The aim of this work is to investigate the properties of approximations obtained by translates of radial basis functions. A natural progression in the discussion starts with an iterative refinement scheme using a strictly positive definite inverse multiquadric. Error estimates for this method are greatly simplified if the inverse multiquadric is replaced by a strictly conditionally positive definite polyharmonic spline. Such error analysis is conducted in a native space generated by the Fourier transform of the basis function. This space can be restrictive when using very smooth basis functions. Some instances are discussed where the native space of can be enlarged by creating a strictly positive definite basis function with comparable approximating properties to , but with a significantly different Fourier transform to . Before such a construction is possible however, strictly positive definite functions in d for d < with compact support must be examined in some detail. It is demonstrated that the dimension in which a function is positive definite can be determined from its univariate Fourier transform. This work is biased towards the computational aspects of interpolation, and the theory is always given with a view to explaining observable phenomena.

The generalization of Hamiltonian mechanics to covariant Hamiltonian eld theory on multiphase space is given an accessible exposition as a practical method covering various topics, and a classical multiphase space BRST formalism is developed for systems with symmetries and applied to a system with secondary constraints (Yang-Mills).

In this thesis we consider three problems related to colourings of graphs and one problem in extremal graph theory. Let $G$ be a connected graph with $n$ vertices and maximum degree $\Delta(G)$. Let $R_k(G)$ denote the graph with vertex set all proper $k$-colourings of $G$ and two $k$-colourings are joined by an edge if they differ on the colour of exactly one vertex. Our first main result states that $R_{\Delta(G)+1}(G)$ has a unique non-trivial component with diameter $O(n^2)$. This result can be viewed as a reconfigurations analogue of Brooks' Theorem and completes the study of reconfigurations of colourings of graphs with bounded maximum degree. A Kempe change is the operation of swapping some colours $a$, $b$ of a component of the subgraph induced by vertices with colour $a$ or $b$. Two colourings are Kempe equivalent if one can be obtained from the other by a sequence of Kempe changes. Our second main result states that all $\Delta(G)$-colourings of a graph $G$ are Kempe equivalent unless $G$ is the complete graph or the triangular prism. This settles a conjecture of Mohar (2007). Motivated by finding an algorithmic version of a structure theorem for bull-free graphs due to Chudnovsky (2012), we consider the computational complexity of deciding if the vertices of a graph can be partitioned into two parts such that one part is triangle-free and the other part is a collection of complete graphs. We show that this problem is NP-complete when restricted to five classes of graphs (including bull-free graphs) while polynomial-time solvable for the class of cographs. Finally we consider a graph-theoretic version formulated by Holroyd, Spencer and Talbot (2007) of the famous Erd\H{o}s-Ko-Rado Theorem in extremal combinatorics and obtain some results for the class of trees.

In some approximation problems, sampling from the target function can be both expensive and time-consuming. It would be convenient to have a method for indicating where the approximation quality is poor, so that generation of new data provides the user with greater accuracy where needed. In this thesis, the author describes a new adaptive algorithm for Radial Basis Function (RBF) interpolation which aims to assess the local approximation quality and adds or removes points as required to improve the error in the specified region. For a multiquadric and Gaussian approximation, one has the flexibility of a shape parameter which one can use to keep the condition number of the interpolation matrix to a moderate size. In this adaptive error indicator (AEI) method, an adaptive shape parameter is applied. Numerical results for test functions which appear in the literature are given for one, two, and three dimensions, to show that this method performs well. A turbine blade design problem form GE Power (Rugby, UK) is considered and the AEI method is applied to this problem. Moreover, a new multilevel approximation scheme is introduced in this thesis by coupling it with the adaptive error indicator. Preliminary numerical results from this Multilevel Adaptive Error Indicator (MAEI) approximation method are shown. These indicate that the MAEI is able to express the target function well. Moreover, it provides a highly efficient sampling.

