A superconducting microwave cavity has been designed to test advanced materials for use in the accelerating structures contained within linear colliders. The electromagnetic design of this cavity produces surface magnetic fields on the sample wafer exceeding the critical limit of Niobium. The ability of this cavity to push up to 4 times the critical field provides, for the first time, a short sample method to reproducibly test these thin films to their ultimate limit. In order for this Wafer Test cavity to function appropriately, the large sapphire at the heart of the cavity must have specific inherent qualities. A second cavity was constructed to test these parameters: dielectric constant, loss tangent, and heat capacity. Several tests were performed and consistent values were obtained. The consequences of these measurements were then applied to the Wafer Cavity, and its performance was evaluated for different power inputs. The Q_0 of the cavity could be as low as 10^7 because of the sapphire heating, therefore removing the ability to measure nano-resistances. However, with additional measurements in a less complex environment, such as the Wafer Test Cavity, the Q_0 could be higher than 10^9.

The (BMN) bulk materials were sintered at 1050°C, 1100°C, 1150°C, 1200°C by the conventional ceramic process, and their microstructure and dielectric properties were investigated by Scanning electron microscopy (SEM), X-ray diffraction (XRD), Raman spectroscopy, Transmission electron microscopy (TEM) (including the X-ray energy dispersive spectrometry EDS and high resolution transmission electron microscopy HRTEM) and dielectric impedance analyzer. We systematically investigated the structure, dielectric properties and voltage tunable property of the ceramics prepared at different sintering temperatures. The XRD patterns demonstrated that the synthesized BMN solid solutions had cubic phase pyrochlore-type structure when sintered at 1050°C or higher, and the lattice parameter (a) of the unit cell in BMN solid solution was calculated to be about 10.56Å. The vibrational peaks observed in the Raman spectra of BMN solid solutions also confirmed the cubic phase pyrochlore-type structure of the synthesized BMN. According to the Scanning Electron Microscope (SEM) images, the grain size increased with increasing sintering temperature. Additionally, it was shown that the densities of the BMN ceramic tablets vary with sintering temperature. The calculated theoretical density for the BMN ceramic tablets sintered at different temperatures is about 6.7521 . The density of the respective measured tablets is usually amounting more than 91% and 5 approaching a maximum value of 96.5% for sintering temperature of 1150°C. The microstructure was investigated by using Scanning Transmission Electron Microscope (STEM), X-ray diffraction (XRD). Combined with the results obtained from the STEM and XRD, the impact of sintering temperature on the macroscopic and microscopic structure was discussed. The relative dielectric constant ( ) and dielectric loss ( ) of the BMN solid solutions were measured to be 161-200 and (at room temperature and 100Hz-1MHz), respectively. The BMN solid solutions have relative high dielectric constant and low dielectric loss. With increasing sintering temperature, the dielectric constant showed the maximum at 1150°C. The leakage current of BMN ceramic material is extraordinary small. When the voltage and thickness of the BMN capacitor are 4000V and 300um, the leakage current amounts only about 0.13-0.65 . The excellent physical and electrical properties make BMN thin films promising for potential tunable capacitor applications.

Sensitivity analysis quantifies the effect that of perturbations of the model inputs have on the model's outputs. Some of the key insights gained using sensitivity analysis are to understand the robustness of the model with respect to perturbations, and to select the most important parameters for the model. MATLODE is a tool for sensitivity analysis of models described by ordinary differential equations (ODEs). MATLODE implements two distinct approaches for sensitivity analysis: direct (via the tangent linear model) and adjoint. Within each approach, four families of numerical methods are implemented, namely explicit Runge-Kutta, implicit Runge-Kutta, Rosenbrock, and single diagonally implicit Runge-Kutta. Each approach and family has its own strengths and weaknesses when applied to real world problems. MATLODE has a multitude of options that allows users to find the best approach for a wide range of initial value problems. In spite of the great importance of sensitivity analysis for models governed by differential equations, until this work there was no MATLAB ordinary differential equation sensitivity analysis toolbox publicly available. The two most popular sensitivity analysis packages, CVODES [8] and FATODE [10], are geared toward the high performance modeling space; however, no native MATLAB toolbox was available. MATLODE fills this need and offers sensitivity analysis capabilities in MATLAB, one of the most popular programming languages within scientific communities such as chemistry, biology, ecology, and oceanogra- phy. We expect that MATLODE will prove to be a useful tool for these communities to help facilitate their research and fill the gap between theory and practice. / Master of Science

