Neste trabalho, estudamos os grupos de tranças em superfícies visando encontrar apresentações para estes grupos em superfícies fechadas orientáveis de gênero g >= 1 ou superfícies fechadas não orientáveis de gênero g >= 2. Uma consequência destas apresentações é resolvermos o problema da palavra, que consiste em encontrar um algoritmo para decidir quando uma dada palavra num grupo definido por seus geradores e suas relações é a palavra trivial / In this work, we find presentations for surface braid groups either in closed orientable surfaces of genus g >= 1 or in closed non-orientable surfaces of genus g >= 2. A consequence of this presentations is to solve the word problem, which consists in finding an algorithm to decide when a given word in a group defined by its generators and its relations is the trivial word

Cette thèse concerne principalement deux objets classiques étroitement liés: d'une part la variété des polynômes complexes unitaires de degré $d>1$ à une variable, et à racines simples (donc de discriminant différent de zéro), et d'autre part, les groupes de tresses d'Artin avec d brins. Le travail présenté dans cette thèse propose une nouvelle approche permettant des calculs cohomologiques explicites à coefficients dans n'importe quel faisceau. En vue de calculs cohomologiques explicites, il est souhaitable d'avoir à sa disposition un bon recouvrement au sens de Čech. L'un des principaux objectifs de cette thèse est de construire un tel recouvrement basé sur des graphes (appelés signatures) qui rappellent les `dessins d'enfant' et qui sont associées aux polynômes complexes classifiés par l'espace de polynômes. Cette décomposition de l'espace de polynômes fournit une stratification semi-algébrique. Le nombre de composantes connexes de chaque strate est calculé dans le dernier chapitre ce cette thèse. Néanmoins, cette partition ne fournit pas immédiatement un recouvrement adapté au calcul de la cohomologie de Čech (avec n'importe quels coefficients) pour deux raisons liées et évidentes: d'une part les sous-ensembles du recouvrement ne sont pas ouverts, et de plus ils sont disjoints puisqu'ils correspondent à différentes signatures. Ainsi, l'objectif principal du chapitre 6 est de ``corriger'' le recouvrement de départ afin de le transformer en un bon recouvrement ouvert, adapté au calcul de la cohomologie Čech. Cette construction permet ensuite un calcul explicite des groupes de cohomologie de Čech à valeurs dans un faisceau localement constant. / This thesis mainly concerns two closely related classical objects: on the one hand, the variety of unitary complex polynomials of degree $ d> 1 $ with a variable, and with simple roots (hence with a non-zero discriminant), and on the other hand, the $d$ strand Artin braid groups. The work presented in this thesis proposes a new approach allowing explicit cohomological calculations with coefficients in any sheaf. In order to obtain explicit cohomological calculations, it is necessary to have a good cover in the sense of Čech. One of the main objectives of this thesis is to construct such a good covering, based on graphs that are reminiscent of the ''dessins d'enfants'' and which are associated to the complex polynomials. This decomposition of the space of polynomials provides a semi-algebraic stratification. The number of connected components in each stratum is counted in the last chapter of this thesis. Nevertheless, this partition does not immediately provide a ''good'' cover adapted to the computation of the cohomology of Čech (with any coefficients) for two related and obvious reasons: on the one hand the subsets of the cover are not open, and moreover they are disjoint since they correspond to different signatures. Therefore, the main purpose of Chapter 6 is to ''correct'' the cover in order to transform it into a good open cover, suitable for the calculation of the Čech cohomology. It is explicitly verified that there is an open cover such that all the multiple intersections are contractible. This allows an explicit calculation of cohomology groups of Čech with values in a locally constant sheaf.

We describe a method for computing rank (and determining quasipositivity) in the free group using dynamic programming. The algorithm is adapted to computing upper bounds on the rank for braids. We test our method on a table of knots by identifying quasipositive knots and calculating the ribbon genus. We consider the possibility that rank is not theoretically computable and prove some partial results that would classify its computational complexity. We then present a method for effectively brute force searching band presentations of small rank and conjugate length.

