We study the relationship between subdivision rules, 3-dimensional manifolds, and circle packings. We find explicit subdivision rules for closed right-angled hyperbolic manifolds, a large family of hyperbolic manifolds with boundary, and all 3-manifolds of the E^3,H^2 x R, S^2 x R, SL_2(R), and S^3 geometries (up to finite covers). We define subdivision rules in all dimensions and find explicit subdivision rules for the n-dimensional torus as an example in each dimension. We define a graph and space at infinity for all subdivision rules, and use that to show that all subdivision rules for non-hyperbolic manifolds have mesh not going to 0. We provide an alternate proof of the Combinatorial Riemann Mapping Theorem using circle packings (although this has been done before). We provide a new definition of conformal for subdivision rules of unbounded valence, show that the subdivision rules for the Borromean rings complement are conformal and show that barycentric subdivision is almost conformal. Finally, we show that subdivision rules can be degenerate on a dense set, while still having convergent circle packings.

Ce manuscrit porte sur les transitions compressible-incompressible dans les équations aux dérivées partielles de la mécanique des fluides. On s'intéresse à deux problèmes : les structures flottantes et les fluides en rotation. Dans le premier problème, l'introduction d'un objet flottant dans les vagues induit une contrainte sur le fluide et les équations gouvernant le mouvement acquièrent une structure compressible-incompressible. Dans le deuxième problème, le mouvement de fluides géophysiques compressibles est influencé par la rotation de la Terre. L'étude de la limite à rotation rapide montre que le champ vectoriel de vitesse tend vers une configuration horizontale et incompressible.Les structures flottantes constituent un exemple particulier d'interaction fluide-structure, où un solide partiellement immergé flotte à la surface du fluide. Ce problème mathématique modélise le mouvement de convertisseurs d'énergie marine. En particulier, on s'intéresse aux bouées pilonnantes, installées proche de la côte où les modèles asymptotiques en eaux peu profondes sont valables. On étudie les équations de Saint-Venant axisymétriques en dimension deux avec un objet flottant à murs verticaux se déplaçant seulement verticalement. Les hypothèses sur le solide permettent de supprimer le problème à bord libre associé avec la ligne de contact entre l'air, le fluide et le solide. Les équations pour le fluide dans le domaine extérieur au solide sont donc écrites comme un problème au bord quasi-linéaire hyperbolique. Celui-ci est couplé avec une EDO non-linéaire du second ordre qui est dérivée de l'équation de Newton pour le mouvement libre du solide. On montre le caractère bien posé localement en temps du système couplé lorsque que les données initiales satisfont des conditions de compatibilité afin de générer des solutions régulières.Ensuite on considère une configuration particulière: le retour à l'équilibre. Il s'agit de considérer un solide partiellement immergé dans un fluide initialement au repos et de le laisser retourner à sa position d'équilibre. Pour cela, on utilise un modèle hydrodynamique différent, où les équations sont linearisées dans le domaine extérieur, tandis que les effets non-linéaires sont considérés en dessous du solide. Le mouvement du solide est décrit par une équation intégro-différentielle non-linéaire du second ordre qui justifie rigoureusement l'équation de Cummins, utilisée par les ingénieurs pour les mouvements des objets flottants. L'équation que l'on dérive améliore l'approche linéaire de Cummins en tenant compte des effets non-linéaires. On montre l'existence et l'unicité globale de la solution pour des données petites en utilisant la conservation de l'énergie du système fluide-structure.Dans la deuxième partie du manuscrit, on étudie les fluides en rotation rapide. Ce problème mathématique modélise le mouvement des flots géophysiques à grandes échelles influencés par la rotation de la Terre. Le mouvement est aussi affecté par la gravité, ce qui donne lieu à une stratification de la densité dans les fluides compressibles. La rotation génère de l'anisotropie dans les flots visqueux et la viscosité turbulente verticale tend vers zéro dans la limite à rotation rapide. Notre interêt porte sur ce problème de limite singulière en tenant compte des effets gravitationnels et compressibles. On étudie les équations de