Surface offsetting is a valuable tool used in Computer Aided Design (CAD). An offset surface is a collection of points that are at a constant distance from another surface. An offset surface is created in CAD by selecting a surface and then specifying the distance that the surface is to be offset. If a surface is selected and a distance of D is specified, then the resulting offset surface should always be distance D from the original surface. The surface offset tool can be used for many applications. Modeling of composites or other layered manufacturing processes rely heavily on offset surfaces. Thin walled parts such as injection molded components are often modeled using the offset tool. Coating processes can also be modeled using the offset tool. Modern CAD systems have surface offsetting tools and are widely used throughout industry. However, CAD systems often fail to produce valid results. The process of surface offsetting can often result in surface self-intersections as well as surface degeneracies. Self-intersections and degeneracies make the surfaces invalid because they are physically impossible to create and CAD systems cannot use these invalid surfaces to represent solid bodies. The surface offset tool is therefore, one of the most challenging CAD tools to implement. The process of avoiding, detecting and removing surface self-intersections is extremely challenging. Much research in the field of CAD is dedicated to the detection and removal of surface self-intersections. However, the methods proposed in the literature all suffer from robustness problems. The purpose of this research is to introduce a method that creates valid offset surfaces and does not suffer from the problem of creating surface self-intersections. This method uses a numerical approach that approximates the offset surface and avoids all self-intersections. Because no self-intersections are created, the method does not require intersection tests of any kind. The value of this method is demonstrated by comparing its results with results from leading CAD systems.

x, 110 p. : ill. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. / This dissertation examines the existence of the self-intersection local time for a superprocess over a stochastic flow in dimensions d ≤ 3, which through constructive methods, gives a Tanaka like representation. The superprocess over a stochastic flow is a superprocess with dependent spatial motion, and thus Dynkin's proof of existence, which requires multiplicity of the log-Laplace functional, no longer applies. Skoulakis and Adler's method of calculating moments is extended to higher moments, from which existence follows. / Committee in charge: Hao Wang, Co-Chairperson, Mathematics; David Levin, Co-Chairperson, Mathematics; Christopher Sinclair, Member, Mathematics; Huaxin Lin, Member, Mathematics; Van Kolpin, Outside Member, Economics

Gegenstand der vorliegenden Dissertation ist das Verhalten sogenannter Selbstüberschneidungslokalzeiten $\\|\\ell_t\\|_p^p$ einer zeitstetigen Irrfahrt $(S_r)_r$ auf dem $d$-dimensionalen Gitter $\\Z^d$. Dabei ist für $p>1$ die Funktion $\\ell_t$ definiert durch $$ \\ell_t(z):=\\int_{0}^{t}\\1_{\\{S_r=z\\}}\\,\\d r\\nonumber $$ und bezeichnet die Aufenthaltsdauer der Irrfahrt bis zum Zeitpunkt $t\\in(0,\\infty)$ im Punkt $z\\in\\Z^d$. Ziel ist es, ein Prinzip großer Abweichungen zu entwickeln, d.h. das Hauptaugenmerk liegt auf dem asymptotischen Verhalten der Wahrscheinlichkeit, dass die Selbstüberschneidungslokalzeiten von ihrem Erwartungswert in erheblichem Maße nach oben abweichen. Mit anderen Worten; es soll das asymptotische Verhalten von $$ \\log\\P(\\|\\ell_t\\|_p^p\\geq r^p_t) $$ genau bestimmt werden, wobei $r_t^p\\in(0,\\infty)$ schneller als der Erwartungswert $\\E[\\|\\ell_t\\|_p^p]$ gegen unendlich streben soll. Dieses Verhalten kann dabei durch $t$, $r_t$ und eine gewisse Variationsformel beschrieben werden. Es wird sich herausstellen, dass es zwei Fälle zu betrachten gilt, in denen sich das probabilistisch beste Verhalten stark unterscheidet; die genaue Position des Phasenübergangs hängt dabei von den Parametern $p$ und $d$ ab. Im Vorgriff auf die Resultate kann man festhalten, dass die nötigen Selbstüberschneidungen in kleinen Dimensionen (im sogenannten subkritischen Fall) über einen großen Bereich erfolgen, aufgrund dessen bei der mathematischen Modellierung eine Reskalierung erforderlich ist. In hohen Dimensionen (dem sogenannten superkritischen Fall) ist dies nicht nötig, da die erforderlichen Selbstüberschneidungen innerhalb eines begrenzten Intervalles erfolgen. Das Interesse an der Untersuchung entstand unter anderem aus der Verbindung zu Modellen der statistischen Mechanik (parabolisches Anderson Modell) und zur Variationsanalysis. In der Vergangenheit wurde eine Vielzahl an Methoden benutzt, um dieses Problem zu lösen. In der vorliegenden Dissertation soll die sogenannte Momentenmethode bestmöglich ausgereizt werden und es wird gezeigt, welche Ergebnisse damit möglich sind.

