Kibble–Slepian formula and generating functions for 2D polynomials

We prove a generalization of the Kibble–Slepian formula (for Hermite polynomials) and its unitary analogue involving the 2D Hermite polynomials recently proved in [16]. We derive integral representations for the 2D Hermite polynomials which are of independent interest. Several new generating functions for 2D q-Hermite polynomials will also be given.     

Euler–MacLaurin formulas via differential operators

Recently there has been a renewed interest in asymptotic Euler–MacLaurin formulas, because of their applications to spectral theory of differential operators. Using elementary means, we recover such formulas for compactly supported smooth functions on intervals, polygons, and three-dimensional polytopes, where the coefficients in the asymptotic expansion are sums of differential operators involving only derivatives of the function in directions normal to the faces of the polytope. Our formulas apply to wedges of any dimension.     

Second Order Optimality Conditions for Problems with Continuously Differentiable Data

In this note, by making use of the concept of unbounded approximate Hessian matrices, we present second order optimality conditions for a constrained mathematical programming problem with data whose gradient map is continuous, but not necessarily locally Lipschitz.     

Approximating exponential and logarithmic functions using polynomial interpolation

This article takes a closer look at the problem of approximating the exponential and logarithmic functions using polynomials. Either as an alternative to or a precursor to Taylor polynomial approximations at the precalculus level, interpolating polynomials are considered. A measure of error is given and the behaviour of the error function is analysed. The results of interpolating polynomials are compared with those of Taylor polynomials.     

一种分解指数算符的简洁方法

提出一种新的分解指数算符的方法,这种方法不仅简单易懂,而且有助于找到更多的算符恒等式.     

MATLAB在光学界面发生反射时的分析与应用

基于MATLAB的计算功能及可视化功能计算并分析了光学中的菲涅耳公式,更加直观的揭示了光波在两介质界面发生反射时的变化规律。使学生更加深入、直观地掌握光波在界面发生变化的条件、实质及布儒斯特角与全反射临界角的物理意义,同时也学会了分析研究方法,且提高了学习的主动性和创造性。     

Stable local trace formula

Langlands program is the central concern of modern number theory. In general, we think that the main tool for studying this program is the trace formula. Arthur-Selberg’s trace formula is successful in solving the endoscopic case of Langlands program. Arthur stabilized the geometric side of the local trace formula in the general case, but did not stabilize the general spectral. This paper aims to solve the general case by different method in the Archimedean case, by directly stabilizing the spectral side of the local trace formula, and obtains the stable local trace formula.     

Numerical study on combustion of H2—O2 in a supersonic reactive expansion in a nozzle

A study on expansion flow inside a nozzle considering full mechanism chemistry of hydrogen and oxygen was carried out. In this study, a full implicit scheme for turbulent reactive flow was obtained by combining the second order TVD scheme of Yee and Harten (1987, Implicit TVD schemes for hyperbolic conservation laws in curvilinear coordinates. American Institute of Aeronautics and Astronautics Journal, 25(2), 266–274) with the efficient implicit lower-upper scheme of Shuen and Yoon (1989, Numerical study of chemically reacting flows using a lower-upper symmetric successive overrelaxation scheme. American Institute of Aeronautics and Astronautics Journal, 27(12), 1752–1756). The species equations, Navier–Stokes equations and turbulence model were implemented in the numerical scheme and solved in conjunction with full detailed finite rate chemistry. The numerical scheme is verified by comparison with experimental results of a converging–diverging nozzle. Effects of inlet pressure, temperature and fuel-oxidant mass ratio on nozzle flow field were studied. Variation of chemical species under different conditions was investigated by considering a chemical mechanism. Results show that increasing inlet pressure increases the rate of reactions due to increasing the concentration of reactants. For lower inlet pressure the radical H increases slightly in the diverging part of the nozzle, while for higher pressures it decreases along the nozzle. Inlet fuel–oxidant mass ratio affects the variation of all species with a greater effect for a near stoichiometric ratio. It was also shown that a higher inlet temperature provides a more enhanced reaction zone in the diverging part of the nozzle.     

The Calculus of Moving Surfaces and Laplace Eigenvalues on an Ellipse with Low Eccentricity

Our goal is to demonstrate the utility of the calculus of moving surfaces (CMS) in boundary variation problems. We discuss the relative advantages of the CMS compared to the alternative approach of interior variations. We illustrate the technique by calculating the two leading terms of a power series for the Laplace eigenvalues on an ellipse with semi-axes 1 + a and 1 + b, where a and b are small. We compare the CMS estimates with those obtained by the conventional finite element method with Richardson extrapolation. The comparison confirms the cubic rate of convergence for the CMS estimates.