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The focus of this thesis is the influence of rotation or a magnetic field on the dynamics of pattern formation by thermal convection in a plane layer (the Rayleigh-Bénard problem). Chapter 1 of the thesis provides a short introduction to the theory of equivalent steady-state and Hopf bifurcations on doubly-periodic lattices in the plane. Known results and new observations for Hopf bifurcations on non-rotating and rotating square lattices are summarised. There is also a detailed presentation of new stability results for the Hopf bifurcation on a square superlattice. In Chapter 2 the effect on convection of an imposed vertical magnetic field is discussed. This leads to a detailed analysis of the simplest three-dimensional Hopf/steady-state mode interaction, where the ratio of the critical wavenumbers for oscillatory and steady convection is 1 : √2. Chapters 3, 4 and 5 contain results on pattern selection in rotating Rayleigh-Bénard convection, specifically for low Prandtl number fields. Calculations in the regime where the onset of convection is oscillatory are performed to determine possible forms of convection at onset. The transition (with increasing Prandtl number) from patterns involving oscillatory rolls to those involving steady rolls is influenced by a second 1 : √2 mode interaction. The resulting amplitude equations contain heteroclinic cycles and bursting behaviour in marked contrast to the magnetoconvection case. Finally, a new asymptotic regime of low Prandtl number and rapid rotation is explored. The relevance of this regime to various sets of experimental results is discussed.

A <i>linear space </i>is an incidence structure consisting of a set of points II and a set of lines L in the power set of P such that any two points are incident with exactly one line. We study those finite linear spaces which admit an automorphism group <i>G </i>which is transitive upon the set of lines of the space. Within the set of all linear spaces lies a particularly important subset: the <i>projective planes. </i>Results exist in the literature [Cam04, CP93] classifying the possible minimal normal subgroups of a group <i>G </i>acting line-transitively on a finite projective plane. We rewrite some of these results to deal with <i>components</i> rather than with minimal normal subgroups. We then prove that, if a group <i>G</i> acts on a projective plane which is not Desarguesian, the <i>G</i> does not contain any components. In order to do this we make use of the classification of finite simple groups; our proof consists of examining the different quasisimple groups given in the classification as possible components of <i>G. </i> We also examine the situation where an almost simple group <i>G </i>with socle <i>PSL</i>(3,<i>q</i>) acs line-transitively on a linear space. This fits into the wider program of examining those almost simple groups which can act line-transitively on linear spaces, a program motivated by the result in [CP01]. We are able to give strong information about the line-transitive actions of <i>G</i>.

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The mechanical properties of very small systems are often strikingly different from the properties of everyday objects. As one considers ever-smaller objects, thermal fluctuations, and then quantum fluctuations start to be important. In this thesis I explain some unusual nanoscale mechanical effects, and predict some new effects. The bulk of the thesis is devoted to calculating the forces between bodies that are closely spaced, but not touching. These van der Waals forces have been studied in detail for bodies in thermal equilibrium. Most of the world is not in thermal equilibrium, and van der Waals forces in this regime are very different from their equilibrium cousins. In contrast to equilibrium forces, nonequilibrium forces are much stronger and may show chemical specificity. There is a <i>friction</i> associated with the van der Waals force between bodies in relative motion. When the bodies are at different temperatures, this friction may be negative. Intermolecular forces with one molecule excited are far stronger than ground-state forces and may be attractive or repulsive. Any optical effect in matter modifies the forces between the constituent molecules. The second part of this thesis is on solitonic kinks in fibrillar materials (e.g. polymers, actin bundles, microtubules, carbon nanotubes). All of these materials support stable kinks, and these kinks play an important role in determining the mechanical properties; often more important than the detailed chemical makeup of the materials.

