Le but de cette thèse est de contribuer à la compréhension des équations de Monge-Ampère complexes paraboliques sur des domaines de Cn. Cette équation a un lien étroit avec le flot de Kähler-Ricci. Notre étude se concentre sur les cas où la condition initiale n'est pas régulière. Nous voulons démontrer l'existence de solutions satisfaisant la continuité jusqu'à la frontière et jusqu'au temps initial. / The aim of this thesis is to make a contribution to understanding parabolic complex Monge-Ampère equations on domains of Cn. Our study is centered around cases where the initial condition is irregular. We want to prove the existence of solutions which satisfies continuity up to the boundary and continuity up to the initial time.

The topic of this thesis concerns computer software for 3D modelling SolidWorks and its use for descriptive geometry problems creation. Descriptions of techniques of work in SolidWorks 3D parametric modeller are a part of this thesis, as well as the work with the software itself and description of problems being solved by it. Thesis also includes options of possible use of the final interactive gadget at basic and elementary schools. Moreover, in conclusion there is described to what extend it is appropriate to use SolidWorks software for descriptive geometry problem solving and when some problems might occur.

L'objet de cette thèse est d'étudier la théorie du transport optimal sur un espace de Wiener abstrait. Les résultats qui se trouvent dans quatre principales parties, portent :Sur la convexité de l'entropie relative. On prolongera des résultats connus en dimension finie, sur l'espace de Wiener muni d'une norme uniforme, à savoir que l'entropie relative est (au moins faiblement) 1-convexe le long des géodésiques induites par un transport optimal sur l'espace de Wiener.Sur les mesures à densité logarithmiquement concaves. Le premier des résultats importants consiste à montrer qu'une inégalité de type Harnack est vraie pour le semi-groupe induit par une telle mesure sur l'espace de Wiener. Le second des résultats obtenus nous fournit une inégalité en dimension finie (mais indépendante de la dimension), contrôlant la différence de deux applications de transport optimal.Sur le problème de Monge. On s'intéressera au problème de Monge sur l'espace de Wiener, muni de plusieurs normes : des normes à valeurs finies, ou encore la pseudo-norme de Cameron-Martin.Sur l'équation de Monge-Ampère. Grâce aux inégalités obtenues précédemment, nous serons en mesure de construire des solutions fortes de l'équation de Monge-Ampère (induite par le coût quadratique) sur l'espace de Wiener, sous de faibles hypothèses sur les densités des mesures considérées / The aim of this PhD is to study the optimal transportation theory in some abstract Wiener space. You can find the results in four main parts and they are aboutThe convexity of the relative entropy. We will extend the well known results in finite dimension to the Wiener space, endowed with the uniform norm. To be precise the relative entropy is (at least weakly) geodesically 1-convex in the sense of the optimal transportation in the Wiener space.The measures with logarithmic concave density. The first important result consists in showing that the Harnack inequality holds for the semi-group induced by such a measure in the Wiener space. The second one provides us a finite dimensional and dimension-free inequality which gives estimate on the difference between two optimal maps.The Monge Problem. We will be interested in the Monge Problem on the Wiener endowed with different norms: either some finite valued norms or the pseudo-norm of Cameron-Martin.The Monge-Ampère equation. Thanks to the inequalities obtained above, we will be able to build strong solutions of the Monge-Ampère (those which are induced by the quadratic cost) equation on the Wiener space, provided the considered measures satisfy weak conditions

In this paper we use the Sobolev steepest descent method introduced by John W. Neuberger to solve the hyperbolic Monge-Ampère equation. First, we use the discrete Sobolev steepest descent method to find numerical solutions; we use several initial guesses, and explore the effect of some imposed boundary conditions on the solutions. Next, we prove convergence of the continuous Sobolev steepest descent to show local existence of solutions to the hyperbolic Monge-Ampère equation. Finally, we prove some results on the Sobolev gradients that mainly arise from general nonlinear differential equations.

The audiological characteristics of the Monge family of Costa Rica were investigated in a sample of fifty-two affected members and twelve unaffected members. Through laboratory analysis by staff personnel from the University of Costa Rica and audiological test results obtained in the present investigation, it was concluded that affected Monge members demonstrate a slowly progressive low frequency sensorineural hearing loss of autosomal dominant transmission. The initial site of lesion appears to be the apical portion of the cochlea, with significant onset occurring during early childhood following normal speech and language acquisition. The rate at which the hearing loss progresses and the frequency regions affected are contingent upon chronological age, culminating in a flat profound hearing impairment by age thirty for all affected members.

