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Jadhav, Rajesh. "Geometric Routing Without Geometry." Kent State University / OhioLINK, 2007. http://rave.ohiolink.edu/etdc/view?acc_num=kent1178080572.
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Fléchelles, Balthazar. "Geometric finiteness in convex projective geometry." Electronic Thesis or Diss., université Paris-Saclay, 2024. http://www.theses.fr/2024UPASM029.
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Cette thèse est consacrée à l’étude des orbivariétés projectives convexes géométriquement finies, et fait suite aux travaux de Ballas, Cooper, Crampon, Leitner, Long, Marquis et Tillmann sur le sujet. Une orbivariété projective convexe est le quotient d’un ouvert convexe et borné d’une carte affine de l’espace projectif réel (appelé aussi ouvert proprement convexe) par un groupe discret de transformations projectives préservant cet ouvert. S’il n’y a pas de segment dans le bord du convexe, on dit que l’orbivariété est strictement convexe, et si de plus il y a un unique hyperplan de support en chaque point du bord, on dit qu’elle est ronde. Suivant Cooper-Long-Tillmann et Crampon-Marquis, on dit qu’une orbivariété strictement convexe est géométriquement finie si son cœur convexe est l’union d’un compact et d’un nombre fini de bouts, appelés pointes, où le rayon d’injectivité est inférieur à une constante ne dépendant que de la dimension. Comprendre la géométrie des pointes est primordial pour l’étude des orbivariétés géométriquement finies. Dans le cas strictement convexe, la seule restriction connue sur l’holonomie des pointes vient d’une généralisation du lemme de Margulis due à Cooper-Long-Tillmann et Crampon-Marquis, qui implique que cette holonomie est virtuellement nilpotente. On donne dans cette thèse une caractérisation de l’holonomie des pointes des orbivariétés strictement convexes et des orbivariétés rondes. En généralisant la méthode de Cooper, qui a produit le seul exemple connu jusqu’ici d’une pointe de variété strictement convexe dont l’holonomie n’est pas virtuellement abélienne, on construit des pointes de variétés strictement convexes et de variétés rondes dont l’holonomie est isomorphe à n’importe quel groupe nilpotent sans torsion de type fini. En collaboration avec M. Islam et F. Zhu, on démontre que dans le cas des groupes relativement hyperboliques sans torsion, les représentations relativement P1-anosoviennes (au sens de Kapovich-Leeb, Zhu et Zhu-Zimmer) qui préservent un ouvert proprement convexe sont exactement les holonomies des variétés rondes géométriquement finies.Dans le cas des orbivariétés projectives convexes non strictement convexes, il n’y a pas pour l’instant de définition satisfaisante de la finitude géométrique. Toutefois, Cooper-Long-Tillmann puis Ballas-Cooper-Leitner ont proposé une définition de pointe généralisée dans ce contexte. Bien qu’ils demandent que l’holonomie des pointes généralisées soit virtuellement nilpotente, tous les exemples connus jusqu’à présent avaient une holonomie virtuellement abélienne. On construit des exemples de pointes généralisées dont l’holonomie peut être n’importe quel groupe nilpotent sans torsion de type fini. On s’autorise également à modifier la définition originale de Cooper-Long-Tillmann en affaiblissant l’hypothèse de nilpotence en une hypothèse naturelle de résolubilité, ce qui nous permet de construire de nouveaux exemples dont l’holonomie n’est pas virtuellement nilpotente.Une orbivariété géométriquement finie qui n’a pas de pointes, c’est-à-dire dont le cœur convexe est compact, est dite convexe cocompacte. On dispose par les travaux de Danciger-Guéritaud-Kassel d’une définition de la convexe cocompacité pour les orbivariétés projectives convexes sans hypothèse de stricte convexité, contrairement au cas géométriquement fini. Ils démontrent que l’holonomie d’une orbivariété projective convexe convexe cocompacte est Gromov hyperbolique si et seulement si la représentation associée est P1-anosovienne. À l’aide de ce résultat, de la théorie de Vinberg et des travaux d’Agol et Haglund-Wise sur les groupes hyperboliques cubulés, on construit en collaboration avec S. Douba, T. Weisman et F. Zhu des représentations P1-anosoviennes pour tout groupe hyperbolique cubulé. Ceci fournit de nouveaux exemples de groupes hyperboliques admettant des représentations anosoviennes<br>This thesis is devoted to the study of geometrically finite convex projective orbifolds, following work of Ballas, Cooper, Crampon, Leitner, Long, Marquis and Tillmann. A convex projective orbifold is the quotient of a bounded, convex and open subset of an affine chart of real projective space (called a properly convex domain) by a discrete group of projective transformations that preserve it. We say that a convex projective orbifold is strictly convex if there are no non-trivial segments in the boundary of the convex subset, and round if in addition there is a unique supporting hyperplane at each boundary point. Following work of Cooper-Long-Tillmann and Crampon-Marquis, we say that a strictly convex orbifold is geometrically finite if its convex core decomposes as the union of a compact subset and of finitely many ends, called cusps, all of whose points have an injectivity radius smaller than a constant depending only on the dimension. Understanding what types of cusps may occur is crucial for the study of geometrically finite orbifolds. In the strictly convex case, the only known restriction on cusp holonomies, imposed by a generalization of the celebrated Margulis lemma proven by Cooper-Long-Tillmann and Crampon-Marquis, is that the holonomy of a cusp has to be virtually nilpotent. We give a complete characterization of the holonomies of cusps of strictly convex orbifolds and of those of round orbifolds. By generalizing a method of Cooper, which gave the only previously known example of a cusp of a strictly convex manifold with non virtually abelian holonomy, we build examples of cusps of strictly convex manifolds and round manifolds whose holonomy can be any finitely generated torsion-free nilpotent group. In joint work with M. Islam and F. Zhu, we also prove that for torsion-free relatively hyperbolic groups, relative P1-Anosov representations (in the sense of Kapovich-Leeb, Zhu and Zhu-Zimmer) that preserve a properly convex domain are exactly the holonomies of geometrically finite round manifolds.In the general case of non strictly convex projective orbifolds, no satisfactory definition of geometric finiteness is known at the moment. However, Cooper-Long-Tillmann, followed by Ballas-Cooper-Leitner, introduced a notion of generalized cusps in this context. Although they only require that the holonomy be virtually nilpotent, all previously known examples had virtually abelian holonomy. We build examples of generalized cusps whose holonomy can be any finitely generated torsion-free nilpotent group. We also allow ourselves to weaken Cooper-Long-Tillmann’s original definition by assuming only that the holonomy be virtually solvable, and this enables us to construct new examples whose holonomy is not virtually nilpotent.When a geometrically finite orbifold has no cusps, i.e. when its convex core is compact, we say that the orbifold is convex cocompact. Danciger-Guéritaud-Kassel provided a good definition of convex cocompactness for convex projective orbifolds that are not necessarily strictly convex. They proved that the holonomy of a convex cocompact convex projective orbifold is Gromov hyperbolic if and only if the associated representation is P1-Anosov. Using these results, Vinberg’s theory and work of Agol and Haglund-Wise about cubulated hyperbolic groups, we construct, in collaboration with S. Douba, T. Weisman and F. Zhu, examples of P1-Anosov representations for any cubulated hyperbolic group. This gives new examples of hyperbolic groups admitting Anosov representations
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Scott, Phil. "Ordered geometry in Hilbert's Grundlagen der Geometrie." Thesis, University of Edinburgh, 2015. http://hdl.handle.net/1842/15948.