In this thesis we study the problem of determining the possible cobordism types of r-fold self-intersection manifolds associated to self-transverse immersions f: M^{n-k} -> \R^n for certain values of n, k, and r. Namely, we study the double-point self-intersection manifolds of immersions M^{n+2} -> \R^{2n+2} and M^{n+4} -> \R^{2n+4}, focusing on the case when $n$ is even. In the case of self-transverse immersions f : M^{n+2} -> \R^{2n+2}, we see that when n is even the double-point self-intersection manifold is a boundary, which is a result originally due to Szucs. In the case of self-transverse immersions f : M^{n+4} -> \R^{2n+4}, we show than when n is even the double-point self-intersection manifold is either a boundary or cobordant to RP^2 x RP^2, which is a new result. We then show that for even n such that the binary expansion of n+4 contains 5 or more 1s, the double-point self-intersection manifold of a self-transverse immersion M^{n+4} -> \R^{2n+4} is necessarily a boundary. We also survey the case when n is odd. We also set up and study the complex versions of the above problems: self-transverse immersions f : M^{2k+2} -> \R^{4k+2} and f : M^{2k+4} -> \R^{4k+4} of stably complex manifolds with a given complex structure on the normal bundle of f$. In these cases, the double-point self-intersection manifold L associated to the immersion inherits a stably complex structure, and we attempt to determine which complex cobordism classes of stably complex manifolds may arise in this way. This is all new work. In the case of self-transverse complex immersions f : M^{2k+2} -> \R^{4k+2}, we show that the first normal Chern number of the double-point self-intersection manifold is a multiple of 2^{\lambda_{k+1}} for some integer \lambda_{k+1}, and provide upper and lower bounds for the value of \lambda_{k+1}. We also determine the exact value of \lambda_{k+1} in certain cases. In the case of self-transverse complex immersions f : M^{2k+4} -> \R^{4k+4}, we identify a large class of stably complex manifolds that may arise as the double-point self-intersection manifold of such an immersion and also identify a class of manifolds that may not. Additionally, in both cases we identify a necessary (and sometimes sufficient) condition for a stably complex manifold of the appropriate dimension to admit a complex immersion of the appropriate codimension.

