Dans cette thèse nous nous intéressons aux propriétés qualitatives des solutions de quelques équations d’ondes en milieux dispersifs ou dispersifs-dissipatifs. Dans le premier chapitre, nous étudions l’explosion de solutions dans le régime log-log et l’existence globale pour le problème de Cauchy de l’équation de Schrödinger L2-critique amortie. Dans un second chapitre, nous considérons l’équation de Schrödinger L2-critique avec un amortissement non linéaire. Selon la puissance du terme d’amortissement, nous montrons l’existence globale ou l’explosion en régime log-log. Dans le troisième chapitre, nous étudions le problème de Cauchy pour l’équation de Kadomtsev-Petviashvili-Burgers-I (KPBI) en deux dimensions,nous montrons que le problème est localement bien posé dans Hs(R2) pour tout s > -½, et que l’existence est globale dans L2(R2) sans aucune condition sur la donnée initiale. Dans le dernier chapitre, nous considèrons l’équation d’Ostrovsky sur le cercle, et nous construisons des mesures invariantes par le flot selon les quantitées conservées par cette équation. / This thesis deals with the qualitative properties of solutions to some wave equations in dispersive or dispersive-dissipative media. In the first chapter, we study the blowup in the log-log regime and global existence of solutions to the Cauchy problem for the L2-critical damped nonlinear Schrödinger equation. In the second chapter, we consider the Cauchy problem for the L2-critical nonlinear Schrödinger equation with a nonlinear damping. According to the power of the damping term, we prove the global existence or the existence of finite time blowup dynamics with a log-log blow-up law. In the third chapter, we study the Cauchy problem for the Kadomtsev-Petviashvili-Burgers-I (KPBI) equations in two dimensions. We show that the problem is locally and globally well posed in Hs(R2) for any s > -½ , and that the existence is global in L2(R2) without any condition on the initial data. In the last chapter, we consider the Ostrovsky equation on the circle. We construct invariant measures under the flow for the conserved quantities of the equation.

Nos últimos anos, váarios pesquisadores tais como: A. Bodin, A. Dimca, A. Durfee, A. Jacquemard, A. Menegon Neto, A. Némethi, A. Pichon, A. Verjovsky, A. Zaharia, D. Siersma, H. A. Hamm, D. Massey, H. Aguilar-Cabrera, H. H. Vui, J. Cisneros, J. Seade, J. Snoussi, L. D. Tráng, L. Paunescu, L. R. Dias, M. A. S. Ruas, M. Oka, M. Tibar, N. Dutertre, R. N. Araújo dos Santos, S. A. Broughton, T. Gaffney, Y. Chen, entre outros, têm apresentado generalizações dos Teoremas de fibrações de Milnor no ambiente real e complexo (e do Teorema de Kurdyka-Orro-Simon, ver por exemplo [Di, KOS]), visando um melhor entendimento de propriedades topológicas locais e globais das singularidades. Nesta direção de pesquisa esses autores tem utilizado várias ferramentas e técnicas de diversas áreas da matemática. O que mostra a riqueza e a complexidade destes estudos e acrescenta, em nossa modesta opinião, um aspecto que é ao mesmo tempo interessante e desafiador. Neste trabalho, mostraremos como estender as fibrações de Milnor em esferas no caso local e global, real e complexo, para uma aplicação C2-semi-algébrica F = (f1, . . . , fp) : RN → Rp e uma variedade W ⊂ RN semi-algébrica com possível singularidade. Com tal objetivo, introduziremos as condições de Milnor (a) e (b) generalizadas" e mostraremos como adaptar a técnica da decomposição open book superior com binding singular, introduzida em [AT, ACT1]. Nossos resultados sugerem que tal estrutura de fibração pode ser um caso particular de algum Teorema estrutural mais geral. Além do mais, considerando π : Rp → Rp-1 a projeção canônica na meta, mostraremos que se F satisfaz tais condições, então G = π o F : RN → Rp-1 também satisfaz e, consequentemente, G também induz em W uma fibração suave localmente trivial. Concluiremos mostrando que após as projeções as fibras destes fibrados são homotopicamente equivalentes e, em seguida, apresentando algumas fórmulas que relacionam a característica de Euler do "link relativo" W ∩ F-1 (0) com a característica de Euler das fibras. / In the last years, several researchers such as: A. Bodin, A. Dimca, A. Durfee, A. Jacquemard, A. Menegon Neto, A. Némethi, A. Pichon, A. Verjovsky, A. Zaharia, D. Siersma, H. A. Hamm, D. Massey, H. Aguilar-Cabrera, H. H. Vui, J. Cisneros, J. Seade, J. Snoussi, L. D. Trang, L. Paunescu, L. R. Dias, M. A. S. Ruas, M. Oka, M. Tibar, N. Dutertre, R. N. Araújo dos Santos, S. A. Broughton, T. Gaffney, Y. Chen, and others, have proven generalizations of Milnor fibrationss Theorems in the real and complex settings (and Kurdyka-Orro-Simons Theorem, see e.g. [Di, KOS]), aiming a better understanding of the local and global topological properties of singularity. In this research branch, these authors have used many different tools and techniques from several areas of Mathematics. This shows the richness and complexities of these studies and adds, in our modest opinion, an aspect that is simultaneously interesting and challenging. In this work, we introduce the generalized Milnors conditions (a) and (b) to show an extension of the Milnor fibration Theorems on spheres in the local and global cases, in the real and complex setting. For this, we consider a C2-semi-algebraic mapping F = (f1, . . . , fp) : RN → Rp , a possible singular semi-algebraic variety W ⊂ RN, and we show how to adapt the technique of Higher open book decomposition with singular binding, introduced by [AT, ACT1], to prove such extension. Our results suggest that such fibration structure may be a particular case of a more general fibration structure. Furthermore, considering : Rp Rp-1 the canonical projection on the target space, we show that if F satisfies the generalized Milnors conditions (a) and (b), then G = π o F : RN → Rp-1 also satisfies these conditions and, hence G also induces on W a smooth locally trivial fibration. Finally, we show that after the projections on the target space, the fibers of these fiber bundles are homotopically equivalent. We conclude by proving some formulae connecting the Euler characteristic of "relative link" W ∩ F-1 (0) with the Euler characteristic of the fibers. Key words and phrases: generalized open book decomposition, fibration structure on semi-algebraic sets, topology of singularity, real and complex Milnors fibrations and, local and global fibration.

