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    Topology: Big decisions

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    Teleportation Topology

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    We discuss the structure of teleportation. By associating matrices to the preparation and measurement states, we show that for a unitary transformation M there is a full teleportation procedure for obtaining M|S> from a given state |S>. The key to this construction is a diagrammatic intepretation of matrix multiplication that applies equally well to a topological composition of a maximum and a minimum that underlies the structure of the teleportation. This paper is a preliminary report on joint work with H. Carteret and S. Lomonaco.Comment: LaTeX document, 16 pages, 8 figures, Talk delivered at the Xth International Conference on Quantum Optics, Minsk, Belaru

    Topology Control in Heterogeneous Wireless Networks: Problems and Solutions

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    Previous work on topology control usually assumes homogeneous wireless nodes with uniform transmission ranges. In this paper, we propose two localized topology control algorithms for heterogeneous wireless multi-hop networks with nonuniform transmission ranges: Directed Relative Neighborhood Graph (DRNG) and Directed Local Spanning Subgraph (DLSS). In both algorithms, each node selects a set of neighbors based on the locally collected information. We prove that (1) the topologies derived under DRNG and DLSS preserve the network connectivity; (2) the out degree of any node in the resulting topology by DLSS is bounded, while the out degree cannot be bounded in DRNG; and (3) the topologies generated by DRNG and DLSS preserve the network bi-directionality

    Algebraic Topology

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    The chapter provides an introduction to the basic concepts of Algebraic Topology with an emphasis on motivation from applications in the physical sciences. It finishes with a brief review of computational work in algebraic topology, including persistent homology.Comment: This manuscript will be published as Chapter 5 in Wiley's textbook \emph{Mathematical Tools for Physicists}, 2nd edition, edited by Michael Grinfeld from the University of Strathclyd
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