Cerebral small vessel disease (SVD) is highly prevalent in older adults and is a predictor of stroke, dementia, and death. SVD is also associated with cognitive dysfunction, gait problems, and urinary incontinence. SVD is diagnosed based on white matter hyperintensities on T2 weighted scans. This thesis investigates the cardiac frequency component of resting state functional magnetic resonance imaging data in young healthy adults, older healthy adults, and older adults with pronounced SVD. A cardiac pulsatility metric is defined, and a tissue type contrast is observed between white matter, grey matter, and cerebrospinal fluid. Aging and disease effects are observed on cardiac pulsatility in white matter. The increased pulsatility may reflect the pathology of venous collagenosis and draining vein stenosis. Developing a better understanding of the etiology of SVD is an important step towards treating the disease.

Non-planar lifting surfaces offer potentially significant gains in aerodynamic efficiency by lowering induced drag. Non-aerodynamic considerations, such as structures can impact the overall efficiency. Here, a panel method and equivalent beam finite element model are used to explore non-planar configurations taking into account the coupling between aerodynamics and structures. A single discipline aerodynamic optimization and a multidisciplinary aerostructural optimization are investigated. Due to the complexity of the design space and the presence of multiple local minima, an augmented Lagrangian particle swarm optimizer is used. The aerodynamic optimum solution found for rectangular lifting surfaces is a box wing, while allowing for sweep and taper yields a joined wing. Adding parasitic drag in the aerodynamic model reduces the size of the non--planar elements. The aerostructural optimal solution found is a winglet configuration when the span is constrained and a wing rake when there is no such constraint.

This dissertation addresses three research challenges in planar lightwave circuit (PLC) optical signal processing. 1. Dynamic localization, a relatively new class of quantum phenomena, has not been demonstrated in any system to date. To address this challenge, the quantum system was mapped to the optical domain using a set of curved, coupled PLC waveguides in aluminum gallium arsenide (AlGaAs). The devices demonstrated, for the first time, exact dynamic localization in any system. These experiments motivate further mappings of quantum phenomena in the optical domain, leading toward the design of novel optical signal processing devices using these quantum-analog effects. 2. The PLC microresonator promises to reduce PLC device size and increase optical signal processing functionality. Microresonators in a parallel cascaded configuration, called "side coupled integrated spaced sequence of resonators" (SCISSORs), could offer very interesting dispersion compensation abilities, if a sufficient number of rings is present to produce fully formed "Bragg" gaps. To date, a SCISSOR with only three rings has been reported in a high-index material system. In this work, one, two, four and eight-ring SCISSORs were fabricated in AlGaAs. The eight-ring SCISSOR succeeded in producing fully formed Bragg peaks, and offers a platform to study interesting linear and nonlinear phenomena such as dispersion compensators and gap solitons. 3. PLCs are ideal candidates to satisfy the projected performance requirements of future microchip interconnects. In addition to data routing, these PLCs must provide over 100-bit switchable delays operating at ~ 1 Tbit/s. To date, no low loss optical device has met these requirements. To address this challenge, an ultrafast, low loss, switchable optically controllable delay line was fabricated in AlGaAs, capable of delaying 126 bits, with a bit-period of 1.5 ps. This successful demonstrator offers a practical solution for the incorporation of optics with microelectronics systems. The three aforementioned projects all employ, in their unique way, the coupling of light between PLC waveguides in AlGaAs. This central theme is explored in this dissertation in both its two- and multi-waveguide embodiments.

