1,994 research outputs found

### On Rational Sets in Euclidean Spaces and Spheres

IFor a positive rational $l$, we define the concept of an $l$-elliptic and an
$l$-hyperbolic rational set in a metric space. In this article we examine the
existence of (i) dense and (ii) infinite $l$-hyperbolic and $l$-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 $l$-ellitpic and $l$-hyperbolic rational infinite sets in unit
spheres and Euclidean spaces for certain values of $l$ 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 $T$ of the $k$-dimensional unit sphere $S^k$
has an antipodal pair if both $x,-x\in T$ for some $x\in S^k$. In this article,
we prove that there does not exist a dense rational set $T\subset S^2$ 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 $S^k$ is equivalent to the existence of a dense $2$-hyperbolic rational set
in $S^k$ which is further equivalent to the existence of a dense 1-elliptic
rational set in the Euclidean space $\mathbb{R}^k$.Comment: 20 page

### On the Coherent Labelling Conjecture of a Polyhedron in Three Dimensions

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 $n$-gons for $n\geq 4$, 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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