>Magister Scientiae - MSc / HIV/AIDS is nowadays considered as the greatest public health disaster of modern time. Its progression has challenged the global population for decades. Through mathematical modelling, researchers have studied different interventions on the HIV pandemic, such as treatment, education, condom use, etc. Our research focuses on different compartmental models with emphasis on the effect of public health education. From the point of view of statistics, it is well known how the public health educational programs contribute towards the reduction of the spread of HIV/AIDS epidemic. Many models have been studied towards understanding the dynamics of the HIV/AIDS epidemic. The impact of ARV treatment have been observed and analysed by many researchers. Our research studies and investigates a compartmental model of HIV with treatment and education campaign. We study the existence of equilibrium points and their stability. Original contributions of this dissertation are the modifications on the model of Cai et al. [1], which enables us to use optimal control theory to identify optimal roll-out of strategies to control the HIV/AIDS. Furthermore, we introduce randomness into the model and we study the almost sure exponential stability of the disease free equilibrium. The randomness is regarded as environmental perturbations in the system. Another contribution is the global stability analysis on the model of Nyabadza et al. in [3]. The stability thresholds are compared for the HIV/AIDS in the absence of any intervention to assess the possible community benefit of public health educational campaigns. We illustrate the results by way simulation The following papers form the basis of much of the content of this dissertation, [1 ] L. Cai, Xuezhi Li, Mini Ghosh, Boazhu Guo. Stability analysis of an HIV/AIDS epidemic model with treatment, 229 (2009) 313-323. [2 ] C.P. Bhunu, S. Mushayabasa, H. Kojouharov, J.M. Tchuenche. Mathematical Analysis of an HIV/AIDS Model: Impact of Educational Programs and Abstinence in Sub-Saharan Africa. J Math Model Algor 10 (2011),31-55. [3 ] F. Nyabadza, C. Chiyaka, Z. Mukandavire, S.D. Hove-Musekwa. Analysis of an HIV/AIDS model with public-health information campaigns and individual with-drawal. Journal of Biological Systems, 18, 2 (2010) 357-375. Through this dissertation the author has contributed to two manuscripts [4] and [5], which are currently under review towards publication in journals, [4 ] G. Abiodun, S. Maku Vyambwera, N. Marcus, K. Okosun, P. Witbooi. Control and sensitivity of an HIV model with public health education (under submission). [5 ] P.Witbooi, M. Nsuami, S. Maku Vyambwera. Stability of a stochastic model of HIV population dynamics (under submission).

