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26
Improved Approximation Algorithms for MAX kCUT and MAX BISECTION
, 1994
"... Polynomialtime approximation algorithms with nontrivial performance guarantees are presented for the problems of (a) partitioning the vertices of a weighted graph into k blocks so as to maximise the weight of crossing edges, and (b) partitioning the vertices of a weighted graph into two blocks ..."
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Cited by 162 (0 self)
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Polynomialtime approximation algorithms with nontrivial performance guarantees are presented for the problems of (a) partitioning the vertices of a weighted graph into k blocks so as to maximise the weight of crossing edges, and (b) partitioning the vertices of a weighted graph into two blocks of equal cardinality, again so as to maximise the weight of crossing edges. The approach, pioneered by Goemans and Williamson, is via a semidefinite relaxation. 1 Introduction Goemans and Williamson [5] have significantly advanced the theory of approximation algorithms. Previous work on approximation algorithms was largely dependent on comparing heuristic solution values to that of a Linear Program (LP) relaxation, either implicitly or explicitly. This was recognised some time ago by Wolsey [11]. (One significant exception to this general rule has been the case of Bin Packing.) The main novelty of [5] is that it uses a SemiDefinite Program (SDP) as a relaxation. To be more precise let...
No eigenvalues outside the support of the limiting spectral distribution of largedimensional sample covariance matrices
 ANNALS OF PROBABILITY 26
, 1998
"... We consider a class of matrices of the form Cn = (1/N)(Rn+σXn)(Rn+σXn) ∗, where Xn is an n × N matrix consisting of independent standardized complex entries, Rj is an n×N nonrandom matrix, and σ> 0. Among several applications, Cn can be viewed as a sample correlation matrix, where information is co ..."
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Cited by 107 (18 self)
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We consider a class of matrices of the form Cn = (1/N)(Rn+σXn)(Rn+σXn) ∗, where Xn is an n × N matrix consisting of independent standardized complex entries, Rj is an n×N nonrandom matrix, and σ> 0. Among several applications, Cn can be viewed as a sample correlation matrix, where information is contained in (1/N)RnR ∗ n, but each column of Rn is contaminated by noise. As n → ∞, if n/N → c> 0, and the empirical distribution of the eigenvalues of (1/N)RnR ∗ n converge to a proper probability distribution, then the empirical distribution of the eigenvalues of Cn converges a.s. to a nonrandom limit. In this paper we show that, under certain conditions on Rn, for any closed interval in R + outside the support of the limiting distribution, then, almost surely, no eigenvalues of Cn will appear in this interval for all n large.
Second Phase Changes in Random MAry Search Trees and Generalized Quicksort: Convergence Rates
, 2002
"... We study the convergence rate to normal limit law for the space requirement of random mary search trees. While it is known that the random variable is asymptotically normally distributed for 3 m 26 and that the limit law does not exist for m ? 26, we show that the convergence rate is O(n ) for ..."
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Cited by 46 (12 self)
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We study the convergence rate to normal limit law for the space requirement of random mary search trees. While it is known that the random variable is asymptotically normally distributed for 3 m 26 and that the limit law does not exist for m ? 26, we show that the convergence rate is O(n ) for 3 m 19 and is O(n ), where 4=3 ! ff ! 3=2 is a parameter depending on m for 20 m 26. Our approach is based on a refinement to the method of moments and applicable to other recursive random variables; we briefly mention the applications to quicksort proper and the generalized quicksort of Hennequin, where more phase changes are given. These results provide natural, concrete examples for which the BerryEsseen bounds are not necessarily proportional to the reciprocal of the standard deviation. Local limit theorems are also derived. Abbreviated title. Phase changes in search trees.
CLT for Linear Spectral Statistics of Large Dimensional Sample Covariance Matrices
, 2003
"... This paper shows their of rate of convergence to be 1/n by proving, after proper scaling, they form a tight sequence. Moreover, if EX 11 =0andEX11 =2, or if X11 and T n are real and EX 11 = 3, they are shown to have Gaussian limits ..."
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Cited by 37 (0 self)
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This paper shows their of rate of convergence to be 1/n by proving, after proper scaling, they form a tight sequence. Moreover, if EX 11 =0andEX11 =2, or if X11 and T n are real and EX 11 = 3, they are shown to have Gaussian limits
Asymptotic semismoothness probabilities
 Mathematics of computation
, 1996
"... Abstract. We call an integer semismooth with respect to y and z if each of its prime factors is ≤ y, and all but one are ≤ z. Such numbers are useful in various factoring algorithms, including the quadratic sieve. Let G(α, β)bethe asymptotic probability that a random integer n is semismooth with res ..."
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Cited by 22 (1 self)
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Abstract. We call an integer semismooth with respect to y and z if each of its prime factors is ≤ y, and all but one are ≤ z. Such numbers are useful in various factoring algorithms, including the quadratic sieve. Let G(α, β)bethe asymptotic probability that a random integer n is semismooth with respect to n β and n α. We present new recurrence relations for G and related functions. We then give numerical methods for computing G,tablesofG, and estimates for the error incurred by this asymptotic approximation. 1.
