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The purpose of this thesis is to show a method, that would transform given Turing machine into propagating scattered context grammar, which language contains all valid computational histories of that particular Turing machine. Afterwards this thesis deals with questions arising from existence of such algorithm, especially in regards to the current knowledge about power of propagating scattered context grammars. Practical examples and implementation of proposed algorithm is also part of this thesis.

In scientific computing, it is very common to visualize the approximate solution obtained by a numerical PDE solver by drawing surface or contour plots of all or some components of the associated approximate solutions. These plots are used to investigate the behavior of the solution and to display important properties or characteristics of the approximate solutions. In this thesis, we consider techniques for drawing such contour plots for the solution of two and three dimensional PDEs. We first present three fast contouring algorithms in two dimensions over an underlying unstructured mesh. Unlike standard contouring algorithms, our algorithms do not require a fine structured approximation. We assume that the underlying PDE solver generates approximations at some scattered data points in the domain of interest. We then generate a piecewise cubic polynomial interpolant (PCI) which approximates the solution of a PDE at off-mesh points based on the DEI (Differential Equation Interpolant) approach. The DEI approach assumes that accurate approximations to the solution and first-order derivatives exist at a set of discrete mesh points. The extra information required to uniquely define the associated piecewise polynomial is determined based on almost satisfying the PDE at a set of collocation points. In the process of generating contour plots, the PCI is used whenever we need an accurate approximation at a point inside the domain. The direct extension of the both DEI-based interpolant and the contouring algorithm to three dimensions is also investigated. The use of the DEI-based interpolant we introduce for visualization can also be used to develop effective Adaptive Mesh Refinement (AMR) techniques and global error estimates. In particular, we introduce and investigate four AMR techniques along with a hybrid mesh refinement technique. Our interest is in investigating how well such a `generic' mesh selection strategy, based on properties of the problem alone, can perform compared with a special-purpose strategy that is designed for a specific PDE method. We also introduce an \`{a} posteriori global error estimator by introducing the solution of a companion PDE defined in terms of the associated PCI.

In scientific computing, it is very common to visualize the approximate solution obtained by a numerical PDE solver by drawing surface or contour plots of all or some components of the associated approximate solutions. These plots are used to investigate the behavior of the solution and to display important properties or characteristics of the approximate solutions. In this thesis, we consider techniques for drawing such contour plots for the solution of two and three dimensional PDEs. We first present three fast contouring algorithms in two dimensions over an underlying unstructured mesh. Unlike standard contouring algorithms, our algorithms do not require a fine structured approximation. We assume that the underlying PDE solver generates approximations at some scattered data points in the domain of interest. We then generate a piecewise cubic polynomial interpolant (PCI) which approximates the solution of a PDE at off-mesh points based on the DEI (Differential Equation Interpolant) approach. The DEI approach assumes that accurate approximations to the solution and first-order derivatives exist at a set of discrete mesh points. The extra information required to uniquely define the associated piecewise polynomial is determined based on almost satisfying the PDE at a set of collocation points. In the process of generating contour plots, the PCI is used whenever we need an accurate approximation at a point inside the domain. The direct extension of the both DEI-based interpolant and the contouring algorithm to three dimensions is also investigated. The use of the DEI-based interpolant we introduce for visualization can also be used to develop effective Adaptive Mesh Refinement (AMR) techniques and global error estimates. In particular, we introduce and investigate four AMR techniques along with a hybrid mesh refinement technique. Our interest is in investigating how well such a `generic' mesh selection strategy, based on properties of the problem alone, can perform compared with a special-purpose strategy that is designed for a specific PDE method. We also introduce an \`{a} posteriori global error estimator by introducing the solution of a companion PDE defined in terms of the associated PCI.

