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Person# Lionel Pournin

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Granular material

A granular material is a conglomeration of discrete solid, macroscopic particles characterized by a loss of energy whenever the particles interact (the most common example would be friction when gr

Experiment

An experiment is a procedure carried out to support or refute a hypothesis, or determine the efficacy or likelihood of something previously untried. Experiments provide insight into cause-and-effe

Computer simulation

Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of, or the outcome of, a real-world or physical system. The reliabi

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Flip-graph connectedness is established here for the vertex set of the 4-dimensional cube. It is found as a consequence, that this vertex set admits 92487256 triangulations, partitioned into 247451 symmetry classes.

Abstract: A simplicial complex C on a d-dimensional configuration of n points is k-regular if its faces are projected from the boundary complex of a polytope with dimension at most d+k. Since C is obviously (n-d-1)-regular, the set of all integers k for which C is k-regular is non-empty. The minimum δ(C) of this set deserves attention because of its link with flip-graph connectivity. This paper introduces a characterization of δ(C) derived from the theory of Gale transforms. Using this characterization, it is proven that δ(C) is never greater than n-d-2. Several new results on flip-graph connectivity follow. In particular, it is shown that connectedness does not always hold for the subgraph induced by 3-regular triangulations in the flip-graph of a point configuration.

A polyhedral subdivision of a d-dimensional point configuration A is k-regular if it is projected from the boundary complex of a polytope with dimension at most d+k. Call γk(A) the subgraph induced by k-regular triangulations in the flip-graph of A. Gel’fand, Kapranov, and Zelevinsky have shown that γ1(A) is connected. It is established here that γ2(A) is connected as well.