No description available.

PART I In a recent paper Wu, T.Y. & Whitney, A.K., the authors studied optimum shape problems in hydrodynamics. These problems are stated in the form of a singular integral equation depending on the unknown shape and an unknown singularity distribution; the shape is then to be determined so that some given performance criterion has to be {maximized/minimized} In the optimum problem to be studied in this part we continue to assume that the state equation is a linear integral equation but we generalize the Wu & Whitney theory in two different ways. This method is applied to functional of quadratic form and a necessary condition for the extremum to be a minimum is derived. PART II The purpose of this part is to evaluate the optimum shape of a two-dimensional hydrofoil of given length and prescribed mean curvature which produces {maximum lift/minimum drag} The problem is discussed in three cases when there is a {full/partial/zero} cavity flow past the hydrofoil. The liquid flow is assumed to be two-dimensional steady, irrotational and incompressible and a linearized theory is assumed. Two-dimensional vortex and source distributions are used to simulate the two dimensional {full/partial/zero} cavity flow past a thin hydrofoil. This method leads to a system of integral equations and these are solved exactly using the Carleman-Muskhelishvili technique. This method is similar to that used by Davies, T.V. We use variational calculus techniques to obtain the optimum shape of the hydrofoil in order to {maximized/minimized} the {lift/drag} coefficient subject to constraints on curvature and given length. The mathematical problem is that of extremizing a functional depending on {? vortex strength/ ? source strength} these three functions are related by singular integral equations. The analytical solution for the unknown shape z and the unknown singularity distribution y has branch-type singularities at the two ends of the hydrofoil. Analytical solution by singular integral equations is discussed and the approximate solution by the Rayleigh-Ritz method is derived. A sufficient condition for the extremum to be a minimum is derived from consideration of the second variation. PART III The purpose of this work is to evaluate the optimum shape of a two-dimensional hydrofoil of given length and prescribed mean curvature which produces minimum drag. A thin hydrofoil of arbitrary shape is in steady, rectilinear, horizontal motion at a depth h beneath the free surface of a liquid. The usual assumptions in problems of this kind are taken as a basis, namely, the liquid is non-viscous and moving two-dimensionally, steadily and without vorticity, the only force acting on it is gravity. With these assumptions together with a linearization assumption we determine the forces, due to the hydrofoil beneath a free surface of the liquid. We use variational calculus techniques similar to those used in Part II to obtain the optimum shape so that the drag is minimized. A sufficient condition for the extremum to be a minimum is derived from consideration of the second variation. In this part some general expressions are established concerning the forces acting on a submerged vortex and source element beneath a free surface using Blasius theorem.

This thesis is concerned with the methods of mean field calculations of the properties of soft matter systems. The first part deals with the application of mean field density functional theory to fluid systems containing small numbers of particles. This is relevant to nucleation studies that can be performed using mean field density functional theory (MFDFT), where the critical clusters that constitute the transition states for phase transitions can be very small. It is also relevant for studies of the behaviour of confined fluids such as fluids in nanopores. The problems in applying MFDFT to small systems are investigated, and modifications to improve the accuracy are identified. These principles are tested on a highly simplified model system of attractive hard rods in one dimension. The second part of the thesis investigates the mean field description of interactions in charged colloidal suspensions within the primitive (PM) model. The phase behaviour of these systems is discussed. In particular, the question of whether experimental observations of coexistence between dense and rarefied phases can be accounted for by mean field theory is discussed. A new approximate method for solving the nonlinear mean field Poisson-Boltzmann equation in the limit of dilute suspensions is proposed. This method is applied to the simple case of charged plates, as well as arrays of spherical colloidal particles. For the latter case, comparisons are made between spherical and cubic Wigner-Seitz cell geometries.

