1,994 research outputs found

    On Rational Sets in Euclidean Spaces and Spheres

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    IFor a positive rational ll, we define the concept of an ll-elliptic and an ll-hyperbolic rational set in a metric space. In this article we examine the existence of (i) dense and (ii) infinite ll-hyperbolic and ll-ellitpic rationals subsets of the real line and unit circle. For the case of a circle, we prove that the existence of such sets depends on the positivity of ranks of certain associated elliptic curves. We also determine the closures of such sets which are maximal in case they are not dense. In higher dimensions, we show the existence of ll-ellitpic and ll-hyperbolic rational infinite sets in unit spheres and Euclidean spaces for certain values of ll which satisfy a weaker condition regarding the existence of elements of order more than two, than the positivity of the ranks of the same associated elliptic curves. We also determine their closures. A subset TT of the kk-dimensional unit sphere SkS^k has an antipodal pair if both x,βˆ’x∈Tx,-x\in T for some x∈Skx\in S^k. In this article, we prove that there does not exist a dense rational set TβŠ‚S2T\subset S^2 which has an antipodal pair by assuming Bombieri-Lang Conjecture for surfaces of general type. We actually show that the existence of such a dense rational set in SkS^k is equivalent to the existence of a dense 22-hyperbolic rational set in SkS^k which is further equivalent to the existence of a dense 1-elliptic rational set in the Euclidean space Rk\mathbb{R}^k.Comment: 20 page

    On the Coherent Labelling Conjecture of a Polyhedron in Three Dimensions

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    In this article we consider an open conjecture about coherently labelling a polyhedron in three dimensions. We exhibit all the forty eight possible coherent labellings of a tetrahedron. We also exhibit that some simplicial polyhedra like bipyramids, Kleetopes, gyroelongated bipyramids are coherently labellable. Also we prove that pyramids over nn-gons for nβ‰₯4n\geq 4, which are not simplicial polyhedra, are coherently labellable. We prove that among platonic solids, the cube and the dodecahedron are not coherently labellable, even though, the tetrahedron, the octahedron and the icosahedron are coherently labellable. Unlike the case of a tetrahedron, in general for a polyhedron, we show that a coherent labelling need not induce a coherent labelling at a vertex. We prove the main conjecture in the affirmative for a certain class of polyhedra which are constructible from tetrahedra through certain types of edge and face vanishing tetrahedron attachments. As a consequence we conclude that a cube cannot be obtained from only these type of tetrahedron attachments. We also give an obstruction criterion for a polyhedron to be not coherently labellable and consequentially show that any polyhedron obtained from a pyramid with its apex chopped off is not coherently labellable. Finally with the suggestion of the affirmative results we prove the main theorem that any simplicial polyhedron is coherently labellable.Comment: 29 pages, 14 figure
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