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Discrete Laplace operators: No free lunch
, 2007
"... Discrete Laplace operators are ubiquitous in applications spanning geometric modeling to simulation. For robustness and efficiency, many applications require discrete operators that retain key structural properties inherent to the continuous setting. Building on the smooth setting, we present a set ..."
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Cited by 28 (0 self)
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Discrete Laplace operators are ubiquitous in applications spanning geometric modeling to simulation. For robustness and efficiency, many applications require discrete operators that retain key structural properties inherent to the continuous setting. Building on the smooth setting, we present a set of natural properties for discrete Laplace operators for triangular surface meshes. We prove an important theoretical limitation: discrete Laplacians cannot satisfy all natural properties; retroactively, this explains the diversity of existing discrete Laplace operators. Finally, we present a family of operators that includes and extends wellknown and widelyused operators.
Vertical Decomposition of Arrangements of Hyperplanes in Four Dimensions
, 1995
"... We show that, for any collection H of n hyperplanes in ! 4 , the combinatorial complexity of the vertical decomposition of the arrangement A(H) of H is O(n 4 log n). The proof relies on properties of superimposed convex subdivisions of 3space, and we also derive some other results concerning them. ..."
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Cited by 8 (5 self)
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We show that, for any collection H of n hyperplanes in ! 4 , the combinatorial complexity of the vertical decomposition of the arrangement A(H) of H is O(n 4 log n). The proof relies on properties of superimposed convex subdivisions of 3space, and we also derive some other results concerning them.
Poisson Power Tesselations
, 1994
"... : We consider generalization of the Voronoi diagram  power diagram, constructed with respect to a Poisson processes with i.i.d. marks (weights). We give first moment of the volume distribution of a typical cell, the probability that a cell is empty, the mean length of a typical edge in the planar c ..."
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Cited by 1 (1 self)
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: We consider generalization of the Voronoi diagram  power diagram, constructed with respect to a Poisson processes with i.i.d. marks (weights). We give first moment of the volume distribution of a typical cell, the probability that a cell is empty, the mean length of a typical edge in the planar case and other geometrical characteristics of this tesselation. AMS 1991 Subject Classification: 60D05, 52C17, 60G55 Keywords: Power diagrams, Voronoi tesselation, Poisson process, Planar tesselations (R'esum'e : tsvp) email: zouev@sophia.inria.fr Unite de recherche INRIA SophiaAntipolis 2004 route des Lucioles, BP 93, 06902 SOPHIAANTIPOLIS Cedex (France) Telephone : (33) 93 65 77 77  Telecopie : (33) 93 65 77 65 Tesselations poissoniennes de la puissance R'esum'e : On consid`ere le diagramme de puissance. Il s'agit d'une g'en'eralisation de la tess'elation de Voronoi construite par rapport `a un processus poissonien marqu'e par des poids i.i.d. Nous donnons les premiers moments ...
Properties of parallelotopes equivalent to Voronoi’s conjecture
, 2003
"... A parallelotope is a polytope whose translation copies fill space without gaps and intersections by interior points. Voronoi conjectured that each parallelotope is an affine image of the Dirichlet domain of a lattice, which is a Voronoi polytope. We give several properties of a parallelotope and pro ..."
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Cited by 1 (0 self)
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A parallelotope is a polytope whose translation copies fill space without gaps and intersections by interior points. Voronoi conjectured that each parallelotope is an affine image of the Dirichlet domain of a lattice, which is a Voronoi polytope. We give several properties of a parallelotope and prove that each of them is equivalent to it is an affine image of a Voronoi polytope. 1
On Traces of dstresses in the Skeletons of Lower Dimensions of Piecewiselinear dmanifolds
, 2001
"... We show how a dstress on a piecewiselinear realization of an oriented (nonsimplicial, in general) dmanifold in R d naturally induces stresses of lower dimensions on this manifold, and discuss implications of this construction to the analysis of selfstresses in spatial frameworks. The mappings w ..."
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We show how a dstress on a piecewiselinear realization of an oriented (nonsimplicial, in general) dmanifold in R d naturally induces stresses of lower dimensions on this manifold, and discuss implications of this construction to the analysis of selfstresses in spatial frameworks. The mappings we construct are not linear, but polynomial. In the 1860–70s J. C. Maxwell described an interesting relationship between selfstresses in planar frameworks and vertical projections of polyhedral 2surfaces. We offer a spatial analog of Maxwell’s correspondence based on our polynomial mappings. By applying our main result we derive a class of threedimensional spider webs similar to the twodimensional spider webs described by Maxwell. We also conjecture an important property of our mappings that is Author: based on the lower bound theorem (g2(d +1) = dim Stress2 ≥ 0) for dpseudomanifolds generically realized in R d+1 [12].
Routing with Guaranteed Delivery . . .
, 2006
"... We propose four simple algorithms for routing on planar graphs using virtual coordinates. These algorithms are superior to existing algorithms in that they are oblivious, work also for nontriangular graphs, and their virtual coordinates are easy to construct. ..."
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We propose four simple algorithms for routing on planar graphs using virtual coordinates. These algorithms are superior to existing algorithms in that they are oblivious, work also for nontriangular graphs, and their virtual coordinates are easy to construct.
Computing SelfSupporting Surfaces by Regular Triangulation
"... and their corresponding power cells are colored in orange. Top right: initial selfsupporting mesh. Spikes appear due to extremely small reciprocal areas. Bottom right: applying our smoothing scheme (5 iterations) improves mesh quality. The power diagrams (black) show how power cell area is distribu ..."
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and their corresponding power cells are colored in orange. Top right: initial selfsupporting mesh. Spikes appear due to extremely small reciprocal areas. Bottom right: applying our smoothing scheme (5 iterations) improves mesh quality. The power diagrams (black) show how power cell area is distributed more evenly. Masonry structures must be compressively selfsupporting; designing such surfaces forms an important topic in architecture as well as a challenging problem in geometric modeling. Under certain conditions, a surjective mapping exists between a power diagram, defined by a set of 2D vertices and associated weights, and the reciprocal diagram that characterizes the force diagram of a discrete selfsupporting network. This observation lets us define a new and convenient parameterization for the space of selfsupporting networks. Based on it and the discrete geometry of this design space, we present novel geometry processing methods including surface smoothing and remeshing which significantly reduce the magnitude of force densities and homogenize their distribution.
Centre de Mathématiques et de Leurs Applications
"... We provide consistent random algorithms for sequential decision under partial monitoring, when the decision maker does not observe the outcomes but receives instead random feedback signals. Those algorithms have no internal regret in the sense that, on the set of stages where the decision maker chos ..."
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We provide consistent random algorithms for sequential decision under partial monitoring, when the decision maker does not observe the outcomes but receives instead random feedback signals. Those algorithms have no internal regret in the sense that, on the set of stages where the decision maker chose his action according to a given law, the average payoff could not have been improved in average by using any other fixed law. They are based on a generalization of calibration, no longer defined in terms of a Voronoï diagram but instead of a Laguerre diagram (a more general concept). This allows us to bound, for the first time in this general framework, the expected average internal, as well as the usual external, regret at stage n by O(n −1/3), which is known to be optimal.