In this dissertation we study The Refractor Problem and its analytic formulation which leads to Monge-Ampere type equation. This problem can be described as follows: Suppose that A and B are two domains of the unit sphere in n dimensions and g and f are two positive functions integrable on A and B respectively. Consider two homogeneous, isotropic media; medium I and medium II, which have different optical densities and assume that from a point O inside medium I, light emanates with intensity g(x); where x is in A. When an incident ray of light hits an interface between two media with different indices of refraction, it splits into two rays a reflected ray that propagates back into medium I and a refracted ray that proceeds into medium II. Consequently, the incident ray loses some of its energy as it proceeds into medium II. By using Fresnel equations, which are consequences of Maxwell's Equations, one can determine precisely how much of the energy is lost due to internal reflection. The problem is to take into account this loss and construct a surface such that all rays emitted from a point O in the first medium, with directions in A are refracted by the surface into media II with directions in B and the prescribed illumination intensity received in the direction m, where m is in B is f(m). We propose a model to this problem. We introduce weak solutions for the problem and prove their existence by using approximation by ellipsoids or hyperboloids depending on whether n1 is less than n2 or n1 is greater than n2. We will also prove that a solution of the problem satisfies a Monge-Ampere type of PDE. / Mathematics

We introduce two models to design near field reflectors in R^3 that solve an inverse problem in radiometry, taking into account the inverse square law of irradiance. The problem leads to a Monge-Ampere type inequality. The surfaces in the first model are strictly convex and require to be far from the source to avoid obstruction. In the second model, the reflectors are neither convex nor concave and do not block the rays even if they are close to the source. / Mathematics

Submitted by Celso Magalhaes (celsomagalhaes@ufrrj.br) on 2017-09-11T11:55:12Z No. of bitstreams: 1 2014 - Nelson Rangel Coutinho Neto.pdf: 1017081 bytes, checksum: 48847c819281a97d7502570c6852b27b (MD5) / Made available in DSpace on 2017-09-11T11:55:12Z (GMT). No. of bitstreams: 1 2014 - Nelson Rangel Coutinho Neto.pdf: 1017081 bytes, checksum: 48847c819281a97d7502570c6852b27b (MD5) Previous issue date: 2014-08-30 / Coordena??o de Aperfei?oamento de Pessoal de N?vel Superior - CAPES / The Descriptive Geometry had its origin during the Napoleonic Age through the mathematician Gaspard Monge. It arrived in Brazil in the early nineteenth century by imposition of D. Jo?o VI. It excelled mainly as part of the curriculum of higher education courses such as Engineering, Architecture and Arts discipline. In high school, teaching Descriptive Geometry is almost extinct, being restricted to only a few more traditional institutions. However, Descriptive Geometry is closely related to the Space Geometry, Analytical Geometry and Differential and Integral Calculus, so that its application in middle school will be of fundamental contribution to the learning of other geometric concepts / A Geometria Descritiva teve sua origem durante a Era Napole?nica atrav?s do matem?tico Gaspard Monge. Chegou ao Brasil no in?cio do s?culo XIX por imposi??o de D. Jo?o VI. Destacou-se basicamente como disciplina integrante de curr?culos de cursos superiores, tais como Engenharia, Arquitetura e Artes. No ensino m?dio, o ensino da Geometria Descritiva est? praticamente extinto, ficando restrito apenas a algumas institui??es mais tradicionais. Entretanto, a Geometria Descritiva se relaciona intimamente com a Geometria Espacial, a Geometria Anal?tica e os C?lculos Diferencial e Integral, de maneira que sua aplica??o no ensino m?dio poder? contribuir para o aprendizado de outros conceitos geom?tricos

