The electromagnetic coupling of a microstrip transmission line (MS-TL) to a metamaterial backward wave Negative-Refractive-Index transmission line (NRI-TL) is the primary investigation of this dissertation. The coupling of forward waves in the MS-TL to the backward waves in the NRI-TL results in the formation of complex modes, characterized by simultaneous phase progression and attenuation along the lossless lines. Through network-theoretic considerations, we investigate the properties of these modes in the complex-frequency plane of the Laplace domain to help unravel the confusion that has existed in the literature regarding the independent excitation of a pair of conjugate complex modes. We show that it is possible to arbitrarily suppress one of the modes over a finite bandwidth and completely eliminate it at a discrete set of frequencies using proper source and load impedances. Hence we use conjugate modes with independent amplitudes in our eigenmode expansion when we analyse various coupling configurations between the two types of lines (MS/NRI-TL coupler). We derive approximate closed-form expression for the scattering parameters of the MS/NRI-TL coupler and these are complemented by design charts that allow the synthesis of a wide range of specifications. Moreover, these expressions reveal that such couplers allow for arbitrary backward coupling levels along with very high-isolation when they are made half a guided wavelength long. The MS/NRI-TL coupler offers some interesting applications which we highlight through the design and testing of a 3-dB power splitter, a high-directivity signal monitor and a compact corporate power divider. We have included design, simulation and experimental data for the fabricated prototypes exhibiting good agreement and thereby justifying the theory that has been developed in this work to explain the coupling between a right-handed MS-TL and a left-handed NRI-TL.

The electromagnetic coupling of a microstrip transmission line (MS-TL) to a metamaterial backward wave Negative-Refractive-Index transmission line (NRI-TL) is the primary investigation of this dissertation. The coupling of forward waves in the MS-TL to the backward waves in the NRI-TL results in the formation of complex modes, characterized by simultaneous phase progression and attenuation along the lossless lines. Through network-theoretic considerations, we investigate the properties of these modes in the complex-frequency plane of the Laplace domain to help unravel the confusion that has existed in the literature regarding the independent excitation of a pair of conjugate complex modes. We show that it is possible to arbitrarily suppress one of the modes over a finite bandwidth and completely eliminate it at a discrete set of frequencies using proper source and load impedances. Hence we use conjugate modes with independent amplitudes in our eigenmode expansion when we analyse various coupling configurations between the two types of lines (MS/NRI-TL coupler). We derive approximate closed-form expression for the scattering parameters of the MS/NRI-TL coupler and these are complemented by design charts that allow the synthesis of a wide range of specifications. Moreover, these expressions reveal that such couplers allow for arbitrary backward coupling levels along with very high-isolation when they are made half a guided wavelength long. The MS/NRI-TL coupler offers some interesting applications which we highlight through the design and testing of a 3-dB power splitter, a high-directivity signal monitor and a compact corporate power divider. We have included design, simulation and experimental data for the fabricated prototypes exhibiting good agreement and thereby justifying the theory that has been developed in this work to explain the coupling between a right-handed MS-TL and a left-handed NRI-TL.

L’objectif de cette thèse est l’étude d’un schéma numérique pour l’approximation des solutions d’équations différentielles doublement stochastiques rétrogrades (EDDSR). Durant les deux dernières décennies, plusieurs méthodes ont été proposées afin de permettre la résolution numérique des équations différentielles stochastiques rétrogrades standards. Dans cette thèse, on propose une extension de l’une de ces méthodes au cas doublement stochastique. Notre méthode numérique nous permet d’attaquer une large gamme d’équations aux dérivées partielles stochastiques (EDPS) nonlinéaires. Ceci est possible par le biais de leur représentation probabiliste en termes d’EDDSRs. Dans la dernière partie, nous étudions une nouvelle méthode des particules dans le cadre des études de protection en neutroniques. / The purpose of this thesis is to study a numerical method for backward doubly stochastic differential equations (BDSDEs in short). In the last two decades, several methods were proposed to approximate solutions of standard backward stochastic differential equations. In this thesis, we propose an extension of one of these methods to the doubly stochastic framework. Our numerical method allows us to tackle a large class of nonlinear stochastic partial differential equations (SPDEs in short), thanks to their probabilistic interpretation. In the last part, we study a new particle method in the context of shielding studies.

