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91
Problems in Computational Geometry
 Packing and Covering
, 1974
"...  reproduced, stored In a retrieval system, or transmlt'ted, In any form or by any means, electronic, mechanical, photocopying, or otherwise, without the prior written permission of the author. ..."
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Cited by 453 (2 self)
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 reproduced, stored In a retrieval system, or transmlt'ted, In any form or by any means, electronic, mechanical, photocopying, or otherwise, without the prior written permission of the author.
The Mathematics Of Eigenvalue Optimization
, 2003
"... Optimization problems involving the eigenvalues of symmetric and nonsymmetric matrices present a fascinating mathematical challenge. Such problems arise often in theory and practice, particularly in engineering design, and are amenable to a rich blend of classical mathematical techniques and contemp ..."
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Cited by 92 (13 self)
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Optimization problems involving the eigenvalues of symmetric and nonsymmetric matrices present a fascinating mathematical challenge. Such problems arise often in theory and practice, particularly in engineering design, and are amenable to a rich blend of classical mathematical techniques and contemporary optimization theory. This essay presents a personal choice of some central mathematical ideas, outlined for the broad optimization community. I discuss the convex analysis of spectral functions and invariant matrix norms, touching briey on semide nite representability, and then outlining two broader algebraic viewpoints based on hyperbolic polynomials and Lie algebra. Analogous nonconvex notions lead into eigenvalue perturbation theory. The last third of the article concerns stability, for polynomials, matrices, and associated dynamical systems, ending with a section on robustness. The powerful and elegant language of nonsmooth analysis appears throughout, as a unifying narrative thread.
Quantum chaotic dynamics and random polynomials
 J. Stat. Phys
, 1996
"... We investigate the distribution of roots of polynomials of high degree with random coefficients which, among others, appear naturally in the context of ”quantum chaotic dynamics”. It is shown that under quite general conditions their roots tend to concentrate near the unit circle in the complex plan ..."
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Cited by 32 (0 self)
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We investigate the distribution of roots of polynomials of high degree with random coefficients which, among others, appear naturally in the context of ”quantum chaotic dynamics”. It is shown that under quite general conditions their roots tend to concentrate near the unit circle in the complex plane. In order to further increase this tendency, we study in detail the particular case of selfinversive random polynomials and show that for them a finite portion of all roots lies exactly on the unit circle. Correlation functions of these roots are also computed analytically, and compared to the correlations of eigenvalues of random matrices. The problem of ergodicity of chaotic wavefunctions is also considered. For that purpose we introduce a family of random polynomials whose roots spread uniformly over phase space. While these results are consistent with random matrix theory predictions, they provide a new and different insight into the problem of quantum ergodicity. Special attention is devoted all over the paper to the role of symmetries in the distribution of roots of random polynomials.
Optimal and nearly optimal algorithms for approximating polynomial zeros
 Comput. Math. Appl
, 1996
"... AbstractWe substantially improve the known algorithms for approximating all the complex zeros of an n th degree polynomial p(x). Our new algorithms save both Boolean and arithmetic sequential time, versus the previous best algorithms of SchSnhage [1], Pan [2], and Neff and Reif [3]. In parallel (N ..."
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Cited by 30 (14 self)
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AbstractWe substantially improve the known algorithms for approximating all the complex zeros of an n th degree polynomial p(x). Our new algorithms save both Boolean and arithmetic sequential time, versus the previous best algorithms of SchSnhage [1], Pan [2], and Neff and Reif [3]. In parallel (NC) implementation, we dramatically decrease the number of processors, versus the parallel algorithm of Neff [4], which was the only NC algorithm known for this problem so far. Specifically, under the simple normalization assumption that the variable x has been scaled so as to confine the zeros of p(x) to the unit disc {x: Ix [ < 1}, our algorithms (which promise to be practically effective) approximate all the zeros of p(x) within the absolute error bound 2b, by using order of n arithmetic operations and order of (b + n)n 2 Boolean (bitwise) operations (in both cases up to within polylogarithmic factors). The algorithms allow their optimal (work preserving) NC parallelization, so that they can be implemented by using polylogarithmic time and the orders of n arithmetic processors or (b + n)n 2 Boolean processors. All the cited bounds on the computational complexity are within polylogarithmic factors from the optimum (in terms of n and b) under both arithmetic and Boolean models of computation (in the Boolean case, under the additional (realistic) assumption that n = O(b)).
Problems and theorems in the theory of multiplier sequences
 Serdica Math. J
, 1996
"... Abstract. The purpose of this paper is (1) to highlight some recent and heretofore unpublished results in the theory of multiplier sequences and (2) to survey some open problems in this area of research. For the sake of clarity of exposition, we have grouped the problems in three subsections, althou ..."
