1 
Forward and inverse problem for nematic liquid crystalsAlHumaidi, Saleh January 2010 (has links)
This thesis starts with an introduction to liquid crystal properties, which are needed to proceed with this research. From the dielectric tensor which appears in the Maxwell equations, we were able to obtain a relationship between the elements on the main diagonal of the dielectric tensor. This relationship has been discussed and illustrated with some examples for both positive and negative birefringence. By introducing a constrain on the Berreman model, we were able to derive a 2x 2 differential equation in matrix form which works for both normal and oblique incidence. This equation gives us a simple and intuitive means to analyze the evolution of light through all sorts of media ie. isotropic, anisotropic with a fixed transmission axis and anisotropic with a twisted transmission axis of anisotropy. One of the objectives of this research was to find the right technique to solve the 2 x 2 dynamic equation. Fortunately, the classic Floquet's theory guarantees the existence of the solution and it gives some of its characteristics. In fact, we were able to solvethe 2x 2 Schrodinger equation by a new method which we called it in this thesis a rotational frame method. The obtained solution is consistent with Floquet's theory and agrees totally with the Jones solutions. Also, this solution allows us to test the Berreman approximation. Finally, in this research we were able to encode the orientation of the optical axis inside a liquid crystal sample, into the potential of the Schrodinger equation. As a consequence of that, solving the inverse problem of the Schrodinger equation that is recovering the potential, is indeed recovering the orientation of the director inside the sample. The Berreman inverse problem and its corresponding linearized problem has been considered in this thesis. In these sections, we give a rigorous derivation for the Frechet derivative.

2 
A study of bulk queueing systemsHolland, W. January 1991 (has links)
This research is concerned with the study of bulk queueing models. A number of M/G/l queues with bulkarrival and/or bulkservice are studied via the imbedded Markov chain technique to derive steadystate models. Results from these models are compared with simulation results after the appropriate numerical analysis, needed to implement these solutions, has been considered. Similar queueing models are considered from a timedependent viewpoint. Because of the difficulties of obtaining timedependent solutions, the emphasis is placed upon the development of approximations to the queueing results. Finally, to demonstrate the practicability of these models, two reallife problems are considered, namely the Gatwick Rapid Transit systems and the proposed Channel Tunnel freight shuttle operation. It is demonstrated that the former is an underutilised system, while the latter is likely to operate very close to its capacity conditions.

3 
Nested backfitting and bandwidth selection of hierarchical bivariate additive modelWang, Yufei January 2009 (has links)
In this thesis, we have given out a comprehensive approach to estimate the pairwise interaction in a bivariate additive model when the additive assumption is untenable. A new "nested backfilling' model fitting approach has been proposed and compared with some earlier approaches. The explicit estimators of each individual term were derived for the hierarchical bivariate additive model fitted by this "nested backfilling' approach. The convergence of the "nested backfilling" approach and the existence of these estimators have been shown depending on the ratio of the bandwidths that were used in the estimation of the effect of the same variable but in different terms. The mean average square error properties of these proposed explicit estimators were investigated. A discussion the pattern left in this bias and variance expressions derived for these estimators, such as mean corrected, GaussSeidel style etc., were provided to facilitate the understanding these properties. Unlike in the pure additive model case, the mean average square error of our model cannot be attributed to each individual variable. The four optimal bandwidths used need to be selected simultaneously to minimize the mean average square error. These estimators were shown worked reasonably well in simulated datasets, regardless of the level of dependence of the covariates.

