DSpace Collection:
https://dspace.lboro.ac.uk/2134/4642
2016-04-30T15:14:42ZCohomology for multicontrolled stratified spaces
https://dspace.lboro.ac.uk/2134/20707
Title: Cohomology for multicontrolled stratified spaces
Authors: Lukiyanov, Vladimir
Abstract: In this thesis an extension of the classical intersection cohomology of Goresky and MacPherson, which we call multiperverse cohomology, is defined for a certain class of depth 1 controlled stratified spaces, which we call multicontrolled stratified spaces. These spaces are spaces with singularities -- this being their controlled structure -- with additional multicontrol data. Multiperverse cohomology is constructed using a cochain complex of tau-multiperverse forms, defined for each case tau of a parameter called a multiperversity. For the spaces that we consider these multiperversities, forming a lattice M, extend the general perversities of intersection cohomology.
Multicontrolled stratified spaces generalise the structure of (the compactifications of) Q-rank 1 locally symmetric spaces. In this setting multiperverse cohomology generalises some of the aspects of the weighted cohomology of Harder, Goresky and MacPherson.
We define two special cases of multicontrolled stratified spaces: the product-type case, and the flat-type case. In these cases we can calculate the multiperverse cohomology directly for cones and cylinders, this yielding the local calculation at a singular stratum of a multicontrolled space. Further, we obtain extensions of the usual Mayer-Vietoris sequences, as well as a partial Kunneth Theorem.
Using the concept a dual multiperversity we are able to obtain a version of Poincare duality for multiperverse cohomology for both the flat-type and the product-type case. For this Poincare duality there exist self-dual multiperversities in certain cases, such as for non-Witt spaces, where there are no self-dual perversities.
For certain cusps, called double-product cusps, which are naturally compactified to multicontrolled spaces, the multiperverse cohomology of the compactification of the double-product cusp for a certain multiperversity is equal to the L2-cohomology, analytically defined, for certain doubly-warped metrics.
Description: A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.2016-01-01T00:00:00ZApproximating solutions of backward doubly stochastic differential equations with measurable coefficients using a time discretization scheme
https://dspace.lboro.ac.uk/2134/20643
Title: Approximating solutions of backward doubly stochastic differential equations with measurable coefficients using a time discretization scheme
Authors: Yeadon, Cyrus
Abstract: It has been shown that backward doubly stochastic differential equations (BDSDEs) provide a probabilistic representation for a certain class of nonlinear parabolic stochastic partial differential equations (SPDEs). It has also been shown that the solution of a BDSDE with Lipschitz coefficients can be approximated by first discretizing time and then calculating a sequence of conditional expectations. Given fixed points in time and space, this approximation has been shown to converge in mean square.
In this thesis, we investigate the approximation of solutions of BDSDEs with coefficients that are measurable in time and space using a time discretization scheme with a view towards applications to SPDEs. To achieve this, we require the underlying forward diffusion to have smooth coefficients and we consider convergence in a norm which includes a weighted spatial integral. This combination of smoother forward coefficients and weaker norm allows the use of an equivalence of norms result which is key to our approach. We additionally take a brief look at the approximation of solutions of a class of infinite horizon BDSDEs with a view towards approximating stationary solutions of SPDEs.
Whilst we remain agnostic with regards to the implementation of our discretization scheme, our scheme should be amenable to a Monte Carlo simulation based approach. If this is the case, we propose that in addition to being attractive from a performance perspective in higher dimensions, such an approach has a potential advantage when considering measurable coefficients. Specifically, since we only discretize time and effectively rely on simulations of the underlying forward diffusion to explore space, we are potentially less vulnerable to systematically overestimating or underestimating the effects of coefficients with spatial discontinuities than alternative approaches such as finite difference or finite element schemes that do discretize space.
Another advantage of the BDSDE approach is that it is possible to derive an upper bound on the error of our method for a fairly broad class of conditions in a single analysis. Furthermore, our conditions seem more general in some respects than is typically considered in the SPDE literature.
Description: A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.2015-01-01T00:00:00ZDetermining the shape of a liquid droplet: from microscopic theory to coarse grained models
https://dspace.lboro.ac.uk/2134/19985
Title: Determining the shape of a liquid droplet: from microscopic theory to coarse grained models
Authors: Hughes, Adam
Abstract: This thesis investigates the wetting of simple liquids using two density functional theory (DFT) models. The first model is a discrete lattice-gas model and the second a continuum DFT model of a hard-sphere reference system with an additional attractive perturbation. The wetting properties of liquids are principally investigated by studying the binding, or interface, potential of the fluid and this thesis presents a method by which a binding potential can be fully calculated from the microscopic DFT.
The binding potentials are used to investigate the behaviour of the model fluid depending on the range to which particle interactions are truncated. Long ranged particle interactions are commonly truncated to increase computational efficiency but the work in this thesis shows that in making this truncation some important aspects of the interfacial phase behaviour are changed. It is demonstrated that in some instances by reducing the interaction range of fluid particles a shift in phase behaviour from wetting to non wetting occurs.
The binding potential is an input to larger scale coarse grained models and this is traditionally given as an asymptotic approximation of the binding potential. By using the full binding potential, calculated from the DFT model, as an input, excellent agreement can be found between the results from the microscopic DFT model and the larger scale models. This is first verified with the discrete lattice-gas model where the discrete nature of the model causes some non-physical behaviour in the binding potentials. The continuum DFT model is then applied which corrects this behaviour.
An adaptation to this continuum model is used to study short ranged systems at high liquid densities at state points below the `Fisher-Widom' line. The form of the decay of the density profiles and binding potentials now switches from monotonic to oscillatory. This model leads to highly structured liquid droplets exhibiting a step-like structure.
Description: A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.2015-01-01T00:00:00ZSpectral properties of integrable Schrodinger operators with singular potentials
https://dspace.lboro.ac.uk/2134/19929
Title: Spectral properties of integrable Schrodinger operators with singular potentials
Authors: Haese-Hill, William
Abstract: The integrable Schrödinger operators often have a singularity on the real line, which creates problems for their spectral analysis. A classical example is the Lamé operator
L = −d^2/dx^2 + m(m + 1)℘(x),
where ℘(z) is the classical Weierstrass elliptic function. We study the spectral properties of its complex regularisations of the form
L = −d^2/dx^2 + m(m + 1)ω^2 ℘(ωx + z_0 ), z_0 ∈ C,
where ω is one of the half-periods of ℘(z). In several particular cases we show that all closed gaps lie on the infinite spectral arc.
In the second part we develop a theory of complex exceptional orthogonal polynomials corresponding to integrable rational and trigonometric Schrödinger operators, which may have a singularity on the real line. In particular, we study the properties of the corresponding complex exceptional Hermite polynomials related to Darboux transformations of the harmonic oscillator, and exceptional Laurent orthogonal polynomials related to trigonometric monodromy-free operators.
Description: A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.2015-01-01T00:00:00Z