Thesis--Universität Basel, 1911. / Vita.

Thesis--Georg-August-Universität, 1912. / Cover-title. "Lebenslauf."

This thesis is concerned with a conjecture of Zilber: that the complex field expanded with the exponential function should be `quasi-minimal'; that is, all its definable subsets should be countable or have countable complement. Our purpose is to study the geometry of this structure and other expansions by holomorphic functions of the complex field without having first to settle any number-theoretic problems, by treating all countable sets on an equal footing. We present axioms, modelled on those for a Zariski geometry, defining a non-first-order class of ``quasi-Zariski'' structures endowed with a dimension theory and a topology in which all countable sets are of dimension zero. We derive a quantifier elimination theorem, implying that members of the class are quasi-minimal. We look for analytic structures in this class. To an expansion of the complex field by entire holomorphic functions $\mathcal{R}$ we associate a sheaf $\mathcal{O}^{\scriptscriptstyle{\mathcal{R}}}$ of analytic germs which is closed under application of the implicit function theorem. We prove that $\mathcal{O}^{\scriptscriptstyle{\mathcal{R}}}$ is also closed under partial differentiation and that it admits Weierstrass preparation. The sheaf defines a subclass of the analytic sets which we call $\mathcal{R}$-analytic. We develop analytic geometry for this class proving a Nullstellensatz and other classical properties. We isolate a condition on the asymptotes of the varieties of certain functions in $\mathcal{R}$. If this condition is satisfied then the $\mathcal{R}$-analytic sets induce a quasi-Zariski structure under countable union. In the motivating case of the complex exponential we prove a low-dimensional case of the condition, towards the original conjecture.

In this thesis an analytical modelling approach is employed to predict and gain insight into the flows induced by turbulent plumes and jets above slender horizontal slots, in otherwise quiescent uniform environments. To supplement the solutions, the effect on the environment of a plume driven by an off-source supply of buoyancy was also considered. The solutions derived provide an advancement on existing idealised models for the jet and plume induced flows, and moreover, complement a number of key advances that have been made in our understanding of plume flows in recent years. The theory of functions of a complex variable, which has not previously been applied in such an application, has been utilised as a fundamental tool throughout the work. This has enabled the entrainment behaviour and geometry of the plumes to be accounted for when developing the induced flow solutions. A novel conformal mapping has been devised specifically to account for the curved perimeter of the contracting lazy plume. This modelling approach is robust in that future developments to aspects of the modelling, for instance, the formulation of a new entrainment closure, can be straightforwardly accounted for using the method. The induced flow solutions exhibit a range of flow patterns which are dependent on the source Richardson number of the plume flow. A pure plume induces a uniform horizontal flow. Forced and lazy plumes correspond to a relative deficit and excess in source buoyancy flux compared to the pure plume, respectively. Generally, forced plumes induce downwardly inclined flows, in contrast to lazy plumes, which induce upwardly inclined flows. Consistent with these solutions, the notionally lazy plume driven by a vertical uniform off-source supply of buoyancy induces an upwardly inclined flow. In addition to an improved understanding of induced flows, our solutions have provided us with insight into the plume flow. Notably, the solution corresponding to the forced plume has led us to fundamentally question existing models describing the plume and, in particular, closures that have been employed to model entrainment. We find that the existing well accepted closures exhibit some form of non-physical flow behaviour.

The object of this paper is to define the elementary transcendental functions of a complex variable by means of integrals, and to discuss their properties.

A presente dissertação tem como propósito apresentar uma proposta didática para o ensino de números complexos e funções de variável complexa, fazendo uso de um objeto de aprendizagem. Para o embasamento teórico, quanto ao processo de construção de conhecimento, referenciamos Vygotsky e Piaget. Também foi feita uma análise das tecnologias como ferramenta para o ensino e, em especial quanto às possibilidades que dizem respeito aos registros dinâmicos de representação semiótica. A metodologia para conceber, realizar e analisar a proposta didática é a Engenharia Didática. Essa metodologia permitiu uma detalhada validação da sequência didática que integra o uso do objeto de aprendizagem "Números Complexos" com animações interativas, vídeos, explicações e exercícios. A sequência foi implementada em um terceiro ano do Ensino Médio. / This work presents a suggestion of a didactical sequence for teaching complex numbers and functions in high school. The sequence supposes the use of the learning object "Complex Numbers" with interactive animations, videos and exercises. As a theoretical frame to support our understanding of the knowledge construction process it was taken into account the Vygotsky´s theory and Piaget´s theory. A discussion about the potential of technologies as a teaching tool is also presented, specially about the possibilities related to dynamical semiotic representation in mathematics. The research methodology used was Didactical Engineering. With this methodology was possible to implemented and validate the didactical sequence.

