Le but de cette thèse est d’étudier profondément le repliement des protéines, au moyendes concepts d’invariance de jauge et d’universalité. La structure de jauge émerge del’équation de Frenet qui est utilisée pour décrire la forme de la chaîne principale de laprotéine. Le principe d’invariance de jauge conduit à une fonctionnelle d’énergieeffective pour une protéine, développée dans le but d’extraire les propriétésuniverselles des protéines repliées durant la phase d’effondrement, et qui estcaractérisée par la loi d’échelle du rayon de giration au niveau tertiaire de la structureprotéique. Dans cette thèse, on étudie l’existence d’une large universalité au niveausecondaire de la structure protéique. La fonctionnelle d’énergie invariante de jaugealliée à l’équation de Frenet discrète conduit à une solution solitonique, identifiéecomme un motif hélice-boucle-hélice dans la protéine. / The purpose of this thesis is to investigate protein folding, by means of the general concepts of gauge invariance and universality. The gauge structure emerges in the Frenet equation which is utilized to describe the shape of protein backbone. The gauge invariance principle leads us an effective energy functional for a protein, which bas been found to catch the universal properties of folded proteins in their collapse phase,characterized by the scaling law of gyration radius on the tertiary level of proteinstructure. In this thesis, the existence of wide universality on the secondary level of protein structure is investigated. The synthesis of the gauge-invariant energy functional with the discrete Frenet equation leads to a soliton solution, which is identified as the helix-loop-helix motif in protein.

A brief explanation of RNA and its general structure on dierent levels is given. The standard continuous Frenet frame is explained. A discrete version of the Frenet frame is explained in detail and constructed for a piecewise linear curve. The results of the application of the discrete Frenet frame to RNA is shown in the form of several distributions. An analysis of these distributions is conducted and gives some results regarding tiny structures in RNA. / <p>Master Degree Project thesis</p>

Orientador: Marcos Benevenuto Jardim / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Cientifica / Made available in DSpace on 2018-08-16T11:14:18Z (GMT). No. of bitstreams: 1 Melo_FabioSilva_M.pdf: 3864560 bytes, checksum: 704d21404c48a187914a0238b121d30e (MD5) Previous issue date: 2010 / Resumo: A teoria local de curvas da Geometria Diferencial no plano e no espaço euclidiano é bem conhecida (vide referências como [4] e [13]). Este trabalho consiste de uma generalização desta teoria usando métricas arbitrárias. Tal generalização é feita substituindo a matriz identidade que define o produto interno usual por outra matriz quadrada, simétrica e positiva definida. Com este novo produto interno, são estudados conceitos como vetor tangente, vetor normal, vetor binormal, fórmulas de Frenet, curvatura e torção / Abstract: The local theory of curves of the Differential Geometry in the Euclidean plane and euclidean space is well known (see references as [4] and [13]). This work consists of a generalization of this theory using arbitrary metrics. Such generalization is made replacing the identity matrix which defines the usual inner product with another square matrix, symmetrical and positive defined. With this new inner product, concepts like tangent vector, normal vector, binormal vector, Frenet's formulas, curvature and torsion are studied / Mestrado / Geometria Diferencial / Mestre em Matemática

Submitted by Jaqueline Silva (jtas29@gmail.com) on 2016-08-30T20:00:18Z No. of bitstreams: 2 Dissertação - Anderson de Azevedo Gomes - 2015.pdf: 2449743 bytes, checksum: 3ff7d4c2fbea731994b5df9da8e7311d (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Jaqueline Silva (jtas29@gmail.com) on 2016-08-30T20:00:30Z (GMT) No. of bitstreams: 2 Dissertação - Anderson de Azevedo Gomes - 2015.pdf: 2449743 bytes, checksum: 3ff7d4c2fbea731994b5df9da8e7311d (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2016-08-30T20:00:30Z (GMT). No. of bitstreams: 2 Dissertação - Anderson de Azevedo Gomes - 2015.pdf: 2449743 bytes, checksum: 3ff7d4c2fbea731994b5df9da8e7311d (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2015-11-06 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / Beginning with the concept plan and how it can be represented in a Vector Plan. Show the identity of vectors and the basic properties of the vector sum with examples and continue with the properties of the Scalar product and how to nd the Angle Between Vectors. After we nd the projection vectors and the distance between a point and a Straight. In the next chapter, I review on Conic Sections. Tract of Ellipse, after Hyperbole and analyze talking about the parable and its utilities. In sequence, show the Parametric equations of a curve. Come to a formula of how to nd a tangent vector and how to nd the length of an arc. Show the usefulness of the Frenet formulas and do the Fundamental Theorem of Plane Curves and their utilities. To end tract Importance of Mathematics in Primary Education with a little history of Dynamic Geometry Software and Software Geogebra, showing how to make some charts in GeoGebra. My goal is to show the beauty of mathematics and attract people to the area. / Começo com o conceito de Plano e como pode ser representado um Vetor no Plano. Mostro a identidade de vetores e as propriedades básicas da soma de vetores com exemplos e continuo com as propriedades do Produto Escalar e como encontrar o Ângulo entre os vetores. Depois, encontramos a projeção de vetores e a distância entre um ponto e uma reta. No próximo capítulo, faço uma revisão sobre as Seções Cônicas. Começo falando da Elipse, depois da Hipérbole e nalizo falando da Parábola e suas utilidades. Em sequência, mostro as Equações Paramétricas de uma curva. Chego a uma fórmula de como encontrar um Vetor Tangente e como encontrar o Comprimento de um Arco. Mostro a utilidade das Fórmulas de Frenet e faço o Teorema Fundamental das Curvas Planas e suas utilidades. Finalizo mostrando como fazer alguns grá cos no GeoGebra e da Importância da Matemática na Formação Básica com um pouco de História dos Softwares de Geometria Dinâmica. Meu objetivo é mostrar a beleza da matemática e atrair pessoas para essa área.

