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42
How many zeros of a random polynomial are real
 Bull. Amer. Math. Soc. (N.S
, 1995
"... Abstract. We provide an elementary geometric derivation of the Kac integral formula for the expected number of real zeros of a random polynomial with independent standard normally distributed coefficients. We show that the expected number of real zeros is simply the length of the moment curve (1, t, ..."
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Cited by 81 (0 self)
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Abstract. We provide an elementary geometric derivation of the Kac integral formula for the expected number of real zeros of a random polynomial with independent standard normally distributed coefficients. We show that the expected number of real zeros is simply the length of the moment curve (1, t,..., t n) projected onto the surface of the unit sphere, divided by π. The probability density of the real zeros is proportional to how fast this curve is traced out. We then relax Kac’s assumptions by considering a variety of random sums, series, and distributions, and we also illustrate such ideas as integral geometry and the FubiniStudy metric. Contents 1.
Random matrix theory
, 2005
"... Random matrix theory is now a big subject with applications in many disciplines of science, engineering and finance. This article is a survey specifically oriented towards the needs and interests of a numerical analyst. This survey includes some original material not found anywhere else. We includ ..."
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Cited by 80 (4 self)
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Random matrix theory is now a big subject with applications in many disciplines of science, engineering and finance. This article is a survey specifically oriented towards the needs and interests of a numerical analyst. This survey includes some original material not found anywhere else. We include the important mathematics which is a very modern development, as well as the computational software that is transforming the theory into useful practice.
Annular noncrossing permutations and partitions, and secondorder asymptotics for random matrices
, 2003
"... ..."
Irreducible Symmetric Group Characters of Rectangular Shape
, 2002
"... this paper we give a new formula for the values of the character . Let be a partition of k n, and let (; 1 ) be the partition obtained by adding n k 1's to . Thus (; 1 ) ` n. De ne the normalized character ) by (n) k where ) denotes the dimension of the chara ..."
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Cited by 24 (1 self)
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this paper we give a new formula for the values of the character . Let be a partition of k n, and let (; 1 ) be the partition obtained by adding n k 1's to . Thus (; 1 ) ` n. De ne the normalized character ) by (n) k where ) denotes the dimension of the character and (n) k = n(n 1) (n k + 1). Thus [5, (7.6)(ii)][8, p. 349] ) is the number f of standard Young tableaux of shape . Identify with its diagram f(i; j) : 1 j i g, and regard the points (i; j) 2 as squares (forming the Young diagram of ). We write diagrams in \English notation," with the rst coordinate increasing from top to bottom and the second coordinate from left to right. Let = ( 1 ; 2 ; : : :) and = ( 1 ; 2 ; : : :), where is the Partially supported by NSF grant #DMS9988459 and by the Isaac Newton Institute for Mathematical Sciences. conjugate partition to . The hook length of the square u = (i; j) 2 is de ned by h(u) = i + j i j + 1; and the FrameRobinsonThrall hook length formula [5, Exam. I.5.2][8, Cor. 7.21.6] states that n! Q u2 h(u) : For w 2 S n let (w) denote the number of cycles of w (in the disjoint cycle decomposition of w). The main result of this paper is the following
All invariant moments of the Wishart distribution
, 2004
"... In this paper, we compute moments of a Wishart matrix variate U of the form E(Q(U)) where Q(u) is a polynomial with respect to the entries of the symmetric matrix u, invariant in the sense that it depends only on the eigenvalues of the matrix u. This gives us in particular the expectedvalue of any ..."
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Cited by 17 (1 self)
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In this paper, we compute moments of a Wishart matrix variate U of the form E(Q(U)) where Q(u) is a polynomial with respect to the entries of the symmetric matrix u, invariant in the sense that it depends only on the eigenvalues of the matrix u. This gives us in particular the expectedvalue of any power of the Wishart matrix U or its inverse U1. For our proofs, we do not rely on traditional combinatorial methods but rather on the interplay between two bases of the space of invariant polynomials in U. This means that all moments can be obtainedthrough the multiplication of three matrices with known entries. Practically, the moments are obtainedby computer with an extremely simple Maple program.
Connection coefficients, matchings, maps and combinatorial conjectures for Jack symmetric functions
 TRANS. AMER. MATH. SOC
, 1996
"... A power series is introduced that is an extension to three sets of variables of the Cauchy sum for Jack symmetric functions in the Jack parameter α. We conjecture that the coefficients of this series with respect to the power sum basis are nonnegative integer polynomials in b, the Jack parameter sh ..."
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Cited by 16 (5 self)
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A power series is introduced that is an extension to three sets of variables of the Cauchy sum for Jack symmetric functions in the Jack parameter α. We conjecture that the coefficients of this series with respect to the power sum basis are nonnegative integer polynomials in b, the Jack parameter shifted by 1. More strongly, we make the MatchingsJack Conjecture, that the coefficients are counting series in b for matchings with respect to a parameter of nonbipartiteness. Evidence is presented for these conjectures and they are proved for two infinite families. The coefficients of a second series, essentially the logarithm of the first, specialize at values 1 and 2 of the Jack parameter to the numbers of hypermaps in orientable and locally orientable surfaces, respectively. We conjecture that these coefficients are also nonnegative integer polynomials in b, andwemake the HypermapJack Conjecture, that the coefficients are counting series in b for hypermaps in locally orientable surfaces with respect to a parameter of nonorientability.
Gaussianization and eigenvalue statistics for random quantum channels (III)
 ANN. APPL. PROBAB
, 2009
"... In this paper, we present applications of the calculus developed in [9], and obtain an exact formula for the moments of random quantum channels whose input is a pure state thanks to gaussianization methods. Our main application is an indepth study of the random matrix model introduced by Hayden a ..."
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Cited by 15 (9 self)
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In this paper, we present applications of the calculus developed in [9], and obtain an exact formula for the moments of random quantum channels whose input is a pure state thanks to gaussianization methods. Our main application is an indepth study of the random matrix model introduced by Hayden and Winter and used recently by Brandao, Horodecki, Fukuda and King to refine the Hastings counterexample to the additivity conjecture in Quantum Information Theory. This model is exotic from the point of view of random matrix theory, as its eigenvalues obey to two different scalings simultaneously. We study its asymptotic behavior and obtain an asymptotic expansion for its von Neumann entropy.