Thesis (MA. (Mathematics)) -- University of Limpopo, 2012 / Topological structures called Yosida frames and related algebraic frames are studied in the realm of Pointfree Topology. It is shown that in algebraic frames regular elements are those for which compact elements are rather below the regular elements, and algebraic frames are regular if and only if every compact element is rather below itself if and only if the frame has the Finite Intersection Property (FIP) and each prime element is minimal. We also show that Yosida frames are those algebraic frames with the Finite Intersection Property and are finitely subfit; that these frames are also those semi-simple algebraic frames with FIP and a disjointification where dim (L)≤ 1; and we prove that in an algebraic frame with FIP, it holds that dom (L) = dim (L). In relation to normality in Yosida frames, we show that in a coherent normal Yosida frame L, the frame is subfit if and only if it is regular if and only if it is zero- dimensional if and only if every compact element is complemented.

The traditional skein relation for the Alexander polynomial involves an oriented knot, K+, with a distinguished positive crossing; a knot K−, obtained by changing the distinguished positive crossing of K+ to a negative crossing; and a link K0, the orientation preserving resolution of the distinguished crossing. We refer to (K+,K−,K0) as the oriented skein triple. A tangle is defined as a pair (B, t) of a 3-dimensional ball B and a collection of disjoint, simple, properly embedded arcs, denoted t. DeWitt Sumners and Claus Ernst developed the tangle model which uses the mathematics of tangles to model DNA-protein binding. The protein is seen as the 3-ball and the DNA bound by the protein as properly embedded curves in the 3-ball. Topoisomerases are proteins that break one segment of DNA allowing a DNA segment to pass through before resealing the break. Effectively, the action of these proteins can be modeled as K− ↔ K+. Recombinases are proteins that cut two segments of DNA and recombine them in some manner. While recombinase local action varies, most are mathematically equivalent to a resolution, i.e. K± ↔ K0. The oriented triple is now viewed as K− = circular DNA substrate, K+ = product of topoisomerase action, K0 = product of recombinase action. The theorem stated in this dissertation gives a relationship between two 2-bridge knots, K+ and K−, that differ by a crossing change and a link, K0 created from the oriented resolution of that crossing. We apply this theorem to difference topology experiments using topoisomerase proteins to study SMC proteins. In recent years, link homology theories have become a popular invariant to develop and study. One such invariant knot Floer homology, was constructed by Peter Ozsváth, Zoltán Szabó, and independently Jacob Rasmussen, denoted by HFK. It is also a refinement of a classical invariant, the Alexander polynomial. The study of DNA knots and links are of great interest to molecular biologists as they are present in many cellular process. The variety of experimentally observed DNA knots and links makes separating and categorizing these molecules a critical issue. Thus, knowing the knot Floer homology will provide restrictions on knotted and linked products of protein action. We give a summary of the combinatorial version of knot Floer homology from known work, providing a worked out example. The thesis ends with reviewing knot Floer homology properties of three particular sub-families of biologically relevant links known as (2, p)- torus links, clasp knots and 3-strand pretzel links.

Cellular life, as we know it, is absolutely dependent on biological membranes; remarkable superstructures made of lipids and proteins. For example, all living cells are surrounded by at least one membrane that protects the cell and holds it together. The proteins that are embedded in the membranes carry out a wide variety of key functions, from nutrient uptake and waste disposal to cellular respiration and communication. In order to function accurately, any integral membrane protein needs to be inserted into the cellular membrane where it belongs, and in that particular membrane it has to attain its proper structure and find partners that might be required for proper function. All membrane proteins have evolved to be inserted in a specific overall orientation, so that e.g. substrate-binding parts are exhibited on the ‘right side’ of the membrane. So, what determines in which way a membrane protein is inserted? Are all membrane proteins inserted just so? The focus of this thesis is on these fundamental questions: how, and when, is the overall orientation of a membrane protein established? A closer look at the inner membrane proteome of the familiar gram-negative bacterium Escherichia coli revealed a small group of proteins that, oddly enough, seemed to be able to insert into the membrane in two opposite orientations. We could show that these dual-topology membrane proteins are delicately balanced, and that even the slightest manipulations make them adopt a fixed orientation in the membrane. Further, we show that these proteins are topologically malleable until the very last residue has been synthesized, implying interesting questions about the topogenesis of membrane proteins in general. In addition, by looking at the distribution of homologous proteins in other organisms, we got some ideas about how membrane proteins might evolve in size and complexity. Structural data has revealed that many membrane bound transporters have internal, inverted symmetries, and we propose that perhaps some of these proteins derive from dual-topology ancestors. / <p>At the time of the doctoral defense, the following paper was unpublished and had a status as follows: Paper 4: Manuscript.</p>

<p>The work presented in this dissertation includes the study of cohomology and cohomological operations within the framework of Persistence. Although Persistence was originally defined for homology, recent research has developed persistent approaches to other algebraic topology invariants. The work in this document extends the field of persistence to include cohomology classes, cohomology operations and characteristic classes. </p><p>By starting with presenting a combinatorial formula to compute the Stiefel-Whitney homology class, we set up the groundwork for Persistent Characteristic Classes. To discuss persistence for the more general cohomology classes, we construct an algorithm that allows us to find the Poincar'{e} Dual to a homology class. Then, we develop two algorithms that compute persistent cohomology, the general case and one for a specific cohomology class. We follow this with defining and composing an algorithm for extended persistent cohomology. </p><p>In addition, we construct an algorithm for determining when a cohomology class is decomposible and compose it in the context of persistence. Lastly, we provide a proof for a concise formula for the first Steenrod Square of a given cohomology class and then develop an algorithm to determine when a cohomology class is a Steenrod Square of a lower dimensional cohomology class.</p> / Dissertation