Analytic continuation of the resolvent of the Laplacian and the dynamical zeta function, Preprint 2008 (arXiv: math.AP:0906.0293
"... Abstract. Let s0 < 0 be the abscissa of absolute convergence of the dynamical zeta function Z(s) for several disjoint strictly convex compact obstacles Ki ⊂ R N, i = 1,..., κ0, κ0 ≥ 3, and let Rχ(z) = χ(−∆D − z 2) −1 χ, χ ∈ C ∞ 0 (R N), be the cutoff resolvent of the Dirichlet Laplacian −∆D in Ω = ..."
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Cited by 19 (3 self)
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Abstract. Let s0 < 0 be the abscissa of absolute convergence of the dynamical zeta function Z(s) for several disjoint strictly convex compact obstacles Ki ⊂ R N, i = 1,..., κ0, κ0 ≥ 3, and let Rχ(z) = χ(−∆D − z 2) −1 χ, χ ∈ C ∞ 0 (R N), be the cutoff resolvent of the Dirichlet Laplacian −∆D in Ω = RN \ ∪ k0 i=1Ki. We prove that there exists σ1 < s0 such that Z(s) is analytic for Re(s) ≥ σ1 and the cutoff resolvent Rχ(z) has an analytic continuation for Im(z) < −σ1, Re(z)  ≥ C> 0. 1.
The Riemann hypothesis for certain integrals of Eisenstein series
 J. Number Theory
"... Abstract. This paper studies the nonholomorphic Eisenstein series E(z, s) for the modular surface PSL(2, Z)\H, and shows that integration with respect to certain nonnegative measures µ(z) gives meromorphic functions Fµ(s) that have all their zeros on the line ℜ(s) = 1 2. For the constant term a0( ..."
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Cited by 8 (2 self)
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Abstract. This paper studies the nonholomorphic Eisenstein series E(z, s) for the modular surface PSL(2, Z)\H, and shows that integration with respect to certain nonnegative measures µ(z) gives meromorphic functions Fµ(s) that have all their zeros on the line ℜ(s) = 1 2. For the constant term a0(y, s) of the Eisenstein series the Riemann hypothesis holds for all values y ≥ 1, with at most two exceptional real zeros, which occur exactly for those y> 4πe −γ = 7.0555+. The Riemann hypothesis holds for all truncation integrals with truncation parameter T ≥ 1. At the value T = 1 this proves the Riemann hypothesis for a zeta function Z2,Q(s) recently introduced by Lin Weng, associated to rank 2 semistable lattices over Q. 1.
Algorithms for the Constrained Design of Digital Filters with Arbitrary Magnitude and Phase Responses
, 1999
"... This thesis presents several new algorithms for designing digital lters subject to speci cations in the frequency domain. Finite impulse response (FIR) as well as innite impulse response (IIR) lter design problems are considered. ..."
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Cited by 5 (0 self)
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This thesis presents several new algorithms for designing digital lters subject to speci cations in the frequency domain. Finite impulse response (FIR) as well as innite impulse response (IIR) lter design problems are considered.
Asymptotic Expansions of the Mergesort Recurrences
 Acta Informatica
, 1998
"... This note provides exact formulae for the mean and variance of the cost of topdown recursive mergesort. These formulae improve upon earlier results of Flajolet and Golin. Key words. analysis of algorithms, mergesort, Dirichlet series, asymptotic expansions. ..."
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Cited by 3 (1 self)
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This note provides exact formulae for the mean and variance of the cost of topdown recursive mergesort. These formulae improve upon earlier results of Flajolet and Golin. Key words. analysis of algorithms, mergesort, Dirichlet series, asymptotic expansions.
Beurling Zeta Functions, Generalised Primes, and Fractal Membranes
, 2004
"... We study generalised prime systems P (1 < p1 ≤ p2 ≤ · · · , with pj ∈ R tending to infinity) and the associated Beurling zeta function ζP(s) = ∏∞ j=1 (1 − p−s j)−1. Under appropriate assumptions, we establish various analytic properties of ζP(s), including its analytic continuation and we charac ..."
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Cited by 3 (0 self)
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We study generalised prime systems P (1 < p1 ≤ p2 ≤ · · · , with pj ∈ R tending to infinity) and the associated Beurling zeta function ζP(s) = ∏∞ j=1 (1 − p−s j)−1. Under appropriate assumptions, we establish various analytic properties of ζP(s), including its analytic continuation and we characterise the existence of a suitable generalised functional equation. In particular, we examine the relationship between a counterpart of the Prime Number Theorem (with error term) and the properties of the analytic continuation of ζP(s). Further we study ‘wellbehaved ’ gprime systems, namely, systems for which both the prime and integer counting function are asymptotically wellbehaved. Finally, we show that there exists a natural correspondence between generalised prime systems and suitable orders on N2. Some of the above results may be relevant to the second author’s theory of ‘fractal