Navier-Stokes-Coriolis anisotropes compressibles avec force gravitationnelle dans la bande infinie horizontale avec une condition au bord de non glissement. Celle-ci et la force de Coriolis donnent lieu à l'apparition des couches d'Ekman proche du bord. Dans ce travail on considère des données initiales bien préparées. On montre un résultat de stabilité des solutions faibles globales pour des lois de pression particulières. La dynamique limite est décrite par une équation quasi-géostrophique visqueuse en dimension deux avec un terme d'amortissement qui tient compte des couches limites. / This manuscript deals with compressible-incompressible transitions arising in partial differential equations of fluid mechanics. We investigate two problems: floating structures and rotating fluids. In the first problem, the introduction of a floating object into water waves enforces a constraint on the fluid and the governing equations turn out to have a compressible-incompressible structure. In the second problem, the motion of geophysical compressible fluids is affected by the Earth's rotation and the study of the high rotation limit shows that the velocity vector field tends to be horizontal and with an incompressibility constraint.Floating structures are a particular example of fluid-structure interaction, in which a partially immersed solid is floating at the fluid surface. This mathematical problem models the motion of wave energy converters in sea water. In particular, we focus on heaving buoys, usually implemented in the near-shore zone, where the shallow water asymptotic models describe accurately the motion of waves. We study the two-dimensional nonlinear shallow water equations in the axisymmetric configuration in the presence of a floating object with vertical side-walls moving only vertically. The assumptions on the solid permit to avoid the free boundary problem associated with the moving contact line between the air, the water and the solid. Hence, in the domain exterior to the solid the fluid equations can be written as an hyperbolic quasilinear initial boundary value problem. This couples with a nonlinear second order ODE derived from Newton's law for the free solid motion. Local in time well-posedness of the coupled system is shown provided some compatibility conditions are satisfied by the initial data in order to generate smooth solutions.Afterwards, we address a particular configuration of this fluid-structure interaction: the return to equilibrium. It consists in releasing a partially immersed solid body into a fluid initially at rest and letting it evolve towards its equilibrium position. A different hydrodynamical model is used. In the exterior domain the equations are linearized but the nonlinear effects are taken into account under the solid. The equation for the solid motion becomes a nonlinear second order integro-differential equation which rigorously justifies the Cummins equation, assumed by engineers to govern the motion of floating objects. Moreover, the equation derived improves the linear approach of Cummins by taking into account the nonlinear effects. The global existence and uniqueness of the solution is shown for small data using the conservation of the energy of the fluid-structure system.In the second part of the manuscript, highly rotating fluids are studied. This mathematical problem models the motion of geophysical flows at large scales affected by the Earth's rotation, such as massive oceanic and atmospheric currents. The motion is also influenced by the gravity, which causes a stratification of the density in compressible fluids. The rotation generates anisotropy in viscous flows and the vertical turbulent viscosity tends to zero in the high rotation limit. Our interest lies in this singular limit problem taking into account gravitational and compressible effects. We study the compressible anisotropic Navier-Stokes-Coriolis equations with gravitational force in the horizontal infinite slab with no-slip boundary condition. Both this condition and the Coriolis force cause the apparition of Ekman layers near the boundary. They are taken into account in the analysis by adding corrector terms which decay in the interior of the domain. In this work well-prepared initial data are considered. A stability result of global weak solutions is shown for power-type pressure laws. The limit dynamics is described by a two-dimensional viscous quasi-geostrophic equation with a damping term that accounts for the boundary layers.

