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The general scope of this thesis is aimed at investigating certain classes of discrete equations through the analysis of certain characteristics of the solutions of these equations. We construct new methods of analysis based on the growth of these characteristics that let us single out known integrable discrete equations from certain class of equations. These integrable discrete equations are discrete analogues of the famous Painleve equations.

This thesis explores and implements techniques for frequency domain modelling and time domain simulation of overhead transmission lines. The popular Vector Fitting algorithm is employed to approximate the frequency domain model using rational functions, and the recursive convolution technique is applied to the rational approximation to generate a time domain form. The frequency domain model is translated into the time domain using delay extraction, modal decomposition, passivity enforcement, and rational approximation. Several approaches to each of these procedures are investigated. The thesis also discusses several choices for the integration method used within the recursive convolution procedure. In order to make the transmission line modeller and simulator easy to use, a Java-based library and partial graphical interface were developed. Specifically, the goal was to develop a platform-independent program that can run either stand-alone or as an applet inside a web page. / October 2005

This thesis explores and implements techniques for frequency domain modelling and time domain simulation of overhead transmission lines. The popular Vector Fitting algorithm is employed to approximate the frequency domain model using rational functions, and the recursive convolution technique is applied to the rational approximation to generate a time domain form. The frequency domain model is translated into the time domain using delay extraction, modal decomposition, passivity enforcement, and rational approximation. Several approaches to each of these procedures are investigated. The thesis also discusses several choices for the integration method used within the recursive convolution procedure. In order to make the transmission line modeller and simulator easy to use, a Java-based library and partial graphical interface were developed. Specifically, the goal was to develop a platform-independent program that can run either stand-alone or as an applet inside a web page.

This thesis explores and implements techniques for frequency domain modelling and time domain simulation of overhead transmission lines. The popular Vector Fitting algorithm is employed to approximate the frequency domain model using rational functions, and the recursive convolution technique is applied to the rational approximation to generate a time domain form. The frequency domain model is translated into the time domain using delay extraction, modal decomposition, passivity enforcement, and rational approximation. Several approaches to each of these procedures are investigated. The thesis also discusses several choices for the integration method used within the recursive convolution procedure. In order to make the transmission line modeller and simulator easy to use, a Java-based library and partial graphical interface were developed. Specifically, the goal was to develop a platform-independent program that can run either stand-alone or as an applet inside a web page.

abstract: The advancement of technology has substantively changed the practices of numerous professions, including teaching. When an instructor first adopts a new technology, established classroom practices are perturbed. These perturbations can have positive and negative, large or small, and long- or short-term effects on instructors’ abilities to teach mathematical concepts with the new technology. Therefore, in order to better understand teaching with technology, we need to take a closer look at the adoption of new technology in a mathematics classroom. Using interviews and classroom observations, I explored perturbations in mathematical classroom practices as an instructor implemented virtual manipulatives as novel didactic objects in rational function instruction. In particular, the instructor used didactic objects that were designed to lay the foundation for developing a conceptual understanding of rational functions through the coordination of relative size of the value of the numerator in terms of the value of the denominator. The results are organized according to a taxonomy that captures leader actions, communication, expectations of technology, roles, timing, student engagement, and mathematical conceptions. / Dissertation/Thesis / Doctoral Dissertation Mathematics Education 2017

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No description available.

Molecular nonlinear optical (NLO) properties are extensively studied using both theory and experiment because of their use in myriad applications. Experimental measurements of the most interesting molecules’ NLO properties are difficult, so experimental data for molecules with desirable NLO properties is scarce. Theoretical tools don’t suffer from the same limitations and can provide significant insights into the physico-chemical phenomena underlying the nonlinear responses, can help in interpreting response behaviour of molecules, and can guide design the materials with desirable response properties. Here, I present my work on developing methods for accurately calculating the NLO properties of molecules using the finite field (FF) approach. The first chapter provides a background for the finite field and electronic structure methods used in this dissertation. Chapter two is a thorough investigation of the finite field method. The limitations of the method are highlighted and the optimal conditions for overcoming its drawbacks and obtaining meaningful and accurate results are described. Chapter three presents the first systematic study of the dependence of optimal field strengths on molecular descriptors. The first protocol for predicting the optimal field for the second hyperpolarizability is presented and successfully tested, and the dependence of the optimal field strength for the first hyperpolarizability on the molecular structure is investigated. Chapter four is an assessment of various DFT functionals in calculating the second hyperpolarizabilities of organic molecules and oligomers. This study shows the limitations of conventional DFT methods and the importance of electron correlation to response properties. In chapter five we present a new method of calculating NLO properties using a rational function model that is shown to be more robust and have lower computational cost than the traditional Taylor expansion. Finally, chapter six includes a summary of the thesis and an overview of future work. / Thesis / Doctor of Philosophy (PhD)

