Disclinations, dislocations and continuous defects: a reappraisal

Disclinations, first observed in mesomorphic phases, are relevant to a number of ill-ordered condensed matter media, with continuous symmetries or frustrated order. They also appear in polycrystals at the edges of grain boundaries. They are of limited interest in solid single crystals, where, owing to their large elastic stresses, they mostly appear in close pairs of opposite signs. The relaxation mechanisms associated with a disclination in its creation, motion, change of shape, involve an interplay with continuous or quantized dislocations and/or continuous disclinations. These are attached to the disclinations or are akin to Nye's dislocation densities, well suited here. The notion of 'extended Volterra process' takes these relaxation processes into account and covers different situations where this interplay takes place. These concepts are illustrated by applications in amorphous solids, mesomorphic phases and frustrated media in their curved habit space. The powerful topological theory of line defects only considers defects stable against relaxation processes compatible with the structure considered. It can be seen as a simplified case of the approach considered here, well suited for media of high plasticity or/and complex structures. Topological stability cannot guarantee energetic stability and sometimes cannot distinguish finer details of structure of defects.Comment: 72 pages, 36 figure

Applications of Hyperbolic Geometry to Continued Fractions and Diophantine Approximation

This dissertation explores relations between hyperbolic geometry and Diophantine approximation, with an emphasis on continued fractions over the Euclidean imaginary quadratic fields, Q(&radic;&minus;d), d = 1, 2, 3, 7, 11, and explicit examples of badly approximable numbers/vectors with an obvious geometric interpretation. The first three chapters are mostly expository. Chapter 1 briefly recalls the necessary hyperbolic geometry and a geometric discussion of binary quadratic and Hermitian forms. Chapter 2 briefly recalls the relation between badly approximable systems of linear forms and bounded trajectories in the space of unimodular lattices (the Dani correspondence). Chapter 3 is a survey of continued fractions from the point of view of hyperbolic geometry and homogeneous dynamics. The chapter discusses simple continued fractions, nearest integer continued fractions over the Euclidean imaginary quadratic fields, and includes a summary of A. L. Schmidt&rsquo;s continued fractions over Q(&radic;&minus;1). Chapters 4 and 5 contain the bulk of the original research. Chapter 4 discusses a class of dynamical systems on the complex plane associated to polyhedra whose faces are two-colorable (i.e. edge-adjacent faces do not share a color). To any such polyhedron, one can associated a right-angled hyperbolic Coxeter group generated by reflections in the faces of a (combinatorially equivalent) right-angled ideal polyhedron in hyperbolic 3-space. After some generalities, we discuss a simpler system, billiards in the ideal hyperbolic triangle. We then discuss continued fractions over Q(&radic;&minus;1) and Q(&radic;&minus;2) coming from the regular ideal right-angled octahedron and cubeoctahedron. Chapter 5 gives explicit examples of numbers/vectors in Rr&times;Cs that are badly approximable over number fields F of signature (r, s) with respect to the diagonal embedding. One should think of these examples as generalizations of real quadratic irrationalities, which we discuss first as our prototype. The examples are the zeros of (totally indefinite anisotropic F-rational) binary quadratic and Hermitian forms (the Hermitian case arises when F is CM). Such forms can be interpreted as compact totally geodesic subspaces in the relevant locally symmetric spaces SL2(OF )\SL2(F&otimes;R)/SO2(R)r &times; SU2(C)s. We discuss these examples from a few different angles: simple arguments stemming from Liouville&rsquo;s theorem on rational approximation to algebraic numbers, arguments using continued fractions (of the sorts considered in chapters 3 and 4) when they are available, and appealing to the Dani correspondence in the general case. Perhaps of special note are examples of badly approximable algebraic numbers and vectors, as noted in 5.10. Chapter 6 considers approximation in Rn (in the boundary &part;Hⁿ of hyperbolic n&minus;space) over &ldquo;weakly Euclidean&rdquo; orders in definite Clifford algebras. This includes a discussion of the relevant background on the &ldquo;SL₂&rdquo; model of hyperbolic isometries (with coefficients in a Clifford algbra) and a discription of the continued fraction algorithm. Some exploration in the case Z3 &sube; R3 is included, along with proofs that zeros of anisotropic rational Hermitian forms are &ldquo;badly approximable,&rdquo; and that the partial quotients of such zeros are bounded (conditional on increasing convergent denominators). Chapter 7 considers simultaneous approximation in Rr &times; Cs as a subset of the boundary of (H2)r &times; (H3)s over a diagonally embedding number field of signature (r, s). A continued fraction algorithm is proposed for norm-Euclidean number fields, but not even convergence is established. Some exploration and experimentation over the norm-Euclidean field Q(&radic;2) is included. Finally, chapter 8 includes some miscellaneous results&nbsp; related to the discrete Markoff spectrum. First, some identities for sums over Markoff numbers are proven (although they are closely related to Mcshane&rsquo;s identity). Secondly, transcendence of certain limits of roots of Markoff forms is established (a simple corollary to [1]). These transcendental numbers are badly approximable with only ones and twos in their continued fraction expansion, and can be written as infinite sums of ratios of Markoff numbers indexed by a path in the tree associated to solutions of the Markoff equation x2 + y2 + z2 = 3xyz. The geometry in this chapter can all be associated to a oncepunctured torus (with complete hyperbolic metric), going back to the observation of H. Cohn [23] v that the Markoff equation is a special case of Fricke&rsquo;s trace identity tr(A)2+tr(B)2+tr(AB)2 = tr(A)tr(B)tr(AB) + tr(ABA-1B-1) + 2, A, B &isin; SL2 in the case that A and B are hyperbolic with parabolic commutator of trace &minus;2 (in particular for the torus associated to the commutator subgroup of SL2(Z)).</p

