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Graphs, Simplicial Complexes and Beyond: Topological Tools for Multi-agent CoordinationMuhammad, Abubakr 16 December 2005 (has links)
In this work, connectivity graphs have been studied as models of local interactions in multi-agent robotic systems. A systematic study of the space of connectivity graphs has been done from a geometric and topological point of view. Some results on the realization of connectivity graphs in their respective configuration spaces have been given. A complexity analysis of networks, from the point of view of intrinsic structural complexity, has been given. Various topological spaces in networks, as induced from their connectivity graphs, have been recognized and put into applications, such as those concerning coverage problems in sensor networks. A framework for studying dynamic connectivity graphs has been proposed. This framework has been used for several applications that include the generation of low-complexity formations as well as collaborative beamforming in sensor networks. The theory has been verified by generating extensive simulations, with the help of software tools of computational homology and semi-definite programming. Finally, several open problems and areas of further research have been identified.
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Topological uniqueness results for the special linear and other classical Lie Algebras.Rees, Michael K. 12 1900 (has links)
Suppose L is a complete separable metric topological group (ring, field, etc.). L is topologically unique if the Polish topology on L is uniquely determined by its underlying algebraic structure. More specifically, L is topologically unique if an algebraic isomorphism of L with any other complete separable metric topological group (ring, field, etc.) induces a topological isomorphism. A local field is a locally compact topological field with non-discrete topology. The only local fields (up to isomorphism) are the real, complex, and p-adic numbers, finite extensions of the p-adic numbers, and fields of formal power series over finite fields. We establish the topological uniqueness of the special linear Lie algebras over local fields other than the complex numbers (for which this result is not true) in the context of complete separable metric Lie rings. Along the way the topological uniqueness of all local fields other than the field of complex numbers is established, which is derived as a corollary to more general principles which can be applied to a larger class of topological fields. Lastly, also in the context of complete separable metric Lie rings, the topological uniqueness of the special linear Lie algebra over the real division algebra of quaternions, the special orthogonal Lie algebras, and the special unitary Lie algebras is proved.
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Commutants of composition operators on the Hardy space of the diskCarter, James Michael 06 November 2013 (has links)
Indiana University-Purdue University Indianapolis (IUPUI) / The main part of this thesis, Chapter 4, contains results on the commutant of a semigroup of operators defined on the Hardy Space of the disk where the operators have hyperbolic non-automorphic symbols. In particular, we show in Chapter 5 that the commutant of the semigroup of operators is in one-to-one correspondence with a Banach algebra of bounded analytic functions on an open half-plane. This algebra of functions is a subalgebra of the standard Newton space. Chapter 4 extends previous work done on maps with interior fixed point to the case of the symbol of the composition operator having a boundary fixed point.
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A topological approach to nonlinear analysisPeske, Wendy Ann 01 January 2005 (has links)
A topological approach to nonlinear analysis allows for strikingly beautiful proofs and simplified calculations. This topological approach employs many of the ideas of continuous topology, including convergence, compactness, metrization, complete metric spaces, uniform spaces and function spaces. This thesis illustrates using the topological approach in proving the Cauchy-Peano Existence theorem. The topological proof utilizes the ideas of complete metric spaces, Ascoli-Arzela theorem, topological properties in Euclidean n-space and normed linear spaces, and the extension of Brouwer's fixed point theorem to Schauder's fixed point theorem, and Picard's theorem.
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Ordered spaces of continuous functions and bitopological spacesNailana, Koena Rufus 11 1900 (has links)
This thesis is divided into two parts: Ordered spaces of Continuous Functions and
the algebras associated with the topology of pointwise convergence of the associated
construct, and Strictly completely regular bitopological spaces.
The Motivation for part of the first part (Chapters 2, 3 and 4) comes from the
recent study of function spaces for bitopological spaces in [44] and [45]. In these
papers we see a clear generalisation of classical results in function spaces ( [14] and
[55]) to bi-topological spaces. The well known definitions of the pointwise topology and
the compact open topology in function spaces are generalized to bitopological spaces,
and then familiar results such as Arens' theorem are generalised. We will use the same
approach in chapters 2, 3 and 4 to formulate analogous definitions in the setting of
ordered spaces. Well known results, including Arens' theorem, are also generalised
to ordered spaces. In these chapters we will also compare function spaces in the
category of topological spaces and continuous functions, the category of bi topological
spaces and bicontinuous functions, and the category of ordered topological spaces and
continuous order-preserving functions. This work has resulted in the publication of
[30] and [31].
Continuing our study of Function Spaces, we oonsider in Chapters 5 and 6 some
Categorical aspects of the construction, motivated by a series of papers which includes
[39], [40], [41] and [50]. In these papers the Eilenberg-Moore Category of algebras of
the monad induced by the Hom-functor on the categories of sets and categories of
topological spaces are classified. Instead of looking at the whole product topology we
will restrict ourselves to the pointwise topology and give examples of the EilenbergMoore Algebras arising from this restriction. We first start by way of motivation, with
the discussion of the monad when the range space is the real line with the usual topology.
We then restrict our range space to the two point Sierpinski space, with the aim
of discovering a topological analogue of the well known characterization of Frames as
the Eilenberg-Moore Category of algebras associated with the Hom-F\mctor of maps
into the Sierpinski space [11]. In this case the order structure features prominently, resulting in the category Frames with a special property called "balanced" and Frame
homomorphisms as the Eilenberg-Moore category of M-algebras. This has resulted
in [34].
The Motivation for the second part comes from [20] and [15]. In [20], J. D. Lawson
introduced the notion of strict complete regularity in ordered spaces. A detailed study
of this notion was done by H-P. A. Kiinzi in [15]. We shall introduce an analogous
notion for bitopological spaces, and then shall also compare the two notions in the categories
of bi topological spaces and bicontinuous functions, and of ordered topological
spaces and continuous order-preserving functions via the natural functors considered
in the previous chapters. We further study the Stone-Cech bicompactification and
Stone-Cech ordered compactification in the two categories. This has resulted in [32] and [33] / Mathematical Sciences / D. Phil. (Mathematics)
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