When increasing magnetic resonance imaging (MRI) resolution into the micron scale, image signal-to-noise ratio (SNR) can be maintained by using small radiofrequency (RF) coils in close proximity to the sample being imaged. Micro-scale RF coils (microcoils) can be easily fabricated on chip and placed adjacent to a sample under test. However, the high series resistance of microcoils limits the SNR due to the thermal noise generated in the copper. Cryo-cooling is a potential technique to reduce thermal noise in microcoils, thereby recovering SNR. In this research, copper microcoils of two different geometries have been cryo-cooled using liquid nitrogen. Quality-factor (Q) measurements have been taken to quantify the reduction in resistance due to cryo-cooling. Image SNR has been compared between identical coils at room temperature and liquid nitrogen temperature. The relationship between the drop in series resistance and the increase in image SNR has been analyzed, and these measurements compared to theory. While cryo-cooling can bring about dramatic increases in SNR, the extremely low temperature of liquid nitrogen is incompatible with living tissue. In general, the useful imaging region of a coil is approximately as deep as the coil diameter, thus cryo-cooling of coils has been limited in the past to larger coils, such that the thickness of a conventional cryostat does not put the sample outside of the optimal imaging region. This research utilizes a scheme of microfluidic cooling (developed in the Texas A&M NanoBio Systems Lab), which greatly reduces the volume of liquid nitrogen required to cryo-cool the coil. Along with a small gas phase nitrogen gap, this eliminates the need for a bulky cryostat. This thesis includes a review of the existing literature on cryo-cooled coils for MRI, as well as a review of planar pair coils and spiral microcoils in MR applications. Our methods of fabricating and testing these coils are described, and the results explained and analyzed. An image SNR improvement factor of 1.47 was achieved after cryo-cooling of a single planar pair coil, and an improvement factor of 4 was achieved with spiral microcoils.

Aquesta tesi es situa en el marc de la teoria qualitativa dels sistemes diferencials en el pla. Cada capítol conté un aspecte diferent. A la introducció, es dóna un resum dels resultats més coneguts i s'hi introdueix la notació que es fa servir al llarg de la tesi. En particular, descrivim el problema de la integrabilitat i alguns resultats sobre la determinació de l'estabilitat d'un punt singular o d'una òrbita periòdica a fi de presentar els darrers capítols. El problema de la integrabilitat es defineix com el problema de trobar la integral primera d'un sistema d'equacions diferencials en el pla i determinar la classe funcional a la qual pertany. Els Capítols 2 i 3 tracten el problema de la integrabilitat.En el Capítol 2 donem un resultat que permet trobar una expressió explícita per a una integral primera d'un cert tipus de sistemes polinomials. Mitjançant un canvi racional de variables, fem correspondre a una equació diferencial lineal homogènia de segon ordre: A2(x) w'(x) + A1(x) w'(x) + A0(x) w(x) = 0, els coeficients de la qual són polinomials, a un sistema diferencial polinomial pla. Provem que aquest sistema té un invariant per a cada solució arbitrària no nul·la w(x) de l'edo de segon ordre, que, quan w(x) és un polinomi, dóna lloc a una corba algebraica invariant. A més, donem una expressió explícita per a una integral primera del sistema construïda a partir de dues solucions independents de l'edo de segon ordre. Aquesta integral primera no és, en general, una funció Liouvilliana. Finalment, verifiquem que tots els exemples coneguts de famílies de sistemes quadràtics amb una corba algebraica invariant de grau arbitràriament alt es poden descriure mitjançant aquesta construcció (mòdul transformacions birracionals).En el Capítol 3, les corbes algebraiques invariants d'un sistema diferencial polinomial pla juguen el paper fonamental. Si un sistema diferencial polinomial pla té una corba algebraica invariant irreductible, aleshores els valors del seu cofactor en cadascun dels punts singulars no degenerats estan determinats. De fet, aquest valor es una combinació lineal a coeficients naturals dels valors propis associats al punt singular no degenerat. Aquests coeficients naturals es poden determinar completament en alguns casos depenent de la natura del punt singular. Així mateix, els punts de l'infinit també es poden tenir en compte. Un cop considerem el sistema en el pla projectiu complex, el grau d'una corba algebraica invariant esdevé un paràmetre del seu cofactor. Si considerem un sistema de grau d, aleshores té d^2 + d + 1 punts singulars (comptats amb la seva multiplicitat) i el cofactor d'una corba algebraica invariant té grau pel cap alt d-1. Procedim