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The focus of this work is on developing optimality criteria corresponding to multiple inference objectives and combining them in compound criteria allowing for finding compromises between different components, especially in the cases of relatively small experiments. In the framework of response surface factorial experiments we take into account the assumption of a potential model misspecification that is expressed in the form of extra polynomial terms that cannot be estimated. Along with obtaining quality estimates of the fitted model parameters, the contamination arising from the model disturbance is desired to be minimised. In addition, in the case of model uncertainty, the model-independent approach of making inference based on `pure error' is to be incorporated. We first present Generalised DP and LP criteria, the components of which correspond to maximising the precision of the fitted model estimates, minimising the joint effect of potentially missed terms and minimising the prediction bias; we also adapt the criteria for use in blocked experiments. In Chapter 5 we develop the Mean Square Error based criteria which, instead of the prediction bias component, include the component minimising the bias of the fitted model parameters that might occur due to the model misspecification. We also provide an alternative way of estimating its value for cases where the originally suggested simulations would be too computationally expensive. An example of a real-life blocked experiment is studied, and we present a set of optimal designs that satised the aims of the experimenters and the restrictions of the experimental setup. Finally, we explore the framework of multistratum experiments; together with adaptation of the MSE-based criteria we provide a flexible design construction and analysis scheme. All of the criteria and experimental settings are accompanied by illustrative examples in order to explore the possible relationship patterns between the criterion components and optimal designs' characteristics, and produce some general practical recommendations.

The problem of designing controllers that are robust with respect to uncertainty leads to questions that are in the areas of operator theory and several complex variables. One direction is the engineering problem of -synthesis, which has led to the study of certain inhomogeneous domains such as the symmetrised polydisc and the tetrablock. The - synthesis problem involves the construction of holomorphic matrix valued functions on the disc, subject to interpolation conditions and a boundedness condition. In more detail, let 1; : : : ; n be distinct points in the disc, and let W1; : : : ;Wn be 2 2 matrices. The -synthesis problem related to the symmetrised bidisc involves nding a holomorphic 2 2 matrix function F on the disc such that F( j) = Wj for all j, and the spectral radius of F( ) is less than or equal to 1 for all in the disc. The -synthesis problem related to the tetrablock involves nding a holomorphic 2 2 matrix function F on the disc such that F( j) = Wj for all j, and the structured singular value (for the diagonal matrices with entries in C) of F( ) is less than or equal to 1 for all in the disc. For the symmetrised bidisc and for the tetrablock, we study the structure of interconnections between the matricial Schur class, the Schur class of the bidisc, the set of pairs of positive kernels on the bidisc subject to a boundedness condition, and the set of holomorphic functions from the disc into the given inhomogeneous domain. We use the theory of reproducing kernels and Hilbert function spaces in these connections. We give a solvability criterion for the interpolation problem that arises from the -synthesis problem related to the tetrablock. Our strategy for this problem is the following: (i) reduce the -synthesis problem to an interpolation problem in the set of holomorphic functions from the disc into the tetrablock; (ii) induce a duality between this set and the Schur class of the bidisc; and then (iii) use Hilbert space models for this Schur class to obtain necessary and su cient conditions for solvability.

We investigate the relationship between a nite group and the set of probabilities associated with evaluating words over the group. Given a nite group, a group element and a word, one may consider the probability that a uniformly random evaluation of the associated verbal mapping yields that particular element of the group. For a xed group, we consider the set of such probabilities obtained by varying over all words and all group elements. It is known that properties of the group are re ected in this associated set of probabilities. For example, if a group is nilpotent then the set of non-zero probabilities associated with that group has a positive lower bound. We seek to further establish the link between a nite group and its set of probabilities. We show how properties of the group, such as nilpotency and verbal subgroup structure are manifested in the properties of its set of probabilities, such as cardinality, the inmum and the corresponding set of accumulation points. We calculate the set of probabilities explicitly for several groups.

We develop a structure theory for nilpotent symplectic alternating algebras. We then give a classification of all nilpotent symplectic alternating algebras of dimension up to 10 over any field F.

Geometric integration concerns the analysis and construction of structure-preserving numerical methods for the long-time integration of differential equations that possess some geometric property, e.g. Hamiltonian or reversible systems. In choosing a structure-preserving method, it is important to consider its efficiency, stability, order, and ability to preserve qualitative properties of the differential system, such as time-reversal symmetry, symplecticity and energy-preservation. Commonly, the symmetric or symplectic Runge--Kutta methods, or the symmetric or G-symplectic linear multistep methods, are chosen as candidates for integration. In this thesis, a class of structure-preserving general linear methods (GLMs) is considered as an alternative choice. The research performed here includes the construction of a set of theoretical tools for analysing derivatives of B-series (a generalisation of Taylor series). These tools are then applied in the development of an a priori theory of parasitism for GLMs, which is used to prove bounds on the parasitic components of the method, and to derive algebraic conditions on the coefficients of the method that guarantee an extension of the time-interval of parasitism-free behaviour. A computational toolkit is also developed to help assist with this analysis, and for other analyses involving the manipulation of B-series and derivative B-series. High-order methods are constructed using a newly developed theory of composition for GLMs, which is an extension of the classical composition theory for one-step methods. A decomposition result for structure-preserving GLMs is also given which reveals that a memory-efficient implementation of these methods can be performed. This decomposition result is explored further, and it is shown that certain methods can be expressed as the composition of several LMMs. A variety of numerical experiments are performed on geometric differential systems to validate the theoretical results produced in this thesis, and to assess the competitiveness of these methods for long-time geometric integrations.

