Nesta Tese desenvolvemos várias abordagens "Darbouxianas"para buscar integrais primeiras (elementares e Liouvillianas) de equações diferenciais ordinárias de segunda ordem (2EDOs) racionais. Os algoritmos (semi-algoritmos) que desenvolvemos seguem a linha do trabalho de Prelle e Singer. Basicamente, os métodos que buscam integrais primeiras elementares são uma extensão da técnica desenvolvida por Prelle e Singer para encontrar soluções elementares de equações diferenciais ordinárias de primeira ordem (1EDOs) racionais. O procedimento que lida com 2EDOs racionais que apresentam integrais primeiras Liouvillianas é baseado em uma extensão ao nosso método para encontrar soluções Liouvillianas de 1EDOs racionais. A ideia fundamental por tras do nosso trabalho consiste em que os fatores integrantes para 1-formas polinomiais geradas pela diferenciação de funções elementares e Liouvillianas são formados por certos polinômios denominados polinômios de Darboux. Vamos mostrar como combinar esses polinômios de Darboux para construir fatores integrantes e, de posse deles, determinar integrais primeiras. Vamos ainda discutir algumas implementações computacionais dos semi-algoritmos.

A presente dissertação é fruto da pesquisa de mestrado realizada sobre o poema Ode marítima, de Álvaro de Campos, heterônimo do poeta português Fernando Pessoa. O poema foi estudado à luz de considerações teóricas que pudessem sustentar o teor artístico do citado poema, para além de um posicionamento estritamente analítico. Em outras palavras, as contribuições teóricas de Walter Benjamin, Philippe Lacoue-Labarthe, Theodor W. Adorno, entre outros, visa propocionar um discurso de pesquisa que seja poético, respeitando a manifestação literária em questão o poema. Nesse sentido, Ode marítima apresentou, ao longo da pesquisa, diferentes temas. Tais temas foram estudados, com o embasamento teórico referido, separadamente. São eles: Imaginação, Cais Absoluto, Sonho, Força e Infância. Para cada tema, um capítulo foi elaborado. Todos os temas pesquisados confluem para a apreensão da poética de Álvaro de Campos uma vez que Ode marítima é um poema singular e fundamental da obra poética do citado heterônimo. Desse modo, a pesquisa realizada não se limita à abordagem do citado poema, mas alcança demais poemas de Álvaro de Campos, de modo que se torna possível conjeturar acerca da poesia deste heterônimo. Partindo, portanto, de um poema, a pesquisa pretende poder dimensionar a obra de Álvaro de Campos, pontuando suas particularidades trazidas pela Ode marítima, mas tendo ciência de que trata-se de uma proposta entre tantas outras válidas e enriquecedoras. / This dissertation is the result of the master's research done about the poem "Ode marítima" by Álvaro de Campos, Fernando Pessoas heteronymous. The poem was studied under theoretical considerations that could sustain the artistic content of that poem. In other words, the theoretical contributions by Walter Benjamin, Philippe Lacoue-Labarthe, Theodor W. Adorno, etc, aims to provide a poetic research discourse, respecting the literary manifestation in question the poem. During the research "Ode marítima" showed different themes. These themes were studied separately. They are: "Imagination," "Absolute Pier", "Dream", "Strength" and "Childhood". For each subject a chapter has been prepared. All subjects studied converge to the apprehension of Álvaro de Campos "Ode marítima" as a poem that contains the singularity of this heteronymous. Thus, the survey reaches other poems by Álvaro de Campos, making possible the conjecture about the poetry of this heteronymous. The research aims to gain the studies of Alvaro de Campos poetry, pointing out its peculiarities, as done about by the "Ode marítima" and being aware that this is one proposal among many others as valid and as enriching as this one

This thesis presents Monza, a system for accelerating the simulation of modelsof physical systems described by ordinary differential equations, using a generalpurpose computer with a PCIe FPGA expansion card. The system allows bothautomatic generation of an FPGA implementation from a model described in theModelica programming language, and simulation of said system.Monza accomplishes this by using a customizable hardware architecture forthe FPGA, consisting of a variable number of simple processing elements. A cus-tom compiler, also developed in this thesis, tailors and programs the architectureto run a specific model of a physical system.Testing was done on two test models, a water tank system and a Weibel-lung,with up to several thousand state variables. The resulting system is several timesfaster for smaller models and somewhat slower for larger models compared to aCPU. The conclusion is that the developed hardware architecture and softwaretoolchain is a feasible way of accelerating model execution, but more work isneeded to ensure faster execution at all times.

