Mesoporous silica crystals have a large variety of structures mainly due to the versatility of their structure template. The configuration and the chemical state of the templating micellar surfactants, together with the kinetic process of silica will determine the final outcome of the synthesis. Increasing the understanding of the complex formation processes involved will enable a possibilityto fine tune the material for specific uses, today focused into the fields of photoniccrystals, drug delivery, catalysis and separation technology. In this thesis emphasis is put on (1) increasing the understanding the formation mechanism yielding the different species of mesoporous silica crystals through an in depth study of quasicrystallinity (2) Characterization and description of the structural complexity through various characterization techniquesand also by studying the kinetic structural transformation phenomenon related to the minimal G- and D-surfaces. (3) The structural studies of the versatile surfactant liquid crystals for establishing a thermodynamically stable basis to evaluate the kinetic mesoporous silica growth processes. Furthermorethe thesis both enlightens the possibilities of and contributes to the developmentof electron microscopy characterization techniques. In these studies, electron microscopy is largely employed in the characterization to give a thorough picture of the mesoporous structures. This is combined with the sample preparation techniques cross-section polishing and ionslicing. Low voltage scanning electron microscopy is utilized for studying the surfaces and cross-sections of various materials at the limit of the resolution. Here, a deep understanding of the electron beam-material interaction is used for a better interpretation of the detected signals. Transmission electron microscopyis combined with electron crystallographic reconstruction to yield a three dimensional structural model. For determination of the quasicrystallinity level for a structure of dodecagonal tiling, revealed in the scope of this study,a phason strain analysis was made. / <p>At the time of the doctoral defense, the following paper was unpublished and had a status as follows: Paper 7: Manuscript.</p>

The intention of this project is to research the potential of emergent technologies for developing sustainable composites for the building industry. It is divided into three parts: Part 1 Identifies emergent practices which are being applied to the development of new sustainable building prototypes such as developments using nanotechnologies, the influence of biomimetics, the development of intelligent interactive systems and the use of digital technologies to generate complex cellular structural systems. Part 2 Outlines existing processes involved in selecting and manufacturing prototypes, as many existing processes have been opportunistically applied to emergent practices to generate innovative sustainable prototypes. Part 3 By synthesising the research gathered in the previous parts of this thesis, this section documents the design process for developing a sustainable building system. The design is based on a minimal surface structure to reduce materiality and to optimise strength and its manufacturing process exploits emergent and existing technologies for its generation.

The intention of this project is to research the potential of emergent technologies for developing sustainable composites for the building industry. It is divided into three parts: Part 1 Identifies emergent practices which are being applied to the development of new sustainable building prototypes such as developments using nanotechnologies, the influence of biomimetics, the development of intelligent interactive systems and the use of digital technologies to generate complex cellular structural systems. Part 2 Outlines existing processes involved in selecting and manufacturing prototypes, as many existing processes have been opportunistically applied to emergent practices to generate innovative sustainable prototypes. Part 3 By synthesising the research gathered in the previous parts of this thesis, this section documents the design process for developing a sustainable building system. The design is based on a minimal surface structure to reduce materiality and to optimise strength and its manufacturing process exploits emergent and existing technologies for its generation.

In this study we provide a new proof of C¹,α boundary regularity for finite perimeter sets with flat boundary which are local minimizers of a variational mean curvature formula. Our proof is provided for curvature term H∈LΩ. The proof is a generalization of Cafarelli and C#243;rdoba's method, and combines techniques from geometric measure theory and the theory of viscosity solutions which have been developed in the last 50 years. We rely on the delicate interplay between the global nature of sets which are variational minimizers of a given functional, and the pointwise local nature of comparison surfaces which satisfy certain PDE. As a heuristic, in our proof we can consider the curvature as an error term which is estimated and controlled at each point of the calculation.

This thesis presents an overview of some approaches to compute Gaussian and mean curvature on discrete surfaces and discusses discrete minimal surfaces. The variety of applications of differential geometry in visualization and shape design leads to great interest in studying discrete surfaces. With the rich smooth surface theory in hand, one would hope that this elegant theory can still be applied to the discrete counter part. Such a generalization, however, is not always successful. While discrete surfaces have the advantage of being finite dimensional, thus easier to treat, their geometric properties such as curvatures are not well defined in the classical sense. Furthermore, the powerful calculus tool can hardly be applied. The methods in this thesis, including angular defect formula, cotangent formula, parallel meshes, relative geometry etc. are approaches based on offset meshes or generalized offset meshes. As an important application, we discuss discrete minimal surfaces and discrete Koenigs meshes.