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The Grundlagen der Geometrie brought Euclid’s ancient axioms up to the standards of modern logic, anticipating a completely mechanical verification of their theorems. There are five groups of axioms, each focused on a logical feature of Euclidean geometry. The first two groups give us ordered geometry, a highly limited setting where there is no talk of measure or angle. From these, we mechanically verify the Polygonal Jordan Curve Theorem, a result of much generality given the setting, and subtle enough to warrant a full verification. Along the way, we describe and implement a general-purpose algebraic language for proof search, which we use to automate arguments from the first axiom group. We then follow Hilbert through the preliminary definitions and theorems that lead up to his statement of the Polygonal Jordan Curve Theorem. These, once formalised and verified, give us a final piece of automation. Suitably armed, we can then tackle the main theorem.
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Liu, Yang, and 劉洋. "Optimization and differential geometry for geometric modeling." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2008. http://hub.hku.hk/bib/B40988077.
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Greene, Michael Thomas. "Some results in geometric topology and geometry." Thesis, University of Warwick, 1997. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.397717.
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Liu, Yang. "Optimization and differential geometry for geometric modeling." Click to view the E-thesis via HKUTO, 2008. http://sunzi.lib.hku.hk/hkuto/record/B40988077.
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Hidalgo, García Marta R. "Geometric constraint solving in a dynamic geometry framework." Doctoral thesis, Universitat Politècnica de Catalunya, 2013. http://hdl.handle.net/10803/134690.
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Geometric constraint solving is a central topic in many fields such as parametric solid modeling, computer-aided design or chemical molecular docking. A geometric constraint problem consists of a set geometric objects on which a set of constraints is defined. Solving the geometric constraint problem means finding a placement for the geometric elements with respect to each other such that the set of constraints holds.
Clearly, the primary goal of geometric constraint solving is to define rigid shapes. However an interesting problem arises when we ask whether allowing parameter constraint values to change with time makes sense. The answer is in the positive. Assuming a continuous change in the variant parameters, the result of the geometric constraint solving with variant parameters would result in the generation of families of different shapes built on top of the same geometric elements but governed by a fixed set of constraints. Considering the problem where several parameters change simultaneously would be a great accomplishment. However the potential combinatorial complexity make us to consider problems with just one variant parameter. Elaborating on work from other authors, we develop a new algorithm based on a new tool we have called h-graphs that properly solves the geometric constraint solving problem with one variant parameter. We offer a complete proof for the soundness of the approach which was missing in the original work.
Dynamic geometry is a computer-based technology developed to teach geometry at secondary school, which provides the users with tools to define geometric constructions along with interaction tools such as drag-and-drop. The goal of the system is to show in the user's screen how the geometry changes in real time as the user interacts with the system. It is argued that this kind of interaction fosters students interest in experimenting and checking their ideas. The most important drawback of dynamic geometry is that it is the user who must know how the geometric problem is actually solved. Based on the fact that current user-computer interaction technology basically allows the user to drag just one geometric element at a time, we have developed a new dynamic geometry approach based on two ideas: 1) the underlying problem is just a geometric constraint problem with one variant parameter, which can be different for each drag-and-drop operation, and, 2) the burden of solving the geometric problem is left to the geometric constraint solver.
Two classic and interesting problems in many computational models are the reachability and the tracing problems. Reachability consists in deciding whether a certain state of the system can be reached from a given initial state following a set of allowed transformations. This problem is paramount in many fields such as robotics, path finding, path planing, Petri Nets, etc. When translated to dynamic geometry two specific problems arise: 1) when intersecting geometric elements were at least one of them has degree two or higher, the solution is not unique and, 2) for given values assigned to constraint parameters, it may well be the case that the geometric problem is not realizable. For example computing the intersection of two parallel lines. Within our geometric constraint-based dynamic geometry system we have developed an specific approach that solves both the reachability and the tracing problems by properly applying tools from dynamic systems theory.
Finally we consider Henneberg graphs, Laman graphs and tree-decomposable graphs which are fundamental tools in geometric constraint solving and its applications. We study which relationships can be established between them and show the conditions under which Henneberg constructions preserve graph tree-decomposability. Then we develop an algorithm to automatically generate tree-decomposable Laman graphs of a given order using Henneberg construction steps.
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Chuang, Wu-yen. "Geometric transitions, topological strings, and generalized complex geometry /." May be available electronically:, 2007. http://proquest.umi.com/login?COPT=REJTPTU1MTUmSU5UPTAmVkVSPTI=&clientId=12498.
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Villa, E. "Methods of geometric measure theory in stochastic geometry." Doctoral thesis, Università degli Studi di Milano, 2007. http://hdl.handle.net/2434/28369.
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All the results of the present thesis have been obtained facing problems related to the study of the so called birth-and-growth stochastic processes, relevant in several real applications, like crystallization processes, tumour growth, angiogenesis, etc. We have introduced a Delta formalism, à la Dirac-Schwartz, for the description of random measures associated with random closed sets in R^d of lower dimensions, such that the usual Dirac delta at a point follows as particular case, in order to provide a natural framework for deriving evolution equations for mean densities at integer Hausdorff dimensions in terms of the relevant kinetic parameters associated to a given birth-and-growth process. In this context connections with the concepts of hazard functions and spherical contact distribution functions, together with local Steiner formulas at first order have been studied and, under suitable general conditions on the resulting random growing set, we may write evolution equations of the mean volume density in terms of the growing rate and of the mean surface density. To this end we have introduced definitions of discrete, continuous and absolutely continuous random closed set, which extend the standard well known definitions for random variables. Further, since in many real applications such as fibre processes, n-facets of random tessellations several problems are related to the estimation of such mean densities, in order to face such problems in the general setting of spatially inhomogeneous processes, we have analyzed an approximation of mean densities for sufficiently regular random closed sets, such that some known results in literature follow as particular cases.
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Persson, Aron. "On the Existence of Electrodynamics on Manifold-like Polyfolds." Thesis, Umeå universitet, Institutionen för fysik, 2019. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-155488.
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This essay examines the question whether the classical theory of electrodynamics can be extended to a spacetime which locally changes dimension and if such an endeavour is mathematically possible. Recent research has developed a new generalisation of smooth manifolds, the so called M-polyfolds, which constitutes a sufficient foundation to make this endeavour a physical plausibility. These M-polyfolds then facilitate the capability to define the velocity of a curve going through a dimensionally shifting spacetime. Moreover, necessary extensions to the theory of M-polyfolds is developed in order to tailor the theory to a more physically focused framework. Concluding the essay, Maxwell’s equations on M-polyfolds are defined.<br>Den här uppsatsen betraktar huruvida klassisk elektrodynamik kan generaliseras till en rumtid som lokalt byter dimension samt om detta är matematiskt möjligt. Nyligen har forskningen utvecklat en generalisering av släta mångfalder, så kallade M-polyfolds, vilka ger oss en tillräcklig grund för att göra detta till en fysikalisk möjlighet. Dessa M-polyfolds möjliggör förmågan att definiera hastigheten av en kurva som går igenom en dimensionellt varierande rumtid. Därutöver utvecklas vissa nödvändiga förlängningar av teorin om M-polyfolds, detta för att skräddarsy teorin till ett mer fysikaliskt ramverk. Därefefter avslutas uppsatsen genom att definiera Maxwells ekvationer på M-polyfolds.
