Apresentamos uma construção para a teoria das hiperfunções no espaço euclidiano seguindo a abordagem de André Martineau baseada em funcionais analíticos e aplicando um teorema de dualidade de Jean-Pierre Serre. Estudamos também o teorema de divisão de hiperfunções por funções reais-analíticas, provado em Kantor e Schapira (1971). No último capítulo, desenvolvemos alguns aspectos da teoria das hiperfunções no toro. / We present the hyperfunction theory on the Euclidean space following André Martineau\'s approach based on analytic functionals and a duality theorem due to Jean- Pierre Serre. We also study a division theorem proved in Kantor and Schapira (1971). In the last chapter, we develop some aspects of hyperfunction theory on the torus.

We study the relationship between geometric properties of toric varieties and combinatorial properties of the corresponding lattice polytopes. In particular, we give a bound for a very ample lattice polytope to be k-normal. Equivalently, we give a new combinatorial bound for the Castelnuovo-Mumford regularity of normal projective toric varieties. We also give a new combinatorial proof for a special case of Reider's Theorem for smooth toric surfaces.

Temporal regularity and metrical organisation are important factors in beat and metre perception. The current thesis aims to investigate neural correlates of beat and metre perception in healthy non-musician volunteers, using functional magnetic resonance imaging (fMRI). In particular, the thesis focuses on determining the role of the Inferior Frontal Gyrus (IFG, in particular BA44/45) in beat and metre perception. The IFG has been proposed to be involved in higher order cognitive processes during various temporal sequencing, such as speech, movement, and music. Previous studies have shown that the temporal processing of rhythm activates auditory and sensorimotor areas, but the role of the IFG in rhythm perception has not yet been fully understood. Study 1 investigated beat perception in complex rhythms, in which the addition of volume accents either enhanced or weakened the beat perception, resulting in Unaccented, Beat Accented or Non-Beat Accented rhythms. Participants (N=14; 6 males) listened to rhythm pairs across all three conditions, and judged whether each rhythm pair was the same or different. Results showed that left IFG activation (BA44) was significantly greater during the Non-Beat Accented condition compared to Beat Accented condition, whereas the right IFG activation showed no significant difference between the two conditions. Study 2 investigated metre perception of a series of isochronous sequences, of which metrical organisation was grouped by 2/4 (C2), 3/4 (C3), or 4/4 (C4) using pitch accents, or remained without metrical grouping (or 1/4, C1). The same participants (N=15; 6 males) listened to the stimuli and indicated the perceived metrical grouping level. Results showed that the activation of bilateral IFG parametrically increased from C2 to C3 to C4. Interestingly, the activation was found to be significantly greater in C1 relative to C2, suggesting that involuntary subjective in C1 may increase the brain response. Converging results from both Study 1 and Study 2 demonstrated, firstly, that the bilateral IFG is involved in rhythm perception in addition to the auditory and sensorimotor areas, including primary and secondary auditory areas, supplementary motor areas, premotor areas, insula, and basal ganglia. Secondly, the left IFG (BA44) in particular was significantly modulated by the rhythmic complexity relating to both temporal regularity and metrical organisation, while showing the suppression during the Beat Accented rhythm condition of Study 1 and the binary pattern (C2) of Study 2. This thesis argues that the left IFG (BA44) may have the role the higher order cognitive processing, such as attention and prediction, in the perception of hierarchical structures in metric rhythms.

In this work, we prove three types of results with the strategy that, together, the author believes these should imply the local version of Hilbert's Fifth problem. In a separate development, we construct a nontrivial topology for rings of map germs on Euclidean spaces. First, we develop a framework for the theory of (local) nonstandard Lie groups and within that framework prove a nonstandard result that implies that a family of local Lie groups that converge in a pointwise sense must then differentiability converge, up to coordinate change, to an analytic local Lie group, see corollary 6.3.1. The second result essentially says that a pair of mappings that almost satisfy the properties defining a local Lie group must have a local Lie group nearby, see proposition 7.2.1. Pairing the above two results, we get the principal standard consequence of the above work which can be roughly described as follows. If we have pointwise equicontinuous family of mapping pairs (potential local Euclidean topological group structures), pointwise approximating a (possibly differentiably unbounded) family of differentiable (sufficiently approximate) almost groups, then the original family has, after appropriate coordinate change, a local Lie group as a limit point. (See corollary 7.2.1 for the exact statement.) The third set of results give nonstandard renditions of equicontinuity criteria for families of differentiable functions, see theorem 9.1.1. These results are critical in the proofs of the principal results of this paper as well as the standard interpretations of the main results here. Following this material, we have a long chapter constructing a Hausdorff topology on the ring of real valued map germs on Euclidean space. This topology has good properties with respect to convergence and composition. See the detailed introduction to this chapter for the motivation and description of this topology.

