Cochran's statistical theorem for outer inverses of matrices and matrix quadratic forms

We extend the matrix version of Cochran's statistical theorem to outer inverses of a matrix. As applications, we investigate the Wishartness and independence of matrix quadratic forms for Kronecker product covariance structures.     

1H and 13C NMR spectroscopic study of oxidation of D,L‐cystine and 3,3′‐dithiobis(propionic acid) with hydrogen peroxide in aqueous solution

The oxidation of symmetrical disulfides [D ,L ‐cystine ( 1 ) and 3,3′‐dithiobis(propionic acid) ( 2 )] with hydrogen peroxide in D₂O–NaOH solution (pH 10–11) was studied by NMR spectroscopy. Assignments of the proton and carbon NMR signals of starting materials ( 1 and 2 ) and products of oxidation are based on conventional 1D NMR methods (DEPT, selective spin decoupling). Formation of C—S bond cleavage products or, in case of 2 , partially oxidized intermediates was not detected. The accelerating effect of Cu²⁺ cations, but not Fe³⁺ cations, on the oxidation rate of 1 in basic medium was demonstrated. Copyright © 2002 John Wiley & Sons, Ltd.     

Structural characterization of polypyrrole in the solid state by high resolution 15N NMR spectroscopy [II]

The solid-state ¹⁵N CP/MAS NMR spectra and ¹⁵N spin-lattice relaxation times (T₁) of doped and dedoped ¹⁵N-labeled polypyrroles prepared by electrochemical polymerization, have been measured by means of high-resolution solid-state ¹⁵N NMR. The ¹⁵N signal of polypyrrole consists of four peaks decomposed by line shape analysis. The four peaks obtained have been assigned to the various structures of polypyrrole. Further, the half-width of the ¹⁵N NMR spectra of polypyrroles is discussed as related to the electrical conductivity. © 1995 John Wiley & Sons, Inc.     

Structure of norbornene-fused 3,1-oxazine double cycloadducts

Pentacyclic isoxazolines were obtained by the cycloaddition of benzonitrile oxide to norbornene-azetidinone-fused 3,1-oxazines. The constitutions of two of the isomers obtained, and the configurations and conformations of all products, were determined by means of ¹H and ¹³C NMR spectroscopy and DNOE experiments.     

On the pivot structure for the weighing matrix W(12,11)

In the present article we concentrate our study on the growth problem for the weighing matrix W(12,11) and show that the unique W(12,11) has three pivot structures. An improved algorithm for extending a k × k (0,+,-) matrix to a W(n,n-1), if possible, has been developed to simplify the proof. For the implementation of the algorithm special emphasis is given to the notions of data structures and parallel processing.     

1H and 13C NMR assignments of abietane diterpenes from Aegiphila lhotzkyana

An NMR study of one new and several known abietane diterpenes isolated from the roots of Aegiphila lhotzkyana is described. In addition to 1D NMR, several 2D shift‐correlated NMR pulse sequences (¹H–¹H‐COSY, NOESY, HMQC and HMBC) were used to establish all the structures, and unambiguously perform the ¹H and ¹³C chemical shift assignments of the new natural diterpene and three derivatives, the NMR data for which have not been reported previously. Revision of current data assignment for teuvincenone H is also suggested. Copyright © 2003 John Wiley & Sons, Ltd.     

Characterization of reduced iso‐α‐acids derived from hops (Humulus lupulus) by NMR

Reduced forms of iso‐α‐acids (isohumulones), used in modern beer brewing were separated and characterized by ¹H and ¹³C NMR spectroscopy. Components from mixtures of rho‐iso‐α‐acids, tetrahydro‐iso‐α‐acids, and hexahydro‐iso‐α‐acids were isolated using high‐performance liquid chromatography (HPLC) and analyzed by use of one‐ and two‐dimensional NMR experiments. The data presented assign the identities of the main peaks in the HPLC traces for the reduced iso‐α‐acids. Previous tentative assignments regarding the cis and trans configurations and the structures of the acyl residues of the reduced iso‐α‐acids were confirmed and extensive NMR assignments were made. Furthermore, the previously unknown stereochemistry in the C‐4 side‐chain of the rho‐ and hexahydro‐iso‐α‐acids was assigned. Copyright © 2003 John Wiley & Sons, Ltd.     

Hybrid organosilazane/organosilylamine telechelic oligomers—I. Two-dimensional NMR spectroscopy characterization of dimethylsilylethylenediamine oligomers

The detailed characterization of multifunctional hybrid organosilazane/organosilylamine telechelic oligomers by IR and ¹H, ¹³C and ²⁹Si NMR spectroscopy in one and two dimenions has been undertaken. The specific multifunctional oligomers, comprising NH/NH₂ or SiCl functionalities depending on the monomer feed-ratio, were prepared from mono- and di-functionally reacted dichlorodimethylsilane (DDS) and mono-, di-, tri- and tetra-functionally reacted ethylenediamine (EDA). Varying the feed-ratio afforded control of the microstructures of the oligomers and the preparation of oligomers with, in some cases, conterminously located silicon–chlorine (SiCl) groups. The combination of the NMR methods with the IR technique has enabled the detailed microstructural characterization of the oligomers and the identification of the functionalities therein. This approach and the knowledge gained from the DDS/EDA system has been applied to the microstructural characterization of other hybrid organosilazane/silylamine preceramic telechelic oligomers.     

Some geometrical aspects of semidefinite linear complementarity problems

This article studies some geometrical aspects of the semidefinite linear complementarity problem (SDLCP), which can be viewed as a generalization of the well-known linear complementarity problem (LCP). SDLCP is a special case of a complementarity problem over a closed convex cone, where the cone considered is the closed convex cone of positive semidefinite matrices. It arises naturally in the unified formulation of a pair of primal-dual semidefinite programming problems. In this article, we introduce the notion of complementary cones in the semidefinite setting using the faces of the cone of positive semidefinite matrices and show that unlike complementary cones induced by an LCP, semidefinite complementary cones need not be closed. However, under R₀-property of the linear transformation, closedness of all the semidefinite complementary cones induced by L is ensured. We also introduce the notion of a principal subtransformation with respect to a face of the cone of positive semidefinite matrices and show that for a self-adjoint linear transformation, strict copositivity is equivalent to strict semimonotonicity of each principal subtransformation. Besides the above, various other solution properties of SDLCP will be interpreted and studied geometrically.