In this paper, we study Bézier surfaces in View the MathML source three-dimensional Minkowski space. In particular, we focus on timelike and spacelike cases for Bézier surfaces. We also deal with the Plateau¿Bézier problem in View the MathML source, obtaining conditions over the control net to be extremal of the Dirichlet function for both timelike and spacelike Bézier surfaces. Moreover, we provide interesting examples showing the behavior of the Plateau¿Bézier problem in View the MathML source and illustrating the relationship between it and the corresponding Plateau¿Bézier problem in the Euclidean space R3.

En 1975 H. Fujimoto a généralisé les résultats d’unicité pour des fonctions holomorphes dus à Nevanlinna pour des applications méromorphes de Cn dans CPN. Il a démontré que pour deux applications méromorphes non linéairement dégénérées f et g de Cn dans CPN, si elles ont les mêmes images réciproques, comptées avec leurs multiplicités, par rapport à (3N + 2) hyperplans de CPN en position générale, alors f g. Depuis, ce problème a été étudié d’une manière intensive par H. Fujimoto, W. Stoll, L. Smiley, M. Ru, G. Dethloff-T.V.Tan, D.D.Thai-S.D.Quang, Chen-Yan et d’autres auteurs. En parallèle avec le développement de la théorie de Nevanlinna, la théorie de distribution des valeurs de l’application de Gauss des surfaces minimales dans Rm a été étudiée d’une manière intensive par R.Osserman, S.S. Chern, F. Xavier, H. Fujimoto, S.J. Kao, M. Ru et d’autres auteurs. Dans cette thèse, nous avons continué d’étudier ces problèmes. Nous avons obtenu les résultats principaux suivants: +) Théorèmes d’unicité avec multiplicités tronquées des applications méromorphes de Cn dans CPN ayant les mêmes images réciproques par rapport è (2N + 2) hyperplans de CPN. +) Théorèmes d’unicité avec multiplicités tronquées des applications méromorphes de Cn dans CPN ayant des cibles mobiles et un ensemble d’identité petit. +) Théorèmes d’unicité avec multiplicités tronquées des applications méromorphes de Cn dans CPN ayant des cibles fixes ou mobiles et satisfaisant des conditions sur les dérivées. +) Théorèmes de ramification de l’application de Gauss de certaines classes de surfaces minimales complètes dans Rm (m = 3,4). / In 1975, H. Fujimoto generalized Nevanlinna’s known results for meromorphic fonctions to the case of meromorphic mappings of Cn into PN(C). He proved that for two linearly nondegenerate meromorphic mappings f and g of C into PN(C). if they have the saine inverse images counted with multiplicities for 3N + 2 hyperplanes in general position in PN(C) then f = g. After that, this problem has been studied intensively by a number of mathematicans as H. Fujimoto, W. Stoll, L. Smiley, M. Ru, G. Dethloff - T. V. Tan, D. D. Thai - S. D. Quang, Chen-Yan and so on. Parallel to the development of Nevanlinna theory, the value distribution theory of the Gauss map of minimal surfaces immersed in Rm vas studied by many mathematicans as R. Osserman, S.S. Chern, F. Xavier, H. Fujimoto, S. J. Kao, M. Ru and many other mathematicans. In this thesis, we continuous studing some problems on these directions. The main goals of the thesis are followings. • Unicity theorems with truncated multiplicities of meromorphic mappings of Cn into PN(C) sharing 2N + 2 fixed hyperplanes.• Unicity theorems with truncated multiplicities of meromorphic mappings of Cn into PN(C) for moving targets, and a small set of identity.

In 1975, H. Fujimoto generalized Nevanlinna's known results for meromorphic fonctions to the case of meromorphic mappings of Cn into PN(C). He proved that for two linearly nondegenerate meromorphic mappings f and g of C into PN(C). if they have the saine inverse images counted with multiplicities for 3N + 2 hyperplanes in general position in PN(C) then f = g. After that, this problem has been studied intensively by a number of mathematicans as H. Fujimoto, W. Stoll, L. Smiley, M. Ru, G. Dethloff - T. V. Tan, D. D. Thai - S. D. Quang, Chen-Yan and so on. Parallel to the development of Nevanlinna theory, the value distribution theory of the Gauss map of minimal surfaces immersed in Rm vas studied by many mathematicans as R. Osserman, S.S. Chern, F. Xavier, H. Fujimoto, S. J. Kao, M. Ru and many other mathematicans. In this thesis, we continuous studing some problems on these directions. The main goals of the thesis are followings. * Unicity theorems with truncated multiplicities of meromorphic mappings of Cn into PN(C) sharing 2N + 2 fixed hyperplanes.* Unicity theorems with truncated multiplicities of meromorphic mappings of Cn into PN(C) for moving targets, and a small set of identity.

