The aim of this thesis is to present and discuss some mathematical and physical problems in the theory of weak, electromagnetic and strong interactions. Our main concern is a parallel development of mathematical and physical concepts and when it is possible, an attempt to bridge the abstract mathematical formulations with physical ideas. A central role in this thesis is played by a general construction scheme, which enables us to calculate explicitly all the mathematical quantities like matrix elements, Clebsch-Gordan series, Clebsch-Gordan coefficients which are necessary for a Grand Unification model construction. In this content, we have followed two basic principles: simplicity and applicability. To meet the first principle, all the construction methods developed are based on first principles and basic concepts of the Lie algebras and its representation theory, like roots and weights. Moreover, the requirement of applicability is met with the implementation of all the algorithms into computer programs. In the physical area, we have concentrated on the problem of mass. The lepton mass spectrum us studied in a theory of weak and electromagnetic interactions, while the mass problem of the SO(10) Grand Unified theory is analysed as a direct application of our Lie group construction scheme.

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Both man-made or natural objects contain repeated geometric elements that can be interpreted as primitive shapes. Plants, trees, living organisms or even crystals, showcase primitives that repeat themselves. Primitives are also commonly found in man-made environments because architects tend to reuse the same patterns over a building and typically employ simple shapes, such as rectangular windows and doors. During my PhD I studied geometric primitives from three points of view: their composition, simulation and autonomous discovery. In the first part I present a method to reverse-engineer the function by which some primitives are combined. Our system is based on a composition function template that is represented by a parametric surface. The parametric surface is deformed via a non-rigid alignment of a surface that, once converged, represents the desired operator. This enables the interactive modeling of operators via a simple sketch, solving a major usability gap of composition modeling. In the second part I introduce the use of a novel primitive for real-time physics simulations. This primitive is suitable to efficiently model volume-preserving deformations of rods but also of more complex structures such as muscles. One of the core advantages of our approach is that our primitive can serve as a unified representation to do collision detection, simulation, and surface skinning. In the third part I present an unsupervised deep learning framework to learn and detect primitives. In a signal containing a repetition of elements, the method is able to automatically identify the structure of these elements (i.e. primitives) with minimal supervision. In order to train the network that contains a non-differentiable operation, a novel multi-step training process is presented. / Graduate

In this thesis is initiated a more systematic geometric exploration of energy shaping. Most of the previous results have been dealt wih particular cases and neither the existence nor the space of solutions has been discussed with any degree of generality. The geometric theory of partial differential equations originated by Goldschmidt and Spencer in late 1960s is utilized to analyze the partial differential equations in energy shaping. The energy shaping partial differential equations are described as a fibered submanifold of a $ k $-jet bundle of a fibered manifold. By revealing the nature of kinetic energy shaping, similarities are noticed between the problem of kinetic energy shaping and some well-known problems in Riemannian geometry. In particular, there is a strong similarity between kinetic energy shaping and the problem of finding a metric connection initiated by Eisenhart and Veblen. We notice that the necessary conditions for the set of so-called $ \lambda $-equation restricted to the control distribution are related to the Ricci identity, similarly to the Eisenhart and Veblen metric connection problem. Finally, the set of $ \lambda $-equations for kinetic energy shaping are coupled with the integrability results of potential energy shaping. The procedure shows how a poor design of closed-loop metric can make it impossible to achieve any flexibility in the character of the possible closed-loop potential function. The integrability results of this thesis have been used to answer some interesting questions about the energy shaping. In particular, a geometric proof is provided which shows that linear controllability is sufficient for energy shaping of linear simple mechanical systems. Furthermore, it is shown that all linearly controllable mechanical control systems with one degree of underactuation can be stabilized using energy shaping feedback. The result is geometric and completely characterizes the energy shaping problem for these systems. Using the geometric approach of this thesis, some new open problems in energy shaping are formulated. In particular, we give ideas for relating the kinetic energy shaping problem to a problem on holonomy groups. Moreover, we suggest that the so-called Fakras lemma might be used for investigating the stabilization condition of energy shaping. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2009-09-02 12:12:55.051

