nÃo hà / O ordenamento jurÃdico brasileiro, à partir de 1998, permitiu que os entes federativos estados e municÃpios pudessem migrar do Regime Geral de PrevidÃncia (RGPS), capitaneado pelo INSS, para um Regime PrÃprio de PrevidÃncia Social (RPPS) tomando o cuidado para os aportes financeiros necessÃrios para a harmonia do equilÃbrio atuarial de modo a garantir, no futuro, o pagamento dos benefÃcios a quem de direito (massa laboral incluÃda nos regimes previdenciÃrios), de sorte que dos 5.509 municÃpios brasileiros, 1.957 instituÃram os seus RPPS e, 55 municÃpios no Estado do CearÃ, dos 184 existentes. O objetivo principal desta pesquisa foi apurar o resultado atuarial dos RPPS instituÃdos nos municÃpios cearenses por meio de um software construÃdo pelo autor e confrontÃ-lo com o resultado atuarial contido nos Demonstrativos de Resultado das AvaliaÃÃes Atuariais â DRRAÂs. De acordo com os cÃlculos realizados os RPPS municipais cearenses apresentam dÃficit atuarial no montante de R$ 3.361.632.976,77, enquanto o valor do resultado atuarial demonstrados nos DRAAÂs apresentam dÃficit atuarial no valor de R$ 10.344.705.187,76. Em ambas as apuraÃÃes o resultado deficitÃrio tem concentraÃÃo nos municÃpios de Fortaleza, CanindÃ, MaracanaÃ, Juazeiro do Norte e Itapipoca. Na 1. e 2. apuraÃÃo os municÃpios de Amontada e Caucaia se mostraram superavitÃrios, acrescentando-se, tambÃm, que os municÃpios de Cruz e Fortim se revelaram superavitÃrios no cÃlculo realizado pelo autor. O dÃficit atuarial do MunicÃpio de Itapipoca apresentou-se preciso, tanto no cÃlculo formulado pelo autor, quanto no demonstrado no DRRA do ente. Conclui-se que os entes previdenciÃrios deficitÃrios nÃo oferecem sistema de estrutura para o acÃmulo de recursos para o pagamento de compromissos definidos nos planos de benefÃcios. / The Brazilian legal system , starting from 1998 revealed that the federative states and municipalities could migrate from the General Provident Fund Scheme ( RGPS ) , headed by the INSS , for its Own Social Security System ( RPPS ) taking care to financial contributions necessary for the harmony of actuarial balance to ensure in the future payment of benefits to those eligible ( work force included in pension schemes ) , so that the 5,509 Brazilian municipalities, 1,957 have instituted their RPPS and 55 towns in State of Cearà , the existing 184 . The main objective of this research was to determine the actuarial results of RPPS established in municipalities of Cearà through a software built by the author and confront him with the actuarial results contained in the statements of income of the Actuarial Reviews - DRRA 's. According to the calculations the Cearà municipal RPPS present actuarial deficit in the amount of R $ 3,361,632,976.77 , while the value of actuarial results demonstrated in the present DRAA 's actuarial deficit of R $ 10,344,705,187.76 . In both calculations the deficit result has concentration in the cities of Fortaleza, CanindÃ, MaracanaÃ, Juazeiro and Itapipoca. In 1. & 2. calculating the municipalities of Amontada and Caucaia proved surplus, also adding to the towns of Cruz and Fortim, the surplus calculation performed by the author. The actuarial deficit of the municipality of Itapipoca presented itself takes both the calculation made by the author, as in shown in the DRRA one. It is concluded that the pension deficit loved not offer structural system for the accumulation of resources for the payment of obligations defined benefit plans. There is evidence that there is no consistency in the figures to the MPAS during transport of the DRAA 's MPAS is recommending the external control bodies investigate that the reason pointed out the differences.