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I denna kvalitativa intervjustudie har fyra gymnasielärare delat med sig av sina uppfattningar om användningen av skrivande och skriftlig korrigerande feedback som ett sätt att utveckla elevers grammatiska korrekthet. Studien visar att deltagarna anser att det är viktigt att integrera grammatikundervisningen med skrivande då det bland annat möjliggör individualisering. Vidare framkommer det att det finns ett större fokus på att rätta lokala fel än globala, och att omfattningen av feedback bör baseras på elevers kunskapsnivå. Lärarnas feedbackstrategier påverkas också av elevers kunskapsnivåer, men även felets komplexitet. Slutligen betonar lärarna att för att den skriftliga korrigerande feedbacken ska vara effektiv behöver eleverna få tid till att bearbeta den.

<p>We investigated 1) how objects come to serve as landmarks in spatial memory and more specifically how they form part of an allocentric cognitive map and 2) how humans encode multiple connected spatial environments. In both sets of experiments, participants performing a virtual driving task incidentally learned the layout of a town and locations of objects or stores in that town. Their spatial memory and recognition memory for the objects or stores were subsequently tested. To assess whether the objects were encoded allocentrically, we developed a new measurement, pointing consistency. We found that when participants had more limited experience of the environment spatial memory for objects at navigationally relevant locations was more consistent across tested viewpoints than for objects at navigationally less relevant locations. When participants’ attention was focused on the appearance of objects, the navigational relevance effect was eliminated, whereas when their attention was focused on the objects’ locations, this effect was enhanced, supporting the hypothesis that when objects are processed in the service of navigation, rather than merely being viewed as objects, they engage qualitatively distinct attentional systems and are incorporated into an allocentric spatial representation. The results were consistent with evidence from the neuroimaging literature that when objects are relevant to navigation, they not only engage the ventral “object processing stream”, but also the dorsal stream and medial temporal lobe memory system classically associated with allocentric spatial memory. Moreover, in the connected environments, our data were more consistent with the formation of local maps, regardless of whether the neighborhoods were learned together or separately. Only when all visible distinctions between neighborhoods were removed did people behave as if they formed one integrated map. These data are broadly consistent with evidence from rodent hippocampal place cell recordings in connected boxes, and with hierarchical models of spatial coding.</p> / Doctor of Philosophy (PhD)

Dans cette thèse nous nous intéressons aux propriétés qualitatives et quantitatives des solutions de quelques équations d'ondes en milieux dispersifs ou dispersifs-dissipatifs. Dans une première partie, nous étudions le problème de Cauchy associé aux équations de Benjamin-Ono généralisées. A l'aide de transformées de jauge, combinées avec des outils d'analyse harmonique, nous prouvons des résultats concernant le caractère localement bien posé pour des données initiales de régularité minimale dans l'échelle des espaces de Sobolev. Dans une seconde partie, nous étudions le problème de Cauchy pour des versions dissipatives des équations de Benjamin-Ono et de Korteweg-de Vries. Nous mettons en évidence l'influence des effets dissipatifs sur ces équations en donnant des résultats optimaux sur leur caractère bien ou mal posé. Ceux-ci sont obtenus en travaillant dans des espaces de type Bourgain adaptés à la partie dispersive-dissipative. Pour finir nous étudions le comportement asymptotique des solutions des équations de KdV dissipatives, lorsque celles-ci existent pour tout temps, en calculant explicitement les premiers termes du développement asymptotique dans de nombreux espaces de Sobolev / This thesis deals with the qualitative and quantitative properties of solutions to some wave equations in dispersive or dispersive-dissipative media. In the first part, we study the Cauchy problem for the generalized Benjamin-Ono equations. By means of gauge transforms combined with some harmonic analysis tools, we prove some local well-posedness results for initial data with minimal regularity in Sobolev spaces. In the second part, we study the Cauchy problem for some dissipative versions of the Benjamin-Ono and Korteweg-de Vries equations. We show the influence of the dissipative effects and prove sharp well and ill-posedness results. This is obtained by working in suitable Bourgain's spaces, adapted to the dispersive-dissipative part of the equation. Finally, we study the asymptotic behavior of solutions to the dissipative KdV equations. We explicitly compute the first terms of the asymptotic expansion in Sobolev spaces

