10,508 research outputs found
A generalization of Alternating Sign Matrices
In alternating sign matrices the first and last nonzero entry in each row and
column is specified to be +1.
Such matrices always exist. We investigate a generalization by specifying
independently the sign of the first and last nonzero entry in each row and
column to be either a +1 or a -1. We determine necessary and sufficient
conditions for such matrices to exist.Comment: 14 page
Optimal transportation, topology and uniqueness
The Monge-Kantorovich transportation problem involves optimizing with respect
to a given a cost function. Uniqueness is a fundamental open question about
which little is known when the cost function is smooth and the landscapes
containing the goods to be transported possess (non-trivial) topology. This
question turns out to be closely linked to a delicate problem (# 111) of
Birkhoff [14]: give a necessary and sufficient condition on the support of a
joint probability to guarantee extremality among all measures which share its
marginals. Fifty years of progress on Birkhoff's question culminate in Hestir
and Williams' necessary condition which is nearly sufficient for extremality;
we relax their subtle measurability hypotheses separating necessity from
sufficiency slightly, yet demonstrate by example that to be sufficient
certainly requires some measurability. Their condition amounts to the vanishing
of the measure \gamma outside a countable alternating sequence of graphs and
antigraphs in which no two graphs (or two antigraphs) have domains that
overlap, and where the domain of each graph / antigraph in the sequence
contains the range of the succeeding antigraph (respectively, graph). Such
sequences are called numbered limb systems. We then explain how this
characterization can be used to resolve the uniqueness of Kantorovich solutions
for optimal transportation on a manifold with the topology of the sphere.Comment: 36 pages, 6 figure
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