Diffraction of light by supersonic waves: The solution of the raman-nath equations

The partial differential equation associated with the system of difference-differential equations of Raman-Nath for the amplitudes of the diffracted light-waves is solved exactly by the method of the separation of the variables. The solution is presented as a double infinite series containing the Fourier coefficients of the even periodic Mathieu functions with periodπ and the corresponding eigenvalues. Considering this solution as a Laurent series in one of the variables, the Laurent coefficients immediately give the exact expressions for the amplitudes of the diffracted light-waves, from which the formulae for the intensities are calculated. The connection between the Raman-Nath method and Brillouin’s Mathieu function method has thus been achieved.     

Regularization of linear difference equations with analytic coefficients and their applications

We propose a regularization method for four-element linear difference equations with analytic coefficients. We study these equations in the class of functions which are holomorphic in the complex plane with a cruciform cut and vanish at infinity. We give several examples illustrating the dependence of the solvability properties of equations on the choice of periodic coefficients. We describe various applications.     

一类非线性微分方程周期解的研究

本文研究了一类最普遍的四阶非线性非自治系统的周期解的存在唯一性与渐近稳定性。我们采用了类比缓交系数的方法,作出了相应的Liapunov函数,对缓交系数作了较为精确的估计,得到了存在唯一渐近稳定的周期解的充分条件。     

Study of convergence of the projection-difference method for hyperbolic equations

We consider the Cauchy problem for an abstract quasilinear hyperbolic equation with variable operator coefficients and a nonsmooth but Bochner integrable free term in a Hilbert space. Under study is the scheme for approximate solution of this problem which is a combination of the Galerkin scheme in space variables and the three-layer difference scheme with time weights. We establish an a priori energy error estimate without any special conditions on the projection subspaces. We give a concrete form of this estimate in the case when discretization in the space variables is carried out by the finite element method (for a partial differential equation) and by the Galerkin method in Mikhlin form.     

Complexity of the Realization of a Linear Boolean Function in the Class of <Emphasis Type="Italic">π</Emphasis>-Schemes

Using Khrapchenko’s method, we obtain the exact lower bound of 40 for the complexity in the class of π-schemes of a linear Boolean function depending substantially on 6 variables. We give a simplified proof of several lower bounds for the complexity of linear Boolean functions which are previously obtained on the basis of the same method.     

概率论中事件的分割思想及其应用

讨论概率论中事件的分割思想,给出一类概率论考研题的新解法,并探讨两个不相关(相关系数为零)的随机变量非独立的判别方法.     

Invariance principles under weak dependence

The aim of this paper is to give a functional form for the central limit theorem obtained by Bradley for strong mxing sequences of random variables, under a certain assumption about the size of the maximal coefficients of correlations. The convergence of the moments of order 2 + δ in the central limit theorem for this class of random variables is also obtained.     

Asymptotics of Green functions and Martin boundaries for elliptic operators with periodic coefficients

We give the asymptotics at infinity of a Green function for an elliptic equation with periodic coefficients on R^d. Basic ingredients in establishing the asymptotics are an integral representation of the Green function and the saddle point method. We also completely determine the Martin compactification of R^d with respect to an elliptic equation with periodic coefficients by using the exact asymptotics at infinity of the Green function.     

The dressing method and separation of variables: The two-dimensional case

Separation of variables is very convenient for obtaining solutions of linear differential equations in explicit form. We use the dressing method to widen the class of such equations. As an example, we dress a two-dimensional linear differential operator, including an operator with constant coefficients.     

Empirical likelihood inferences for semiparametric instrumental variable models

This paper studies the empirical likelihood inferences for a class of semiparametric instrumental variable models. We focus on the case that some covariates are endogenous variables, and some auxiliary instrumental variables are available. An instrumental variable based empirical likelihood method is proposed, and it is shown that the proposed empirical log-likelihood ratio is asymptotically chi-squared. Then, the confidence intervals for the regression coefficients are constructed. Some simulation studies are undertaken to assess the finite sample performance of the proposed empirical likelihood procedure.     

Quantum Pieri rules for isotropic Grassmannians

We study the three point genus zero Gromov-Witten invariants on the Grassmannians which parametrize non-maximal isotropic subspaces in a vector space equipped with a nondegenerate symmetric or skew-symmetric form. We establish Pieri rules for the classical cohomology and the small quantum cohomology ring of these varieties, which give a combinatorial formula for the product of any Schubert class with certain special Schubert classes. We also give presentations of these rings, with integer coefficients, in terms of special Schubert class generators and relations.     