de la manera següent: prenem un polinomi de grau d-1 amb els seus d(d+1)/2 coeficients arbitraris i suposem que és el cofactor d'una corba algebraica invariant irreductible de grau n. Aleshores, imposem totes les condicions que ens donen els punts singulars no degenerats. En el cas general, imposem d^2 + d +1 condicions i, així, podem determinar completament el cofactor i el grau de la corba, l'existència de la qual es pot determinar resolent un sistema d'equacions lineal, o trobem una condició d'incompatibilitat. D'aquesta manera, en general, podem determinar l'existència de totes les corbes algebraiques invariants d'un sistema.El Capítol 4 tracta sobre l'estabilitat d'una òrbita periòdica d'un sistema diferencial pla. Suposem que f(x,y)=0 és una corba invariant irreductible amb cofactor que conté l'òrbita periòdica. Provem que les integrals sobre l'òrbita periòdica de la divergència i del cofactor coincideixen. Així, podem decidir sobre l'estabilitat de l'òrbita periòdica mitjançant la integració del cofactor sobre aquesta. En el Capítol 5, donem una aplicació dels resultats descrits en els Capítols 3 i 4. Considerem els sistemes quadràtics amb un cicle límit algebraic coneguts fins al moment de la redacció d'aquesta tesi. Aquest cicles límit algebraics estan continguts en corbes algebraiques invariants de graus 2, 4, 5 i 6 i algunes d'aquestes famílies de sistemes quadràtics son birracionalment equivalents. Aplicant el mètode descrit en el Capítol 3, mostrem que la corba algebraica invariant que conté el cicle límit es l'única corba algebraica invariant del sistema. Aprofitem aquest resultat per provar que aquests sistemes no tenen integral primera Liouvilliana. I aplicant la formula donada en el Capítol 4, provem que aquests cicles límit algebraics son hiperbòlics. El Capítol 6 tracta sobre l'estudi i les propietats de la funció període associada a un punt singular amb part lineal de tipus centre-focus. Com que el punt singular és sempre monodròmic, donada una secció transversal al flux amb el punt singular com a extrem, podem definir l'aplicació de Poincaré i la funció període associades a la secció. Diem que el punt és isòcron si podem trobar una secció tal que la seva funció període associada és constant. Aquesta definició generalitza la definició usual donada per centres a punts singulars qualssevol amb part lineal de tipus centre-focus. Caracteritzem aquesta propietat mitjançant simetries de Lie i formes normals, tot generalitzant els procediments coneguts per centres. Així mateix, donem un exemple d'una família de sistemes depenent d'un paràmetre real, tals que el seu origen és un punt singular amb part lineal de tipus centre-focus i que mai no és un punt isòcron. / Esta tesis se sitúa en el marco de la teoría cualitativa de los sistemas diferenciales en el plano. Cada capítulo contiene un aspecto distinto. En la introducción, se da un resumen de los resultados conocidos y se presenta la notación usada durante el resto de la tesis. En particular, se describe el problema de la integrabilidad y algunos resultados referentes a la determinación de la estabilidad de un punto singular o una órbita periódica con el fin de introducir los últimos capítulos. Definimos el problema de la integrabilidad como el problema de encontrar una integral primera para un sistema diferencial plano y determinar la clase funcional a la cual ésta debe pertenecer. Los Capítulos 2 y 3 tratan sobre el problema de la integrabilidad. En el Capítulo 2, obtenemos un resultado que permite encontrar una expresión explícita para una integral primera para un cierto tipo de sistema polinomial. Mediante un cambio racional de variable, hacemos corresponder una ecuación diferencial lineal homogénea de segundo orden: A2 (x) w'(x) + A1(x) w'(x) + A0(x) w(x) = 0, cuyos coeficientes son polinomios, a un sistema diferencial polinomial en el plano. Probamos que dicho sistema tiene un invariante para cada solución arbitraria no nula w(x) de la edo de segundo orden, que, en caso que w(x) sea un polinomio, da lugar a una curva algebraica invariante. Además, damos una expresión explícita de una integral primera para el sistema construida a partir de dos soluciones independientes de la edo de segundo orden. Esta integral primera no es, en general, una función Liouvilliana. Finalmente, verificamos que todos los ejemplos conocidos de familias de sistemas cuadráticos con una curva algebraica invariante de grado arbitrariamente alto se pueden describir mediante esta construcción (módulo transformaciones birracionales).En el Capítulo 3, las curvas algebraicas invariantes de un sistema diferencial plano polinomial juegan un papel fundamental. Si una curva algebraica invariante e irreducible existe para un sistema polinomial plano, entonces los valores de su cofactor en cada punto singular no degenerado están determinados. De hecho, este