Simplification of the expansions of viscous terms in basic aerodynamic equations in non-orthogonal curvilinear coordinate system

The application of non-orthogonal curvilinear coordinate system to the calculation of the flow field inside the channel, with complex boundary geometry, can effectively simplify the work of designing the calculation program and improve the accuracy of calculation [1]. Therefore, it is obviously necessary to expand the viscous terms, i.e. viscous force, rate of work done by viscous stress and dissipation, in basic aerodynamic equations in the nonorthogonal curvilinear system [2]. However, each of these expansions consistes of tens or even hundreds of algebraic terms. The expansions of these three viscous terms discribed in this paper are considerably simplified by analysing their order of magnitude.     

Polynomial interpolation in several variables,cubature formulae,and ideals[*]Supported by the National Science Foundation under Grant DMS-9802265.

We discuss polynomial interpolation in several variables from a polynomial ideal point of view. One of the results states that if I is a real polynomial ideal with real variety and if its codimension is equal to the cardinality of its variety, then for each monomial order there is a unique polynomial that interpolates on the points in the variety. The result is motivated by the problem of constructing cubature formulae, and it leads to a theorem on cubature formulae which can be considered an extension of Gaussian quadrature formulae to several variables. This revised version was published online in June 2006 with corrections to the Cover Date.     

On generalized invariant cubature formulae

An important quality criterion of cubature formulae is their algebraic or trigonometric degree of exactness. The invariant theory is a powerful tool to construct cubature formulae of a given degree. In this paper, a quantitative expression is established for the classical invariant cubature formulas (ICFs). Motivated by this expression (or structure), we generalize the concept of ICFs and extend the famous Sobolev's Theorem on ICFs. The transformations allowed are no longer just orthogonal transformations. We illustrate the concepts and the constructions of the generalized ICFs by several examples.     

Modified Implicit–Explicit BDF Methods for Nonlinear Parabolic Equations

Implicit–explicit multistep methods for nonlinear parabolic equations were recently analyzed. If the implicit scheme is one of the backward differentiation formulae (BDF) of order up to six, then the corresponding implicit–explicit method of the same order is stable provided the stability constant is less than a specific scheme-dependent constant. Based on BDF, implicit methods are constructed such that the corresponding implicit–explicit scheme of the same order exhibits improved stability properties.     

Computer search for energy-consuming organic compounds with desired physicochemical parameters

A new method for computer searching for energy-consuming organic compounds (EOC) with desired physicochemical properties has been developed. The method is based on a specific multilevel system of efficient parametec and quasi-structural selection criteria specially oriented to the search for EOC. This method makes it possible to perform the search in the area of practically significant empirical formulae (not less than 15 atoms).Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 9, pp. 2168–2176, September, 1996.     

Finite extinction time for some perturbed Hamilton-Jacobi equations

In this paper we study initial value problems likeu _t–R¦u¦^m+u^q=0 in ⁿ× ₊, u(·,0⁺)=u_o(·) in ^N, whereR > 0, 0 <q < 1,m 1, andu _o is a positive uniformly continuous function verifying –R¦u _o¦^m+u ₀ ^q 0 in ^N . We show the existence of the minimum nonnegative continuous viscosity solutionu, as well as the existence of the function t(·) defined byu(x, t) > 0 if 0<t<t (x) andu(x, t)=0 ift t (x). Regularity, extinction rate, and asymptotic behavior of t(x) are also studied. Moreover, form=1 we obtain the representation formulau(x, t)=max{([(u _o(x – t))^1–q –(1–q)t]₊)^1/(1–q): ¦¦R}, (x, t) ₊ ^N+1 .Partially supported by the DGICYT No. 86/0405 project.     

Variational formulae for Fuchsian groups over families of algebraic curves

We study the problem of understanding the uniformizing Fuchsian groups for a family of plane algebraic curves by determining explicit first variational formulae for the generators.     

Positivity of the Weights of Interpolatory Quadrature Formulae with Jacobi Abscissae

We consider interpolatory quadrature formulae, relative to the Legendre weight function w(t) = 1 on [–1, 1], having as nodes the zeros of the nth degree Jacobi polynomial P _n ^{(, )} plus the points 1 and –1. We show that in specific domains of and gb the weights of these formulae are almost all positive, exceptions occurring only with the weights corresponding to 1 and –1.     

广义Bernoulli数和广义高阶Bernoulli数

定义了广义Bernoulli数和广义高阶Bernoulli数，建立了它们的递推公式和有关性质，从而推广了Bernoulli数和高阶Bernoulli数。