This thesis is broadly divided in two parts. In the first part we give a survey of various distances between metric spaces, namely the uniform distance, Lipschitz distance, Hausdor distance and the Gramoz Hausdor distance. Here we talk about only the most basic of their properties and give a few illustrative examples. As we wish to study collections of metric measure spaces, which are triples (X; d; m) consisting of a complete separable metric space (X; d) and a Boral probability measure m on X, there are discussions about some distances between them. Among the three that we discuss, the transportation and distortion distances were introduced by Sturm. The later, denoted by 2, on the space X2 of all metric measure spaces having finite L2-size is the focus of the second part of this thesis. The second part is an exposition based on the work done by Sturm. Here we prove a number of results on the analytic and geometric properties of (X2; 2). Beginning by noting that (X2; 2) is a non-complete space, we try to understand its completion. Towards this end, the notion of a gauged measure space is useful. These are triples (X; f; m) where X is a Polish space, m a Boral probability measure on X and f a function, also called a gauge, on X X that is symmetric and square integral with respect to the product measure m2. We show that, Theorem 1. The completion of (X2; 2) consists of all gauged measure spaces where the gauges satisfy triangle inequality almost everywhere. We denote the space of all gauged measure spaces by Y. The space X2 can be embedded in Y and the transportation distance 2 extends easily from X2 to Y. These two spaces turn out to have similar geometric properties. On both these spaces 2 is a strictly intrinsic metric; i.e. any two members in them can be joined by a shortest path. But more importantly, using a description of the geodesics in these spaces, the following result is proved. Theorem 2. Both (X2; 2) and (Y; 2) have non-negative curvature in the sense of Alexandrov.

A class of parametric optimal control problems for semilinear parabolic equations is considered. Using recent regularity results for solutions of such equations, sufficient conditions are derived under which the solutions to optimal control problems are locally Lipschitz continuous functions of the parameter in the L1-norm. It is shown that these conditions are also necessary, provided that the dependence of data on the parameter is sufficiently strong.

Consider a domain Ω ⊂ RN with Lipschitz boundary and let d(x) = dist(x, ∂Ω). It is well known for p ∈ (1, ∞) that u ∈ W1,p 0 (Ω) if and only if u/d ∈ Lp (Ω) and ∇u ∈ Lp (Ω). Recently a new characterization appeared: it was proved that u ∈ W1,p 0 (Ω) if and only if u/d ∈ L1 (Ω) and ∇u ∈ Lp (Ω). In the author's bachelor thesis the condition u/d ∈ L1 (Ω) was weakened to the condition u/d ∈ L1,p (Ω), but only in the case N = 1. In this master thesis we prove that for N ≥ 1, p ∈ (1, ∞) and q ∈ [1, ∞) we have u ∈ W1,p 0 (Ω) if and only if u/d ∈ L1,q (Ω) and ∇u ∈ Lp (Ω). Moreover, we present a counterexample to this equivalence in the case q = ∞. 1

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第19468号 / 理博第4128号 / 新制||理||1594(附属図書館) / 32504 / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)准教授 矢野 孝次, 教授 上田 哲生, 教授 重川 一郎 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

In this thesis we address the multi-armed bandit (MAB) problem with stochastic rewards and correlated arms. Particularly, we investigate the case when the expected rewards are a Lipschitz function of the arm, and the learning to rank problem, as viewed from a MAB perspective. For the former, we derive a problem specific lower bound and propose both an asymptotically optimal algorithm (OSLB) and a (pareto)optimal, algorithm (POSLB). For the latter, we construct the regret lower bound and determine its closed form for some particular settings, as well as propose two asymptotically optimal algorithms PIE and PIE-C. For all algorithms mentioned above, we present performance analysis in the form of theoretical regret guarantees as well as numerical evaluation on artificial datasets as well as real-world datasets, in the case of PIE and PIE-C. / <p>QC 20160223</p>