It is proved that Taub-Bolt infillings are double-valued whereas Taub-Nut and Eguchi-Hanson infillings are unique in arbitrary dimensions. In the case of trivial bundles, there are two or no Schwarzschild infillings. The condition of whether a particular type of infilling exists can be expressed as a limitation on squashing through a functional dependence on dimension in each case. The case of the Eguchi-Hanson metric is solved in arbitrary dimension. The Taub-Nut and the Taub-Bolt are solved in four dimensions and methods for higher dimensions are discussed. For the case of Schwarzschild in arbitrary dimension, thermodynamic properties of the two infilling black-hole solutions are discussed and analytic formulae for their masses are obtained using higher order hypergeometric functions. Convexity of the infilling solutions and isoperimetric inequalities involving the volume of the boundary and the volume of the infilling solutions are investigated. In particular, analogues of Minkowski’s celebrated inequality in flat space are found and discussed. In Chapters 3, the Dirichlet problem is studied for an <i>SU </i>(2) x <i>U</i>(1)-invariant <i>S³</i> boundary within the class of self-dual Taub-Nut-(anti) de Sitter metrics. Including complex ones there can be a total of three solutions for the infilling although there will be a unique real solution or no real solution depending on the boundary data - the two radii of the <i>S³</i>. Exact solutions of the infilling geometries are obtained making its possible to find their Euclidean actions as analytic functions of the two radii of the <i>S³</i>-boundary. The case of L < 0 is investigated further. For reasonable squashing of the <i>S³</i>, all three infilling solutions have real-valued actions which possess a “cusp catastrophe” structure with a “catastrophe manifold” that shows that the unique real positive-definite solution dominates. The necessary and sufficient condition for the existence of the positive-definite solution is found as a condition on the two radii of the <i>S³</i>. In Chapter 4, the same boundary-value problem is studied for the Taub-Bolt-anti-de Sitter metrics. Such metrics are obtained from the two-parameter Taub-NUT-anti de Sitter family. The condition of regularity results in two bifurcated one-parameter family. It is found that <i>any</i> axially symmetric <i>S³</i>-boundary can be filled in with at least one solution coming from each of these two branches. The infillings appear or disappear catastrophically in pairs as the values of the two radii of <i>S³</i> are varied; this happens simultaneously for both branches. It is found that the total number in independent infillings is two, six or ten. When the two radii are of the same order and large this number is two. In the isotropic limit, i.e., for round <i>S³</i> this holds for small radii as well. In Chapter 5, the Dirichlet problem is studied within Euclideanised Schwarzschild-anti de Sitter and anti de Sitter metrics, i.e., for an <i>S¹</i> x <i>Sⁿ</i> boundary. For such boundary data there exist two or no black-holes and always a unique anti de Sitter solution. The black holes have strictly positive and negative specific heats (and hence locally thermodynamically stable and unstable respectively). It is shown that for any radius of the cavity, the larger hole can be globally thermodynamically stable above a critical temperature by demonstrating that a phase transition occurs from hot AdS to Schwarzschild-AdS within the cavity. This gives the Hawking-Page phase transition in the infinite cavity limit. It is found that the case of five dimensions is special in that the masses of the two black holes, and hence other quantities of classical and semi-classical interest, can be obtained exactly as functions of cavity radius and temperature. It is also possible in this case to obtain the minimum temperature (below which no black holes exist) and the critical temperature for phase transition as analytic functions of cavity-radius. In Chapter 6, cosmological and instanton solutions are found for <i>CP¹</i> and <i>CP²</i> sigma models coupled to gravity with a possible cosmological constant.

In chapter 3 the relation between two different classes of perturbative non-BPS bi-spinor states of heterotic string theory and certain non-perturbative non-BPS D-brane states of the dual type I' theory is exhibited. The domains of stability of these states as well as their decay products in both theories are determined and shown to agree with the duality map. In chapter 4 the effects of the non-BPS D-instanton in type I theory and its M-theory origin is described. The starting point is the tree-level amplitude for the scattering of two gauge particles in the Hořava-Witten formulation of M-theory. At low momenta this exactly reproduces the corresponding tree-level scattering amplitude of the <I>E</I><SUB>8</SUB> x <I>E</I><SUB>8</SUB> heterotic string theory. After compactification to nine dimensions this amplitude is used to describe the scattering of two massive <I>SO</I>(16) spinor states. The non-BPS D-instanton component of this amplitude is explicitly determined from this expression. In chapter 5 the renormalization group method is used to study tachyon condensation on bosonic D25-brane. The decay of the D25-brane is controlled by a nearby IR fixed point representing D24-branes. The boundary entropy corresponding to the D24-brane tension is calculated in leading order in perturbation theory and agrees with the expected result to an accuracy of 8%. Multicritical behaviour of the IR theory suggests that the end point of the flow represents a configuration of two D24-branes. An analogy with Kondo physics is discussed. Chapter 6 ongoing developments in the context of little string theory and matrix theory.

