In his 1947 work "Theory of Braids" Emil Artin asked whether the braid group remained unchanged when one considered classes of braids under linkhomotopy, allowing each strand of a braid to pass through itself but not through other strands. We generalize Artin's question to string links over orientable surface M and show that under link-homotopy surface string links form a group PBn(M), which is isomorphic to a quotient of the surface pure braid group PBn(M). Surface braid groups and their properties are an area of active research by González-Meneses, Paris and Rolfsen, Goçalves and Guaschi, and our work explores the geometric and visual beauty of this subject. We compute a presentation of PBn(M) in terms of the generators and relations and discuss the orderability of the group in the case when the surface in question is a unit disk D. / Science, Faculty of / Mathematics, Department of / Graduate

This work constructs and compares different kinds of categorified cohomology of a locally contractible topological space X. Fix a commutative ring k of characteristic 0 and also denote by k the differential graded category with a single object and endomorphisms k. In the Morita model structure k is weakly equivalent to the category of perfect chain complexes over k. We define and compute derived global sections of the constant presheaf k considered as a presheaf of dg-categories with the Morita model structure. If k is a field this is done by showing there exists a suitable local model structure on presheaves of dg-categories and explicitly sheafifying constant presheaves. We call this categorified Cech cohomology Morita cohomology and show that it can be computed as a homotopy limit over a good (hyper)cover of the space X. We then prove a strictification result for dg-categories and deduce that under mild assumptions on X Morita cohomology is equivalent to the category of homotopy locally constant sheaves of k-complexes on X. We also show categorified Cech cohomology is equivalent to a category of ∞-local systems, which can be interpreted as categorified singular cohomology. We define this category in terms of the cotensor action of simplicial sets on the category of dg-categories. We then show ∞-local systems are equivalent to the category of dg-representations of chains on the loop space of X and find an explicit method of computation if X is a CW complex. We conclude with a number of examples.

We investigate generalizations of coherent states as a means of representing the dynamics of excitations of the superconducting ground state. We also analyse the propagation of generalized coherent state wave packets under the Bogoliubov-de Gennes Hamiltonian. The excitations of the superconducting ground state are superpositions of electron and hole quasi-particles described by the Bogoliubov-de Gennes equations, that can only exist at energies outside the band gap. A natural generalization relevant to the excitations of the superconducting ground state is the tensor product of canonical and spin coherent states. This state will quickly become de-localized on phase space under evolution by the Bogoliubov-de Gennes Hamiltonian due to the opposite velocities of the quasi-spin components. We therefore define the electron-hole coherent states which remain localised on phase space over longer times. We show that the electron-hole coherent states though entangled retain many defining features of coherent states. We analyse the propagation of both product and electron hole coherent states in a superconductor with a spatially homogeneous superconducting band gap. The dispersion relation indicates that wavepackets defined on the band gap have a zero group velocity, but we will show that interference effects can create states on the band gap that propagate at the Fermi velocity. We also consider the two semiclassical, short wavelength regimes, hbar→0$ and the large Fermi energy limit mu→infinity. In general these limits produce behaviour analogous to the canonical coherent states except for isolated cases. Finally we analyse the dynamics of the Andreev Reflection of a Gaussian wavepacket incident on a discontinuous normal-superconducting interface. We show that restricting the energy bandwidth of the incident state inside the superconducting band gap precludes the wavepacket from fully entering the superconducting region. We again consider the two semiclassical regimes.

This dissertation applies policy improvement and successive approximation or value iteration to a general class of Markov decision processes with discounted costs. In particular, a class of Markov decision processes, called piecewise-linear, is studied. Piecewise-linear processes are characterized by the property that the value function of a process observed for one period and then terminated is piecewise-linear if the terminal reward function is piecewise-linear. Partially observable Markov decision processes have this property. It is shown that there are e-optimal piecewise-linear value functions and piecewise-constant policies which are simple. Simple means that there are only finitely many pieces, each of which is defined on a convex polyhedral set. Algorithms based on policy improvement and successive approximation are developed to compute simple approximations to an optimal policy and the optimal value function. / Business, Sauder School of / Graduate