valor es una combinación lineal a coeficientes naturales de los valores propios asociados al punto singular no degenerado. Estos coeficientes naturales se pueden determinar completamente según la naturaleza del punto singular. Además, también podemos considerar los puntos del infinito. Una vez que el sistema se considera en el plano proyectivo complejo, el grado de una curva algebraica invariante deviene un parámetro de su cofactor. Si consideramos un sistema de grado d, entonces tiene d^2 + d + 1 puntos singulares (contados con su multiplicidad) y el cofactor de una curva algebraica invariante es un polinomio de grado a lo sumo d-1. Procedemos de la manera siguiente: tomamos un polinomio de grado d-1 con sus d(d+1)/2 coeficientes arbitrarios y suponemos que es el cofactor de una curva algebraica invariante e irreducible de grado n. Entonces, imponemos todas las condiciones dadas por los puntos singulares no degenerados. En el caso general, imponemos d^2 + d + 1 condiciones y, en consecuencia, determinamos completamente el cofactor y el grado de la curva, cuya existencia puede ser determinada resolviendo un sistema lineal de ecuaciones, o mostramos una condición de incompatibilidad. Por tanto, podemos determinar la existencia de todas las curvas algebraicas invariantes para un sistema general. El Capítulo 4 trata sobre la estabilidad de una órbita periódica de un sistema diferencial plano. Suponemos que f(x,y)=0 es una curva invariante e irreducible con cofactor que contiene la órbita periódica. Probamos que las integrales sobre la órbita periódica de la divergencia y del cofactor coinciden. De aquí que podamos deducir la estabilidad de una órbita periódica mediante la integración del cofactor sobre ésta. En el Capítulo 5, describimos una aplicación de los resultados dados en los Capítulos 3 y 4. Consideramos los sistemas cuadráticos con un ciclo límite algebraico conocidos hasta la redacción de esta tesis. Estos ciclos límite algebraicos están contenidos en curvas algebraicas invariantes de grados 2, 4, 5 y 6 y algunas de estas familias de sistemas cuadráticos son birracionalmente equivalentes. Aplicando el método descrito en el Capítulo 3, mostramos que no existe ninguna curva algebraica invariante excepto la que contiene el ciclo límite. Aprovechamos este resultado para mostrar que estos sistemas no tienen integral primera Liouvilliana. Y, aplicando la formula dada en el Capítulo 4, probamos que estos ciclos límite algebraicos son hiperbólicos. El Capítulo 6 trata sobre el estudio de las propiedades de la función periodo asociada a un punto singular con parte lineal de tipo centro-foco. Dada una sección transversal al flujo con dicho punto singular por extremo, podemos definir la aplicación de Poincaré y la función periodo asociadas a esta sección puesto que este punto es siempre monodrómico. Decimos que este punto es isócrono si podemos encontrar una sección tal que la función periodo asociada a ella sea constante. Esta definición generaliza la definición usual dada para centros a cualquier punto singular con parte lineal de tipo centro-foco. Caracterizamos esta propiedad mediante simetrías de Lie y formas normales, generalizando los procedimientos conocidos para centros. Además, damos un ejemplo de una familia de sistemas que dependen de un parámetro real, tales que el origen es un punto singular con parte lineal de tipo centro-foco y que nunca es un punto isócrono. / This thesis is situated in the framework of the qualitative theory of differential systems in the plane. Each chapter contains a different topic. In the introduction, a summary of known results is given and the notation used through the rest of the memory is presented. In particular, we describe the integrability problem and some results concerning the determination of the stability of a singular point or a periodic orbit in order to introduce the latest chapters. We define the integrability problem as the problem of finding a first integral for a planar differential system and determining the functional class it must belong to. Chapters 2 and 3 are concerned with the integrability problem. In Chapter 2, we obtain a result which allows to find an explicit expression for a first integral of a certain type of polynomial system. By means of a rational change of variable, we make correspond the homogenous second order linear differential equation: A2 (x) w'(x) + A1(x) w'(x) + A0(x) w(x) = 0, whose coefficients are polynomials, to a planar polynomial differential system. We prove that this system has an invariant for each arbitrary nonnull solution w(x) of the second-order ode, which, in case w(x) is a polynomial, gives rise to an invariant algebraic curve. In addition, we give an explicit expression of a first integral for the system constructed from two independent solutions of the second order ode. This first integral is not, in