Morphing aircraft offer many benefits. However, the design of stiff yet flexible structures still provides many obstacles to fully exploring and realizing morphing structures. Due to this, many morphing challenges remain open. Topology optimization is a type of structural optimization that optimizes the material layout of a structure based on imposed boundary conditions and load paths. This type of optimization is promising for solving morphing design challenges but many of the optimized structures are not suited for traditional manufacturing and material arrangements. Multi-material additive manufacturing is an emerging technology that can produce a single structure with many different materials integrated in custom geometries. This could be the solution to realizing topology optimized structures. Despite the rich amount of current research in morphing aircraft, many challenges still remain open and topology of morphing structures could provide the solution to these morphing challenges. / Master of Science / Morphing aircraft offer many benefits. However, the design of stiff yet flexible structures still provides many obstacles to fully exploring and realizing morphing structures. Due to this, many morphing challenges remain open. Topology optimization is a type of structural optimization that optimizes the material layout of a structure based on imposed boundary conditions and load paths. This type of optimization is promising for solving morphing design challenges but many of the optimized structures are not suited for traditional manufacturing and material arrangements. Multi-material additive manufacturing is an emerging technology that can produce a single structure with many different materials integrated in custom geometries. This could be the solution to realizing topology optimized structures. Despite the rich amount of current research in morphing aircraft, many challenges still remain open and topology of morphing structures could provide the solution to these morphing challenges.

This thesis covers improvements on the upperbounds for ropelength of a specific class of algebraic knots.

In algebraic topology, we work to classify objects. My research aims to build a better understanding of one important notion of classification of differentiable manifolds called cobordism. Cobordism is an equivalence relation, and the equivalence classes in cobordism form a graded ring, with operations disjoint union and Cartesian product. My dissertation studies this graded ring in two ways: 1. by attempting to find preferred class representatives for each class in the ring. 2. by computing the image of the ring under an interesting ring homomorphism called the Witten Genus.

This thesis investigates the topological reconstruction problem, which is inspired by the reconstruction conjecture in graph theory. We ask how much information about a topological space can be recovered from the homeomorphism types of its point-complement subspaces. If the whole space can be recovered up to homeomorphism, it is called reconstructible. In the first part of this thesis, we investigate under which conditions compact spaces are reconstructible. It is shown that a non-reconstructible compact metrizable space must contain a dense collection of 1-point components. In particular, all metrizable continua are reconstructible. On the other hand, any first-countable compactification of countably many copies of the Cantor set is non-reconstructible, and so are all compact metrizable h-homogeneous spaces with a dense collection of 1-point components. We then investigate which non-compact locally compact spaces are reconstructible. Our main technical result is a framework for the reconstruction of spaces with a maximal finite compactification. We show that Euclidean spaces ℝⁿ and all ordinals are reconstructible. In the second part, we show that it is independent of ZFC whether the Stone-Čech remainder of the integers, ω*, is reconstructible. Further, the property of being a normal space is consistently non-reconstructible. Under the Continuum Hypothesis, the compact Hausdorff space ω* has a non-normal reconstruction, namely the space ω*\{p} for a P-point p of ω*. More generally, the existence of an uncountable cardinal κ satisfying κ = κ^<κ implies that there is a normal space with a non-normal reconstruction. The final chapter discusses the Stone-Čech compactification and the Stone-Čech remainder of spaces ω*\{x}. Assuming the Continuum Hypothesis, we show that for every point x of ω*, the Stone-Čech remainder of ω*{x} is an ω₂-Parovičenko space of cardinality 2^{2^c} which admits a family of 2^c disjoint open sets. This implies that under 2^c = ω₂, the Stone-Čech remainders of ω*\{x} are all homeomorphic, regardless of which point x gets removed.

The goal of this paper is to establish the dependency of the topology of a simple Lie group, specifically any of the special linear groups, on its underlying group structure. The intimate relationship between a Lie group's topology and its algebraic structure dictates some necessary topological properties, such as second countability. However, the extent to which a Lie group's topology is an "algebraic phenomenon" is, to date, still not known.

This paper contains a discussion of topological dimension theory. Original proofs of theorems, as well as a presentation of theorems and proofs selected from Ryszard Engelking's Dimension Theory are contained within the body of this endeavor. Preliminary notation is introduced in Chapter I. Chapter II consists of the definition of and theorems relating to the small inductive dimension function Ind. Large inductive dimension is investigated in Chapter III. Chapter IV comprises the definition of covering dimension and theorems discussing the equivalence of the different dimension functions in certain topological settings. Arguments pertaining to the dimension o f Jn are also contained in Chapter IV.

