A note on the efficiency of the conjugate gradient method for a class of time‐dependent problems

We discuss the efficiency of the conjugate gradient (CG) method for solving a sequence of linear systems; Auⁿ⁺¹ = uⁿ, where A is assumed to be sparse, symmetric, and positive definite. We show that under certain conditions the Krylov subspace, which is generated when solving the first linear system Au¹ = u⁰, contains the solutions {uⁿ} for subsequent time steps. The solutions of these equations can therefore be computed by a straightforward projection of the right‐hand side onto the already computed Krylov subspace. Our theoretical considerations are illustrated by numerical experiments that compare this method with the order‐optimal scheme obtained by applying the multigrid method as a preconditioner for the CG‐method at each time step. Copyright © 2007 John Wiley & Sons, Ltd.     

Theoretical calculation of the shock compression properties of liquid H2 + D2 mixtures

Based on liquid variational perturbation theory with quantum mechanics correction, the effective exp-6 potential is adopted to compute the shock Hugoniot of liquid H₂ + D₂ mixtures at different molar rations. An examination of the confidence of the above computation is performed by comparing experiments and calculations, in which similar calculation procedure used for H₂ + D₂ is adopted for H₂ and D₂ each, since no experimental data are available to conduct this kind of comparison. Good agreement in both comparisons is found. This fact may look as if an indirect positive verification of calculation procedure was used here at least in the pressure and temperature domain covered by the experimental data of H₂ and D₂ used for comparison, numerically nearly up to 20 GPa and 10⁴ K. Project supported by the Science and Technology Foundation of China Academy of Engineering Physics (Grant No. 97Z011).     

概率度量空间中压缩映象的不动点定理

本文通过引入压缩映象集序列的概念,推广并综合了张石生以及V.M.Sehger和A.T.Bharucha-Reid中某些主要定理的结果。     

Photonic band structures of two-dimensional photonic crystals with deformed lattices

Using the plane-wave expansion method, we have calculated and analysed the changes of photonic band structures arising from two kinds of deformed lattices, including the stretching and shrinking of lattices. The square lattice with square air holes and the triangular lattice with circular air holes are both studied. Calculated results show that the change of lattice size in some special ranges can enlarge the band gap, which depends strongly on the filling factor of air holes in photonic crystals; and besides, the asymmetric band edges will appear with the broken symmetry of lattices.     

End‐grafting copolymers of styrene and 4‐vinylpyridine on an interacting solid surface

Copolymers of styrene and 4‐vinylpyridine with a styrene fraction f varying from 1 to 0 were grafted onto a silicon substrate in the melt. The grafting reaction and the stability of the grafted chains were investigated by Fourier transform infrared and X‐ray photoelectron spectroscopy. The thickness and surface morphology of the grafted copolymer layers were characterized with ellipsometry and atomic force microscopy (AFM). The copolymer chains were successfully grafted to the surface of the silicon substrate by a reaction between the hydroxyl groups of the nitroxide moiety at the end of the copolymers and the silanol groups on the surface of the silicon wafer. A measurement of the thickness of the grafted copolymer layers showed that the ratio of grafted‐layer thickness to the unperturbed chain radius of gyration decreased with the increasing fraction of 4‐vinylpyridine in the copolymer; this indicated that the grafted layer was strongly attracted to the substrate. In addition, an accelerated grafting process was observed at grafting times ranging from 48 to 72 h for pure poly(4‐vinylpyridine) and copolymers with f values of 0.3 and 0.5. AFM observation revealed that the grafted layers densely and homogeneously covered the silicon substrate. © 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 1332‐1343, 2005     

妊娠晚期羊水中微量元素测定及其与胎儿生长发育关系的研究

对１０２名足月正常妊娠第一胎的产妇妊娠晚期羊水中６种微量元素（锌、铜、锰、铁、钙、镁）首次进行了测定，并通过对出生新生儿生长发育的监测探讨其与胎儿生长发育的关系。妊娠晚期羊水中６种微量元素含量（±Ｓ／×１０－５）分别为：Ｚｎ２．２９±１．７７，Ｃｕ０．１４±０．１４，Ｍｎ０．１３±０．２３，Ｆｅ３．６９±２．７０，Ｃａ８２．５４±７４．１８，Ｍｇ１８．７１±１１．０９．男女新生儿其母羊水中微量元素含量比较无显著差异，Ｐ≥０．０５．与文献报道中期妊娠羊水中所含微量元素比较有增加趋势．６种微量元素相互关系中．与锌、钙呈正相关外，锌与铜、锰、铁、镁呈负相关．低体重组与低身长组新生儿孕母羊水中锌含量明显低于高体重组和高身长组，有显著差异，Ｐ≤０．０１．铁在高身长组含量增高，与低身长组比较有显著差异，Ｐ≤０．０５．锰于低体重组含量升高，与高体重组比较有显著差异，Ｐ≤０．０１．镁与铜于低身长组含量升高，与高身长组比较有显著差异，Ｐ分别≤０．０５与０．０１．可见高铜、高锰、高镁可致胎儿生长发育迟缓．     

多重经济增长路径,多重稳态解和分歧

该文给出一个具有可解内生生育率的Cass-Koopmans（C-K）经济增长模型．证明参数满足时，模型存在多重经济增长路径和多重稳态解；当时，经济增长路径和稳态解唯一，即描述模型的微分动力系统出现分歧．文中讨论了多重经济增长路径的几何形态、位置关系和主要结果的经济意义．