A monogenic quasigroup is one generated by a single element, and as such is not just non-associative but in general non power associative. We show that monogenic quasigroups or loops with various specified characteristics can exist, by demonstrating constructions and by giving examples. This often involves completing an associated partial latin square, or demonstrating that a completion is possible. It is shown that for every order n ≥ 4 there are monogenic quasigroups generated by each of any m ≤ n of their elements, and similarly for monogenic loops (n ≥ 6 , 2 ≤ m ≤ n −1). Any element in a quasigroup must have its powers unambiguous and distinct (called good) up to some degree j ≥ 2, and unambiguous but not necessarily distinct (called clear) up to some degree k ≥ j. The conditions for the existence of a quasigroup of order n having a generator with a good j th and clear k th power are determined. A monogenic quasigroup may be said to be g-good if every element has a good g th power. An algorithm for finding examples based on diagonally cyclic latin squares is developed, and a computer program used to find comprehensive solutions for g ≤ 16 and odd orders n ≤ 95 (and patchily to g = 17, n = 111), with particular reference to the lowest n affording a solution for any g. A maximally non power associative quasigroup has every element with all its bracketings up to some length distinct. A diagonally cyclic quasigroup of order 23 with all 23 products of length ≤ 5 distinct for every element is displayed,as is one of order 63 with 63 of the 65 bracketings up to length 6 distinct for each element. Properties of direct products of monogenic quasigroups, and the significance of parastrophy and isotopy, are considered. The existence or not of monogenic versions of particular types of quasigroups and loops (for example, totally symmetric, inverse property, entropic, Bol and Moufang, among others) is also explored.

In this thesis we consider four families of nonlinear recurrences which can be shown to either fit into Fomin and Zelevinsky's framework of cluster algebras, or the more general setting of Laurent phenomenon algebras given recently by Lam and Pylyavskyy. It then follows that each family of recurrences we study possesses the Laurent property. Our main interest lies in the linearisability and Liouville integrability of the maps defined by these families. We prove that three of the families are linearisable. Firstly, we study examples arising in the context of cluster algebras and provide a detailed survey of recent results of Fordy and Hone, with the aim to develop the understanding of Liouville integrability for odd order examples of this type. Following this, we extend the results of Heideman and Hogan, to show that their family of nonlinear recurrences is linearisable for general initial data. The third order example from this family of recurrences admits a different generalisation of a new family of nonlinear recurrences for which we also show the general case to be linearisable. We also present a connection with the dressing chain which provides a generating function for the first integrals for recurrences of this type. Lastly we study a family of Somos-type recurrences which is not linearisable. However we present the method of finding the Lax representation from which we can generate first integrals and show that the examples of recurrences studied here, arising in the context of cluster algebras, are Liouville integrable.

Inner ideals of simple locally finite dimensional Lie algebras over an algebraically closed field of characteristic 0 are described. In particular, it is shown that a simple locally finite dimensional Lie algebra has a non-zero proper inner ideal if and only if it is of diagonal type. Regular inner ideals of diagonal type Lie algebras are characterized in terms of left and right ideals of the enveloping algebra. Regular inner ideals of finitary simple Lie algebras are described. Inner ideals of some finite dimensional Lie algebras are studied. Maximal inner ideals of simple plain locally finite dimensional Lie algebras are classified.

Riemann-Hilbert analysis has become an essential tool in integrability for handling the most difficult asymptotic problems. This thesis demonstrates exactly this, by applying techniques in Riemann-Hilbert analysis to problems in random matrix theory and quantum information. Using an orthogonal polynomial approach, we generate the asymptotics of a. partition function of a random unitary matrix model with essential singularitics in the weight. Then turning our attention to a related partition function of a random Hermitian matrix model, again with essential singularities in the weight, we show that a double scaling limit exists and that the asymptotics are described by a Painleve III transcendent in this double scaling limit. Following this, we use ideas in Riemann-Hilbert theory related to integrable operators to rigorously calculate the entanglement entropy of a disjoint subsystem in a quantum spin chain. We then illustrate the implications of this result, by showing how the methods can be applied to entropy calculations of other disjoint subsystems in the quantum spin chain.

