52,538 research outputs found
A categorical approach to the maximum theorem
Berge's maximum theorem gives conditions ensuring the continuity of an
optimised function as a parameter changes. In this paper we state and prove the
maximum theorem in terms of the theory of monoidal topology and the theory of
double categories.
This approach allows us to generalise (the main assertion of) the maximum
theorem, which is classically stated for topological spaces, to
pseudotopological spaces and pretopological spaces, as well as to closure
spaces, approach spaces and probabilistic approach spaces, amongst others. As a
part of this we prove a generalisation of the extreme value theorem.Comment: 45 pages. Minor changes in v2: this is the final preprint for
publication in JPA
Harmonic functions on hyperbolic graphs
We consider admissible random walks on hyperbolic graphs. For a given
harmonic function on such a graph, we prove that asymptotic properties of
non-tangential boundedness and non-tangential convergence are almost everywhere
equivalent. The proof is inspired by the works of F. Mouton in the cases of
Riemannian manifolds of pinched negative curvature and infinite trees. It
involves geometric and probabilitistic methods.Comment: 14 page
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