Nous nous intéressons dans ce mémoire à des méthodes de parallélisation par découplage du système dynamique. Plusieurs applications numériques de nos jours conduisent à des systèmes dynamiques de grande taille et nécessitent des méthodes de parallélisation en conséquence pour pouvoir être résolues sur les machines de calcul à plusieurs processeurs. Notre but est de trouver une méthode numérique à la fois consistante et stable pour réduire le temps de la résolution numérique. La première approche consiste à découpler le système dynamique en sous-systèmes contenant des sous-ensembles de variables indépendants et à remplacer les termes de couplage par l’extrapolation polynomiale. Une telle méthode a été introduite sous le nom de schéma C (p, q, j), nous améliorons ce schéma en introduisant la possibilité à utiliser des pas de temps adaptatifs. Cependant, notre étude montre que cette méthode de découplage ne peut satisfaire les propriétés numériques que sous des conditions très strictes et ne peut donc pas s’appliquer aux problèmes raides présentant des couplages forts entre les sous-systèmes. Afin de pouvoir répondre à cette problématique de découplage des systèmes fortement couplés, on introduit le deuxième axe de recherche, dont l’outil principal est la réduction d’ordre du modèle. L’idée est de remplacer le couplage entre les sous-ensembles de variables du système par leurs représentations sous forme réduite. Ces sous-systèmes peuvent être distribués sur une architecture de calcul parallèle. Notre analyse du schéma de découplage résultant nous conduit à définir un critère mathématique pour la mise à jour des bases réduites entre les sous-systèmes. La méthode de réduction d’ordre du modèle utilisée est fondée sur la décomposition orthogonale aux valeurs propres (POD). Cependant, ne disposant pas à priori des données requises pour la construction de la base réduite, nous proposons alors un algorithme de construction incrémentale de la base réduite permettant de représenter le maximum des dynamiques des solutions présentes dans l’intervalle de simulation. Nous avons appliqué la méthode proposée sur les différents systèmes dynamiques tels que l’exemple provenant d’une EDP et celui provenant de l’équation de Navier Stokes. La méthode proposée montre l’avantage de l’utilisation de l’algorithme de découplage basé sur la réduction d’ordre. Les solutions numériques sont obtenues avec une bonne précision comparées à celle obtenue par une méthode de résolution classique tout en restant très performante selon le nombre de sous-systèmes définis. / In this thesis, we are interested in parallelization algorithm for solving dynamical systems. Many industrial applications nowadays lead to large systems of huge number of variables. A such dynamical system requires parallel method in order to be solved on parallel computers. Our goal is to find a robust numerical method satisfying stability and consistency properties and suitable to be implemented in parallel machines. The first method developed in this thesis consists in decoupling dynamical system into independent subsystems and using polynomial extrapolation for coupled terms between subsystems. Such a method is called C(p; q; j).We have extended this numerical scheme to adaptive time steps. However, this method admits poor numerical properties and therefore cannot be applied in solving stiff systems with strong coupling terms.When dealing with systems whose variables are strongly coupled, contrary to the technique of using extrapolation for coupled terms, one may suggest to use reduced order models to replace those terms and solve separately each independent subsystems. Thus, we introduced the second approach consisting in using order reduction technique in decoupling dynamical systems. The order reduction method uses the Proper Orthogonal Decomposition. Therefore, when constructing reduced order models, we do not have all the solutions required for the POD basis, then we developed a technique of updating the POD during the simulation process. This method is applied successfully to solve different examples of dynamical systems : one example of stiff ODE provided from PDE and the other was the ODE system provided from the Nervier-Stokes equations. As a result, we have proposed a robust method of decoupling dynamical system based on reduced order technique. We have obtained good approximations to the reference solution with appropriated precision. Moreover, we obtained a great performance when solving the problem on parallel computers.

