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295
Optimization and Dynamical Systems
, 1994
"... researchers in the areas of optimization, dynamical systems, control systems, signal processing, and linear algebra. The motivation for the results developed here arises from advanced engineering applications and the emergence of highly parallel computing machines for tackling such applications. The ..."
Abstract

Cited by 143 (18 self)
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researchers in the areas of optimization, dynamical systems, control systems, signal processing, and linear algebra. The motivation for the results developed here arises from advanced engineering applications and the emergence of highly parallel computing machines for tackling such applications. The problems solved are those of linear algebra and linear systems theory, and include such topics as diagonalizing a symmetric matrix, singular value decomposition, balanced realizations, linear programming, sensitivity minimization, and eigenvalue assignment by feedback control. The tools are those, not only of linear algebra and systems theory, but also of differential geometry. The problems are solved via dynamical systems implementation, either in continuous time or discrete time, which is ideally suited to distributed parallel processing. The problems tackled are indirectly or directly concerned with dynamical systems themselves, so there is feedback in that dynamical systems are used to understand and optimize dynamical systems. One key to the new research results has been
Random Matrix Theory and ζ(1/2 + it)
, 2000
"... We study the characteristic polynomials Z(U,#)of matrices U in the Circular Unitary Ensemble (CUE) of Random Matrix Theory. Exact expressions for any matrix size N are derived for the moments of and Z/Z # , and from these we obtain the asymptotics of the value distributions and cumulants of the re ..."
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Cited by 85 (15 self)
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We study the characteristic polynomials Z(U,#)of matrices U in the Circular Unitary Ensemble (CUE) of Random Matrix Theory. Exact expressions for any matrix size N are derived for the moments of and Z/Z # , and from these we obtain the asymptotics of the value distributions and cumulants of the real and imaginary parts of log Z as N ##. In the
Operator algebras and conformal field theory  III. Fusion of positive energy representations of LSU(N) using bounded operators
, 1998
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The Asymptotics of the Laplacian on a Manifold With Boundary
, 1990
"... : Let P be a secondorder differential operator with leading symbol given by the metric tensor on a compact Riemannian manifold with boundary. We compute the asymptotics of the heat equation for Dirichlet, Neumann, and mixed boundary conditions. x1 Statement of results Let M m be a compact Riemann ..."
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Cited by 63 (21 self)
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: Let P be a secondorder differential operator with leading symbol given by the metric tensor on a compact Riemannian manifold with boundary. We compute the asymptotics of the heat equation for Dirichlet, Neumann, and mixed boundary conditions. x1 Statement of results Let M m be a compact Riemannian manifold with boundary @M: Let V be a smooth vector bundle over M equipped with a connection r V : Let E be an endomorphism of V: Define P = \Gamma(\Sigma i;j g ij r V i r V j +E) : C 1 (V ) ! C 1 (V ): Every second order elliptic operator on M with leading symbol given by the metric tensor can be put in this form. Let f 2 C 1 (M): If @M = ;; then as t ! 0 + ; T r L 2 (fe \GammatP ) ' t \Gammam=2 \Sigma n t n an (f; P ) where n = 0; 1; 2; ::: ranges over the nonnegative integers. If @M 6= ;; we must impose suitable boundary conditions. Let OE 2 C 1 (V ): Dirichlet boundary conditions are BOE = OEj @M = 0: Let OE ;N be the covariant derivative of OE with respect ...
Natural Operations In Differential Geometry
, 1993
"... CONTENTS PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 CHAPTER I. MANIFOLDS AND LIE GROUPS . . . . . . . . . . . . . . . . 4 1. Differentiable manifolds . . . . . . . . . . . . . . . . . . . . . 4 2. Submersions and immersions . . . . . . . . . . . . . . . . . . 11 3. Vector ..."