Les traitements en radiothérapie consistent à irradier le patient avec desfaisceaux de particules énergétiques (typiquement des photons) ciblant la tumeur. Cesparticules sont transporté à travers le milieu et y dépose de l'énergie. Cette énergiedéposée, appelée la dose, est responsable des effets biologiques des radiations.Ce travail a pour but de développer des méthodes numériques de calcul etd'optimisation de la dose qui sont compétitives en termes de coût de calcul et de précisionpar rapport à des méthodes de référence.Le mouvement des particules est d'abord étudié via un système d'équationscinétiques linéaires. Cependant, résoudre directement ces systèmes est numériquementtrop coûteux pour des applications médicales. Pour palier ce coût de calcul, la méthodemoment, et en particulier les modèles Mn, est utilisée. Ces équations aux moments sontnon linéaires et valides sous une condition appelée réalisabilité.Les schémas numériques standards pour les équations aux moments sontcontraints par des conditions de stabilité qui se trouvent être très restrictives lorsque lemilieu contient des zones sous-denses. Des schémas numériques inconditionnellementstables et adaptés aux équations aux moments (préservant la réalisation) sontdéveloppés. Ces schémas se révèlent compétitifs en termes de coûts de calcul par rapportaux approches de référence. Finalement, ces méthodes sont appliquées dans uneprocédure d'optimisation visant à maximiser la dose dans la tumeur et la minimiser dansles tissus sains. / Radiotherapy treatments consists in irradiating the patient with beams ofenergetic particles (typically photons) targeting the tumor. Such particles are transportedthrough the medium and deposit energy in the medium. This deposited energy is the socalleddose, responsible for the biological effect of the radiations.The present work aim to develop numerical methods for dose computation andoptimization that are competitive in terms of computational cost and accuracy compared toreference method.The motion of particles is first studied through a system of linear transport equationsat the kinetic level. However, solving directly such systems is numerically too costly formedical application. Instead, the moment method is used with a special focus on the Mnmodels. Those moment equations are non-linear and valid under a condition calledrealizability.Standard numerical schemes for moment equations are constrained by stabilityconditions which happen to be very restrictive when the medium contains low densityregions. Inconditionally stable numerical schemes adapted to moment equations(preserving the realizability property) are developped. Those schemes are shown to becompetitive in terms of computational costs compared to reference approaches. Finallythey are applied to in an optimization procedure aiming to maximize the dose in the tumorand to minimize the dose in healthy tissues.

This thesis consists of five articles mainly devoted to problems in dynamical systems and ergodic theory. We consider non-uniformly hyperbolic two dimensional systems and limit distributions of point measures which are absolutely continuous with respect to the Lebesgue measure. Let $f_{a_0}(x)=a_0-x^2$ be a quadratic map where the parameter $a_0\in(1,2)$ is chosen such that the critical point $0$ is pre-periodic (but not periodic). In Papers A and B we study skew-products $(\th,x)\mapsto F(\th,x)=(g(\th),f_{a_0}(x)+\al s(\th))$, $(\th,x)\in S^1\times\real$. The functions $g:S^1\to S^1$ and $s:S^1\to[-1,1]$ are the base dynamics and the coupling functions, respectively, and $\al$ is a small, positive constant. Such quadratic skew-products are also called Viana maps. In Papers A and B we show for several choices of the base dynamics and the coupling function that the map $F$ has two positive Lyapunov exponents and for some cases we further show that $F$ admits also an absolutely continuous invariant probability measure. In Paper C we consider certain Bernoulli convolutions. By showing that a specific transversality property is satisfied, we deduce absolute continuity of the to these Bernoulli convolutions associated distributions. In Papers D and E we consider sequences of real numbers in the unit interval and study how they are distributed. The sequences in Paper D are given by the forward iterations of a point $x\in[0,1]$ under a piecewise expanding map $T_a:[0,1]\to[0,1]$ depending on a parameter $a$ contained in an interval $I$. Under the assumption that each $T_a$ admits a unique absolutely continuous invariant probability measure $\mu_a$ and that some technical conditions are satisfied, we show that the distribution of the forward orbit $T_a^j(x)$, $j\ge1$, is described by the distribution $\mu_a$ for Lebesgue almost every parameter $a\in I$. In Paper E we apply the ideas in Paper D to certain sequences which are equidistributed in the unit interval and give a geometrical proof of an old result by Koksma.

<p>This thesis deals with the Open Mapping Theorem for analytic functions on domains in the complex plane: A non-constant analytic function on an open subset of the complex plane is an open map.</p><p>As applications of this fundamental theorem we study Schwarz’s Lemma and its consequences concerning the groups of conformal automorphisms of the unit disk and of the upper halfplane.</p><p>In the last part of the thesis we indicate the first steps in hyperbolic geometry.</p> / <p>Denna uppsats behandlar satsen om öppna avbildningar för analytiska funktioner på domäner i det komplexa talplanet: En icke-konstant analytisk funktion på en öppen delmängd av det komplexa talplanet är en öppen avbildning.</p><p>Som tillämpningar på denna fundamentala sats studeras Schwarz’s lemma och dess konsekvenser för grupperna av konforma automorfismer på enhetsdisken och på det övre halvplanet.</p><p>I uppsatsens sista del antyds de första stegen inom hyperbolisk geometri.</p>