This is a masters thesis (20p) in computer science at the University of Linköping. This thesis will give an introduction to what a physics engine is and what it consist of. It will put some engines under the magnifying glass and test them in a couple of runtime tests. Two cutting edge commercial physics engines have been examined, trying to predict the future of physics engines. From the research and test results, an interface for physics engine independency has been implemented for a company called Craft Animations in Gothenburg, Sweden.

Philosophiae Doctor - PhD / This study focused on developing an intervention programme for school health nurses on guiding adolescents in their decision-making on reproductive health. A review of literature shows that this role is necessary, as there is a great need to reduce adolescents' morbidity and mortality rates due to poor decision-making about their reproductive health. School health nurses are strategically positioned to perform this role in ensuring that adolescents are well guided in making responsible decisions about their reproductive health. However, available evidence shows that school health nurses have not been performing this role in the school health service, especially in Nigeria. The provision of guidance for adolescents, on making decisions about their reproductive health is an adaptive role of school health nurses. The literature further shows that school health nurses require adequate preparation before they can perform this role. The study was conducted in the secondary school environment of Ijebu Ode local government area of Nigeria. Work role performance theory, adult learning principles, and experiential learning constituted the theoretical point of departure for this study. The paradigmatic assumptions revolved around intrepretivism/constructionism using the qualitative methodological approach. Semi-structured interviews and focus groups were the means of obtaining information from study participants for the study. The Intervention Design and Development model of Rothman and Thomas (2013) was used to design the study. The participants for the study were eight school health nurses, five school teachers, thirty-six adolescents, and one school health coordinator. They were all purposively selected. The data collected was analysed manually using inductive content analysis. The main findings from the interviews show that school health nurses have a poor awareness of their role and responsibilities, a lack of knowledge on how adolescents make their decisions, a lack of adequate knowledge on how to guide adolescents in their decision-making. The findings also show that there is poor interpersonal communication between school health nurses, and adolescents, and between school health nurses and members of the teaching staff. The findings further show that there are insufficient continuous professional development programmes. Results from integrative reviews regarding the types of intervention programmes that had been developed for school health nurses at different times in the past focused on role orientation, knowledge and skills acquisition, and mutual interaction between school health nurses and adolescents, and members of the teaching staff. The findings reflect a gap in how school health nurses provide guidance to school adolescents in decision-making on their reproductive health, hence the need for a training programme that will assist them in discharging this function effectively. A training programme was designed and developed for school health nurses to assist them on guiding adolescents in their decision making about their reproductive health. The training programme was pilot tested with observational methods, an interview being used as a means of assessing the quality and outcomes of the training programme. The results of the pilot test show the participants' satisfaction with the organisation and the quality of the training workshop. Participants indicated that they had gained more knowledge and understanding of adolescent reproductive issues, and their decision-making processes. They also said that they had gained more interpersonal skills, and greater communication skills. Some expressed the conviction that they had gained more confidence in their ability to communicate with the teaching staff. Some also expressed their readiness to apply the skills obtained during the training to their practice area. It is recommended that the training programme be fully evaluated in phase five of the Intervention Design and Development model of Rothman and Thomas, which will enable full dissemination and implementation of the programme (Rothman and Thomas, 2013). It is further recommended that the training programme be disseminated to end users (school health nurses) by sensitizing the necessary stake-holders on the need to use the training programme for school health nurses in their respective school contexts.