4 
Existence theorems for periodic solutions to partial differential equations with applications in hydrodynamicsBagri, Gurjeet S. January 2010 (has links)
The thesis looks at a number of existence theorems that prove the existence of smallamplitude periodic solutions to systems of partial differential equations. The existence theorems we consider are the Hopf bifurcation theorem, the Lyapunov centre theorem, the WeinsteinMoser theorem, and extensions of these theorems; the HopfIooss bifurcation theorem, the LyapunovIooss centre theorem and the WeinsteinMoserIooss theorem, respectively. The theorems have been derived so that they are applicable to functional analytical problems, and have been represented in a coherent and uniform manner in order to bridge the fundamental structure common to them all. Applications of these theorems, in this standardised form, are then applied in a systematic way to two particular hydrodynamical problems; the water wave problem and the NavierStokes equations. The classic water wave problem concerns the irrotational flow of a perfect fluid of unit density, subject to the forces of gravity and surface tension. We apply the LyapunovIooss centre theorem to prove the existence of doublyperiodic waves; a doublyperiodic wave is a travelling wave that possess spatially periodic profiles in two different horizontal directions. Fundamental to our approach is the spatial dynamics formulation. The spatial dynamics formulation involves formulating a system of partial differential equations, defined on some spatial domain, as a dynamical system where one of the unbounded spatial variables plays the role of time. We catalogue a variety of parameter values for which it is possible to obtain doubly periodic waves, and we conclude with an existence result for doubly periodic waves under specific parameter restrictions. The NavierStokes equations in an exterior domain models the flow of an incompressible, viscous fluid past an obstacle. We apply the HopfIooss bifurcation theorem to the defining equations to determine the existence of timeperiodic waves. Our approach involves a careful examination of the Oseen problem to which we apply a 'cutoff' technique. This technique is used to constructs a solution to the Oseen problem using the respective solutions to the Oseen problem on a bounded domain and free space (the existence of which are well established). Timeperiodic solutions are established using the HopfIooss bifurcation theorem provided certain spectral conditions are met. The verification of the conditions may only be possible numerically, and so beyond the scope of our investigation.

5 
On problems in the estimation of statistical parameters from samples grouped in broad categoriesHowie, A. J. January 1952 (has links)
No description available.

6 
An investigation of response variance in sample surveysO'Muircheartaigh, Colm Aongus January 2000 (has links)
The dissertation considers response variance in sample surveys in the broader context of survey quality and survey error. Following a historical review of the evolution of both the terms and the concepts a brief overview is given of earlier research in the area. The principal content of the dissertation draws on investigations carried out by the author over the last thirty years. There are three separate strands of argument, each associated with a particular approach to the analysis. First there is the descriptive (simple diagnostic) orientation of establishing the circumstances under which (or if) response variance arises, the associated issue of how it should be accommodated in analysis  primarily estimating the impact on the variance of univariate statistics  and an assessment of its likely order of magnitude. Second, there is the modelassisted orientation which attempts to decompose the effects into their constituent parts: one approach is to incorporate the correlating source (cluster or interviewer for example) as a term or terms in other models that we are estimating so that the effect is incorporated into the estimation of these models; the other is to model the response error itself  in doing this we are trying to decompose it into its constituent parts. Third, and most radical, is to view error as information. By conceptualizing the process that generated the errors as a substantive process rather than as a set of nuisance effects we can extract from the results of the process information about both the process and the subject matter. Any particular piece of analysis may include any combination of these three approaches. The dissertation draws on special studies incorporated into a number of major sample surveys. Two principal data sets are involved. The first arises from a special investigation of response error carried out in conjunction with the World Fertility Survey; the second is the reinterview data set from the Current Population Survey carried out by the US Bureau of the Census. Four other surveys are used; an absenteeism survey in Ireland, two crosssectional British surveys (one on Noise Annoyance, the other on Physical Handicap), and a British panel survey (the British Household Panel Survey).

7 
Statistical analysis of multivariate bilinear time series modelsStensholt, B. K. January 1989 (has links)
In the last thirty years there has been extensive research in the analysis of linear time series models. In analyzing univariate and multivariate time series the assumption of linearity is, in many cases, unrealistic. With this in view, recently, many nonlinear models for the analysis of time series have been proposed, mainly for univariate series. One class of models proposed which has received considerable interest, is the class of bilinear models. In particular has the theory of univariate bilinear time series been considered in a number of papers (d. Granger and Andersen (1978), Subba Rao (1981) and Bhaskara Rao et. al. (1983) and references therein); these models are analogues of the bilinear systems as proposed and studied previously by control theorists. Recently several analytic properties of these time series models have been investigated, and their estimation and applications have been reported in Subba Rao and Gabr (1983). But it is important to study the relationship between two or more time series, also 10 the presence of nonlinearity. Therefore, multivariate generalizations of the bilinear models have been considered by Subba Rao (1985) and Stensholt and Tj(llstheim (1985, 1987). Here we consider some theoretical aspects of multivariate bilinear time series models (such as strict and second order stationarity, ergodicity, invertibility, and, for special cases. strong consistency of least squares estimates). The theory developed is illustrated with simulation results. Two applications to real bivariate data (minkmuskrat data and "housing startshouses soldll data) and the FORTRAN programs developed in this project are also included.