Estuda-se numericamente uma placa circular homogênea com furo centrado sob estado plano de deformação. A placa está fixa ao longo do contorno interno e está sob compressão radial uniforme ao longo do contorno externo. O material da placa é elástico-linear e anisotrópico. Apresenta-se a solução analítica do problema, a qual satisfaz as equações governantes de equilíbrio, no contexto da elasticidade linear clássica. Esta solução prediz o comportamento espúrio da auto-intersecção em uma região central da placa. Para evitar este comportamento, utiliza-se uma teoria que propõe encontrar um campo de deslocamento que minimize a energia potencial total do corpo sujeito à restrição de injetividade local para o campo da deformação correspondente. Esta teoria, juntamente com o método das penalidades interiores, permite encontrar uma solução numérica que preserva a injetividade. Esta solução corresponde a um campo de deslocamento radialmente simétrico. Estuda-se a possibilidade de encontrar uma solução rotacionalmente simétrica do problema restrito, em que o campo de deslocamento possua as componentes radial e tangencial, ambas funções somente do raio. Os resultados desta última modelagem mostram que a componente tangencial é nula, indicando que o campo de deslocamento é, de fato, radialmente simétrico. Mostra-se também que a solução do problema do anel converge para a solução do problema de um disco sem furo à medida que o raio do furo tende a zero. / This work concerns a numerical study of a homogeneous circular plate with a centered hole that is under a state of plane strain. The plate is fixed at its inner surface and is under uniform radial compression at its outer surface. The plate is linear, elastic, and anisotropic. An analytical solution for this problem, which satisfies the governing equations of equilibrium, is presented in the context of classical linear elasticity. This solution predicts the spurious behavior of self-intersection in a central region of the plate. To avoid this behavior, a constrained minimization theory is used. This theory concerns the search for a displacement field that minimizes the total potential energy of the body, which is a quadratic functional from the classical linear theory, subjected to the constraint of local injectivity for the associated deformation field. This theory together with an interior penalty method and a standard finite element methodology yield a numerical solution, which is radially symmetric, that preserves injectivity. Here, it is investigated the possibility of finding a rotationally symmetric solution to the constrained problem; one for which the associated displacement field has radial and tangential components, which are both functions of the radius only. The numerical results show, however, that the tangential component is zero. It is also shown that, as the radius of the hole tends to zero, the corresponding sequence of solutions tends to the solution of a solid disk.

Estuda-se numericamente uma placa circular homogênea com furo centrado sob estado plano de deformação. A placa está fixa ao longo do contorno interno e está sob compressão radial uniforme ao longo do contorno externo. O material da placa é elástico-linear e anisotrópico. Apresenta-se a solução analítica do problema, a qual satisfaz as equações governantes de equilíbrio, no contexto da elasticidade linear clássica. Esta solução prediz o comportamento espúrio da auto-intersecção em uma região central da placa. Para evitar este comportamento, utiliza-se uma teoria que propõe encontrar um campo de deslocamento que minimize a energia potencial total do corpo sujeito à restrição de injetividade local para o campo da deformação correspondente. Esta teoria, juntamente com o método das penalidades interiores, permite encontrar uma solução numérica que preserva a injetividade. Esta solução corresponde a um campo de deslocamento radialmente simétrico. Estuda-se a possibilidade de encontrar uma solução rotacionalmente simétrica do problema restrito, em que o campo de deslocamento possua as componentes radial e tangencial, ambas funções somente do raio. Os resultados desta última modelagem mostram que a componente tangencial é nula, indicando que o campo de deslocamento é, de fato, radialmente simétrico. Mostra-se também que a solução do problema do anel converge para a solução do problema de um disco sem furo à medida que o raio do furo tende a zero. / This work concerns a numerical study of a homogeneous circular plate with a centered hole that is under a state of plane strain. The plate is fixed at its inner surface and is under uniform radial compression at its outer surface. The plate is linear, elastic, and anisotropic. An analytical solution for this problem, which satisfies the governing equations of equilibrium, is presented in the context of classical linear elasticity. This solution predicts the spurious behavior of self-intersection in a central region of the plate. To avoid this behavior, a constrained minimization theory is used. This theory concerns the search for a displacement field that minimizes the total potential energy of the body, which is a quadratic functional from the classical linear theory, subjected to the constraint of local injectivity for the associated deformation field. This theory together with an interior penalty method and a standard finite element methodology yield a numerical solution, which is radially symmetric, that preserves injectivity. Here, it is investigated the possibility of finding a rotationally symmetric solution to the constrained problem; one for which the associated displacement field has radial and tangential components, which are both functions of the radius only. The numerical results show, however, that the tangential component is zero. It is also shown that, as the radius of the hole tends to zero, the corresponding sequence of solutions tends to the solution of a solid disk.