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Lokteva, Elizaveta. "On Smooth Knots and Tangent Lines." Thesis, Uppsala universitet, Algebra och geometri, 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-354484.
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Collin, Jan-Ola. "The Existence of Riemannian Metrics on Real Vector Bundles." Thesis, Umeå universitet, Institutionen för matematik och matematisk statistik, 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-151964.
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In this thesis we present a self-contained proof of the existence of Riemannian metrics on real vector bundles.<br>I denna uppsats presenterar vi ett självständigt bevis på existensen av Riemannskametriker på reella vektorbuntar.
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Söderman, Andreas, and Landin Fredrik. "Surfplattans roll i geometriundervisningen : En litteraturstudie om surfplattans positiva effekter i geometriklassrummet." Thesis, Jönköping University, Matematikdidaktisk forskning, 2021. http://urn.kb.se/resolve?urn=urn:nbn:se:hj:diva-52281.
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Hedlund, William. "K-Theory and An-Spaces." Thesis, Uppsala universitet, Algebra och geometri, 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-414082.
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Deigård, Patrik. "Liouville’s equation on simply connected domains." Thesis, Uppsala universitet, Algebra och geometri, 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-419483.
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Bucht, Erik. "Konstruktionen av en regelbunden 17-hörning." Thesis, Uppsala universitet, Algebra och geometri, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-326056.
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Alyounes, Noraldeen. "Elliptiska kurvor och kryptografi." Thesis, Uppsala universitet, Algebra och geometri, 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-404527.
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Pitkälä, Elisa. "Kvadratiska rester." Thesis, Uppsala universitet, Algebra och geometri, 2019. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-388185.
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Pöder, Balkeståhl Sebastian. "Simple homotopy type of the Hamiltonian Floer complex." Licentiate thesis, Uppsala universitet, Matematiska institutionen, 2019. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-393298.
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For an aspherical symplectic manifold M, closed or with convex contact boundary, and with vanishing first Chern class, a Floer chain complex is defined for Hamiltonians linear at infinity with coefficients in the group ring of the fundamental group of M. For two non-degenerate Hamiltonians of the same slope continuation maps are shown to be simple homotopy equivalences. As a corollary the number of contractible Hamiltonian orbits of period 1 can be bounded from below.
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Franklin, Gustav. "Removing cusps from Legendrian front projections." Thesis, Uppsala universitet, Algebra och geometri, 2019. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-395818.
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Asplund, Johan. "Fiber Floer cohomology and conormal stops." Licentiate thesis, Uppsala universitet, Matematiska institutionen, 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-403462.
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Dario, Douglas Francisco. "Geometrias não euclidianas: elíptica e hiperbólica no ensino médio." Universidade Tecnológica Federal do Paraná, 2014. http://repositorio.utfpr.edu.br/jspui/handle/1/862.
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Este trabalho tem como objetivo colaborar na inserção do ensino das Geometrias Não
Euclidianas no ensino médio. Para tanto, fizemos uma pesquisa bibliográfica sobre o
surgimento de tais Geometrias, em seguida apresentamos uma sequência de
conteúdos para o ensino das Geometrias Elíptica e Hiperbólica, abordando os
principais tópicos elencados pelas Diretrizes Curriculares do Estado do Paraná,
comparando-as sempre que possível com a Geometria Euclidiana. Esclarecemos que
onde citamos Geometria Elíptica, estamos realmente tratando da Geometria da
Superfície Esférica, para que este trabalho fique compatível com as Diretrizes
Curriculares do Estado do Paraná. Apesar de haver algumas proposições e suas
provas, em grande parte do trabalho não há teoria e demonstrações com o rigor
exigido pela matemática, buscamos apenas apresentar os principais conceitos e usar
uma linguagem que possa ser compreendida por qualquer profissional que esteja
disposto a compreender e depois de estudar, ensinar estas geometrias. Em novembro
de 2013, na XVII Semana da Matemática e III Encontro de Ensino de Matemática do
Câmpus de Pato Branco – PR da UTFPR, aplicamos um minicurso com parte deste
conteúdo. Ao final do minicurso aplicamos um questionário sobre o conhecimento
inicial do tema e a atual situação de ensino destas geometrias. Tal questionário visou
identificar o interesse sobre o tema e sobre a real possibilidade de inserção destas
geometrias nas salas de aula, cujos resultados encontram-se no texto.<br>This work aims to contribute in including teaching of Non-Euclidean Geometry in high
school. For this, a bibliographic research was made about the appearance of such
geometries and introduce content for teaching of Elliptical and Hyperbolic Geometries, addressing the main topics listed by Curriculum Guidelines of Paraná, comparing them with Euclidean Geometry. Clarify that where quoted elliptic geometry, we are really dealing with Surface Spherical Geometry, for that this work be compatible with the Curriculum Guidelines of the State of Paraná. Although there are some propositions and their proofs, in most part of the work there aren´t theoretical studies and statements with all rigors mathematics requires, we seek to show the main concepts
and use a language that can be understood by any person who is willing to understand
and after studying, teach these geometries in school. In November 2013, during the
XVII Semana de Matemática and III Encontro de Ensino de Matemática Câmpus de
Pato Branco – PR of UTFPR, a mini-course was applied with part of this content to
some participants. At the end of the mini-course a questionnaire was applied inquiring the basic knowledge, the current teaching situation of these geometries and aim to identify the interest in this issue and the real possibility of inclusion in the classrooms, the results can be found in the following work.
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Gouvea, Flavio Roberto [UNESP]. "Um estudo de fractais geométricos através de caleidoscópios e softwares de geometria dinâmica." Universidade Estadual Paulista (UNESP), 2005. http://hdl.handle.net/11449/91080.
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Made available in DSpace on 2014-06-11T19:24:53Z (GMT). No. of bitstreams: 0
Previous issue date: 2005-08-31Bitstream added on 2014-06-13T19:11:45Z : No. of bitstreams: 1
gouvea_fr_me_rcla.pdf: 3114009 bytes, checksum: 7cfd768795cfd2d4315b640578fa631f (MD5)<br>Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)<br>Neste trabalho abordamos um tema pouco explorado nos cursos de graduação em Matemática, que é a Geometria Fractal, resgatando conceitos básicos da Geometria Euclidiana, utilizando caleidoscópios e softwares educacionais. Assim, foram tecidas algumas considerações a respeito da utilização de computadores na sala de aula, através de um estudo que investigou: Que contribuições pode trazer, para o ensinoaprendizagem de Geometria, um estudo de Fractais Geométricos através de caleidoscópios e softwares de Geometria Dinâmica ?. Foram elaboradas atividades e aplicadas a alunos da Licenciatura em Matemática (do 1º e 2º semestres) da Unesp de Rio Claro, que participaram de um Curso de Extensão. A utilização de materiais diferentes do tradicional, como o caleidoscópio e o computador (este último como elemento inserido no contexto educacional), e a contextualização da Geometria contribuíram para o estabelecimento de um ambiente de aprendizagem agradável e participativo. Nosso estudo mostrou uma maneira inovadora de obterem-se fractais geométricos: através de bases caleidoscópicas, o que enseja um grande estudo sobre espelhos e caleidoscópios, e traz em si a oportunidade de estudarem-se muitos conceitos geométricos (reflexão, simetrias, transformações geométricas, bissetriz, mediatriz, seqüências, etc.). Apresentamos, ainda, alguns aspectos pedagógicos e matemáticos relacionados à aplicabilidade dos Fractais Geométricos no processo de construção de conceitos geométricos, por meio da interação aluno-aluno, aluno-computador e alunoprofessor, tendo como pano de fundo a resolução de problemas. Dessa forma, nosso estudo proporcionou para os alunos uma maior relação com os conceitos fundamentais de Geometria Euclidiana e Geometria Fractal, além de uma alternativa metodológica inerente ao ensino da Geometria.<br>In this work we approached a theme little explored in the degree courses in Mathematics, that it is the Fractal Geometry ransoms basic concepts of the Euclidian Geometry, using kaleidoscopic and educational softwares. At his, are some woven considerations respect the use computers in the classroom, through a study that enquired: What contributions can bring, for teaching-learning of Geometry, a study of the geometrical fractals that include kaleidoscopic and softwares of Dynamic Geometry? Activities were elaborated and applied to students of the degree in mathematics (of the 1st and 2nd semesters) of Unesp de Rio Claro, who participated in a Course of Extension. The use of different materials from the traditional as the kaleidoscopic and computer (this last one as element inserted in the education context), and the contextualization of the Geometry contributed to the establishment of an environment of the pleasing learning and interest. Our study showed an innovator way of they be obtained fractal geometrics: through of kaleidoscopic bases, that wish a great study with mirrors and kaleidoscopic, and bring in itself the opportunity of they be studied many geometric concepts (reflection, symmetric, geometric transformations, bisector, mediate, etc). We presented, still, some pedagogic and mathematic aspects related to the applicability of Fractal Geometrics in the process of construction of geometrical concepts, through the interaction student-student, student-computer and student-teacher using as backdrop the problem solve. Of this form, our study it provided for the students a bigger relation with the basic concepts of Euclidean Geometry and Fractal Geometry, beyond inherent a metodology alternative to the teaching of Geometry.