This work was motivated by the famous Plateau\'s Problem which concerns the existence of a minimizing set of the area functional with prescribed boundary. In order to solve the Plateau\'s Problem, we make use of different theories: the theory of varifolds, currents and locally finite perimeter sets (Caccioppoli sets). Working on the Caccioppoli sets theory, it is straightforward to prove the existence of a minimizing set in some classical problems as the isoperimetric and Plateau\'s problems. If we switch the problem to find the regularity that we can extract of some minimizing set, we come across complicated ideas and tools. Although, the Plateau\'s Problem and other classical problems are well settled. Because of that, we have extensively studied the almost minimizing condition ((; r)-minimizing sets) considered by Maggi ([?]) which subsumes some classical problems. We focused on the regularity theory extracted from this almost minimizing condition. / This work was motivated by the famous Plateau\'s Problem which concerns the existence of a minimizing set of the area functional with prescribed boundary. In order to solve the Plateau\'s Problem, we make use of different theories: the theory of varifolds, currents and locally finite perimeter sets (Caccioppoli sets). Working on the Caccioppoli sets theory, it is straightforward to prove the existence of a minimizing set in some classical problems as the isoperimetric and Plateau\'s problems. If we switch the problem to find the regularity that we can extract of some minimizing set, we come across complicated ideas and tools. Although, the Plateau\'s Problem and other classical problems are well settled. Because of that, we have extensively studied the almost minimizing condition ((; r)-minimizing sets) considered by Maggi ([?]) which subsumes some classical problems. We focused on the regularity theory extracted from this almost minimizing condition.

本研究中討論了非線性微分方程式之解的正則性。在這之中發現了一些有趣的現象，得到了方程式解可以做任意次的微分，並且得到對該解任意次微分後其值趨近到無限大時之爆破速率、爆破常數及當其值遞減至零時的爆破速率、爆破常數。 / In this paper we work with the regularity of solutions for the non-linear ordinary differential equation u''-u^p=0 for some well-defined functions u^p. We have found some interesting phenomena, u belongs to C^q for any q in positive integer, blow-up constant, blow-up rate, null point and decay rate of u^(n) are obtained in this work, through that we get the characterization for these equations in this case.

貨幣需求在總體經濟學中一直扮演著重要的角色。同時，貨幣與貨幣性資 產間替代之研究亦是一般貨幣經濟學者關心的課題之一。因為瞭解了這層 關係後，對於如何定義貨幣、貨幣總計數之衡量及貨幣政策的制定等，將 會有所助益。傳統的貨幣需求分析，實質貨幣餘額需求為實質所得( 產 出 )，預期通貨膨脹率與名目利率的函數。但實證結果顯示，使用這些變 數對於貨幣需求的預測或是制定、評估貨幣政策時，並不十分有用。近來 ，許多學者嘗試以符合個體基礎的方式來估計貨幣需求。然而，大部份的 實證結果亦是令人沮喪。本論文將回顧估計貨幣需求的一般化個體─經濟 計量方法，並嘗試使用較新的個體─經濟計量模型─漸近理想化模型( The Asymptotically Id eal Model)來估計台灣地區的貨幣需求。同時也 討論下列問題：ぇ貨幣性資產間的替代性╱互補性。えAIM 與Translog貨 幣需求系統之比較。ぉ效用最大化條件之比較。お一階AIM貨幣需求系統 之動態分析。實證研究的主要結果如下：LTL、HTL貨幣需求系統，不但違 反滿足效用函數彎曲性的必要條件且與需求法則相違背。由一階 AIM貨幣 需求系統估計之彈性值發現，活期儲蓄存款加郵局存簿儲金與活期存款、 郵局劃撥儲金呈現淨互補的關係，印證了交易性存款與儲蓄性存款彼此替 代性不高的現象。就滿足效用最大化條件而言，一階AIM貨幣需求系統滿 足Regular ity條件的情況仍優於LTL、HTL貨幣需求系統。就一階AIM貨幣 性資產成長率之模擬而言，通貨淨額加支票存款之實際與模擬成長率配適 的最佳，其餘次之。

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.