La notion d’entropie d’intrication a eu un profond impact sur la physique théorique, particulièrement depuis ces dix dernières années. D’abord introduite afin expliquer l’entropie des trous noirs, son champ d’application s’est par la suite ouvert à une grande variété de domaines de recherche, de la matière condensée à la gravitation quantique, de l’information quantique à la théorie quantique des champs. Dans ce contexte scientifique effervescent, l’entropie d’intrication apparait comme un outil central et doit donc intensivement être étudiée. A l’origine de cette thèse se trouve le désir de mieux comprendre cette entropie. D’intéressants développements concernant les effets de bord sur l’entropie d’intrication ont vu le jour récemment. Nous proposons donc d’explorer comment le bord d’un espace affecte l’entropie, en particulier dans la situation où la surface d’intrication intersecte ce bord. Nous présentons des calculs explicites de l’entropie d’intrication en espace plat avec bords. Nous montrons que des termes induits par ces bords apparaissent dans l’entropie et nous soulignons le rôle prépondérant que jouent les conditions aux bords. Nous étudions ensuite la contribution de bord dans le terme logarithmique de l’entropie d’intrication en dimensions trois et quatre. Nous calculons en premier lieu ce terme en théorie des champs pour la théorie N = 4 de Yang-Mills, puis nous répétons ce calcul de manière holographique. Nous montrons que ces deux méthodes de calcul donnent le même résultat, si du côté théorie des champs les conditions aux bords préservent la moitié de la supersymétrie et que du côté gravité l’extension du bord dans le bulk est une surface minimale. / The entanglement entropy has had a tremendous and profound impact on theoretical physics, particularly since the last decade. First introduced in an attempt to explain black holes entropy, it has then found applications in a wide range of research areas, from condensed matter physics to quantum gravity, from quantum information to quantum field theory. In this exciting scientific context, the entanglement entropy has thus emerged as a useful and pivotal tool, and as such justifies the need to be intensively studied. At the heart of this thesis therefore lies the desire to better understand the entanglement entropy. Interesting developments during the recent years concern the boundary effects on the entanglement entropy. This dissertation proposes to explore the question of how the presence of spacetime boundaries affects the entropy, specifically in situations where the entangling surface intersects these boundaries. We present explicit calculations of entanglement entropy in flat spacetime with plane boundaries. We show that boundary induced terms appear in the entropy and we emphasize the prominent role of the boundary conditions. We then study the boundary contribution to the logarithmic term in the entanglement entropy in three and four dimensions. We perform the field theoretic computation of this boundary term for the free N = 4 super-gauge multiplet and then repeat the same calculation holographically. We show that these two calculations are in agreement provided that on the field theory side one chooses the boundary conditions which preserve half of the full supersymmetry and that on the gravity side the extension of the boundary in the bulk is minimal.

In the work we modify the well-known minimal surface problem to a very special form, where the exponent two is replaced by a general positive parameter. To the modified problem we define four notions of solution in nonreflexive Sobolev space and in the space of functions of bounded variation. We examine the relationships between these notions to show that some of them are equivalent and some are weaker. After that we look for assumptions needed to prove the existence of solution to the problem in the sense of definitions provided. We outline that in the setting of spaces of functions of bounded variation the solution exists for any positive finite parameter and that if we accept some restrictions on the parameter then the solution exists in the Sobolev space, too. We also provide counterexample indicating that if the domain is non-convex, the solution in Sobolev space need not exist. Powered by TCPDF (www.tcpdf.org)

Made available in DSpace on 2015-05-15T11:46:04Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 816279 bytes, checksum: 28c5081e37dbd539abb463a0ed89b87c (MD5) Previous issue date: 2012-07-27 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this paper we studied surfaces with constant mean curvature and surfaces with constant Gaussian curvature in the Sol space which are invariant under the action of two one-parameter subgroups of isometries of the ambient space. Furthermore, we classify the surfaces that satisfy a relationship of type k1 = mk2, where k1 and k2 are the principal curvatures of the surface and m ∈ R. / O presente trabalho aborda um estudo das superfícies com curvatura média constante e das superfícies com curvatura Gaussiana constante no espaço Sol que são invariantes sob a ação de dois grupos a 1-parâmetro de isometrias do espaço ambiente. Além disso, classificamos as superfícies que satisfazem uma relação do tipo k1 = mk2, onde k1 e k2 são as curvaturas principais da superfície e m ∈ R.

In this dissertation, we study a characterization of the flat equatorial disk and the critical catenoid in the unitary ball B3 of R3 in terms of its second fundamental form. The work developed here is based on the article “A gap theorem for free boundary minimal surfaces in the three-ball” by Lucas Ambrozio and Ivaldo Nunes. / Nesta dissertação, estudamos uma caracterização do disco equatorial plano e do catenóide crítico na bola unitária B3 do R3 em termos da sua segunda forma fundamental. O trabalho aqui desenvolvido baseia-se no artigo “A gap theorem for free boundary minimal surfaces in the three-ball” de Lucas Ambrozio e Ivaldo Nunes.