general, a Liouvillian function. Finally, we verify that all the known examples of families of quadratic systems with an invariant algebraic curve of arbitrarily high degree can be described by this construction (modulus birrational transformations). In Chapter 3, invariant algebraic curves of a planar polynomial differential system play the fundamental role. If an irreducible invariant algebraic curve for a planar polynomial differential system exists, then the values of its cofactor at each non-degenerate singular point are determined. In fact, this value is a linear combination with natural coefficients of the eigenvalues associated to the non-degenerate singular point. These natural coefficients can be completely determined in some cases depending on the nature of the singular point. Moreover, the points at infinity can also be taken into account. Once the system is considered in the projective complex plane, the degree of an invariant algebraic curve becomes a parameter of its cofactor. If we consider a system of degree d, then it has d^2 + d + 1 singular points (counted with multiplicity) and the cofactor of an invariant algebraic curve is a polynomial of degree at most d-1. We proceed as follows: we take a polynomial of degree d-1 with its d(d+1)/2 arbitrary coefficients and we assume that it is the cofactor of an irreducible invariant algebraic curve of degree n. Then, we impose all the conditions given by the non-degenerate singular points. In the general case, we impose d^2 + d + 1 conditions and, hence, we completely determine the cofactor and the degree of the curve, whose existence can be determined by solving a linear system of equations, or we show an incompatibility condition. Therefore, we can determine the existence of all the invariant algebraic curves of a general system.Chapter 4 is about the stability of a periodic orbit of a planar differential system. We assume that f(x,y)=0 is a real irreducible invariant curve with cofactor which contains the periodic orbit. We prove that the integrals over the periodic orbit of the divergence and the cofactor coincide. Hence, we can decide the stability of a periodic orbit by means of the integration of the cofactor over it. In Chapter 5, we describe an application of the results given in Chapters 3 and 4. We consider the quadratic systems with an algebraic limit cycle known until the composition of this thesis. These algebraic limit cycles are contained in invariant algebraic curves of degrees 2, 4, 5 and 6 and there are some of these families of quadratic systems which are birrationally equivalent one to the other. Applying the method given in Chapter 3, we show that there is no other irreducible invariant algebraic curve that the one which contains the limit cycle. We take profit from this result to show that these systems have no Liouvillian first integral. And applying the formula given in Chapter 4, we prove that these algebraic limit cycles are hyperbolic.Chapter 6 is devoted to the study of the properties of the period function associated to a singular point with linear part of centre-focus type. Given a section through the flow with such a singular point as endpoint, we can define the Poincaré map and the period function associated to this section since this point is always monodromic. We say that this point is isochronous if we can find a section such that the period function associated to it is constant. This definition generalizes the usual definition given for centres to any singular point with linear part of centre-focus type. We characterize this property by means of Lie symmetries and normal forms, generalizing the known procedures for centres. Moreover, we provide an example of a family of systems depending on a real parameter, such that the origin is a singular point with linear part of centre-focus type and which is never an isochronous point. / Cette thèse de doctorat traite sur la théorie qualitative des systèmes différentiels planaires. Chaque chapitre contient un sujet différent. Dans l'introduction, un sommaire des résultats connus est donné et la notation utilisée dans le reste du mémoire est présentée. En particulier, nous décrivons le problème de l'intégrabilité et quelques résultats concernant la détermination de la stabilité d'un point singulier ou d'une orbite périodique afin de présenter les derniers chapitres. Nous définissons le problème de l'intégrabilité comme le problème de trouver une intégrale première pour un système différentiel planaire et de déterminer la classe fonctionnelle à la quelle elle doit appartenir. Les Chapitres 2 et 3 traitent du problème de l'intégrabilité. Au Chapitre 2, nous obtenons un résultat permettant de trouver une expression explicite pour une intégrale première d'un certain type de système polynomial. Au moyen d'un changement rationnel de variables, nous faisons correspondre