Les progrès constants en miniaturisation des transistors et l’augmentation des fréquences des signaux utilisés sont les principales tendances dans l’évolution des circuits électroniques. Avec ces évolutions apparaissent de nombreux effets indésirables qui perturbent le comportement des systèmes électroniques et sont soupçonnés d’être responsables de la majorité des dégradations de signaux dans les systèmes en haute fréquence. Des retards de propagation indésirables sont ainsi introduits par la présence des interconnexions, et la diaphonie, phénomène dû aux couplages entre lignes d’interconnexions, peut éventuellement provoquer des commutations non désirées des transistors. La prise en compte des interconnexions, dès les premières phases de conception d'un système, est par conséquent devenue une nécessité ces dernières années. Mais la simulation temporelle d’un réseau d’interconnexions est très gourmande en temps de calcul, ce qui impacte la durée globale de conception. Le remplacement des modèles électriques, décrivant précisément les interconnexions, par des modèles plus simples est primordial pour limiter les coûts de calcul. Une méthode de réduction d'ordre des modèles peut alors être employée pour effectuer cette opération efficacement. Le modèle final doit en effet décrire assez précisément certains aspects importants du modèle original et conserver les propriétés importantes du réseau d'interconnexions. Cette démarche permettra aux concepteurs d’effectuer des simulations temporelles rapides et d’étudier les paramètres d’intégrité du signal tel que le retard, le temps de montée, le dépassement….L'objectif de cette thèse est d’établir un nouvel outil de réduction de complexité des modèles de réseaux d'interconnexions. Différentes descriptions initiales des systèmes d'interconnexions sont envisagées : modèles circuits (fonctions de transfert) ou mesures fréquentielles. L’approche développée repose sur l’utilisation des fonctions orthogonales de Müntz-Laguerre et de Kautz afin de décrire mathématiquement, de manière précise, le système d'origine. Un opérateur linéaire, lié à ces fonctions de base, est ensuite appliqué pour déterminer un modèle rationnel de moindre complexité. La technique proposée est comparée à d'autres méthodes de la littérature d’abord sur des exemples académiques. Tout le potentiel de la méthode est ensuite illustré par sa mise en œuvre sur des réseaux d'interconnexions. / The ongoing progress in transistor miniaturization and a continuous frequency increase are the main trends in the present day evolution of electronic circuits. A number of undesired effects are intrinsic to these developments and are suspected to be responsible for most of the flawed signals present in high frequency systems. Parasitic delays are thus introduced by the presence of interconnect lines and crosstalk due to coupling may lead to undesired switching events in transistor circuits. Accounting for the presence of interconnect lines, at a very early stage in the design flow has become unavoidable in recent years. However, time domain simulations of massively coupled interconnect networks may be computationally costly and have a tremendous impact on the overall duration of the design process. Replacing complex, high order circuit models by more compact surrogates is thus necessary. Model order reduction is an effective way to derive such surrogates. The final model must mimic certain aspects of the original model with sufficient accuracy and preserve the interconnect network’s most important properties. This approach enables designers to account for the undesired effects of interconnect lines such as, delays, rise-times and overshoots while maintaining the overall duration of time-domain simulations within acceptable limits. The aim of this thesis is to create a new model order reduction tool applicable to complex interconnect networks. Different initial representations were considered – circuit models (transfer functions) or frequency domain measurements. The proposed approach uses orthogonal basis functions such as Müntz-Laguerre and Kautz to build an accurate mathematical representation of the original system .A linear operator, related to these functions, is subsequently used to derive a simplified model. The technique is first compared to other approaches using examples available in literature, its full potential being demonstrated on coupled interconnect models.