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.     

How to Express a Parallel Sum of k Matrices

We present an alternative expression for the parallel sum of k Hermitian nonnegative definite matrices by using the Moore-Penrose inverse of a block matrix.     

Synthesis, spectroscopic studies and crystal structure of a polyoxoanion cluster incorporating para-bromophenylimido ligand, (Bu4N)2[Mo6O18(NC6H4Br-p)]

An organic-inorganic hybrid compound, (Bu₄N)₂[Mo₆O₁₈(NAr)] (Ar = p-BrC₆H₄) has been synthesized via the DCC dehydrating protocol of the reaction of [α-Mo₈O₂₆]⁴⁻ with 4-bromoaniline hydrochloride in anhydrous acetonitrile, which has been characterized by UV-Vis spectra, ¹H NMR and single crystal X-ray diffraction study. By comparing the UV-Vis spectra, which were used to monitor the reaction, the optimum preparative condition for this compound was also determined. This compound crystallizes in the monoclinic space group P2₁/n, which is featured in a terminal para-bromophenylimido group linked to an Mo atom of a hexamolybdate cluster by a Mo-N triply bond. In addition, there are π-π dimerization of the neighboring cluster anions though the parallel aromatic rings in their crystals.     

Synthesis,crystal structure and magnetic property of a binuclear iron(III) citrate complex

The compound (Hql)₂[Fe₂(cit)₂(H₂O)₂]·4H₂O (1) [ql = quinoline, cit^4– = C(O^–)(CO^– ₂)(CH₂CO^– ₂)₂], prepared by reacting ferric nitrate, sodium citrate and quinoline in a molar ratio of 1:1:1 in aqueous solution, was characterized by density measurements, elementary analysis, i.r., X-ray crystallography and magnetic measurements. The X-ray crystallography results reveal that the molecule (1) consists of a binuclear iron(III) citrate anionic complex [Fe₂(cit)₂(H₂O)₂]^2– and two protonated quinolines [Hql]⁺. The anionic complex has a centro-symmetric structure, in which two Fe³⁺ ions are bridged by two ₂-alkoxo groups of the two deprotonated citrate ligands. The other coordination sites of the two slightly distorted octahedra are completed by all the carboxylate groups of the two cit^4– ligands in a monodentate mode, and two coordinated water molecules. Magnetic measurements indicate that the two Fe³⁺ ions are antiferromagnetically coupled below 200 K. A least-squares fit of variable-temperature (1.5–291 K) molar susceptibility data to a dimer model gave the coupling constant J/k = –6.35(7) K and Landé factor g = 2.052(9), where the spin-only Heisenberg–Dirac–van Vleck Hamiltonian is expressed as H = –2J S ₁ S ₂.     

Approximation properties for solutions to non‐Lipschitz stochastic differential equations with Lévy noise

In this paper, we consider the non‐Lipschitz stochastic differential equations and stochastic functional differential equations with delays driven by Lévy noise, and the approximation theorems for the solutions to these two kinds of equations will be proposed respectively. Non‐Lipschitz condition is much weaker condition than the Lipschitz one. The simplified equations will be defined to make its solutions converge to that of the corresponding original equations both in the sense of mean square and probability, which constitute the approximation theorems. Copyright © 2014 John Wiley & Sons, Ltd.     

Aliphatic Organoimido Derivatives of Polyoxometalates Containing a Bioactive Ligand

A series of aliphatic organoimido derivatives of hexamolybdate based on amantadine, namely (nBu₄N)₂[Mo₆O₁₈(?NC₁₀H₁₅)] ( 1 ), (nBu₄N)₂ {cis‐[Mo₆O₁₇(?NC₁₀H₁₅)₂]} ( 2 ), (nBu₄N)₂{trans‐[Mo₆O₁₇(?NC₁₀H₁₅)₂]} ( 3 ), and (nBu₄N)₂[Mo₆O₁₆(?NC₁₀H₁₅)₃] ( 4 ), was synthesized in reasonable yield by dehydration with N,N′‐dicyclohexylcarbodiimide (DCC). They were characterized by IR and UV/Vis spectroscopy, elemental analysis, ESI mass spectrometry, and single‐crystal X‐ray structure analysis. The spectral and structural similarities and differences between monosubstituted, cis‐disubstituted, and trans‐disubstituted organoimido derivatives were elucidated and may provide guidance for related work on organoimido‐functionalized Lindqvist‐type polyoxometalates. In addition, trans‐disubstituted and polysubstituted derivatives containing aliphatic organoimido ligands have not yet been reported, and the crystal structure of the trans isomer may lead us to a deeper understanding of disubstituted derivatives. Furthermore, proliferation and morphology of MCF‐7 cells were studied with compound 1 . The present results show that the DCC‐dehydrating protocol could be an efficient approach to covalently graft bioactive ligands such as amantadine onto POMs and enhance their application in clinical cancer treatment.     

Solving Some Linear Equations Over Alternative Rings

One of the fundamental research problems in algebra theory is to solve equations over the given algebraic structures. The purpose of this note is to solve some basic linear equations over alternative rings. In particular, we derive necessary and sufficient conditions for the equations ax = b, xa = b and axa = b to be solvable over an alternative ring, respectively, and give general solutions of these equations. In addition, we solve the three linear equations ax ? ya = b, ax ? xa = b and x ? axa = b over alternative rings, respectively, where the coefficient element a is idempotent or involutory.