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ON MOMENTS OF A POLYTOPE
"... Abstract. We show that the multivariate generating function of appropriately normalized moments of a measure with homogeneous polynomial denisity supported on a compact polytope P ⊂ R d is a rational function. Its denominator is the product of linear forms dual to the vertices of P raised to the pow ..."
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Abstract. We show that the multivariate generating function of appropriately normalized moments of a measure with homogeneous polynomial denisity supported on a compact polytope P ⊂ R d is a rational function. Its denominator is the product of linear forms dual to the vertices of P raised to the power equal to the degree of the density function. Using this, we solve the inverse moment problem for the set of, not necessarily convex, polytopes having a given set S of vertices. Under a weak nondegeneracy assumption we also show that the uniform measure supported on any such polytope is a linear combination of uniform measures supported on simplices with vertices in S. 1.
Bipartite rigidity
"... We develop a bipartite rigidity theory for bipartite graphs parallel to the classical rigidity theory for general graphs, and define for two positive integers k, l the notions of (k, l)rigid and (k, l)stress free bipartite graphs. This theory coincides with the study of Babson–Novik’s balanced sh ..."
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We develop a bipartite rigidity theory for bipartite graphs parallel to the classical rigidity theory for general graphs, and define for two positive integers k, l the notions of (k, l)rigid and (k, l)stress free bipartite graphs. This theory coincides with the study of Babson–Novik’s balanced shifting restricted to graphs. We establish bipartite analogs of the cone, contraction, deletion, and gluing lemmas, and apply these results to derive a bipartite analog of the rigidity criterion for planar graphs. Our result asserts that for a planar bipartite graph G its balanced shifting, Gb, does not contain K3,3; equivalently, planar bipartite graphs are generically (2, 2)stress free. We also discuss potential applications of this theory to Jockusch’s cubical lower bound conjecture and to upper bound conjectures for embedded simplicial complexes.
Converse SturmHurwitzKellogg theorem and related results
, 2008
"... We prove that if V n is a Chebyshev system on the circle and f(x) is a continuous function with at least n + 1 sign changes then there exists an orientation preserving diffeomorphism of S 1 that takes f to a function L 2orthogonal to V. We also prove that if f(x) is a function on the real projectiv ..."
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We prove that if V n is a Chebyshev system on the circle and f(x) is a continuous function with at least n + 1 sign changes then there exists an orientation preserving diffeomorphism of S 1 that takes f to a function L 2orthogonal to V. We also prove that if f(x) is a function on the real projective line with at least four sign changes then there exists an orientation preserving diffeomorphism of RP 1 that takes f to the Schwarzian derivative of a function on RP 1. We show that the space of piecewise constant functions on an interval with values ±1 and at most n + 1 intervals of constant sign is homeomorphic to ndimensional sphere. To V. I. Arnold for his 70th birthday 1 Introduction and formulation of results The classic four vertex theorem asserts that the curvature of a plane oval (strictly convex smooth closed curve) has at least four extrema. Discovered