A presente dissertação consiste em um estudo de apresentações da demonstração do Segundo Teorema de Incompletude de Gödel. Considera, com especial atenção, aquelas feitas por Shoefield no Mathematical Logic e por Hilbert e Bernays no Grundlagen der Mathematik. Como resultado, obtém-se uma análise das condições de derivabilidade e considerações sobre como é possível demonstrá-las.

Submitted by (lucia.rodrigues@ufrpe.br) on 2017-03-28T14:00:56Z No. of bitstreams: 1 Carlos Bino de Souza.pdf: 2371603 bytes, checksum: d2f0bb2e430fc899161fe573fbae4e50 (MD5) / Made available in DSpace on 2017-03-28T14:00:56Z (GMT). No. of bitstreams: 1 Carlos Bino de Souza.pdf: 2371603 bytes, checksum: d2f0bb2e430fc899161fe573fbae4e50 (MD5) Previous issue date: 2015-08-26 / Euclid wrote a book in 13 volumes called Elements where systematized all the mathematical knowledge of his time. In this work, the 5 postulates of Euclidean geometry were presented. For several years, the 5th Postulate was frequently asked, this inquiries it was discovered that there are several other possible geometries, including hyperbolic geometry. Beltrimi proved that hyperbolic geometry is consistent if Euclidean geometry is consistent. Hilbert showed that Euclidean geometry is consistent if the arithmetic is consistent and presented an axiomatic system that capped the gaps in Euclid’s axiomatic system. Poincaré created a model, called the Poincaré disk, to represent the plan of hyperbolic geometry. The objective of this work is to show that the Poincaré disk model is consistent with reference Axioms Hilbert, replacing only the Axioms of Parallel to "On a point outside a line passes through the two parallel straight lines given", by constructions of Euclidean geometry. / Euclides escreveu uma obra em 13 volumes chamada de Elementos onde sistematizava todo o conhecimento matemático do seu tempo. Nesta obra, foram apresentados os 5 postulados da Geometria Euclidiana. Durante vários anos, o 5o Postulado foi muito questionado, desses questionamentos descobriu-se a existência de várias outras Geometrias possíveis, entre elas a Geometria Hiperbólica. Beltrimi provou que a Geometria Hiperbólica é consistente se a Geometria Euclidiana é consistente. Hilbert mostrou que a Geometria Euclidiana é consistente se a Aritmética é consistente e apresentou um sistema axiomático que preencheu as lacunas do sistema axiomático de Euclides. Poincaré criou um Modelo, chamado de Disco de Poincaré, para representar o plano da Geometria Hiperbólica. O objetivo deste trabalho é mostrar que o Modelo de Disco de poincaré é consistente, tomando como referência os Axiomas de Hilbert, substituindo apenas os Axiomas das Paralelas para "Por um ponto fora de uma reta passam duas retas paralelas à reta dada", através de construções da Geometria Euclidiana.

Orientador: Prof. Dr. Vinicius Cifú Lopes / Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Mestrado Profissional em Matemática em Rede Nacional, 2018. / Esse trabalho apresenta o desenvolvimento do método axiomático a partir de Euclides e sua obra os Elementos. Apresenta as discussões sobre eventuais erros lógicos, sobre o postulado das paralelas e como essa reflexão levou matemáticos a chegarem nas geometrias não euclidianas e independência de axiomas. A consolidação do método axiomático na matemática vem com David Hilbert e sua magistral obra Fundamentos de Geometria, a qual nos deu suporte para elaborar uma nova axiomática para a geometria euclidiana plana, mais moderna e sem apresentar erros lógicos cometidos por Euclides. Ao formalizar tais axiomas com uma linguagem de primeira ordem, deparamo-nos com alguns problemas com os axiomas de continuidade, que não são formalizáveis em primeira ordem. Por fim, apresentamos um modelo para a geometria euclidiana e um modelo para uma geometria não euclidiana. / This study presents the development of the axiomatic method based on Euclides and his work Elements. It discusses possible logical errors, the postulate of parallels and how those led mathematicians to conceive non-euclidean geometries and the independence of axioms. The consolidation of the axiomatic method in mathematics is attributed to David Hilbert and his magistral work Foundations of Geometry. It established a new axiomatic for the plane euclidean geometry, more modern and without logical errors as seen in Euclid's work. By formalizing such axioms in a first order language, we find issues with the continuity axioms as they are not formalizable in that order. Lastly, we present a model for the euclidean geometry and another for the non-euclidean geometry.

Let H be a separable Hilbert space over the complex field. The class S := {N|M : N is normal on H and M is an invariant subspace for Ng of subnormal operators. This notion was introduced by Halmos. The minimal normal extension Ň of a subnormal operator S was introduced by σ (S) and then Bram proved that Halmos. Halmos proved that σ(Ň) (S) is obtained by filling certain number of holes in the spectrum (Ň) of the minimal normal extension Ň of a subnormal operator S. Let σ (S) := σ (Ň) be the spectrum of the minimal normal extension Ň of S; which is called the normal spectrum of a subnormal operator S: This notion is due to Abrahamse and Douglas. We give several well-known characterization of subnormality. Let C* (S1) and C* (S2) be the C*- algebras generated by S1 and S2 respectively, where S1 and S2 are bounded operators on H: Next we give a characterization for subnormality which is purely C - algebraic. We also establish an intrinsic characterization of the normal spectrum for a subnormal operator, which enables us to answer the fol-lowing two questions. Let II be a *- representation from C* (S1) onto C* (S2) such that II(S1) = S2. If S1 is subnormal, then does it follow that S2 is subnormal? What is the relation between σ (S1) and σ (S2)? The first question was asked by Bram and second was asked by Abrahamse and Douglas. Answers to these questions were given by Bunce and Deddens.

