Differential and numerical models of hysteretic systems with stochastic and deterministic inputs

Many deterministic models with hysteresis have been developed in the areas of economics, finance, terrestrial hydrology and biology. These models lack any stochastic element which can often have a strong effect in these areas. In this work stochastically driven closed loop systems with hysteresis type memory are studied. This type of system is presented as a possible stochastic counterpart to deterministic models in the areas of economics, finance, terrestrial hydrology and biology. Some price dynamics models are presented as a motivation for the development of this type of model. Numerical schemes for solving this class of stochastic differential equation are developed in order to examine the prototype models presented. As a means of further testing the developed numerical schemes, numerical examination is made of the behaviour near equilibrium of coupled ordinary differential equations where the time derivative of the Preisach operator is included in one of the equations. A model of two phenotype bacteria is also presented. This model is examined to explore memory effects and related hysteresis effects in the area of biology. The memory effects found in this model are similar to that found in the non-ideal relay. This non-ideal relay type behaviour is used to model a colony of bacteria with multiple switching thresholds. This model contains a Preisach type memory with a variable Preisach weight function. Shown numerically for this multi-threshold model is a pattern formation for the distribution of the phenotypes among the available thresholds.

Financial modelling with 2-EPT probability density functions

The class of all Exponential-Polynomial-Trigonometric (EPT) functions is classical and equal to the Euler-d’Alembert class of solutions of linear differential equations with constant coefficients. The class of non-negative EPT functions defined on [0;1) was discussed in Hanzon and Holland (2010) of which EPT probability density functions are an important subclass. EPT functions can be represented as ceAxb, where A is a square matrix, b a column vector and c a row vector where the triple (A; b; c) is the minimal realization of the EPT function. The minimal triple is only unique up to a basis transformation. Here the class of 2-EPT probability density functions on R is defined and shown to be closed under a variety of operations. The class is also generalised to include mixtures with the pointmass at zero. This class coincides with the class of probability density functions with rational characteristic functions. It is illustrated that the Variance Gamma density is a 2-EPT density under a parameter restriction. A discrete 2-EPT process is a process which has stochastically independent 2-EPT random variables as increments. It is shown that the distribution of the minimum and maximum of such a process is an EPT density mixed with a pointmass at zero. The Laplace Transform of these distributions correspond to the discrete time Wiener-Hopf factors of the discrete time 2-EPT process. A distribution of daily log-returns, observed over the period 1931-2011 from a prominent US index, is approximated with a 2-EPT density function. Without the non-negativity condition, it is illustrated how this problem is transformed into a discrete time rational approximation problem. The rational approximation software RARL2 is used to carry out this approximation. The non-negativity constraint is then imposed via a convex optimisation procedure after the unconstrained approximation. Sufficient and necessary conditions are derived to characterise infinitely divisible EPT and 2-EPT functions. Infinitely divisible 2-EPT density functions generate 2-EPT Lévy processes. An assets log returns can be modelled as a 2-EPT Lévy process. Closed form pricing formulae are then derived for European Options with specific times to maturity. Formulae for discretely monitored Lookback Options and 2-Period Bermudan Options are also provided. Certain Greeks, including Delta and Gamma, of these options are also computed analytically. MATLAB scripts are provided for calculations involving 2-EPT functions. Numerical option pricing examples illustrate the effectiveness of the 2-EPT approach to financial modelling.

Magnetic field-induced ferroelectric switching in multiferroic aurivillius phase thin films at room temperature

Single-phase multiferroic materials are of considerable interest for future memory and sensing applications. Thin films of Aurivillius phase Bi 7Ti3Fe3O21 and Bi6Ti 2.8Fe1.52Mn0.68O18 (possessing six and five perovskite units per half-cell, respectively) have been prepared by chemical solution deposition on c-plane sapphire. Superconducting quantum interference device magnetometry reveal Bi7Ti3Fe 3O21 to be antiferromagnetic (TN = 190 K) and weakly ferromagnetic below 35 K, however, Bi6Ti2.8Fe 1.52Mn0.68O18 gives a distinct room-temperature in-plane ferromagnetic signature (Ms = 0.74 emu/g, μ0Hc =7 mT). Microstructural analysis, coupled with the use of a statistical analysis of the data, allows us to conclude that ferromagnetism does not originate from second phase inclusions, with a confidence level of 99.5%. Piezoresponse force microscopy (PFM) demonstrates room-temperature ferroelectricity in both films, whereas PFM observations on Bi6Ti2.8Fe1.52Mn0.68O18 show Aurivillius grains undergo ferroelectric domain polarization switching induced by an applied magnetic field. Here, we show for the first time that Bi6Ti2.8Fe1.52Mn0.68O18 thin films are both ferroelectric and ferromagnetic and, demonstrate magnetic field-induced switching of ferroelectric polarization in individual Aurivillius phase grains at room temperature.