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In this thesis, we introduce two different methods for determining noise covariance matrices in order to improve the stability and accuracy in state estimation and output prediction of discrete-time linear time varying (LTV) and nonlinear state space systems. The first method is based on the auto-covariance least squares (ALS) method, where the noise covariance matrices can be estimated by establishing a linear relationship between noise covariances and correlations of innovation sequence, hence solving a linear least squares problem. For LTV systems, we propose a new ALS algorithm that does not involve any approximations in the formulation. Our new ALS algorithm has fewer parameters to determine and can provide more accurate noise covariance estimation even when the historical output measurement window is not sufficiently long, comparing to an existing method. In addition to the noise covariance estimates, our ALS algorithm can also provide the estimate of the initial state error covariance, which is required by most state estimation methods. For higher-order systems, we also provide a much faster and less memory demanding formulation by splitting large Kronecker products with sums of smaller Kronecker or Schur products. For nonlinear systems, we have to approximate nonlinear parts as time-varying matrices by linearizing the nonlinear function around current state estimates. In addition to the extended Kalman Filter (EKF), our ALS algorithm also uses moving horizon estimation (MHE) to estimate the system state. MHE guarantees stability, is able to add state constraints and provides more accurate state estimates and local linearizations around the current state than the EKF. The second method is based on expectation maximization (EM), where the noise covariance matrices are determined by recursively maximizing the likelihood of covariance matrices, given output measurements. In our method, the noise covariance matrices are estimated using a semi-definite programming (SDP) solver, so that the results are more accurate and guaranteed to be positive definite. We propose a new EM algorithm that, combined with MHE and full information estimation (FIE) rather than a Kalman-based filter/smoother, allows the addition of state constraints, provides stable and more accurate estimates, so that the performance of noise covariance estimation can be significantly improved. Finally, we apply our noise covariance estimation methods to ocean wave prediction for the control of a wave energy converter (WEC), in order to approach optimal efficiency of wave energy extraction. We use a state space model representation for an autoregressive (AR) process, combined with noise covariance estimation, to simulate wave height forecasting based on data recorded at Galway Bay, Ireland. The simulation returns good wave predictions. Compared to existing wave prediction methods, our model has fewer parameters to tune and is able to provide more stable and accurate wave predictions by using a Kalman-based filter combined with the ALS or EM method.

The study of Complex Systems focuses on how interactions of constituents within a system, individually or grouped into clusters, produce behavioral patterns locally or globally and how these interact with the external environment. Over the last few decades the study of Complex Systems has gone through a growing rate of interest and today, given a sufficiently big set of data, we are able to construct comprehensive models describing emerging characteristics and properties of complex phenomena transcending the different domains of physical, biological and social sciences. The use of network theory has shown, amongst others, a particular t in describing statical and dynamical correlations of complex data sets because its ability to deal not only with deterministic quantities but also with probabilistic methods. A complex system is generally an open system flexible in adapting to variable external conditions in the way that it exchanges information with environment and adjusts its internal structure in the process of self-organization. Moreover, it has been shown how real world phenomena that are represented by complex systems display interesting statistical properties such as power-law distributions, long-range interactions, scale invariance, criticality, multifractality and hierarchical structure. In the era of big data where effort is largely put to collect large data sets carrying relevant information about given phenomena to be studied and analysed, the interesting field of quantitative semantics, e.g. dealing with information expressed in natural language, is becoming more and more relevant particularly in the social sciences. However, recent studies are expanding these techniques to become a tool for structuring and organising information across a number of disparate disciplines. In this Thesis I propose a methodology that (i) extracts a structured complex data set from large corpora of descriptive language sources and efficiently exploits the power of quantitative semantics techniques to map the essence of a complex phenomena into a network representation, and (ii) combines such induced knowledge network with a graph theoretical framework utilising a number of graph theory tools to study the emerging properties of complex systems. Thus, leveraging on developments in Computational Linguistics and Network Theory, the proposed approach builds a graph representation of knowledge, which is analyzed with the aim of observing correlations between any two nodes or across clusters of nodes and highlights emerging properties by means of both topological structure analysis and dynamic evolution, i.e. the change in connectivity. Under this framework I will provide two real-world applications: - The fist application deals with the creation of a structured network of biomedical concepts starting from an unstructured corpus of biological text-based data set (peer reviewed articles) and next it retrieves known pathophysiological Mode of Actions by applying a stochastic random-walk measure and finds new ones by meaningfully selecting and aggregating contributions from known bio-molecular interactions. By exploiting the proposed graph-theoretic model, this approach has proven to be an innovative way to find emergent mechanism of actions aimed at drug repurposing where existing biologic compounds originally intended to deal with certain pathophysiologic actions are redirected for treating other type of clinical indications. - The second application consists of a representation of a finnancial and economic system through a network of interacting entities and to devise a novel semantic index influenced by the topological properties of agglomerated information in a semantic graph. I have shown how it is possible to fully capture the dynamical aspects of the phenomena under investigation by identifying clusters carrying in uflential information and tracking them over time. By computing graph-based statistics over such clusters I turn the evolution of textual information into a mathematically well-defined, multivariate time series, where each time series encodes the evolution of particular structural, topological and semantic properties of the set of concepts previously extracted and filtered. Eventually an autoregressive model with vectorial exogenous inputs is defined, which linearly mixes previous values of an index with the evolution of other time series induced by the semantic information in the graph. The methodology brie y described above concludes the contribution of my research work in the field of Complex Systems and it has been instrumental in successfully defining a graph-theoretical model for the study of drug repurposing [1] and the construction of a framework for the analysis of financial and economic unstructured data (see chapter 6).