Submitted by Rosivalda Pereira (mrs.pereira@ufma.br) on 2017-06-13T17:29:47Z No. of bitstreams: 1 GeovanCampos.pdf: 458133 bytes, checksum: 442b76b0f10e2ef37624745cce5924a3 (MD5) / Made available in DSpace on 2017-06-13T17:29:47Z (GMT). No. of bitstreams: 1 GeovanCampos.pdf: 458133 bytes, checksum: 442b76b0f10e2ef37624745cce5924a3 (MD5) Previous issue date: 2016-03-31 / In this work, we talk about the article "Area-Minimizing Projective Planes in 3- Manifolds" due to Hubert Bray, Simon Brendle, Michael Eichmair and Andr´e Neves. In this article they consider a compact Riemannian 3-manifold (M; g) with positive scalar curvature and an embedded projective plane. In these conditions they prove a higher estimate of curvature, in term of infimum of the scalar curvature of (M; g), for the area of the projective plane that has the smallest area within the class of all surfaces Σ ⊂ M homeomorphic to projective plane. Furthermore, they prove that this inequality is great. More precisely, they get that if this equality hold in (M 3; g), so M is isometric to the three-dimensional projective space RP3 with constant sectional curvature. / Neste trabalho, dissertamos sobre o artigo "Area-minimizing Projective Planes in 3-Manifolds" devido a Hubert Bray, Simon Brendle, Michael Eichmair e André Neves. Neste artigo eles consideram uma 3-variedades Riemannianas compactas (M³, g) com curvatura escalar positiva e que admitem planos projetivos mergulhados. Nestas condições eles provam uma estimativa superior, em termo do ínfimo da curvatura escalar de (M; g), para a área do plano projetivo que possui a menor área dentro da classe de todas as superfícies Σ ⊂ M homeomorfas ao plano projetivo. Além disso, eles provam que esta desigualdade é ótima. Mais precisamente, eles obtém que se a igualdade ocorre então a variedade Riemanniana (M³, g) é isométrica ao espaço projetivo tridimensional RP3 coma métrica de curvatura seccional constante.

Dans cette thèse, nous étudions la structure d’un morphisme harmonique F d’une variété riemannienne M4 dans une surface N2 au voisinage d’un point critique mO. Si mO est un point I critique isolé ou si M4 est compact sans bord, nous montrons que F est pseudo-Holomorphe par rapport à une structure presque hermitienne definie dans un voisinage de mO. Si M4 est compact sans bord, les fibres singuliers de F sont des surfaces minimales avec points de branchement. Ensuite, nous étudions des exemples de morphismes harmoniques due a Burel de (S4, gk,l) dans S2 où (gk,I) est une famille de métriques conforme à la métrique canonique. Nous construisons tout d’abord une application semi-Conforme Φk,l de S4 dans S2 en composant deux applications semi-Conformes régulières, F de S4 dans S3 et Φk,i, de S3 dans S2. II résulte de Baird-Eells que le fibres régulier de øk,l pour tout k, I sont minimales. Si [k] = [l] = 1, l’ensemble des points critiques est donnée par l’image réciproque du pâle nord: il consiste en deux 2-Sphères ayant deux points d’intersection. Si k, I 6= 1 l’ensemble des points critiques sont les images réciproques du pôle nord (de la même façon que pour k = t = 1 deux sphères, mais avec une multiplicité I) ainsi que la pré-Image du pôle sud (un tore) avec multiplicité k. Enfin, nous étudions une construction due à Baird-Ou de morphismes harmoniques d’une ensembles ouverts de (S2×S2, can) dans S2. Nous vérifions qu’ils sont holomorphe par rapport à une des quatre structures complexes canoniques hermitiennes. / In this thesis, we are interested in harmonic morphisms between Riemannian manifolds (Mm, g) and (Nn, h) for m > n. Such a smooth map is a harmonic morphism if it pulls back local harmonic functions to local harmonic functions: if ƒ : V → ℝ is a harmonic function on an open subset V on N and Φ-1(V) is non-Empty, then the composition ƒ ∘ Φ : Φ-1(V) → ℝ is harmonic. The conformal transformations of the complex plane are harmonic morphisms. In the late 1970's Fuglede and Ishihara published two papers ([Fu]) and ([Is]), where they discuss their results on harmonic morphisms or mappings preserving harmonic functions. They characterize non-Constant harmonic morphisms F : (M,g) → (N,h) between Riemannian manifolds as those harmonic maps, which are horizontally conformal, where F horizontally conformal means : for any x ∈ M with dF(x) ≠ 0, the restriction of dF(x) to the orthogonal complement of kerdF(x) in TxM is conformal and surjective. This means that we are dealing with a special class of harmonic maps.