L'objectif de cette thèse est l'étude de la représentation probabiliste des différentes classes d'EDPSs non-linéaires(semi-linéaires, complètement non-linéaires, réfléchies dans un domaine) en utilisant les équations différentielles doublement stochastiques rétrogrades (EDDSRs). Cette thèse contient quatre parties différentes. Nous traitons dans la première partie les EDDSRs du second ordre (2EDDSRs). Nous montrons l'existence et l'unicité des solutions des EDDSRs en utilisant des techniques de contrôle stochastique quasi- sure. La motivation principale de cette étude est la représentation probabiliste des EDPSs complètement non-linéaires. Dans la deuxième partie, nous étudions les solutions faibles de type Sobolev du problème d'obstacle pour les équations à dérivées partielles inteégro-différentielles (EDPIDs). Plus précisément, nous montrons la formule de Feynman-Kac pour l'EDPIDs par l'intermédiaire des équations différentielles stochastiques rétrogrades réfléchies avec sauts (EDSRRs). Plus précisément, nous établissons l'existence et l'unicité de la solution du problème d'obstacle, qui est considérée comme un couple constitué de la solution et de la mesure de réflexion. L'approche utilisée est basée sur les techniques de flots stochastiques développées dans Bally et Matoussi (2001) mais les preuves sont beaucoup plus techniques. Dans la troisième partie, nous traitons l'existence et l'unicité pour les EDDSRRs dans un domaine convexe D sans aucune condition de régularité sur la frontière. De plus, en utilisant l'approche basée sur les techniques du flot stochastiques nous démontrons l'interprétation probabiliste de la solution faible de type Sobolev d'une classe d'EDPSs réfléchies dans un domaine convexe via les EDDSRRs. Enfin, nous nous intéressons à la résolution numérique des EDDSRs à temps terminal aléatoire. La motivation principale est de donner une représentation probabiliste des solutions de Sobolev d'EDPSs semi-linéaires avec condition de Dirichlet nul au bord. Dans cette partie, nous étudions l'approximation forte de cette classe d'EDDSRs quand le temps terminal aléatoire est le premier temps de sortie d'une EDS d'un domaine cylindrique. Ainsi, nous donnons les bornes pour l'erreur d'approximation en temps discret. Cette partie se conclut par des tests numériques qui démontrent que cette approche est effective. / The objective of this thesis is to study the probabilistic representation (Feynman-Kac for- mula) of different classes ofStochastic Nonlinear PDEs (semilinear, fully nonlinear, reflected in a domain) by means of backward doubly stochastic differential equations (BDSDEs). This thesis contains four different parts. We deal in the first part with the second order BDS- DEs (2BDSDEs). We show the existence and uniqueness of solutions of 2BDSDEs using quasi sure stochastic control technics. The main motivation of this study is the probabilistic representation for solution of fully nonlinear SPDEs. First, under regularity assumptions on the coefficients, we give a Feynman-Kac formula for classical solution of fully nonlinear SPDEs and we generalize the work of Soner, Touzi and Zhang (2010-2012) for deterministic fully nonlinear PDE. Then, under weaker assumptions on the coefficients, we prove the probabilistic representation for stochastic viscosity solution of fully nonlinear SPDEs. In the second part, we study the Sobolev solution of obstacle problem for partial integro-differentialequations (PIDEs). Specifically, we show the Feynman-Kac formula for PIDEs via reflected backward stochastic differentialequations with jumps (BSDEs). Specifically, we establish the existence and uniqueness of the solution of the obstacle problem, which is regarded as a pair consisting of the solution and the measure of reflection. The approach is based on stochastic flow technics developed in Bally and Matoussi (2001) but the proofs are more technical. In the third part, we discuss the existence and uniqueness for RBDSDEs in a convex domain D without any regularity condition on the boundary. In addition, using the approach based on the technics of stochastic flow we provide the probabilistic interpretation of Sobolev solution of a class of reflected SPDEs in a convex domain via RBDSDEs. Finally, we are interested in the numerical solution of BDSDEs with random terminal time. The main motivation is to give a probabilistic representation of Sobolev solution of semilinear SPDEs with Dirichlet null condition. In this part, we study the strong approximation of this class of BDSDEs when the random terminal time is the first exit time of an SDE from a cylindrical domain. Thus, we give bounds for the discrete-time approximation error.. We conclude this part with numerical tests showing that this approach is effective.

