Large-scale Stein and Lyapunov equations, Smith method, and applications

We consider the solution of large-scale Lyapunov and Stein equations. For Stein equations, the well-known Smith method will be adapted, with $A_k = A^{2^k}$ not explicitly computed but in the recursive form $A_k = A_{k-1}^{2}$ , and the fast growing but diminishing components in the approximate solutions truncated. Lyapunov equations will be first treated with the Cayley transform before the Smith method is applied. For algebraic equations with numerically low-ranked solutions of dimension n, the resulting algorithms are of an efficient O(n) computational complexity and memory requirement per iteration and converge essentially quadratically. An application in the estimation of a lower bound of the condition number for continuous-time algebraic Riccati equations is presented, as well as some numerical results.     

Structured backward error for palindromic polynomial eigenvalue problems

A detailed structured backward error analysis for four kinds of palindromic polynomial eigenvalue problems (PPEP)

Synthesis,crystal structure and magnetic studies of cis-configuration copper(II)—M(II) (M = Ba,Ca) complexes of novel Schiff bases derived from salicylaldehyde and dipeptides

The novel heteronuclear complexes [M(CuL)₂]₂ · nH₂O [M = Ba (1), Ca (2); H₂L = N-salicylideneglycylglycine] were synthesized and characterized, and the crystal structure of complex (1) was determined by X-ray diffraction. The entire structure is held together by an extensive network of H-bonds and – interactions. Variable-temperature magnetic susceptibility measurements (75–300 K) revealed the occurrence of an intramolecular antiferromagnetic interaction in (1) and (2).     

A continuation BSOR-Lanczos–Galerkin method for positive bound states of a multi-component Bose–Einstein condensate

We develop a continuation block successive over-relaxation (BSOR)-Lanczos–Galerkin method for the computation of positive bound states of time-independent, coupled Gross–Pitaevskii equations (CGPEs) which describe a multi-component Bose–Einstein condensate (BEC). A discretization of the CGPEs leads to a nonlinear algebraic eigenvalue problem (NAEP). The solution curve with respect to some parameter of the NAEP is then followed by the proposed method. For a single-component BEC, we prove that there exists a unique global minimizer (the ground state) which is represented by an ordinary differential equation with the initial value. For a multi-component BEC, we prove that m identical ground/bound states will bifurcate into m different ground/bound states at a finite repulsive inter-component scattering length. Numerical results show that various positive bound states of a two/three-component BEC are solved efficiently and reliably by the continuation BSOR-Lanczos–Galerkin method.     

电感耦合等离子体质谱间接法测定蛋白质含量

尺寸排阻色谱分离牛血清白蛋白(BSA),超氧化物歧化酶(SOD)和金属硫蛋白(MT)等3种标准蛋白的混合物后,在线使用电感耦合等离子体质谱测定这三种蛋白中的S,并根据每种蛋白质含有的S原子数,计算出3种蛋白质的相对含量。尺寸排阻色谱的流动相为0.1mol/LTris-HAc。向电感耦合等离子体质谱的六级杆加入反应气O2,使之与S反应生成SO ,间接测定32S16O 而避开直接测量32S时存在的严重干扰。测量结果与天平称量所得结果一致。方法的精密度好,每个蛋白质峰面积的RSD<3%(n=3)。BSA、SOD和MT的检出限分别为14、52和27pmol。     

Platycoside O, a new triterpenoid saponin from the roots of Platycodon grandiflorum

A new unusual minor triterpenoid saponin, platycoside O (1), was isolated from the 75% EtOH extract obtained from the roots of Platycodon grandiflorum, together with four known saponins: platycoside M-3 (2), platycoside J (3), platycoside F (4) and platycoside B (5). The structure of 1 was determined as 3-O-β-D-glucopyranosyl-(1→6)-β-D-glucopyranosyl-2β,3β,16α,23-tetrahydroxyolean-12-en-24-methoxyl, 24-oxo-28-oic acid 28-O-β-D-xylopyranosyl-(1→4)-α-L-rhamnopyranosyl-(1→2)-α-L-arabinopyranoside on the basis of spectral analysis and chemical evidence.     

基于布朗棘轮的单通道分离体系的模拟研究

分子在微尺度受限空间内的扩散行为是微观化学的重要研究领域. 本文利用布朗棘轮效应构建了一种微型单通道分离器件模型, 并基于随机行走理论对其中分子的扩散分离运动进行了模拟研究, 阐明了该体系的分离机理并考察了不同条件对分子扩散分离运动的影响. 模拟结果表明, 通过调节驱动力对各组分粒子的作用周期, 可以控制粒子与分离通道两端势垒发生不同程度的相互作用, 从而在单分离通道内实现粒子向不同方向的有效分离. 同时, 也给出了利用该分离体系进行分离的组分粒子本身的扩散运动与受外力驱动运动的相对大小需满足的条件. 通过调整分离器件的结构参数, 可在实现最佳分离效果的同时节约时间成本. 本文提出的模拟方法对开发微型分离器件及优化操作参数等具有参考意义.     

On the convergence of Ritz pairs and refined Ritz vectors for quadratic eigenvalue problems

For a given subspace, the Rayleigh-Ritz method projects the large quadratic eigenvalue problem (QEP) onto it and produces a small sized dense QEP. Similar to the Rayleigh-Ritz method for the linear eigenvalue problem, the Rayleigh-Ritz method defines the Ritz values and the Ritz vectors of the QEP with respect to the projection subspace. We analyze the convergence of the method when the angle between the subspace and the desired eigenvector converges to zero. We prove that there is a Ritz value that converges to the desired eigenvalue unconditionally but the Ritz vector converges conditionally and may fail to converge. To remedy the drawback of possible non-convergence of the Ritz vector, we propose a refined Ritz vector that is mathematically different from the Ritz vector and is proved to converge unconditionally. We construct examples to illustrate our theory.