This thesis was originally motivated by considering vector space analogues of problems in extremal set theory, but our main results concern colouring a graph that is intimately related to these vector space analogues. The vertices of the <em>q</em>-Kneser graph are the <em>k</em>-dimensional subspaces of a vector space of dimension <em>v</em> over F_<em>q</em>, and two <em>k</em>-subspaces are adjacent if they have trivial intersection. The new results in this thesis involve colouring the <em>q</em>-Kneser graph when <em>k</em>=2. There are two cases. When <em>k</em>=2 and <em>v</em>=4, the chromatic number is <em>q</em>²+<em>q</em>. If <em>k</em>=2 and <em>v</em>>4, the chromatic number is (<em>q</em>^(v-1)-1)/(<em>q</em>-1). In both cases, we characterise the minimal colourings. We develop some theory for colouring the <em>q</em>-Kneser graph in general.

Thesis (doctoral)--Friedrich-Wilhelms-Universität zu Berlin, 1882. / Vita.

This thesis investigates various rigidity and flexibility phenomena of convex projective structures on hyperbolic manifolds, particularly in dimension 3. Let M[superscipt n] be a finite volume hyperbolic n-manifold where [mathematical equation] and [mathematical symbol] be its fundamental group. Mostow rigidity tells us that there is a unique conjugacy class of discrete faithful representation of [mathematical symbol] into PSO(subscript n, 1). In light of this fact we examine when this representations can be non-trivially deformed into the larger Lie group of PGL[subscript n+1](R) as well as the relationship between these deformations and convex projective structures on M. Specifically, we show that various two-bridge knots do not admit such deformations into PGL[subscript 4](R) satisfying certain boundary conditions. We subsequently use this result to show that certain orbifold surgeries on amphicheiral knot complements do admit deformations. / text

No description available.

We present a uniform framework generalising and extending the classical theories of projective differential geometry, c-projective geometry, and almost quaternionic geometry. Such geometries, which we call \emph{projective parabolic geometries}, are abelian parabolic geometries whose flat model is an R-space $G\cdot\mathfrak{p}$ in the infinitesimal isotropy representation $\mathbb{W}$ of a larger self-dual symmetric R-space $H\cdot\mathfrak{q}$. We also give a classification of projective parabolic geometries with $H\cdot\mathfrak{q}$ irreducible which, in addition to the aforementioned classical geometries, includes a geometry modelled on the Cayley plane $\mathbb{OP}^2$ and conformal geometries of various signatures. The larger R-space $H\cdot\mathfrak{q}$ severely restricts the Lie-algebraic structure of a projective parabolic geometry. In particular, by exploiting a Jordan algebra structure on $\mathbb{W}$, we obtain a $\mathbb{Z}^2$-grading on the Lie algebra of $H$ in which we have tight control over Lie brackets between various summands. This allows us to generalise known results from the classical theories. For example, which riemannian metrics are compatible with the underlying geometry is controlled by the first BGG operator associated to $\mathbb{W}$. In the final chapter, we describe projective parabolic geometries admitting a $2$-dimensional family of compatible metrics. This is the usual setting for the classical projective structures; we find that many results which hold in these settings carry over with little to no changes in the general case.

This thesis examines the rank of elliptic curves. We first examine the correspondences between projective space and affine space, and use the projective point at infinity to establish the group law on elliptic curves. We prove a section of Mordell’s Theorem to establish that the abelian group of rational points on an elliptic curve is finitely generated. We then use homomorphisms established in our proof to find a formula for the rank, and then provide examples of computations.

Orientador: Claudina Izepe Rodrigues / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-19T19:28:06Z (GMT). No. of bitstreams: 1 Moraes_JoseGalhardoLeitede_M.pdf: 2235311 bytes, checksum: bd14e5445f2f952cca204379e998c41a (MD5) Previous issue date: 2012 / Resumo: Esta dissertação tem por objetivo apresentar o estudo das Cônicas e suas propriedades, mediante a perspectiva da Geometria Projetiva, bem como propor o software livre GeoGebra como uma alternativa para visualização dos Teoremas de Geometria Projetiva e das propriedades das Cônicas. O trabalho inicia-se com uma introdução histórica do desenvolvimento da projeção, na arte, e da Geometria Projetiva. Em seguida é apresentada a base teórica para o estudo das Cônicas e suas propriedades, o que é tratado em seguida. Por fim, são apresentadas algumas construções, que podem ser executadas no software livre, de geometria dinâmica, GeoGebra / Abstract: This dissertation has for objective to present the study of the Conical , and its properties, through the perspective of Projective Geometry, moreover to present free software GeoGebra as an alternative for visualization of the Theorems of Projective Geometry and the properties of the Conical. The work starts with a historical introduction about the development of the projection, in the art, and of Projective Geometry. Next is presented the theoretical basis for the study of conic sections and their properties, which is treated soon after. To finish, some constructions are presented, that can be executed in software of dynamic geometry GeoGebra / Mestrado / Matematica / Mestre em Matemática

No description available.

Yes / The Desargues property is well known in the context of projective geometry. An analogous property is presented in the context of both classical and Quantum Physics. In a classical context, the Desargues property implies that two logical circuits with the same input, show in their outputs selective correlations. In general their outputs are uncorrelated, but if the output of one has a particular value, then the output of the other has another particular value. In a quantum context, the Desargues property implies that two experiments each of which involves two successive projective measurements, have selective correlations. For a particular set of projectors, if in one experiment the second measurement does not change the output of the rst measurement, then the same is true in the other experiment.

In this thesis we study projective perspective on planar euclidean geometry. First we take an euclidean construction and transform it into the projective language. Then we discover and show principles of this transformation. We show equivalence between complex points I, J and some euclidean structures. Moreover we study conics, triangles, polygons and circles. We build this thesis on examples. 1