The moral justification of punishment is the fundamental concern of this thesis. It is argued that a moral response to crime has to be a civilised response; therefore, the notion of "civility" is defined and discussed. Punishment is then defended in such a way that it accords with being a civilised response to crime. It is argued that in order to be such a response, and thereby qualify as a moral response, punishment must have a certain structure, i.e. it must fulfil seven necessary conditions, which, it is argued, together constitute the sufficient condition for morally justified punishment. In arguing for each of the necessary conditions, different onedimensional theories of punishment are dealt with (retributivism, utilitarianism, deterrence theory, rehabilitationism, a paternalistic theory of punishment, and restitutionalism}, indicating that each fulfils some of the criteria for morally justified punishment. None of the onedimensional theories fulfils all the necessary conditions, however, and hence none on its own fulfils the sufficient condition for morally justified punishment. This is not to argue that a straightforward theory could never on its own fulfil the conditions for morally justified punishment, but I have not been able to conceive how this could be done. The theory I here present is therefore a hybrid approach, incorporating elements of all the above-mentioned theories into a unitary theory. In doing so, it fulfils all the necessary conditions for being a civilised response to crime, thereby fulfilling the sufficient condition too, and hence providing a morally defensible account of punishment. Finally, the question of how this theory can be put into practice is addressed. Because the objective of punishment ought to be a civilised response, thereby benefiting both society as a whole and those being punished and rehabilitated, the thesis may be seen as a progressive synthesis of the various approaches examined. / Philosophy, Practical & Systematic Theology / D. Litt. et Phil. (Philosophy)

Nous étudions la convergence de systèmes dynamiques vers des équilibres. En particulier, nous nous intéressons à deux types d'équilibres. D'une part, les solutions d'inéquations variationnelles sous contraintes qui interviennent aussi dans la résolution de problèmes d'optimisation hiérarchique. D'autre part l'état stable d'un système dynamique, c'est à dire l'état où l'énergie du système est nulle. Cette thèse est divisée en deux parties principales, chacune focalisée sur la recherche d'un de ces équilibres. Dans la première partie nous étudions une classe d'algorithmes explicite-implicites pour résoudre certaines inéquations variationnelles sous contraintes. Nous introduisons un algorithme proximal-gradient pénalisé, "splitting forward-backward penalty scheme". Ensuite, nous prouvons sa convergence ergodique faible vers un équilibre dans le cas général d'un opérateur maximal monotone, et sa convergence forte vers l'unique équilibre si l'opérateur est de plus fortement monotone. Nous appliquons aussi notre algorithme pour résoudre des problèmes d'optimisation sous contrainte ou hiérarchique dont les fonctions objectif et de pénalisation sont formées d'une partie lisse et d'une autre non lisse. En effet, nous démontrons la convergence faible de l'algorithme vers un optimum hiérarchique lorsque l'opérateur est le sous-différentiel d'une fonction convexe semi-continue inférieurement et propre. Nous généralisons ainsi plusieurs algorithmes connus et nous retrouvons leurs résultats de convergence en affaiblissant les hypothèses utilisées dans nombre d'entre eux.Dans la deuxième partie, nous étudions l'action d'un contrôle interne local sur la stabilisation indirecte d'un système dynamique couplé formé de trois équations d'ondes, le système de Bresse. Sous la condition d'égalité des vitesses de propagation des ondes, nous montrons la stabilité exponentielle du système. En revanche, quand les vitesses sont différentes, nous prouvons sa stabilité polynomiale et nous établissons un nouveau taux de décroissance polynomial de l'énergie. Ceci étend des résultats présents dans la littérature au sens où le contrôle est localement distribué (et non pas appliqué à tout le domaine) et nous améliorons le taux de décroissance polynomial de l'énergie pour des conditions au bord de type Dirichlet et Dirichlet-Neumann. / We study the convergence of dynamical systems towards equilibria. In particular, we are interested in two types of equilibria. On one hand solutions of constrained variational inequations that are also involved in the resolution of hierarchical optimization problems. On the other hand the stable state of a dynamical system, i.e. the state when the energy of the system is zero. The thesis is divided into two parts, each focused on one of these equilibria. In the first part, we study a class of forward-backward algorithms for solving constrained variational inequalities. We consider a splitting forward-backward penalty scheme. We prove the weak ergodic convergence of the algorithm to an equilibrium for a general maximal monotone operator, and the strong convergence to the unique equilibrium if the operator is an addition strongly monotone. We also apply our algorithm for solving constrained or hierarchical optimization problems whose objective and penalization functions are formed of a smooth and a non-smooth part. In fact, we show the weak convergence to a hierarchical optimum when the operator is the subdifferential of a closed convex proper function. We then generalize several known algorithms and we find their convergence results by weakening assumptions used in a number of them. In the second part, we study the action of a locally internal dissipation law in the stabilization of a linear dynamical system coupling three wave equations, the Bresse system. Under the equal speed wave propagation condition we show that the system is exponentially stable. Otherwise, when the speeds are different, we prove the polynomial stability and establish a new polynomial energy decay rate. This extends results presented in the literature in the sense that the dissipation law is locally distributed (and not applied in the whole domain) and we improve the polynomial energy decay rate with both types of boundary conditions, Dirichlet and Dirichlet-Neumann.