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Riedel, Gårding Elias. "Geometric algebra, conformal geometry and the common curves problem." Thesis, KTH, Skolan för teknikvetenskap (SCI), 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-210866.
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This bachelor’s thesis gives a thorough introduction to geometric algebra (GA), an overview of conformal geometric algebra (CGA) and an application to the processing of single particle data from cryo-electron microscopy (cryo-EM). The geometric algebra over the vector space Rp;q, i.e. the Clifford algebra over an orthogonal basis of the space, is a strikingly simple algebraic construction built from the geometric product, which generalizes the scalar and cross products between vectors. In terms of this product, a host of algebraically and geometrically meaningful operations can be defined. These encode linear subspaces, incidence relations, direct sums, intersections and orthogonal complements, as well as reflections and rotations. It is with good reason that geometric algebra is often referred to as a universal language of geometry. Conformal geometric algebra is the application of geometric algebra in the context of the conformal embedding of R3 into the Minkowski space R4;1. By way of this embedding, linear subspaces of R4;1 represent arbitrary points, lines, planes, point pairs, circles and spheres in R3. Reflections and rotations in R4;1 become conformal transformations in R3: reflections, rotations, translations, dilations and inversions. The analysis of single-particle cryo-electron microscopy data leads to the common curves problem. By a variant of the Fourier slice theorem, this problem involves hemispheres and their intersections. This thesis presents a rewriting, inspired by CGA, into a problem of planes and lines.
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White, Edward C. Jr. "Polar - legendre duality in convex geometry and geometric flows." Thesis, Atlanta, Ga. : Georgia Institute of Technology, 2008. http://hdl.handle.net/1853/24689.
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Oliveira, Vivianne Tasso Perugini de 1975. "Geometria do táxi : pelas ruas de uma cidade aprende-se uma geometria diferente." [s.n.], 2014. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306859.
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Previous issue date: 2014<br>Resumo: Neste trabalho apresentamos o estudo sobre a Geometria do Táxi, uma Geometria não-Euclidiana de fácil compreensão e muito próxima do cotidiano das pessoas, uma vez que tem uma ampla gama de aplicações em situações relacionadas à geografia urbana. A Geometria do Táxi é uma geometria muito semelhante à Geometria Euclidiana, diferindo desta apenas pela definição de distância. Enquanto que, na Geometria Euclidiana, a distância entre dois pontos é o comprimento do segmento de reta que os une, podendo ser obtida com o auxílio do Teorema de Pitágoras, na Geometria do Táxi, a distância entre dois pontos é o comprimento do menor caminho percorrido por linhas horizontais e verticais de um ponto a outro. Esse pequeno detalhe sob o ponto de vista matemático, apresenta grandes diferenças, principalmente nas figuras geométricas que estão relacionadas à distância. Abordamos esse aspecto sob a forma de exemplos e apresentamos no final do trabalho uma sugestão de atividades pedagógicas para serem trabalhadas em sala de aula<br>Abstract: In this paper we present the study of the Taxicab Geometry, a non-Euclidean Geometry of easy understanding and very close to people's daily lives, as it has a wide range of applications in situations related to urban geography. The Taxicab Geometry is a geometry very similar to Euclidian Geometry, differing only by the definition of distance. While in Euclidean Geometry the distance between two points is the length of the line that unites them, which can be obtained with the help of the Pythagorean Theorem, in the Taxicab Geometry the distance between two points is the length of the shortest path travelled by horizontal and vertical lines from one point to another. This small detail, from the mathematical point of view, presents major differences, particularly in the geometric figures that are related to distance. We cover this aspect in the form of examples and present in the end of the work a suggestion of pedagogical activities to be used in class<br>Mestrado<br>Matemática em Rede Nacional<br>Mestra em Matemática em Rede Nacional
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Oliveira, Selma Souza de [UNESP]. "Temas regionais em atividades de geometria: uma proposta na formação continuada de professores de Manaus (AM)." Universidade Estadual Paulista (UNESP), 2004. http://hdl.handle.net/11449/91118.
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oliveira_ss_me_rcla.pdf: 2198783 bytes, checksum: 153b6044438facff703d9c872f622b97 (MD5)<br>Esta pesquisa tem como objetivo discutir uma proposta de trabalho, a partir de temas regionais como uma alternativa para o ensino da Geometria. Foi desenvolvida em um ambiente de reflexão e investigação, caracterizado, pela pesquisadora, como Laboratório de Ensino e Aprendizagem de Matemática. Desenvolveu-se um Estudo de Caso com enfoque qualitativo sob forma de um curso de Geometria. Em cenários para investigação e atividades de caráter aberto, foram investigados que conhecimentos geométricos os professores em formação continuada de Manaus (AM) poderiam obter a partir de imagens da Amazônia. Discutiu-se também a viabilidade desta proposta. A análise dos dados obtidos mostrou a importância de um trabalho com Geometria que estabeleça conexões com a realidade de alunos e professores de uma determinada região. Os resultados deste estudo apontaram que a existência de um ambiente para reflexão, investigação e discussão é uma necessidade urgente nas escolas daquela realidade.<br>The objective of this research was to discuss an alternative proposal for teaching geometry that is based on regional themes. It was developed in an environment of reflection and investigation, characterized by the researcher as a Mathematics Teaching and Learning Laboratory. A case study was conducted, from a qualitative research perspective, of a geometry course being proffered to teachers engaged in continuing education in Manaus, Amazonas. In different landscapes of investigation, and open-ended activities, we investigated the knowledge of geometry that these teachers could acquire when they observed images of the Amazon. The viability of this proposal is also discussed. The analysis of the data obtained pointed to the importance of working with geometry in a way that establishes links with the reality of the region where the students and teachers are from. Results suggest that an environment for reflection, investigation, and discussion is urgently needed in the school in this region.
