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991.
We prove that rational data of bounded input length are uniformly distributed (in the classical sense of H. Weyl, in [42]) with respect to the probability distribution of condition numbers of numerical analysis. We deal both with condition numbers of linear algebra and with condition numbers for systems of multivariate polynomial equations. For instance, we prove that for a randomly chosen n\times n rational matrix M of bit length O(n 4 log n) + log w , the condition number k(M) satisfies k(M) ≤ w n 5/2 with probability at least 1-2w -1 . Similar estimates are established for the condition number μ_ norm of M. Shub and S. Smale when applied to systems of multivariate homogeneous polynomial equations of bounded input length. Finally, we apply these techniques to estimate the probability distribution of the precision (number of bits of the denominator) required to write approximate zeros of systems of multivariate polynomial equations of bounded input length. March 7, 2001. Final version received: June 7, 2001.  相似文献   
992.
In [4] Dias da Silva and Hamidoune obtained a lower bound for the degree of the minimal polynomial of the Kronecker sum of two matrices in terms of the degrees of the minimal polynomials of the matrices. We determine the pairs of matrices for which this bound is reached.  相似文献   
993.
We characterize limits for orthogonal Szegö polynomials of fixed degree k, with respect to certain measures on the unit circle which are weakly convergent to a sum of m<k point masses. Such measures arise, for example, as a convolution of point masses with an approximate identity. It is readily seen that the underlying measures in two recently-proposed methods for estimating the m frequencies, θj, of a discrete-time trigonometric signal using Szegö polynomials are of this form. We prove existence of Szegö polynomial limits associated with a general class of weakly convergent measures, and prove that for convolution of point masses with the Poisson kernel, which underlies one of the recently-proposed methods, the limit has as a factor the Szegö polynomial with respect to a related measure, which we specify. Since m of the zeros approach the eiθj, this result uniquely characterizes the limit. A similar result is obtained for measures consisting of point masses with additive absolutely continuous part.  相似文献   
994.
Denote by πn the set of all algebraic polynomials of degree at most n with complex coefficients. An inequality of I. Schur asserts that the first derivative of the transformed Tchebycheff polynomial has the greatest uniform norm in [?1, 1] among all f ∈ ??n, where (1) Here we show that this extremal property of persists in the wider class of polynomials f ∈ πn which vanish at ±1, and for which there exist n ? 1 points separating the zeros of and such that for j = 1, …, n ? 1. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)  相似文献   
995.
996.
Estimation of the mean function in nonparametric regression is usefully separated into estimating the means at the observed factor levels—a one-way layout problem—and interpolation between the estimated means at adjacent factor levels. Candidate penalized least squares (PLS) estimators for the mean vector of a one-way layout are expressed as shrinkage estimators relative to an orthogonal regression basis determined by the penalty matrix. The shrinkage representation of PLS suggests a larger class of candidate monotone shrinkage (MS) estimators. Adaptive PLS and MS estimators choose the shrinkage vector and penalty matrix to minimize estimated risk. The actual risks of shrinkage-adaptive estimators depend strongly upon the economy of the penalty basis in representing the unknown mean vector. Local annihilators of polynomials, among them difference operators, generate penalty bases that are economical in a range of examples. Diagnostic techniques for adaptive PLS or MS estimators include basis-economy plots and estimates of loss or risk.  相似文献   
997.
For a random sample from a population having a finite k-th moment, it is shown that the constancy of regression of polynomial statistics of order k in the mean implies that all higher moments exist and are uniquely determined by the first k moments. This result is utilized to give a moment's approach to some characterization results.  相似文献   
998.
Kiselman's method permitting one to estimate the volume of sublevel sets of a plurisubharmonic function f in C n with the aid of the Lelong number of f is applied to polynomial lemniscates in C n . In particular, it yields estimates from below of the volume of polynomial lemniscates which completes recent results of Cuyt, Driver and Lubinsky.  相似文献   
999.
In an earlier paper, we used the absolute grading on Heegaard Floer homology HF+ to give restrictions on knots in S3 which admit lens space surgeries. The aim of the present article is to exhibit stronger restrictions on such knots, arising from knot Floer homology. One consequence is that the non-zero coefficients of the Alexander polynomial of such a knot are ±1. This information can in turn be used to prove that certain lens spaces are not obtained as integral surgeries on knots. In fact, combining our results with constructions of Berge, we classify lens spaces L(p,q) which arise as integral surgeries on knots in S3 with |p|?1500. Other applications include bounds on the four-ball genera of knots admitting lens space surgeries (which are sharp for Berge's knots), and a constraint on three-manifolds obtained as integer surgeries on alternating knots, which is closely to related to a theorem of Delman and Roberts.  相似文献   
1000.
In this paper, the asymptotic optimality of the cross validation bandwidth selector for the local polynomial fitting under strongly mixing dependence is obtained. The asymptotic normality of the bandwidth selected by the cross-validation method is derived, which is an extension of W. Härdle, P. Hall, and J. S. Marron (1988, J. Am. Statist. Assoc.83, 86–101).  相似文献   
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