In this thesis, we study the existence of pathwise random periodic solutions to both the semilinear stochastic differential equations with linear multiplicative noise and the semilinear stochastic partial differential equations with linear multiplicative noise in a Hilbert space. We identify them as the solutions of coupled forward-backward infinite horizon stochastic integral equations in general cases, and then perform the argument of the relative compactness of Wiener-Sobolev spaces in C([0, T],L2Ω,Rd)) or C([0, T],L2(Ω x O)) and Schauder's fixed point theorem to show the existence of a solution of the coupled stochastic forward-backward infinite horizon integral equations.

Doctor of Philosophy / Department of Physics / Dean Zollman / As part of a collaborative study of the science preparation of elementary school teachers, we investigated the quality of students’ reasoning and explored the relationship between sophistication of reasoning and the degree to which the courses were considered inquiry oriented. To probe students’ reasoning, we developed open-ended written content questions with the distinguishing feature of applying recently learned concepts in a new context. We devised a protocol for developing written content questions that provided a common structure for probing and classifying students’ sophistication level of reasoning. In designing our protocol, we considered several distinct criteria, and classified students’ responses based on their performance for each criterion. First, we classified concepts into three types: Descriptive, Hypothetical, and Theoretical and categorized the abstraction levels of the responses in terms of the types of concepts and the inter-relationship between the concepts. Second, we devised a rubric based on Bloom’s revised taxonomy with seven traits (both knowledge types and cognitive processes) and a defined set of criteria to evaluate each trait. Along with analyzing students’ reasoning, we visited universities and observed the courses in which the students were enrolled. We used the Reformed Teaching Observation Protocol (RTOP) to rank the courses with respect to characteristics that are valued for the inquiry courses. We conducted logistic regression for a sample of 18 courses with about 900 students and reported the results for performing logistic regression to estimate the relationship between traits of reasoning and RTOP score. In addition, we analyzed conceptual structure of students’ responses, based on conceptual classification schemes, and clustered students’ responses into six categories. We derived regression model, to estimate the relationship between the sophistication of the categories of conceptual structure and RTOP scores. However, the outcome variable with six categories required a more complicated regression model, known as multinomial logistic regression, generalized from binary logistic regression. With the large amount of collected data, we found that the likelihood of the higher cognitive processes were in favor of classes with higher measures on inquiry. However, the usage of more abstract concepts with higher order conceptual structures was less prevalent in higher RTOP courses.

A better way to improve understanding and quantification of nitrous oxide (N₂O) emitted from intensive maize cropping systems is to develop an advanced emissions measurement method This study developed an open path (OP) method to measure N₂O emissions from four adjacent maize plots managed by tillage practices of no-till (NT) and chisel plow (ChP), and different nitrogen (N) treatments from 2014 to 2016. Anhydrous ammonia (220 kg NH₃-N ha^-1) was applied in once or equally split (full vs. split rate) and applied in different timing (Fall vs. Spring). The spring N application occurred either before planting (pre-plant) or in season (side-dress). Emissions measurements were conducted by using the OP method (the scanning OP Fourier transform infrared spectrometry (OP-FTIR) + the gas point-sampling system + a backward Lagrangian stochastic (bLS) dispersion model) and static closed chamber methods. The performance and feasibility of the OP measurements were assessed by a sensitivity analysis, starting with errors associated with the OP-FTIR for calculating N₂O concentrations, and then errors associated with the bLS model for estimating N₂O emissions. The quantification of N₂O concentrations using the OP-FTIR spectrum was influenced by ambient humidity, temperature, and the path length between a spectrometer and a retro-reflector. The optimal quantitative method mitigated these ambient interference effects on N₂O quantification. The averaged bias of the calculated N₂O concentrations from the spectra acquired from wide ranges of humidity (0.5 – 2.0 % water vapor content), temperature (10 – 35 °C), and path length (100 – 135 meters) was 1.4 %. The precision of the OP-FTIR N₂O concentrations was 5.4 part per billion(3σ) in a stationary flow condition for a 30-minute averaging period. The emissions measurement from multiple sources showed that the field of interest was likely interfered by adjacent fields. Fields with low emission rates were more sensitive to the adjacent fields with high emissions, resulting in substantial biases and uncertainties. The minimum detection limit of the N₂O emission rates was 1.2 µg m^-2 s^-1 (MDL; 3σ). The OP measurements showed that the NT practice potentially reduced N₂O emission compared with ChP. Under the long-term NT treatments, the split-N rate application (110 kg NH₃-N ha^-1 in the fall and spring) resulted in lower N₂O emissions than the full application (220 kg NH₃-N ha^-1 in the fall). The management of NT coupled with split-N rate application minimized N₂O emissions among treatments in this study, resulting in N₂O-N losses of 3.8, 13.2, and 6.6 N kg ha^-1 over 9-, 35-, and 20-days after the spring NH₃ application in 2014, 2015, and 2016, respectively. The spring pre-plant N application in 2015 also resulted in higher N₂O emissions than the spring side-dress application in 2016, and the increased N₂O-N loss was corresponding to lower N recovery efficiency in 2015 measurements. A comparison of chamber and OP measurements showed that soil N₂O emissions were likely underestimated by 10x without considering the wind-induced effect on gas transport at the ground-atmospheric interface. This study showed that the OP method provides a great opportunity to study agricultural N₂O emissions as well as management optimization for the sustainability of the agroecosystems.