l'équation linéaire du deuxième degré: A2(x) w''(x) + A1(x) w'(x) + A0(x) w(x) = 0, dont les coefficients sont des polynômes, à un système différentiel polynomial planaire. Nous montrons que ce système a un invariant pour chaque solution arbitraire w(x) différent de zéro de l'équation considérée, qui, dans le cas où le w(x) serait un polynôme, est une courbe algébrique invariante. De plus, nous donnons une expression explicite d'une intégrale première pour le système construite à partir de deux solutions indépendantes de l'edo du deuxième degré. Cette intégrale première n'est pas, en général, une fonction de Liouville. En conclusion, nous vérifions que tous les exemples connus des familles des systèmes quadratiques avec une courbe algébrique invariante de degré arbitrairement élevé peuvent être décrits par cette construction (modulo des transformations birationnelles). Au Chapitre 3, les courbes algébriques invariantes d'un système différentiel polynomial planaire jouent le rôle fondamental. Si une courbe algébrique invariante et irréductible existe pour un système différentiel polynomial planaire, alors les valeurs de son cofacteur à chaque point singulier non dégénéré sont déterminées. En fait, cette valeur est une combinaison linéaire avec des coefficients naturels des valeurs propres associées au point singulier non dégénéré. Ces coefficients naturels peuvent être complètement déterminés dans certains cas selon la nature du point singulier. De plus, les points à l'infini peuvent également être pris en considération. Une fois que le système est considéré dans le plan projectif complexe, le degré d'une courbe algébrique invariante devient un paramètre de son cofacteur. Si nous considérons un système de degré d, alors il y a d^2 + d + 1 points singuliers (comptés avec sa multiplicité) et le cofacteur d'une courbe algébrique invariante est un polynôme de degré au plus d-1. Nous opérons comme suit: nous prenons un polynôme de degré d-1 avec ses d(d+1)/2 coefficients arbitraires et nous supposons que c'est le cofacteur d'une courbe algébrique invariante et irréductible de degré n. Nous imposons alors toutes les conditions données par les points singuliers non dégénérés. Dans le cas général, nous imposons d^2 + d + 1 conditions et, par conséquent, nous déterminons complètement le cofacteur et le degré de la courbe, dont l'existence peut être déterminée en résolvant un système linéaire d'équations, ou bien nous prouvons l'incompatibilité d'une condition. Par conséquent, nous pouvons déterminer l'existence de toutes les courbes algébriques invariantes d'un système général. Le sujet du Chapitre 4 est la stabilité d'une orbite périodique d'un système différentiel planaire. Nous supposons que f(x,y)=0 est une courbe invariante irréductible réelle avec cofacteur qui contient l'orbite périodique. Nous montrons que les intégrales sur l'orbite périodique de la divergence et le cofacteur coïncident. Par conséquent, nous pouvons déterminer la stabilité d'une orbite périodique en intégrant le cofacteur sur celle-ci.Dans le Chapitre 5, nous décrivons une application des résultats donnés aux Chapitres 3 et 4. Nous considérons les systèmes quadratiques avec un cycle limite algébrique connus jusqu'alors. Ces cycles limites algébriques sont contenus dans des courbes algébriques invariantes de degrés 2, 4, 5 et 6 et il existe certaines de ces familles de systèmes quadratiques qui sont birationnellement équivalentes. Appliquant la méthode exposée au Chapitre 3, nous prouvons qu'il n'y a aucune autre courbe algébrique invariante et irréductible différente à celle qui contient le cycle limite. Ceci nous permet de prouver que ces systèmes n'ont aucune intégrale première de Liouville. En appliquant la formule donnée au Chapitre 4, nous montrons que ces cycles limites algébriques sont hyperboliques. Le Chapitre 6 est consacré à l'étude des propriétés de la fonction de période associée à un point singulier dont la partie linéaire est de type centre-foyer. Etant donnée une section du flux avec tel point singulier comme point final, nous pouvons définir l'application de Poincaré et la fonction de période associée à cette section puisque ce point est toujours monodromique. Nous disons que ce point est isochronique si nous pouvons trouver une section telle que la fonction de période associée à elle est constante. Cette définition généralise la définition habituelle donnée pour des centres à n'importe quel point singulier dont la partie linéaire est de type centre-foyer. Nous caractérisons cette propriété au moyen des symétries de Lie et des formes normales, généralisant les procédures connues pour des centres. De plus, nous donnons un exemple d'une famille de systèmes avec un paramètre réel, telle que l'origine est un point singulier dont la partie linéaire est de type centre-foyer et qui n'est jamais un point isochronique.

