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.     

The optimization of silica gel synthesis from chemical bottle waste using response surface methodology

To reduce the amount of hazardous chemical bottle waste in the environment, we report the optimization research of silica extraction in chemical bottle waste into silica gel. Alkali fusion and sol–gel process were utilised to prepare silica gel effectively. The alkali fusion process was carried out by adding sodium hydroxide to produce sodium silicate. Afterwards, silica gel was prepared by the sol–gel method using hydrochloric acid. Box-Behnken Design (BBD) was applied to Optimisation factors the poptimiseactors affecting the silica recovery. The factors that optimised mass ratio, particle size, and temperature. The optimum recovery of silica gel was obtained by SiO₂: NaOH mass ratio of 1:3, the particle size of 63–74 µm, and a temperature of 800 °C. The purity of silica gel optimum is 63.74% characterised using X-ray fluorescence. The structure of silica gel is the appearance of amorphous peaks at 2θ 20-30° characterised using an x-ray diffractogram. The silica gel surface was characterises using scanning electron microscopy-energy dispersive x-ray. It showed an irregular surface and characteristic showed that silica gel had a radius of 15.74 nm and a specific surface area of 297.08 m².     

Electrophoresis simulations using Chebyshev pseudo-spectral method on a moving mesh

We present the implementation and demonstration of the Chebyshev pseudo-spectral method coupled with an adaptive mesh method for performing fast and highly accurate electrophoresis simulations. The Chebyshev pseudo-spectral method offers higher numerical accuracy than all other finite difference methods and is applicable for simulating all electrophoresis techniques in channels with open or closed boundaries. To improve the computational efficiency, we use a novel moving mesh scheme that clusters the grid points in the regions with poor numerical resolution. We demonstrate the application of the Chebyshev pseudo-spectral method on a moving mesh for simulating nonlinear electrophoretic processes through examples of isotachophoresis (ITP), isoelectric focusing (IEF), and electromigration-dispersion in capillary zone electrophoresis (CZE) at current densities as high as 1000 A/m. We also show the efficacy of our moving mesh method over existing methods that cluster the grid points in the regions with large concentration gradients. We have integrated the adaptive Chebyshev pseudo-spectral method in the open-source SPYCE simulator and verified its implementation with other electrophoresis simulators.     

Synthesis and absolute configuration of (1S,8S)-as-hydrindacene-1,8-diol as determined by the circular dichroism exciton chirality method

2,3,6,7-Tetrahydro-as-indacene-1,8-dione 4 was prepared in 4 steps starting from 2-methyl-furan by modification of a literature procedure. Appliance of Noyori’s asymmetric transfer hydrogenation, resulted in (1S,8S)-1,2,3,6,7,8-hexahydro-as-indacene-1,8-diol 5 in high yield (81%) and excellent enantioselectivity (>99% ee) or (8S)-8-hydroxy-3,6,7,8-tetrahydro-2H-as-indacen-1-one 6 in moderate yield (58%) and equally high enantioselectivity (98.5% ee), depending on the conditions. The asymmetric reduction was expected to yield the (S)-alcohols using the (S,S)-Ts-DPEN ligand, which was confirmed by the appliance of the exciton chirality method on the corresponding bis(p-dimethylamino)benzoate 7.     

GRECP/MRD‐CI calculations on the Hg atom and HgH molecule

Multireference single‐ and double‐excitation configuration interaction (MRD‐CI) calculations of transition energies for the Hg atom and spectroscopic constants for the HgH molecule are carried out with the generalized relativistic effective core potential (GRECP) method. A new selection criterium for the reference configurations is discussed. The calculated spectroscopic constants are compared with experimental data and results of calculations of other groups. Improvement of accuracy is mainly observed for bond lengths from the GRECP/MRD‐CI calculations (without applying the T = 0 correction) with respect to the results of other groups. Analysis of the quality of the approximations employed is carried out. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

Reactivity ratios in conventional and nickel-mediated radical copolymerization of methyl methacrylate and functionalized methacrylate monomers