M-theory is well-known but not well-understood. It arises as an umbrella theory that unifies the various perturbative string theories into a single nonperturbative theory. In its strong coupling phase M-theory does not possess string states but rather M2-branes and M5-branes. The purpose of this thesis is to explore the properties of multiple coincident M2- and M5-branes. It is based on the author’s papers [1, 2] (in collaboration with Neil Lambert), [3] (in collaboration with Imtak Jeon and Neil Lambert) and [4]. We begin with a review of the construction of three-dimensional J\f = 8 and J\i = 6 super-symmetric Chern-Simons-matter theories. These include the BLG and ABJM models of multiple M2-branes and our focus will be on their formulation in terms of 3-algebras. We then examine the coupling of multiple M2-branes to the background 3-form and 6-form gauge fields of eleven-dimensional supergravity. In particular we show in detail how a natural generalisation of the Myers flux-terms, along with the resulting curvature of the background metric, leads to mass terms m the effective field theory. Working to lowest nontrivial order in fermions, we demonstrate the supersymmetric invariance of the four-derivative order corrected Lagrangian of the Euclidean BLG theory and determine the theory’s higher derivative corrected supersymmetry transformations. The supersymmetry algebra is also shown to close on the scalar and gauge fields. We also consider periodic arrays of M2-branes in the ABJM model in the spirit of a circle compactification to D2-branes in type IIA string theory. The result is a curious formulation of three-dimensional maximally supersymmetric Yang-Mills theory. Upon further T-duality on a transverse torus we obtain a non-manifest-Lorentz-invariant description of five-dimensional maxi¬mally supersymmetric Yang-Mills which can be viewed as an M-theory description of M5-branes on T3. After reviewing work to describe multiple M5-branes using 3-algebras we show how the re-sulting novel system of equations reduces to one-dimensional motion on instanton moduli space. Quantisation leads to the previous light-cone proposal of the (2,0) theory, generalised to include a potential that arises on the Coulomb branch as well as couplings to background gauge and self-dual 2-form fields.

This thesis reports the theoretical study of several cold atom ratchet systems. In particular the focus of the work is the determination of the ratchet current as a function of the ratchet parameters through analysis of the system symmetries and through numerical simulation. Ratchets are devices that exhibit directed motion in the absence of net forces. It is necessary to drive them away from thermal equilibrium so as to not violate the second law of thermodynamics. Currents are generated when the symmetries of the ratchet do not forbid it, a consequence of Curie's principle. An analysis of the symmetries will help determine for what parameters currents will be generated we perform such analyses in our investigations. The ratchets studied are modelled on the experimentally realised implementation of cold atoms in a driven optical lattice. Through the parameters of the driving and the optical lattice itself, we control the breaking of the symmetries and thus the generation of atomic currents. The precise relationship between current and ratchet parameters is explored by numerical simulation. In experiments the driving is achieved through a phase-modulation of the optical lattice beams. In numerical simulations we include the driving force directly in the equations of motion. We verify theoretically and numerically that the two approaches are equivalent. We have modelled the dynamics of atoms in light-fields through semiclassical and quantum treatments. The semiclassical treatment results in stochastic differential equations for the external degrees of freedom. These are simulated using the Monte-Carlo technique. For the fully quantum treatment we apply a stochastic trajectory method to simulate the master equation. We perform a comparison between different treatments for an over-damped ratchet.

We present here two calculations based on the <i>phase-space of interactions</i> treatment of neural network models. As a way of introduction we begin by discussing the type of neural network models we wish to study, and the analytical techniques available to us from the branch of disordered systems in statistical mechanics. We then detail a neural network which models a <i>content addressable memory</i>, and sketch the mathematical methods we shall use. The model is a mathematical realisation of a neural network with its synaptic efficacies optimised in its phase space of interactions through some <i>training function</i>. The first model looks at how the basin of attraction of such a content addressable memory can be enlarged by the use of noisy external fields. These fields are used separately during the training and retrieval phases, and their influences compared. Expressed in terms of the number of memory patterns which the network's dynamics can retrieve with a microscopic initial overlap, we shall show that content addressability can be substantially improved. The second calculation concerns the use of <i>dual distribution functions</i> for two networks with different constraints on their synapses, but required to store the same set of memory patterns. This technique allows us to see how the two networks accommodate the demands imposed on them, and whether they arrive at radically different solutions. The problem we choose is aimed at, and eventually succeeds in, resolving a paradox in the sign-constrained model.

This thesis examines some methods of solution of systems with dynamical symmetry by considering the Coulomb and isotropic potentials on a sphere. It is shown that the classical solutions provide cloned orbits which is a criterion for dynamical symmetry. The extra constants of the motion, which are the generators of the symmetry groups, are found and it is shown that the groups are 80(4) and SUM respectively. However, the form of the commutation relations of the quantum-mechanical operators prevents the direct use of group representation theory. An indirect technique, which Pauli used to solve the usual Coulomb problem, is employed to derive the energy eigenvalues and eigenfunctions of both systems. This technique makes use of the matrix elements of the operators in a basis of energy and angular momentum eigenstates. This is shown to be equivalent to a method of Schrodinger for solving a special class of differential equations. The systems above are generalised to N dimensions and solved by this method. For systems with dynamical symmetry the Schrodinger equation is separable in more than one set of coordinates. This is equivalent to choosing different bases of eigenstates. The sets of coordinates are found and the equations are separated in them but neither they nor the corresponding algebra of matrix elements has been solved.