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Bresciani, Giulio. "Topics in Anabelian Geometry." Doctoral thesis, Scuola Normale Superiore, 2018. http://hdl.handle.net/11384/85735.
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Bråmå, Erik. "Strain Energy of Bézier Surfaces." Thesis, Linköpings universitet, Matematik och tillämpad matematik, 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-145645.
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Bézier curves and surfaces are used to great success in computer-aided design and finite element modelling, among other things, due to their tendency of being mathematically convenient to use. This thesis explores the different properties that make Bézier surfaces the strong tool that it is. This requires a closer look at Bernstein polynomials and the de Castiljau algorithm. To illustrate some of these properties, the strain energy of a Bézier surface is calculated. This demands an understanding of what a surface is, which is why this thesis also covers some elementary theory regarding parametrized curves and surface geometry, including the first and second fundamental forms.
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Valério, José Carlos. "Introdução à geometria hiperbólica." Universidade Federal de Juiz de Fora (UFJF), 2017. https://repositorio.ufjf.br/jspui/handle/ufjf/5405.
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Previous issue date: 2017-05-04<br>CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior<br>Na presente dissertação será introduzido o desenvolvimento histórico da Geometria Hiperbólica. Será apresentado o quinto postulado de Euclides, de acordo com o ponto de vista dos Axiomas de Hilbert, correlacionando-os com os resultados da Geometria Neutra. Serão apresentados e provados alguns resultados da Geometria Hiperbólica, no que diz respeito às propriedades das retas paralelas, dos triângulos generalizados e seus critérios de congruência. Por fim, serão discutidas as propriedades que são válidas tanto para a Geometria Euclidiana quanto Hiperbólica, enfatizando que a principal diferença entre elas é o postulado das paralelas.<br>In the present dissertation we will introduce the historical development of the hyperbolic geometry. We will present Euclid’s fifth postulate from the Hilbert’s axioms point of view and we will correlate them with results of the Neutral Geometry. We will present and prove some results of the Hyperbolic Geometry, regarding the properties of the parallel lines, and the generalized triangles and their congruence criteria. At last, we will discuss the proprieties which are valid in both Euclidean and Hyperbolic Geometry, and we will emphasize that the main difference between them is the parallel postulate.
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Davis, Robert Tucker. "Geometric Build-up Solutions for Protein Determination via Distance Geometry." TopSCHOLAR®, 2009. http://digitalcommons.wku.edu/theses/102.
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Proteins carry out an almost innumerable amount of biological processes that are absolutely necessary to life and as a result proteins and their structures are very often the objects of study in research. As such, this thesis will begin with a description of protein function and structure, followed by brief discussions of the two major experimental structure determination methods. Another problem that often arises in molecular modeling is referred to as the Molecular Distance Geometry Problem (MDGP). This problem seeks to find coordinates for the atoms of a protein or molecule when given only a set of pair-wise distances between atoms. To introduce the complexities of the MDGP we begin at its origins in distance geometry and progress to the specific sub-problems and some of the solutions that have been developed. This is all in preparation for a discussion of what is known as the Geometric Build-up (GBU) Solution. This solution has lead to the development of several algorithms and continues to be modified to account for more and different complexities. The culmination of this thesis, then, is a new algorithm, the Revised Updated Geometric Build-up, that is faster than previous GBU’s while maintaining the accuracy of the resulting structure.
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Avery, Steven. "Noncommutative Geometry." Scholarship @ Claremont, 2005. https://scholarship.claremont.edu/hmc_theses/167.
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We develop noncommutative field theory, starting from a very basic background and explore recent and important results in classical noncommutative field theory. The background section is of interest because it presents mathematical and physical interpretations of differential geometry together in a coherent way, not seen in most of the literature. We present several interesting examples that resulted from recent research in the field.
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Staib, Armando. "Geometria hiperbólica = uma proposta para o desenvolvimento de atividades utilizando o software livre NonEuclid." [s.n.], 2010. http://repositorio.unicamp.br/jspui/handle/REPOSIP/307014.
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Previous issue date: 2010<br>Resumo: Este trabalho trata do ensino das Geometrias Hiperbólica e Euclidiana utilizando softwares de Geometria Dinâmica, em especial o software NonEuclid. O objetivo deste trabalho é ser uma proposta de atividades em Geometria Hiperbólica com o uso do software. O computador introduz uma diversidade dinâmica ao estudo, proporcionando ao aluno, verificar, conjecturar e investigar. As figuras planas podem ser manipuladas e transformadas de diferentes maneiras mantendo as suas propriedades geométricas. Elaboramos algumas atividades de Geometria Hiperbólica utilizando o software NonEuclid para alunos da graduação em matemática e fizemos também atividades que relacionam ambas as geometrias. Os futuros professores precisam saber mais do que irão lecionar e, em geometria, a utilização dos softwares de Geometria Dinâmica contribuem na evolução gradual da aprendizagem de ambas Geometrias: Hiperbólica e Euclidiana, potencializando as habilidades dos alunos pela visualização, experimentação e compreensão das propriedades geométricas<br>Abstract: This work deals with the teaching of Euclidian and Hyperbolic Geometry using software in the Dynamic Geometry area, especially the software by the name of NonEuclid". The objective of this work is to be a proposal for activities in Hyperbolic Geometry using this software. The computer introduces a dynamic diversity to the study, allowing students to examine, investigate and conjecture in this area. The plane figures can be manipulated and processed in different ways while maintaining their geometric properties. We can prepare some activities in Hyperbolic Geometry using the software NonEuclid for graduate students in mathematics and related activities that we also both geometries. Future teachers need to know more than material they present to their students, the use of Dynamic Geometry software contributes to the gradual evolution of learning of geometry, both Euclidean and Hyperbolic. This increases the students' abilities to visualize and experiment and therefore their understanding of geometric properties<br>Mestrado<br>Mestre em Matemática
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Li, Shiyue. "Tropical Derivation of Cohomology Ring of Heavy/Light Hassett Spaces." Scholarship @ Claremont, 2017. http://scholarship.claremont.edu/hmc_theses/104.
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The cohomology of moduli spaces of curves has been extensively studied in classical algebraic geometry. The emergent field of tropical geometry gives new views and combinatorial tools for treating these classical problems. In particular, we study the cohomology of heavy/light Hassett spaces, moduli spaces of heavy/light weighted stable curves, denoted as $\calm_{g, w}$ for a particular genus $g$ and a weight vector $w \in (0, 1]^n$ using tropical geometry. We survey and build on the work of \citet{Cavalieri2014}, which proved that tropical compactification is a \textit{wonderful} compactification of the complement of hyperplane arrangement for these heavy/light Hassett spaces. For $g = 0$, we want to find the tropicalization of $\calm_{0, w}$, a polyhedral complex parametrizing leaf-labeled metric trees that can be thought of as Bergman fan, which furthermore creates a toric variety $X_{\Sigma}$. We use the presentation of $\overline{\calm}_{0,w}$ as a tropical compactification associated to an explicit Bergman fan, to give a concrete presentation of the cohomology.
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Oliveira, Selma Souza de. "Temas regionais em atividades de geometria : uma proposta na formação continuada de professores de Manaus (AM) /." Rio Claro : [s.n.], 2004. http://hdl.handle.net/11449/91118.