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Cited by 15 (3 self)
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Abstract. The purpose of this paper is (1) to highlight some recent and heretofore unpublished results in the theory of multiplier sequences and (2) to survey some open problems in this area of research. For the sake of clarity of exposition, we have grouped the problems in three subsections, although several of the problems are interrelated. For the reader’s convenience, we have included the pertinent definitions, cited references and related results, and in several instances, elucidated the problems by examples.
The Zeros Of The Daubechies Polynomials
, 1996
"... . To study wavelets and filter banks of high order, we begin with the zeros of Bp (y). This is the binomial series for (1 \Gamma y) \Gammap , truncated after p terms. Its zeros give the p \Gamma 1 zeros of the Daubechies filter inside the unit circle, by z + z \Gamma1 = 2 \Gamma 4y. The filter has p ..."
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Cited by 14 (0 self)
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. To study wavelets and filter banks of high order, we begin with the zeros of Bp (y). This is the binomial series for (1 \Gamma y) \Gammap , truncated after p terms. Its zeros give the p \Gamma 1 zeros of the Daubechies filter inside the unit circle, by z + z \Gamma1 = 2 \Gamma 4y. The filter has p additional zeros at z = \Gamma1, and this construction makes it orthogonal and maximally flat. The dilation equation leads to orthogonal wavelets with p vanishing moments. Symmetric biorthogonal wavelets (generally better in image compression) come similarly from a subset of the zeros of Bp (y). We study the asymptotic behavior of these zeros. Matlab shows a remarkable plot for p = 70. The zeros approach a limiting curve j4y(1 \Gamma y)j = 1 in the complex plane, which is jz \Gamma z \Gamma1 j = 2. All zeros have jyj 1=2, and the rightmost zeros approach y = 1=2 (corresponding to z = \Sigmai ) with speed p \Gamma1=2 . The curve j4y(1 \Gamma y)j = (4ßp) 1=2p j1 \Gamma 2yj 1=p...
When Are Two Numerical Polynomials Relatively Prime?
, 1997
"... Let a and b be two polynomials having numerical coefficients. We consider the question: when are a and b relatively prime? Since the coefficients of a and b are approximant, the question is the same as: when are two polynomials relatively prime, even after small perturbations of the coefficients? In ..."
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Cited by 13 (3 self)
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Let a and b be two polynomials having numerical coefficients. We consider the question: when are a and b relatively prime? Since the coefficients of a and b are approximant, the question is the same as: when are two polynomials relatively prime, even after small perturbations of the coefficients? In this paper we provide a numeric parameter for determining that two polynomials are prime, even under small perturbations of the coefficients. Our methods rely on an inversion formula for Sylvester matrices to establish an effective criterion for relative primeness. The inversion formula can also be used to approximate the condition number of a Sylvester matrix. 1
Coefficients and roots of Ehrhart polynomials
 In Integer points in polyhedra—geometry, number theory, algebra, optimization, volume 374 of Contemp. Math
, 2005
"... Abstract. The Ehrhart polynomial of a convex lattice polytope counts integer points in integral dilates of the polytope. We present new linear inequalities satisfied by the coefficients of Ehrhart polynomials and relate them to known inequalities. We also investigate the roots of Ehrhart polynomials ..."
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Cited by 13 (3 self)
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Abstract. The Ehrhart polynomial of a convex lattice polytope counts integer points in integral dilates of the polytope. We present new linear inequalities satisfied by the coefficients of Ehrhart polynomials and relate them to known inequalities. We also investigate the roots of Ehrhart polynomials. We prove that for fixed d, there exists a bounded region of C containing all roots of Ehrhart polynomials of dpolytopes, and that all real roots of these polynomials lie in [−d, ⌊d/2⌋). In contrast, we prove that when the dimension d is not fixed the positive real roots can be arbitrarily large. We finish with an experimental investigation of the Ehrhart polynomials of cyclic polytopes and 0/1polytopes. 1.
Asymptotic Properties Of HeineStieltjes And Van Vleck Polynomials
 J. APPROX. THEORY
"... We study the the asymptotic behavior of the zeros of polynomial solutions of a class of generalized Lamé differential equations, when their coeffients satisfy ertain asymptotic conditions. The limit distribution is described by an equilibrium measure in presence of an external field, generated by ch ..."
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Cited by 12 (4 self)
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We study the the asymptotic behavior of the zeros of polynomial solutions of a class of generalized Lamé differential equations, when their coeffients satisfy ertain asymptotic conditions. The limit distribution is described by an equilibrium measure in presence of an external field, generated by charges at the singular points of the equation. Moreover, a case of nonpositive charges is onsidered, which leads to an equilibrium with a nonconvex external field.