<p><b>In this dissertation we present the quantitative response of an epidemic of the so-called SIR-type, in a population consisting of a local component and a migrant component. Each component can be divided into three classes, the susceptible individuals, usually denoted by S, who are uninfected but may contract the disease, infected individuals (I) who are infected and can spread the disease to the susceptible individuals and the class (R) of recovered individuals. If a susceptible individual becomes infected, it moves into the infected class. An infected individual, at recovery, moves to the class R. Firstly we develop a model describing two interacting populations with vaccination. Assuming the vaccination rate in both groups or components are constant, we calculate a threshold parameter and we call it a vaccination reproductive number. This invariant determines whether the disease will die out or becomes endemic on the (in particular, local) population. Then we present the stability analysis of equilibrium points and the effect of vaccination. Our primary finding is that the behaviour of the disease free equilibrium depend on the vaccination rates of the combined population. We show that the disease free equilibrium is locally asymptotically stable if the vaccination reproductive number is less than one. Also our stability analysis show that the global stability of the disease free equilibrium depends on the basic reproduction number, not the vaccination reproductive number. If the vaccination reproductive number is greater than one, then the disease free equilibrium is unstable and there exists three endemic equilibrium points in our model. Two of these three endemic equilibria are so-called boundary equilibrium points, which means that the infection is only in one group of the population. The third one which we focus on is the general endemic point for the whole system. We derive a threshold condition that determines whether the endemic equilibria is locally asymptotically stable or not. Secondly, by assuming that the rate of vaccination in the migrant population is constant, we apply optimal control theory to find an optimal vaccination strategy in the local population. Our numerical simulation shows the effectiveness of the control strategy. This model is suitable for modeling the real life situation to control many communicable diseases. Models similar to the model used in the main contribution of our dissertation do exist in the literature. In fact, our model can be regarded as being in-between those of [Jia et al., Theoretical Population Biology 73 (2008) 437-448] and [Piccolo and Billings, Mathematical and Computer Modeling 42 (2005) 291-299]. Nevertheless our stability analysis is original, and furthermore we perform an optimal control study whereas the two cited papers do not. The essence of chapter 5 and 6 of this dissertation is being prepared for publication.</b></p>

<p><b>In this dissertation we present the quantitative response of an epidemic of the so-called SIR-type, in a population consisting of a local component and a migrant component. Each component can be divided into three classes, the susceptible individuals, usually denoted by S, who are uninfected but may contract the disease, infected individuals (I) who are infected and can spread the disease to the susceptible individuals and the class (R) of recovered individuals. If a susceptible individual becomes infected, it moves into the infected class. An infected individual, at recovery, moves to the class R. Firstly we develop a model describing two interacting populations with vaccination. Assuming the vaccination rate in both groups or components are constant, we calculate a threshold parameter and we call it a vaccination reproductive number. This invariant determines whether the disease will die out or becomes endemic on the (in particular, local) population. Then we present the stability analysis of equilibrium points and the effect of vaccination. Our primary finding is that the behaviour of the disease free equilibrium depend on the vaccination rates of the combined population. We show that the disease free equilibrium is locally asymptotically stable if the vaccination reproductive number is less than one. Also our stability analysis show that the global stability of the disease free equilibrium depends on the basic reproduction number, not the vaccination reproductive number. If the vaccination reproductive number is greater than one, then the disease free equilibrium is unstable and there exists three endemic equilibrium points in our model. Two of these three endemic equilibria are so-called boundary equilibrium points, which means that the infection is only in one group of the population. The third one which we focus on is the general endemic point for the whole system. We derive a threshold condition that determines whether the endemic equilibria is locally asymptotically stable or not. Secondly, by assuming that the rate of vaccination in the migrant population is constant, we apply optimal control theory to find an optimal vaccination strategy in the local population. Our numerical simulation shows the effectiveness of the control strategy. This model is suitable for modeling the real life situation to control many communicable diseases. Models similar to the model used in the main contribution of our dissertation do exist in the literature. In fact, our model can be regarded as being in-between those of [Jia et al., Theoretical Population Biology 73 (2008) 437-448] and [Piccolo and Billings, Mathematical and Computer Modeling 42 (2005) 291-299]. Nevertheless our stability analysis is original, and furthermore we perform an optimal control study whereas the two cited papers do not. The essence of chapter 5 and 6 of this dissertation is being prepared for publication.</b></p>

In this paper we view learning as an unconstrained non-linear minimization problem in which the objective function is defined by the negative log-likelihood function and the search space by the parameter space of an origin constrained product unit neural spatial interaction model. We consider Alopex based global search, as opposed to local search based upon backpropagation of gradient descents, each in combination with the bootstrapping pairs approach to solve the maximum likelihood learning problem. Interregional telecommunication traffic flow data from Austria are used as test bed for comparing the performance of the two learning procedures. The study illustrates the superiority of Alopex based global search, measured in terms of Kullback and Leibler's information criterion.