by Wong Hon-chuen. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 83-87). / Abstracts in English and Chinese. / Acknowledgements --- p.i / List of Figures --- p.v / List of Table --- p.viii / Abstract --- p.ix / Contents / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Algorithm Outline --- p.7 / Chapter 2.1 --- Assumptions --- p.7 / Chapter 2.2 --- Algorithm Outline --- p.8 / Chapter 3 --- Obstacle Detection --- p.11 / Chapter 3.1 --- Introduction --- p.11 / Chapter 3.2 --- Image Processing --- p.14 / Chapter 3.3 --- Coordinate Transformation --- p.14 / Chapter 3.4 --- Example --- p.20 / Chapter 4 --- Real-time Construction of Configuration Space --- p.22 / Chapter 4.1 --- Introduction --- p.22 / Chapter 4.2 --- Configuration Space --- p.23 / Chapter 4.3 --- Type-A Contact --- p.26 / Chapter 4.4 --- Type-B Contact --- p.27 / Chapter 4.5 --- Inverse Mapping Method --- p.29 / Chapter 4.6 --- Simulation --- p.31 / Chapter 5 --- Motion Planning and Re-Construction of C-space --- p.34 / Chapter 5.1 --- Introduction --- p.34 / Chapter 5.2 --- Path Planning --- p.36 / Chapter 5.3 --- Update of C-space --- p.41 / Chapter 5.4 --- Re-planning of Robot Path --- p.44 / Chapter 6 --- Implementation and Experiments --- p.55 / Chapter 6.1 --- Introduction --- p.55 / Chapter 6.2 --- Architecture of the Mobile Robot System --- p.55 / Chapter 6.3 --- Algorithm Implementation --- p.56 / Chapter 6.4 --- Experiment --- p.58 / Chapter 6.4.1 --- Experiment on a Fixed Unknown Environment --- p.58 / Chapter 6.4.2 --- Experiment on a Dynamic Unknown Environment --- p.70 / Chapter 7 --- Conclusions --- p.81 / References --- p.83

A hierarchical representation for configuration space is presented, along with an algorithm for searching that space for collision-free paths. The detail of the algorithm are presented for polygonal obstacles and a moving object with two translational and one rotational degrees of freedom.

Thesis (Ph. D.)--University of Oregon, 2007. / Typescript. Includes vita and abstract. Includes bibliographical references (leaf 76). Also available for download via the World Wide Web; free to University of Oregon users.

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The main theme of this thesis is higher algebraic structures that come from operads and props. The first chapter is an introduction to the mathematical framework needed for the content of this thesis. The chapter does not contain any new results. The second chapter is concerned with the construction of a configuration space model for a particular 2-colored differential graded operad encoding the structure of two A∞ algebras with two A∞ morphisms and a homotopy between the morphisms. The cohomology of this operad is shown to be the well-known 2-colored operad encoding the structure of two associative algebras and of an associative algebra morphism between them. The third chapter is concerned with deformation quantization of (potentially) infinite dimensional (quasi-)Poisson manifolds. Our proof employs a variation on the transcendental methods pioneered by M. Kontsevich for the finite dimensional case. The first proof of the infinite dimensional case is due to B. Shoikhet. A key feature of the first proof is the construction of a universal L∞ structure on formal polyvector fields. Our contribution is a simplification of B. Shoikhet proof by considering a more natural configuration space and a simpler choice of propagator. The result is also put into a natural context of the dg Lie algebras coming from graph complexes; the L∞ structure is proved to come from a Maurer-Cartan element in the oriented graph complex. The fourth chapter also deals with deformation quantization of (quasi-)Poisson structures in the infinite dimensional setting. Unlike the previous chapter, the methods used here are purely algebraic. Our main theorem is the possibility to deformation quantize quasi-Poisson structures by only using perturbative methods; in contrast to the transcendental methods employed in the previous chapter. We give two proofs of the theorem via the theory of dg operads, dg properads and dg props. We show that there is a dg prop morphism from a prop governing star-products to a dg prop(erad) governing (quasi-)Poisson structures. This morphism gives a theorem about the existence of a deformation quantization of (quasi-)Poisson structure. The proof proceeds by giving an explicit deformation quantization of super-involutive Lie bialgebras and then lifting that to the dg properad governing quasi-Poisson structures. The prop governing star-products was first considered by S.A. Merkulov, but the properad governing quasi-Poisson structures is a new construction. The second proof of the theorem employs the Merkulov-Willwacher polydifferential functor to transfer the problem of finding a morphism of dg props to that of finding a morphism of dg operads.We construct an extension of the well known operad of A∞ algebras such that the representations of it in V are equivalent to an A∞ structure on V[[ħ]]. This new operad is also a minimal model of an operad that can be seen as the extension of the operad of associative algebras by a unary operation. We give an explicit map of operads from the extended associative operad to the operad we get when applying the Merkulov-Willwacher polydifferential functor to the properad of super-involutive Lie bialgebras. Lifting this map so as to go between their respective models gives a new proof of the main theorem.

