In this thesis, we write explicitly the open book decompositions of links of quotient surface singularities that support the corresponding unique Milnor fillable contact structures. The page-genus of these Milnor open books are minimal among all Milnor open books supporting the corresponding unique Milnor fillable contact structures. That minimal page-genus is called Milnor genus. In this thesis we also investigate whether the Milnor genus is equal to the support genus for links of quotient surface singularities. We show that for many types of the quotient surface singularities the Milnor genus is equal to the support genus of the corresponding contact structure. For the remaining we are able to find an upper bound for the support genus which would be a step forward in understanding these contact structures.

This is a thesis on general relativity. It analyzes dynamical properties of Einstein's field equations in cosmology and in the vicinity of spacetime singularities in a number of different situations. Different techniques are used depending on the particular problem under study; dynamical systems methods are applied to cosmological models with spatial homogeneity; Hamiltonian methods are used in connection with dynamical systems to find global monotone quantities determining the asymptotic states; Fuchsian methods are used to quantify the structure of singularities in spacetimes without symmetries. All these separate methods of analysis provide insights about different facets of the structure of the equations, while at the same time they show the relationships between those facets when the different methods are used to analyze overlapping areas. The thesis consists of two parts. Part I reviews the areas of mathematics and cosmology necessary to understand the material in part II, which consists of five papers. The first two of those papers uses dynamical systems methods to analyze the simplest possible homogeneous model with two tilted perfect fluids with a linear equation of state. The third paper investigates the past asymptotic dynamics of barotropic multi-fluid models that approach a `silent and local' space-like singularity to the past. The fourth paper uses Hamiltonian methods to derive new monotone functions for the tilted Bianchi type II model that can be used to completely characterize the future asymptotic states globally. The last paper proves that there exists a full set of solutions to Einstein's field equations coupled to an ultra-stiff perfect fluid that has an initial singularity that is very much like the singularity in Friedman models in a precisely defined way. / <p>Status of the paper "Perfect Fluids and Generic Spacelike Singularities" has changed from manuscript to published since the thesis defense.</p>

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The Master paper consists of introduction and two parts: a) the singularities of an alternation of the pupils‘ evaluation in the context of the learning paradigm; b) the investigation of the ruling singularities of parents and teachers into the evaluation of the pupils. The conclusions and the recommendations for the teachers and parents are provided in the end of the paper. In the primary school on the basis of the humanistic ideas the grading evaluation was rejected. Further it was made according to the individual progress evaluation the criteria of which might have been not completely clear to parents as well as it might raise unclearness to the teachers. The negative ruling possibility might possibly become higher because of the unclearness in the evaluation of the pupils‘ progress. Therefore, the question turns out: which of the ruling singularities of parents and teachers (positive, neutral or negative) are oriented towards the basic changes of the students‘ progress evaluation? The question and search for the answer is the problematic basis of this Master‘s work. Goal of the analysis: to analyze the ruling singularities of parents and teachers of the primary school towards the evaluation of the pupils by grading and towards the idiographic way of evaluation. The fundamental goals of the work: to clarify the impact of the nature of the ruling singularities of parents and teachers upon the progress and the achievement evaluation system in the primary school regarding the... [to full text]

Let M be an analytic manifold over the real or complex field, J be a coherent and everywhere non-zero ideal sheaf over M, E be a reduced SNC divisor and Θ an involutive singular distribution everywhere tangent to E. The main objective of this work is to obtain a resolution of singularities for the ideal sheaf J that preserves some ''good" properties of the singular distribution Θ. More precisely, the R-monomial property : the existence of local monomial first integrals. This problem arises naturally when we study the ''interaction" between a variety and a foliation and, thus, is also related with the problem of monomialization of maps and of ''quasi-smooth" resolution of families of ideal sheaves.- The first result is a global resolution if the ideal sheaf J is invariant by the singular distribution Θ;- The second result is a global resolution if the the singular distribution Θ has leaf dimension 1;- The third result is a local uniformization if the the singular distribution Θ has leaf dimension 2;We also present two applications of the previous results. The first application concerns the resolution of singularities in families, either of ideal sheaves or vector fields. For the second application, we apply the results to a dynamical system problem motivated by a question of Mattei.

Errata inserted opposite p.177. / Bibliography: p. 173-177. / x, 177 p. : ill. ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / The classical dynamical structure of cosomological models in which the matter content of the universe consists of a scalar field with arbitrary non-negative potential is analyzed in full. (abstract) / Thesis (Ph.D.)--University of Adelaide, Dept. of Physics and Mathematical Physics, 1996?

Thesis (Ph.D.)--Michigan State University. Dept. of Mathematics, 2008. / Title from PDF t.p. (viewed on July 24, 2009) Includes bibliographical references (p. 75-78). Also issued in print.