Electrical impedance tomography (EIT) is an emerging medical imaging technique that aims to reconstruct the internal conductivity distribution of a subject from electrical measurements obtained on the skin. In this thesis we explore the promising application of EIT to the respiratory monitoring of humans. We pay particular focus to the forward problem, highlighting the need to have an accurately known external boundary shape and electrode positions on a reconstruction model. A theoretical study of uniqueness results of EIT with an unknown external boundary shape is presented. A novel sensitivity study of the external boundary shape is presented as well as results from a reconstruction algorithm to account for errors in electrode position with simulated data in 3D. We also demonstrate results of a shape correction algorithm from a pilot study of lung EIT with data collected using the fEITER system, and MR images used to inform the external boundary shape of healthy subjects. After image co-registration of the resulting dynamic 3D EIT reconstruction images with the lung-segmented MR image, we outline a novel mutual information performance criterion to measure the quality of reconstructed images. We also outline the computation of the forward problem of the complete electrode model in 3D using high order polynomial finite elements and present convergence results in 2D for the continuum, point and complete electrode model. Our numerical study demonstrates that the convergence rate of the forward problem is independent of the polynomial approximation order for the complete electrode model and there is no global convergence for the point electrode model in the energy norm. Reconstructed conductivity images can be difficult to interpret at the bedside. Moreover clinicians would like clinically meaningful indices, such as regional lung compliance, to determine the pathologies of patients in real time. By modelling the respiratory system as a coupled time dependent system of simple mechanical functional units, we propose a novel methodology to couple mechanical ventilation and EIT. The mechanical properties of the lungs are estimated through an inverse coefficient problem on coupled ODEs, with the measurable data being the time series of pressure at airway opening and interior air volume data. We present results with simulated data as well as a discussion on extensions and limitations to the mechanical models. Finally we present a theoretical discussion of anisotropic EIT. It is well known that any diffeomorphism fixing points on the boundary gives rise to a conductivity with the same electrical measurements on the skin, generating a large class of conductivities that are electrically equivalent. We define novel classes of anisotropic media with constraints on their eigenspace: prescribed eigenvalues, prescribed orthogonal coordinates, prescribed eigenvectors, fibrous and layered conductivities. By drawing analogies with elasticity theory, we discuss how these constraints on the eigenspace restrict the set of diffeomorphisms fixing points on the boundary, and present two uniqueness results for anisotropic conductivities with prescribed eigenvalues and prescribed eigenvectors.

First-principle modelling of chemical processes and their unit operations has been of great interest in the chemical process, as well as the control and allied industries over the past decades. This is because it offers the opportunity to develop virtual representations (models) of real process systems, which can be used to describe and predict the dynamic behaviour of those systems. These models are based on the fundamentals of the transport phenomena of fluid dynamics (involving momentum transfer), mass transfer, and energy transfer of the systems they describe. A first-principle model of a semi-batch rhodium dissolution chemical process has been developed. It describes the dynamic behaviour of two exothermic reactions, occurring simultaneously in a semi-batch process. The dissolution of 29 kg of solid crude rhodium sponge (Rh) into 546 L of a solution of hydrochloric acid (HCl(aq)), to produce a solution of aqueous rhodium(III) chloride (RhCl3.H2O), as well as the reaction of chlorine (Cl2(aq)) with water (H2O(l)) to produce some more HCl(aq) in the reactor. The model was formulated as a system of explicit ordinary differential equations (ODEs), which demonstrated some good and stable qualitative tracking of the temperature and pressure data of the real reactor. The molar responses of all chemical species, as well as the heats of reactions, showed to be consistent with the description of the process, and no negative values of those variables were generated. Estimates of the key parameters of heat and mass transfer coefficients, arrhenius constants, and activation energies of reactions were assumed and tuned to satisfaction by trial-and-error, but not optimised. This is because during simulations, the numerical solver would often fail to integrate the equations, due to the appearance of large derivatives in some model equations whenever those parameters varied, thereby stopping simulations. Finally, the model was validated with a set of data from 45 batches. For all simulations done, the simulated temperature responses showed better prediction of data than the simulated pressure responses did, with an average percentage accuracy of 80% against 60 percent, respectively. / Dissertation (MSc)--University of Pretoria, 2017. / Anglo American Platinum / BluESP (Pty) Ltd / Chemical Engineering / MSc / Unrestricted

