We introduce a new class of Conley-Morse chain maps for the purpose of comparing the qualitative structure of flows across multiple scales. Conley index theory generalizes classical Morse theory as a tool for studying the dynamics of flows. The qualitative structure of a flow, given a Morse decomposition, can be stored algebraically as a set of homology groups (Conley indices) and a boundary map between the indices (a connection matrix). We show that as long as the qualitative structures of two flows agree on some, perhaps coarse, level we can construct a chain map between the corresponding chain complexes that preserves the relations between the (coarsened) Morse sets. We present elementary examples to motivate applications to data analysis.

This work investigates pathways of conformational transitions in ubiquitous RNA structural motifs. In our lab, we have developed multi-scale structural datamining techniques for identification of three-dimensional structural patterns in high-resolution crystal structures of globular RNA. I have applied these techniques to identify variations in the conformations of RNA double-helices and tetraloops. The datamined structural information is used to propose reaction coordinates for conformational transitions involved in double-strand helix propagation and tetraloop folding in RNA. I have also presented an algorithm to identify stacked RNA bases. In this work, experimentally derived thermodynamic evaluation of the conformations has been used to as an additional parameter to add detail to RNA structural transitions. RNA conformational transitions help control processes in small systems such as riboswitches and in large systems such as ribosomes. Adopting functional conformations by globular RNA during a folding process also involves structural transitions. RNA double-helices and tetraloops are common, ubiquitous structural motifs in globular RNA that independently fold in to a thermodynamically stable conformation. Folding models for these motifs are proposed in this work with probable intermediates ordered along the reaction coordinates. We hypothesize that frequently observed structural states in crystals structures are analogous in conformation to stable thermodynamic â on-pathwayâ folded states. Conversely, we hypothesize that conformations that are rarely observed are improbable folding intermediates, i.e., these conformational states are â off-pathwayâ states. In general on-pathway states are assumed to be thermodynamically more stable than off-pathway states, with the exception of kinetic traps. Structural datamining shows that double helices in RNA may propagate by the â stack-ratchetâ mechanism proposed here instead of the commonly accepted zipper mechanism. Mechanistic models for RNA tetraloop folding have been proposed and validated with experimentally derived thermodynamic data. The extent of stacking between bases in RNA is variable, indicating that stacking may not be a two-state phenomenon. A novel algorithm to define and identify stacked bases at atomic resolution has also been presented in this work.

In his thesis we are concerned with certain numerical invariants of homotopy type akin to the Lusternik-Schnirelmann category and cocategory. In a series of papers I. Bernstein, T. Ganea, and P.J. Hilton developed the concepts of the category and weak category of a topological space. They also considered the related concepts of conilpotency and cup product length of a space and the weak category of a map. Later T. Ganea gave another definition of category and weak category (which we shall write as G-cat and G-wcat) in terms of vibrations and cofibrations and hence this dualizes easily in the sense of Eckmann-Hilton. We find the relationships between these invariants and then find various examples of spaces which show that the invariants are all different except cat and G-cat. The results are contained in the following theorem. The map $e:B -> OmegaSigma B$ is the natural embedding. All the invariants are normalized so as to take the value 0 on contractible spaces. THEOREM Let B have the homotopy type of a simply connected CW-complex, then $cat B = G-cat B geq G-wcat B geq wcat B geq wcat e geq conil B geq cup-long B$ and furthermore all the inequalities can occur. All the examples are spaces of the form $B = S^qcup_alpha e^n$ where $alphain pi_{n-1} (S^q)$. When B is of this form, we obtain conditions for the category and the weak categories of B to be less than or equal to one of the terms of Hopf invariants of $alpha$. We use these conditions to prove the examples. We then prove the dual theorem concerning the relationships between the invariants cocategory, weak cocategory, nilpotency and Whitehead product length. THEOREM Let A be countable CW-complex, then $cocat A geq wcocat A geq nil A geq W-long A$ and furthermore all the inequalities can occur. The proof is not dual to the first theorem, though the examples we use to show that the inequalities can exist are all spaces with two non-zero homotopy groups. The most interesting of these examples is the space A with 2 non-zero homotopy groups, $mathbb Z$ in dimension 2 and ${mathbb Z}_4$ in dimension 7 with k-invariant $u^4 in H^8(mathbb Z, 2; {mathbb Z}_4)$. This space is not an H-space, but has weak cocategory 1. The condition $wcocat A leq 1$ is equivalent to the fact that d is homotopic to 0 in the fibration $D -d-> A -e-> OmegaSigma A$. In order to show that wcocat A = 1 we have to calculate to cohomology ring of $OmegaSigma K(mathbb Z,2)$. The method we use to do this is the same as that used to calculate the cohomology ring of $OmegaSigma S^{n+1}$ using James' reduced product construction. Finally we show that for the above space A the fibration $Omega A -g-> A^S -f-> A$ has a retraction $ ho$ such that $ hocirc g$ is homotopic to 1 even though A is not an H-space.

