Explicit and Implicit Continuation Algorithms for Strongly Monotone Variational Inequalities with Box Constraints

In this paper we discuss the variational inequality problems VIP(X, F), where F is assumed to be a strongly monotone mapping from ⁿ to ⁿ, and the feasible set X = [l, u] has the form of box constraints. Based on the Chen-Harker-Kanzow smoothing functions, first we present an explicit continuation algorithm (ECA) for solving VIP(X, F). The ECA possesses main features as follows: (a) at each iteration, it yields a new iterative point by solving a system of equations in ^{(n + s)} with a parameter and nonsingular Jacobian matrix, where s = |{j: - < l _j < u _j < +}|, (b) it generates a sequence of iterative points in the interior of the feasible set X. Secondly we give an implicit continuation algorithm (ICA) for solving VIP(X,F), the prime character of the ICA is that it solves only one, rather than a series of, system of nonlinear equations to obtain a solution of VIP(X,F). The two proposed algorithms are shown to possess strongly global convergence. Finally, some preliminary numerical results of the two algorithms are reported.     

Saturation theorems for interpolation and the Bernstein-Schnabl operator

We shall study properties of box spline operators: cardinal interpolation, convolution, and the Bernstein-Schnabl operator. We prove the saturation theorem.

     

A NEW SMOOTHING APPROXIMATION METHOD FOR SOLVING BOX CONSTRAINED VARIATIONAL INEQUALITIES

In this paper, we first give a smoothing approximation function of nonsmooth system based on box constrained variational inequalities and then present a new smoothing approximation algorithm. Under suitable conditions,we show that the method is globally and superlinearly convergent. A few numerical results are also reported in the paper.     

含有不可数个无界变差点的一维连续函数

在单位区间[0,1]上构造了图像长度为无穷的一维连续函数.该函数含有不可数个但Lebesgue测度为0的无界变差点.所有无界变差点组成的集合中每一点皆为该集合的聚点.     

Microwave-Assisted Extraction and High-Performance Liquid Chromatography for Simultaneous Determination of Osthole and Isoimperatorin in Libanotis buethorimensis

Osthole and isoimperatorin are two of the major bioactive components in Libanotis buethorimensis, an effective traditional Chinese medicine. A method of microwave-assisted extraction followed by high performance liquid chromatography was developed for simultaneous determination of osthole and isoimperatorin in Libanotis buethorimensis. The conditions for extraction were formulated by using the response surface method, and the chromatographic condition was optimized. The calibration curves were linear from 10.0 to 800.0 μg mL^?1 for osthole and from 1.0 to 200.0 μg mL^?1 for isoimperatorin. Results showed that the proposed method is reliable and a potential tool for the quality assessment of Libanotis buethorimensis materials.     

Determination of carboxylic acids in non-alcoholic beer samples by an ultrasonic-assisted dispersive micro-solid phase extraction based on Ni/Cu-Al layered double hydroxide nanocomposites followed by gas chromatography

Magnetically separable layered double hydroxide Ni/CuAl-LDH nanocomposites were synthesized and employed as ultrasonic-assisted dispersive micro-solid phase extraction (UA-D-μSPE) sorbent to extract several carboxylic acids (namely propionic, butyric, pentanoic, hexanoic, heptanoic, octanoic, and decanoic) from non-alcoholic beer samples. Ni/CuAl-LDH sorbent was characterized by Fourier transform-infrared (FTIR) spectroscopy, transmission electron microscopy (TEM), and vibrating sample magnetometry (VSM). Effective variables such as amount of sorbent (mg), pH and ionic strength of sample solution, volume of eluent solvent (μL), vortex, and ultrasonic times (min) were investigated via fractional factorial design (FFD). The significant variables were optimized by a Box–Behnken design and combined by a desirability function (DF). Under optimized conditions, the calibration graphs of analytes were linear in a concentration range of 0.05–100 μg/mL and had correlation coefficients more than 0.997. The limits of detection and quantification were in the ranges of 16–40 μg/L and 53–133 μg/L, respectively. This procedure was successfully employed in the determination of target analytes in spiked beer samples, and the relative mean recoveries ranged from 87 to 110%.     

Hydration of nitriles: an examination in terms of No Barrier Theory

Nitriles pose an interesting problem to the explanatory powers of organic chemistry because, despite the favorable overall thermodynamics of hydrolysis to the corresponding amides, the reactions are inherently slow. The rate determining step is hydration of the nitrile to give the imidic acid, which quickly tautomerizes to the amide. In terms of Marcus Theory, the intrinsic barriers for acid and base‐catalyzed hydration are higher for nitriles than for amides, which are themselves slow reactions. It is remarkable that hydration of a nitrile, for which the free energy change is close to zero except for polyhaloacetonitriles, is much slower than hydration of an amide, which is energetically unfavorable. This can be explained by No Barrier Theory in terms of the high energetic cost of the geometrical distortions in “one thing at a time” corner species. There are no experimental equilibrium constants for this initial hydration step, so we have determined them computationally. The free energy change for the initial hydration is small; it is the fast and energetically downhill second step, tautomerization to the amide, which makes the overall hydrolysis of nitrile to amide thermodynamically favorable. Very few of the pK_a values needed in the acid and base‐catalyzed mechanisms are known, so we used linear free energy relations and treat the parent pK_a values as adjustable parameters. This procedure leads to pK_a values in accord with expectation based on such data as are available and permits calculation of rate constants in satisfactory agreement with experiment. Copyright © 2013 John Wiley & Sons, Ltd.     

本刊英文版Vol.30(2014),No.12论文摘要(英文)北大核心CSCD

Regularity of the Inverse of a Homeomorphism with Finite Inner Distortion Chang Yu GUO Abstract Let f:Ω→f（Ω）Rⁿ be a W^1,1-homeomorphism with L¹-integrable inner distortion.We show that finiteness of min{lip_f（x）,k_f（x）},for every x∈Ω\E,implies that f^-1∈W^1,nand has finite distortion,provided that the exceptional set E hasσ-finite H¹-measure.Moreover,f has finite distortion,differentiable a.e.and the Jacobian J_f>0 a.e.