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.     

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.     

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.     

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.     

Performance characteristics of suspended particulate reagent-iminodiacetate as a binding agent for diffusive gradients in thin films

The performance characteristics of an alternative binding agent, suspended particulate reagent-iminodiacetate (SPR-IDA), for use with DGT methodology were investigated. The parameters investigated during this study included gel hydration, blank levels, elution factor (f_e), capacity, the effects of pH on the binding of trace metals by DGT. The novel application of this resin for use as a quantitative standard for laser ablation ICP-MS was also evaluated. To further constrain the results for the SPR-IDA binding agent, parallel experiments were performed using resin gel containing Chelex 100, which has been widely reported in the literature. Hydration results showed that the SPR-IDA resin gel reached a stable dimension and weight within ∼30 min and was dimensionally stable for ≤6 months. The measured DGT blanks for the SPR-IDA resin were 0.0023, 0.15, 0.21, 0.0033 and 0.011 ng disc⁻¹ for Co, Ni, Cu, Cd and Pb, respectively. The elution factor differed for the two resin types with the Chelex 100 recoveries slightly lower than previous reports and the SPR-IDA resin showing on average ∼5-9% better recoveries than DGT containing Chelex 100. The measured capacity of DGT discs containing the SPR-IDA binding agent was 0.26 mg Cd, similar to the calculated value of 0.29 mg Cd, indicating the entire resin layer was available for metal uptake.Both resin types performed equally well when deployed in 1 mM NaNO₃ solutions with DGT measurements of ∼100% of direct solution measurements for Co, Ni and Cd. However, DGT measurements of Cu and Pb systematically decreased with increasing solution pH down to ∼50% of solution values at pH 8.0, due to artifacts resulting from colloid formation during the addition of the metals. This was remedied by adding the metals as dilute salt standards and addition of Mg(NO₃)₂ to eliminate adsorption to the container walls. In the latter experiments, DGT measured concentrations of Co, Ni, Cu, Cd and Pb were in agreement with solution concentrations. Deployment of DGT in solutions with increasing concentrations of trace metals yielded linear results, suggesting that quantitative analysis using simplified laser ablation techniques should be possible using this newly characterized SPR-IDA resin gel.     

Length scale effects in epoxy: The dependence of elastic moduli measurements on spherical indenter tip radius

Significant increases in the measured elastic moduli with decreasing indentation depth have been previously found in various polymers by indentation tests with a Berkovich tip at micro-to nanometer length scales. These increases in the determined elastic moduli were related to second order displacement gradients which increase with decreasing depth when a conical tip is applied. When a spherical tip is applied, such depth dependence should not be present as the second order displacement gradients remain essentially unchanged with indentation depth. However, these gradients should be proportional to the radius of the spherical tip. To examine the notion of second order displacement gradient dependence in measurements of elastic moduli, indentation experiments are conducted on epoxy with spherical tips of different nominal radii. Accounting for tip imperfections, an increase in the determined elastic moduli is found with decreasing tip radius, which corroborates the notion of second order displacement gradient dependence.     

Hierarchical Implicit Shape Modeling

In this paper, a new hierarchical approach for object detection is proposed. Object detection methods based on Implicit Shape Model (ISM) efficiently handle deformable objects, occlusions and clutters. The structure of each object in ISM is defined by a spring like graph. We introduce hierarchical ISM in which structure of each object is defined by a hierarchical star graph. Hierarchical ISM has two layers. In the first layer, a set of local ISMs are used to model object parts. In the second layer, structure of parts with respect to the object center is modeled by global ISM. In the proposed approach, the obtained parts for each object category have high discriminative ability. Therefore, our approach does not require a verification stage. We applied the proposed approach to some datasets and compared the performance of our algorithm to comparable methods. The results show that our method has a superior performance.