The first part of this thesis is devoted to symmetrizations. Symmetrizations are tranformations of functions that preserve many properties of functions and enhance their symmetry. In the calculus of variation they are a simple and powerful tool to prove that minimizers of functionals are symmetric functions. In this work, the approximation of symmetrizations by simpler symmetrizations is investigated: The existence of a universal approximating sequence is proved, sufficient conditions for deterministic and random sequences to be approximating are given. These approximation methods are then used to prove some symmetry properties of critical points obtained by minimax methods: For example if there is a solution obtained by the mountain pass theorem, then there is a symmetric solution with the same energy. This part ends with a study of the properties of anisotropic symmetrizations i.e. symmetrizations performed with respect to noneuclidean norms. The second part is devoted to L^1 estimates.  In general, the second derivative of the solution of the Poisson equation with L^1 data fails to be in L^1. Recently it was proved that if the data is a L^1 divergence-free vector-field, then even if in general it is false that the second derivative of the solution is in L^1, all the consequences thereof by Sobolev embeddings hold. Elementary proofs of such results, as well as a generalization with a second order operator replacing the divergence, are given. / La première partie de cette thèse est consacrée aux symétrisations. Les symétrisations sont des transformations de fonctions qui préservent de nombreuses propriétés des fonctions et qui améliorent leur symétrie. Elles sont un outil simple et puissant pour montrer dans le calcul des variations que les minimiseurs de certaines fonctionnelles sont des fonctions symétriques. Dans ce travail, nous étudions l'approximation des symétrisations par des symétrisations plus simples. Nous prouvons l'existence d'une suite approximante universelle et nous donnons des conditions suffisantes pour que des suites déterministes et aléatoires soient approximantes. Nous utilisons ensuite ces méthodes d'approximation pour prouver des propriétés de symétrie de points critiques obtenus par des méthodes de minimax. Par exemple, s'il y a une solution obtenue par le théorème du col, alors il y a une solution symétrique de même énergie. Nous achevons cette partie par une étude des symétrisations anisotropes (symétrisations par rapport à des normes non euclidiennes). La seconde partie est consacrée aux estimations L^1. En général, les dérivées secondes de la solution de l'équation de Poisson avec des données L^1 ne sont pas dans L^1. Recemment, on a prouvé que si les données sont un champ de vecteurs L^1 à divergence nulle, même si en général les dérivées secondes ne sont toujours pas dans L^1, toutes les conséquences qui en suivraient par les injections de Sobolev sont vraies. Nous donnons des preuves élémentaires de ces résultats, avec une extension où la divergence est remplacée par un opérateur différentiel du second ordre.

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.

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.

This thesis consists of six scientific papers, an introduction and a summary. All six papers concern the boundary behavior of non-negative solutions to partial differential equations. Paper I concerns solutions to certain p-Laplace type operators with variable coefficients. Suppose that u is a non-negative solution that vanishes on a part Γ of an Ahlfors regular NTA-domain. We prove among other things that the gradient Du of u has non-tangential limits almost everywhere on the boundary piece Γ, and that log|Du| is a BMO function on the boundary.  Furthermore, for Ahlfors regular NTA-domains that are uniformly (N,δ,r0)-approximable by Lipschitz graph domains we prove a boundary Harnack inequality provided that δ is small enough.  Paper II concerns solutions to a p-Laplace type operator with lower order terms in δ-Reifenberg flat domains. We prove that the ratio of two non-negative solutions vanishing on a part of the boundary is Hölder continuous provided that δ is small enough. Furthermore we solve the Martin boundary problem provided δ is small enough. In Paper III we prove that the boundary type Riesz measure associated to an A-capacitary function in a Reifenberg flat domain with vanishing constant is asymptotically optimal doubling. Paper IV concerns the boundary behavior of solutions to certain parabolic equations of p-Laplace type in Lipschitz cylinders. Among other things, we prove an intrinsic Carleson type estimate for the degenerate case and a weak intrinsic Carleson type estimate in the singular supercritical case. In Paper V we are concerned with equations of p-Laplace type structured on Hörmander vector fields. We prove that the boundary type Riesz measure associated to a non-negative solution that vanishes on a part Γ of an X-NTA-domain, is doubling on Γ. Paper VI concerns a one-phase free boundary problem for linear elliptic equations of non-divergence type. Assume that we know that the positivity set is an NTA-domain and that the free boundary is a graph. Furthermore assume that our solution is monotone in the graph direction and that the coefficients of the equation are constant in the graph direction. We prove that the graph giving the free boundary is Lipschitz continuous.

We investigate stability for certain geometric and functional inequalities and address the regularity of the free boundary for a problem arising in optimal transport theory. More specifically, stability estimates are obtained for the relative isoperimetric inequality inside convex cones and the Gaussian log-Sobolev inequality for a two parameter family of functions. Thereafter, away from a ``small" singular set, local C^{1,\alpha} regularity of the free boundary is achieved in the optimal partial transport problem. Furthermore, a technique is developed and implemented for estimating the Hausdorff dimension of the singular set. We conclude with a corresponding regularity theory on Riemannian manifolds. / text