Submitted by Cássia Santos (cassia.bcufg@gmail.com) on 2017-04-17T11:15:23Z No. of bitstreams: 2 Dissertação - Rodrigo Miyasaki - 2017.pdf: 9172139 bytes, checksum: 8a55c1474b190e44bc6643b51239ccdd (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2017-04-17T11:52:07Z (GMT) No. of bitstreams: 2 Dissertação - Rodrigo Miyasaki - 2017.pdf: 9172139 bytes, checksum: 8a55c1474b190e44bc6643b51239ccdd (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2017-04-17T11:52:07Z (GMT). No. of bitstreams: 2 Dissertação - Rodrigo Miyasaki - 2017.pdf: 9172139 bytes, checksum: 8a55c1474b190e44bc6643b51239ccdd (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2017-03-21 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This paper proposes to make use of GeoGebra Software as a tool in the study of flat curves. The Software was chosen because it is a powerful tool in teaching learning at all levels of Mathematics because it works with Algebra and Geometry, allowing the user to perform algebraic and graphical operations in the same interface. The topics covered include the parameterized flat curves up to the study of the Frenet benchmark, making Geogebra an important tool for the study and visualization of the concepts covered. With the help of the Software we can better analyze the curves through their graphs. / Este trabalho tem a proposta de fazer uso do Software GeoGebra como uma ferramenta no estudo das curvas planas. O Software foi escolhido pois é uma ferramenta poderosa no ensino aprendizagem em todos os níveis da Matemática, pois trabalha com Álgebra e Geometria, permitindo com que o usuário realize operações algébricas e gráficas na mesma interface. Os temas abordados incluem as curvas planas parametrizadas até o estudo do Referencial de Frenet, fazendo do Geogebra uma importante ferramenta para o estudo e visualização dos conceitos abordados. Com o auxílio do Software podemos analisar melhor as curvas através de seus gráficos.

This thesis work presents a model predictive control (MPC) based trajectory planner forhigh speed lane change and low speed parking scenarios of autonomous four wheel steering(4WS) vehicle. A four wheel steering vehicle has better low speed maneuverabilityand high speed stability compared with normal front wheel steering(FWS) vehicles. TheMPC optimal trajectory planner is formulated in a curvilinear coordinate frame (Frenetframe) minimizing the lateral deviation, heading error and velocity error in a kinematicdouble track model of a four wheel steering vehicle. Using the proposed trajectory planner,simulations show that a four wheel steering vehicle is able to track different type ofpath with lower lateral deviations, less heading error and shorter longitudinal distance. / I detta avhandlingsarbete presenteras en modellbaserad prediktiv kontroll (MPC) -baseradbanplaneringsplan f¨or h¨oghastighetsbanan och l°aghastighetsparametrar f¨or autonomtfyrhjulsdrift (4WS). Ett fyrhjulsdrivna fordon har b¨attre man¨ovrerbarhet med l°ag hastighetoch h¨oghastighetsstabilitet j¨amf¨ort med vanliga fr¨amre hjulstyrningar (FWS). MPC-optimalbanplanerare ¨ar formulerad i en kr¨okt koordinatram (Frenet-ram) som minimerar sidof¨orl¨angningen,kursfel och hastighetsfel i en kinematisk dubbelsp°armodell av ett fyrhjulsstyrda fordon.Med hj¨alp av den f¨oreslagna banaplaneraren visar simuleringar att ett fyrhjulsstyrfordonkan sp°ara olika typer av banor med l¨agre sidof¨orl¨angningar, mindre kursfel ochkortare l¨angsg°aende avst°and.

