Frequency of digits in the Lüroth expansion

In this note we consider the Lüroth expansion of a real number, and we study the Hausdorff dimension of a class of sets defined in terms of the frequencies of digits in the expansion. We also study the speed at which the approximants obtained from the Lüroth expansion converge. In addition, we describe the multifractal properties of the level sets of the Lyapunov exponent, which measures the exponential speed of approximation obtained from the approximants. Finally, we describe the relation of the Lüroth expansion with the continued fraction expansion and the β-expansion. We remark that our work is still another application of the theory of dynamical systems to number theory.     

Construction of an asymptotic expansion for the solution of a linear system of equations of neutral type with a small delay in the absence of a boundary layer

In this paper we simplify the algorithm for constructing the asymptotic expansion for the solution of a linear system of neutral type at a large distance from the origin. After using the Laplace transformation to determine the asymptotic expansion near the initial point, we succeed in reducing the problem of determining the initial conditions to the computation of the residues of certain functions for which we have recurrence formulas.Translated from Matematicheskie Zametki, Vol. 6, No. 1, pp. 109–113, July, 1969.     

An Infinite Asymptotic Expansion for the Extreme Zeros of the Pollaczek Polynomials

In this paper, we first establish an integral expression for the Pollaczek polynomials P_n ( x ; a , b ) from a generating function. By applying a canonical transformation to the integral and carrying out a detailed analysis of the integrand, we derive a uniform asymptotic expansion for P_n (cosθ; a , b ) in terms of the Airy function and its derivative, in descending powers of n . The uniformity is in an interval next to the turning point , with M being a constant. The coefficients of the expansion are analytic functions of a parameter that depends only on t where , and not on the large parameter n . From the expansion of the polynomials we obtain an asymptotic expansion in powers of n ^−1/3 for the largest zeros. As a special case, a four-term approximation is provided for comparison and illustration. The method used in this paper seems to be applicable to more general situations.     

Bounds for visual cryptography schemes

In this paper, we investigate the best pixel expansion of various models of visual cryptography schemes. In this regard, we consider visual cryptography schemes introduced by Tzeng and Hu (2002) [13]. In such a model, only minimal qualified sets can recover the secret image and the recovered secret image can be darker or lighter than the background. Blundo et al. (2006) [4] introduced a lower bound for the best pixel expansion of this scheme in terms of minimal qualified sets. We present another lower bound for the best pixel expansion of the scheme. As a corollary, we introduce a lower bound, based on an induced matching of hypergraph of qualified sets, for the best pixel expansion of the aforementioned model and the traditional model of visual cryptography scheme realized by basis matrices. Finally, we study access structures based on graphs and we present an upper bound for the smallest pixel expansion in terms of strong chromatic index.     

关于边界层方法

本文指出传统的边界层方法(包括匹配法和Vi?ik—Lyusternik方法)的不足:不能作出边界层项的渐近展开式.提出多重尺度构造边界层项的方法,得到符合实情的结果.又与Levinson所用的方法比较,本方法能更简单地导出后一方法给出的边界层项的渐近展开式.又应用此方法研究现有的关于奇异摄动的某些成果,指出这些成果的局限性,并在一般情况下作出解的渐近展开式.     

Expansion of vectors by powers of a matrix

In this paper, we investigate the problem of expansion of any d-dimensional vector in powers of a dilation matrix M, where a dilation matrix is an integer matrix of size d × d with all modules of its eigenvalues more than one. We consider this expansion as a multidimensional system of numeration, where we take the matrix as the base of the system of numeration and a special set of vectors as the set of digits. We give a constructive method of expansion of integer vectors in powers of a dilation matrix and prove the existence of expansion for any real vector. Bibliography: 4 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 355, 2008, pp. 199–218.     

Local theory of extrapolation methods

In this paper we discuss the theory of one-step extrapolation methods applied both to ordinary differential equations and to index 1 semi-explicit differential-algebraic systems. The theoretical background of this numerical technique is the asymptotic global error expansion of numerical solutions obtained from general one-step methods. It was discovered independently by Henrici, Gragg and Stetter in 1962, 1964 and 1965, respectively. This expansion is also used in most global error estimation strategies as well. However, the asymptotic expansion of the global error of one-step methods is difficult to observe in practice. Therefore we give another substantiation of extrapolation technique that is based on the usual local error expansion in a Taylor series. We show that the Richardson extrapolation can be utilized successfully to explain how extrapolation methods perform. Additionally, we prove that the Aitken-Neville algorithm works for any one-step method of an arbitrary order s, under suitable smoothness.     

Edgeworth expansion for the pre-averaging estimator

In this paper, we study the Edgeworth expansion for a pre-averaging estimator of quadratic variation in the framework of continuous diffusion models observed with noise. More specifically, we obtain a second order expansion for the joint density of the estimators of quadratic variation and its asymptotic variance. Our approach is based on martingale embedding, Malliavin calculus and stable central limit theorems for continuous diffusions. Moreover, we derive the density expansion for the studentized statistic, which might be applied to construct asymptotic confidence regions.     

An integral expansion for analytic functions based upon the remainder values of the Taylor series expansions

In this work, by using a special property in the integral representation of the remainder value of the Taylor series expansion, we introduce a new expansion for analytic functions. We also give several interesting consequences of this expansion formula as well as some practical examples in order to illustrate the subject presented here.     