A Julia set for a complex function 𝑓 is the set of all points in the complex plane where the iterates of 𝑓 do not form a normal family. A picture of the Julia set for a function can be generated with a computer by coloring pixels (which we consider to be small squares) based on the behavior of the point at the center of each pixel. We consider the accuracy of computer generated pictures of Julia sets. Such a picture is said to be accurate if each colored pixel actually contains some point in the Julia set. We extend previous work to show that the pictures generated by an algorithm for the family λe² are accurate, for appropriate choices of parameters in the algorithm. We observe that the Julia set for meromorphic functions with polynomial Schwarzian derivative is the closure of those points which go to infinity under iteration, and use this as a basis for an algorithm to generate pictures for such functions. A pixel in our algorithm will be colored if the center point becomes larger than some specified bound upon iteration. We show that using our algorithm, the pictures of Julia sets generated for the family λtan(z) for positive real λ are also accurate. We conclude with a cautionary example of a Julia set whose picture will be inaccurate for some apparently reasonable choices of parameters, demonstrating that some care must be exercised in using such algorithms. In general, more information about the nature of the function may be needed. / Ph. D.

Nous nous intéressons à la modélisation de formes complexes de type structures arborescences, formes lacunaires ou surfaces rugueuses. Ces formes sont intéressantes de par leurs propriétés physiques particulières :objets légers, économie de matière, résistance mécanique, absorption acoustique importante. Les modèles basés sur le concept de la géométrie fractale permettent de générer de telles formes et notamment les formes auto-similaires. A partir des travaux de Barnsley sur les systèmes itérés de fonctions, Tosan et al, ont proposé une extension, Boundary Controled Iterated Funcions Systems (BCIFS) pour contrôler plus facilement les formes et faciliter leur description. Nous nous intéressons aux propriétés différentielles des formes décrites par BCIFS. Nous proposons une définition plus générale d'espace tangent qui permet de caractériser le comportement de cas non-classiquement différentiables.Nous montrons que l'étude du comportement différentiel peut alors se faire simplement par analyse des valeurs propres et vecteurs propres généralisés des opérateurs de subdivision. Il devient alors possible de contrôler ces propriétés différentielles. Nous présentons une application de nos résultats, en proposant une méthode pour construire des raccords entre deux structures définies par des processus de subdivision différents. Cette méthode est appliquée pour la construction d'un raccord entre une surface de subdivision de Doo-Sabin(schéma dual) et une surface de subdivision de Catmull-Clark (schéma primal) / The fractal geometry is a relatively new branch of mathematics that studies complex objects of non-integer dimensions. It finds applications in many branches of science as objects of such complex structure often poses interesting properties. In 1988 Barnsley presented the Iterative Func-tion System (IFS) model that allows modelling complex fractal shapes with only a limited set of contractive transformations. Later many other models were based on the IFS model such as Language-Restricted IFS,Projective IFS, Controlled IFS and Boundary Controlled IFS. The lastto allow modelling complex shapes with control points and specific topol-ogy. These models cover classical geometric models such as B-splines and subdivision surfaces as well as fractal shapes.This thesis focuses on the analysis of the differential behaviour of the shapes described with Controlled IFS and Boundary Controlled IFS. Wederive the necessary and sufficient conditions for differentiability for ev-erywhere dense set of points. Our study is based on the study of the eigenvalues and eigenvectors of the transformations composing the IFS. We apply the obtained conditions to modelling curves in surfaces. We describe different examples of differential behaviour presented in shapes modelled with Controlled IFS and Boundary Controlled IFS. We also use the Boundary Controlled IFS to solve the problem of connecting different subdivision schemes. We construct a junction between Doo-Sabin and Catmull-Clark subdivision surfaces and analyse the differential behaviour of the intermediate surface