Hair, as a vital component of the human's appearance, plays an important role in producing digital characters. However, the generation of realistic hairstyles usually needs professional digital artists and/or complex hardware, and the procedure is often time-consuming due to its numerous numbers, and diverse hairstyles. Thus, automatic capture of real-world hairstyles with easy input can greatly benefit the production pipeline. State-of-the-art 3D hair modeling systems require either multi-view images or a single- view image with complementary synthetic 3D hair models. For the multi-view image based 3D hair reconstruction, the capture systems are often made of a large number of cameras, projectors, light sources, and are usually in the indoor environment, which prevents popular use of the methods. On the contrary, single-view image based methods only use simple capture devices, e.g.; a handheld camera. However, a front view containing a face is often required and the resulting 3D hair strand reconstruction quality is compromised. Meanwhile, several hairstyles can not be easily modeled, such as braids and kinky hairstyles (afro-textured hairs), even though they are very common in real life. In this dissertation, we implement a single-view imaged based 3D hair modeling system, where our hair reconstruction is done through 2D hair analysis and 3D strands creation, which benefits from both traditional image processing techniques and the strengths of machine learning. Our 2D hair analysis is used to learn the attributes of input hairs, including 2D hair strands, detailed hairstyle patterns, and the corresponding parametric representation (which includes braids and kinky hairs), and braid structures. Simultaneously 3D hair strands are generated using deep-learning models. Our method is different from previous methods as our generated hair models can be modified by controlling the attributes and parameters we learned from the 2D hair recognition/analysis system. Our system does not require a face to be shown in the input image and to our best knowledge, our work is the first work that can reconstruct 3D braided hair and kinky hair given a single-view image. Qualitatively and quantitatively assessments indicate that our system can generate a variety of realistic 3D hairstyle models.

En undersökning kring flätandet som tradition, språklighet och keramisk metod.

No description available.

A configuration space is a space whose points represent the possible states of a given physical system. As such they appear naturally both in theoretical physics and technical applications. For an example of the former, in analytical mechanics, the Lagrangian and Hamiltonian formulations of classical mechanics depend heavily on the use of a physical system’s configuration space for the description of its kinematical and dynamical behavior, and importantly, its evolution in time. As an example of a technical application, consider robotics, where the space of possible configurations of the mechanical linkages that make up a robot is an important tool in motion planning. In this case it is of particular interest to study the singularities of these mechanical linkages, to see if a given configuration is singular or not. This can be done with the help of configuration spaces and their topological properties. Arguably, the simplest configuration space possible arises when the system is just a collection of point-like particles in a plane. Despite its simplicity, the corresponding configuration space has substantial complexity and is of great interest in mathematics, physics and technology: For instance, it arises naturally in the mathematical modelling of robots performing tasks in a warehouse. In this thesis we go through the mathematics necessary to study the behaviour of paths in this space, which corresponds to motions of the particles. We use the theory of groups, algebraic topology, and manifolds to examine the properties of the configuration space of point-like particles in a plane. An important role in the discussion will be played by braids, which are certain collections of curves, interlaced in three-space. They are connected to many different topics in algebra, geometry, and mathematical physics, such as representation theory, the Yang-Baxter equation and knot theory. They are also important in their own right. Here we focus on their relation to configurations of points.

Short, J-section columns fabricated from a textile composite are tested in axial compression to study the modes of failure with and without local buckling occuring.The textile preform architecture is a 2x2, 2-D triaxial braid with a yarn layup of [0 deg 18k/+-64 deg 6k] 39.7% axial. The preform was resin transfer molded with 3M PR500 epoxy resin. Finite element analyses (FEA) of the test specimens are conducted to assess intra- and inter- laminar progressive failure models. These progressive failure models are then implemented in a FEA of a circular fuselage frame of the same cross section and material for which test data was available. This circular frame test article had a nominal radius of 120 inches, a forty-eight degree included angle, and was subjected to a quasi-static, radially inward load, which represented a crash type loading of the frame. The short column test specimens were cut from some of the fuselage frames. The branched shell finite element model of the frame included geometric nonlinearity and contact of the load platen of the testing machine with the frame. Intralaminar progressive failure is based on a maximum in-plane stress failure criterion followed by a moduli degradation scheme. Interlaminar progressive failure was implemented using an interface finite element to model delamination initiation and the progression of delamination cracks. Inclusion of both the intra- and inter- laminar progressive failure models in the FEA of the frame correlated reasonably well with the load-displacement response from the test through several major failure events. / Master of Science

A graph configuration space is generated by the movement of a finite number of robots on a graph. These configuration spaces of points in a graph are topologically interesting objects. By using local, combinatorial properties, we define a new classification of graphs whose configuration spaces are pseudomanifolds with boundary. In algebraic topology, graph configuration spaces are closely related to classical braid groups, which can be described as fundamental groups of configuration spaces of points in the plane. We examine this relationship by finding a presentation for the fundamental group of one graph configuration space. / Master of Science

A representation is a group homomorphism whose image is a group of invertible matrices. Representations and their associated matrices are analyzed through well-established techniques from linear algebra. We characterize representations by a unique decomposition into irreducible representations just as we characterize the decomposition of matrices into their eigenspaces. Through the study of these representations, we uncover mathematical relationships that underlie groups with physical applications. Due to physical symmetries, we study how the irreducible representations of groups that embody the actions of even the most basic rotations are utilized in the computation of irreducible representations groups that reflect more complicated mechanics, like the Poincar\'e Group. Further, we utilize the representations of the abstract braid group to gain key insights into understanding the behavior of anyonic systems in quantum mechanics. Finally, we explore the behavior of Fibonacci anyons for ways to understand to illustrate the underlying braid relations.