Discrete Geometry

The workshop on Discrete Geometry was attended by 53 participants, many of them young researchers. In 13 survey talks an overview of recent developments in Discrete Geometry was given. These talks were supplemented by 16 shorter talks in the afternoon, an open problem session and two special sessions. Mathematics Subject Classification (2000): 52Cxx. Abstract regular polytopes: recent developments. (Peter McMullen) Counting crossing-free configurations in the plane. (Micha Sharir) Geometry in additive combinatorics. (József Solymosi) Rigid components: geometric problems, combinatorial solutions. (Ileana Streinu) • Forbidden patterns. (János Pach) • Projected polytopes, Gale diagrams, and polyhedral surfaces. (Günter M. Ziegler) • What is known about unit cubes? (Chuanming Zong) There were 16 shorter talks in the afternoon, an open problem session chaired by Jesús De Loera, and two special sessions: on geometric transversal theory (organized by Eli Goodman) and on a new release of the geometric software Cinderella (Jürgen Richter-Gebert). On the one hand, the contributions witnessed the progress the field provided in recent years, on the other hand, they also showed how many basic (and seemingly simple) questions are still far from being resolved. The program left enough time to use the stimulating atmosphere of the Oberwolfach facilities for fruitful interaction between the participants

An Introduction to Geometric Topology

This book provides a self-contained introduction to the topology and geometry of surfaces and three-manifolds. The main goal is to describe Thurston's geometrisation of three-manifolds, proved by Perelman in 2002. The book is divided into three parts: the first is devoted to hyperbolic geometry, the second to surfaces, and the third to three-manifolds. It contains complete proofs of Mostow's rigidity, the thick-thin decomposition, Thurston's classification of the diffeomorphisms of surfaces (via Bonahon's geodesic currents), the prime and JSJ decomposition, the topological and geometric classification of Seifert manifolds, and Thurston's hyperbolic Dehn filling Theorem

Robust and affordable localization and mapping for 3D reconstruction. Application to architecture and construction

La localización y mapeado simultáneo a partir de una sola cámara en movimiento se conoce como Monocular SLAM. En esta tesis se aborda este problema con cámaras de bajo coste cuyo principal reto consiste en ser robustos al ruido, blurring y otros artefactos que afectan a la imagen. La aproximación al problema es discreta, utilizando solo puntos de la imagen significativos para localizar la cámara y mapear el entorno. La principal contribución es una simplificación del grafo de poses que permite mejorar la precisión en las escenas más habituales, evaluada de forma exhaustiva en 4 datasets. Los resultados del mapeado permiten obtener una reconstrucción 3D de la escena que puede ser utilizada en arquitectura y construcción para Modelar la Información del Edificio (BIM). En la segunda parte de la tesis proponemos incorporar dicha información en un sistema de visualización avanzada usando WebGL que ayude a simplificar la implantación de la metodología BIM.Departamento de Informática (Arquitectura y Tecnología de Computadores, Ciencias de la Computación e Inteligencia Artificial, Lenguajes y Sistemas Informáticos)Doctorado en Informátic

Industrial Robotics

This book covers a wide range of topics relating to advanced industrial robotics, sensors and automation technologies. Although being highly technical and complex in nature, the papers presented in this book represent some of the latest cutting edge technologies and advancements in industrial robotics technology. This book covers topics such as networking, properties of manipulators, forward and inverse robot arm kinematics, motion path-planning, machine vision and many other practical topics too numerous to list here. The authors and editor of this book wish to inspire people, especially young ones, to get involved with robotic and mechatronic engineering technology and to develop new and exciting practical applications, perhaps using the ideas and concepts presented herein

Hadron models and related New Energy issues

The present book covers a wide-range of issues from alternative hadron models to their likely implications in New Energy research, including alternative interpretation of lowenergy reaction (coldfusion) phenomena. The authors explored some new approaches to describe novel phenomena in particle physics. M Pitkanen introduces his nuclear string hypothesis derived from his Topological Geometrodynamics theory, while E. Goldfain discusses a number of nonlinear dynamics methods, including bifurcation, pattern formation (complex GinzburgLandau equation) to describe elementary particle masses. Fu Yuhua discusses a plausible method for prediction of phenomena related to New Energy development. F. Smarandache discusses his unmatter hypothesis, and A. Yefremov et al. discuss Yang-Mills field from Quaternion Space Geometry. Diego Rapoport discusses theoretical link between Torsion fields and Hadronic Mechanic. A.H. Phillips discusses semiconductor nanodevices, while V. and A. Boju discuss Digital Discrete and Combinatorial methods and their likely implications in New Energy research. Pavel Pintr et al. describe planetary orbit distance from modified Schrödinger equation, and M. Pereira discusses his new Hypergeometrical description of Standard Model of elementary particles. The present volume will be suitable for researchers interested in New Energy issues, in particular their link with alternative hadron models and interpretation

Aerial Vehicles

This book contains 35 chapters written by experts in developing techniques for making aerial vehicles more intelligent, more reliable, more flexible in use, and safer in operation.It will also serve as an inspiration for further improvement of the design and application of aeral vehicles. The advanced techniques and research described here may also be applicable to other high-tech areas such as robotics, avionics, vetronics, and space