This paper is an exposition of the theory of the hyperspaces 2^X and C(X) of a topological space X. These spaces are obtained from X by collecting the nonempty closed and nonempty closed connected subsets respectively, and are topologized by the Vietoris topology. The paper is organized in terms of increasing specialization of spaces, beginning with T1 spaces and proceeding through compact spaces, compact metric spaces and metric continua. Several basic techniques in hyperspace theory are discussed, and these techniques are applied to elucidate the topological structure of hyperspaces.

The results from this dissertation are a computation of ultrapowers by supercompactness measures and concepts related to such measures. The second chapter gives an overview of the basic ideas required to carry out the computations. Included are preliminary ideas connected to measures, and the supercompactness measures. Order type results are also considered in this chapter. In chapter III we give an alternate characterization of 2 using the notion of iterated ordinal measures. Basic facts related to this characterization are also considered here. The remaining chapters are devoted to finding bounds fwith arguments taking place both inside and outside the ultrapowers. Conditions related to the upper bound are given in chapter VI.

Engineers have utilized CAE technique as an analysis tool to refine the engineering design over decades. However, CAE alone is not the key to open the door for the final goal. In order to achieve the practical solution to the real-time engineering problem, we need to integrate CAD, CAE and optimization techniques into a single framework. / In the optimization algorithm part, apart from the general parametric steepest descent (ST) algorithm, we also study the least square (LSQ) based optimization algorithm. As a result, we can solve the problem arisen from the variant dimensional sizes of the different design variables by using the weighted sensitivity information. / In the problem of the structural optimizations, three categories of the approaches can be identified: size, shape and topology optimizations. For size optimization, explicit dimensions are usually chosen as the design variables, for example, the thickness of a beam or the diameter of a cylinder. For shape optimization, the shape related parameters of the geometrical boundary are always considered to be the design variables, like the positions of the control points for a Bezier curve. However, these two methods are lack of the capability to handle the topological changes of the geometry. On the contrary, topology optimization is the generalization of size and shape optimizations, which offers a more flexible and powerful tool to determine the best layout of the materials and the topology to the design problem, and it is becoming increasingly important in the conceptual design phase. In other words, topology optimization gives one the inspiration for the locations where we put holes to reach the best design. / In this thesis, we put forward the algebraic level set (ALS) model with the consideration of the constructive solid geometry (CSG) model so that it is consistent with half-space primitive concept in CSG. Based on general shape derivative, we propose the general shape design sensitivity analysis (SDSA) formulations for general geometric primitives that are represented implicitly, such as line and circle primitives in two-dimensional space and plane primitive in three-dimensional space. We then extend the relevant formulations into corresponding parametrically represented primitives as they are widely used in today's mainstream CAD systems. / The material density method and the boundary-variation method are the popular methods adopted in both academia and industrial community. Even though the former method is dominant in industry, the latter method is more preferable these years owing to its boundary description nature. Undoubtedly, the level set based method is the most promising technique of the boundary-variation type. Scientists successfully developed the optimization algorithms based on the level set method (LSM) in the past few years. With the implicit representation of the LSM, topological changes of the design can be handled easily and the geometrical complexity is then reserved. / The numerical examples for the design optimization problem are successfully implemented with both the implicit geometric representation (2D cases) and the parametric geometric representation (3D cases), which proves the feasibility of the proposed framework. The results show that both shape and topology optimizations of a design could be accomplished in a natural way. / The optimal result given by conventional topology optimization usually involves tedious post-processing to form CAD geometry. Using our parameterizations with basic primitives and the proposed optimization algorithms, we can deliver comparatively complicated shapes with rich topological information. Therefore, the detail design could be conducted directly later. / Zhang, Jiwei. / "December 2010." / Adviser: Yu Michael Wang. / Source: Dissertation Abstracts International, Volume: 73-04, Section: B, page: . / Thesis (Ph.D.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 119-129). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [201-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese.

Orientador: Elíris Cristina Rizziolli / Banca: Edivaldo Lopes da Silva / Banca: João Peres Vieira / Resumo: Neste trabalho tratamos do objeto matemático Grupo Topológico. Para este desenvolvimento, abordamos elementos básicos de Grupo e Espaço Topológico / Abstract: In this work we consider the mathematical object Topological Group. For this development, we discuss the basic elements of the Group and Topological Space / Mestre