8 
Interpretable and fast dimension reduction of multivariate dataEnki, Doyo Gragn January 2010 (has links)
The main objective of this thesis is to propose new techniques to simplify the interpretation of newly formed `variables' or components, while reducing the dimensionality of multivariate data. Most attention is given to the interpretation of principal components, although one chapter is devoted to that of factors in factor analysis. Sparse principal components are proposed, in which some of the component loadings are made exactly zero. One approach is to make use of the idea of correlation biplots, where orthogonal matrix of sparse loadings is obtained from computing the biplot factors of the product of principal component loading matrix and functions of their variances. Other approachesin volve clustering of variablesa s a preprocessings tep, so that sparse components are computed from the data or correlation matrix of each cluster. New clustering techniques are proposed for this purpose. In addition, a penalized varimax approach is proposed for simplifying the interpretation of factors in factor analysis, especially for factor solutions with considerably different sum of squares. This is done by adding a penalty term to the ordinary varimax criterion. Data sets of varying sizes, both synthetic and real, are used to illustrate the proposed methods, and the results are compared with those of existing ones. In the case of principal component analysis, the resulting sparse components are found to be more interpretable (sparser) and explain higher cumulative percentage of adjusted variance compared to their counterparts from other techniques. The penalized varimax approach contributes in finding a factor solution with simple structures which are not revealed by the standard varimax solution. The proposed methods are very simple to understand and involve fast algorithms compared to some of the existing methods. They contribute much to the interpretation of components in a reduced dimension while dealing with dimensionality reduction of multivariate data.

9 
Fixed and variable timestepping numerical methods for dynamical systemsChristodoulou, Nikolaos Styllianou January 2002 (has links)
This thesis is concerned with the numerical solution of dynamical systems by fixed and variable timestepping methods. Chapter I, reports on current work in the field and states principal results. Chapter 2, briefly reviews dynamical systems theory for ordinary differential equations. In Chapter 3, standard numerical methods for continuation of solution branches are summarised. Chapter 4, continues the discussion on bifurcations and spurious solutions for numerical methods. The mechanism by which the presence of spurious numerical solutions degrades the numerical approximation of an attractor of the underlying system is studied. Further, an investigation into how well real bifurcations in the family of dynamical systems are approximated as the stepsize varies is carried out. In general, the preservation of bifurcation structures and stability under numerical simulations is discussed. In addition, the behaviour of numerical solutions generated by a RungeKutta method applied to a dynamical system whose analytical solution undergoes a Hopf bifurcation is investigated. Hopf bifurcation results for the numerical solution are presented and analysed. In Chapter 5, the stability stepsize constraints are discussed further. In particular, it is proved that for any dynamical system with locally Lipschitz I, trajectories of solutions neither cross nor merge in phase space. A necessary condition to stop merging or crossing of trajectories in numerical simulations is derived using linear theory. Finally, in Chapter 6, a phase space error control "PS8 error control" is introduced which bounds the truncation error at each step by a fraction of the solution arc length over the corresponding time interval. It is shown that this error control can be incorporated within a standard algorithm as an additional constraint at negligible additional computational cost. Numerical results are given to demonstrate that the new error control has positive effects on the linear stability properties around true fixed points and moreover, prevents spurious fixed points that might otherwise be allowed by the adaptive algorithm. Also,prevents spurious fixed points that might otherwise be allowed by the adaptive algorithm. Also, since stepsize selection is nontrivial for phase space error controls as they are not based on a simple error estimate, a new stepsize selection scheme is introduced which leads to stable stepsizes (with fast linear convergence to a constant value) near fixed points. Numerical simulations that illustrate and confirm the analysis, as regards the dynamics of the numerical solution and the stepsize sequences near to stable and saddle points, are also presented.

10 
Calculating ice–water interfacial free energy by molecular simulationHandel, Richard James January 2009 (has links)
This study presents a calculation of the free energy of the ice–water interface using molecular simulation. The method used is an adaptation of the cleaving method, introduced by Broughton and Gilmer, and subsequently enhanced by Davidchack and Laird. The calculation is direct in the sense that an interface is formed during the simulation: isolated ice and water systems are transformed, via a sequence of reversible steps, into a single system of ice and water in contact. The method is essentially computational, that is, it does not correspond to any possible physical experiment, since nonphysical potential energies are introduced (and subsequently removed) during the transformation process. The adaptation of the method to water presented significant challenges, notably the avoidance of hysteresis during the transformation, and the devising of an ‘external’ energy potential to control the position and orientation of water molecules. The results represent the first direct calculation by simulation of the solid–liquid interfacial free energy for a model of a molecular (as opposed to atomic) system.

Page generated in 0.0933 seconds