O projeto de pesquisa propôs avaliar a metodologia de Simulação de Escalas Adaptativas (SAS) para descrever escoamentos turbulentos e não-reativos utilizando estudos de casos característicos, amplamente documentados, os quais possuem comportamentos do escoamento distintos presentes em diferentes regiões de uma câmara de combustão. O primeiro estudo de caso foi a análise do escoamento sobre um degrau, em que foi avaliada a capacidade do modelo Simulação de Escalas Adaptativas, frente aos modelos de Navier-Stokes com Média de Reynolds (RANS) e Simulação de Grandes Escalas (LES) e aos dados experimentais, em prever a distribuição de pressão, ponto de recolamento e de perfis de velocidade ao longo do domínio após a separação. Pode-se notar que o modelo SAS apresentou resultados praticamente idênticos aos resultados obtidos pelo modelo RANS com relação à distribuição de pressão e a posição ponto de recolamento. Porém, os perfis de velocidade apresentaram algumas discrepâncias com relação aos perfis de velocidade dos modelos RANS e LES e dos resultados experimentais. Um segundo estudo de caso foi a análise do escoamento através de um turbilhonador, em que a capacidade do modelo SAS foi avaliada, comparando seus resultados com os resultados do modelo de Navier-Stokes Não-Estacionárias com Média de Reynolds (URANS) e com os dados experimentais, em prever perfis de velocidade em regiões de recirculação presentes neste estudo de caso. Pode-se observar que ambos os modelos conseguiram prever as principais estruturas de recirculação do escoamento, porém, os perfis de velocidade apresentaram significativas discrepâncias com relação aos dados experimentais. Em seguida, foram feitas comparações entre os modelos SAS e URANS com relação à previsão da precessão central de vórtice e de estruturas de vórtices, das quais foi observado que o modelo SAS apresenta uma maior capacidade para prever estas estruturas em relação ao modelo URANS. / The research project aimed to evaluate the Scale-Adaptive Simulation (SAS) methodology to describe turbulent and non-reactive flows using characteristic, widely documented, case studies, which have distinct flow behaviors present in different regions of a chamber of combustion. The first case study was the analysis of a flow over a backward-facing step, from which the Scale-Adaptive Simulation (SAS) model capacity was evaluated, compared to the Reynolds Averaged Navier-Stokes (RANS) and Large-Eddy Simulation (LES) models and experimental data, in order to predict the pressure distribution, reattachment point and velocity profiles throughout the domain after separation. It can be noticed that the SAS model presented results almost identical to the results obtained by the RANS model in relation to the pressure distribution and reattachment position. However, the velocity profiles presented some discrepancies in respect to RANS and LES velocity profiles and the experimental results. A second case study was the analysis of the flow through a swirler, from which the capacity of the SAS model was evaluated, comparing its results to the results of the Unsteady Reynolds Averaged Navier-Stokes (URANS) model and with the experimental data, to predict velocity profiles in recirculation regions present in this case study. It can be observed that both models were able to predict the main recirculation structures of the flow, however, the velocity profiles presented significant discrepancies in relation to the experimental data. Then, comparisons were made between the SAS and URANS models in respect to the prediction of vortex precession vortex core and vortex structures, from which it was observed that the SAS model presents a greater capacity to predict these structures in relation to the URANS model.