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19
Percolation in the Hyperbolic Plane
, 2000
"... Following is a study of percolation in the hyperbolic plane H 2 and on regular tilings in the hyperbolic plane. The processes discussed include Bernoulli site and bond percolation on planar hyperbolic graphs, invariant dependent percolations on such graphs, and PoissonVoronoiBernoulli percolation ..."
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Cited by 43 (4 self)
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Following is a study of percolation in the hyperbolic plane H 2 and on regular tilings in the hyperbolic plane. The processes discussed include Bernoulli site and bond percolation on planar hyperbolic graphs, invariant dependent percolations on such graphs, and PoissonVoronoiBernoulli percolation. We prove the existence of three distinct nonempty phases for the Bernoulli processes. In the first phase, p ∈ (0, pc], there are no unbounded clusters, but there is a unique infinite cluster for the dual process. In the second phase, p ∈ (pc, pu), there are infinitely many unbounded clusters for the process and for the dual process. In the third phase, p∈[pu, 1), there is a unique unbounded cluster, and all the clusters of the dual process are bounded. We also study the dependence of pc in the PoissonVoronoiBernoulli percolation process on the intensity of the underlying Poisson process.
UNIQUENESS FOR THE SIGNATURE OF A PATH OF BOUNDED VARIATION AND THE REDUCED PATH GROUP
, 2006
"... Abstract. We introduce the notions of treelike path and treelike equivalence between paths and prove that the latter is an equivalence relation for paths of finite length. We show that the equivalence classes form a group with some similarity to a free group, and that in each class there is one sp ..."
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Cited by 29 (6 self)
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Abstract. We introduce the notions of treelike path and treelike equivalence between paths and prove that the latter is an equivalence relation for paths of finite length. We show that the equivalence classes form a group with some similarity to a free group, and that in each class there is one special tree reduced path. The set of these paths is the Reduced Path Group. It is a continuous analogue to the group of reduced words. The signature of the path is a power series whose coefficients are definite iterated integrals of the path. We identify the paths with trivial signature as the treelike paths, and prove that two paths are in treelike equivalence if and only if they have the same signature. In this way, we extend Chen’s theorems on the uniqueness of the sequence of iterated integrals associated with a piecewise regular path to finite length paths and identify the appropriate extended meaning for reparameterisation in the general setting. It is suggestive to think of this result as a noncommutative analogue of the result that integrable functions on the circle are determined, up to Lebesgue null sets, by their Fourier coefficients. As a second theme we give quantitative versions of Chen’s theorem in the case of lattice paths and paths with continuous derivative, and as a corollary derive results on the triviality of exponential products in the tensor algebra. 1.
The number of unbounded components in the Poisson Boolean model of continuum percolation in hyperbolic space
, 2007
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unknown title
, 2013
"... Morphological processing of univariate Gaussian distributionvalued images based on ..."
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Morphological processing of univariate Gaussian distributionvalued images based on
Morphological processing of univariate Gaussian
, 2013
"... distributionvalued images based on Poincare upperhalf plane representation ..."
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distributionvalued images based on Poincare upperhalf plane representation
Riemannian Mathematical Morphology
"... This paper introduces mathematical morphology operators for realvalued images whose support space is a Riemannian manifold. The starting point consists in replacing the Euclidean distance in the canonic quadratic structuring function by the Riemannian distance used for the adjoint dilation/erosion ..."
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This paper introduces mathematical morphology operators for realvalued images whose support space is a Riemannian manifold. The starting point consists in replacing the Euclidean distance in the canonic quadratic structuring function by the Riemannian distance used for the adjoint dilation/erosion. We then extend the canonic case to a most general framework of Riemannian operators based on the notion of admissible Riemannian structuring function. An alternative paradigm of morphological Riemannian operators involves an external structuring function which is parallel transported to each point on the manifold. Besides the definition of the various Riemannian dilation/erosion and Riemannian opening/closing, their main properties are studied. We generalize also some results on Lasry–Lions regularization for nonsmooth images on Cartan–Hadamard manifolds. Theoretical connections with previous works on adaptive morphology and manifold shape morphology are also considered. From a practical viewpoint, various useful image embedding into Riemannian manifolds are formalized, with some illustrative examples of morphological processing realvalued 3D surfaces.