Stieltjes Polynomials and Related Quadrature Formulae for a Class of Weight Functions

Consider a (nonnegative) measure with support in the interval such that the respective orthogonal polynomials, above a specific index , satisfy a three-term recurrence relation with constant coefficients. We show that the corresponding Stieltjes polynomials, above the index , have a very simple and useful representation in terms of the orthogonal polynomials. As a result of this, the Gauss-Kronrod quadrature formulae for have all the desirable properties, namely, the interlacing of nodes, their inclusion in the closed interval (under an additional assumption on ), and the positivity of all weights. Furthermore, the interpolatory quadrature formulae based on the zeros of the Stieltjes polynomials have positive weights, and both of these quadrature formulae have elevated degrees of exactness.

     

On the uniform convergence of the Fourier series for one spectral problem with a spectral parameter in a boundary condition

In this paper, we investigate the uniform convergence of the Fourier series expansions in terms of eigenfunctions for the spectral problem where λ is a spectral parameter, q(x) is a real‐valued continuous function on the interval [0,1], and a₁,b₀,b₁,c₁,d₀, and d₁ are real constants that satisfy the conditions Copyright © 2015 John Wiley & Sons, Ltd.     

求解“余数问题”的算法研究

求解"余数问题"可归结为一次同余式组x≡ri(mod pi)或一次不定方程组x=pixi+ri的求解.当方程的个数n与模pi(i=1,2,…,n)较大时,用同余式理论和孙子定理求解的过程非常繁琐.为此,运用试算分析法和辗转相除法,给出了求解上述问题的两种通用的计算机算法和程序.通过实践证明,该算法具有计算步骤简便,求解灵活快速,通用性强等优点.     

Recursive Formulae for Enumeration of LM-Conjugated Circuits in Structurally Related Benzenoid Hydrocarbons

The linearly independent and minimal conjugated (LM-conjugated) circuits of benzenoid hydrocarbons play the central role in the conjugated circuit model. For a general case, the enumeration of LM-conjugated circuits may be tedious as it requires construction of all Kekule structures. In our previous work, a recursive method for enumeration of LM-conjugated circuits of benzenoid hydrocarbons was established. In this paper, we further extend the recursive formulae for enumerations of LM-conjugated circuits for both catacondensed benzenoid hydrocarbons and some families of structurally related pericondensed benzenoid hydrocarbons.     

非线性中立型控制系统的镇定与控制

本文利用Riccati矩阵方程的对称正定解构造正定二次型V函数,运用L yapunov分解等价法,给出了非线性中立型定常控制系统的镇定与次优控制。同时得到了滞后量与非线性项的界限的估计公式。     

Partial fraction decompositions and trigonometric sum identities

The partial fraction decomposition method is explored to establish several interesting trigonometric function identities, which may have applications to the evaluation of classical multiple hypergeometric series, trigonometric approximation and interpolation.

     

支付交易费的回望期权定价

以Black-Scholes模型为基础,通过对回望期权的研究,结合有交易费的欧式期权的定价公式,运用证券组合技术与无套利原理,建立了支付交易费的回望期权定价模型.通过对方程化简和分析,运到PDE相关方法化为Cauchy问题,得出定价公式.     

An Asymptotic Approach to the Torsion Problem in Thin Walled Beams

A rather straightforward derivation of the Γ-limit of the torsion problem as the thickness goes to zero is obtained for generic thin walled beams. The limit stresses are evaluated and the distributional nature of one of the stress components is clarified.      

Propagation of nonparaxial vector hollow Gaussian beams through a circular aperture

Based on the vectorial Rayleith-Sommerfeld formulae, the nonparaxial propagation properties of the vector hollow Gaussian beams (HGBs) through a circular aperture are studied in detail. We describe the derivation of the integral expressions of the propagation of nonparaxial vector HGBs through a circular aperture. The derived expression is independent the approximation of paraxial and far field, which are valid for either far and near field and for the systems in which aperture radius is comparable to or even smaller than wavelength. And it is also strict integral formula for the light field on the axis. Numerical calculation results indicate that there is no difference between derived formulae and the Collins formulae in the situation of paraxial approximation. Using the formula deduced, we calculate the propagation properties of HGBs. The calculated results indicate that the propagation field of vector hollow Gaussian beams is asymmetric in near field, while the propagation field is symmetric in far field. These research results could well shed light on the further understanding of the vectorial property of HGBs through a circular aperture, and would play a guiding role in the practical application of HGBs.     

Solution of certain weakly singular integral equations

^** Email: alok{at}math.iisc.ernet.in Direct function theoretic methods are developed to handle twoweakly singular integral equations with their kernels havinglogarithmic singularity. The present methods avoid the occurrenceof higher-order (or strong) singularities, like the Cauchy typesingularity in the representation of the solutions of such integralequations.