Tout p-cycle tropical VT de Rn, on attache naturellement un courant fermé (p, p) dimensionnel d'ordre 0 sur (C)n, noté Tp n(VT). Un tel "courant tropical" T p n(VT) ne saurait etre le courant d'intégration sur un quelconque sous-ensemble analytique de (C)n du fait qu'il a pour support l'ensemble log-1(VT) (C)n, où l'application Log désigne la multivluation (Z1, ..., Zn) 7! (logIZ1I, ..., logIZnI). On donne des conditions suffisantes (de nature locale) sur un p-cycle tropical VT pour le courant tropical T p n(VT) qui lui est associé soit" fortement extrémal" dans D0p, p((C)n). En particulier, si une telle condition s'avère remplie pour un p-cycle tropical effectif, alors le courant tropical qui lui est attaché est extrémal dans le cône des courants fermés de bidimension (p, p) sur (C)n. On explique ensuite comment prolonger ces courants tropicaux et les propirétés d'extrémilité dont ils héritent à l'espace projectif CPn. On montre également comment définir le produit de tels courants tropicaux pour en déduire une théorie de l'intersection entre cycles tropicaux. pour opérer ces calculs, on établit une formule pour la mesure de Monge Ampère réelle associée à un polynôme tropical. De plus, comme un tel courant tropical attaché à un p-cycle tropical VT s'obtient en moyennisant des courants d'intégration sur des variétés toriques, on met en correspondance théorie de l'intersection dans le cadre torique et théorie de l'intersection dans le cadre tropical. On explicite enfin certains liens entre les problèmes relevant de l'approximation (an sens ensembliste, pour la métrique de Hausdorff) des cycles tropicaus de Rn par les amibes de cycles algébriques de (C)n et l'approximation (ans sens faible) des courants tropicaux associées par des multiples positifs de courants d'intégration sur de tels cycles algébriques. on explique en quoi ces questions d'approximation se trouvent reliées à une formulation forte de la célèbre confecture de Hodge. / To a tropical p-cycle VT in Rn, we naturally assoicate a closed (p, p)-dimensional current of order zero on (C)n denoted bu T p n(VT). Such e "tropical current" T p n(VT) cannot be an integration current along any analytic set since its support has the form log -1(VT) (C)n, where log is the coordinate-wise valuation with log(I.I). We provide sufficient (local) conditions on a tropical p-cycle such that its associated tropical is "strongly extremal" in Dop, p((C)n). In particular, if these conditions hokd for the effective cycles, then the associated current are extremal in the cone of strongly positive closed currents of bidimension (p, p) on (C)n. Nexte we explain how to extend the currents and extremality results to CPn. Further, we demonstrate how to use the intersection theory of currents to derive an intersection theory for the inderlying tropical cycles. The explicit calculations will be established by using e formula for the real Monge-Ampère measure of a tropical polynomial. Moreoer, since such tropical currents are obtained by an averaging of integration currents on toric sets, an equality between toric intersection multipmicities and the tropical multiplicities is readily settled. Finally, we explain certain relations between approximation problems of tropical cycles by amoebas of algebraic cycles and approximations of the associated currents by positive multiples of integration currents along analytic cycles. Il will be discussed haw these approximtion problems are related to a stronger formulation of the celebrated hodge conjecture.

Cette thèse est consacrée à l’étude du problème de transport de Monge pour le coût quadratique en géométrie sous-Riemannienne et des conditions essentielles à l’obtention des résultats d’existence et et d’unicité de solutions. Ces travaux consistent à étendre ces résultats au cas des structures sous-Riemanniennes admettant des géodésiques minimisantes singulières. Dans une première partie, on développe des techniques inspirées de travaux de Cavalletti et Huesmann pour d’obtenir des résultats significatifs pour des structures de rang 2 en dimension 4. Dans une deuxième partie, on étudie des outils analytiques de la h-semiconcavité de la distance sousriemannienne et on montre comment ce type de régularité peut aboutit à l’obtention d’existence et d’unicité de solutions dans un cas général. / This thesis is devoted to the study of the Monge transport problem for the quadratic cost in sub-Riemannian geometry and the essential conditions to obtain existence and uniqueness of solutions. These works consist in extending these results to the case of sub-Riemannian structures admitting singular minimizing geodesics. In a first part, we develop techniques inspired by works by Cavalletti and Huesmann in order to obtain significant results for structures of rank 2 in dimension 4. In a second part, we study analytical tools of the h-semiconcavity of the sub-Riemannian distance and we show how this type of regularity can lead to the well-posedness of the Monge problem in general cases.