Gegenstand der vorliegenden Dissertation ist das Verhalten sogenannter Selbstüberschneidungslokalzeiten $\\|\\ell_t\\|_p^p$ einer zeitstetigen Irrfahrt $(S_r)_r$ auf dem $d$-dimensionalen Gitter $\\Z^d$. Dabei ist für $p>1$ die Funktion $\\ell_t$ definiert durch $$ \\ell_t(z):=\\int_{0}^{t}\\1_{\\{S_r=z\\}}\\,\\d r\\nonumber $$ und bezeichnet die Aufenthaltsdauer der Irrfahrt bis zum Zeitpunkt $t\\in(0,\\infty)$ im Punkt $z\\in\\Z^d$. Ziel ist es, ein Prinzip großer Abweichungen zu entwickeln, d.h. das Hauptaugenmerk liegt auf dem asymptotischen Verhalten der Wahrscheinlichkeit, dass die Selbstüberschneidungslokalzeiten von ihrem Erwartungswert in erheblichem Maße nach oben abweichen. Mit anderen Worten; es soll das asymptotische Verhalten von $$ \\log\\P(\\|\\ell_t\\|_p^p\\geq r^p_t) $$ genau bestimmt werden, wobei $r_t^p\\in(0,\\infty)$ schneller als der Erwartungswert $\\E[\\|\\ell_t\\|_p^p]$ gegen unendlich streben soll. Dieses Verhalten kann dabei durch $t$, $r_t$ und eine gewisse Variationsformel beschrieben werden. Es wird sich herausstellen, dass es zwei Fälle zu betrachten gilt, in denen sich das probabilistisch beste Verhalten stark unterscheidet; die genaue Position des Phasenübergangs hängt dabei von den Parametern $p$ und $d$ ab. Im Vorgriff auf die Resultate kann man festhalten, dass die nötigen Selbstüberschneidungen in kleinen Dimensionen (im sogenannten subkritischen Fall) über einen großen Bereich erfolgen, aufgrund dessen bei der mathematischen Modellierung eine Reskalierung erforderlich ist. In hohen Dimensionen (dem sogenannten superkritischen Fall) ist dies nicht nötig, da die erforderlichen Selbstüberschneidungen innerhalb eines begrenzten Intervalles erfolgen. Das Interesse an der Untersuchung entstand unter anderem aus der Verbindung zu Modellen der statistischen Mechanik (parabolisches Anderson Modell) und zur Variationsanalysis. In der Vergangenheit wurde eine Vielzahl an Methoden benutzt, um dieses Problem zu lösen. In der vorliegenden Dissertation soll die sogenannte Momentenmethode bestmöglich ausgereizt werden und es wird gezeigt, welche Ergebnisse damit möglich sind.

Title: Billiard time machine Author: Jindřich Dolanský Department: Institute of Theoretical Physics Supervisor: doc. RNDr. Jiří Langer, CSc. Supervisor's e-mail address: Jiri.Langer@mff.cuni.cz Abstract: In this work we investigate a simple interacting system of an elastic particle in the non-relativistic spacetime with a nontrivial causal structure realized by a worm- hole with a time shift. We require that standard local physical laws hold, and search for their globally consistent solutions, i.e, we assume the validity of the principle of self-consistency. If there were nontrivial set of initial conditions which would violate this principle, the system would be logically inconsistent. We show that the investigated system is not inconsistent in this sense, i.e., that all standard initial conditions have a globally consistent evolution. Even for the so called dangerous initial conditions which threaten to result into the paradoxical situation a consistent solution exists. In this case, the paradoxical collision-free trajectory is superseded by a special consistent self-colliding trajectory. Moreover, we demonstrate that more than one globally consistent evolution exists for a wide class of initial conditions. Thus, the evolution of the described system is not unique due to the nontrivial causal structure...