Isogeometric analysis (IGA) is a numerical method for solving partial differential equations (PDE). In this master thesis we explain a concept of IGA with special emphasis on problems on closed domains created by a single NURBS patch. For them we show a process how to modify the NURBS basis to ensure the highest possible continuity of the function space. Then we solve the minimal surface problem using two different Newton type methods. The first one is based on the classical approach using PDE, in the second one we use unique advantages of IGA to directly minimize the area functional.

Cette thèse s'inscrit dans l'étude des sous-variétés minimales et à courbure moyenne constante et de l'influence de la géométrie de la variété ambiante sur les solutions de ce problème.Dans le premier chapitre, en suivant les idées de F. Almgren, on propose une généralisation de la notion d'hypersurface de courbure moyenne constante à toutes codimensions. En dimension n-k on définie les sous-variétés à courbure moyenne constante comme les points critiques de la fonctionnelle de k-volume des bords des variétés minimales de dimension k+1. On prouve l'existence dans une variété riemannienne compacte de dimension n de sous-variétés à courbure moyenne constante de codimension n-k pour tout k < n qui sont des perturbations des sphères géodésiques de petit volume.Dans le deuxième chapitre, on s'intéresse aux surfaces minimales à bords libres dans la boule unité de l'espace euclidien de dimension 3, c'est-à-dire aux surfaces minimales plongées dans la boule unité dont le bord rencontre la sphère unité orthogonalement. On démontre l'existence de deux famille géométriquement distinctes de telles surfaces qui sont indexées par un entier n assez grand, qui représente le nombre de composantes connexes du bord de ces surfaces. Nous donnons en particulier une deuxième preuve d'un résultat de A. Fraser et R. Schoen concernant l'existence de telles surfaces.Un des résultats fondamentaux de la théorie des surfaces à courbure moyenne constante est le théorème de Hopf qui affirme que les seules sphères topologiques à courbure moyenne constante dans l'espace euclidien de dimension 3 sont les sphères rondes. Dans le troisième chapitre, on propose une construction dans une variété riemannienne de dimension 3 d'une famille de sphères topologiques à courbure moyenne constante qui ne sont pas convexes et dont la courbure moyenne est très grande. / The subject of this thesis is the study of minimal and constant mean curvature submanifolds and of the influence of the geometry of the ambient manifold on the solutions of this problem.In the first chapter, following the ideas of F. Almgren, we propose a generalization of the notion of hypersurface with constant mean curvature to all codimensions. In codimension n-k we define constant mean curvature submanifolds as the critical points of the functional of the k - dimensional volume of the boundaries of k+1 - dimensional minimal submanifolds. We prove the existence in compact n-dimensional manifolds of n-k codimensional submanifolds with constant mean curvature for all k<n which are perturbations of geodesic spheres of small volume.In the second chapter, we consider free boundary minimal surfaces in the unit ball of the three dimensional Euclidean space, i.e. minimal surfaces embedded in the unit ball and which meet the unit sphere orthogonally. We prove the existence of two geometrically distinct families of such surfaces parametrized by an integer n large enough, which represents the number of the boundary components. In particular, we give an independent proof of the result of A. Fraser and R. Schoen concerning the existence of such surfaces.One of the fundamental results of the theory of constant mean curvature surfaces is the Hopf's theorem which asserts that the only topological spheres with constant mean curvature in the Euclidean 3-space are round spheres. In the third chapter, we propose a construction in a three dimensional Riemannian manifold of a family of nonconvex topological spheres with large constant mean curvature.

No description available.