Historical studies of quaternion mathematics have usually placed Sir William Rowan Hamilton's "discovery" of quaternions within the context of the history of modern vector analysis. Exemplary of this technique is the seminal study A History of Vector Analysis by Michael Crowe (1967), in which Hamilton's development of quaternions is seen as an important precursor to the eventual development of contemporary vector calculus. Within Crowe's account, the reader also finds the story of two transitional figures: Peter Guthrie Tait (1831-1901) and William Kingdon Clifford (1845- 1879). Tait is described as a propagator of Hamiltonian methods - someone who wrote about them more succinctly than did Hamilton, and someone who applied them to various topical problems in dynamics. Meanwhile, Clifford is described as a secondary, minor figure - a transitional character whose development of bi-quaternions figures not at all in Crowe's historiography. This thesis redresses those categorizations by effectively "stopping the clock" at 1880, before the "modern" conception of vector analysis had emerged. Following a brief account of the state of British mathematics and science in the first half of the century (1800-1850), the present study focuses on the motivations behind Tait and Clifford's respective engagements with and uses of quaternion mathematics in the second half of the 19th-century. Using the analytical metaphor of "terrains of knowledge" (which is inspired in part by the Wittgensteinian metaphor of language games, and the Strong Program account of finitism in scientific knowledge), I aim to describe the environments - philosophical, institutional, political, and religious - within which Tait and Clifford worked. By describing those "terrains of knowledge", the historian is able to explain why Tait and Clifford, two actors who lived in a similar time and similar place, engaged with the conceptual artifacts of "quaternions" in divergent ways. In the case of Tait, the crucial "terrains of knowledge" to consider in identifying the conceptual environment requisite for him to have used quaternions in the manner that he did includes Cambridge and Belfast mathematics, the University of Edinburgh as an institution in flux (1840-1870), the "science of energy" (1850-1870), and Presbyterian politics and Tait's attack on secularism. In Clifford's case, the salient "terrains of knowledge" include the University of Cambridge and the morphing of symbolical algebra (1860-1870), non-Euclidean geometries in Britain, Clifford's Darwinism, and University College, London as a secularist urban educational institution. When combined, these terrains constitute the varied intellectual environments within which each actor engaged with "quaternion" mathematics, and within which each actor found the resources needed to justify and render meaningful his respective view of that particular concept.

The 3A-axes are one of four famous families of vectors which in union span the acclaimed Monster algebra. Existing in 4-dimensional subalgebras generated by a pair of 2A-axes, they are idempotents of length 8/5. Inside the Monster algebra, these idempotents have a special association with, and are indexed by the 3A-elements of the Monster. It is therefore paramount to understand these axes in order to further understand the Monster. This thesis sets out to uncover the many properties and profound consequences of the 3A-axes. We present three main accomplishments. The first is an axiomatic approach. Properties of the 3A-axes in the Monster algebra are first proven. These properties are then axiomatized as the definition of what we call a standard 3A-axis. The second is on the (2A,3A)-configurations. This is the study of subalgebras of the Monster algebra generated by a 2A- and 3A-axis. The algebra products between a 2A- and 3A-axis for three new cases are discovered. We also present the structures of several subalgebras generated by a 2A- and 3A-axis for the very first time. The third central result is the successful formulation of a methodology for determining all values of inner products between two 3A-axes contained in a very prominent Majorana algebra. There has been much interest especially in Majorana theory in this algebra associated with A12. The inner product classification achieved in this thesis contributes towards this open topic notably in the study of linear spans of axes.

The parabolic algebra was introduced by Katavolos and Power, in 1997, as the weak∗-closed operator algebra acting on L2(R) that is generated by the translation and multiplication semigroups. In particular, they proved that this algebra is reflexive, in the sense of Halmos, and is equal to the Fourier binest algebra, that is, to the algebra of operators that leave invariant the subspaces in the Volterra nest and its analytic counterpart. We prove that a similar result holds for the corresponding algebras acting on Lp(R), where 1 < p < ∞. It is also shown that the reflexive closures of the Fourier binests on Lp(R) are all order isomorphic for 1 < p < ∞. The weakly closed operator algebra on L2(R) generated by the one-parameter semigroups for translation, dilation and multiplication by eiλx, λ ≥ 0, is shown to be a reflexive operator algebra with invariant subspace lattice equal to a binest. This triple semigroup algebra, Aph, is antisymmetric, it has a nonzero proper weakly closed ideal generated by the finite-rank operators, and its unitary automorphism group is R. Furthermore, the 8 choices of semigroup triples provide 2 unitary equivalence classes of operator algebras, with Aph and (Aph)∗ being chiral representatives. In chapter 4, we consider analogous operator norm closed semigroup algebras. Namely, we identify the norm closed parabolic algebra Ap with a semicrossed product for the action on analytic almost periodic functions by the semigroup of one-sided translations and we determine its isometric isomorphism group. Moreover, it is shown that the norm closed triple semigroup algebra AphG+ is the triple semi-crossed product Ap ×v G+, where v denotes the action of one-sided dilations. The structure of isometric automorphisms of AphG+ is determined and AphG+ is shown to be chiral with respect to isometric isomorphisms. Finally, we consider further results and state open questions. Namely, we show that the quasicompact algebra QAp of the parabolic algebra is strictly larger than the algebra CI + K(H), and give a new proof of reflexivity of certain operator algebras,generated by the image of the left regular representation of the Heisenberg semigroup H+.