Multicomponent polymer systems comprised of two or more chemically different polymer moieties provide an effective way to attain the desired properties from a limited palette of commodity polymers. Variations in macromolecular topologies often result in unique and unusual properties leading to novel applications. This dissertation addresses the effect of topology on properties of two multicomponent polymers systems: blends and polyrotaxanes. Blends of cyclic and linear polymers were compared to their topological counterparts, polyrotaxanes, in which cyclic components are threaded onto the linear polymer chains. The first part of the dissertation focuses on the synthesis and purification of cyclic polymers derived from linear (polyoxyethylene) (POE). Cyclic POEs of different cycle sizes were synthesized and then purified from their linear byproducts by inclusion complexation with alpha-cyclodextrin. Polystyrene was threaded through the resulting cycles by in situ free radical polymerization of styrene monomer in the presence of an excess of POE cycles. A bulky free radical initiator was utilized to endcap the polystyrene molecule at the two ends to prevent dethreading of cyclic moieties. In the second part of the dissertation, phase behavior, morphology and dynamics of cyclic POE and polystyrene blends were compared to linear POE and polystyrene blends. Advanced solid-state NMR techniques and differential scanning calorimetry were employed for this purpose. Cyclic POE was found to be much more miscible with polystyrene when compared to linear POE, resulting in nanometer-sized domains and significantly reduced mobilities of the cyclic POE components in the blends. The unusual behavior of cyclic POE in the blends was attributed to topological as well as end-group effects with the topological effects being predominant. Polyrotaxanes composed of polystyrene and cyclic POE components exhibited cyclic POE domain sizes similar to that of physical blends. Cyclic POE dynamics in polyrotaxanes were considerably hindered, however, due to the threaded architecture. Surface segregation studies of cyclic POE/polystyrene blends and polyrotaxanes did not show segregation of POE to the surface because of the improved miscibility and the topological constraints present in these systems.

Thesis (Ph. D.)--Rutgers University, 2007. / "Graduate Program in Electrical and Computer Engineering." Includes bibliographical references (p. 140-144).

x, 72 p. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. / The theory of equivariant homology and cohomology was first created by Bredon in his 1967 paper and has since been developed and generalized by May, Lewis, Costenoble, and a host of others. However, there has been a notable lack of computations done. In this paper, a version of the Serre spectral sequence of a fibration is developed for RO ( G )-graded equivariant cohomology of G -spaces for finite groups G . This spectral sequence is then used to compute cohomology of projective bundles and certain loop spaces. In addition, the cohomology of Rep( G )-complexes, with appropriate coefficients, is shown to always be free. As an application, the cohomology of real projective spaces and some Grassmann manifolds are computed, with an eye towards developing a theory of equivariant characteristic classes. / Adviser: Daniel Dugger

Let n > 1. We prove a homological stability theorem for the diffeomorphism groups of (4n+1)-dimensional manifolds, with respect to forming the connected sum with (2n-1)-connected, (4n+1)-dimensional manifolds that are stably parallelizable. Our techniques involve the study of the action of the diffeomorphism group of a manifold M on the linking form associated to the homology groups of M. In order to study this action we construct a geometric model for the linking form using the intersections of embedded and immersed Z/k-manifolds. In addition to our main homological stability theorem, we prove several results regarding disjunction for embeddings and immersions of Z/k-manifolds that could be of independent interest.

A set-theoretical point of view to study algebraic numbers has been introduced. We extend a result of Navarro-Bermudez concerning shift invariant measures in the Cantor space which are topologically equivalent to shift invariant measures which correspond to some algebraic integers. It is known that any transcendental numbers and rational numbers in the unit interval are not binomial. We proved that there are algebraic numbers of degree greater than two so that they are binomial numbers. Algebraic integers of degree 2 are proved not to be binomial numbers. A few compositive relations having to do with algebraic numbers on the unit interval have been studied; for instance, rationally related, integrally related, binomially related, B1-related relations. A formula between binomial numbers and binomial coefficients has been stated. A generalized algebraic equation related to topologically equivalent measures has also been stated.

Thesis(M. Sc. (Mathematics)) -- University of Limpopo, 2021 / A Cs´ asz´ ar frame is a point free version of syntopogenous space, itself a concept that is attributed to ´ Akos Cs´ asz´ ar [14]. In his two papers, Chung ([12] and [13]) characterised few types of Cs´ asz´ ar frames and extended Hong’s construction [21] to the Cauchy completions in Cs´ asz´ ar frames. From his results, we anchored objectives of our study on the actions of certain frame homomorphisms on proximal Cs´ asz´ ar frames, as well as co-reflective subcategories of Cauchy complete Cs´ asz´ ar frames. We conclude the dissertation by constructing the compactification of proximal Cs´ asz´ ar frames by applying the methods of Banaschewski and Mulvey [7]. We introduce a weak notion of connectedness of Cs´ asz´ ar frames and show, following the approach of Baboolal and Banaschewski [4], that most of the standard results on connectedness are do-able in the setting of Cs´asz'ar.