Thesis (MSc (Applied Mathematics))--Stellenbosch University, 2008. / A popular method for interpolating multidimensional scattered data is using radial basis functions. In this thesis we present the basic theory of radial basis function interpolation and also regard the solvability and stability of the method. Solving the interpolant directly has a high computational cost for large datasets, hence using numerical methods to approximate the interpolant is necessary. We consider some recent numerical algorithms. Software to implement radial basis function interpolation and to display the 3D interpolants obtained, is developed. We present results obtained from using our implementation for radial basis functions on GIS and 3D face data as well as an image warping application.

La prédiction de l'émission infrarouge des jets de propulseurs composites aluminisés nécessite principalement trois étapes : le calcul des grandeurs aérothermochimiques du jet, la conversion de ces grandeurs en propriétés optiques (coefficient d'absorption, coefficient de diffusion, fonction de phase) puis la résolution de l'équation de transfert radiatif. Cette thèse,essentiellement consacrée à cette troisième étape, propose de nouvelles voies pour l'application des modèles de bande aux cas de milieux biphasiques et diffusants.D'une part, nous avons étendu ces modèles aux cas de milieux caractérisés par un déséquilibre thermique entre gaz et particules. D'autre part, nous avons proposé une méthode de séparation de la luminance en deux contributions, appelées respectivement non diffusée et diffusée, à partir d'une idée originale de Liu et al. La contribution non diffusée est solution de l'équation de transfert radiatif obtenue en ignorant les effets de la diffusion. Par conséquent, elle peut être résolue par une formulation en modèles de bande. Cette approche permet de réduire les erreurs de corrélations spectrales au seul terme de luminance diffusée.Nous avons montré l'intérêt de ces approches par comparaison avec une résolution de l'équation de transfert radiatif en raie par raie, sur des milieux représentatifs de situations de télédétection de jets. / Prediction of infrared emission of exhaust plumes from aluminized composite rocket, follows mainly three steps : calculating aero-thermo-chemical values in the plume, converting those valuesto optical properties (absorption coefficient, scattering coefficient and phase function) and resolving the radiative transfer equation. This thesis is mostly devoted to this third step, and proposes new ways to use band models on two-phases and scattering media.Firstly, we extended band models to cases with thermic non equilibrium between gas and particles. Secondly, we proposed a method consisting in splitting radiance in two parts, one called un-scattered and the other scattered, from an original idea of Liu et al. The un-scattered part is solution of the radiative transfer equation obtained by ignoring scattering. As a result, the unscattered radiance can be found by using band models. By this approach, errors on spectral correlations are only present on the scattered radiance.We show the interest of thoses approches by comparing them with a line by line resolutionof the radiative transfer equation, on media representative of remote sensing cases of rocket exhaust plumes.

