132,451 research outputs found
Max-Flow on Regular Spaces
The max-flow and max-coflow problem on directed graphs is studied in the
common generalization to regular spaces, i.e., to kernels or row spaces of
totally unimodular matrices. Exhibiting a submodular structure of the family of
paths within this model we generalize the Edmonds-Karp variant of the classical
Ford-Fulkerson method and show that the number of augmentations is
quadratically bounded if augmentations are chosen along shortest possible
augmenting paths
Geometry and Analysis of Dirichlet forms
Let be a regular, strongly local Dirichlet form on
and the associated intrinsic distance. Assume that the topology induced by
coincides with the original topology on , and that is compact,
satisfies a doubling property and supports a weak -Poincar\'e
inequality. We first discuss the (non-)coincidence of the intrinsic length
structure and the gradient structure. Under the further assumption that the
Ricci curvature of is bounded from below in the sense of
Lott-Sturm-Villani, the following are shown to be equivalent:
(i) the heat flow of gives the unique gradient flow of ,
(ii) satisfies the Newtonian property,
(iii) the intrinsic length structure coincides with the gradient structure.
Moreover, for the standard (resistance) Dirichlet form on the Sierpinski
gasket equipped with the Kusuoka measure, we identify the intrinsic length
structure with the measurable Riemannian and the gradient structures. We also
apply the above results to the (coarse) Ricci curvatures and asymptotics of the
gradient of the heat kernel.Comment: Advance in Mathematics, to appear,51p
The surface diffusion and the Willmore flow for uniformly regular hypersurfaces
We consider the surface diffusion and Willmore flows acting on a general
class of (possibly non-compact) hypersurfaces parameterized over a uniformly
regular reference manifold possessing a tubular neighborhood with uniform
radius. The surface diffusion and Willmore flows each give rise to a
fourth-order quasilinear parabolic equation with nonlinear terms satisfying a
specific singular structure. We establish well-posedness of both flows for
initial surfaces that are -regular and parameterized over a
uniformly regular hypersurface. For the Willmore flow, we also show long-term
existence for initial surfaces which are -close to a sphere, and
we prove that these solutions become spherical as time goes to infinity.Comment: 22 page
Maximal Hamiltonian tori for polygon spaces
We study the poset of Hamiltonian tori for polygon spaces. We determine some
maximal elements and give examples where maximal Hamiltonian tori are not all
of the same dimension.Comment: 15 pages, Latex, 1 figur
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