An inner product lemma

Given the operator product BA in which both A and B are symmetric positive‐definite operators, for which symmetric positive‐definite operators C is BA symmetric positive‐definite in the C inner product 〈x, y〉C? This question arises naturally in preconditioned iterative solution methods, and will be answered completely here. Copyright © 2004 John Wiley & Sons, Ltd.     

二维椭圆型方程反问题中优化算法的比较

近年来，决定椭圆型方程系数反问题在地磁、地球物理、冶金和生物等实际问题上有着广泛的应用．本文讨论了二维的决定椭圆型方程系数反问题的数值求解方法．由误差平方和最小原则，这个反问题可化为一个变分问题，并进一步离散化为一个最优化问题，其目标函数依赖于要决定的方程系数．本文着重考察非线性共轭梯度法在此最优化问题数值计算中的表现，并与拟牛顿法作为对比．为了提高算法的效率我们适当选择加快收敛速度的预处理矩阵．同时还考察了线搜索方法的不同对优化算法的影响．数值实验的结果表明，非线性共轭梯度法在这类大规模优化问题中相对于拟牛顿法更有效．     

Differentiability of the Value Function of Nonclassical Optimal Growth Models

We consider an optimal growth (multi-sector) model with nonconvex technology. Using the Clarke results on generalized gradients, we prove that the value function has left and right derivatives with respect to the initial capital stock, without requiring supermodularity assumptions.     

A new method for determination of uric acid by the lactic acid-acetone-BrO(3)(-)-Mn(2+)-H(2)SO(4) oscillating reaction using the analyte pulse perturbation technique

A new analytical method for the determination of uric acid (UA) by the perturbation of UA on the Belousov-Zhabotinsky oscillating reaction is proposed. The method is based on the linear relationship between the changes in the oscillating period and the concentration of UA. The calibration curve is linear over the range of 2.0 × 10⁻⁵ to 5.0 × 10⁻⁴ M, with a detecting limit of 3.28 × 10⁻⁶ M. The method features good precision (R.S.D.: 3.59%) and excellent throughput (10 samples h⁻¹). The possible mechanism of the perturbation of UA on the oscillating reaction is discussed.     

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.     

Chemical applications of group theory and topology

The following procedure is described for investigating the qualitative dynamics of simple chemical systems: 1) A so-called influence diagram is generated representing the relationships between the reference reactants (phase-determining intermediates); 2) This influence diagram is used to generate a truth table indicating possible transitions between state vectors representing the signs of the time derivatives of of the reference reactant concentrations; 3) The truth table is used to determine a state transition diagram representing the flow topology around unstable equilibrium points; 4) The characteristic equation of the adjacency matrix of the influence diagram is solved in order to determine the presence of such unstable equilibrium points. The two types of qualitative dynamics possible for chemical systems containing two reference reactants and one feedback circuit are bifurcation between two attracting regions (bistability) and limit cycle oscillation. However, in two reference reactant systems oscillation requires an additional self-activating loop to generate the unstable equilibrium point required for its realization. Bistability and limit cycle oscillation are also two of the possible types of qualitative dynamics for chemical systems containing three reference reactants. However, chemical systems with three reference reactants and two or more feedback circuits can also contain interlocking limit cycles, which can lead to toroidal oscillations or chaos. The influence diagrams are given for the systems exhibiting these various types of dynamic behavior along with a summary of the important properties of all 729 possible influences for simple chemical systems containing three reference reactants.     

Use of short columns and high flow rates for rapid gradient reversed-phase chromatography

Summary Rapid analyses were performed using reversed-phase liquid chromatography with short (20–100 mm) columns swept by fast yet shallow gradients, and the results compared with those obtained with 150 mm columns and slow gradients. The resolution losses incurred with shorter columns were minimised by employing elevated flow rates, to ensure that comparable mean retention factors were experienced by individual analytes during gradients run on different columns. This conserves gradient steepness. High quality performance was obtained with turn-around times of 5–10 minutes. An overall 5-fold enhancement in the rate of information generation was obtained. The relevance of instrumental parameters and of column and packing dimensions, upon the potential for improved performance is discussed. Some implications for the rapidly developing technique of capillary electrochromatography are briefly indicated.     

Computerized optimization of gradient elution conditions with ternary solvent mobile phases in RPLC

Summary In this work, an optimization procedure for gradient RPLC separation, using ternary mobile phases, is described. This procedure requires eight preliminary experiments in gradient elution mode to predict the retention surface for each solute over the whole triangular space. This is followed by computerized calculations to determine the best ternary gradient elution profile with respect to both selectivity and analysis time. The efficiency of this procedure from the point of view of rapidity and of accuracy, is illustrated for the specific separation of twelve phenyl urea herbicides.     

Polymerization of MMA by oscillating zirconocene catalysts, diastereomeric zirconocene mixtures, and diastereospecific metallocene pairs

Characteristics of methyl methacrylate (MMA) polymerization using oscillating zirconocene catalysts, (2-Ph-Ind)₂ZrX₂ (X = Cl, 1; X = Me, 2), mixtures of rac- and meso-zirconocene diastereomers, (SBI)ZrMe₂ [3, SBI = Me₂Si(Ind)₂] and (EBI)ZrMe₂ [4, EBI = C₂H₄(Ind)₂], as well as diastereospecific metallocene pairs, rac-4/Cp₂ZrMe₂ (5) and rac-4/CGCTiMe₂ [6, CGC = Me₂Si(Me₄C₅)(t-BuN)], are reported. MMA polymerization using the chloride catalyst precursor 1 activated with a large excess of the modified methyl aluminoxane is sluggish, uncontrolled, and produces atactic PMMA. On the other hand, the polymerization by a 2/1 ratio of 2/B(C₆F₅)₃ or 2/Ph₃CB(C₆F₅)₄ is controlled and produces syndiotactic PMMA. Mixtures of diastereomeric ansa-zirconocenes 3 or 4 containing various rac/meso ratios, when activated with B(C₆F₅)₃, yield bimodal PMMA; this behavior is attributed to the meso-diastereomer that, in its pure form, affords bimodal, syndio-rich atactic PMMA. For MMA polymerization using diastereospecific metallocene pairs, rac-4/5 and rac-4/6, the isospecific catalyst site dominates the polymerization events under the conditions employed in this study, and the aspecific and syndiospecific sites are largely nonproductive, thereby forming only highly isotactic PMMA.     

Effect of mobile phase density gradients on efficiency in packed column supercritical fluid chromatography

Summary The efficiency of packed columns was measured as a function of flow rate, temperature, outlet density, and the density differential across the column, unsing pure carbon dioxide as the mobile phase. Although density differentials are often blamed for a serious loss in efficiency in packed column supercritical fluid chromatography, the results show that efficiency was not a function of the density differential. Peak shapes suggest that apparent loss in efficiency is actually due to inadequate solubility of the solute in carbon dioxide.