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No description available.

We consider Livsic regularity for Lie group valued cocycles over: a class of piecewise expanding maps of the interval, namely Lasota-Yorke maps; uniformly hyperbolic toral maps with singularities and a class of nonuniformly expanding interval maps. As applications of the results we prove stable ergodicity theorems for compact Lie group extension of Lasota-Yorke maps and uniformly hyperbolic toral maps with singularities. Additionally we consider conditions for the ergodicity and weak-mixing of finite group extensions of hyperbolic basic sets given in terms of periodic data and cohomological equations. We also consider stable ergodicity results for a class of nonconnected compact Lie group extensions of hyperbolic basic sets.

Recently, control techniques that adopt the geometrical structure and physical properties of dynamical systems have gained a lot of interest. In this thesis, we address nonlinear and robust control problems for systems represented by port-controlled Hamiltonian (PCH) models using the interconnection and damping assignment passivity-based control(IDA-PBC) methodology, which is the most notable technique facilitating the PCH framework. In this thesis, a novel constructive framework to simplify and solve the partial differential equations (PDEs) associated with IDA-PBC for a class of underactuated mechanical systems is presented. Our approach focuses on simplifying the potential energy PDEs to shape the potential energy function which is the most important procedure in the stabilization of mechanical systems. The simplification is achieved by parametrizing thedesired inertia matrix that shapes the kinetic energy function, thus achieving total energy shaping. The simplification removes some constraints (conditions and assumptions) that have been imposed in recently developed methods in literature, thus expanding the class of systems for which the methods can be applied including the separable PCH systems(systems with constant inertia matrix) and non-separable PCH systems (systems with non-constant inertia matrix). The results are illustrated through software simulations and hardware experiments on real engineering applications. We also propose an integral control and adaptive control schemes to improve the robustness of the IDA-PBC method in presence of uncertainty. We first provide some results for the case of fully-actuated mechanical systems, and then extend those results to underactuated systems which are more complex. Integral action control on both the passive and non-passive outputs in the IDA-PBC construction, a strategy to ensure the robustness of the systems by preserving its stability in face of external disturbances, is introduced, establishing the input-to-state stability (ISS) property. The results are applied to both the separable and non-separable PCH systems and illustrated via several simulations. The extension to the non-separable case exhibits more complicated design as we need to take into account the derivative of the inertia matrix. Finally, the IDA-PBC method is employed to solve an important nonlinear phenomenon called ‘pull-in’ instability associated with the electrostatically actuated microelectromechanical systems (MEMSs). The control construction is an output-feedback controller that ensures global asymptotic stability and avoids velocity measurement which may not be practically available. Furthermore, the integral, adaptive and ISS control schemes proposed in this thesis for mechanical systems are extended to facilitate the stabilization of electromechanical systems which exhibit strong coupling between different energy domains.

We study the homotopy types of gauge groups of principal U(n)-bundles associated to pseudo Real vector bundles in the sense of Atiyah [Ati66]. We provide satisfactory homotopy decompositions of these gauge groups into factors in which th homotopy groups are well known. Therefore, we substantially build upon the low dimensional homotopy groups as provided in [BHH10].

This thesis is a wide-ranging investigation of convergence properties in topological spaces, primarily Fréchet-Urysohn and radial spaces. The former are spaces such that every point in a closure of a subset is the limit of a sequence from within that set. The latter is a generalisation, defined by replacing 'sequence' with 'transfinite sequence'. Although not all spaces have these properties, they form a large enough class to encompass many important examples of spaces. These convergence properties can and should be studied locally and structurally. The first is achieved by removing the quantification over points in the definitions. For the second, we introduce the notion of spokes for points in topological spaces, which are sub-spaces for which the point has a descending neighbourhood base. In Chapter I, we introduce several convergence properties, and recall how they are connected and characterised by particular quotient maps. We also introduce p-character and quasi-isolation, to give our results full generality by not assuming the T1-axiom. Our main focus is the development of the theory of spokes in Chapter II. Here, we study how spokes can be used to approximate the neighbourhood base of a radial point and how (transfinite) sequences converge. We prove several characterisation theorems for radial and Fréchet-Urysohn points and their relationship with independently-based points, which are described through nests. We also use spokes in productivity problems and variants of the Fréchet-Urysohn property. In the final chapter, we demonstrate how properties of spokes manifest in different settings. For example, in compactifications of locally-compact spaces, spoke structures at the point-at-infinity reflect into the compact structure of the original space. Other examples are obtained by dualities, characterising radiality in ring spectra or Stone spaces algebraically. Such results justify using internal structures to investigate convergence properties and the author wishes to continue this line of investigation for the foreseeable future.

Ends and coends can be described as objects which are universal amongst extranatural transformations. We describe a cate- gorification of this idea, extrapseudonatural transformations, in such a way that bicodescent objects are the objects which are universal amongst such transfor- mations. We recast familiar results about coends in this new setting, providing analogous results for bicodescent objects. In particular we prove a Fubini theorem for bicodescent objects. The free cocompletion of a category C is given by its category of presheaves [C^op ,Set]. If C is also monoidal then its category of presheaves can be pro- vided with a monoidal structure via the convolution product of Day. This monoidal structure describes [C^op ,Set] as the free monoidal cocompletion of C. Day’s more general statement, in the V-enriched setting, is that if C is a promonoidal V-category then [C^op ,V] possesses a monoidal structure via the convolution product. We define promonoidal bicategories and go on to show that if A is a promonoidal bicategory then the bicategory of pseudofunctors Bicat(A^op ,Cat) is a monoidal bicategory.

No description available.

Floer theory is a rich collection of tools for studying symplectic manifolds and their Lagrangian submanifolds with the help of holomorphic curves. Its origins lie in estimating the numbers of equilibria in Hamiltonian dynamics, and more recently it has become a major component of the Homological Mirror Symmetry conjecture. This work presents several new computations in Floer theory which combine the use of geometric symmetries, naturally arising in various contexts, with advanced algebraic structures related to Floer theory, like the string maps and the Fukaya category. The three main directions of our study are: the Floer cohomology for a pair of commuting symplectomorphisms; the Fukaya algebra of a Lagrangian submanifold invariant under a circle action; and rigidity properties of non-monotone Lagrangian submanifolds based on the use of low-area versions of the string maps. In each of the three mentioned setups we provide concrete applications of our general results to the study of symplectic manifolds. For example, we prove that Dehn twists in most projective hypersurfaces have infinite order in the symplectic mapping class group; prove that the real projective space split-generates the Fukaya category of the complex projective space and therefore must intersect any other Lagrangian submanifold that is nontrivial in that Fukaya category; and we exhibit a continuous family of Lagrangian tori in the complex projective plane that cannot be made disjoint from the standard Clifford torus by a Hamiltonian isotopy.

In this thesis our goal is to develop the equivariant version of Hochschild cohomology. In the equivariant world there is given a group G which acts on objects. First naive object which can be considered is a G-algebra, that is, an associative algebra A on which G acts via algebra automorphisms. In our work we consider two more general situations. In the first case we develop a cohomology theory for oriented algebras and in the second case we develop a cohomology theory for Green functors.