This thesis presents an investigation into the behaviour of a laser beam of finite diameter in a plasma with respect to forces and optical properties, which lead to self-focusing of the beam. The transient setting of ponderomotive nonlinearity in a collisionless plasma has been studied, and consequently the self- focusing of the pulse, and the focusing of the plasma wave occurs. The description of a self-focusing mechanism of laser radiation in the plasma due to nonlinear forces acting on the plasma in the lateral direction, relative to the laser has been investigated in the non-relativistic regime. The behaviour of the laser beams in plasma, which is the domain of self-focusing at high or moderate intensity, is dominated by the nonlinear force. The investigation of self-focusing processes of laser beams in plasma result from the relativistic mass and energy dependency of the refractive index at high laser intensities. Here the relativistic effects are considered to evaluate the relativistic self-focusing lenghts for the neodymium glass radiation, at different plasma densities of various laser intensities. A sequence of code in C++ has been developed to explore in depth self-focusing over a wide range of parameters. The nonlinear plasma dielectric function to relativistic electron motion will be derived in the latter part of this thesis. From that, one can obtain the nonlinear refractive index of the plasma and estimate the importance of relativistic self-focusing as compared to ponderomotive non-relativistic self-focusing, at very high laser intensities. When the laser intensity is very high, pondermotive self-focusing will be dominant. But at some point, when the oscillating velocity of the plasma electron becomes very large, relativistic effects will also play a role in self-focusing. A numerical and theoretical study of the generation and propagation of oscillation in the semiclassical limit of the nonlinear paraxial equation is presented in this thesis. In a general setting of both dimension and nonlinearity, the essential differences between the 'defocusing' and 'focusing' cases hence is identified. Presented in this thesis are the nonlinearity and dispersion effects involved in the propagation of solitions which can be understood by using a numerical routines were implemented through the use of the mathematica program, and results give a very clear idea of this interesting phenomena / Doctor of Philosophy (PhD)

This research focuses on improvements and application of 1D finite elements for nonlinear dynamic analysis of reinforced concrete frames under extreme loadings. The concept of force interpolation is adopted for the element formulation and a solution scheme developed based on a total secant stiffness approach that provides good convergence characteristics. The geometrical nonlinearities including 2nd order P-Delta effects as well as catenary action are considered in the element formulation. It is shown that geometrical nonlinearities may have a significant effect on member (structure) response within extreme loading scenarios. In the analysis of structures subjected to extreme loadings, accurately modelling of the post peak response is vital and, in this respect, the objectivity of the solution with softening must be maintained. The softening of concrete under compression is taken into account, and the objectivity preserved, by adopting a nonlocal damage model for the compressive concrete. The capability of nonlocal flexibility-based formulation for capturing the post-peak response of reinforced concrete beam-columns is demonstrated by numerical examples. The 1D frame element model is extended for the modelling of 3D framed structures using a simplified torque-twist model that is developed to take account of interaction between normal and tangential forces at the section level. This simplified model can capture the variation of element torsional stiffness due to presence of axial force, bending moment and shear and is efficient and is shown to provide a reasonable degree of accuracy for the analysis of 3D reinforced concrete frames. The formulations and solution algorithms developed are tested for static and dynamic analysis of reinforced concrete framed structures with examples on impact analysis of beams, dynamic analysis of frames and progressive collapse assessment of frames taken from the literature. The verification shows that the formulation is very efficient and is capable of modelling of large scale framed structures, under extreme loads, quickly and with accuracy.

This thesis presents the nonlinear in-plane behaviour of circular fixed arches subjected to thermal loading only. Due to the nonlinear prebuckling behaviour of arches and its effects, classical buckling theory which is founded on geometric prebuckling linearity can not predict the in-plane buckling of arches accurately. Based on a nonlinear formulation of the strain and displacement relationship and considering constant thermal distributions only, virtual work formulations are used to establish the differential equations of in-plane equilibrium and the statical boundary conditions, from which the nonlinear equilibrium equations are derived in closed form and which are suitable to use in design. By considering the adjacent buckled configurations, the differential buckling equilibrium equations are formulated from the principle of virtual work as well, and the analytical solutions for the nonlinear buckling of fixed arches are obtained. It is shown that nonlinear elastic buckling of a fixed in the plane of it curvature can not occur when it is subjected to thermal loading only, except if the arch is as a straight column. By using the algebraic representation of nonlinear in-plane equilibrium derived in this thesis, the elastic response of fixed arches at elevated temperatures and the attainment of first yield are examined in detail. The arch deflects transversely without bound in the elastic range at elevated temperatures, whereas it will yield first at the top extreme fibre of the cross section at the supports when a critical temperature is reached. The influence of several parameters such as the included angle is also considered. Based on the models of stress distributions at cross sections, the spread of yield both through the cross section and along the length of the arch is studied. It is indicated that the progress of yielding causes the first two hinges to form at the supports of the fixed arches, and then moment redistribution leads to the generation of the third hinge at the crown with an increase of temperature. Thus nonlinear plastic hinge analysis can be applied to the arch analysis under thermal loading.