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This PhD thesis consists of a summary and five papers which all deal with asymptotic problems on certain homogeneous spaces. In Paper I we prove asymptotic equidistribution results for pieces of large closed horospheres in cofinite hyperbolic manifolds of arbitrary dimension. All our results are given with precise estimates on the rates of convergence to equidistribution. Papers II and III are concerned with statistical problems on the space of n-dimensional lattices of covolume one. In Paper II we study the distribution of lengths of non-zero lattice vectors in a random lattice of large dimension. We prove that these lengths, when properly normalized, determine a stochastic process that, as the dimension n tends to infinity, converges weakly to a Poisson process on the positive real line with intensity 1/2. In Paper III we complement this result by proving that the asymptotic distribution of the angles between the shortest non-zero vectors in a random lattice is that of a family of independent Gaussians. In Papers IV and V we investigate the value distribution of the Epstein zeta function along the real axis. In Paper IV we determine the asymptotic value distribution and moments of the Epstein zeta function to the right of the critical strip as the dimension of the underlying space of lattices tends to infinity. In Paper V we determine the asymptotic value distribution of the Epstein zeta function also in the critical strip. As a special case we deduce a result on the asymptotic value distribution of the height function for flat tori. Furthermore, applying our results we discuss a question posed by Sarnak and Strömbergsson as to whether there in large dimensions exist lattices for which the Epstein zeta function has no zeros on the positive real line.

This thesis deals with the Open Mapping Theorem for analytic functions on domains in the complex plane: A non-constant analytic function on an open subset of the complex plane is an open map. As applications of this fundamental theorem we study Schwarz’s Lemma and its consequences concerning the groups of conformal automorphisms of the unit disk and of the upper halfplane. In the last part of the thesis we indicate the first steps in hyperbolic geometry. / Denna uppsats behandlar satsen om öppna avbildningar för analytiska funktioner på domäner i det komplexa talplanet: En icke-konstant analytisk funktion på en öppen delmängd av det komplexa talplanet är en öppen avbildning. Som tillämpningar på denna fundamentala sats studeras Schwarz’s lemma och dess konsekvenser för grupperna av konforma automorfismer på enhetsdisken och på det övre halvplanet. I uppsatsens sista del antyds de första stegen inom hyperbolisk geometri.

In dieser Arbeit werden zwei Klassen von Evolutionsgleichungen in einem Matrixraum-Setting studiert: Hyperbolische Erhaltungsgleichungen und aktive skalare Gleichungen. Für erstere wird untersucht, wann man Oszillationen mit Hilfe polykonvexen Maßen ausschließen kann; für Zweitere wird mit Hilfe von Oszillationen gezeigt, dass es unendlich viele periodische schwache Lösungen gibt.

One of the major problem faced by banks is how to manage the risk exposure in large portfolios. According to Basel II regulation banks has to measure the risk using Value-at-Risk with confidence level 99%. However, this regulation does not specify the way to calculate Valueat- Risk. The easiest way to calculate Value-at-Risk is to assume that portfolio returns are normally distributed. Altough, this is the most common way to calculate Value-at-Risk, there exists also other methods. The previous crisis shows that the regular methods are unfortunately not always enough to prevent bankruptcy. This paper is devoted to compare the classical methods of estimating risk with other methods such as Cornish-Fisher Expansion (CFVaR) and assuming generalized hyperbolic distribution. To be able to do this study, we estimate the risk in a large portfolio consisting of ten stocks. These stocks are chosen from the NASDAQ 100-list in order to have highly liquid stocks (bluechips). The stocks are chosen from different sectors to make the portfolio welldiversified. To investigate the impact of dependence between the stocks in the portfolio we remove the two most correlated stocks and consider the resulting eight stock portfolio as well. In both portfolios we put equal weight to the included stocks. The results show that for a well-diversified large portfolio none of the risk measures are violated. However, for a portfolio consisting of only one highly volatile stock we prove that we have a violation in the classical methods but not when we use the modern methods mentioned above.