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Random matrix theory and Lfunctions at s =1=2
 Comm. Math Phys
"... random matrix theory, number theory, Lfunctions Recent results of Katz and Sarnak [9,10] suggest that the lowlying zeros of families of Lfunctions display the statistics of the eigenvalues of one of the compact groups of matrices U (N), O(N) or U Sp (2 N). We here explore the link between the valu ..."
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Cited by 41 (8 self)
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random matrix theory, number theory, Lfunctions Recent results of Katz and Sarnak [9,10] suggest that the lowlying zeros of families of Lfunctions display the statistics of the eigenvalues of one of the compact groups of matrices U (N), O(N) or U Sp (2 N). We here explore the link between the value distributions of the Lfunctions within these families at the central point s = 1/2 and those of the characteristic polynomials Z (U, 3) of matrices U with respect to averages over SO (2N) and U Sp (2N) at the corresponding point 3 = 0, using techniques previously developed for U (N) in [7]. For any matrix size N we find exact expressions for the moments of Z (U, 0) for each ensemble, and hence calculate the asymptotic (large N) value distributions for Z (U, 0) and log Z (U, 0). The asymptotic results for the integer moments agree precisely with the few corresponding values known for Lfunctions. The value distributions suggest consequences for the nonvanishing of Lfunctions at the central point. 1
Autocorrelation of random matrix polynomials
 COMMUN. MATH. PHYS
, 2003
"... We calculate the autocorrelation functions (or shifted moments) of the characteristic polynomials of matrices drawn uniformly with respect to Haar measure from the groups U(N), O(2N) and USp(2N). In each case the result can be expressed in three equivalent forms: as a determinant sum (and hence in t ..."
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Cited by 37 (20 self)
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We calculate the autocorrelation functions (or shifted moments) of the characteristic polynomials of matrices drawn uniformly with respect to Haar measure from the groups U(N), O(2N) and USp(2N). In each case the result can be expressed in three equivalent forms: as a determinant sum (and hence in terms of symmetric polynomials), as a combinatorial sum, and as a multiple contour integral. These formulae are analogous to those previously obtained for the Gaussian ensembles of Random Matrix Theory, but in this case are identities for any size of matrix, rather than largematrix asymptotic approximations. They also mirror exactly the autocorrelation formulae conjectured to hold for Lfunctions in a companion paper. This then provides further evidence in support of the connection between Random Matrix Theory and the theory of Lfunctions.
Inversion of the Pieri formula for Macdonald polynomials
 Adv. Math
"... We give the explicit analytic development of Macdonald polynomials in terms of “modified complete ” and elementary symmetric functions. These expansions are obtained by inverting the Pieri formula. Specialization yields similar developments for monomial, Jack and Hall–Littlewood symmetric functions. ..."
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Cited by 36 (13 self)
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We give the explicit analytic development of Macdonald polynomials in terms of “modified complete ” and elementary symmetric functions. These expansions are obtained by inverting the Pieri formula. Specialization yields similar developments for monomial, Jack and Hall–Littlewood symmetric functions.
Projectively equivariant symbol calculus
, 1999
"... The spaces of linear differential operators Dλ(R n) acting on λdensities on R n and the space Pol(T ∗ R n) of functions on T ∗ R n which are polynomial on the fibers are not isomorphic as modules over the Lie algebra Vect(R n) of vector fields of R n. However, these modules are isomorphic as sl(n + ..."
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Cited by 35 (2 self)
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The spaces of linear differential operators Dλ(R n) acting on λdensities on R n and the space Pol(T ∗ R n) of functions on T ∗ R n which are polynomial on the fibers are not isomorphic as modules over the Lie algebra Vect(R n) of vector fields of R n. However, these modules are isomorphic as sl(n + 1, R)modules where sl(n + 1, R) ⊂ Vect(R n) is the Lie algebra of infinitesimal projective transformations. In addition, such an sln+1equivariant bijection is unique (up to normalization). This leads to a notion of projectively equivariant quantization and symbol calculus for a manifold endowed with a (flat) projective structure. We apply the sln+1equivariant symbol map to study the Vect(M)modules D k λ (M) of kthorder linear differential operators acting on λdensities, for an arbitrary manifold M and classify the quotientmodules D k λ (M)/Dℓ λ (M). 1
Appendix  Projective Geometry for Machine Vision
, 1992
"... Introduction The idea for this Appendix arose from our perception of a frustrating situation faced by vision researchers. For example, one is interested in some aspect of the theory of perspective image formation such as the epipolar line. The interested party goes to the library to check out a boo ..."
