Current high-resolution wavenumber processors are either applicable only to linear equi-spaced arrays or require a manifold search over the ideal array response. As a result of this they exhibit a loss of resolution, bias and increased variance in bearing estimates when the receiving array is subjected to spatial sampling errors. This thesis has addressed the nature of these problems and proposed signal processing algorithms which are robust to the spatial sampling errors. A review of current super-resolution methods is included, explaining why each exhibits a performance degradation. An eigen decomposition of the array cross-spectral matrix is shown to retain information about the spatial sampling process which can be then made available after suitable processing. The required procedure involves rotating all the principal eigenvectors in the signal subspace until they have elements of equal magnitude. Gradient search techniques are derived which can be applied to solve the resulting non-linear equations. A novel matrix notation is introduced which allows the non-linear equations to be written more concisely, this in turn leads to their solution by constrained Lagrangian optimization and several new algorithms are proposed. The minimization equations are then reformulated as a maximization which enables all the eigenvectors to be rotated simultaneously, as opposed to' pairwisc or individually in the minimization case. These formulae are generalized to allow rotating vectors without pre-calculation of the signal subspace but using the cross-spectral and data matrices directly. Extensive simulations have been performed comparing the new methods with similar previous work, MUSIC and the Cramer-Rao lower bound. Air acoustic experiments on a 16 element array have also been performed to verify the practical implementation and evaluate the algorithms performance with a deformed line array using real data.

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1-D Multivariable system theory has been developed richly over the past fifty years using various approaches. The classical approach includes the matrix fraction description (MFD), the state-space approach etc., while the behavioural approach is relatively new. Nowadays, however there is an enormous need to develop this theory for systems where information depends on more than one independent variable i.e. the n-D system theory (n ≥ 2), due to the vast number of applications for these kind of systems. By contrast to the 1-D system theory, the n-D system theory is less developed and its main aspects are not yet complete, where generalising the results from 1-D to n-D has proved to be not straight forward nor smooth. This could be attributed to the n-D polynomial matrices which are the basic elements used in the analysis of n-D systems. n-D polynomial matrices are more difficult to manipulate when compared to the 1-D polynomial matrices used in the analysis of 1-D systems, because the ring of n-D polynomials to which their elements belong does not possess many of the favourable properties which the ring of 1-D polynomials possesses. The work proposed in this thesis considers the Rosenbrock system matrix and the matrix fraction description approaches to the study of n-D systems.

By studying the representation theory of a certain infinite p-group and using the generalised characters of Hopkins, Kuhn and Ravenei we find useful ways of understanding the rational Morava E-theory of the classifying spaces of general linear groups over finite fields. Making use of the well understood theory of formal group laws we establish more subtle results integrally, building on relevant work of Tanabe. In particular, we study in detail the cases where the group has dimension less than or equal to the prime p at which the E-theory is localised.

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Let \(G\) be a simple linear algebraic group over the algebraically closed field \(k\). Assume \(p\) = char \(k\) > 0 is good for \(G\) and that \(G\) is defined and split over the prime field \(\char{bbold10}{0x46}_p\). For a power \(q\) of \(p\), we write \(G(q)\) for the Chevalley group consisting of the \(\char{bbold10}{0x46}_q\)-rational points of \(G\). Let \(F : G \rightarrow G\) be the standard Frobenius morphism such that \(G^F\)= \(G(q)\). Let \(B\) be an \(F\)-stable Borel subgroup of \(G\); write \(U\) for the unipotent radical of \(B\) and \(\char{eufm10}{0x75}\) for its Lie algebra. We note that \(U\) and \(\char{eufm10}{0x75}\) are \(F\)-stable and that \(U(q)\) is a Sylow \(p\)-subgroup of \(G(q)\). We study the adjoint orbits of \(U\) and show that the conjugacy classes of \(U(q)\) are in correspondence with the \(F\)-stable adjoint orbits of \(U\). This allows us to deduce results about the conjugacy classes of \(U(q)\). We are also interested in the adjoint orbits of \(B\) in \(\char{eufm10}{0x75}\) and the \(B(q)\)-conjugacy classes in \(U(q)\). In particular, we consider the question of when \(B\) acts on a \(B\)-submodule of \(\char{eufm10}{0x75}\) with a Zariski dense orbit. For our study of the adjoint orbits of \(U\) we require the existence of \(B\)-equivariant isomorphisms of varieties \(U/M \rightarrow\) \(\char{eufm10}{0x75}\)/\(\char{eufm10}{0x6d}\), where \(M\) is a unipotent normal subgroup of \(B\) and \(\char{eufm10}{0x6d}\) = Lie\(M\). We define relative Springer isomorphisms which are certain maps of the above form and prove that they exist for all \(M\).