Aquesta tesi es situa en el marc de la teoriaqualitativadelssistemesd’equacionsdiferencials en el pla. Cada capítol conté un aspectediferent, però en totsells es tractenproblemes, la soluciódelsqualsestà basada en el rol que hi juguen les simetriesdiscretes i continues (reversibilitat o simetries de Lie) de campsvectorialsplans. A la introducció, es dóna un resumdelsresultatsmésconeguts i s’hiintrodueix la notació que es fa servir al llarg de la tesi. En el segon i tercer capítol, s’aborda el problema de trobarl’expressió explícita del canvilinealitzant o orbitalmentlinealitzantd#un camp vectorial suau a partir del coneixementd’uncommutador, en el cas de la linealització, o una simetria de Lie, en el cas de la linealització orbital. Cada capítol finalitzaambexemplesil.lustratius del procedimentconstructiudelscanvis. Al Capítol 5 s’apliquenelsresultatsdelscapítolsanteriors, combinatsamblinealitzacionsDarbouxianes. Concretament, es considera un sistema quadràtictipusLotka-Volterra i es caracteritzen les selles linealitzables i orbitalmentlinealitzablesmitjançant la troballadelscanvislinealitzants o orbitalmentlinealitzants. En el sisè capítol, s’utilitzal’existènciad’unàlgebra de simetriespuntuals de Lie per donar informació sobre l’existència i localitzaciód’òrbitesperiòdiques. En particular, quanl’àlgebra de simetriespuntuals de Lie d’unaequació diferencial escalar de segónordreautònoma i suau té dimensiómajor o igual a dos, definim les anomenadesfuncionsfonamentals que enspermeten estudiar les òrbitesperiòdiques al pla de fases. En el cas particular d’equacionspolinomials de Liénard, mostrem la no existència de cicles límit en tot el pla de fases. Finalment, al Capítol 7 es relacionen elssistemes reversibles amb el problema del centre aixícomamb el problema de la integrabilitat analítica. Consideremsistemesd’equacionsdiferencialsanalíticsamb centres degenerats i mostrem que poden transformar-se, després d’un reescalat del temps, en un sistema lineal i reversible. El coneixement de integralsprimeresens proporciona un procediment per caracteritzar, en alguns casos, la condició de reversibilitat del centre degenerat. D’altra banda, relacioneml’existència de integralsprimeresanalítiquesamb la reversibilitat orbital analítica en el cas de singularitatsdèbils no degenerades.

The suspension system of a vehicle is conventionally designed such that the spring-damper element is configured in the vertical direction, and the longitudinal connection between the vehicle chassis and wheels is always very stiff compared to the vertical one. This mechanism can isolate vibrations and absorb shocks efficiently in the vertical direction but cannot attenuate the longitudinal impacts caused by road obstacles. In order to overcome such a limitation, a planar suspension system (PSS) is proposed. This novel vehicle suspension system has a longitudinal spring-damper strut between the vehicle chassis and wheel. The dynamic performance, including ride comfort, pitch dynamics, handling characteristics and total dynamic behaviour, of a mid-size passenger vehicle equipped with such planar suspension systems is thoroughly investigated and compared with those of a conventional vehicle. To facilitate this investigation, various number of vehicle models are developed considering the relative longitudinal motions of wheels with respect to the chassis. A 4-DOF quarter-car model is used to conduct a preliminary study of the ride quality, and a pitch plane half-car model is employed to investigate the pitch dynamics in both the frequency and time domain. A 5-DOF yaw plane single-track half-car model along with a pitch plane half-car model is proposed to carry out the handling performance study, and also an 18-DOF full-car model is used to perform total dynamics study. In addition to these mathematical models, virtual full-car models are constructed in Adams/car to validate the proposed mathematical models. For the sake of prediction of the tire-ground interaction force, a radial-spring tire model is modified by adding the tire damping to generate the road excitation forces due to road disturbances in the vertical and longitudinal directions. A dynamic 2D tire friction model based on the LuGre friction theory is modified to simulate the dynamic frictional interaction in the tire-ground contact pitch. The ride quality of a PSS vehicle is evaluated in accordance with the ISO 2631 and compared with that of a conventional vehicle. It is shown that the PSS system exhibits good potential to attenuate the impact and isolate the vibration due to road excitations in both the vertical and longitudinal directions, resulting in improved vehicles’ ride and comfort quality. The relatively soft longitudinal strut can absorb the longitudinal impact and, therefore, can protect the components. The investigation of handling performance including the steady-state handling characteristics, transient and frequency responses in various scenarios demonstrates that the PSS vehicle is directionally stable and generally has comparable handling behaviour to a similar conventional vehicle. The application of PSS in vehicles can enhance the understeer trend, i.e. the understeer becomes more understeer, neutral steer becomes slightly understeer, and oversteer becomes less oversteer. The total dynamic behaviour combining the bounce, pitch, roll and the longitudinal dynamics under various scenarios such as differential brake-in-turn and asymmetric obstacle traversing was thoroughly investigated. Simulation results illustrate that the PSS vehicle has a relatively small roll angle in a turning manoeuvre. In some cases such as passing road potholes, the PSS vehicle has a better directional stability.