We prove that the non-k-equal configuration space of a graph has a discretized model, analogous to the discretized model for configurations on graphs. We apply discrete Morse theory to the latter to give an explicit combinatorial formula for the ranks of homology and cohomology of configurations of two points on a tree. We give explicit presentations for homology and cohomology classes as well as pairings for ordered and unordered configurations of two and three points on a few simple trees, and show that the first homology group of ordered and unordered configurations of two points in any tree is generated by the first homology groups of configurations of two points in three particular graphs, K_{1,3}, K_{1,4}, and the trivalent tree with 6 vertices and 2 vertices of degree 3, via graph embeddings.

Em um curso tradicional de Química Quântica, os modelos estudados para ilustrar algumas das ferramentas da Mecânica Quântica relevantes para a compreensão da estrutura da matéria no nível atômico e molecular são apresentados no que se convencionou chamar, numa apresentação mais formal, de representação da posição. Nesta representação, o estado do sistema é descrito por uma função de onda dependente das posições das partículas que o constituem. Isso leva o estudante de química a uma concepção distorcida de que na natureza os estados dos sistemas devem ser obrigatoriamente descritos em termos das posições de suas partículas. Aqui mostramos que essa não é a única forma de abordar quanticamente a descrição de um sistema físico. Uma outra forma é servir-se da representação do momento, onde a função de estado depende do momento de cada uma das partículas. Existem dois caminhos para se obter as funções de estado na representação do momento. Uma delas é fazer-se a transformada de Fourier das funções de estado na representação da posição, e a outra é buscar resolver a equação de Schrödinger diretamente na representação do momento. Neste trabalho, foram discutidas essas duas abordagens para os modelos mais comuns estudados num curso de Química Quântica, sendo eles: a partícula na caixa, o oscilador harmônico, o átomo de hidrogênio, o átomo de hélio, o íon-molécula de hidrogênio (H2 +) e a molécula de hidrogênio (H2). Buscou-se mostrar uma perspectiva diferente na descrição desses sistemas bem como uma abordagem matemática distinta da usual e, também, as dificuldades, principalmente matemáticas, de sua aplicação e ensino num curso de Química Quântica. / In a conventional course in Quantum Chemistry, the models usually presented to illustrate the use of some quantum mechanical tools that are relevant for a comprehension of the structure of matter at the atomic and molecular levels are approached in a way that has been termed, in a more formal presentation, as position representation. In this representation, the state of a system is described by a wavefunction that is dependent on the positions of all particles that define the system. As a consequence of this presentation, chemistry students assimilate a distorted conception that in nature the state of a system must necessarily be described in terms of particles positions. Here we show that this is not the only way to approach quantum mechanically the description of a physical system. In an alternative way, known as momentum representation, the state function is expressed in a way that it is explicitly dependent on the momentum of each particle. There are two ways to obtain wavefunctions in the momentum representation. In of them, use is made of a Fourier transform of the wavefunctions in the position representation, and in the other one, an attempt is made to solve Schroedinger´s equation directly in the momentum representation. In this work, we have discussed these two approaches by examining the most common models studied in a Quantum Chemistry course, namely: the particle in a box, the harmonic oscillator, the hydrogen atom, the helium atom, the hydrogen molecular ion, and the hydrogen molecule. We have tried to show a different physical perspective in the description of these systems as well as a distinct mathematical approach than the usual one, and also the difficulties, mainly mathematical, of applying and teaching this representation in a Quantum Chemistry course.