Proteins in nature fold to one dominant native structure. Despite being a heavily studied field, predicting the native structure from the amino acid sequence and modeling the folding process can still be considered unsolved problems. In this thesis I present a new approach to this problem with methods borrowed from theoretical physics. In the first part I show how it is possible to use a discrete Frenet frame to define the discrete curvature and torsion of the main chain of the protein. This method is then extended to the side chains as well. In particular I show how to use the discrete Frenet frame to produce a statistical distribution of angles that works in similar fashion as the commonly used Ramachandran plot and side chain rotamers. The discrete Frenet frame displays a gauge symmetry, in the choice of basis vectors on the normal plane, that is reminiscent of features of Abelian-Higgs theory. In the second part of the thesis I show how this similarity with Abelian-Higgs theory can be translated into an effective energy for a protein. The loops of the proteins are shown to correspond to solitons so that the whole protein can be constructed by gluing together any number of solitons. I present results of simulating proteins by minimizing the energy, starting from a real line or straight helix, where the correct native fold is attained. Finally the model is shown to display the same phase structure as real proteins.

In order to perform a specific function, a protein needs to fold into the proper structure. Prediction the protein structure from its amino acid sequence has still been unsolved problem. The main focus of this thesis is to develop new approach on the protein structure modeling by means of differential geometry and integrable theory. The start point is to simplify a protein backbone as a piecewise linear polygonal chain, with vertices recognized as the central alpha carbons of the amino acids. Frenet frame and equations from differential geometry are used to describe the geometric shape of the protein linear chain. Within the framework of integrable theory, we also develop a general geometrical approach, to systematically derive Hamiltonian energy functions for piecewise linear polygonal chains. These theoretical studies is expected to provide a solid basis for the general description of curves in three space dimensions. An efficient algorithm of loop closure has been proposed.

HOLANDA, Felipe D’Angelo. Introdução à geometria diferencial das curvas planas. 2015. 64 f. Dissertação (Mestrado em Matemática em Rede Nacional) – Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2015. / Submitted by Erivan Almeida (eneiro@bol.com.br) on 2015-09-14T17:46:48Z No. of bitstreams: 1 2015_dis_fdholanda.pdf: 2177390 bytes, checksum: 53286a68fd72b70cba214a2700429d7c (MD5) / Approved for entry into archive by Rocilda Sales(rocilda@ufc.br) on 2015-09-15T13:11:15Z (GMT) No. of bitstreams: 1 2015_dis_fdholanda.pdf: 2177390 bytes, checksum: 53286a68fd72b70cba214a2700429d7c (MD5) / Made available in DSpace on 2015-09-15T13:11:15Z (GMT). No. of bitstreams: 1 2015_dis_fdholanda.pdf: 2177390 bytes, checksum: 53286a68fd72b70cba214a2700429d7c (MD5) Previous issue date: 2015 / The intention of this work is to address in basic form and introductory study of Differential Geometry, which in turn has started his studies with Planas curves. It will require a knowledge of Differential Calculus, Integral and Analytic Geometry for better understanding of this work, because as its name says in Differential Geometry comes from the joint study of geometry involving Calculation. So we discuss sub-themes as smooth curves, tangent vector, arc length through formulas of Frenet, evolutas curves and involute and conclude with some important theorems, as the fundamental theorem of plane curves, Jordan 's theorem and the theorem of four vertices. What basically is, Chapter 1, 4 and 6 of the book Introduction to Plane Curves Hilário Alencar and Walcy Santos. / A intenção desse trabalho será de abordar de forma básica e introdutória o estudo da Geometria Diferencial, que por sua vez tem seus estudos iniciados com as Curvas Planas. Será necessário um conhecimento de Cálculo Diferencial, Integral e Geometria Analítica para melhor compreensão desse trabalho, pois como seu próprio nome nos transparece Geometria Diferencial vem de uma junção do estudo da Geometria envolvendo Cálculo. Assim abordaremos subtemas como curvas suaves, vetor tangente, comprimento de arco passando por fórmulas de Frenet, curvas evolutas e involutas e finalizaremos com alguns teoremas importantes, como o teorema fundamental das curvas planas, teorema de Jordan e o teorema dos quatro vértices. O que, basicamente representa, o capítulo 1, 4 e 6 do livro Introdução às Curvas Planas de Hilário Alencar e Walcy Santos.

The present study focuses on developing a Frenet-Serret frame and the Apollonian Window. In the first part of the study Apollonian disks are generated for first four generations by developing visual basic codes in excel. For the second part of the study, three orthonormal basis vectors, namely, tangent, normal, and binormal vectors have been calculated for the tangent points of Apollonian discs for the first three generations. Equations of the Normal, Osculating and Rectifying planes and Taylor Series approximation have been calculated for specific theta. Because Apollonian Window consists of planar curves with constant curvature, torsion is nowhere present. The planar Frenet-Serret Equations for the first three generations for the Apollonian Window is also shown.