The influence of turbulence on the orientation state of a dilute suspension of stiff fibers at high Reynolds number in a planar contraction is investigated. High speed imaging and LDV techniques are used to quantify fiber orientation distribution and turbulent characteristics. A nearly homogenous, isotropic grid generated turbulent flow is introduced at the contraction inlet. Flow Reynolds number and inlet turbulent characteristics are varied in order to determine their effects on orientation distribution. The orientation anisotropy is shown to be accurately modelled by a Fokker-Planck type equation. Results show that rotational diffusion is highly influenced by inlet turbulent characteristics and decays exponentially with convergence ratio. Furthermore, the effect of turbulent energy production in the contraction is shown to be negligible. Also, the results show that the flow Reynolds number has negligible effect on the development of orientation anisotropy, and the influence of turbulence on fiber rotation is negligible for $mathrm{Pe_r}>$ 10. It was concluded that inertia induced fiber motion played a negligible role in the experiments.

In the research reported in this dissertation, the concept of classic conjugate imaging, a non-tomographic nuclear medicine technique, is modified such that activity of a radiopharmaceutical distribution in the striata can be estimated. A mathematical model is developed that extended the application of classic conjugate imaging to estimation of two distinct and aligned activity distributions. Error analysis of the mathematical model is performed to characterize the accuracy of the model and to benchmark the limitations of the model. Phantom experiments are performed to demonstrate the practical application of the model and to evaluate its accuracy. A Monte Carlo simulation model of conjugate imaging of activity uptake in the striata of a primate is developed to evaluate the accuracy of the modified conjugate imaging technique as applied in the use of a dedicate conjugate imaging system. In addition, the simulation model is used to determine and characterize the shielding design of the small field of view gamma cameras comprising the dedicated conjugate imaging system. The application of scatter correction is investigated to address the downscatter of high-energy photon emissions into the photopeak window and the inclusion of scattered primary photons in the photopeak window. In this dissertation, it is shown that the modified conjugate imaging technique developed can be used to estimate accurately activity uptake in each of two distinct and aligned activity distributions. The accuracy of the technique is shown to be comparable to that of clinical quantitative SPECT. The modified conjugate imaging technique used with the dedicated conjugate imaging system may, therefore, be a viable quantitative nuclear medicine technique for activity estimation of radiopharmaceutical uptake in the striata of Parkinsonian and schizophrenic patients. The portability and low cost relative to SPECT systems make a dedicated conjugate imaging system advantageous for clinics with Parkinsonian and schizophrenic patients, who are unable to travel due to physical or mental limitation.

The phenomenon of polarization interaction in certain nonlinear materials is presented, and the design of an all-optical logic device based on this concept is described. An efficient two-dimensional finite-difference time-domain code for studying third-order nonlinear optical phenomena is discussed, in which both the slowly varying and the rapidly varying components of the electromagnetic fields are considered. The algorithm solves the vector form Maxwell s equations for all field components and uses the nonlinear constitutive relation in matrix form as the equations required to describe the nonlinear system. The stability of the code is discussed and its accuracy demonstrated through the simulation of the self-phase modulation effect observed in Kerr media. Finally, the code is used to simulate polarization mixing in photonic crystal-based line defect and coupled resonator optical waveguides.