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Cited by 55 (17 self)
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CONTENTS PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 CHAPTER I. MANIFOLDS AND LIE GROUPS . . . . . . . . . . . . . . . . 4 1. Differentiable manifolds . . . . . . . . . . . . . . . . . . . . . 4 2. Submersions and immersions . . . . . . . . . . . . . . . . . . 11 3. Vector fields and flows . . . . . . . . . . . . . . . . . . . . . 16 4. Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5. Lie subgroups and homogeneous spaces . . . . . . . . . . . . . 41 CHAPTER II. DIFFERENTIAL FORMS . . . . . . . . . . . . . . . . . . . 49 6. Vector bundles . . . . . . . . . . . . . . . . . . . . . . . . 49 7. Differential forms . . . . . . . . . . . . . . . . . . . . . . . 61 8. Derivations on the algebra of differential forms and the FrolicherNijenhuis bracket . . . . . . . . . . . . .
The twoeigenvalue problem and density of jones representation of braid groups
 Commun. Math. Phys
, 2002
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On the characteristic polynomial of a random unitary matrix
 Comm. Math. Phys
, 2001
"... Abstract: We present a range of fluctuation and large deviations results for the logarithm of the characteristic√polynomial Z of a random N × N unitary matrix, as N →∞. First 12 we show that ln Z / ln N, evaluated at a finite set of distinct points, is asymptotically a collection of i.i.d. complex n ..."
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Cited by 42 (11 self)
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Abstract: We present a range of fluctuation and large deviations results for the logarithm of the characteristic√polynomial Z of a random N × N unitary matrix, as N →∞. First 12 we show that ln Z / ln N, evaluated at a finite set of distinct points, is asymptotically a collection of i.i.d. complex normal random variables. This leads to a refinement of a recent central limit theorem due to Keating and Snaith, and also explains the covariance structure of the eigenvalue counting function. Next we obtain a central limit theorem for ln Z in a Sobolev space of generalised functions on the unit circle. In this limiting regime, lowerorder terms which reflect the global covariance structure are no longer negligible and feature in the covariance structure of the limiting Gaussian measure. Large deviations results for ln Z/A, evaluated at a finite set of distinct points, can be obtained for √ ln N ≪ A ≪ ln N. For higherorder scalings we obtain large deviations results for ln Z/A evaluated at a single point. There is a phase transition at A = ln N (which only applies to negative deviations of the real part) reflecting a switch from global to local conspiracy.
Conformally equivariant quantization: Existence and uniqueness
"... We prove the existence and the uniqueness of a conformally equivariant symbol calculus and quantization on any conformally flat pseudoRiemannian manifold (M,g). In other words, we establish a canonical isomorphism between the spaces of polynomials on T ∗ M and of differential operators on tensor de ..."
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Cited by 42 (5 self)
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We prove the existence and the uniqueness of a conformally equivariant symbol calculus and quantization on any conformally flat pseudoRiemannian manifold (M,g). In other words, we establish a canonical isomorphism between the spaces of polynomials on T ∗ M and of differential operators on tensor densities over M, both viewed as modules over the Lie algebra o(p + 1,q + 1) where p + q = dim(M). This quantization exists for generic values of the weights of the tensor densities and compute the critical values of the weights yielding obstructions to the existence of such an isomorphism. In the particular case of halfdensities, we obtain a conformally invariant starproduct.
The Kinematic Formula In Riemannian Homogeneous Spaces
, 1993
"... Let G be a Lie group and K a compact subgroup of G. Then the homogeneous space G/K has an invariant Riemannian metric and an invariant volume form# G . Let M and N be compact submanifolds of G/K, and I(M # gN) an "integral invariant" of the intersection M # gN . Then the integral (1) ..."
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Cited by 37 (2 self)
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Let G be a Lie group and K a compact subgroup of G. Then the homogeneous space G/K has an invariant Riemannian metric and an invariant volume form# G . Let M and N be compact submanifolds of G/K, and I(M # gN) an "integral invariant" of the intersection M # gN . Then the integral (1) Z G I(M # gN)# G (g) is evaluated for a large class of integral invariants I. To give an informal definition of the integral invariants I considered, let X # G/K be a submanifold, h X the vector valued second fundamental form of X in G/K. Let P be an "invariant polynomial" in the components of the second fundamental form of h X . Then the integral invariants considered are of the form I P (X) = Z X P(h X )# X . If P # 1 then I P (M #gN) = Vol(M #gN ). In this case the integral (1) is evaluated for all G, K, M and N . For P of higher degree the integral (1) is evaluated when G is unimodular and G is transitive on the set on tangent spaces of each of M and N . Then, given P,...