We determine the high energy asymptotic density of the eigenvalues of the scat- tering matrix associated with the operators H0 = −∆ and H = (i∇ + A)2 + V (x), where V : Rd → R is a smooth short-range real-valued electric potential and A = (A1, . . . , Ad) : Rd → Rd is a smooth short-range magnetic vector-potential. Two cases are considered. The first case is where the magnetic vector-potential is non-zero. The spectral density of the associated scattering matrix in this case is expressed as an integral solely in terms of the magnetic vector-potential A. The second case considered is where the magnetic vector-potential is identically zero. Again the spectral density of the scattering matrix is expressed as an integral, this time in terms of the poten- tial V . These results share similar characteristics to results pertaining to semiclassical asymptotics for pseudodifferential operators.

The work reported in Part I of this thesis concerns ionisation and spin exchange. The familiar time-dependent perturbation equations are derived from the time-dependent Schrodinger wave equation. Coriolis coupling is explicitly taken account of, the relevant coupled equations are set up for several M + X systems and are solved numerically. The results are presented in §1.10 and §I.11 where the solutions arc compared with the Landau-Zenar approximation and with the numerical solution of a simple two-state problem. In Part II elastic and inelastic differential cross sections for two crossing monotonic repulsive potentials are calculated. Two model potentials are used and the scattered amplitudes are evaluated as a partial wave summation. The S-matrix elements are calculated using the Landau-Zener-Stuer-Kelberg approximation and phase shifts are evaluated analytically by using a straight line approximation to the trajectory. The results, which are interpreted in an analysis due to Ford and ?heeler, are presented in §11.6. In Part III the quadrupole-quadrupole mechanism for fine structure transitions in heavy atoms induced by collision with H2 IID and D2 is considered. Transition probabilities were evaluated using first order time-dependent perturbation theory. The quadrupole-quadrupole term of the nailtipolar expansion of the electrostatic interaction potential is evaluated in two co-ordinate systems. An analytic expression for the transition probabilities is obtained in terms of atomic and molecular matrix elements, atomic and molecular quantum numbers and an integral ever the collision trajectory. The matrix elements were evaluated using best available wave functions. Transition probabilities as functions of impact parameter and velocity were obtained by numerically integrating the trajectory integrals. The probability functions were numerically integrated first over impact parameter and finally over the Boltzmann velocity distribution to obtain rate constants as functions of temperature. The rate constants so calculated were compared with experimental values and the results are presented in §111.5.

The goal of this thesis is to obtain a superfield formulation of local supersymmetry, and to construct via this formalism a model of spontaneous local supersymmetry breakdown. In the first chapter, the superfield method and some globally supersymmetric models are reviewed. These include Lagrangians for massive interacting chiral multiplets, and models for both massive and massless vector multiplets. In particular, the globally supersymmetric extension of the Higgs mechanism, due to Fayet, is described in detail. This model will form the basis of a locally supersymmetric model incorporating spontaneous supersymmetry breakdown in the third chapter. None of this work is original. The second chapter is devoted to gauging supersymmetry without superfields. The earliest supergravity theories (those not involving matter coupling) are reviewed. The fiber bundle approach is described, and shown to be ambiguous. An alternative algebraic scheme for dealing with gravitational symmetries is given. Superfield supergravity in two dimensions forms the subject matter of the third chapter. A brief glimpse of a one-dimensional locally supersymmetric theory (the spinning particle) is given. Its two-dimensional analogue, the spinning string, is obtained first without recourse to superfields, and then via an elegant superfield Ansatz due to Howe. It is shown how to derive this Ansatz and its transformation. Finally, a locally supersymmetric version of the Fayet model is given. The generalised Higgs mechanism works to remove the Goldstone spinor, but via a gauge field (the gravitino) which is forced to be non-dynamical in two dimensions. The methods of the third chapter are extended to four dimensions in the fourth chapter. The corresponding vielbein is derived, and shown not to transform covariantly without the addition of new terms. An attempt is made to find these terms, and it is argued that no additions can render the vielbein covariant. Consequently the approach of the third chapter proves inapplicable to four dimensions, and no matter-supergravity coupling can be obtained in this way. Three appendices on the history of anticommiting variables, the use of differential forms, and on some useful identities, complete the thesis.