MSc (Mathematics) / Department of Mathematics

Die zu erwartenden Temperaturen und Regenmengen der folgenden Tage bis Stunden sind heutzutage eine der wichtigsten Informationen. Diese Kenntnis ist nicht nur von allgemeinem Interesse. Insbesondere Bereiche wie die Landwirtschaft und Forstwirtschaft sind die zu erwartenden Regenmengen selbst über einen langen Zeitraum von Wochen von besonderen Interesse um zum Beispiel die Ernte oder den Schutz von Pflanzen zu planen. Daher ist die Fähigkeit, das Wetter zuverlässig und schnell für ausreichend lange Zeiträume vorher zu sagen, wesentlich. Die Zuverlässigkeit der Wettervorhersage, oder genau genommen der numerischen Wettervorhersage, hängt von mehreren Faktoren ab. Einer dieser Faktoren ist die Detailliertheit der Atmosphärenmodelle. Während die ersten numerischen Experimente die Atmosphäre als eine Schicht trockenen idealen Gases betrachteten, beinhalten aktuelle Modelle die Feuchte, Wolken, Niederschlag und Strahlung. Mit jedem zusätzlichen Detail steigt natürlich der Simulationsaufwand. Daher müssen parallel zur verbesserten Modellierung auch die numerischen Verfahren erweitert werden. Im allgemeinen sind die Atmosphärenmodelle Systeme nichtlinearer hyperbolischer Differentialgleichungen (PDEs). Insbesondere beinhalten die Modelle Wellen unterschiedlicher Ausbreitungsgeschwindigkeit, welche nahezu nicht gedämpft werden. Diese unterschiedlichen Geschwindigkeiten sind die Grundlage für den Mehrskalencharakter der Atmosphärenmodelle. Eine effektive numerische Methode muss daher die unterschiedlichen Skalen adäquat behandeln. Die Entwicklung und Analyse numerischer Mehrskalenverfahren zur Lösung von Systemen hyperbolischer Differentialgleichungen ist herausfordernd. Beispiele für hyperbolische Systeme beginnen bei der einfachen skalaren linearen Advektionsgleichung, der Wellengleichung und enden bei nichtlinearen Systemen wie den Flachwassergleichungen oder den (reibungsfreien) Eulergleichungen. Letztere sind die Grundlage für alle Atmosphärenmodelle. Viele hyperbolische PDEs besitzen eine additive Struktur, wobei die Aufteilung gerade den Zeitskalen entsprechen. Wir gehen von einer angepassten Diskretisierung im Raum, in der Regel eine Finite-Volumen Diskretisierung, aus. Diese Diskretisierung erhält die additive Struktur des kontinuierlichen Problems in der (ortsdiskretisierten) gewöhnlichen Differentialgleichung (ODE). Daher entwickeln wir eine neue numerische Methode zur Lösung gewöhnlicher Differentialgleichungen, welche die additive Struktur und gleichzeitig die zugehörigen Zeitskalen ausnutzt. Die Analyse von Splittingverfahren ist herausfordernd sowohl in der Entwicklung der Ordnungsbedingungen als auch der Stabilitätskriterien. Jeder Mehrskalenansatz kombiniert die unterschiedlichen Zeitskalen auf unterschiedliche Art und Weise. Daher gibt es keine einheitliche Ordnungs- und Stabilitätstheorie. Wir entwickeln die Ordnungsbedingungen auf klassischem Wege, durch Differentiation der numerischen Lösung. Die Aufteilung der rechten Seite in schnelle und langsame Terme führt auf zusätzliche Koeffizienten und Kombinationen der elementaren Differentiale. Im Vergleich zu klassischen Verfahren hat jedes elementare Differential unterschiedliche nicht-klassische Koeffizienten, ohne erkennbare Struktur. Dieser Strukturverlust erschwert die numerische Lösung zusätzlich. Analytische Lösungen gibt es nur in Sonderfällen. Wir entwickeln und analysieren eine neue Klasse von Mehrskalen methoden, welche mit den Integrator der schnellen Skale parametriert ist. Dieser neue Ansatz erlaubt die Verallgemeinerung der Ausgangsmethode und vereinfacht etliche Schritte in der Herleitung der Ordnungsbedinungen. Zusätzlich hat die Verallgemeinerung auch den Vorteil, die Ordnungsbedingungen des Gesamtverfahrens und die Struktur des darunter liegenden Lösers der schnellen Zeitskale zu assoziieren. Wir untersuchen ebenfalls die lineare Stabilität der neuen Methoden. Aufgrund der Aufteilung in langsame und schnelle Terme gibt es viele verschiedene Modellprobleme. Wir leiten ein Modellproblem auf Basis eines vereinfachten hyperbolischer PDEs her. Auf Basis dieses Stabilitsproblems konstruieren wir die neuen Methoden und untersuchen ihre Effizienz anhand zweier nichtlineare Benchmarkprobleme. Analog zur Herleitung der Ordnungsbedingungen vereinheitlichen wir die Konstruktion der Stabilitätsfunktionen und heben im nachhinein die Unterschiede aufgrund des fast-scale integrators hervor. Gute numerische Methoden führen nicht nur zu einem kleinen Fehler, sondern haben auch ein großes Stabilitätsgebiet. Daher optimizieren wir die