Let H be a separable Hilbert space over the complex field. The class S := {N|M : N is normal on H and M is an invariant subspace for Ng of subnormal operators. This notion was introduced by Halmos. The minimal normal extension Ň of a subnormal operator S was introduced by σ (S) and then Bram proved that Halmos. Halmos proved that σ(Ň) (S) is obtained by filling certain number of holes in the spectrum (Ň) of the minimal normal extension Ň of a subnormal operator S. Let σ (S) := σ (Ň) be the spectrum of the minimal normal extension Ň of S; which is called the normal spectrum of a subnormal operator S: This notion is due to Abrahamse and Douglas. We give several well-known characterization of subnormality. Let C* (S1) and C* (S2) be the C*- algebras generated by S1 and S2 respectively, where S1 and S2 are bounded operators on H: Next we give a characterization for subnormality which is purely C - algebraic. We also establish an intrinsic characterization of the normal spectrum for a subnormal operator, which enables us to answer the fol-lowing two questions. Let II be a *- representation from C* (S1) onto C* (S2) such that II(S1) = S2. If S1 is subnormal, then does it follow that S2 is subnormal? What is the relation between σ (S1) and σ (S2)? The first question was asked by Bram and second was asked by Abrahamse and Douglas. Answers to these questions were given by Bunce and Deddens.

No description available.

We investigate optimal partnership of rearrangement-invariant Banach func- tion spaces for the Hilbert transform and the Riesz potential. We establish sharp theorems which characterize optimal action of these operators on such spaces. These results enable us to construct optimal domain (i.e. the largest) and op- timal range (i.e. the smallest) partner spaces when the other space is given. We illustrate the obtained results by non-trivial examples involving Generalized Lorentz-Zygmund spaces with broken logarithmic functions. The method is pre- sented in such a way that it should be easily adaptable to other appropriate operators. 1

The curvature of a contraction T in the Cowen-Douglas class is bounded above by the curvature of the backward shift operator. However, in general, an operator satisfying the curvature inequality need not be contractive. In this thesis, we characterize a slightly smaller class of contractions using a stronger form of the curvature inequality. Along the way, we find conditions on the metric of the holomorphic Hermitian vector bundle E corresponding to the operator T in the Cowen-Douglas class which ensures negative definiteness of the curvature function. We obtain a generalization for commuting tuples of operators in the Cowen-Douglas class. Secondly, we obtain an explicit formula for the curvature of the jet bundle of the Hermitian holomorphic bundle E f on a planar domain Ω. Here Ef is assumed to be a pull-back of the tautological bundle on gr(n, H ) by a nondegenerate holomorphic map f :Ω →Gr (n, H ). Clearly, finding relationships amongs the complex geometric invariants inherent in the short exact sequence 0 → Jk(Ef ) → Jk+1(Ef ) →J k+1(Ef )/ Jk(Ef ) → 0 is an important problem, whereJk(Ef ) represents the k-th order jet bundle. It is known that the Chern classes of these bundles must satisfy c(Jk+1(Ef )) = c(Jk(Ef )) c(Jk+1(Ef )/ Jk(Ef )). We obtain a refinement of this formula: trace Idnxn ( KJk(Ef )) - trace Idnxn ( KJk-1(Ef ))= KJk(Ef )/ Jk-1(Ef )(z).

Tropical geometry is an area of mathematics that interfaces algebraic geometry and combinatorics. The main object of study in tropical geometry is the tropical variety, which is the combinatorial counterpart of a classical variety. A classical variety is converted into a tropical variety by a process called tropicalization, thus reducing the problems of algebraic geometry to problems of combinatorics. This new tropical variety encodes several useful information about the original variety, for example an algebraic variety and its tropical counterpart have the same dimension. In this thesis, we look at the some of the computational aspects of tropical algebraic geometry. We study a generalization of Grobner basis theory of modules which unlike the standard Grobner basis also takes the valuation of coefficients into account. This was rst introduced in (Maclagan & Sturmfels, 2009) in the settings of polynomial rings and its computational aspects were first studied in (Chan & Maclagan, 2013) for the polynomial ring case. The motivation for this comes from tropical geometry as it can be used to compute tropicalization of varieties. We further generalize this to the case of modules. But apart from that it has many other computational advantages. For example, in the standard case the size of the initial submodule generally grows with the increase in degree of the generators. But in this case, we give an example of a family of submodules where the size of the initial submodule remains constant. We also develop an algorithm for computation of Grobner basis of submodules of modules over Z=p`Z[x1; : : : ; xn] that works for any weight vector. We also look at some of the important applications of this new theory. We show how this can be useful in efficiently solving the submodule membership problem. We also study the computation of Hilbert polynomials, syzygies and free resolutions.

In the first chapter of this report, our aim is to introduce harmonic maps between Riemann surfaces using the Energy integral of a map. Once we have the desired prerequisites, we move on to show how to continuously deform a given map to a harmonic map (i.e., find a harmonic map in its homotopy class). We follow J¨urgen Jost’s approach using classical potential theory techniques. Subsequently, we analyze the additional conditions needed to ensure a certain uniqueness property of harmonic maps within a given homotopy class. In conclusion, we look at a couple of applications of what we have shown thus far and we find a neat proof of a slightly weaker version of Hurwitz’s Automorphism Theorem. In the second chapter, we introduce the concept of minimal surfaces. After exploring a few examples, we mathematically formulate Plateau’s problem regarding the existence of a soap film spanning each closed, simple wire frame and discuss a solution. In conclusion, a partial result (due to Rad´o) regarding the uniqueness of such a soap film is discussed.