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Orientador: Geraldo Perez<br>Banca: Carmen Lúcia Brancaglion Passos<br>Banca: Miriam Godoy Penteado<br>Resumo: Esta pesquisa tem como objetivo discutir uma proposta de trabalho, a partir de temas regionais como uma alternativa para o ensino da Geometria. Foi desenvolvida em um ambiente de reflexão e investigação, caracterizado, pela pesquisadora, como Laboratório de Ensino e Aprendizagem de Matemática. Desenvolveu-se um Estudo de Caso com enfoque qualitativo sob forma de um curso de Geometria. Em "cenários para investigação" e atividades de caráter aberto, foram investigados que conhecimentos geométricos os professores em formação continuada de Manaus (AM) poderiam obter a partir de imagens da Amazônia. Discutiu-se também a viabilidade desta proposta. A análise dos dados obtidos mostrou a importância de um trabalho com Geometria que estabeleça conexões com a realidade de alunos e professores de uma determinada região. Os resultados deste estudo apontaram que a existência de um ambiente para reflexão, investigação e discussão é uma necessidade "urgente" nas escolas daquela realidade.<br>Abstract: The objective of this research was to discuss an alternative proposal for teaching geometry that is based on regional themes. It was developed in an environment of reflection and investigation, characterized by the researcher as a Mathematics Teaching and Learning Laboratory. A case study was conducted, from a qualitative research perspective, of a geometry course being proffered to teachers engaged in continuing education in Manaus, Amazonas. In different "landscapes of investigation", and open-ended activities, we investigated the knowledge of geometry that these teachers could acquire when they observed images of the Amazon. The viability of this proposal is also discussed. The analysis of the data obtained pointed to the importance of working with geometry in a way that establishes links with the reality of the region where the students and teachers are from. Results suggest that an environment for reflection, investigation, and discussion is urgently needed in the school in this region.<br>Mestre
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Lam, Tsui-ling. "A study on how secondary three students make geometric conjectures using "Sketchpad" a graphic geometry software." Click to view the E-thesis via HKUTO, 2005. http://sunzi.lib.hku.hk/hkuto/record/B35711590.
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Eklund, David. "Topics in computation, numerical methods and algebraic geometry." Doctoral thesis, KTH, Matematik (Avd.), 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-25941.
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This thesis concerns computation and algebraic geometry. On the computational side we have focused on numerical homotopy methods. These procedures may be used to numerically solve systems of polynomial equations. The thesis contains four papers. In Paper I and Paper II we apply continuation techniques, as well as symbolic algorithms, to formulate methods to compute Chern classes of smooth algebraic varieties. More specifically, in Paper I we give an algorithm to compute the degrees of the Chern classes of smooth projective varieties and in Paper II we extend these ideas to cover also the degrees of intersections of Chern classes. In Paper III we formulate a numerical homotopy to compute the intersection of two complementary dimensional subvarieties of a smooth quadric hypersurface in projective space. If the two subvarieties intersect transversely, then the number of homotopy paths is optimal. As an application we give a new solution to the inverse kinematics problem of a six-revolute serial-link mechanism. Paper IV is a study of curves on certain special quartic surfaces in projective 3-space. The surfaces are invariant under the action of a finite group called the level (2,2) Heisenberg group. In the paper, we determine the Picard group of a very general member of this family of quartics. We have found that the general Heisenberg invariant quartic contains 320 smooth conics and we prove that in the very general case, this collection of conics generates the Picard group.<br>QC 20101115
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Reis, Joana D'Arc da Silva [UNESP]. "Geometria esférica por meio de materiais manipuláveis." Universidade Estadual Paulista (UNESP), 2006. http://hdl.handle.net/11449/91152.
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reis_jds_me_rcla.pdf: 2063932 bytes, checksum: b3a3de600d7749d917237284e24bd8f6 (MD5)<br>Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)<br>Governo do Estado de São Paulo<br>Esta pesquisa tem como objetivo identificar materiais manipuláveis e descrever o seu uso em um processo de ensino e aprendizagem de Geometria Esférica. Para isso, foi desenvolvido um curso de extensão universitária sobre Geometria Esférica utilizando tais materiais e, desse modo, investigar esta utilização em um ambiente natural de sala de aula. Primeiramente, foram feitos estudos nos livros e dissertações que abordam as Geometrias Não-Euclidianas, bem como uma pesquisa sobre os recursos pedagógicos disponíveis que pudessem ser utilizados neste contexto, tais como softwares de geometria dinâmica, caleidoscópios, além de outros materiais manipuláveis. Após esta etapa, fizemos um estudo piloto para verificar a adequação e o encadeamento na aplicação das atividades. Em seguida, elaboramos e aplicamos o curso de extensão intitulado Geometria Esférica que foi direcionado a alunos do 3° ao 8° semestres da Graduação em Matemática da UNESP de Rio Claro. Os sujeitos de nossa pesquisa foram dez alunos deste programa de formação. Os dados coletados foram analisados qualitativamente, buscando compreender como estes materiais manipuláveis podem colaborar na aquisição de conceitos e propriedades básicas da Geometria Esférica. De acordo com os resultados, acreditamos que esta pesquisa pode auxiliar na busca por propostas alternativas para o ensino de Geometria, possibilitando uma melhor experiência de aprendizagem do futuro professor, enquanto aluno de graduação.<br>This research aims to identify handling materials and to describe their use in a teaching learning process of Spherical Geometry. For this, we developed a course on Spherical Geometry for students of higher education using those materials and, thus, investigate this use in a natural classroom environment. First, we studied books and dissertations about Non-Euclidean Geometries, as well as, we had done a search about available pedagogic sources that could be used in this context, such as softwares of dynamic geometry, kaleidoscope, besides others handling materials. After this stage, we made a pilot study to verify the adaptation and chaining in the application of the activities. Following, we elaborated and applied the course entitled Spherical Geometry that was addressed to the math students at the third to the eighth semesters of UNESP College, at Rio Claro city. The subjects of our research were ten students from this institution. The collected material were analyzed qualitatively, in order to understand how these handle materials can collaborate in the acquisition of concepts and basic proprieties of the Spherical Geometry. According to our results, we think that this research can assist in a search for alternatives purposes to the Geometry teaching, making possible a better experience of learning for the future teacher, while graduated student at a college.
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Reis, Joana D'Arc da Silva. "Geometria esférica por meio de materiais manipuláveis /." Rio Claro : [s.n.], 2006. http://hdl.handle.net/11449/91152.