In this thesis, geometry is used as a basis for conducting experiments aimed at growing and arranging inorganic minerals on curved interfaces. Mineralisation is directed using crystalline and liquid-crystalline metallic soaps and surfactant/water systems as templates.¶ A review of the history, syntheses, structure and liquid crystallinity of metallic soaps and other amphiphiles is presented as a foundation to understanding the interfacial architectures in mesostructured template systems in general.¶ In this study, a range of metallic soaps of varying chain length and cation type are synthesised and characterised to find potentially useful templates for mineral growth. These include alkaline-earth, transition metal, heavy metal and lanthanide soaps. These are systematically characterised using a variety of analytical techniques, including chemical analyses, x-ray diffraction (XRD) infrared spectroscopy (IR) and differential scanning calorimetry (DSC). Their molecular and crystal structures are studied using transmission electron microscopy (TEM), cryo-TEM, electron diffraction (ED), electron paramagnetic spin resonance (EPR), absorption spectroscopy (UV-VIS), high resolution laser spectroscopy, atomic force microscopy (AFM), nuclear magnetic resonance spectroscopy, scanning electron microscopy (SEM), electron dispersive x-ray analysis (EDXA), thermal gravimetric analysis (TGA) and magnetic measurements. Models for the molecular and crystal structures of metallic soaps are proposed. The soaps are predominantly lamellar crystalline or liquid crystalline lamellar rotor phases with tilted and/or untilted molecular constituents. These display evidence of varying degrees of headgroup organisation, including superstructuring and polymerisation. A single crystal structure is presented for a complex of pyridine with cobalt soap. Simple models for their structure are discussed in terms of their swelling properties in water and oils. Experiments are also presented to demonstrate the sorbent properties of aluminium soaps on oil spills.¶ The thermotropic liquid crystallinity of alkaline earth, transition metal, heavy metal and lanthanide soaps is investigated in detail. This is done to assess their suitability as templates, and to document their novel thermotropic behaviour, particularly the relatively unknown lanthanide soaps. Liquid crystalline behaviours are studied using high-temperature XRD (HTXRD), hot-stage optical microscopy and DSC. Models for a liquid crystalline phase progression from crystals to anisotropic liquids are discussed in terms of theories of self-assembly and interfacial curvature. The terminology required for this is drawn from various nomenclature systems for amphiphilic crystals and liquid crystals. General agreement with previous studies is reported for known soaps, while liquid crystallinity is demonstrated in the lanthanide and some non-lanthanide soaps for the first time. A general phase progression of crystalline lamellar through liquid crystalline lamellar to non-lamellar liquid crystalline is discussed in terms of models concerned with the molecular and crystal structures of the soaps and their phase transitions via headgroup and chain re-arrangements.¶ Experiments aimed at guiding growth of metal sulfides using metallic soaps as templates are described, and a model for this growth is discussed. Metal sulfides have been successfully grown by reacting crystalline and liquid crystalline transition metal and heavy metal soaps with H2S gas at room temperature and at elevated temperature. These have been characterised using XRD, TEM, ED and IR. Sulfide growth is demonstrated to be restricted and guided by the reacting soap template architecture. Zinc, cadmium, indium and lead soaps formed confined nanoparticles within the matrix of their reacting soap template. In contrast, curved and flat sheet-like structures, some resembling sponges were found in the products of sulfided iron, cobalt, nickel, copper, tin and bismuth soaps. A model to explain this behaviour is developed in terms of the crystal and liquid crystal structures of the soaps and the crystal structures of the metal sulfide particles.¶ Liquid crystalline iron soaps have been subjected to controlled thermal degradation yielding magnetic iron oxide nanoparticles. Some XRD and TEM evidence has been found for formation of magnetic mesostructures in heat-treated iron soaps. Models for the molecular and liquid crystalline structure of iron soaps, their thermotropic phase progression and eventual conversion to these magnetic products are discussed. Systematic syntheses of mesoporous silicates from sheeted clays are discussed.¶The templates that have been used are cationic surfactants and small, organic molecular salts. Experiments are reported where a cooperative self-assembly of surfactant/water/kanemite plus or minus salt and oils yields 'folded sheet materials' (FSM'S). Templating of kanemite has also been achieved using cobalt cage surfactants. A theoretical prediction of the specific surface areas and specific volumes of homologous sets of FSM's gave excellent agreement with measured values. The geometry and topology of the mesostructures are discussed. A theoretical model is also discussed regarding the curvature found in the sheets of natural clays , and results of templating clays and silica using metallic soaps are presented. Experiments and a model for low temperature nucleation and growth of microporous silicalite-1 are described in terms of silica templating by water clathrates.¶ Finally, the problem of finding minimal surface descriptions of crystal networks is addressed. Combinatoric methods are used to disprove the existence of possible embeddings of type I and II clathrate networks in non-self intersecting periodic minimal surfaces. The crystal network of the clathrate silicate, melanophlogite is successfully embedded in the WI-10 self-intersecting surface. Details of a previously unreported, genus-25 periodic surface with symmetry Im3m are discussed.