Depuis l'avènement des premiers appareils imageurs par rayonnement ionisant initié par les prix Nobel Godfrey Newbold Hounsfield et Allan MacLeod Cormack en 1979, le besoin en de nouvelles techniques d'imagerie non invasives n'a cessé de croître. Ces techniques s'appuient sur les propriétés de pénétration dans la matière des rayonnements X et gamma pour détecter une structure cachée sans avoir à détruire le milieu exposé. Elles sont employées dans de nombreux domaines allant de l'imagerie médicale au contrôle non destructif en passant par le contrôle environnemental. Cependant les techniques utilisées jusqu'à maintenant subissent de fortes dégradations dans la qualité des mesures et des images reconstruites. Généralement approchées par un bruit, ces dégradations exigent d'être compensées ou corrigées par des dispositifs de collimation et de filtrage souvent coûteux. Ces dégradations sont principalement dues aux phénomènes de diffusion qui peuvent constituer jusqu'à 80 % du rayonnement émis en imagerie biomédicale. Dès les années 80 un nouveau concept a vu le jour pourcontourner cette difficulté : la tomographie Compton. Cette nouvelle approche propose de mesurer le rayonnement dit diffusé en se plaçant dans des gammes d'énergie (140−511 keV) où l'effet Compton est le phénomène de diffusion prépondérant. L'exploitation de tels dispositifs d'imagerie nécessite une compréhension profonde des interactions rayonnement/matière afin de proposer un modèle, cohérent avec les données mesurées, indispensable à la reconstruction d'images. Dans les systèmes d'imagerie conventionnels (qui mesurent le rayonnement primaire), la transformée de Radon définie sur les lignes droites est apparue comme le modèle naturel. Mais en tomographie Compton, l'information mesurée est liée à l'énergie de diffusion et ainsi à l'angle de diffusion.Ainsi la géométrie circulaire induite par le phénomène de diffusion rend la transformée de Radon classique inadaptée. Dans ce contexte, il devient nécessaire de proposer des transformées de type Radon sur des variétés géométriques plus larges.L'étude de la transformée de Radon sur de nouvelles diversités de courbes devient alors nécessaire pour répondre aux besoins d'outils analytiques de nouvelles techniques d'imagerie. Cormack, lui-même, fut le premier à étendre les propriétés de la transformée de Radon classique à une famille de courbes du plan. Par la suite plusieurs travaux ont été menés dans le but d'étudier la transformée de Radon définie sur différentes variétés de cercles, des sphères, des lignes brisées pour ne citer qu'eux. En 1994 S.J. Norton proposa la première modalité de tomography Compton modélisable par une transformée de Radon sur lesarcs de cercle, la CART1. En 2010 Nguyen et Truong établirent l'inversion de la transformée de Radon sur les arcs de cercle, CART2, permettant de modéliser la formation d'image dans une nouvelle modalité de tomographie Compton. La géométrie des supports d'intégration impliqués dans de nouvelles modalitésde tomographie Compton les conduirent à démontrer l'invertibilité de la transformée de Radon définie sur une famille de courbes de type Cormack, appelée C_alpha. Ils illustrèrent la procédure d'inversion dans le cadre d'une nouvelle transformée, la CART3 modélisant une nouvelle modalité de tomographie Compton.En nous basant sur les travaux de Cormack et de Truong et Nguyen, nous proposons d'établir plusieurs propriétés de la transformée de Radon définie sur la famille C_alpha et plus particulièrement sur C1. Nous avons ainsi démontré deux formules d'inversion qui reconstruisent l'image d'origine via sa décompositionharmonique circulaire et celle de sa transformée et qui s'apparentent à celles établies par Truong and Nguyen. Nous avons enfin établi la bien connue rétroprojection filtrée ainsi que la décomposition en valeurs singulières dans le cas alpha = 1. L'ensemble des résultats établis dans le cadre de cette étude apporte des réponses concrètes a / Since the advent of the first ionizing radiation imaging devices initiated by Godfrey Newbold Hounsfield and Allan MacLeod Cormack, Nobel Prizes in 1979, the requirement for new non-invasive imaging techniques has grown. These techniques rely upon the properties of penetration in the matter of X and gamma radiation for detecting a hidden structure without destroying the illuminated environment. They are used in many fields ranging from medical imaging to non-destructive testing through. However, the techniques used so far suffer severe degradation in the quality of measurement and reconstructed images. Usually approximated by a noise, these degradations require to be compensated or corrected by collimating devices and often expensive filtering. These degradation is mainly due to scattering phenomena which may constitute up to 80% of the emitted radiation in biological tissue. In the 80's a new concept has emerged to circumvent this difficulty : the Compton scattering tomography (CST).This new approach proposes to measure the scattered radiation considering energy ranges ( 140-511 keV) where the Compton effect is the phenomenon of leading broadcast. The use of such imaging devices requires a deep understanding of the interactions between radiation and matter to propose a modeling, consistent with the measured data, which is essential to image reconstruction. In conventional imaging systems (which measure the primary radiation) the Radon transformdefined on the straight lines emerged as the natural modeling. But in Compton scattering tomography, the measured information is related to the scattering energy and thus the scattering angle. Thus the circular geometry induced by scattering phenomenon makes the classical Radon transform inadequate.In this context, it becomes necessary to provide such Radon transforms on broader geometric manifolds.The study of the Radon transform on new manifolds of curves becomes necessary to provide theoretical needs for new imaging techniques. Cormack, himself, was the first to extend the properties of the conventional Radon transform of a family of curves of the plane. Thereafter several studies have been done in order to study the Radon transform defined on different varieties of circles, spheres, broken lines ... . In 1994 S.J. Norton proposed the first modality in Compton scattering tomography modeled by a Radon transform on circular arcs, the CART1 here. In 2010, Nguyen and Truong established the inversion formula of a Radon transform on circular arcs, CART2, to model the image formation in a new modality in Compton scattering tomography. The geometry involved in the integration support of new modalities in Compton scattering tomography lead them to demonstrate the invertibility of the Radon transform defined on a family of Cormack-type curves, called C_alpha. They illustrated the inversion procedure in the case of a new transform, the CART3, modeling a new modeling of Compton scattering tomography. Based on the work of Cormack and Truong and Nguyen, we propose to establish several properties of the Radon transform on the family C_alpha especially on C1. We have thus demonstrated two inversion formulae that reconstruct the original image via its circular harmonic decomposition and itscorresponding transform. These formulae are similar to those established by Truong and Nguyen. We finally established the well-known filtered back projection and singular value decomposition in the case alpha = 1. All results established in this study provide practical problems of image reconstruction associated with these new transforms. In particular we were able to establish new inversion methods for transforms CART1,2,3 as well as numerical approaches necessary for the implementation of these transforms. All these results enable to solve problems of image formation and reconstruction related to three Compton scattering tomography modalities.In addition we propose to improve models and algorithms es