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Orientador: Claudemir Murari<br>Banca: Henrique Lazari<br>Banca: Ruy Madsen Barbosa<br>Resumo: Esta pesquisa tem como objetivo identificar materiais manipuláveis e descrever o seu uso em um processo de ensino e aprendizagem de Geometria Esférica. Para isso, foi desenvolvido um curso de extensão universitária sobre Geometria Esférica utilizando tais materiais e, desse modo, investigar esta utilização em um ambiente natural de sala de aula. Primeiramente, foram feitos estudos nos livros e dissertações que abordam as Geometrias Não-Euclidianas, bem como uma pesquisa sobre os recursos pedagógicos disponíveis que pudessem ser utilizados neste contexto, tais como softwares de geometria dinâmica, caleidoscópios, além de outros materiais manipuláveis. Após esta etapa, fizemos um estudo piloto para verificar a adequação e o encadeamento na aplicação das atividades. Em seguida, elaboramos e aplicamos o curso de extensão intitulado "Geometria Esférica" que foi direcionado a alunos do 3° ao 8° semestres da Graduação em Matemática da UNESP de Rio Claro. Os sujeitos de nossa pesquisa foram dez alunos deste programa de formação. Os dados coletados foram analisados qualitativamente, buscando compreender como estes materiais manipuláveis podem colaborar na aquisição de conceitos e propriedades básicas da Geometria Esférica. De acordo com os resultados, acreditamos que esta pesquisa pode auxiliar na busca por propostas alternativas para o ensino de Geometria, possibilitando uma melhor experiência de aprendizagem do futuro professor, enquanto aluno de graduação.<br>Abstract: This research aims to identify handling materials and to describe their use in a teaching learning process of Spherical Geometry. For this, we developed a course on Spherical Geometry for students of higher education using those materials and, thus, investigate this use in a natural classroom environment. First, we studied books and dissertations about Non-Euclidean Geometries, as well as, we had done a search about available pedagogic sources that could be used in this context, such as softwares of dynamic geometry, kaleidoscope, besides others handling materials. After this stage, we made a pilot study to verify the adaptation and chaining in the application of the activities. Following, we elaborated and applied the course entitled "Spherical Geometry" that was addressed to the math students at the third to the eighth semesters of UNESP College, at Rio Claro city. The subjects of our research were ten students from this institution. The collected material were analyzed qualitatively, in order to understand how these handle materials can collaborate in the acquisition of concepts and basic proprieties of the Spherical Geometry. According to our results, we think that this research can assist in a search for alternatives purposes to the Geometry teaching, making possible a better experience of learning for the future teacher, while graduated student at a college.<br>Mestre
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Leung, Hoi-cheung, and 梁海翔. "Enhancing students' ability and interest in geometry learning through geometric constructions." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2011. http://hub.hku.hk/bib/B48367746.
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Students nowadays are relatively confident in directly applying geometrical theorems and theories. Nevertheless, it has been a common phenomenon that students are not confident in constructing geometric proofs. They lack the confidence and sufficient experience and knowledge in conducting deductive geometrical proofs. To some students, they treat proofs simply as another type of examination questions which they can tackle by repeated drillings.
Students make use of straightedges and compasses to construct different geometry figures in geometric constructions. Through geometric constructions, we can train our prediction and logical thinking skills when investigating the properties of geometric figures. Geometric constructions provide students with hands-on experience to geometry learning which requires students to have more in-depth thinking.
This is an empirical study on the implementation of geometric construction workshops among junior secondary students in Hong Kong. Results have shown that students enjoyed the construction tasks during the workshops. Analysis has implied that geometric constructions help improve students’ ability in constructing geometric proofs and to raise their interests in geometry learning.<br>published_or_final_version<br>Education<br>Master<br>Master of Education
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Miller, Richard A. "Geometric algebra| An introduction with applications in Euclidean and conformal geometry." Thesis, San Jose State University, 2014. http://pqdtopen.proquest.com/#viewpdf?dispub=1552269.
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<p>This thesis presents an introduction to geometric algebra for the uninitiated. It contains examples of how some of the more traditional topics of mathematics can be reexpressed in terms of geometric algebra along with proofs of several important theorems from geometry. We introduce the conformal model. This is a current topic among researchers in geometric algebra as it is finding wide applications in computer graphics and robotics. The appendices provide a list of some of the notational conventions used in the literature, a reference list of formulas and identities used in geometric algebra along with some of their derivations, and a glossary of terms. </p>
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Terra, Neto Platão Gonçalves. "Possibilidades na conversão entre registros de geometria plana." reponame:Biblioteca Digital de Teses e Dissertações da UFRGS, 2016. http://hdl.handle.net/10183/151181.
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Nesta pesquisa, que consiste de um estudo de caso, elaboramos uma sequência didática que prevê atividades que devem ser resolvidas de duas maneiras distintas. Uma das maneiras utiliza conceitos de Geometria Plana – como Teorema de Pitágoras e semelhanças – e a outra maneira utiliza conceitos de Geometria Analítica – como equações de reta e cálculos de área via determinantes. Para analisar os dados coletados, com a aplicação desta sequência, a Teoria de Registros de Representação Semiótica foi utilizada. Duval (2009), autor da teoria, trata sobre a importância dos registros em Ensino de Matemática, sobre a conversão de um registro em outro e sobre a necessidade de utilização de mais de um registro como um meio de entender o modo matemático de pensar. Como meio de dar um suporte a nossa pesquisa, em nossa revisão bibliográfica, procuramos produções recentes, nas quais foram utilizadas a mesma teoria sob o aspecto da conversão, e analisamos também se os livros didáticos de Matemática, do terceiro ano do Ensino Médio, contemplam atividades que incentivem a utilização de mais de um registro para resolução de atividades. Esta sequência foi aplicada em uma turma de alunos do terceiro ano, de uma escola de Ensino Médio Técnico integrado e sua estrutura foi inspirada na Investigação Matemática de Ponte (2006). Nesta pesquisa, os registros, majoritariamente utilizados pelos alunos, foram os de Geometria Plana – Figural – e de Geometria Analítica – Gráfico – e verificamos que os alunos conseguiram, quando solicitados, articular a utilização destes dois tipos de registro.<br>In this case study we elaborate a didactic sequence that predicts activities that should be solved in two different ways. One of them uses the concepts of plane geometry – such as the Pythagorean theorem and similarities – and the other uses the concepts of analytic geometry – such as the equations of a line and area calculations. To analyze the data assembled with the application of this sequence we used The Theory of Registers of Semiotic Representation. Duval (2009), the author of this theory, addresses the importance of registers in Mathematics Teaching, the conversion of one register to another, and the need to use more than one register as a way to understand the mathematical way of thinking. To support our research, we looked in our bibliographical review for recent articles that made use of the same theory under the conversion aspect, and we also analyzed whether third year high school mathematics textbooks offer activities that encourage the use of more than one register in the solution of activities. This sequence was applied in a class of third-year students, from an integrated technical high school and its structure was inspired by Ponte’s Mathematical Investigation (2006). In this research, the registers most used by the students were those of plane geometry – figure – and of analytic geometry – graph – and we verified that the students, on request, achieved to articulate the use of these two types of registers.
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DURIGHETTO, Sara. "Classical and Derived Birational Geometry." Doctoral thesis, Università degli studi di Ferrara, 2019. http://hdl.handle.net/11392/2488324.
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In the field of algebraic geometry, the study of birational transforma-
tions and their properties plays a primary role. In this, there are two different
approaches: the classical one due to the Italian school who focuses on the Cremona
group and a modern one which utilizes instruments like derived categories and
semiorthogonal decompositions.
About the Cremona group, that is the group of birational self-morphisms of P^n, we
do not know much in general and we focus on the complex case. We know a set of
generators only in dimension n = 2. Moreover, we do not have a classication of
curves and linear systems in P^2 up to Cremona transformations. Among the known
results there are: irreducible curves and curves with two irreducible components. In
this thesis we approach tha case of a conguration of lines in the projective plane.
The last theorem lists the known contractible configurations.
From a categorical point of view, the semiorthogonal decompositions of the derived
category of a variety provide some useful invariants in the study of the variety.
