15,114 research outputs found

    Some characterizations of spheres and elliptic paraboloids II

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    We show some characterizations of hyperspheres in the (n+1)(n+1)-dimensional Euclidean space En+1{\Bbb E}^{n+1} with intrinsic and extrinsic properties such as the nn-dimensional area of the sections cut off by hyperplanes, the (n+1)(n+1)-dimensional volume of regions between parallel hyperplanes, and the nn-dimensional surface area of regions between parallel hyperplanes. We also establish two characterizations of elliptic paraboloids in the (n+1)(n+1)-dimensional Euclidean space En+1{\Bbb E}^{n+1} with the nn-dimensional area of the sections cut off by hyperplanes and the (n+1)(n+1)-dimensional volume of regions between parallel hyperplanes. For further study, we suggest a few open problems.Comment: 10 page

    Enumerations of finite topologies associated with a finite graph

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    The number of topologies and non-homeomorphic topologies on a fixed finite set are now known up to n=18n=18, n=16n=16 but still no complete formula yet (Sloane). There are one to one correspondence among topologies, preorder and digraphs. In this article, we enumerate topologies and non-homeomorphic topologies whose underlying graph is a given finite graph.Comment: 2 figure

    Double Submission, Double Publication

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