In this thesis we consider an open question of Feige that asks whether there always exists a constantly Lipschitz bijection of an n2 -point subset of Z2 onto a regular grid [n] × [n] for every n ∈ N. We relate this question to an already resolved problem of the existence of a bounded positive measurable density in R2 that is not the Jacobian of any bilipschitz map. This problem was resolved by Burago and Kleiner [1], and independently, by McMullen [12]. We present the work of Burago and Kleiner, analyze its relation to Feige's problem and sug- gest a continuous formulation of Feige's question in a special case. Then we present the Burago-Kleiner density, make several observation about the properties of this density, and after that we construct a density that is everywhere nonrealizable as the Jacobian of a bilipschitz map. Subsequently, we discuss our continuous variant of Feige's question, provide several observation concerning it, and finally, we try to use the everywhere nonrealizable density constructed before to answer our continuous variant of Feige's question. However, this last task still remains incomplete. 1

<p>Using the finite element software FEMLAB® solutions are computed to Dirichlet problems for the Infinity-Laplace equation ∆∞(<i>u</i>) ≡ <i>u</i>²_x<i>u</i>_xx+ 2<i>u</i>_x<i>u</i>_y<i>u</i>_xy+<i>u</i>²_y<i>u</i>_yy= 0. For numerical reasons ∆<i>q</i>(<i>u</i>) = div (|▼<i>u</i>|<i>q</i>▼<i>u</i>)<i> = </i>0, which (formally) approaches as ∆∞(<i>u</i>) = 0 as <i>q</i> → ∞, is used in computation. A parametric nonlinear solver is used, which employs a variant of the damped Newton-Gauss method. The analysis of the experiments is done using the known theory of solutions to Dirichlet problems for ∆∞(<i>u</i>) = 0, which includes AMLEs (Absolutely Minimizing Lipschitz Extensions), sets of uniqueness, critical segments and lines of singularity. From the experiments one main conjecture is formulated: For Dirichlet problems, which have a non-constant boundary function, it is possible to predict the structure of the lines of singularity in solutions in the Infinity-Laplace case by examining the corresponding Laplace case.</p>

Dans cette thèse, nous explorons premièrement la notion de directionnalité lors de la conception de repères d'ondelettes du plan. Cette propriété, qui semble essentielle pour la vision biologique, donne lieu à une meilleure représentation des contours d'objets dans les décompositions d'images utilisant ces repères. Elle génère en outre une redondance supplémentaire qui exploitée à bon escient, permet par exemple de réduire les effets d'un bruit additif (gaussien). Nous montrons également comment cette directionnalité, généralement perçue comme un paramètre figé, peut être adaptée localement aux éléments d'une image. Nous définissons ainsi le concept d'analyse d'images multisélective. Dans ce cadre, des règles de récurrence héritées d'une analyse multirésolution circulaire associent des ondelettes d'une certaine sélectivité angulaire pour générer des ondelettes de plus faible directionnalité jusqu'à l'obtention d'une ondelette totalement isotrope. Dans le cas d'un repère d'ondelettes linéaire, ces différents niveaux de sélectivité ont la possibilité de s'ajuster localement au contenu d'une image. Nous constatons par ailleurs que cette adaptabilité fournit de meilleures reconstructions que les méthodes à sélectivité fixe lors d'approximations non linéaires d'images. Cette thèse traite ensuite du problème de l'analyse de données représentées sur la sphère. Il a été établi précédemment [AV99] que la transformée continue en ondelettes (CWT) s'étend à cet espace par l'emploi d'une dilatation stéréographique respectant la compacité de S2. Dans certain cas, il est utile de réduire la redondance de cette transformée, ne fut ce que pour faciliter le traitement des données dans l'espace multi-échelle généré. Nous étudions par conséquent comment créer des repères sphériques semi-continus, où seule l'échelle est échantillonnée, et totalement discrétisés. Nous tirons parti dans ce dernier cas de grilles sphériques équi-angulaires et des règles de quadrature associées pour obtenir des conditions suffisantes à la reconstruction des fonctions analysées. Les capacités d'analyse et de synthèse de repères d'ondelettes DOG sont également testées sur des exemples de données sphériques. Une dernière partie de ce document est dédiée à l'étude d'un objet physique étonnant : la couronne solaire. Cette couche extérieure du soleil est observée depuis 1996 par l'expérience EIT à bord du satellite SoHO dans différentes longueurs d'onde de l'ultraviolet lointain. La compréhension physique des multiples phénomènes apparaissant dans la couronne solaire passe par le traitement automatique des images EIT. Dans cette tâche, nous nous limitons à deux problèmes particuliers. Nous utilisons premièrement la CWT et sa capacité à analyser la régularité locale d'une image pour gommer les traces laissées par les rayons cosmiques, majoritairement non solaires, sur les enregistrements EIT. Deuxièmement, la couronne solaire contient des éléments de faible taille (<60arcsec) nommés points brillants (ou BPs pur Bright points). Ceux-ci trouvent leur origine dans l'échauffement local du plasma coronal sous l'action du champs magnétique solaire. En abordant une approche similaire à celle développée en [Bij99], nous étudions comment sélectionner et caractériser ces BPS en décomposant une image en ses objets constitutifs. Ces derniers sont issus de tubes de maxima dans la description multi-échelle de l'image, c'est-à-dire d'une généralisation discrète des lignes de maxima de la CWT.