Dans les catégories de foncteurs entre catégories abéliennes, les foncteurs additifs jouent un rôle privilégié dans plusieurs domaines de l'algébre. Cependant il existe de nombreux foncteurs trés intéressants qui ne sont pas additifs. Par exemple, le produit tensoriel de groupes abéliens définit un foncteur $T^2: Ab \to Ab$ donné par $T^2(G)=G \otimes G$ qui n'est pas additif mais polynomial de degré deux. Les foncteurs polynomiaux ont été introduits par Eilenberg et MacLane pour les foncteurs entre catégories de modules. De nombreux exemples de foncteurs polynomiaux apparaissent naturellement en topologie algébrique. En particulier, l'homologie stable de familles de groupes à coefficients donnés par des foncteurs polynomiaux peut être interprétée en termes d'homologie des foncteurs. Dans les cas favorables, cette homologie des foncteurs est accessible et fournit ainsi des calculs explicites des valeurs stables des homologies à coefficients tordus. Ce mémoire comporte deux parties. La première concerne l'étude de la structure des foncteurs polynomiaux et la seconde concerne le calcul de l'homologie stable d'une famille de groupes à coefficients donnés par un foncteur polynomial.

Given a C*-algebra $A$ endowed with an action $\alpha$ of $\R$ and an $\alpha$-invariant trace $\tau$, there is a canonical dual trace $\widehat \tau$ on the crossed product $A \rtimes_\alpha \R$. This dual trace induces (as would any suitable trace) a real-valued homomorphism $\widehat \tau_* : K_0(A \rtimes_\alpha \R) \to \R$ on the even $K$-theory group. Recall there is a natural isomorphism $\phi_\alpha^i : K_i(A) \to K_{i+1}(A \rtimes_\alpha \R)$, the Connes-Thom isomorphism. The attraction of describing $\widehat \tau_* \circ \phi_\alpha^1$ directly in terms of the generators of $K_1(A)$ is clear. Indeed, the paper where the isomorphisms $\{\phi_\alpha^0,\phi_\alpha^1\}$ first appear sees Connes show that $\widehat \tau_* \phi_\alpha^1[u] = \frac{1}{2 \pi i} \tau(\delta(u) u^*)$, where $\delta = \frac{d}{dt} \big|_{t=0} \alpha_t(\cdot)$ and $u$ is any appropriate unitary. A careful proof of the aforementioned result occupies a central place in this thesis. To place the result in its proper context, the right-hand side is first considered in its own right, i.e., in isolation from mention of the crossed-product. A study of 1-parameter dynamical systems and exterior equivalence is undertaken, with several useful technical results being proven. A connection is drawn between a lemma of Connes on exterior equivalence and projections, and a quantum-mechanical theorem of Bargmann-Wigner. An introduction to the Connes-Thom isomorphism is supplied and, in the course of this introduction, a refined version of suspension isomorphism $K_1(A) \to K_0(\susp A)$ is formulated and proven. Finally, we embark on a survey of unbounded traces on C*-algebras; when traces are allowed to be unbounded, there is inevitably a certain amount of hard, technical work needed to resolve various domain issues and justify various manipulations. / Graduate / 0280

Thesis (Ph. D.)--State University of New York at Binghamton, of Department of Mathematical Sciences, 2007. / Includes bibliographical references.

