On the finite termination of the Douglas-Rachford method for the convex feasibility problem

In this paper we apply the Douglas–Rachford (DR) method to solve the problem of finding a point in the intersection of the interior of a closed convex cone and a closed convex set in an infinite-dimensional Hilbert space. For this purpose, we propose two variants of the DR method which can find a point in the intersection in a finite number of iterations. In order to analyse the finite termination of the methods, we use some properties of the metric projection and a result regarding the rate of convergence of fixed point iterations. As applications of the results, we propose the methods for solving the conic and semidefinite feasibility problems, which terminate at a solution in a finite number of iterations.     

Absolute configuration and conformation analysis of 1-phenylethanol by matrix-isolation infrared and vibrational circular dichroism spectroscopy combined with density functional theory calculation

The absolute configuration and conformation of 1-phenylethanol (1-PhEtOH) have been determined by matrix-isolation infrared (IR) and vibrational circular dichroism (VCD) spectroscopy combined with quantum chemical calculations. Quantum chemical calculations have identified that there are three conformers, namely, I, II, and III, in which characteristic intramolecular interactions are found. The IR spectrum-conformation correlation for 1-PhEtOH has been developed by the Ar matrix-isolation IR measurement and used for the assignments of the observed IR bands. In a dilute CCl(4) solution, 1-PhEtOH exists predominantly as conformer I along with a trace amount of conformer II. By considering conformations and intermolecular hydrogen-bonding in the spectral simulation for (S)-1-PhEtOH, we have successfully reproduced the VCD spectrum of (-)-1-PhEtOH observed in a dilute CS(2) solution. Thus, (-)-1-PhEtOH is of S-configuration and conformer I in the dilute solution. The same method has been applied to analyze the VCD spectra measured in the liquid state of (-)-1-PhEtOH. The absolute configuration of 1-PhEtOH in the condensed phase is enabled by identifying VCD bands that are insensitive to conformational changes and intermolecular interactions. The present work provides a combinatorial procedure for determination of both the absolute configuration and the conformation of chiral molecules in a dilute solution and condensed phase.     

Electrochemical reduction of carbon dioxide catalyzed by cofacial dinuclear metalloporphyrin

Cofacial dinuclear metalloporphyrins exhibited a catalytic activity for the electrochemical reduction of carbon dioxide. The cofacial dinuclear porphyrin was automatically generated by mixing a cationic cobalt porphyrin (CoTMPyP) and an anionic metalloporphyrin (MTPPS) in solution. The redox system of this complex was examined by electrochemical methods. According to the cyclic voltammogram, the catalytic active species was generated at −1.8V vs. Ag/Ag⁺, which was considered to be a monovalent cobalt porphyrin, Co(I)TMPyP. The catalytic activity of the dinuclear complex was two times greater than that of the mononuclear one because the anionic porphyrin acted as an electron mediator.     

On V-orthogonal projectors associated with a semi-norm

For any n ×  p matrix X and n ×  n nonnegative definite matrix V, the matrix X(X′V X)⁺ X′V is called a V-orthogonal projector with respect to the semi-norm , where (·)⁺ denotes the Moore-Penrose inverse of a matrix. Various new properties of the V-orthogonal projector were derived under the condition that rank(V X) =  rank(X), including its rank, complement, equivalent expressions, conditions for additive decomposability, equivalence conditions between two (V-)orthogonal projectors, etc.     

On the necessary and sufficient condition for the extended Wedderburn–Guttman theorem

Let A be a u by v matrix, and let M and N be u by p and v by q matrices, where p may not be equal to q or rank(M^′AN)<min(p,q). Recently, Galantai [A. Galantai, A note on the generalized rank reduction, Acta Math. Hungarica 116 (2007) 239–246] presented what he claimed to be the necessary and sufficient condition for rank(A-AN(M^′AN)^-M^′A)=rank(A)-rank(AN(M^′AN)^-M^′A) to hold. This rank subtractivity formula along with the condition under which it holds is called the extended Wedderburn–Guttman theorem. In this paper, we show that some of Galantai’s assertions are incorrect.     

Rational synthesis of A2B-type meso-triarylsubporphyrins

Rational synthesis of A(2)B-type meso-arylsubporphyrins has been accomplished by the condensation of triethylamine-tri-N-tripyrromethene-borane with acid chlorides. These subporphyrins are useful for evaluations of the intrinsic substituent effects and the influences of substitution patterns, A(3)-type versus A(2)B-type substitution.     

Generalized canonical correlation analysis with missing values

Generalized canonical correlation analysis is a versatile technique that allows the joint analysis of several sets of data matrices. The generalized canonical correlation analysis solution can be obtained through an eigenequation and distributional assumptions are not required. When dealing with multiple set data, the situation frequently occurs that some values are missing. In this paper, two new methods for dealing with missing values in generalized canonical correlation analysis are introduced. The first approach, which does not require iterations, is a generalization of the Test Equating method available for principal component analysis. In the second approach, missing values are imputed in such a way that the generalized canonical correlation analysis objective function does not increase in subsequent steps. Convergence is achieved when the value of the objective function remains constant. By means of a simulation study, we assess the performance of the new methods. We compare the results with those of two available methods; the missing-data passive method, introduced in Gifi’s homogeneity analysis framework, and the GENCOM algorithm developed by Green and Carroll. An application using world bank data is used to illustrate the proposed methods.