Reasoning with sorted-pareto dominance and other qualitative and partially ordered preferences in soft constraints

In decision making problems where we need to choose a particular decision or alternative from a set of possible choices, we often have some preferences which determine if we prefer one decision over another. When these preferences give us an ordering on the decisions that is complete, then it is easy to choose the best or one of the best decisions. However it often occurs that the preferences relation is partially ordered, and we have no best decision. In this thesis, we look at what happens when we have such a partial order over a set of decisions, in particular when we have multiple orderings on a set of decisions, and we present a framework for qualitative decision making. We look at the different natural notions of optimal decision that occur in this framework, which gives us different optimality classes, and we examine the relationships between these classes. We then look in particular at a qualitative preference relation called Sorted-Pareto Dominance, which is an extension of Pareto Dominance, and we give a semantics for this relation as one that is compatible with any order-preserving mapping of an ordinal preference scale to a numerical one. We apply Sorted-Pareto dominance to a Soft Constraints setting, where we solve problems in which the soft constraints associate qualitative preferences to decisions in a decision problem. We also examine the Sorted-Pareto dominance relation in the context of our qualitative decision making framework, looking at the relevant optimality classes for the Sorted-Pareto case, which gives us classes of decisions that are necessarily optimal, and optimal for some choice of mapping of an ordinal scale to a quantitative one. We provide some empirical analysis of Sorted-Pareto constraints problems and examine the optimality classes that result.

Beyond simple imaging with energetic beams – nanopatterning, correlative microscopy and statistical analysis of inclusions

Electron microscopy (EM) has advanced in an exponential way since the first transmission electron microscope (TEM) was built in the 1930’s. The urge to ‘see’ things is an essential part of human nature (talk of ‘seeing is believing’) and apart from scanning tunnel microscopes which give information about the surface, EM is the only imaging technology capable of really visualising atomic structures in depth down to single atoms. With the development of nanotechnology the demand to image and analyse small things has become even greater and electron microscopes have found their way from highly delicate and sophisticated research grade instruments to key-turn and even bench-top instruments for everyday use in every materials research lab on the planet. The semiconductor industry is as dependent on the use of EM as life sciences and pharmaceutical industry. With this generalisation of use for imaging, the need to deploy advanced uses of EM has become more and more apparent. The combination of several coinciding beams (electron, ion and even light) to create DualBeam or TripleBeam instruments for instance enhances the usefulness from pure imaging to manipulating on the nanoscale. And when it comes to the analytic power of EM with the many ways the highly energetic electrons and ions interact with the matter in the specimen there is a plethora of niches which evolved during the last two decades, specialising in every kind of analysis that can be thought of and combined with EM. In the course of this study the emphasis was placed on the application of these advanced analytical EM techniques in the context of multiscale and multimodal microscopy – multiscale meaning across length scales from micrometres or larger to nanometres, multimodal meaning numerous techniques applied to the same sample volume in a correlative manner. In order to demonstrate the breadth and potential of the multiscale and multimodal concept an integration of it was attempted in two areas: I) Biocompatible materials using polycrystalline stainless steel and II) Semiconductors using thin multiferroic films. I) The motivation to use stainless steel (316L medical grade) comes from the potential modulation of endothelial cell growth which can have a big impact on the improvement of cardio-vascular stents – which are mainly made of 316L – through nano-texturing of the stent surface by focused ion beam (FIB) lithography. Patterning with FIB has never been reported before in connection with stents and cell growth and in order to gain a better understanding of the beam-substrate interaction during patterning a correlative microscopy approach was used to illuminate the patterning process from many possible angles. Electron backscattering diffraction (EBSD) was used to analyse the crystallographic structure, FIB was used for the patterning and simultaneously visualising the crystal structure as part of the monitoring process, scanning electron microscopy (SEM) and atomic force microscopy (AFM) were employed to analyse the topography and the final step being 3D visualisation through serial FIB/SEM sectioning. II) The motivation for the use of thin multiferroic films stems from the ever-growing demand for increased data storage at lesser and lesser energy consumption. The Aurivillius phase material used in this study has a high potential in this area. Yet it is necessary to show clearly that the film is really multiferroic and no second phase inclusions are present even at very low concentrations – ~0.1vol% could already be problematic. Thus, in this study a technique was developed to analyse ultra-low density inclusions in thin multiferroic films down to concentrations of 0.01%. The goal achieved was a complete structural and compositional analysis of the films which required identification of second phase inclusions (through elemental analysis EDX(Energy Dispersive X-ray)), localise them (employing 72 hour EDX mapping in the SEM), isolate them for the TEM (using FIB) and give an upper confidence limit of 99.5% to the influence of the inclusions on the magnetic behaviour of the main phase (statistical analysis).

Cost savings from relaxation of operational constraints on a power system with high wind penetration

Wind energy is predominantly a nonsynchronous generation source. Large-scale integration of wind generation with existing electricity systems, therefore, presents challenges in maintaining system frequency stability and local voltage stability. Transmission system operators have implemented system operational constraints (SOCs) in order to maintain stability with high wind generation, but imposition of these constraints results in higher operating costs. A mixed integer programming tool was used to simulate generator dispatch in order to assess the impact of various SOCs on generation costs. Interleaved day-ahead scheduling and real-time dispatch models were developed to allow accurate representation of forced outages and wind forecast errors, and were applied to the proposed Irish power system of 2020 with a wind penetration of 32%. Savings of at least 7.8% in generation costs and reductions in wind curtailment of 50% were identified when the most influential SOCs were relaxed. The results also illustrate the need to relax local SOCs together with the system-wide nonsynchronous penetration limit SOC, as savings from increasing the nonsynchronous limit beyond 70% were restricted without relaxation of local SOCs. The methodology and results allow for quantification of the costs of SOCs, allowing the optimal upgrade path for generation and transmission infrastructure to be determined.