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.     

Trends in the CNDO/2 molecular orbital study of electronic structure in some neurotransmitter drugs

An investigation of electronic structure in some neurotransmitter drugs has been made using the CNDO/2 semi-empirical molecular orbital method.The electronic structure has been conveniently characterized by the electronic parameters nett atomic population (NAP) and bond index (BI). A variation of these electronic parameters with respect to conformation has been studied and has been found unlikely to exceed 0.1 e in most. cases. Further, the useful extent to which the electronic parameters of some commonly occurring functional groups may be regarded as conformationally invariant has been demonstrated. Also presented are (i) a discussion on the intramolecular close-approach of functional groups — the interaction between terminal —COO^? and ?NH₃⁺ groups of α-ω anaino acids is explicitly considered; (ii) an enquiry into the extension of ‘standard’ (idealized) geometry models in the elucidation of electronic structure.The implication of the results and observations presented here are briefly discussed with reference to classical and quantum structure-activity studies of drug molecules.     

19-Electron arenecyclopentadienyl complexes of ruthenium

A series of are necyc lope ntadienyl complexes,i. e., [Ru(⁵-c₅R₅)(⁶- are ne)]⁺ (1, R= H, arene = C₆H₆; 2, R = Me, arme = C₆H₆; 3, R = H, arctic = C₆H₃Me₃; 4, R = Me, arene = C₆H₃Me₃; 5, R = H, arene = C₆Me₆; 6, R = Me, arene = C₆Me₆) was studied by cyclic voltammetry. These compounds are capable of both oxidation and reduction. The reduction potential values depend on the number of methyl groups in the complex. Reduction of benzene complexes I and 2 by sodium amalgam in THF leads to the formation of decomplexation products, the addition of hydrogen to benzene, and dimerization of the benzene ligands. Both chemical and electrochemical reductions of mesitylene complexes3 and4 result in dimeric products [(⁵-C₅R₅)Ru(-⁵;⁵-Me₃H₃C₆H₃Me₃)Ru(⁵-C₅R₅)] (14, R = H; 15, R = Me). The action of sodium amalgam on compound5 gives products of hydrogen addition to both hexamethylbenzene (17) and cyclopentadienyl (18) ligands along with the major product, the dimer [⁵-C₅H₅)Ru(-⁵; ⁵-Me₆C₆C₆Me₆)Ru(⁵-C₅H₅)] (16). In contrast to5, its permcthylated analog 6 is only capable of adding hydrogen to the hexamethylbenzene ligand.Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 7, pp. 1691–1697, July, 1996.     

A study of the polarographic behaviour of meso-tetra(4-trimethylammoniumphenyl)porphine and its copper and magnesium complexes and its analytical applications

The investigation of the electrochemical reduction and the adsorption of meso-tetra(4-trimethylammoniumphenyl)porphine (T(4-TMAP)P) at a mercury electrode in alkaline solution shows that the overall reduction involves three two-electron steps, of which the first step is reversible and the latter two are irreversible. In addition, T(4-TMAP)P and its metal complexes of Cu(II) and Mg(II) can be strongly adsorbed on the surface of a mercury electrode. The adsorption phenomena have been utilized as a preconcentration step for the determination of trace amounts of the two ions by single sweep polarography. For copper, the detection limit is 1 × 10^–8 mol dm^–3, for magnesium, 1 × 10^–7 mol dm^–3, the latter being limited by the reagent blank. The proposed method was applied to the determination of Cu and Mg in various types of samples (chemicals, hair and liver tissues) with satisfactory results.     

On the affinity of cytosine towards electrophiles

Ab initio SCF computations on the intrinsic preferences of the H⁺, CH ₃ ⁺ and C₂H ₅ ⁺ cations towards the two principal sites of protonation or alkylation on cytosine, N₃ or O₂, show that this preference undergoes a continuous modification with the increase in size and complexity of the cation. N₃ is the preferred site of fixation of H⁺, O₂ the preferred site of C₂H ₅ ⁺ , while CH ₃ ⁺ has no marked preference. The exchange repulsion term of the binding energy appears responsible for the preference of C₂H ₅ ⁺ for O₂.This work was supported by the Ligue Francaise contre le Cancer and the National Foundation for Cancer Research (USA)     

Seawater—A test of multicomponent electrolyte solution theories. II. Enthalpy of mixing and dilution of the major sea salts

In a continuing effort to predict the physicochemical properties of seawater from the properties of single aqueous electrolyte solutions, the pairwise heats of mixing at constant molal ionic strength,I=1.0 ional, have been determined for the six possible pairs of salts from the set (NaCl, Na₂SO₄, MgCl₂, MgSO₄) at 30°C. In addition, heats of dilution for two aqueous solutions formed from these salts and havingI=1.0 ional have been determined at 30°C. In order to present the most thermodynamically consistent results, it was found necessary to apply a correction term to the relative apparent equivalent enthalpies given in the literature at 30°C. These correction terms derived from a consideration of published results on heats of dilution at very low concentrations. Further, in order to make predictions for seawater at 25°C, it was deemed desirable to refit existing heat-capacity data. The heats relative apparent equivalent enthalpies for the two mixtures mentioned as well as for seawater. The estimates are based on the theoretical equation of Reilly and Wood for charge-asymmetric mixtures which derives from the work of Friedman. In the most applicable cases, the estimates agree with experimental relative apparent equivalent enthalpies to within 5%. In general, the results substantiate the theoretical equation.Taken in part from the Ph.D. dissertation of W. H. Leung, University of Miami, Miami, Florida 33149.     

Structure of palladium quinoline-8-selenolate Pd(C9H6NSe)2

The molecular and crystal structure of palladium quinoline-8-selenolate Pd(C₉H₆NSe)₂ has been determined by X-ray structural analysis. The structures of the five-membered metallocycles of palladium 8-hydroxy-, 8-mercapto-, and 8-hydroselenoquinolinates of one type are compared. __________ Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 3, pp. 396–402, March, 2006.     

连续一级反应的热动力学研究

本文把时间比法引入热动力学研究,建立了连续一级反应的热动力学研究法,并用来研究了一个连续反应体系的热动力学,验证了该方法的正确性.