450 research outputs found
An index theorem for Wiener--Hopf operators
We study multivariate generalisations of the classical Wiener--Hopf algebra,
which is the C-algebra generated by the Wiener--Hopf operators, given by
the convolutions restricted to convex cones. By the work of Muhly and Renault,
this C-algebra is known to be isomorphic to the reduced C-algebra of a
certain restricted action groupoid. In a previous paper, we have determined a
composition series of this C-algebra, and compute the -theory
homomorphisms induced by the `symbol' maps given by the subquotients of the
composition series in terms of the analytical index of a continuous family of
Fredholm operators. In this paper, we obtain a topological expression for these
index maps in terms of geometric-topological data naturally associated to the
underlying convex cone. The resulting index formula is expressed in the
framework of Kasparov's bivariant -theory. Our proof relies heavily on
groupoid methods.Comment: 46 pages, 1 figure; last version prior to publication, journal
reference adde
A Fixed Parameter Tractable Approximation Scheme for the Optimal Cut Graph of a Surface
Given a graph cellularly embedded on a surface of genus , a
cut graph is a subgraph of such that cutting along yields a
topological disk. We provide a fixed parameter tractable approximation scheme
for the problem of computing the shortest cut graph, that is, for any
, we show how to compute a approximation of
the shortest cut graph in time .
Our techniques first rely on the computation of a spanner for the problem
using the technique of brick decompositions, to reduce the problem to the case
of bounded tree-width. Then, to solve the bounded tree-width case, we introduce
a variant of the surface-cut decomposition of Ru\'e, Sau and Thilikos, which
may be of independent interest
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