Orientador: Marcelo José Saia / Banca: Claudio Aguinaldo Buzzi / Banca: João Nivaldo Tomazella / Resumo: Neste trabalho estudaremos a equivalência de contato a fim de compreender seu papel na Cr-classificação dos germes Cr-estáveis, 0 ≤ r ≤ ∞. No caso r = ∞, este é um resultado clássico em Teoria de Singularidades provado por J. Mather [9]. Baseados nos artigos [12] e [11] de T. Nishimura, o principal objetivo deste trabalho é mostrar que existe uma versão do resultado de Mather que diz respeito à Cr-A-classificação de germes Cr-estáveis, 0 ≤ r ≤ ∞. / Abstract: In this work we study the contact equivalence in order to understand its role in the Cr-classification of Cr-stable map germes, 0 ≤ r ≤ ∞. In the case r = ∞, this is a classical result in Singularity Theory proved by J. Mather [9]. Based on the T. Nishimura's papers [12] and [11], the main goal of this report is to show that there exists a version of the Mather's result that is about the Cr-A-classification of Cr-stable map germs, 0 ≤ r ≤ ∞. / Mestre

La modélisation numérique via la méthode des éléments finis utilise classiquement des fonctions de forme polynomiale qui de par leur régularité représentent difficilement des évolutions singulières telles que celles observées dans les phénomènes de localisation en mécanique. Pour pallier cette difficulté, ces travaux de thèse ont eu pour objectif de proposer un nouveau support d'approximation adaptatif couplant la méthode de représentation par ondelettes à la méthode des éléments finis classique. Dans le domaine du traitement du signal, la méthode des ondelettes montre un réel potentiel pour traiter les phénomènes singuliers. L'étude porte sur la création d'un support de discrétisation hybride, associant une interpolation polynomiale et une interpolation en ondelettes exprimée via la fonction d'échelle de l'ondelette de Daubechies. Ce couplage permet de représenter la partie régulière de la réponse via le support polynomial et les éventuelles singularités à l'aide du support en ondelettes. L'adaptation du support hybride est effectuée via l'apport multirésolution, qui ajuste le support en fonction de l'importance des singularités observées. Une méthodologie de détection et d'enrichissement automatique est réalisée ayant pour objectif d'obtenir le support optimum. L'ondelette de Daubechies n'étant connue qu'en des points discrets, une méthode d'intégration particulière est proposée. Une modification de l'interpolation naturellement non nodale de l'ondelette est également introduite, de manière à pouvoir imposer des conditions limites classiques nodales. Une illustration de la méthode et de son implémentation informatique est présentée via une étude académique 1D. / The numerical modelling with the finite element method conventionally uses functions of polynomial form which, by their regularity, hardly represent singular evolutions such as those observed in the phenomena of localization in mechanics. To solve the issue, the aim of this thesis was to propose a new adaptive approximation support coupling the wavelet representation with the classical finite element method. In the field of signal processing, the wavelet method shows a real capacity to treat singular phenomena. This research study deals with the creation of a hybrid discretisation support, including a polynomial interpolation and a wavelet interpolation formulated with the scaling function of the Daubechies wavelet. The regular part of the solution is represented with the polynomial support and the singularities are visualised with the wavelet support. The adaptation of the hybrid support is carried out with the multiresolution contribution, which adjusts the support according to the importance of observed singularities. An automatic detection and enrichment method is carried out in order to obtain the optimum support. The Daubechies wavelet being known only in discrete points, a particular integration method is proposed. A modification of the not nodal naturally interpolated wavelet interpolation is also introduced, in order to impose classical nodal boundary conditions. An illustration of the method and its computer implementation is presented via a 1D academic study.

An algebraic variety is called rationally connected if two generic points can be connected by a curve isomorphic to the projective line. The output of the minimal model program applied to rationally connected variety is variety admitting Mori fiber spaces over a rationally connected base. In this thesis we study the birational geometry of a particular class of rationally connected Mori fiber spaces: Fano fibrations over the projective line. We construct examples of Fano fibrations with a unique Mori fiber space in their birational classes. We prove that these examples are not birational to varieties of Fano type, thus answering the question of Cascini and Gongyo. That is we prove that the classes of rationally connected varieties and varieties of Fano type are not birationally equivalent. To construct the examples we use the techniques of birational rigidity. A Mori fiber space is called birationally rigid if there is a unique Mori fiber space structure in its birational class. The birational rigidity of smooth varieties admitting a del Pezzo fibration of degrees 1 and 2 is a well studied question. Unfortunately it is not enough to study smooth del Pezzo fibrations as there are fibrations which do not have smooth or even smoothable minimal models. In the case of fibrations of degree 2 we know that there is a minimal model with 2-Gorenstein singularities. These singularities are degenerations of the simplest terminal quotient singularity: singular points of the type 1/2(1,1,1). We give first examples of birationally rigid del Pezzo fibrations with 2-Gorenstein singularities. We then apply this result to study finite subgroups of the Cremona group of rank three. We then study the birational geometry of Fano fibrations from a different side. Using the reduction to characteristic 2 method we prove that double covers of Pn-bundles over Pm branched over a divisor of sufficiently high degree are not stably rational. For a del Pezzo fibration Y→P1 of degree 2 such that X is smooth there is a double cover Y→ X, where X is a P2-bundle over P1. In this case a stronger result holds: a very general Y with Pic(Y)≅Z⊕Z is not stably rational. We discuss the proof of this statement.