Acoustic regularity encoding has been associated with a decrease of the neural response to repeated stimulation underlying the representation of auditory objects in the brain. The present thesis encloses two studies that sought to assess the neural correlates of acoustic regularity encoding in the human auditory system, by means of analyzing auditory evoked potentials. Study I was conducted at the Cognitive Neuroscience Research Group, at the Faculty of Psychology of the University of Barcelona (Barcelona, Catalonia, Spain), under the direct supervision of Dr. Carles Escera. This study aimed to explore the dynamics of adaptation of the auditory evoked potentials to probabilistic stimuli embedded in a complex sequence of sounds. The main outcome of this study was the demonstration that the amplitude of auditory evoked potentials adapts to the complex history of stimulation with different time constants concurrently: it adapts faster to local and slower to global probabilities of stimulation. This study also showed that auditory evoked potential amplitudes correlate with stimulus expectancy as defined by a combination of local and global stimulus probabilities. Study II was conducted at the Institute of Child Health (ICH), at the University College of London (UCL; London, United Kingdom), under the direct supervision of Dr. Torsten Baldeweg. This study aimed to explore the influence of timing predictability in the neural adaptation to probabilistic stimuli. The main outcome of this study was the demonstration that timing predictability enhances the repetition-related modulation of the auditory evoked potentials amplitude, being essential for repetition effects at early stages of the auditory processing hierarchy. / La codificació de regularitats acústiques està associada amb la reducció de la resposta neuronal a l’estimulació repetida, essent la base de la representació dels objectes auditius al cervell. La present tesi doctoral inclou dos estudis que exploren els correlats neuronals de la codificació de regularitats acústiques al sistema auditiu humà, mitjançant l’anàlisi dels potencials evocats auditius. L’objectiu del primer estudi, realitzat al Grup de Recerca en Neurociència Cognitiva de la Facultat de Psicologia de la Universitat de Barcelona (UB) i sota la supervisió directa del Dr. Carles Escera, va ser el d’explorar les dinàmiques d’adaptació dels potencials evocats auditius a estímuls probabilístics en una complexa seqüència de sons. El resultat principal d’aquest estudi va ser la demostració de que l’amplitud dels potencials evocats auditius s’adapta a la historia complexa d’estimulació amb diferents constants temporals simultàniament: s’adapta més ràpidament a probabilitats d’estimulació locals que globals. Aquest estudi també va mostrar que l’amplitud dels potencials evocats auditius correlaciona amb l’expectància d’un estímul definida com a una combinació de probabilitats locals i globals d’estimulació. L’objectiu del segon estudi, realitzat al Institute of Child Health (ICH), de l’University College of London (UCL), sota la supervision directa del Dr. Torsten Baldeweg, va ser el d’explorar la influència de la predictabilitat temporal en l’adaptació de l’activitat neuronal a estímuls probabilístics. El resultat principal d’aquest estudi va ser la demostració que la predictabilitat temporal intensifica la modulació de l’amplitud dels potencials evocats auditius a la repetició dels estímuls, essent esencial pels efectes que la repetició exerceix en etapes primerenques de la jerarquía de processament auditiu.

The miscible displacement equations provide the mathematical model for simulating the displacement of a mixture of oil and miscible fluid in underground reservoirs during the Enhance Oil Recovery(EOR) process. In this thesis, I propose a stable numerical scheme combining a mixed finite element method and space-time discontinuous Galerkin method for solving miscible displacement equations under low regularity assumption. Convergence of the discrete solution is investigated using a compactness theorem for functions that are discontinuous in space and time. Numerical experiments illustrate that the rate of convergence is improved by using a high order time stepping method. For petroleum engineers, it is essential to compute finely detailed fluid profiles in order to design efficient recovery procedure thereby increase production in the EOR process. The method I propose takes advantage of both high order time approximation and discontinuous Galerkin method in space and is capable of providing accurate numerical solutions to assist in increasing the production rate of the miscible displacement oil recovery process.