Continuous-time systems are usually modelled by differential equations arising from physical laws. However, the use of these models in practice requires discretisation. In this thesis we consider sampled-data models for linear and nonlinear systems. We study some of the issues involved in the sampling process, such as the accuracy of the sampled-data models, the artifacts produced by the particular sampling scheme, and the relations to the underlying continuous-time system. We review, extend and present new results, making extensive use of the delta operator which allows a clearer connection between a sampled-data model and the underlying continuous-time system. In the first part of the thesis we consider sampled-data models for linear systems. In this case exact discrete-time representations can be obtained. These models depend, not only on the continuous-time system, but also on the artifacts involved in the sampling process, namely, the sample and hold devices. In particular, these devices play a key role in determining the sampling zeros of the discrete-time model. We consider robustness issues associated with the use of discrete-time models for continuous-time system identification from sampled data. We show that, by using restricted bandwidth frequency domain maximum likelihood estimation, the identification results are robust to (possible) under-modelling due to the sampling process. Sampled-data models provide a powerful tool also for continuous-time optimal control problems, where the presence of constraints can make the explicit solution impossible to find. We show how this solution can be arbitrarily approximated by an associated sampled-data problem using fast sampling rates. We also show that there is a natural convergence of the singular structure of the optimal control problem from discrete- to continuous-time, as the sampling period goes to zero. In Part II we consider sampled-data models for nonlinear systems. In this case we can only obtain approximate sampled-data models. These discrete-time models are simple and accurate in a well defined sense. For deterministic systems, an insightful observation is that the proposed model contains sampling zero dynamics. Moreover, these correspond to the same dynamics associated with the asymptotic sampling zeros in the linear case. The topics and results presented in the thesis are believed to give important insights into the use of sampled-data models to represent linear and nonlinear continuous-time systems. / PhD Doctorate

This thesis applies several standard nonlinear quantifiers to EEG analysis to examine both human primary generalised epilepsy (PGE) and rat models of human epilepsy. We analysed rat EEG, and then used the analysed data, in parallel with an impedance recording, to better understand the events during experiments. Next, the nonlinear analysis of EEG was used to attempt to model the behaviour of the impedance data. This modeling did not yield a useful predictive tool, so we recommend the continued recording of impedance data as a means of augmenting EEG recordings. The analyses were also applied to human data, and showed differences between the PGE and control groups in apparently normal EEG. We then attempted to use these differences to detect the presence of PGE in an unclassified subject – a diagnostic tool. This was done using a feed-forward neural network. We found that the inter-group differences were exploitable and facilitated the diagnosis of PGE in previously unknown subjects. The extent to which this is useful as a diagnostic tool should be assessed by further trials. Finally, the analyses were used to examine data from a paralysed human subject, in an attempt to identify the mental task being performed by that subject. This was not successful, suggesting that the same analyses that were useful in discriminating between PGE and control were not useful in detecting the mental state of the subject. It was also apparent that the presence of EMG (in an unparalysed state) assisted task-classification.

