, and the original eigenvalue problem is projected to this small subspace. With this projection, the size of the eigenvalue problem is reduced significantly, and then the small dimensional eigenvalue problem can be easily and rapidly worked out by employing a traditional solver. Combined with a restarting strategy, this can be used to solve large dimensional eigenvalue problem much more rapidly and precisely. So far as we know, the Implicitly Restarted Arnoldi iteration(IRA) is considered as one of the most effective asymmetric eigenvalue solvers. In order to improve the efficiency of linear stability analysis in collapsible channel flows, an IRA method is employed to the linear stability analysis of collapsible channel flows in FBM. A Frontal Solver, which is an efficient solver of large sparse linear system, is also used to replace the process of shift-and-invert transformation. After applying these two efficient solvers, the new eigenvalue solver of collapsible channel flows---Arnoldi method with a Frontal Solver(AR-F), not only gets rid of the restriction of memory storage, but also reduces the computational time observably. Some validating and testing work have been done to variety of meshes. The AR-F can solve the eigenvalues with largest real parts very quickly, and can also solve the large scale eigenvalue problems, which cannot be solved by the QZ Algorithm, whose results have been proved to be correct with the unsteady simulations. Compared with the traditional QZ Algorithm, not only a great deal of elapsed time is saved, but also the increasing rate of the operation numbers is dropped to $O(n)$ from $O(n^3)$ of QZ Algorithm. With the powerful AR-F, the stability problems of refined meshes in collapsible channel flows are no long a barrier to the study. So AR-F is used to solve the eigenvalue problems from two refined meshes of the two different boundary conditions(pressure-driven system and flow-driven system), and the two neutral curves obtained are both revised and extended. This is the first time that IRA is made use of in the problem of fluid-structure interaction, and this is also a critical footstone to adopt a three dimensional model over FBM. Recently, the energy analysis and the energetics are the centre of research in collapsible channel flow. Because the linear stability analysis is much more accurate and faster than the unsteady simulation, the energy solutions from eigenpairs are also achieved in this thesis. The energy analysis with eigenpairs has its own advantages: the accuracy, the timing, the division, any mode and any point. In order to analyze the energy from eigenpairs much more clearly, the energy results with different initial solutions are presented first, then the energy solutions with eigenpairs are validated with those presented by Liu et al. in the pressure-driven system. By using the energy analysis with eigenpairs, much more energy results in flow-dirven system are obtained and analyzed.

This thesis identifies and explores a link between the theory of linear viscoelasticity and the spectral theory of Sturm-Liouville problems. The thesis is divided into five chapters. Chapter 1 gives a brief account of the relevant parts of the theory of linear viscoelasticity and lays the foundation for making the link with spectral theory. Chapter 2 is concerned with the construction of approximate Dirichlet series for completely monotonic functions. The chapter introduces various connections between non-negative measures, orthogonal polynomials, moment problems, and the Stieltjes continued fraction. Several interlacing properties for discrete relaxation and retardation times are also proved. The link between linear viscoelasticity and spectral theory is studied in detail in Chapter 3. The stepwise spectral functions associated with some elementary viscoelastic models are derived and their Sturm-Liouville potentials are explicitly found by using the Gelfand-Levitan method for inverse spectral problems. Chapter 4 presents a new family of exact solutions to the nonlinear integrodifferential A-equation, which is the main equation in a recent method proposed by Barry Simon for solving inverse spectral problems. Starting from the A-amplitude A(t) = A(t, 0) which is determined by the spectral function, the solution A(t, x) of the A-equation identifies the potential q(x) as A(0, x). Finally, Chapter 5 deals with two numerical approaches for solving an inverse spectral problem with a viscoelastic continuous spectral function. In the first approach, the A-equation is solved by reducing it to a system of Riccati equations using expansions in terms of shifted Chebyshev polynomials. In the second approach, the spectral function is approximated by stepwise spectral functions whose potentials, obtained using the Gelfand-Levitan method, serve as approximations for the underlying potential

The concept of causality has been widely studied in econometrics and statistics since 1969, when C. J. Granger published his paper "Investigating causal relations by econometric models and cross-spectral methods". The intuitive basis for his definition of causality is the following: time series Y is causing time series X if the use of the additional information provided by Y improves the forecast of series X. In the present thesis we focus on combining Granger's causality concept with the Singular Spectrum Analysis (SSA) technique. SSA is founded on the idea of transforming the time series into a multidimensional trajectory form (Hankel matrices), Singular Value Decomposition with subsequent projection to a lower-dimensional subspace and diagonal averaging. The main aim of the present thesis is to study the causality concept through SSA prism in details and suggest a novel causality measure, which can be used outside the stationary autoregressive class, which is the framework for Granger's original causality concept. We first apply standard statistical tests directly to simulated data to assess the improvement of forecast quality of bivariate multidimensional SSA (MSSA) of time series X and Y compared with SSA of time series X only. Although the results of performance of these tests are reasonably conclusive, the simulation method is time consuming and, thus, more theoretical understanding is desirable. We solve a fundamental scaling problem of the MSSA approach by introducing so-called linearized MSSA. The linearized MSSA approach shows a way towards a causality measure, calculated from the forecast linear recurrence formula (LRF) coefficients. We finally analyze SSA and (non-linear) bivariate MSSA approach in terms of first order stability analysis under perturbations leading to the construction of a valid suitable measure of causality. The construction of the measure requires some simplifying assumptions in the stability analysis whose validity we verify for both simulated and real data.

We prove that the operator product expansion (OPE), which is usually thought of as an asymptotic short distance expansion, actually converges at arbitrary finite distances within perturbative quantum field theory. The result is derived for the massive scalar field with $\varphi^{4}$-interaction on Euclidean spacetime. This constitutes a generalisation of an earlier result by Hollands and Kopper, which states that the OPE of exactly two quantum fields converges. We also show that the OPE coefficients satisfy factorisation conditions for certain configurations of the spacetime arguments. Such conditions are known to encode information on the algebraic structure of the underlying quantum field theory. Both results rely on modified versions of the renormalisation group flow equations, which allow us to derive explicit bounds on the remainder of these expansions. Within this framework, we also derive a new formula for the perturbation of OPE coefficients, i.e. an equation relating coefficients at a given perturbation order to those of lower order. By iteration of this formula, a new constructive method for the computation of OPE coefficients in perturbation theory is obtained, which only requires the coefficients of the free theory as initial data. Finally, we investigate a strategy to restrict renormalisation ambiguities in quantum field theory via the condition that the OPE coefficients depend analytically on the coupling constant(s) of the respective model. We apply this strategy to the computation of the vacuum expectation value of the stress energy operator in the two dimensional Gross-Neveu model and we obtain a unique prediction for the non-perturbative contribution to this expectation value, which is of the order $\exp(-2\pi/g^{2})$ (here $g$ is the coupling constant). We discuss the possibility that a similar effect, if present in the Standard Model of particle physics, could account for the ''unnatural'' smallness of the cosmological constant.