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Cited by 34 (0 self)
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Introduction The idea for this Appendix arose from our perception of a frustrating situation faced by vision researchers. For example, one is interested in some aspect of the theory of perspective image formation such as the epipolar line. The interested party goes to the library to check out a book on projective geometry filled with hope that the necessary mathematical machinery will be directly at hand. These expectations are quickly dashed. Upon opening the book, the expectant reader finds the presentation dominated by endless observations about harmonic relations and a few chapters which explore the minutiae of Pappus' theorem. Finally, as a last cruel twist of irony, the book ends in triumph with a rather exhilarating discourse on the conic pencil. All of the material is presented in the form of theorems defined on points, lines and conics without the use of coordinates, except perhaps for a quick pause to define barycentric coordinates just to taunt the reader. Dejected, the vis
The WellPosed Problem
 Foundations of Physics
, 1973
"... distributions obtained from transformation groups, using as our main example the famous paradox of Bertrand. Bertrand's problem (Bertrand, 1889) was stated originally in terms of drawing a straight line "at random" intersecting a circle. It will be helpful to think of this in a more ..."
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Cited by 33 (0 self)
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distributions obtained from transformation groups, using as our main example the famous paradox of Bertrand. Bertrand's problem (Bertrand, 1889) was stated originally in terms of drawing a straight line "at random" intersecting a circle. It will be helpful to think of this in a more concrete way; presumably, we do no violence to the problem (i.e., it is still just as "random") if we suppose that we are tossing straws onto the circle, without specifying how they are tossed. We therefore formulate the problem as follows. A long straw is tossed at random onto a circle; given that it falls so that it intersects the circle, what is the probability that the chord thus defined is longer than a side of the inscribed equilateral triangle? Since Bertrand proposed it in 1889 this problem has been cited to generations of students to demonstrate that Laplace's "principle of indifference" contains logical inconsistencies. For, there appear to be many ways of defining "equally possibl
Magic squares and matrix models of Lie algebras
 Adv. Math
"... Abstract. This paper is concerned with the description of exceptional simple Lie algebras as octonionic analogues of the classical matrix Lie algebras. We review the TitsFreudenthal construction of the magic square, which includes the exceptional Lie algebras as the octonionic case of a constructio ..."
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Abstract. This paper is concerned with the description of exceptional simple Lie algebras as octonionic analogues of the classical matrix Lie algebras. We review the TitsFreudenthal construction of the magic square, which includes the exceptional Lie algebras as the octonionic case of a construction in terms of a Jordan algebra of hermitian 3 × 3 matrices (Tits) or various plane and other geometries (Freudenthal). We present alternative constructions of the magic square which explain its symmetry, and show explicitly how the use of split composition algebras leads to analogues of the matrix Lie algebras su(3), sl(3) and sp(6). We adapt the magic square construction to include analogues of su(2), sl(2) and sp(4)
The Convergence Of Numerical Transfer Schemes In Diffusive Regimes I: DiscreteOrdinate Method
"... . In highly diffusive regimes, the transfer equation with anisotropic boundary conditions has an asymptotic behavior as the mean free path ffl tends to zero that is governed by a diffusion equation and boundary conditions obtained through a matched asymptotic boundary layer analysis. A numerical sch ..."
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Cited by 30 (17 self)
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. In highly diffusive regimes, the transfer equation with anisotropic boundary conditions has an asymptotic behavior as the mean free path ffl tends to zero that is governed by a diffusion equation and boundary conditions obtained through a matched asymptotic boundary layer analysis. A numerical scheme for solving this problem has a ffl \Gamma1 contribution to the truncation error that generally gives rise to a nonuniform consistency with the transfer equation for small ffl, thus degrading its performance in diffusive regimes. In this paper we show that whenever the discreteordinate method has the correct diffusion limit, both in the interior and at the boundaries, its solutions converge to the solution of the transport equation uniformly in ffl. Our proof of the convergence is based on an asymptotic diffusion expansion and requires error estimates on a matched boundary layer approximation to the solution of the discreteordinate method. Key words. transfer equation, diffusion appro...