Copolymerization of an excess of methyl methacrylate (MMA) relative to 2-hydroxyethyl methacrylate (HEMA) was carried out in toluene at 80 °C according to both conventional and controlled Ni-mediated radical polymerizations. Reactivity ratios were derived from the copolymerization kinetics using the Jaacks method for MMA and integrated conversion equation for HEMA (r_MMA = 0.62 ± 0.04; r_HEMA = 2.03 ± 0.74). Poly(ethylene glycol) α-methyl ether, ω-methacrylate (PEGMA, M_n = 475 g mol⁻¹) was substituted for HEMA in the copolymerization experiments and reactivity ratios were also determined (r_MMA = 0.75 ± 0.07; r_PEGMA ∼ 1.33). Both the functionalized comonomers were consumed more rapidly than MMA indicating the preferred formation of heterogeneous bottle-brush copolymer structures with bristles constituted by the hydrophilic (macro)monomers. Reactivity ratios for nickel-mediated living radical polymerization were comparable with those obtained by conventional free radical copolymerization. Interactions between functional monomers and the catalyst (NiBr₂(PPh₃)₂) were observed by ¹H NMR spectroscopy.     

Kinetics of CO oxidation on the surface of heterophase catalysts. Modeling by the Monte Carlo method

The Monte Carlo method has been used to simulate CO oxidation on a lattice consisting of various alternating patches: M1, where s(CO)>s(O₂) and M2, where s(CO)2). The reaction is shown to proceed over all the surface at low temperature as CO_ads spillover from M1 to M2 and backwards.     

Electronic structure and conformational flexibility of 1,2-dihydropyridine, 1,2- and 1,6-dihydropyrimidines,and their ylide derivatives

Molecular and electronic structures of 1,2-dihydropyridine, 1,2- and 1,6-dihydropyrimidine, and their oxo, imino, and methylene derivatives were studied by the semiempirical quantum-chemical AM1 method. In all compounds, the heterocycle exhibits a high conformational flexibility. The transition from a planar equilibrium conformation to a distorted sofa conformation with the =C-NH-C-C(N)= torsion angle of ±20° causes an increase in the energy by less than 1.7 kcal mol^–1. All molecules have similar -electronic structures, which, apparently, determines the similarity in their conformational behavior. The bending strain and the nonaromatic character of the cyclic -system are the factors that stabilize the nonplanar conformation of the ring in unsubstituted dihydroazines and ylide derivatives, respectively.Translated fromIzvestiva Akademii Nauk. Seriya Khimicheskaya, No. 8, pp. 1934–1937, August, 1996.     

A modified steepest descent method with applications to maximizing likelihood functions

In maximizing a non-linear function G(), it is well known that the steepest descent method has a slow convergence rate. Here we propose a systematic procedure to obtain a 1–1 transformation on the variables , so that in the space of the transformed variables, the steepest descent method produces the solution faster. The final solution in the original space is obtained by taking the inverse transformation. We apply the procedure in maximizing the likelihood functions of some generalized distributions which are widely used in modeling count data. It was shown that for these distributions, the steepest descent method via transformations produced the solutions very fast. It is also observed that the proposed procedure can be used to expedite the convergence rate of the first derivative based algorithms, such as Polak-Ribiere, Fletcher and Reeves conjugate gradient methods as well.     

Grouped coordinate minimization using Newton's method for inexact minimization in one vector coordinate

Letf(x,y) be a function of the vector variablesx R ⁿ andy R ^m. The grouped (variable) coordinate minimization (GCM) method for minimizingf consists of alternating exact minimizations in either of the two vector variables, while holding the other fixed at the most recent value. This scheme is known to be locally,q-linearly convergent, and is most useful in certain types of statistical and pattern recognition problems where the necessary coordinate minimizers are available explicitly. In some important cases, the exact minimizer in one of the vector variables is not explicitly available, so that an iterative technique such as Newton's method must be employed. The main result proved here shows that a single iteration of Newton's method solves the coordinate minimization problem sufficiently well to preserve the overall rate of convergence of the GCM sequence.The authors are indebted to Professor R. A. Tapia for his help in improving this paper.