<p>During the last decade, potential theory and p-harmonic functions have been developed in the setting of doubling metric measure spaces supporting a <em>p</em>-Poincar´e inequality. This theory unifies, and has applications in several areas of analysis, such as weighted Sobolev spaces, calculus on Riemannian manifolds and Carnot groups, subelliptic differential operators and potential theory on graphs.</p><p>In this thesis we investigate the double obstacle problem for p-harmonic functions on metric spaces. We show the existence and uniqueness of solutions and their continuity when the obstacles are continuous. Moreover the solution is p-harmonic in the open set where it does not touch the continuous obstacles. The boundary regularity of the solutions is also studied.</p><p>Furthermore we study two kinds of convergence problems for the solutions. First we let the obstacles vary and fix the boundary values and show the convergence of the solutions. Second we consider an increasing sequence of open sets, with union Ω, and fix the obstacles and the boundary values. We show that the solutions of the obstacle problems in these sets converge to the solution of the corresponding problem in Ω.</p> / <p>Låt oss börja med att betrakta följande situation: Vi vill förflytta oss från en plats vid ena sidan av en äng till en viss punkt på andra sidan ängen. På båda sidor om ängen finns skogsområden som vi inte får gå in i. Ängen är tyvärr inte homogen utan består av olika sorters mark som vi har noggrant beskrivet på en karta. Vi vill göra förflyttningen på smidigast sätt, men då ängen inte är homogen ska vi förmodligen inte gå rakaste vägen utan ska anpassa vägen optimalt efter terrängen. Detta är ett exempel på ett dubbelhinderproblem där hindren är skogsområdena på sidorna som vi måste hålla oss utanför.</p><p>Mer abstrakt vill man minimiera energin hos funktioner som tar vissa givna randvärden (de givna start- och slutpunkterna i exemplet ovan) och som håller sig mellan ett undre och ett övre hinder. I denna avhandling studeras detta dubbelhinderproblem i väldigt allmänna situationer.</p><p>För att kunna lösa hinderproblemet krävs det att vi tillåter ickekontinuerliga lösningar och då visas i avhandlingen att hinderproblemet är entydigt lösbart. Ett huvudresultat i avhandlingen är att om våra hinder är kontinuerliga så blir även lösningen kontinuerlig. Vidare visas diverse konvergenssatser som visar hur lösningarna varierar när hindren eller området i vilket problemet löses varierar.</p><p>Hinderproblem har utöver eget intresse viktiga tillämpningar i potentialteorin, bland annat för att studera motsvarande energiminimeringsproblem utan hinder.</p>

Let F0 be a fixed k-uniform hypergraph, and let H be a given k-uniform hypergraph on n vertices. An F0-packing of H is a family F of edge-disjoint copies of F0 which are subhypergraphs in H. Let nF0(H) denote the maximum size |F| of an F0-packing F of H. It is well-known that computing nF0(H) is NP-hard for nearly any choice of F0. In this thesis, we consider the special case when F0 is a linear hypergraph, that is, when no two edges of F0 overlap in more than one vertex. We establish for z > 0 and n &ge n0(z) sufficiently large, an algorithm which, in time polynomial in n, constructs an F0-packing F of H of size |F| ≥ nF0(H) - znk. A central direction in our proof uses so-called fractional F0-packings of H which are known to approximate nF0(H). The driving force of our argument, however, is the use and development of several tools within the theory of hypergraph regularity.

It is the main goal of this thesis to study the regularity of solutions for a nonlinear elliptic system coming from population segregation, and the free boundary problem that is obtained in the limit as the competition parameter goes to infinity [mathematical symbol]. The main results are existence and Hölder regularity of solutions of the elliptic system, characterization of the limit as a free boundary problem, and Lipschitz regularity at the boundary for the limiting problem. / text

Szemerédi's Regularity Lemma is powerful tool in Graph Theory, yielding many applications in areas such as Extremal Graph Theory, Combinatorial Number Theory and Theoretical Computer Science. Strong hypergraph extensions of graph regularity techniques were recently given by Nagle, Rödl, Schacht and Skokan, by W.T. Gowers, and subsequently, by T. Tao. These extensions have yielded quite a few non-trivial applications to Extremal Hypergraph Theory, Combinatorial Number Theory and Theoretical Computer Science. A main drawback to the hypergraph regularity techniques above is that they are highly technical. In this thesis, we consider a less technical version of hypergraph regularity which more directly generalizes Szemeredi's regularity lemma for graphs. The tools we discuss won't yield all applications of their stronger relatives, but yield still several applications in extremal hypergraph theory (for so-called linear or simple hypergraphs), including algorithmic ones. This thesis surveys these lighter regularity techiques, and develops three applications of them.