Thesis (Ph.D.)--University of Wollongong. / Typescript. Includes bibliographical references: leaf 169-179.

Thesis (Ph. D.)--University of Oregon, 2008. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 71-72). Also available online in Scholars' Bank; and in ProQuest, free to University of Oregon users.

Made available in DSpace on 2014-06-11T19:27:09Z (GMT). No. of bitstreams: 0 Previous issue date: 2014-02-18Bitstream added on 2014-06-13T20:16:07Z : No. of bitstreams: 1 000753998.pdf: 885291 bytes, checksum: e06a634b31c2012fc0b6d5e72ec13aa3 (MD5) / Este trabalho é espelhado no livro “Teoria do Índice” [1] de Daciberg Lima Gonçalves e José Carlos de Souza Kiihl, publicado em 1983 no 14o Colóquio Brasileiro de Matemática pelo IMPA. Para a leitura deste trabalho é necessário uma familiaridade prévia com Topologia Algébrica, na qual indicamos [2] e [3] para consulta. Inicialmente apresentaremos alguns pré-requisitos algébricos e topológicos necessários para o desenvolvimento do trabalho e a seguir estudaremos: pontos fixos de aplicações contínuas de X em X, em que X é um espaço topológico; Grau de Brouwer de aplicações contínuas de Sn em Sn (ou respectivamente (Bn+1; Sn) em (Bn+1; Sn)); Grau Local de uma aplicação contínua f de V em Sn em torno de um ponto Q 2 Sn, em que V Sn é um aberto e f1(Q) é um compacto e Índices dos Pontos Fixos de uma aplicação contínua de V em Sn, em que V Rn é um aberto / This work is based on the book titled “Teoria do Índice” [1] by Daciberg Lima Gonçalves and José Carlos de Souza Kiihl , published in 1983 in the 14o Brazilian Math Colloquium held by IMPA . In order to perform the reading of this work, a basic acquaintance from the algebraic topology is needed, on which we can indicate the following [2] and [3] references. Firstly, for the development of the work, some previous necessary algebraic and topological requirements are shown and the next topics will be studied: fixed points of continuous maps from X to X, where X is a topological space, Brouwer’s degree of continuous maps from Sn to Sn ( or respectively (Bn+1; Sn) to (Bn+1; Sn)), Local Degree of continuous maps from V to Sn around a point Q 2 Sn, where V Sn is an open set and f1(Q) is a compact set and Fixed Points Index of continuous maps from V to Sn, where V Rn is an open set

The main goal of this thesis is to construct a tensor product on the 2-category BFC-A of braided fusion categories containing a symmetric fusion category A. We achieve this by introducing the new notion of Z(A)-crossed braided categories. These are categories enriched over the Drinfeld centre Z(A) of the symmetric fusion category. We show that Z(A) admits an additional symmetric tensor structure, which makes it into a 2-fold monoidal category. ByTannaka duality, A= Rep(G) (or Rep(G; w)) for a finite group G (or finite super-group (G,w)). Under this identication Z(A) = VectG[G], the category of G-equivariant vector bundles over G, and we show that the symmetric tensor product corresponds to (a super version of) to the brewise tensor product. We use the additional symmetric tensor product on Z(A) to define the composition in Z(A)-crossed braided categories, whereas the usual tensor product is used for the monoidal structure. We further require this monoidal structure to be braided for the switch map that uses the braiding in Z(A). We show that the 2-category Z(A)-XBF is equivalent to both BFC=A and the 2-category of (super)-G-crossed braided categories. Using the former equivalence, the reduced tensor product on BFC-A is dened in terms of the enriched Cartesian product of Z(A)-enriched categories on Z(A)-XBF. The reduced tensor product obtained in this way has as unit Z(A). It induces a pairing between minimal modular extensions of categories having A as their Mueger centre.