An optimal modification of the LIML estimation for many instruments and persistent heteroscedasticity

We consider the estimation of coefficients of a structural equation with many instrumental variables in a simultaneous equation system. It is mathematically equivalent to the estimating equations estimation or a reduced rank regression in the statistical multivariate linear models when the number of restrictions or the dimension of estimating equations increases with the sample size. As a semi-parametric method, we propose a class of modifications of the limited information maximum likelihood (LIML) estimator to improve its asymptotic properties as well as the small sample properties for many instruments and persistent heteroscedasticity. We show that an asymptotically optimal modification of the LIML estimator, which is called AOM-LIML, improves the LIML estimator and other estimation methods. We give a set of sufficient conditions for an asymptotic optimality when the number of instruments or the dimension of the estimating equations is large with persistent heteroscedasticity including a case of many weak instruments.     

Borel summability of divergent solutions for singular first-order partial differential equations with variable coefficients. Part II

Under a certain restriction, singular first-order linear partial differential equations of nilpotent type with two variables are divided into two classes. In the previous paper Part I, we dealt with the one class, and comprehended that there was a close affinity between the Borel summability of divergent solutions and global analytic continuation properties for coefficients. In this Part II, we give a similar consideration on the other class. More precise global estimates than those given in Part I for coefficients will be required to prove the Borel summability of divergent solutions.     

An algebra of Stein operators

We build upon recent advances on the distributional aspect of Stein's method to propose a novel and flexible technique for computing Stein operators for random variables that can be written as products of independent random variables. We show that our results are valid for a wide class of distributions including normal, beta, variance-gamma, generalized gamma and many more. Our operators are kth degree differential operators with polynomial coefficients; they are straightforward to obtain even when the target density bears no explicit handle. As an application, we derive a new formula for the density of the product of k independent symmetric variance-gamma distributed random variables.     

Exponential sums and singular hypersurfaces

We give upper bounds for the absolute value of exponential sums in several variables attached to certain polynomials with coefficients in a finite field. This bounds are given in terms of invariants of the singularities of the projective hypersurface defined by its highest degree form. For exponential sums attached to the reduction modulo a power of a large prime of a polynomial f with integer coefficients and veryfying a certain condition on the singularities of its highest degree form, we give a bound in terms of the dimension of the Jacobian quotient . Received: 3 November 1997     

Reduction in the Resonance Error in Numerical Homogenization II: Correctors and Extrapolation

This paper is the follow-up of Gloria (Math Models Methods Appl Sci 21(8):1601–1630, 2011). One common drawback among numerical homogenization methods is the presence of the so-called resonance error, which roughly speaking is a function of the ratio \(\frac{\varepsilon }{\rho }\), where \(\rho \) is a typical macroscopic lengthscale and \(\varepsilon \) is the typical size of the heterogeneities. In the present work, we make a systematic use of regularization and extrapolation to reduce this resonance error at the level of the approximation of homogenized coefficients and correctors for general non-necessarily symmetric stationary ergodic coefficients. We quantify this reduction for the class of periodic coefficients, for the Kozlov subclass of almost-periodic coefficients, and for the subclass of random coefficients that satisfy a spectral gap estimate (e.g., Poisson random inclusions). We also report on a systematic numerical study in dimension 2, which demonstrates the efficiency of the method and the sharpness of the analysis. Last, we combine this approach to numerical homogenization methods, prove the asymptotic consistency in the case of locally stationary ergodic coefficients, and give quantitative estimates in the case of periodic coefficients.     

On linear relations for L-values over real quadratic fields

In this paper, we give a method to construct a classical modular form from a Hilbert modular form. By applying this method, we can get linear formulas which relate the Fourier coefficients of the Hilbert and classical modular forms. The paper focuses on the Hilbert modular forms over real quadratic fields. We will state a construction of relations between the special values of L-functions, especially at 0, and arithmetic functions. We will also give a relation between the sum of squares functions with underlying fields \(\mathbb {Q}(\sqrt{D})\) and \(\mathbb {Q}\).     

On stability of interconnected finite difference systems

We give a method of construction of Lyapunov functions in the form of a linear form with respect to moduli of variables, for which there exist Krasovskii constants in the case of asymptotic stability, for linear systems with constant coefficients and some types of nonlinear systems of finite-difference equations. An application of the above functions as components of a vector Lyapunov function allowed us to obtain conditions on asymptotic stability for interrelated finite-difference systems.Translated from Dinamicheskie Sistemy, No. 8, pp. 68–71, 1989.