Forsythe and Leibler presented the first research, in 1950, showing how a matrix could be inverted using Monte Carlo (MC) methods. West and Sobol extended this research by presenting MC algorithms to give statistical estimates for the elements of the inverse matrix, or for the components of the solution vector of a system of linear algebraic equations (SLAE). This algorithm uses a Markov chain MC method to generate a rough approximation to the inverse matrix and then rapidly improves the accuracy of the rough inverse using an iterative refinement scheme. Further results are presented comparing the performance of the sparse hybrid MC algorithm with other methods for producing inverse matrices.

this thesis we investigate questions on convolution operators and function algebras, as well as generalising a result from from function algebras to the noncommutuative setting. After detailing the necessary background in Chapter 1, Chapter 2 deals with convolution operators on Banach lattices. \~le start by describing certain Banach lattices of functions defined on a locally compact abelian group G which generalises a class of Banach lattices introduced in Johansson (2008).

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This thesis is devoted to providing a detailed description and construction of intertwining operators related to L2-type spaces in terms of representations of the affine group Aff. We review numerous connections between unitary operators, provide decompositions of the wavelet transforms from the affine group. The group naturally acts by the quasi- regular representation on the space L2 (R) of square integrable functions on the real line. The Hardy space H2(R) is an irreducible invariant subspace under such an action. A eo- adjoint representation of this group spatially splits into irreducible components supported on the orbits, which turns out to be half-lines and {0}. The intertwining operator between quasi-regular and co-adjoint representations turns out to be the Fourier transform. This provides a background for wavelets technique in the theory of complex operators. We analyze the construction and origin of unitary operators describing the structure of the space of continuous wavelet transforms inside the space of all square integrable function on Aff with the left Haar measuer dv- from the viewpoint of induced representations. We show that these operators are intertwining operators among pairs of induced representations of the affine group Aft. A characterization of the space of wavelet transforms using the Cauchy-Riemann type equations is given.

We investigate an alternative formulation of quantum field theory that elevates the Wilson- Zimmermann operator product expansion (OPE) to an axiom of the theory. We observe that the information contained in the OPE coefficients may be straightforwardly repackaged into "vertex operators". This way of formulating quantum field theory has quite obvious similarities to the theory of vertex algebras. As examples of this framework, we discuss the free massless boson in D dimensions and the massless Thirring model. We set up perturbation theory for vertex algebras. We discuss a general theory of perturbations of vertex algebras, which is similar to the Hochschild cohomology describing the deformation theory of ordinary algebras. We pass on to a more explicit discussion by looking at perturbations of the free massless boson in D dimensions. The perturbations we consider correspond to some interaction Lagrangian P(<p) = A Cp if. We construct the perturbations by exploiting the associativity of the vertex operators and the field equation in perturbative form. We develop a set of graphical rules that display the vertex operators as certain multiple series reminiscent of the hypergeometric series.

The Boltzmann equation (BE) is a commonly used tool for the study of non-equilibrium many particle systems. It has been introduced in 1872 by Ludwig Boltzmann and has been widely generalized throughout the years. Today it is commonly used in physical applications, from the study of ordinary fluids to problems in particle Cosmology where Quantum Field Theoretical techniques are essential. Despite its numerous experimental successes, the conceptual basis of the BE is not entirely clear. For instance, it is well known that it is not a fundamental equation of physics like, say, the Heisenberg equation (HE). A natural question then arises whether it is possible to derive the BE from physical first principles, i.e. the Heisenberg equation in Quantum Field Theory. In this work we attempted to answer this question and succeeded in deriving the BE from the HE, thus further clarifying its conceptual status. In particular, the results we have obtained are as follows. Firstly, we establish the non-perturbative validity of what we call the "pre-Boltzmann equation". The crucial point here is that this latter equation is equivalent to the Heisenberg equation. Secondly, we proceed to consider various limits of the pre-Boltzmann equation, namly the "low density" and the "weak coupling" limits, to obtain two equations that can be considered as generalizations of the BE. These limits are always taken together with the "long time" limit, which allows us to interpret the BE as an appropriate long time limit of the HE. The generalization we obtain consists in additional "correction" terms to the usual Boltzmann collision factor, and can be associated to multiple particle scattering. Unlike the pre-Boltzmann equation, these latter results are only valid pertubatively. Finally, we briefly consider the possibility to extend these results beyond said limits and outline some important aspects in this case.