An approximation for the complex normal probability integral

An analytic approximation to the complex normal probability integral, (x+iy)=(2)^–1/2 _–t ^x exp[–(t–iy)²/2]dt, is given together with a study of the error in the approximation.     

Combinatorial 5/6-approximation of Max Cut in graphs of maximum degree 3

The best approximation algorithm for Max Cut in graphs of maximum degree 3 uses semidefinite programming, has approximation ratio 0.9326, and its running time is Θ(n^3.5logn); but the best combinatorial algorithms have approximation ratio 4/5 only, achieved in O(n²) time [J.A. Bondy, S.C. Locke, J. Graph Theory 10 (1986) 477–504; E. Halperin, et al., J. Algorithms 53 (2004) 169–185]. Here we present an improved combinatorial approximation, which is a 5/6-approximation algorithm that runs in O(n²) time, perhaps improvable even to O(n). Our main tool is a new type of vertex decomposition for graphs of maximum degree 3.     

On optimal simultaneous rational approximation to with being some kind of cubic algebraic function

It is shown that each rational approximant to (ω,ω²)^τ given by the Jacobi–Perron algorithm (JPA) or modified Jacobi–Perron algorithm (MJPA) is optimal, where ω is an algebraic function (a formal Laurent series over a finite field) satisfying ω³+kω-1=0 or ω³+kdω-d=0. A result similar to the main result of Ito et al. [On simultaneous approximation to (α,α²) with α³+kα-1=0, J. Number Theory 99 (2003) 255–283] is obtained.     

On Distribution of Error of Linear Wavelet Estimator of Probability Density

We establish the asymptotic normality of the squared L ²-norm of the approximation error of a linear wavelet estimator of the density of a distribution. The calculations are based on the smallness of correlations between the coefficients of the high-frequency part of the multiresolution expansion of the estimator.Supported by the FCT Foundation (Portugal) in the framework of the project Probability and Statistics (2000–2002), Centro de Matematica, Universidade da Beira Interior.__________Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 2, pp. 184–207, April–June, 2005.     

On the Approximation of Quantile Processes by Kiefer Processes

We prove that the best possible almost sure rate of uniform approximation of a uniform quantile process by a normed Kiefer process is O(n ^–1/4(log n)^1/2× (log log n)^1/4).     

Vibrational frequency shifts in the stretched polypropylene molecule

The GF matrix method is used to compute the changes in the skeletal vibration frequencies for a simplified model of isotactic polypropylene stretched by a loadf. The series of frequencies obtained atf=0 are identified with those experimentally observed. A linear approximation is used for the deformation potential energy atf=0–2·10^–4 dyneperchain. The results of the calculations are presented in graph form.A. F. Ioffe Physicotechnical Institute, Academy of Sciences of the USSR, Leningrad. Translated from Mekhanika Polimerov, Vol. 4, No. 4, pp. 586–594, July–August, 1968.     

Improvement of approximations for the distributions of multinomial goodness-of-fit statistics under nonlocal alternatives

Cressie and Read (J. Roy. Statist. Soc. B 46 (1984) 440–464) introduced the power divergence statistics, R^a, as multinomial goodness-of-fit statistics. Each R^a has a limiting noncentral chi-square distribution under a local alternative and has a limiting normal distribution under a nonlocal alternative. Taneichi et al. (J. Multivariate Anal. 81 (2002) 335–359) derived an asymptotic approximation for the distribution of R^a under local alternatives. In this paper, using multivariate Edgeworth expansion for a continuous distribution, we show how the approximation based on the limiting normal distribution of R^a under nonlocal alternatives can be improved. We apply the expansion to the power approximation for R^a. The results of numerical investigation show that the proposed power approximation is very effective for the likelihood ratio test.     

Local and Global Approximation Theorems for Positive Linear Operators

In this paper we bridge local and global approximation theorems for positive linear operators via Ditzian–Totik moduliω²_φ(f, δ) of second order whereby the step-weightsφare functions whose squares are concave. Both direct and converse theorems are derived. In particular we investigate the situation for exponential-type and Bernstein-type operators.     