Using the finite element software FEMLAB® solutions are computed to Dirichlet problems for the Infinity-Laplace equation ∆∞(u) ≡ u2xuxx + 2uxuyuxy + u2yuyy = 0. For numerical reasons ∆q(u) = div (|▼u|q▼u) = 0, which (formally) approaches as ∆∞(u) = 0 as q → ∞, is used in computation. A parametric nonlinear solver is used, which employs a variant of the damped Newton-Gauss method. The analysis of the experiments is done using the known theory of solutions to Dirichlet problems for ∆∞(u) = 0, which includes AMLEs (Absolutely Minimizing Lipschitz Extensions), sets of uniqueness, critical segments and lines of singularity. From the experiments one main conjecture is formulated: For Dirichlet problems, which have a non-constant boundary function, it is possible to predict the structure of the lines of singularity in solutions in the Infinity-Laplace case by examining the corresponding Laplace case.

In this thesis we prove several results on the structure of solutions to optimal transportation problems. The second chapter represents joint work with Robert McCann and Micah Warren; the main result is that, under a non-degeneracy condition on the cost function, the optimal is concentrated on a $n$-dimensional Lipschitz submanifold of the product space. As a consequence, we provide a simple, new proof that the optimal map satisfies a Jacobian equation almost everywhere. In the third chapter, we prove an analogous result for the multi-marginal optimal transportation problem; in this context, the dimension of the support of the solution depends on the signatures of a $2^{m-1}$ vertex convex polytope of semi-Riemannian metrics on the product space, induce by the cost function. In the fourth chapter, we identify sufficient conditions under which the solution to the multi-marginal problem is concentrated on the graph of a function over one of the marginals. In the fifth chapter, we investigate the regularity of the optimal map when the dimensions of the two spaces fail to coincide. We prove that a regularity theory can be developed only for very special cost functions, in which case a quotient construction can be used to reduce the problem to an optimal transport problem between spaces of equal dimension. The final chapter applies the results of chapter 5 to the principal-agent problem in mathematical economics when the space of types and the space of available goods differ. When the dimension of the space of types exceeds the dimension of the space of goods, we show if the problem can be formulated as a maximization over a convex set, a quotient procedure can reduce the problem to one where the two dimensions coincide. Analogous conditions are investigated when the dimension of the space of goods exceeds that of the space of types.