Polynomial neural networks (NNs-Hp) have recently demonstrated high expressivity and efficiency across several tasks. However, a theoretical explanation toward such success is still unclear, especially when compared to the classical neural networks. Neural tangent kernel (NTK) is a powerful tool to analyze the training dynamics of neural networks and their generalization bounds. The study on NTK has been devoted to typical neural network architectures, but is incomplete for NNs-Hp. In this work, we derive the finite-width NTK formulation for NNs-Hp, and prove their equivalence to the kernel regression predictor with the associated NTK, which expands the application scope of NTK. Based on our results, we elucidate the difference of NNs-Hp over standard neural networks with respect to extrapolation and spectral bias. Our two key insights are that when compared to standard neural networks, a) NNs-Hp are able to fit more complicated functions in the extrapolation region; and b) NNs-Hp admit a slower eigenvalue decay of the respective NTK. Our empirical results provide a good justification for a deeper understanding of NNs-Hp / Polynomiska neurala nätverk (NNs-Hp) har nyligen visat hög uttrycksförmåga och effektivitet över flera uppgifter. En teoretisk förklaring till sådan framgång är dock fortfarande oklar, särskilt jämfört med de klassiska neurala nätverken. Neurala tangentkärnor (NTK) är ett kraftfullt verktyg för att analysera träningsdynamiken i neurala nätverk och deras generaliseringsgränser. Studien om NTK har ägnats åt typiska neurala nätverksarkitekturer, men är ofullständig för NNs-Hp. I detta arbete härleder vi NTK-formuleringen med ändlig bredd för NNs-Hp och bevisar deras likvärdighet med kärnregressionsprediktorn med den associerade NTK, vilket utökar tillämpningsomfånget för NTK. Baserat på våra resultat belyser vi skillnaden mellan NNs-Hp jämfört med standardneurala nätverk med avseende på extrapolering och spektral bias. Våra två viktiga insikter är att jämfört med vanliga neurala nätverk, a) NNs-Hp kan passa mer komplicerade funktioner i extrapolationsregionen; och b) NNs-Hp medger en långsammare egenvärdesavklingning av respektive NTK. Våra empiriska resultat ger en bra motivering för en djupare förståelse av NNs-Hp.

In this work is analysed the tangent bundle geometry order 3. Those bundles are defined like 3 - jet space. Co - ordinates transformation formulas of those bundles are received, how the object of linear connection inducted affine connections is demonstrated. In this work the theorem how the object of linear connection of tangent bundle inducted linear connection of tangent bundle order 3 is proved.

This work is focused on thermodynamic modelling of isobaric vapour-liquid-liquid equilibrium (VLLE) (homogeneous) and (heterogeneous) for binary, ternary and quaternary systems. This work uses data for organic/aqueous systems; historically these mixtures were used in the production of penicillin and were required to be separated by continuous fractional distillation. Modelling of the separation required phase equilibrium data to be available so that predictions could be made for equilibrium stage temperatures, vapour compositions, liquid compositions and any phase splitting occurring in the liquid phase. Relevant data became available in the literature and work has been carried out to use relevant theories in correlating and predicting as was originally required in the distillation equilibrium stage modelling. All the modelling carried out was at atmospheric pressure. The modelling has been done using an Equation of State, specifically Peng Robinson Styrjek Vera (PRSV), combined with the activity coefficient model UNIversal QUAsi Chemical (UNIQUAC) through Wong Sandler mixing rules (WSMR). The success of all correlations and predictions was justified by minimizing the value of the Absolute Average Deviation (AAD) as defined within the thesis. Initially the integral Area Method and a method called Tangent Plane Intersection (TPI) were used in the prediction of liquid-liquid equilibrium (LLE) binary systems. This work used a modified 2-point search, suggested a 3-point search and has successfully applied both of these methods to predict VLLE for binary systems. It was discovered through the application of the TPI on ternary VLLE systems that the method was strongly sensitive to initial values. This work suggested and tested a Systematic Initial Generator (SIG) to provide the TPI method with realistic initial values close to the real solution and has demonstrated the viability of the SIG on improving the accuracy of the TPI results for the ternary systems investigated. In parallel with the TPI another method the Tangent Plane Distance Function (TPDF) was also investigated. This method is based on the minimisation of Gibbs free energy function related to the Gibbs energy surface. This method consistently showed it was capable of predicting VLLE for both ternary and quaternary systems as demonstrated throughout this work. The TPDF method was found to be computationally faster and less sensitive to the initial values. Some of the methods investigated in this work were also found to be applicable as phase predictors and it was discovered that the TPDF and the SIG methods were successful in predicting the phase regions; however the TPI method failed in identifying the 2 phase region. Applying the techniques described to newly available quaternary data has identified the strengths and weaknesses of the methods. This work has expanded the existing knowledge and developed a reliable model for design, operation and optimisation of the phase equilibria required for prediction in many separation processes. Currently available modelling simulation packages are variable in their predictions and sometimes yield unsatisfactory predictions. Many of the current uses of VLLE models are particularly focused on Hydrocarbon/Water systems at high pressure. The work described in this thesis has demonstrated that an EOS with suitable mixing rules can model and predict data for polar organic liquids at atmospheric and below atmospheric pressure and offers the advantage of using the same modelling equations for both phases.