ABSTRACT The quest to model wave propagation in various physical systems has produced a large set of diverse nonlinear equations. Nonlinear singular integro-differential equations rank amongst the intricate nonlinear wave equations available to study the classical problem of wave propagation in physical systems. Integro-differential equations are characterized by the simultaneous presence of integration and differentiation in a single equation. Substantial interest exists in nonlinear wave equations that are amenable to the Inverse Scattering Transform (IST). The IST is an adroit mathematical technique that delivers analytical solutions of a certain type of nonlinear equation: soliton equation. Initial value problems of numerous physically significant nonlinear equations have now been solved through elegant and novel implementations of the IST. The prototype nonlinear singular integro-differential equation receptive to the IST is the Intermediate Long Wave (ILW) equation, which models one-dimensional weakly nonlinear internal wave propagation in a density stratified fluid of finite total depth. In the deep water limit the ILW equation bifurcates into a physically significant nonlinear singular integro-differential equation known as the 'Benjamin-Ono' (BO) equation; the shallow water limit of the ILW equation is the famous Korteweg-de Vries (KdV) equation. Both the KdV and BO equations have been solved by dissimilar implementations of the IST. The Modified Korteweg-de Vries (MKdV) equation is a nonlinear partial differential equation, which was significant in the historical development of the IST. Solutions of the MKdV equation are mapped by an explicit nonlinear transformation known as the 'Miura transformation' into solutions of the KdV equation. Historically, the Miura transformation manifested the intimate connection between solutions of the KdV equation and the inverse problem for the one-dimensional time independent Schroedinger equation. In light of the MKdV equation's significance, it is natural to seek 'modified' versions of the ILW and BO equations. Solutions of each modified nonlinear singular integro-differential equation should be mapped by an analogue of the original Miura transformation into solutions of the 'unmodified' equation. In parallel with the limiting cases of the ILW equation, the modified version of the ILW equation should reduce to the MKdV equation in the shallow water limit and to the modified version of the BO equation in the deep water limit. The Modified Intermediate Long Wave (MILW) and Modified Benjamin-Ono (MBO) equations are the two nonlinear singular integro-differential equations that display all the required attributes. Several researchers have shown that the MILW and MBO equations exhibit the signature characteristic of soliton equations. Despite the significance of the MILW and MBO equations to soliton theory, and the possible physical applications of the MILW and MBO equations, the initial value problems for these equations have not been solved. In this thesis we use the IST to solve the initial value problems for the MILW and MBO equations on the real-line. The only restrictions that we place on the initial values for the MILW and MBO equations are that they be real-valued, sufficiently smooth and decay to zero as the absolute value of the spatial variable approaches large values.

In computer graphics a projection describes the mapping of scene geometry to the screen. While linear projections such as perspective and orthographic projection are common, increasing applications are being found for nonlinear projections, which do not necessarily map straight lines in the scene to straight lines on the screen. Nonlinear projections occur in reflections and refractions on curved surfaces, in art, and in visualisation. This thesis presents a new nonlinear projection technique called a trilinear projection that is based on the trilinear interpolation of surface normals used in Phong shading. Trilinear projections can be combined to represent more complicated nonlinear projections. Nonlinear projections have previously been implemented with ray tracing, where rays are generated by the nonlinear projections and traced into the scene. However for performance reasons, most current graphics software uses scanline rendering, where a scene point is imaged on a screen as a function of the projection parameters. The techniques developed in this thesis are of this nature. This thesis presents several algorithms used in trilinear projection: 1. An algorithm to analytically determine which screen locations image a given scene point. 2. An algorithm that correctly connects projected vertices. Each scene point may be imaged multiple times, which means a projected scene triangle may form from one to four different shapes of from two to nine vertices. Once connected, the projected shapes may be rendered with standard scanline algorithms. 3. An algorithm to more accurately render the curved edges between projected vertices. 4. A scene-space edge-clipping algorithm that handles continuity issues for projected shapes across composite projections. The trilinear projection technique is demonstrated in two different application areas: visualisation, and reflections and refractions. Specifically, various nonlinear projections that are congruent with pre-existing visualisation techniques are implemented with trilinear projections and a method for approximating the reflections and refractions on curved surfaces with trilinear projections is presented. Finally, the performance characteristics of the trilinear projection is explored over various parameter ranges and compared with a naive ray tracing approach.

<p>A model is developed for the Quanser torsion system available at Control Systems Research Laboratory at Chulalongkorn University. The torsion system is a laboratory equipment that is designed for the study of position control. It consists of a DC motor that drives three inertial loads that are coupled in series with the motor, and where all components are coupled to each other through torsional springs.</p><p>Several nonlinearities are observed and the most significant one is an offset in the input signal, which is compensated for. Experiments are carried out under feedback as the system is marginally stable. Different input signals are tested and used for system identification. Linear black-box state-space models are then identified using PEM, N4SID and a subspace method made for closed-loop identification, where the last two are the most successful ones. PEM is used in a second step and successfully enhances the parameter estimates from the other algorithms.</p>

Martin Schetzen. / "July 6, 1962." "Submitted to the Department of Electrical Engineering, M.I.T., May 20, 1961, in partial fulfillment of the requirements for the degree of Doctor of Science." / Bibliography: p. 53. / Army Signal Corps Contract DA 36-039-sc-78108 Department of the Army Task 3-99-20-001 and Project 3-99-00-000. Army Signal Corps Contract DA-SIG-36-039-61-G14.

George Zames. / "Submitted to the Department of Electrical Engineering, M.I.T., August 22, 1960, in partial fulfillment of the requirements for the degree of Doctor of Science." / Bibliography: p.76. / Army Signal Corps Contract DA 36-039-sc-78108. Dept. of the Army Task 3-99-20-001 and Project 3-99-00-000.