372,098 research outputs found
Duality and separation theorems in idempotent semimodules
We consider subsemimodules and convex subsets of semimodules over semirings
with an idempotent addition. We introduce a nonlinear projection on
subsemimodules: the projection of a point is the maximal approximation from
below of the point in the subsemimodule. We use this projection to separate a
point from a convex set. We also show that the projection minimizes the
analogue of Hilbert's projective metric. We develop more generally a theory of
dual pairs for idempotent semimodules. We obtain as a corollary duality results
between the row and column spaces of matrices with entries in idempotent
semirings. We illustrate the results by showing polyhedra and half-spaces over
the max-plus semiring.Comment: 24 pages, 5 Postscript figures, revised (v2
Associahedron, cyclohedron, and permutohedron as compactifications of configuration spaces
As in the case of the associahedron and cyclohedron, the permutohedron can
also be defined as an appropriate compactification of a configuration space of
points on an interval or on a circle. The construction of the compactification
endows the permutohedron with a projection to the cyclohedron, and the
cyclohedron with a projection to the associahedron. We show that the preimages
of any point via these projections might not be homeomorphic to (a cell
decomposition of) a disk, but are still contractible. We briefly explain an
application of this result to the study of knot spaces from the point of view
of the Goodwillie-Weiss manifold calculus.Comment: 27 pages The new version gives a more detailed exposition for the
projection from the cyclohedron to the associahedron as maps of
compactifications of configuration spaces. We also develop a similar picture
for the projection from the permutohedron to the cyclohedron/associahedro
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