Solution of nonstationary boundary layer problems by fastest descent method

We consider the formation of a boundary layer above a semi-infinite plate that moves with velocity V_t8=At ⁿ in a viscous incompressible fluid. The problem is solved by Meksyn's asymptotic method. First, second, and third approximation equations are obtained. A numerical calculation is performed for various n, and especially interesting results are obtained for –0.5 < n < 0. The numerical solution of the third approximation equation is consistent with other known results.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 65, pp. 51–55, 1988.     

New Cyclic Difference Sets with Singer Parameters Constructed from d-Homogeneous Functions

In this paper, for a prime power q, new cyclic difference sets with Singer para- meters ((q ⁿ –1/q–1), (q ^n–1–1/q–1), (q ^n–2–1/q–1)) are constructed by using q-ary sequences (d-homogeneous functions) of period q ⁿ –1 and the generalization of GMW difference sets is proposed by combining the generation methods of d-form sequences and extended sequences. When q is a power of 3, new cyclic difference sets with Singer parameters ((q ⁿ –1/q–1), (q ^n–1–1/q–1), (q ^n–2–1/q–1)) are constructed from the ternary sequences of period q ⁿ –1 with ideal autocorrelation introduced by Helleseth, Kumar, and Martinsen.     

Distribution of Cardinalities of Row Spaces of Boolean Matrices of Order n

Let R(A) denote the row space of a Boolean matrix A of order n. We show that if n 7, then the cardinality |R(A)| (2^n–1 - 2^n–5, 2^n–1 - 2^n–6) U (2^n–1 - 2^n–6, 2^n–1). This result confirms a conjecture in [1].AMS Subject Classification (1991): 05B20 06E05 15A36Support partially by the Postdoctoral Science Foundation of China.Dedicated to Professor Chao Ko on the occasion of his 90th birthday     

A New Family of Cyclic Difference Sets with Singer Parameters in Characteristic Three

We construct a new family of cyclic difference sets with parameters ((3 ^d – 1)/2, (3 ^{d – 1} – 1)/2, (3 ^{d – 2} – 1)/2) for each odd d. The difference sets are constructed with certain maps that form Jacobi sums. These new difference sets are similar to Maschietti's hyperoval difference sets, of the Segre type, in characteristic two. We conclude by calculating the 3-ranks of the new difference sets.     

Approximation to continuous functions by Cesàro means of double Fourier series and conjugate series

We study the rate of uniform approximation to continuous functions ƒ(x, y), 2π-periodic in each variable, in Lipschitz classes Lip(α, β) and in Zygmund classes Z(α, β), 0 < α, β 1, by Cesàro means σ_mn^γδ(ƒ) of positive orders of the rectangular partial sums of double Fourier series. The rate of uniform approximation to the conjugate functions ^1,0, ^0,1 and ^1,1 by the corresponding Cesàro means is also discussed in detail. The difference between the classes Lip(α, β) and Z(α, β), similar to the one-dimensional case, appears again when max(α, β) = 1. (Compare Theorems 2 and 3 with Theorems 4 and 5.) One surprising result is the following: The uniform approximation rate by σ_mn^{γδ 1,0} to ^1,0 is worse in general than that by σ_mn^{γδ 1,1} to ^1,1 for ƒ ε Lip(1, 1). In fact, the appearance of an extra factor [log(n + 2)]² in the former case is unavoidable (see Theorem 6). All approximation rates we obtain, with one exception, are shown to be exact. Two conjectures are also included.     

Normal Approximation Rate of the Kernel Smoothing Estimator in a Partial Linear Model

By establishing the asymptotic normality for the kernel smoothing estimatorβ_nof the parametric componentsβin the partial linear modelY=X′β+g(T)+, P. Speckman (1988,J. Roy. Statist. Soc. Ser. B50, 413–456) proved that the usual parametric raten^−1/2is attainable under the usual “optimal” bandwidth choice which permits the achievement of the optimal nonparametric rate for the estimation of the nonparametric componentg. In this paper we investigate the accuracy of the normal approximation forβ_nand find that, contrary to what we might expect, the optimal Berry–Esseen raten^−1/2is not attainable unlessgis undersmoothed, that is, the bandwidth is chosen with faster rate of tending to zero than the “optimal” bandwidth choice.     