The main purpose of this paper is the study of transformations in topological space and relationships between special types of transformations.

abstract: Investigation into research literature was conducted in order to understand the impacts of traditional concrete construction and explore recent advancements in 3D printing technologies and methodologies. The research project focuses on the relationship between computer modeling, testing, and verification to reduce concrete usage in flexural elements. The project features small-scale and large-scale printing applications modelled by finite element analysis software and printed for laboratory testing. The laboratory testing included mortar cylinder testing, digital image correlation (DIC), and four pointbending tests. Results demonstrated comparable performance between casted, printed solid, and printed optimized flexural elements. Results additionally mimicked finite element models regarding failure regions. / Dissertation/Thesis / Masters Thesis Engineering 2020

Using topology for feature analysis in flow fields faces several problems. First of all, not all features can be detected using topology based methods. Second, while in flow feature analysis the user is interested in a quantification of feature parameters like position, size, shape, radial velocity and other parameters of feature models, many of these parameters can not be determined using topology based methods alone. Additionally, in some applications it is advantageous to regard the vector field as a superposition of several, possibly simple, features. As topology based methods are quite sensitive to superposition effects, their precision and usability is limited in these cases. In this paper, topology based analysis and visualization of flow fields is estimated and compared to other feature based approaches demonstrating these problems.

"A topological ring can be defined as a Hausdorff space which is a ring whose operations are continuous in both variables simultaneously. The purpose of this paper is to present a development of a branch of topological rings, normed rings. Chapter I presents the algebraic concepts and Chapter II, the topological concepts which are basic to the understanding of the remaining chapters. Chapter III is devoted to theory of Banach spaces and linear transformations. Chapter IV gives a treatment of normed rings, particularly the rings of continuous functions of a compact topological space. The latter part of this chapter presents a characterization of these rings"--Introduction. / Typescript. / "August, 1955." / "Submitted to the Graduate Council of Florida State University in partial fulfillment of the requirements for the degree of Master of Arts." / Advisor: M. J. Walsh, Professor Directing Paper. / Includes bibliographical references (leaf 51).

In 1984, T. Kobayashi gave a classification of the genus two 3-manifolds with a nontrivial torus decomposition. The intent of this study is to extend this classification to the genus two, torally bounded 3-manifolds with a separating non-trivial torus decomposition. These 3-manifolds are also known as the tunnel-1 generalized satellite knot exteriors. The main result of the study is a full decomposition of the exterior of a tunnel-1 satellite knot in an arbitrary 3-manifold. Several corollaries are drawn from this classification. First, Schubert's 1953 results regarding the existence and uniqueness of a core component for satellite knots in the 3-sphere is extended to tunnel-1 satellite knots in arbitrary 3-manifolds. Second, Morimoto and Sakuma's 1991 classification of tunnel-1 satellite knots in the 3-sphere is extended to a classification of the tunnel-1 satellite knots in lens spaces. Finally, for these knot exteriors, a result of Eudave-Muñoz in 1994 regarding the relative position of tunnels and decomposing tori is recovered.

Indiana University-Purdue University Indianapolis (IUPUI) / Topology Optimization is a technique that allows for the obtaining structures which maximize the use of the material. This is done by intelligently deciding the binary distribution of solid material and void, in a discretized given space. Several researchers have provided methods to tackle binary topology optimization. New ef- forts are focused on extending the application for multi-phase optimizations. At the industrial level, several components designed are made up of more than one material to reduce weight and production costs. The objective of this work is to implement the algorithm of Hybrid Cellular Automaton for multi-material topology optimiza- tion. The commonly used interpolation rule SIMP, which allows to relate the design variables to the mechanical properties of the material, is replaced by ordered SIMP interpolation function. The multiple volume constraints are applied sequentially, starting with the most elastic material. When a constraint is satisfied, the elements assigned to this material remain passive by a defined number of iterations to promote the convergence of the solution. Examples are shown for static and dynamic loads. The work demonstrates the versatility of algorithms based on control systems to solve problems of multiple phases and transient response fields.