The main purpose of this thesis is to calculate the integral cohomology ring of the symmetric square of quaternionic projective space, which has been an open problem since computations with symmetric squares were first proposed in the 1930's. The geometry of this particular case forms an essential part of the thesis, and unexpected results concerning two universal Pin(4) bundles are also included. The cohomological computations involve a commutative ladder of long exact sequences, which arise by decomposing the symmetric square and the corresponding Borel space in compatible ways. The geometry and the cohomology of the configuration space of unordered pairs of distinct points in quaternionic projective space, and of the Thom space MPin(4), also feature, and seem to be of independent interest.

Em um curso tradicional de Química Quântica, os modelos estudados para ilustrar algumas das ferramentas da Mecânica Quântica relevantes para a compreensão da estrutura da matéria no nível atômico e molecular são apresentados no que se convencionou chamar, numa apresentação mais formal, de representação da posição. Nesta representação, o estado do sistema é descrito por uma função de onda dependente das posições das partículas que o constituem. Isso leva o estudante de química a uma concepção distorcida de que na natureza os estados dos sistemas devem ser obrigatoriamente descritos em termos das posições de suas partículas. Aqui mostramos que essa não é a única forma de abordar quanticamente a descrição de um sistema físico. Uma outra forma é servir-se da representação do momento, onde a função de estado depende do momento de cada uma das partículas. Existem dois caminhos para se obter as funções de estado na representação do momento. Uma delas é fazer-se a transformada de Fourier das funções de estado na representação da posição, e a outra é buscar resolver a equação de Schrödinger diretamente na representação do momento. Neste trabalho, foram discutidas essas duas abordagens para os modelos mais comuns estudados num curso de Química Quântica, sendo eles: a partícula na caixa, o oscilador harmônico, o átomo de hidrogênio, o átomo de hélio, o íon-molécula de hidrogênio (H2 +) e a molécula de hidrogênio (H2). Buscou-se mostrar uma perspectiva diferente na descrição desses sistemas bem como uma abordagem matemática distinta da usual e, também, as dificuldades, principalmente matemáticas, de sua aplicação e ensino num curso de Química Quântica. / In a conventional course in Quantum Chemistry, the models usually presented to illustrate the use of some quantum mechanical tools that are relevant for a comprehension of the structure of matter at the atomic and molecular levels are approached in a way that has been termed, in a more formal presentation, as position representation. In this representation, the state of a system is described by a wavefunction that is dependent on the positions of all particles that define the system. As a consequence of this presentation, chemistry students assimilate a distorted conception that in nature the state of a system must necessarily be described in terms of particles positions. Here we show that this is not the only way to approach quantum mechanically the description of a physical system. In an alternative way, known as momentum representation, the state function is expressed in a way that it is explicitly dependent on the momentum of each particle. There are two ways to obtain wavefunctions in the momentum representation. In of them, use is made of a Fourier transform of the wavefunctions in the position representation, and in the other one, an attempt is made to solve Schroedinger´s equation directly in the momentum representation. In this work, we have discussed these two approaches by examining the most common models studied in a Quantum Chemistry course, namely: the particle in a box, the harmonic oscillator, the hydrogen atom, the helium atom, the hydrogen molecular ion, and the hydrogen molecule. We have tried to show a different physical perspective in the description of these systems as well as a distinct mathematical approach than the usual one, and also the difficulties, mainly mathematical, of applying and teaching this representation in a Quantum Chemistry course.