Computer models of the geometry of the real world have a tendency to assume that the shapes and positions of objects can be described exactly. However, real surfaces are subject to irregularities such as bumps and undulations and so do not have perfect, mathematically definable forms. Engineers recognise this fact and so assign tolerance specifications to their designs. This thesis develops a representation of geometric tolerance and uncertainty in assemblies of rigid parts. Geometric tolerances are defined by tolerance zones which are regions in which the real surface must lie. Parts in an assembly can slop about and so their positions are uncertain. Toleranced parts and assemblies of toleranced parts are represented by networks of tolerance zones and datums. Each arc in the network represents a relationship implied by the tolerance specification or by a contact between the parts. It is shown how all geometric constraints can be converted to an algebraic form. Useful results can be obtained from the network of tolerance zones and datums. For example it is possible to determine whether the parts of an assembly can be guaranteed to fit together. It is also possible to determine the maximum slop that could occur in the assembly assuming that the parts satisfy the tolerance specification. Two applications of this work are (1) tolerance checking during design and (2) analysis of uncertainty build-up in a robot assembly plan. I n the former, a designer could check a proposed tolerance specification to make sure that certain design requirements are satisfied. In the latter, knowledge of manufacturing tolerances of parts being manipulated can be used to determine the constraints on the positions of the parts when they are in contact with other parts.

In this project we introduce the general idea of geometric quantization and demonstratehow to apply the process on a few examples. We discuss how to construct a line bundleover the symplectic manifold with Dirac’s quantization conditions and how to determine if we are able to quantize a system with the help of Weil’s integrability condition. To reducethe prequantum line bundle we employ real polarization such that the system does notbreak Heisenberg’s uncertainty principle anymore. From the prequantum bundle and thepolarization we construct the sought after Hilbert space.

published_or_final_version / abstract / Physics / Master / Master of Philosophy

We present new approaches to define and analyze geometric graphs. The region-counting distances, introduced by Demaine, Iacono and Langerman, associate to any pair of points (p,q) the number of items of a dataset S contained in a region R(p,q) surrounding (p,q). We define region-counting disks and circles, and study the complexity of these objects. Algorithms to compute epsilon-approximations of region-counting distances and approximations of region-counting circles are presented. We propose a definition of the locality for properties of geometric graphs. We measure the local density of graphs using the region-counting distances between pairs of vertices, and we use this density to define local properties of classes of graphs. We illustrate the locality by introducing the local diameter of geometric graphs: we define it as the upper bound on the size of the shortest path between any pair of vertices, expressed as a function of the density of the graph around those vertices. We determine the local diameter of several well-studied graphs such as the Theta-graph, the Ordered Theta-graph and the Skip List Spanner. We also show that various operations, such as path and point queries using geometric graphs as data structures, have complexities which can be expressed as local properties. A family of proximity graphs, called Empty Region Graphs (ERG) is presented. The vertices of an ERG are points in the plane, and two points are connected if their neighborhood, defined by a region, does not contain any other point. The region defining the neighborhood of two points is a parameter of the graph. This family of graphs includes several known proximity graphs such as Nearest Neighbor Graphs, Beta-Skeletons or Theta-Graphs. We concentrate on ERGs that are invariant under translations, rotations and uniform scaling of the vertices. We give conditions on the region defining an ERG to ensure a number of properties that might be desirable in applications, such as planarity, connectivity, triangle-freeness, cycle-freeness, bipartiteness and bounded degree. These conditions take the form of what we call tight regions: maximal or minimal regions that a region must contain or be contained in to make the graph satisfy a given property. We show that every monotone property has at least one corresponding tight region; we discuss possibilities and limitations of this general model for constructing a graph from a point set. We introduce and analyze sigma-local graphs, based on a definition of locality by Erickson, to illustrate efficient construction algorithm on a subclass of ERGs.

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