7,666 research outputs found
Geometry of canonical self-similar tilings
We give several different geometric characterizations of the situation in
which the parallel set of a self-similar set can be described
by the inner -parallel set of the associated
canonical tiling , in the sense of \cite{SST}. For example,
if and only if the boundary of the
convex hull of is a subset of , or if the boundary of , the
unbounded portion of the complement of , is the boundary of a convex set. In
the characterized situation, the tiling allows one to obtain a tube formula for
, i.e., an expression for the volume of as a function of
. On the way, we clarify some geometric properties of canonical
tilings.
Motivated by the search for tube formulas, we give a generalization of the
tiling construction which applies to all self-affine sets having empty
interior and satisfying the open set condition. We also characterize the
relation between the parallel sets of and these tilings.Comment: 20 pages, 6 figure
Mutually Unbiased Bases and The Complementarity Polytope
A complete set of N+1 mutually unbiased bases (MUBs) forms a convex polytope
in the N^2-1 dimensional space of NxN Hermitian matrices of unit trace. As a
geometrical object such a polytope exists for all values of N, while it is
unknown whether it can be made to lie within the body of density matrices
unless N=p^k, where p is prime. We investigate the polytope in order to see if
some values of N are geometrically singled out. One such feature is found: It
is possible to select N^2 facets in such a way that their centers form a
regular simplex if and only if there exists an affine plane of order N. Affine
planes of order N are known to exist if N=p^k; perhaps they do not exist
otherwise. However, the link to the existence of MUBs--if any--remains to be
found.Comment: 18 pages, 3 figure
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