Compactness in

This paper is concerned with compactness for some topologies on the collection of bounded linear operators on Banach spaces. New versions of the Eberlein–Šmulian theorem and Day's lemma in the collection are established. Also we obtain a partial solution of the dual problem for the quasi approximation property, that is, it is shown that for a Banach space X if X^** is separable and X^* has the quasi approximation property, then X has the quasi approximation property.     

A new difference scheme with high accuracy and absolute stability for solving convection–diffusion equations

In this paper, we use a semi-discrete and a padé approximation method to propose a new difference scheme for solving convection–diffusion problems. The truncation error of the difference scheme is O(h⁴+τ⁵). It is shown through analysis that the scheme is unconditionally stable. Numerical experiments are conducted to test its high accuracy and to compare it with Crank–Nicolson method.     

Average CaseL∞-Approximation in the Presence of Gaussian Noise

We consider the average caseL_∞-approximation of functions fromC^r([0, 1]) with respect to ther-fold Wiener measure. An approximation is based onnfunction evaluations in the presence of Gaussian noise with varianceσ²>0. We show that the n th minimal average error is of ordern^{−(2r+1)/(4r+4)} ln^1/2 n, and that it can be attained either by the piecewise polynomial approximation using repetitive observations, or by the smoothing spline approximation using non-repetitive observations. This completes the already known results forL_q-approximation withq<∞ andσ0, and forL_∞-approximation withσ=0.     

Regularized Resolvent of Sums of Commuting Operators

For B ₁ and B ₂ commuting linear operators on a Banach space such that B ₁ generates a bounded strongly continuous semigroup and –B ₂ generates an exponentially decaying strongly continuous holomorphic semigroup, it is shown that (B ₁ – B ₂)^–1 B ₂ ^–r and (B ₁ – B ₂)^–1(–B ₁)^–r are bounded and everywhere defined, for any r > 0. Density of domains may also be removed. The results are applied to various abstract Cauchy problems.     

A sharpening of the Johnson bound for binary linear codes and the nonexistence of linear codes with preparata parameters

We show fort>3 the nonexistence of binary [2 ^t –2, 2 ^t –2t–1, 5] codes.     

Constrained near-minimax approximation by weighted expansion and interpolation using Chebyshev polynomials of the second,third, and fourth kinds

Well known results on near-minimax approximation using Chebyshev polynomials of the first kind are here extended to Chebyshev polynomials of the second, third, and fourth kinds. Specifically, polynomial approximations of degreen weighted by (1–x ²)^1/2, (1+x)^1/2 or (1–x)^1/2 are obtained as partial sums of weighted expansions in Chebyshev polynomials of the second, third, or fourth kinds, respectively, to a functionf continuous on [–1, 1] and constrained to vanish, respectively, at ±1, –1 or +1. In each case a formula for the norm of the resulting projection is determined and shown to be asymptotic to 4^–2logn +A +o(1), and this provides in each case and explicit bound on the relative closeness of a minimax approximation. The constantA that occurs for the second kind polynomial is markedly smaller, by about 0.27, than that for the third and fourth kind, while the latterA is identical to that for the first kind, where the projection norm is the classical Lebesgue constant _n . The results on the third and fourth kind polynomials are shown to relate very closely to previous work of P.V. Galkin and of L. Brutman.Analogous approximations are also obtained by interpolation at zeros of second, third, or fourth kind polynomials of degreen+1, and the norms of resulting projections are obtained explicitly. These are all observed to be asymptotic to 2^–1logn +B +o(1), and so near-minimax approximations are again obtained. The norms for first, third, and fourth kind projections appear to be converging to each other. However, for the second kind projection, we prove that the constantB is smaller by a quantity asymptotic to 2^–1log2, based on a conjecture about the point of attainment of the supremum defining the projection norm, and we demonstrate that the projection itself is remarkably close to a minimal (weighted) interpolation projection.All four kinds of Chebyshev polynomials possess a weighted minimax property, and, in consequence, all the eight approximations discussed here coincide with minimax approximations when the functionf is a suitably weighted polynomial of degreen+1.