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16
BERNSTEIN POLYNOMIALS, BERGMAN KERNELS AND TORIC KÄHLER VARIETIES
, 2008
"... We show that the classical Bernstein polynomials BN(f)(x) on the interval [0, 1] (and their higher dimensional generalizations on the simplex Σm ⊂ R m) may be expressed in terms of Bergman kernels for the FubiniStudy metric on CP m: BN(f)(x) is obtained by applying the Toeplitz operator f(N −1 Dθ ..."
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Cited by 7 (3 self)
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We show that the classical Bernstein polynomials BN(f)(x) on the interval [0, 1] (and their higher dimensional generalizations on the simplex Σm ⊂ R m) may be expressed in terms of Bergman kernels for the FubiniStudy metric on CP m: BN(f)(x) is obtained by applying the Toeplitz operator f(N −1 Dθ) to the FubiniStudy Bergman kernels. The expression generalizes immediately to any toric Kähler variety and Delzant polytope, and gives a novel definition of Bernstein ‘polynomials ’ B h N(f) relative to any toric Kähler variety. They uniformly approximate any continuous function f on the associated polytope P with all the properties of classical Bernstein polynomials. Upon integration over the polytope one
COUNTEREXAMPLES TO OKOUNKOV’S LOGCONCAVITY CONJECTURE
, 2007
"... Abstract. We give counterexamples to Okounkov’s logconcavity conjecture for Littlewood ..."
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Cited by 7 (1 self)
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Abstract. We give counterexamples to Okounkov’s logconcavity conjecture for Littlewood
Convergence rates of random walk on irreducible representations of finite groups
 J. Theoret. Probab
"... Abstract. Random walk on the set of irreducible representations of a finite group is investigated. For the symmetric and general linear groups, a sharp convergence rate bound is obtained and a cutoff phenomenon is proved. As a related result, an asymptotic description of Plancherel measure of the fi ..."
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Cited by 6 (3 self)
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Abstract. Random walk on the set of irreducible representations of a finite group is investigated. For the symmetric and general linear groups, a sharp convergence rate bound is obtained and a cutoff phenomenon is proved. As a related result, an asymptotic description of Plancherel measure of the finite general linear groups is given. 1.
ASYMPTOTICS FOR THE NUMBER OF WALKS IN A WEYL CHAMBER OF TYPE B
, 906
"... Abstract. We consider lattice walks in R k confined to the region 0 < x1 < x2... < xk with fixed (but arbitrary) starting and end points. The walks are required to be ”reflectable”, that is, we assume that the number of paths can be counted using the reflection principle. The main result is ..."
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Cited by 2 (1 self)
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Abstract. We consider lattice walks in R k confined to the region 0 < x1 < x2... < xk with fixed (but arbitrary) starting and end points. The walks are required to be ”reflectable”, that is, we assume that the number of paths can be counted using the reflection principle. The main result is an asymptotic formula for the total number of walks of length n for a general class of walks as n tends to infinity. As applications, we find the asymptotics for the number of knoncrossing tangled diagrams on the set {1, 2,...,n} as n tends to infinity, the asymptotics for the number of kvicious walkers subject to a wall restriction in the random turns model, and we rederive known asymptotics for the number of kvicious walkers subject to a wall restriction in the lock step model. 1.
Bernstein measures on convex polytopes
, 805
"... We define the notion of Bernstein measures and Bernstein approximations over general convex polytopes. This generalizes wellknown Bernstein polynomials which are used to prove the Weierstrass approximation theorem on one dimensional intervals. We discuss some properties of Bernstein measures and ap ..."
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We define the notion of Bernstein measures and Bernstein approximations over general convex polytopes. This generalizes wellknown Bernstein polynomials which are used to prove the Weierstrass approximation theorem on one dimensional intervals. We discuss some properties of Bernstein measures and approximations, and prove an asymptotic expansion of the Bernstein approximations for smooth functions which is a generalization of the asymptotic expansion of the Bernstein polynomials on the standard msimplex obtained by AbelIvan and Hörmander. These are different from the BergmanBernstein approximations over Delzant polytopes recently introduced by Zelditch. We discuss relations between Bernstein approximations defined in this paper and Zelditch’s BergmanBernstein approximations. 1
A spectral analogue of the Meinardus theorem on asymptotics of the number of partitions
, 2007
"... ..."
unknown title
, 705
"... Abstract. We show that the classical Bernstein polynomials BN(f)(x) on the interval [0, 1] (and their higher dimensional generalizations on the simplex Σm ⊂ R m) may be expressed in terms of Bergman kernels for the FubiniStudy metric on CP m: BN(f)(x) is obtained by applying the Toeplitz operator f ..."
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Abstract. We show that the classical Bernstein polynomials BN(f)(x) on the interval [0, 1] (and their higher dimensional generalizations on the simplex Σm ⊂ R m) may be expressed in terms of Bergman kernels for the FubiniStudy metric on CP m: BN(f)(x) is obtained by applying the Toeplitz operator f(N −1 Dθ) to the FubiniStudy Bergman kernels. The expression generalizes immediately to any toric Kähler variety and Delzant polytope, and gives a novel definition of Bernstein ‘polynomials ’ B h N(f) relative to any toric Kähler variety. They uniformly approximate any continuous function f on the associated polytope P with all the properties of classical Bernstein polynomials. Upon integration over the polytope one
unknown title
, 2008
"... Random walks of partons in SU(Nc) and classical representations of color charges in QCD at small x ..."
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Random walks of partons in SU(Nc) and classical representations of color charges in QCD at small x
ASYMPTOTICS OF MATRIX INTEGRALS AND TENSOR INVARIANTS OF COMPACT LIE GROUPS
, 2006
"... Abstract. In this paper we give an asymptotic formula for a matrix integral which plays a crucial role in the approach of Diaconis et al. to random matrix eigenvalues. The choice of parameter for the asymptotic analysis is motivated by an invariant theoretic interpretation of this type of integral. ..."
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Abstract. In this paper we give an asymptotic formula for a matrix integral which plays a crucial role in the approach of Diaconis et al. to random matrix eigenvalues. The choice of parameter for the asymptotic analysis is motivated by an invariant theoretic interpretation of this type of integral. For arbitrary regular irreducible representations of arbitrary connected semisimple compact Lie groups, we obtain an asymptotic formula for the trace of certain operators on the space of tensor invariants, thus extending a result of Biane on the dimension of these spaces. 1.
Asymptotics for Walks in a Weyl chamber of Type B (extended abstract) †
"... We consider lattice walks in R k confined to the region 0 < x1 < x2... < xk with fixed (but arbitrary) starting and end points. The walks are required to be ”reflectable”, that is, we assume that the number of paths can be counted using the reflection principle. The main result is an asympt ..."
Abstract
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We consider lattice walks in R k confined to the region 0 < x1 < x2... < xk with fixed (but arbitrary) starting and end points. The walks are required to be ”reflectable”, that is, we assume that the number of paths can be counted using the reflection principle. The main result is an asymptotic formula for the total number of walks of length n with fixed but arbitrary starting and end point for a general class of walks as the number n of steps tends to infinity. As applications, we find the asymptotics for the number of knoncrossing tangled diagrams on the set {1, 2,..., n} as n tends to infinity, and asymptotics for the number of kvicious walkers subject to a wall restriction in the random turns model as well as in the lock step model. Asymptotics for all of these objects were either known only for certain special cases, or have only been partially determined.