Conjugate-gradient method for computing the Moore-Penrose inverse and rank of a matrix

A conjugate-gradient method is developed for computing the Moore-Penrose generalized inverseA of a matrix and the associated projectors, by using the least-square characteristics of both the method and the inverseA . Two dual algorithms are introduced for computing the least-square and the minimum-norm generalized inverses, as well asA . It is shown that (i) these algorithms converge for any starting approximation; (ii) if they are started from the zero matrix, they converge toA ; and (iii) the trace of a sequence of approximations multiplied byA is a monotone increasing function converging to the rank ofA. A practical way of compensating the self-correcting feature in the computation ofA is devised by using the duality of the algorithms. Comparison with Ben-Israel's method is made through numerical examples. The conjugate-gradient method has an advantage over Ben-Israel's method.After having completed the present paper, the author received from Professor M. R. Hestenes his paper entitledPseudo Inverses and Conjugate Gradients. This paper treated the same subject and appeared in Communications of the ACM, Vol. 18, pp. 40–43, 1975.     

方酸作桥联配体的双核铕螯合物的电致发光

利用方酸作为桥联配体合成了一种双核铕鳌合物Eu_2（DBM）_4（Sq）Phen_2 ．用TPD作空穴传输材料、Eu_2（DBM）_4（Sq）Phen_2作发光材料和载流子传输材料、 8-羟基喹啉和铝（AlQ）作电子传输材料，设计了不同电致发光电特性，结果表明 Eu_2（DBM）_4（Sq）phen_2是一种同时具有空穴和电子传输能力的红色电致发光 材料，在器件结构为ITO/TPD,50nm/Eu_2（DBM）_4（Sq）Phen2，20nm/ALQ，50nm/LiF， 1nm/Al,200nm时，获得了在16V,6.9mA下有最大亮度91cd／m~2的电致发光器件.     

Hydrido-Carbonyl Rhenium Clusters with a Square Geometry of the Metal Core. Synthesis and X-Ray Characterization of the Novel [Re4(μ-H)3(CO)16]− Anion

Three and tetranuclear ring clusters have been obtained by treatment of [Re₂(CO)₈(THF)₂] with carbonyl-rhenates containing two terminal hydrides. The reaction with [ReH₂(CO)₄]^- provided a selective route to the previously known [Re₃(-H)₂(CO)₁₂]^- triangular cluster anion 1. The reaction with [Re₂H₂(-H)(CO)₈]^- gave the novel [Re₄(-H)₃(CO)₁₆]^- anion 2, containing a rare example of a puckered-square metal cluster. Protonation of 1 is known to afford the neutral [Re₃(-H)₃(CO)₁₂] species 3. Analogously the reaction of 2 with a strong acid afforded the previously known square metal clusters [Re₄(-H)₄(CO)₁₆] 4. The reaction could not be reversed by treatment with bases. Photolysis of 4 gave the unsaturated complex [Re₂(-H)₂(CO)₈] 5: this is the reverse of the dimerization reaction, that in THF at room temperature produces 4 from 5. Thermal treatment (reflux in cyclohexane for 24 h) left 4 almost unchanged. A single crystal X-ray analysis of [NEt₄]2 showed a s/e/s/s (e=eclipsed, s=staggered) conformation of the Re(CO)₄ units, leading to a puckered geometry of the ring, at variance with the square-planar geometry of 4 (all eclipsed). Two of the three hydrides of 2 have been located as bridging the Re–Re edges from inside the metal ring, as previously observed in 4. Density functional computations indicated a puckered conformation as the most stable for both 2 and 4, with very low activation energies for ring inversion (6.6 and 2.2 kcal·mol^-1, respectively), but ruled out solid state fluxionality for 4, whose observed planar geometry must be attributed to packing stabilization.     

Simultaneous kinetic spectrophotometric determination of cyanide and thiocyanate using the partial least squares (PLS) regression

The partial least squares (PLS-1) calibration model based on spectrophotometric measurement, for the simultaneous determination of CN⁻ and SCN⁻ ions is described. The method is based on the difference in the rate of the reaction between CN⁻ and SCN⁻ ions with chloramine-T in a pH 4.0 buffer solution and at 30 °C. The produced cyanogen chloride (CNCl) reacts with pyridine and the product condenses with barbituric acid and forms a final colored product. The absorption kinetic profiles of the solutions were monitored by measuring absorbance at 578 nm in the time range 20-180 s after initiation of the reaction with 2 s intervals. The experimental calibration matrix for partial least squares (PLS-1) calibration was designed with 31 samples. The cross-validation method was used for selecting the number of factors. The results showed that simultaneous determination could be performed in the range 10.0-900.0 and 50.0-1200.0 ng mL⁻¹ for CN⁻ and SCN⁻ ions, respectively. The proposed method was successfully applied to the simultaneous determination of cyanide and thiocyanate in water samples.     

2-[(卤代苯胺基)羰基]苯甲酸与Cu(Ⅱ)、Ni(Ⅱ)、Co(Ⅱ)的配合物研究

本文报道了4种2-[(卤代苯胺(?))羰基]苯甲酸与Cu(Ⅱ)、Ni(Ⅱ)、Co(Ⅱ)配合物的制备,并通过元素分析、热重分析、红外光谱和电子光谱分析以及磁化率测定对它们进行了表征.结果证明,配合物是通过羧酸根上羟基氧原子和酰胺羰基上氧原子配位成键,除Cu(Ⅱ)配合物分子为平面正方形结构外.其余均为八面体结构,只是扭曲程度不同.并对它们的配位场参数进行了计算.     