This thesis covers the development of the Reverberation Chamber as a measurement tool for cell phone tests in electronic production. It also covers the development of the Scattered Field Chamber as a measurement tool for simulations of real propagation environments. The first part is a more ”general knowledge about Reverberation Chambers”-part that covers some important phenomena like unstirred power and position dependence that might occour in a small Reverberation Chamber used for cell phone tests. Knowing how to deal with these phenomenas, give the possibility to use the chamber as a measurement tool for production tests even though it is too complex for a simple test of the antenna function. The second part shows how to alter some important propagation parameters inside the chamber to fit some real world propagation environments. The 3D plane wave distribution, the polarization and the amplitude statistics of the plane waves are all altered with simple techniques that are implementable all together. A small, shielded anechoic box with apertures is used to control 3D plane wave distribution and polarization. Antennas that introduce unstirred power in the chamber are used to control the statistics.

This work introduces and discusses generalized scattered context grammars that are based upon sequences of productions whose left-hand sides are formed by nonterminal strings, not just single nonterminals. It places two restrictions on the derivations in these grammars. More specifically, let k be a constant. The first restriction requires that rewriting all symbols occurs within the first k symbols of the first continuous block of nonterminals in the sentential form during every derivation step. The other restriction defines the derivations over sentential forms containing no more than k occurrences of nonterminals. As its main result, the thesis demonstrates that both restrictions decrease the generative power of these grammars to the power of context-free grammars.

The scattered field from an object, when illuminated with ultrasound, is useful in the reconstruction of it's cross section - a problem broadly classified as 'tomography'. In many situations of medical imaging, we will be interested in getting to know the location and the extent of growth of the inhomogeneity. The Maximum Likelihood (ML) estimation of the location and the shape parameters (of scale and orientation angle), has been done along with the corresponding CR bounds, for the case of weakly scattering objects, where the Fourier Diffraction Theorem(FDT) holds. It has been found that the a-priori information of a reference object function helps in drastic reduction of the number of receivers and illuminations required. For a polygonal object, the shape is specified, when the corner locations are known. We have formulated the problem as, estimation of the frequencies of sum of undamped sinusoids. The result is a substantial reduction in the number of illuminations and receivers required. For acoustically soft and rigid polygons, where the FDT does not hold, the necessary theory is developed to show the dependence of the scattered field on the corner location, using an On Surface Radiation Condition(OSRC). The corner locations are estimated along similar lines, to the one adopted for the weakly scattering objects.