A -functional and the rate of convergence of some linear polynomial operators

We show that the -functional

where , is equivalent to the rate of convergence of a certain linear polynomial operator. This operator stems from a Riesz-type summability process of expansion by Legendre polynomials. We use the operator above to obtain a linear polynomial approximation operator with a rate comparable to that of the best polynomial approximation.

     

Mosco approximation of integrands and integral functionals

It is shown that, given a lower semicontinuous convex integrandf satisfying a suitable integrability condition, there exists a sequence of Lipschitz simple integrands which Mosco converges tof and such that the sequence of conjugate integrands Mosco converges tof ^*. Moreover, this sequence can be chosen so that the sequence of associated integral functionals, respectively defined onL ¹(X) andL (X ^*), Mosco converges as well.We wish to thank Professor Erik J. Balder for interesting remarks on the first version of this work.     

Constrained minimax approximation and optimal preconditioned for Toeplitz matrices

A good preconditioner is extremely important in order for the conjugate gradients method to converge quickly. In the case of Toeplitz matrices, a number of recent studies were made to relate approximation of functions to good preconditioners. In this paper, we present a new result relating the quality of the Toeplitz preconditionerC for the Toeplitz matrixT to the Chebyshev norm (f– g)/f, wheref and g are the generating functions forT andC, respectively. In particular, the construction of band-Toeplitz preconditioners becomes a linear minimax approximation problem. The case whenf has zeros (but is nonnegative) is especially interesting and the corresponding approximation problem becomes constrained. We show how the Remez algorithm can be modified to handle the constraints. Numerical experiments confirming the theoretical results are presented.     

An application of the generalized alternating polynomials to the numerical solution of Fredholm integral equations

Generalized alternating polynomials have been introduced by the author earlier. In the present paper their indirect analogue is constructed for numerical solution of the Fredholm linear integral equations. Although the proposed method is a particular case of the general projection scheme, its valuable feature is the presence of a sequence of parameters, which, for sufficiently smooth kernels and inhomogeneous terms, serves as an indicator of the quality of approximation.     

Diffusion in Lorentz lattice gas automata with backscattering

The probability of first return to the initial intervalx and the diffusion tensorD _x are calculated exactly for a ballistic Lorentz gas on a Bethe lattice or Cayley tree. It consists of a moving particle and a fixed array of scatterers, located at the nodes, and the lengths of the intervals between scatterers are determined by a geometric distribution. The same values forx andD _x apply also to a regular space lattice with a fraction of sites occupied by a scatterer in the limit of a small concentration of scatterers. If backscattering occurs, the results are very different from the Boltzmann approximation. The theory is applied to different types of lattices and different types of scatterers having rotational or mirror symmetries.     

Electron energy level crossings in the time-dependent Born-Oppenheimer approximation

Summary We summarize the results of a mathematical study of the time-dependent Born-Oppenheimer approximation near crossings of two non-degenerate electron energy surfaces. We illustrate our techniques by relatively simple examples that contain the essential ingredients of the general cases. We discuss all generic types of crossings of two non-degenerate electron energy surfaces.Supported in part by the National Science Foundation under Grant number DMS-8801360     

Self-consistent-field – Hartree–Fock method with finite nuclear mass corrections

We have upgraded a Self-consistent-field – Hartree–Fock routine to include a finite nuclear mass correction for molecules developed in our laboratory. The new routine can handle isotopomers without calculating any nuclear kinetic energy matrix element. Tests on H₂, LiH, HF, F₂, and H₂O isotopomers indicate the equivalence of our correction to the standard diagonal adiabatic correction. A further original application to C₂H₆ illustrates the usefulness of the method for polyatomic molecules. The resulting molecular orbitals carry the nuclear mass signature, exemplified with Koopmans ionization potentials.From the Proceedings of the 28th Congreso de Químicos Teóricos de Expresión Latina (QUITEL 2002)     

Application of Short-wave Near-infrared Reflectance Spectroscopy in Controlling Extract of Fructus Cnidii Using Supercritical Carbon Dioxide

Fructus cnidii （Chinese name shechuangzi） is the fruit produced by Cnidium monnieri （L.） Cusson （Umbelliferae）. It is a perennial herb that is used to treat skin-related diseases and gynecopathyell. Recent pharmacological studies have revealed crude extracts or components isolated from fructus cnidii possess antiallergic, antipruritic, antidermatophytic, antibacterial, antifungal, and antiosteoporotic activities. Osthole and imperatorin are the major compounds present in shechuangzi. They are often used as standards for the evaluation of the quality of shechuangzi products.     

三级相互作用对聚亚甲基链末端距四次矩的影响

本文用三级相互作用近似理论导出了高分子链末端距的四次矩的计算公式。计算聚亚甲基链的末端距四次矩,并同二级相互作用的结果进行了比较。对于聚亚甲基链,三级相互作用近似的四次矩小于二级相互作用近似的四次矩。