Bundles and morphisms between bundles are defined in the category of Fr¨olicher spaces (earlier known as the category of smooth spaces, see [2], [5], [9], [6] and [7]). We show that the sections of Fr¨olicher bundles are Fr¨olicher smooth maps and the fibers of Fr¨olicher bundles have a Fr¨olicher structure. We prove in detail that the tangent and cotangent bundles of a n-dimensional pseudomanifold are locally diffeomorphic to the even-dimensional Euclidian canonical F-space R2n. We define a bilinear form on a finite-dimensional pseudomanifold. We show that the symplectic structure on a cotangent bundle in the category of Fr¨olicher spaces exists and is (locally) obtained by the pullback of the canonical symplectic structure of R2n. We define the notion of symplectomorphism between two symplectic pseudomanifolds. We prove that two cotangent bundles of two diffeomorphic finite-dimensional pseudomanifolds are symplectomorphic in the category of Frölicher spaces.

Cette thèse porte sur l'étude de la géométrie de l'espace de spécialisation φ : (X, 0) → (C, 0) d'un germe de singularité analytique complexe (X, 0) sur son cône tangent (CX,0 , 0) du point de vue de l'équisingularité à la Whitney. L'application φ nous donne une famille plate des germes avec section tel que pour chaque t =! 0 le germe φ−1 (t) est isomorphe à (X, 0) et la fibre spéciale est isomorphe au cône tangent. Le but est de établir des conditions sur les strates de la stratification de Whitney minimale de (X, 0) qui assurent l'équisingularité du germe et son cône tangent, generalisant ainsi le résultat de Lê et Teissier pour les hypersurfaces de C3 qui prouve que l'absence des tangentes exceptionnelles est suffisant. Dans ce travail on montre que cette condition est nécessaire et suffisante dans le cas général pour la strate de codimension zero. L'un des ingrédients clés dans la preuve est la théorie de la dépendance integrale sur des ideaux et des modules développé par Teissier, Lejeune, Gaffney, Kleiman, etc, qu'on rappelle au troisième chapitre et où l'on obtient des résultats spécifiques pour cette situation. Les deux premiers chapitres correspondent aux préliminaires, on commence par rappeller la modification de Nash et l'espace conormal d'un espace analytique plongé dans ses versions absolues et relatives à un morphisme et on donne une description explicite de la relation entre le conormal (Nash) relatif de φ : (X, 0) → (C, 0) et le conormal (Nash) de (X, 0). Dans le deuxième chapitre on définit le diagram normal/conormal, l'auréole du germe (X, 0), les cônes exceptionnelles, et on énonce les résultats principaux correspondant à l'équisingularité à la Whitney en incluant la caractérisation des conditions de Whitney en termes du diagramme normal/conormal.