Adaptive-order rational Arnoldi-type methods in computational electromagnetism

In this paper, we will consider moment-matching methods for the application of model order reduction to Maxwell’s equations. Since the main difficulty of moment-matching methods results from the adequate determination of a set of expansion points, we will introduce an adaptive expansion point selection on the basis of a suitable approximation of an upper bound of the output moment-matching error. Furthermore, we give some remarks about structure- and passivity preserving adaptive-order rational Arnoldi methods for Maxwell’s equations. Numerical examples indicate the reliability of the proposed algorithm.     

RANDOM WEIGHTING APPROXIMATION IN LINEAR REGRESSION MODELS

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Asymptotic Expansion for the Null Distribution of the <Emphasis Type="Italic">F</Emphasis>-statistic in One-way ANOVA under Non-normality

In this paper we derive the asymptotic expansion of the null distribution of the F-statistic in one-way ANOVA under non-normality. The asymptotic framework is when the number of treatments is moderate but sample size per treatment (replication size) is small. This kind of asymptotics will be relevant, for example, to agricultural screening trials where large number of cultivars are compared with few replications per cultivar. There is also a huge potential for the application of this kind of asymptotics in microarray experiments. Based on the asymptotic expansion we will devise a transformation that speeds up the convergence to the limiting distribution. The results indicate that the approximation based on limiting distribution are unsatisfactory unless number of treatments is very large. Our numerical investigations reveal that our asymptotic expansion performs better than other methods in the literature when there is skewness in the data or even when the data comes from a symmetric distribution with heavy tails.     

Isoperimetric inequalities in simplicial complexes

In graph theory there are intimate connections between the expansion properties of a graph and the spectrum of its Laplacian. In this paper we define a notion of combinatorial expansion for simplicial complexes of general dimension, and prove that similar connections exist between the combinatorial expansion of a complex, and the spectrum of the high dimensional Laplacian defined by Eckmann. In particular, we present a Cheeger-type inequality, and a high-dimensional Expander Mixing Lemma. As a corollary, using the work of Pach, we obtain a connection between spectral properties of complexes and Gromov’s notion of geometric overlap. Using the work of Gundert and Wagner, we give an estimate for the combinatorial expansion and geometric overlap of random Linial-Meshulam complexes.     

有关调和数的更多渐近展开公式

基于指数型完全Bell多项式,建立了一个一般调和数渐近展开式,并给出展开式中系数的相应递推关系.由生成函数方法进一步推导出这些系数的具体表达式.另外,我们建立了两个在对数项里只含有奇数或偶数次幂项的lacunary调和数渐近展开式,     

Modelling of an underground waste disposal site by upscaling

In this paper, we study the global behaviour of an underground waste disposal in order to have an accurate upscaled model suitable for the computations involved in safety assessment processes. We start from a detailed model describing the transport of pollutant leaking from a high number of units. Using the method of homogenization, going to the limit, we obtain first a macroscopic model where the sources are now appearing globally. Then we compute a first‐order matched asymptotic expansion and we give the error estimates for this approximation. Copyright © 2004 John Wiley & Sons, Ltd.     

Hecke-Siegel's pull-back formula for the Epstein zeta function with a harmonic polynomial

In this paper, we discuss the generalization of the Hecke's integration formula for the Epstein zeta functions. We treat the Epstein zeta function as an Eisenstein series come from a degenerate principal series. For the Epstein zeta function of degree two, Siegel considered the Hecke's formula as the constant term of a certain Fourier expansion of the Epstein zeta function and obtained the other Fourier coefficients as the Dedekind zeta functions with Grössencharacters of a real quadratic field. We generalize this Siegel's Fourier expansion to more general Eisenstein series with harmonic polynomials. Then we obtain the Dedekind zeta functions with Grössencharacters for arbitrary number fields.     

Error Expansion for FEM and Superconvergence Under Natural Assumption

In this paper, we derive the error expansion for finite element method under natural assumption and discuss the superconvergence as a special case of error expansion.     

Filtering with Marked Point Process Observations via Poisson Chaos Expansion

We study a general filtering problem with marked point process observations. The motivation comes from modeling financial ultra-high frequency data. First, we rigorously derive the unnormalized filtering equation with marked point process observations under mild assumptions, especially relaxing the bounded condition of stochastic intensity. Then, we derive the Poisson chaos expansion for the unnormalized filter. Based on the chaos expansion, we establish the uniqueness of solutions of the unnormalized filtering equation. Moreover, we derive the Poisson chaos expansion for the unnormalized filter density under additional conditions. To explore the computational advantage, we further construct a new consistent recursive numerical scheme based on the truncation of the chaos density expansion for a simple case. The new algorithm divides the computations into those containing solely system coefficients and those including the observations, and assign the former off-line.     

Asymptotic derivation of frictionless contact models for elastic rods on a foundation with normal compliance

Using asymptotic methods we derive some models for elastic rods in frictionless contact with a foundation with normal response. Starting from the three-dimensional problem we characterize the first terms of an asymptotic expansion of the solution taking the diameter of cross section as small parameter. Then we prove the convergence as this diameter tends to zero. In this way, we obtain and we mathematically justify a simplified model generalizing the best known classical models of such frictionless contact problems.