This thesis concerns sets of points in the finite projective space PG(n,q) that are combinatorially identical to quadrics. A quadric is the set of points of PG(n,q) whose coordinates satisfy a quadradic equation, and the term quadral is used in this thesis to mean a set of points with all the combinatorial properties of a quadric. Most of the thesis concerns the characterisation of certain sets of subspaces associated with quadrals. Characterisations are proved for the external lines of an oval cone in PG(3,q), of a non-singular quadric in PG(4,q), q even, and of a large class of cones in PG(n,q), q even. Characterisations are also proved for the planes meeting the non-singular quadric of PG(4,q) in a non-singular conic, and for the tangents and generator lines of this quadric for q odd. The second part of the thesis is concerned with the intersection of ovoids of PG(3,q). A new bound is proved on the number of points two ovoids can share, and configurations of secants and external lines that two ovoids can share are determined. The structure of ovoidal fibrations is discussed, and this is used to prove new results on the intersection of two ovoids sharing all of their tangents. / Thesis (Ph.D.) -- University of Adelaide, School of Mathematical Sciences, 2008

Thesis (Ph. D.)--University of Hong Kong, 1988. / Also availalbe in microfilm.

This thesis concerns sets of points in the finite projective space PG(n,q) that are combinatorially identical to quadrics. A quadric is the set of points of PG(n,q) whose coordinates satisfy a quadradic equation, and the term quadral is used in this thesis to mean a set of points with all the combinatorial properties of a quadric. Most of the thesis concerns the characterisation of certain sets of subspaces associated with quadrals. Characterisations are proved for the external lines of an oval cone in PG(3,q), of a non-singular quadric in PG(4,q), q even, and of a large class of cones in PG(n,q), q even. Characterisations are also proved for the planes meeting the non-singular quadric of PG(4,q) in a non-singular conic, and for the tangents and generator lines of this quadric for q odd. The second part of the thesis is concerned with the intersection of ovoids of PG(3,q). A new bound is proved on the number of points two ovoids can share, and configurations of secants and external lines that two ovoids can share are determined. The structure of ovoidal fibrations is discussed, and this is used to prove new results on the intersection of two ovoids sharing all of their tangents. / Thesis (Ph.D.) -- University of Adelaide, School of Mathematical Sciences, 2008

Thesis (Ph.D.) -- University of Adelaide, School of Mathematical Sciences, Discipline of Pure Mathematics, 2008. / "Submitted May 2008" Bibliography: pages 191-195. Also available in print form.

Thesis (doctoral)--Friedrich-Wilhelms-Universität zu Berlin, 1877. / Vita.

Akademisk afhandling--Lund.

Inaug.-diss.--Strassburg. / Lebenslauf.

Using pencils of quadrics, we study a construction of torsors of Jacobians of hyperelliptic curves twice of which is Pic^1. We then use this construction to study the arithmetic invariant theory of the actions of SO2n+1 and PSO2n+2 on self-adjoint operators and show how they facilitate in computing the average order of the 2-Selmer groups of Jacobians of hyperelliptic curves with a rational Weierstrass point, and the average order of the 2-Selmer groups of Jacobians of hyperelliptic curves with a rational non-Weierstrass point, over arbitrary number fields. / Mathematics

Inaug.-diss.-Münster i. W.

Submitted by Edilaine Cláudia Lima da Silva (edilaine.clsilva@gmail.com) on 2018-09-20T14:32:59Z No. of bitstreams: 1 ESTUDO DE CONICAS E QUADRICAS-CONSTRUÇÕES COM O USO DO GEOGEBRA - VERSÃO FINAL - (EDILAINE CLAUDIA LIMA DA SILVA - OK).pdf: 29127272 bytes, checksum: acfd2392a1e7dcb2bd20c58c69c1c5b8 (MD5) / Approved for entry into archive by Elza Mitiko Sato null (elzasato@ibilce.unesp.br) on 2018-09-20T17:21:30Z (GMT) No. of bitstreams: 1 silva_ecl_me_sjrp.pdf: 2655426 bytes, checksum: a1fdb09f1acb62ebdade4446ec1f8083 (MD5) / Made available in DSpace on 2018-09-20T17:21:30Z (GMT). No. of bitstreams: 1 silva_ecl_me_sjrp.pdf: 2655426 bytes, checksum: a1fdb09f1acb62ebdade4446ec1f8083 (MD5) Previous issue date: 2018-08-24 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Este trabalho tem como propósito estudar cônicas e quádricas que podem ser representadas algebricamente por equações do segundo grau em duas e três variáveis, respectivamente. Em particular, a temática de cônicas foi objeto de estudo dos gregos bem antes do início da era cristã, muito embora sob uma perspectiva meramente geométrica. As cônicas e as superfícies de revolução obtidas a partir destas possuem inúmeras aplicações práticas em várias áreas do conhecimento humano, sendo, portanto, um conceito interdisciplinar. Vale salientar que os Parâmetros Curriculares Nacionais (PCN’s), sugerem a investigação de temas e eixos transversais que possam ser discutidas em várias disciplinas ao longo da vida escolar do estudante. Atividades didáticas que exploram os elementos fundamentais associados a cada uma das cônicas foram propostas para serem desenvolvidas junto aos estudantes do ensino médio. No intuito de se diferenciar das formas tradicionais de ensino, procura-se fazer uso das denominadas novas tecnologias, em especial do software de matemática dinâmica Geogebra, que é capaz de trabalhar conteúdos de geometria, álgebra, cálculo e estatística e, em particular, simular construções geométricas baseadas em régua e compasso. Os inúmeros recursos de visualização em 2D e 3D, aliados a animação de objetos matemáticos, permite que os jovens estudantes possam ter níveis de abstração e enxergar relações entre objetos no espaço difíceis de serem obtidas por meios convencionais. Ademais, essa ferramenta permite aos estudantes explorar, investigar, conjecturar e com isso despertar e estimular o interesse dos mesmos pela construção de seu saber matemático, tornando-os agentes nesse processo. / The purpose of this work is to study conics and quadrics that can be represented algebraically by equations of the second degree in two and three variables, respectively. In particular, the concepts of conics was studied by the Greeks before the beginning of the Christian era, albeit from a purely geometric perspective. The conics and surfaces of revolution obtained from these have numerous practical applications in several areas of human knowledge, being, therefore, an interdisciplinary concept. It should be noted that the National Curricular Parameters (PCN’s) suggest the investigation of themes and transversal axes that can be discussed in various disciplines throughout the student’s school life. Didactic activities that explore the fundamental elements associated to each of the conics were proposed to be developed with high school students. In order to differentiate itself from the traditional forms of teaching, we try to make use of the so-called new technologies, especially Geogebra dynamic mathematics software, which is able to work with geometry, algebra, calculus and statistics content and, in particular, simulate geometric constructions based on ruler and compass. This software has numerous 2D and 3D visualization capabilities, coupled with the animation of mathematical objects, enable young students to have levels of abstraction and to see relationships between objects in space that are difficult to obtain by conventional means. In addition, this tool allows students to explore, investigate, conjecture and thereby awaken and stimulate their interest in building their mathematical knowledge, making them agents in this process.