Following the work of Clemens-Griffiths about the complex cubic threefold, we
want to characterize the obstructions to the rationality of a variety X of dimension
n. The idea is to collect the component of a semiorthogonal decomposition which
are not equivalent to the derived category of a variety of dimension at least n-1. In
this way we defined the so called Griffiths-Kuznetsov component of X. In this thesis
we study the case of surfaces on an arbitrary field, we define that component and
show that it is a birational invariant. It appears clearly that the Griffiths-Kuznetsov
component vanishes only if the surface is rational.<br>Nell'ambito della geometria algebrica, lo studio delle trasformazioni
birazionali e delle loro proprietà riveste un ruolo di importanza primaria. In questo,
si affiancano l'approccio classico della scuola italiana che si concentra sul gruppo
di Cremona e quello più moderno che utilizza strumenti come categorie derivate e
decomposizioni semiortogonali.
Del gruppo di Cremona Cr_n, cioé il gruppo degli automorfismi birazionali di P^n, in
generale non si conosce molto e ci si concentra sul caso complesso. Si conosce un
insieme di generatori solo nel caso di dimensione 2. Inoltre non é ancora nota una
classicazione tramite trasformazioni di Cremona delle curve e dei sistemi lineari di
P^2. Tra i casi noti ci sono: le curve irriducibili e quelle formate da due componenti
irriducibili. In questa tesi ci si approccia al caso di una configurazione di d rette
nel piano proiettivo. Il teorema finale fornisce condizioni necessarie o sufficienti alla
contraibilità.
Da un punto di vista categoriale invece, le decomposizioni semiortogonali della cat-
egoria derivata di una varietà ci forniscono degli invarianti utili nello studio della
varietà. Seguendo l'approccio di Clemens-Griffiths riguardante la cubica complessa
di dimensione 3, si vuole caratterizzare le ostruzioni alla razionalità di una varietà X
di dimensione n. L'idea è di raccogliere le componenti di una decomposizione ortog-
onale che non sono equivalenti a categorie derivate di varietà di dimensione almeno
n-1 e in questo modo definire quella che chiamiamo componente di Griffiths-
Kuznetsov di X. In questa tesi si studia il caso delle superci geometricamante
razionali su un campo arbitrario, si definisce tale componente e si mostra che essa è
un invariante birazionale. Si vede anche che la componente di Griffiths-Kuznetsov
è nulla solo se la supercie è razionale.
44
Bjurulf, Anders. "Chip geometry : methods to impact the geometry of market chips /." Uppsala : Swedish University of Agricultural Sciences, 2006. http://diss-epsilon.slu.se/archive/00001251/.
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Thesis (doctoral)--Swedish University of Agricultural Sciences, 2006.<br>Thesis documentation sheet inserted. Appendix reprints four papers and manuscripts, two co-authored with others. Includes bibliographical references. Also issued electronically via World Wide Web in PDF format; online version lacks appendix.
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Björklund, Johan. "Knots and Surfaces in Real Algebraic and Contact Geometry." Doctoral thesis, Uppsala universitet, Matematiska institutionen, 2011. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-156908.
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This thesis consists of a summary and three articles. The thesis is devoted to the study of knots and surfaces with additional geometric structures compared to the classical smooth structure. In Paper I, real algebraic rational knots in real projective space are studied up to rigid isotopy and we show that two real rational algebraic knots of degree at most 5 are rigidly isotopic if, and only if, their degree and encomplexed writhe are equal. We also show that any smooth irreducible knot which admits a plane projection with less than or equal to four crossings has a rational parametrization of degree at most 6. Furthermore, an explicit construction of rational knots of a given degree with arbitrary encomplexed writhe (subject to natural restrictions) is presented. In Paper II, we construct an invariant of parametrized generic real algebraic surfaces in real projective space which generalizes the Brown invariant of immersed surfaces from smooth topology. The invariant is constructed using the self intersection, which is a real algebraic curve with points of three local characters: an intersection of two real sheets, an intersection of two complex conjugate sheets or a Whitney umbrella. The Brown invariant was expressed through a self linking number of the self intersection by Kirby and Melvin. We extend their definition of this self linking number to the case of parametrized generic real algebraic surfaces. In Paper III, we give a combinatorial description of the Legendrian differential graded algebra associated to a Legendrian knot in the product of a punctured Riemann surface with the real line. As an application we show that for any nonzero homology class h, and for any integer k there exist k Legendrian knots all representing h which are pairwise smoothly isotopic through a formal Legendrian isotopy but which lie in mutually distinct Legendrian isotopy classes.
46
Silva, Adriane Renófio da. "Aspectos da geometria neutra /." Rio Claro, 2015. http://hdl.handle.net/11449/131891.
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Orientador: Alice Kimie Miwa Libardi<br>Banca: Thaís Fernanda Mendes Monis<br>Banca: Edson de Oliveira<br>Resumo: Neste trabalho estudamos alguns aspectos da Geometria Neutra, assim chamada porque não é assumido o Axioma das Paralelas. São apresentados resultados possíveis de serem demonstrados assumindo alguns Axiomas de Incidência, Ordem, Congruência e Medida. Demonstramos o Teorema de Saccheri-Legendre e mostramos que nesta geometria não se pode garantir a existência de retângulos. Não nos preocupamos em construir uma teoria axiomática, no sentido exato da palavra<br>Abstract: In this work we study some aspects of Neutral Geometry, so called because it is not assumed the Axiom of Parallels. We present results which are possible to be demonstrated assuming some axioms Incidence, Betweenness, Congruence and Measure are developed. We demonstrate the Saccheri-Legendre theorem and show that this geometry can not guarantee the existence of rectangles. We are not interested to construct an axiomatic theory, in the strict sense of the word<br>Mestre
47
Bassan, André Roberto. "Observações sobre geometria sintética /." Rio Claro, 2015. http://hdl.handle.net/11449/132066.
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Orientador: Alice Kimie Miwa Libardi<br>Banca: Sérgio Roberto Nobre<br>Banca: Edson de Oliveira<br>Resumo: O objetivo deste trabalho é apresentar alguns resultados da Geometria Euclidiana no plano, que são vistos no ensino fundamental e médio sob ponto de vista sintético, ou seja, não serão assumidos os axiomas métricos. Como aplicação faremos algumas construções, usando as ferramentas desenvolvidas<br>Abstract: The objective of this work is to present some results of Euclidean geometry which are given in elementary and high school from the synthetic point of view, that is we will not assume the metric axioms. As an application we will make some constructions using the developed tools<br>Mestre
48
Poggiali, Dario. "Parallel geometry processing." Zürich : ETH, Eidgenössische Technische Hochschule Zürich, cgl Computer Graphics Laboratory, 2008. http://e-collection.ethbib.ethz.ch/show?type=dipl&nr=393.
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Rennie, Adam Charles. "Noncommutative spin geometry." Title page, contents and introduction only, 2001. http://web4.library.adelaide.edu.au/theses/09PH/09phr4163.pdf.
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Li, Shiyan. "Geometry of belief." School of Computer Science and Software Engineering - Faculty of Informatics, 2007. http://ro.uow.edu.au/theses/81.
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Usually, the researchers of traditional belief change theories (e.g., AGM theory) assume that the knowledge of the agents which have the lower priorities should fully accept the knowledge of those higher priority ones in the process of belief revision. These kinds of theories are called prioritized belief change theories. On the contrary, in the discussion of non-prioritized belief change theories (e.g., Konieczny and Pino-P{\'e}rez's merging theory), the belief changes happen among the agents which have the same priorities. In this dissertation, we provide a new style of epistemic states and the belief change operations on this kind of epistemic states such that the prioritized or non-prioritized characteristics of belief change operators will be determined only by the properties of agents' knowledge.