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168
Orbihedra Of Nonpositive Curvature
 Progress in Mathematics
, 1995
"... . A 2dimensional orbihedron of nonpositive curvature is a pair (X; \Gamma), where X is a 2dimensional simplicial complex with a piecewise smooth metric such that X has nonpositive curvature in the sense of Alexandrov and Busemann and \Gamma is a group of isometries of X which acts properly disc ..."
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Cited by 155 (7 self)
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. A 2dimensional orbihedron of nonpositive curvature is a pair (X; \Gamma), where X is a 2dimensional simplicial complex with a piecewise smooth metric such that X has nonpositive curvature in the sense of Alexandrov and Busemann and \Gamma is a group of isometries of X which acts properly discontinuously and cocompactly. By analogy with Riemannian manifolds of nonpositive curvature we introduce a natural notion of rank 1 for (X; \Gamma) which turns out to depend only on \Gamma and prove that, if X is boundaryless, then either (X; \Gamma) has rank 1, or X is the product of two trees, or X is a thick Euclidean building. In the first case the geodesic flow on X is topologically transitive and closed geodesics are dense. 1. Introduction The idea of considering curvature bounds on metric spaces belongs to Alexandrov [Ale], Busemann [Bus] and Wald [Wal]. Busemann initiated the theory of spaces of nonpositive curvature. Later, Bruhat and Tits [BrTi] showed that there is a natural ...
Probability on trees and networks
, 2011
"... with Yuval PeresA love and respect of trees has been characteristic of mankind since the beginning of human evolution. Instinctively, we understood the importance of trees to our lives before we were able to ascribe reasons for our dependence on them. — America’s Garden Book, James and Louise BushB ..."
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Cited by 87 (8 self)
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with Yuval PeresA love and respect of trees has been characteristic of mankind since the beginning of human evolution. Instinctively, we understood the importance of trees to our lives before we were able to ascribe reasons for our dependence on them. — America’s Garden Book, James and Louise BushBrown, rev. ed. by The New York Botanical Garden, Charles Scribner’s
Spectral Theory Of Elliptic Operators On NonCompact Manifolds
, 1992
"... preliminaries Let H be a complex Hilbert space, A a densely dened linear operator in H (the domain of A will be denoted D(A)). Suppose that A has a closure A or, equivalently, that the adjoint operator A is densely dened (see e.g. [32]). We shall denote by GA the graph of A i.e. the set of pairs ..."
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Cited by 61 (9 self)
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preliminaries Let H be a complex Hilbert space, A a densely dened linear operator in H (the domain of A will be denoted D(A)). Suppose that A has a closure A or, equivalently, that the adjoint operator A is densely dened (see e.g. [32]). We shall denote by GA the graph of A i.e. the set of pairs fu; Aug; u 2 D(A). Then G A = GA , i.e. the graph of A is the closure of the graph of A. Moreover A = A = (A ) . Now let A + be another densely dened linear operator in H. DEFINITION 1.1. A + is called formally adjoint to A if (1:1) (Au; v) = (u; A + v); u 2 D(A); v 2 D(A + ); where (; ) is the scalar product in H. If A = A + then A is called symmetric or formally self{adjoint. Note that since A; A + are densely dened, both A and A + have closures. DEFINITION 1.2. Let A; A + be as in Denition 1.1. Then the minimal and the maximal operator for A are dened as follows: A min = A = A ; A max = (A + ) : Note that both A min and A max are...
Contractions in the 2Wasserstein Length Space and Thermalization of Granular Media
, 2004
"... An algebraic decay rate is derived which bounds the time required for velocities to equilibrate in a spatially homogeneous flowthrough model representing the continuum limit of a gas of particles interacting through slightly inelastic collisions. This rate is obtained by reformulating the dynamical ..."
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Cited by 54 (19 self)
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An algebraic decay rate is derived which bounds the time required for velocities to equilibrate in a spatially homogeneous flowthrough model representing the continuum limit of a gas of particles interacting through slightly inelastic collisions. This rate is obtained by reformulating the dynamical problem as the gradient flow of a convex energy on an infinitedimensional manifold. An abstract theory is developed for gradient flows in length spaces, which shows how degenerate convexity (or even nonconvexity) — if uniformly controlled — will quantify contractivity (limit expansivity) of the flow.
Efficient Computation of IsometryInvariant Distances between Surfaces
"... We present an efficient computational framework for isometryinvariant comparison of smooth surfaces. We formulate the GromovHausdorff distance as a multidimensional scaling (MDS)like continuous optimization problem. In order to construct an efficient optimization scheme, we develop a numerical ..."
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Cited by 52 (18 self)
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We present an efficient computational framework for isometryinvariant comparison of smooth surfaces. We formulate the GromovHausdorff distance as a multidimensional scaling (MDS)like continuous optimization problem. In order to construct an efficient optimization scheme, we develop a numerical tool for interpolating geodesic distances on a sampled surface from precomputed geodesic distances between the samples. For isometryinvariant comparison of surfaces in the case of partially missing data, we present the partial embedding distance, which is computed using a similar scheme. The main idea is finding a minimumdistortion mapping from one surface to another, while considering only relevant geodesic distances. We discuss numerical implementation issues and present experimental results that demonstrate its accuracy and efficiency.
On the convergence of metric and geometric properties of polyhedral surfaces
 GEOMETRIAE DEDICATA
, 2005
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Universal bounds for hyperbolic Dehn surgery
 Annals of Math
, 2005
"... Abstract. This paper gives a quantitative version of Thurston’s hyperbolic Dehn surgery theorem. Applications include the first universal bounds on the number of nonhyperbolic Dehn fillings on a cusped hyperbolic 3manifold, and estimates on the changes in volume and core geodesic length during hyp ..."
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Cited by 39 (2 self)
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Abstract. This paper gives a quantitative version of Thurston’s hyperbolic Dehn surgery theorem. Applications include the first universal bounds on the number of nonhyperbolic Dehn fillings on a cusped hyperbolic 3manifold, and estimates on the changes in volume and core geodesic length during hyperbolic Dehn filling. The proofs involve the construction of a family of hyperbolic conemanifold structures, using infinitesimal harmonic deformations and analysis of geometric limits. 1.
The geometry of dynamical triangulations
 Lecture Notes in Physics m50
, 1997
"... The express purpose of these Lecture Notes is to go through some aspects of the simplicial quantum gravity model known as the Dynamical Triangulations approach. Emphasis has been on lying the foundations of the theory and on illustrating its subtle and often unexplored connections with many distinct ..."
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Cited by 35 (6 self)
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The express purpose of these Lecture Notes is to go through some aspects of the simplicial quantum gravity model known as the Dynamical Triangulations approach. Emphasis has been on lying the foundations of the theory and on illustrating its subtle and often unexplored connections with many distinct mathematical fields ranging from global riemannian geometry, moduli theory, number theory, and topology. Our exposition will concentrate on these points so that graduate students may find in these notes a useful exposition of some of the rigorous results one can establish in this field and hopefully a source of inspiration for new exciting problems. We also illustrate the deep and beautiful interplay between the analytical aspects of dynamical triangulations and the results of MonteCarlo simulations. The techniques described here are rather novel and allow us to address successfully many high points of great current interest in the subject of simplicial quantum gravity while requiring very lit1 tle in the way of fancy field theoretical arguments. As a consequence, these