Measurements by Bernhoeft et al [1] of the small-angle neutron scattering in the high-temperature superconductor YBa<SUB>2</SUB>Cu<SUB>3</SUB>O<SUB>7</SUB>-δ showed increased scattering in the vicinity of the superconducting transition temperature of the order of 100-150 mbarns sr<SUP>-1</SUP> atom<SUP>-1</SUP>. This thesis describes further investigations of the temperature dependence of the small-angle neutron scattering from two high-temperature superconductors YBa<SUB>2</SUB>Cu<SUB>3</SUB>O<SUB>7</SUB>-δ and Bi<SUB>2</SUB>Sr<SUB>2</SUB>CaCu<SUB>2</SUB>O<SUB>8</SUB> where the two predominant problems with these measurements were identified and addressed; namely the large temperature independent 'background' scattering from the superconductor samples, and the drifting of the detector's efficiency. The first part of this thesis is concerned with the preparation, characterisation and preliminary small-angle neutron scattering measurement of YBa<SUB>2</SUB>Cu<SUB>3</SUB>O<SUB>7</SUB>-δ and Bi<SUB>2</SUB>Sr<SUB>2</SUB>CaCu<SUB>2</SUB>O<SUB>8</SUB> with the small-angle neutron scattering instruments at NIST, U.S.A. and Saclay, France. In these measurements no increase in scattering was observed on the absolute scale of that previously observed [1], the upper limit of scattering being 20-30 mbarns st<SUP>-1</SUP> atom<SUP>-1</SUP> for YBa<SUB>2</SUB>Cu<SUB>3</SUB>O<SUB>7</SUB>-δ and 30 mbarns st<SUP>-1</SUP> atom<SUP>-1</SUP> for Bi<SUB>2</SUB>Sr<SUB>2</SUB>CaCu<SUB>2</SUB>O<SUB>8</SUB>. This however does not rule out the possibility that the small-angle neutron scattering observed at T<SUB>c </SUB>(=90K) at Q=0.035A<SUP>-1</SUP> in reference [1] scales with the background as the percentage change here is of the order of 0.5%±0.1% whereas in the measurements presented in this thesis the change is of the order of 1%±0.2%. Further analysis of the scattering at NIST and measurements made at Saclay, France, showed that for both materials there was considerable variation in the wave-vector dependence of the fractional difference in the differential scattering cross-section when two temperatures were compared.

In many real world optimization problems it is essential or desirable to determine the global minimum of the objective function. The subject of this dissertation is a new class of methods that tackle such problems. In particular, we have in mind problems where function evaluations are expensive and no additional information is available. The methods employ radial basis functions that have been proved to be useful for interpolation problems. Examples include thin plate splines and multiquadrics. Specifically, in each iteration, radial basis function interpolation is used to define a utility function. A maximizer of this function is chosen to be the next point where the objective function is evaluated. Relations to similar optimization methods are established, and a general framework is presented that combines these methods and our methods. A large part of the dissertation is devoted to the convergence theory. We show that convergence can be achieved for most types of basis functions without further assumptions on the objective function. For other types, however, a similar results could not be obtained. This is due to the properties of the so-called native space that is associated with a basis function. In particular, it is of interest whether this space contains sufficiently smooth functions with compact support. In order to address this question, we present two approaches. First, we establish a characterization of the native space in terms of generalized Fourier transforms. For many types, for example thin plate splines, this helps to derive conditions on the smoothness of a function that guarantee that it is in the native space. For other types, for example multiquadrics, however, we show that the native space does not contain a nonzero function with compact support. The second approach we present gives slightly weaker results, but it employs some new theory using interpolation on an infinite regular grid.

Chapter 3 reviews anomalies in field theories and the anomaly inflow mechanism in space-filling intersecting brane configurations. We review and discuss dynamics at the intersection domain of the <i>I</i>1 system (whose intersection domain is (1 + 1)-dimensional). We also present a supergravity solution of intersecting <i>D7</i>-branes over (5 + 1) dimensions i.e. the <i>I</i>5 system, and discuss their connections to the conifold in F-theory. Chapter 4 is concerned with the study of entropies of small black holes in IIB string theory, in the presence of higher derivative corrections. In chapter 5 we study <i>D</i>-brane inflation in the warped throat, where a <i>D</i>3-brane open-string modulus plays the role of the inflation. We review the computation of non-perturbative corrections to the inflation potential from holomorphically-embedded stacks of <i>D7-</i>branes, and derive explicit expressions for the case of two non-intersecting sets of Kuperstein embedded <i>D7</i>’s. We present explicit models that satisfies <i>angular stability</i>, and demonstrate how models with an inflation scale exceeding that of SUSY breaking are constructed.