Methodenparameter im Hinblick auf die Größe des Stabilitätsgebiets. Unsere neuen Methoden besitzen sowohl reelle, als auch rationale Parameter. Die Lösung des gemischten ganzzahligen-reellen Optimierungsproblem vereinfachen wir durch die Auswahl einzelner rationaler Parameter. Dadurch erhalten wir allerdings einige tausend unabhängige Teilprobleme. Zum Abschluss analysieren wir die Effizienz der neuen Methoden anhand zweier nichtlinearer Benchmarkprobleme und vergleichen die Genauigkeit und Stabilität mit Referenzverfahren. / The expected temperatures and rainfall in the next days to hours is one of the most important information nowadays. This knowledge is not only of general interest. Disciplines like agriculture and forestry the knowledge of the rain is even more important for a time span of weeks to plan the harvest or protect the plants. Therefore, the possibility to forecast the weather reliably and fast is very important nowadays. The reliability of weather forecast, or more accurate the numerical weather forecast, depends on several factors. One factor is the complexity of atmosphere models. Whereas the first numerical experiments treat the atmosphere as dry ideal gas with one layer, recent models incorporate the humidity, clouds, precipitation and radiation. But every higher detail in the model come at higher costs for simulation. Hence the development of finer grained models also require more advanced numerical methods to solve them. The atmosphere models are in general a nonlinear hyperbolic set of partial differential equations (PDEs). In particular the models consist of several waves, traveling with different speeds with nearly no damping. Roughly speaking these varying velocities lead to the multiscale nature of the atmosphere models and a suitable numerical method should respect the different time scales. The development and analysis of multirate methods for hyperbolic systems remains a challenging problem. Examples for class of hyperbolic systems of PDEs range from the scalar and linear advection equation, the wave equation to nonlinear systems like the shallow water equations or the (inviscid) Euler equations, which are the basis for the atmosphere models. The hyperbolic PDEs often have an additive split structure, which in turn account for the different time scales. We assume a suitable, often finite volume, discretization in space. Hence we retain the additive splitting from the continuous problem in the semi-discretized ordinary differential equation (ODE). Hence we develop a new numerical method which accounts for the additive split structure and the multiscale nature. The development of splitting methods is challenging in the analysis of the order conditions and the stability criteria. In particular the interaction between the fast and slow scales render the order conditions often complicated and unstructured. Furthermore every multiscale approach combines the scales in a different way, which is why there is no unified order condition theory. With these challenges in mind we derive the order conditions in a classical way by differentiation of the numerical method. The splitting in a fast and a slow right hand side leads to several combinations of elementary differentials. And every differential has different non-standard coefficients, without any structure between these combinations. This loss in structure renders the numerical solutions of the order conditions quite complicated, and the analytical solutions are only possible in rare cases. We develop a new class of multirate methods, which is parameterized by the fast scale solver. That new approach allows for a better generalization and simplifies several steps by unification. Nevertheless this new type of generalization has the advantage to associate the order conditions of the complete (macro scale) method with the structure of the underlying (micro scale) integrator. The second challenge is the analysis of the (linear) stability of multirate methods. We also analyze the (linear) stability of the newly developed methods. Due to the splitting structure there are many different model problems possible. We deduce a model problem from a simplified system of hyperbolic PDEs. On top of these stability model problems we will construct the new methodss. In analogy to the analysis of the order conditions, we unify the construction of the stability functions and highlight the differences due to the different fast scale integrators afterwards. A good method does not only lead to low errors, but also has a large stability area. Hence we optimize the method parameters with respect to the stability area. In our case, the parameter set contains rational and real parameters. We circumvent the solution of a mixed-integer optimization problem by considering only some rational parameters and optimize for them independently. Nevertheless, we obtain several thousand sub problems. Finally we consider two nonlinear benchmark problems. With these problems we analyze the accuracy and stability again and compare the efficiency with two reference multiscale methods.