A presente dissertação tem como propósito apresentar uma proposta didática para o ensino de números complexos e funções de variável complexa, fazendo uso de um objeto de aprendizagem. Para o embasamento teórico, quanto ao processo de construção de conhecimento, referenciamos Vygotsky e Piaget. Também foi feita uma análise das tecnologias como ferramenta para o ensino e, em especial quanto às possibilidades que dizem respeito aos registros dinâmicos de representação semiótica. A metodologia para conceber, realizar e analisar a proposta didática é a Engenharia Didática. Essa metodologia permitiu uma detalhada validação da sequência didática que integra o uso do objeto de aprendizagem "Números Complexos" com animações interativas, vídeos, explicações e exercícios. A sequência foi implementada em um terceiro ano do Ensino Médio. / This work presents a suggestion of a didactical sequence for teaching complex numbers and functions in high school. The sequence supposes the use of the learning object "Complex Numbers" with interactive animations, videos and exercises. As a theoretical frame to support our understanding of the knowledge construction process it was taken into account the Vygotsky´s theory and Piaget´s theory. A discussion about the potential of technologies as a teaching tool is also presented, specially about the possibilities related to dynamical semiotic representation in mathematics. The research methodology used was Didactical Engineering. With this methodology was possible to implemented and validate the didactical sequence.

A presente dissertação tem como propósito apresentar uma proposta didática para o ensino de números complexos e funções de variável complexa, fazendo uso de um objeto de aprendizagem. Para o embasamento teórico, quanto ao processo de construção de conhecimento, referenciamos Vygotsky e Piaget. Também foi feita uma análise das tecnologias como ferramenta para o ensino e, em especial quanto às possibilidades que dizem respeito aos registros dinâmicos de representação semiótica. A metodologia para conceber, realizar e analisar a proposta didática é a Engenharia Didática. Essa metodologia permitiu uma detalhada validação da sequência didática que integra o uso do objeto de aprendizagem "Números Complexos" com animações interativas, vídeos, explicações e exercícios. A sequência foi implementada em um terceiro ano do Ensino Médio. / This work presents a suggestion of a didactical sequence for teaching complex numbers and functions in high school. The sequence supposes the use of the learning object "Complex Numbers" with interactive animations, videos and exercises. As a theoretical frame to support our understanding of the knowledge construction process it was taken into account the Vygotsky´s theory and Piaget´s theory. A discussion about the potential of technologies as a teaching tool is also presented, specially about the possibilities related to dynamical semiotic representation in mathematics. The research methodology used was Didactical Engineering. With this methodology was possible to implemented and validate the didactical sequence.

All procedures in the literature for decomposing symmetric second rank (stress) tensor and symmetric fourth rank (elastic constant) tensor are elaborated and compared which have many engineering and scientific applications for anisotropic materials. The decomposition methods for symmetric second rank tensors are orthonormal tensor basis method, complex variable representation and spectral method. For symmetric fourth rank (elastic constant) tensor, there are four mainly decomposition methods namely as, orthonormal tensor basis, irreducible, harmonic decomposition and spectral. Those are applied to anisotropic materials possessing various symmetry classes which are isotropic, cubic, transversely isotropic, tetragonal, trigonal and orthorhombic. For isotropic materials, an expression for the elastic constant tensor different than the traditionally known form is given. Some misprints found in the literature are corrected. For comparison purposes, numerical examples of each decomposition process are presented for the materials possessing different symmetry classes. Some applications of these decomposition methods are given. Besides, norm and norm ratio concepts are introduced to measure and compare the anisotropy degree for various materials with the same or di&curren / erent symmetries. For these materials,norm and norm ratios are calculated. It is suggested that the norm of a tensor may be used as a criterion for comparing the overall e&curren / ect of the properties of anisotropic materials and the norm ratios may be used as a criterion to represent the anisotropy degree of the properties of materials. Finally, comparison of all methods are done in order to determine similarities and differences between them. As a result of this comparison process, it is proposed that the spectral method is a non-linear decomposition method which yields non-linear orthogonal decomposed parts. For symmetric second rank and fourth rank tensors, this case is a significant innovation in decomposition procedures in the literature.

Orientador: Edmundo Capelas de Oliveira / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-04T02:33:55Z (GMT). No. of bitstreams: 1 Camargo_RubensdeFigueiredo_M.pdf: 815551 bytes, checksum: c2c6d5deb1da34ce6eef3781c5acc1a8 (MD5) Previous issue date: 2005 / Resumo: Este trabalho versa sobre variaveis complexas, em particular sobre o teorema integral de Cauchy, suas consequencias e aplicações. Como consequencia do teorema integral de Cauchy temos o teorema dos residuos, peça chave para o desenvolvimento deste trabalho. Nas aplicações nos concentramos no estudo das transformadas integrais como metodologia na resolução de equações diferenciais parciais, em particular no calculo da inversão das transformadas de Laplace, Fourier e Hankel, bem como na justa posição das transformadas. Para inversão da justa posição das transformadas nos concentramos no metodo de Cagniard e algumas de suas variações / Abstract: This work is about complex variables, in particular about Cauchy¿s integral theorem and its consequences and applications. We have, as consequences of Cauchy¿s integral theorem, Cauchy¿s theorem and the residue theorem, a keynote to the development of this work. As for the applications, our main objective was to study the integral transforms as a method to solve partial differential equations and, specifically, the inversion of the Laplace, Fourier and Hankel transforms, in the same way, the juxtaposition of transforms. In order to invert the juxtaposition of transforms our main concern was to study Cagniard¿s method and some of its variations / Mestrado / Matematica Aplicada / Mestre em Matemática