THE CORRELATION EFFECT ON THE CHAIN MOTION CONTRIBUTION TO IONIC TRANSPORT IN POLYMER ELECTROLYTES

The ionic transport process in polymer electrolytes (such as polyethylene oxide) wassimulated numerically on a two dimensional square lattice where charge carriers areaccommodated by the lattice sites connected randomly with available bonds to represent theamorphous chain configuration. Following the dynamic bond percolation theory(DBPT),the chainmotion contribution to the ionic conduction was incorporated via periodical renewal of the randombond configuration. To check and extend the prediction made by DBPT employing global abruptbond renewal,spatial correlation of the bond reassignment was introduced to the system by: 1)regional bond renewal and 2) organized bond motion. It is found that the difference between thediffusivities simulated involving regional bond renewal and those of DBPT becomes negligiblewhen the bond renewal rate approaches the carrier hopping rate.     

可调移动重叠分离图法用于高效液相色谱多台阶梯度分离条件优化的研究

计算机辅助高效液相色谱（HPLC）分离条件优化可以低成本、快速地得到优化的分离条,因而已较为广泛地用于复杂样品的分离分析。基于移动重叠分离图方法,又发展了一种新型的多台阶梯度分离条件的优化方法可调移动重叠分离图法。该方法通过预测不同流动相条件下各组分的保留时间、峰宽和分离度,绘制出对于样品中各组分的重叠分离区域图。在对当前台阶流动相组成进行优化的同时,考虑其对后面一到两个台阶上流出组分保留的影响,实时地重新绘制对于后面台阶上流出组分的重叠分离区域图。通过观察当前台阶流动相条件对当前台阶和后面台阶上流出组分分离的影响,综合考虑样品中所有组分的分离情况,找到更接近全局最优的分离条件。通过扫描的方法对优化得到的分离条件进行微调,能够进一步提高分离效果。采用文献数据对可调移动重叠分离图法的应用加以说明,在二元流动相体系下,证明了该方法在HPLC方法建立方面的优越性。     

不可逆电极过程方波伏安法动力学研究 Ⅰ.平行催化体系

本文报道了在不可逆电极过程的强催化体系中，当采用快速方波扫描时，正、逆向电流将出现峰形。依据实验结果，提出了“迁出－进入”的等浓度面理论模型，推导了并讨论了电流方程式。结果表明，文中提出的理论能较好地说明实验结果。     

Optimizing control of simulated moving beds--linear isotherm

A new optimization based adaptive control strategy for simulated moving beds (SMBs) is proposed. A linearized reduced order model, which accounts for the periodic nature of the SMB process, is used for online optimization and control. The manipulated variables are the four inlet flow rates, the outputs are the raffinate and extract concentrations. Concentration measurements at the raffinate and extract outlets are used as the feedback information. The state estimate from the periodic Kalman filter is used for the prediction of the outlet concentrations over a chosen horizon. Predicted outlet concentrations are the basis for the calculation of the optimal input adjustments, which maximize the productivity and minimize the desorbent consumption subject to constraints on product purities. The realization of this concept is discussed and the implementation on a virtual eight column SMB platform is assessed, in the case of binary linear systems. For a whole series of typical plant disturbances it is shown that the proposed approach is effective in minimizing off-spec products and in achieving optimal SMB operation, also in the case where there are significant model uncertainties.     

Optimal convergence properties of kernel density estimators without differentiability conditions

Let X ₁, X ₂, ..., X _n be independent observations from an (unknown) absolutely continuous univariate distribution with density f and let % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% GabmOzayaajaGaaiikaiaadIhacaGGPaGaeyypa0Jaaiikaiaad6ga% caWGObGaaiykamaaCaaaleqabaGaeyOeI0IaaGymaaaakmaaqadaba% Gaam4saiaacUfadaWcgaqaaiaacIcacaWG4bGaeyOeI0Iaamiwamaa% BaaaleaacaWGPbaabeaakiaacMcaaeaacaWGObGaaiyxaaaaaSqaai% aadMgacqGH9aqpcaaIXaaabaGaamOBaaqdcqGHris5aaaa!5356!\[\hat f(x) = (nh)^{ - 1} \sum\nolimits_{i = 1}^n {K[{{(x - X_i )} \mathord{\left/ {\vphantom {{(x - X_i )} {h]}}} \right. \kern-\nulldelimiterspace} {h]}}} \] be a kernel estimator of f(x) at the point x, \s-<x<, with h=h _n (h _n O and nh _n , as n) the bandwidth and K a kernel function of order r. Optimal rates of convergence to zero for the bias and mean square error of such estimators have been studied and established by several authors under varying conditions on K and f. These conditions, however, have invariably included the assumption of existence of the r-th order derivative for f at the point x. It is shown in this paper that these rates of convergence remain valid without any differentiability assumptions on f at x. Instead some simple regularity conditions are imposed on the density f at the point of interest. Our methods are based on certain results in the theory of semi-groups of linear operators and the notions and relations of calculus of finite differences.This research was supported in part by grants from the Natural Sciences and Engineering Research Council of Canada and the University of Alberta Central Research Fund.