Organic‐Inorganic Hybrids: Preparation and Structural Characterization of (Bu4N)2[Mo6O17(NAr)2] and (Bu4N)2[Mo6O18(NAr)] (Ar = o‐CH3C6H4)

In the presence of N, N′‐dicyclohexylcarbodiimide (DCC), (Bu₄N)₂[Mo₆O₁₈(NAr)] ( 1 ) and (Bu₄N)₂[Mo₆O₁₇(NAr)₂] ( 2 ), Ar = o‐CH₃C₆H₄, have been synthesized via the reaction of [α‐Mo₈O₂₆]^4— with o‐toluidine. If the hydrochloride salt of o‐toluidine was added into the reactive mixture, only the monofunctionalized imido derivative of [Mo₆O₁₉]^2— was obtained; the bifunctionalized derivative of [Mo₆O₁₉]^2— was exclusively synthesized in the presence of non‐protonated o‐toluidine. The molecular and crystal structures of the hybrid compounds 1 and 2 were determined by X‐ray single crystal diffraction, and their UV, IR and NMR spectra were compared. Additionally, a possible reaction mechanism was proposed.     

The chemistry of organoimido derivatives of polyoxometalates

The chemistry of organoimido derivatives of polyoxometalates (POMs) has received increasing interest. In recent decades, a great development of the synthetic chemistry of organoimido derivatives of POMs using different imido-releasing reagents has taken place, particularly the novel DCC-dehydrating protocol to prepare organoimido derivatives of POMs was developed recently by us and co-workers. Organoimido substituted POMs as valuable building blocks can construct novel nanostructured organic-inorganic hybrid molecular materials in well-developed common organic synthesis methods, such as Pd-catalyzed carbon-carbon coupling and esterification. Therefore, the reactive chemistry of organoimido derivatives of POMs stands for the fascinating future of the chemistry of organoimido derivatives of POMs.     

Preparation, characterization, and pharmacodynamics of exenatide-loaded poly(DL-lactic-co-glycolic acid) microspheres

Exenatide (synthetic exendin-4), a 39-amino acid peptide, was encapsulated in poly(DL-lactic-co-glycolic acid) (PLGA) microspheres as a sustained release delivery system for the therapy of type 2 diabetes mellitus. The microspheres were prepared by a double-emulsion solvent evaporation method and the particle size, surface morphology, drug encapsulation efficiency, in vitro release profiles and in vivo hypoglycemic activity were evaluated. The results indicated that the morphology of the exenatide PLGA microspheres presented as a spherical shape with smooth surface, and the particle sizes distributed from 5.8 to 13.6 μm. The drug encapsulation efficiency tested by micro-bicinchoninic acid (BCA) assay was influenced by certain parameters such as inner and outer aqueous phase volume, PLGA concentration in oil phase, polyvinyl alcohol (PVA) concentrations in outer aqueous phase. Moreover, in vitro release behaviors were also affected by some parameters such as polymer type, PLGA molecular, internal aqueous phase volume, PLGA concentration. The pharmacodynamics in streptozotocin (STZ)-induced diabetic mice suggested that, exenatide microspheres have a significant hypoglycemic activity within one month, and its controlling of plasma glucose was similar to that of exenatide solution injected twice daily with identical exenatide amount. In conclusion, this microsphere could be a well sustained delivery system for exenatide to treat type 2 diabetes mellitus.     

Some rank equalities and inequalities for Kronecker products of matrices

A set of rank equalities and inequalities are established for block matrices consisting of Kronecker products. Various consequences are also given.     

Ranks of Solutions of the Matrix Equation AXB = C

The purpose of this article is to solve two problems related to solutions of a consistent complex matrix equation AXB = C : (I) the maximal and minimal ranks of solution to AXB = C , and (II) the maximal and minimal ranks of two real matrices X ₀ and X ₁ in solution X = X ₀ + iX ₁ to AXB = C . As applications, the maximal and minimal ranks of two real matrices C and D in generalized inverse (A + iB)^- = C + iD of a complex matrix A + iB are also examined.     

Rank equalities related to outer inverses of matrices and applications

A matrix X is called an outer inverse for a matrix A if XAX=X. In this paper, we present some basic rank equalities for difference and sum of outer inverses of a matrix, and apply them to characterize various equalities related to outer inverses, Moore-Penrose inverses, group inverses, Drazin inverses and weighted Moore-Penrose inverses of matrices.     

A disjoint idempotent decomposition for linear combinations produced from two commutative tripotent matrices and its applications

A square matrix A of order n is said to be tripotent if A ³?=?A. In this note, we give a nine-term disjoint idempotent decomposition for the linear combination of two commutative tripotent matrices and their products. Using the decomposition, we derive some closed-form formulae for the eigenvalues, determinant, rank, trace, power, inverse and group inverse of the linear combinations. In particular, we show that the linear combinations of two commutative tripotent elements and their products can produce 3⁹?=?19,683 tripotent elements.     

Distributions of eigenvalues and inertias of some block Hermitian matrices consisting of orthogonal projectors

A complex square matrix A is called an orthogonal projector if A ²?=?A?=?A*, where A* is the conjugate transpose of A. In this article, we first give some formulas for calculating the distributions of real eigenvalues of a linear combination of two orthogonal projectors. Then, we establish various expansion formulas for calculating the inertias, ranks and signatures of some 2?×?2 and 3?×?3, as well as k?×?k block Hermitian matrices consisting of two orthogonal projectors. Many applications of the formulas are presented in characterizing interval distributions of numbers of eigenvalues, and nonsingularity of these block Hermitian matrices. In addition, necessary and sufficient conditions are given for various equalities and inequalities of these block Hermitian matrices to hold.     

Band gap energy and bowing parameter of In-rich InAlN films grown by magnetron sputtering

The crystal structure, band gap energy and bowing parameter of In-rich In_xAl_1−xN (0.7 < x < 1.0) films grown by magnetron sputtering were investigated. Band gap energies of In_xAl_1−xN films were obtained from absorption spectra. Band gap tailing due to compositional fluctuation in the films was observed. The band gap of the as-grown InN measured by optical absorption method is 1.34 eV, which is larger than the reported 0.7 eV for pure InN prepared by molecular beam epitaxy (MBE) method. This could be explained by the Burstein-Moss effect under carrier concentration of 10²⁰ cm⁻³ of our sputtered films. The bowing parameter of 3.68 eV is obtained for our In_xAl_1−xN film which is consistent with the previous experimental reports and theoretical calculations.     

On Additive Decomposition of the Hermitian Solution of the Matrix Equation AXA* = B

The decomposition of a Hermitian solution of the linear matrix equation AXA* = B into the sum of Hermitian solutions of other two linear matrix equations A₁X₁A^*₁ = B₁{A_{1}X_{1}A^{*}_{1} = B_{1}} and A₂X₂A^*₂ = B₂{A_{2}X_{2}A^*_{2} = B_{2}} are approached. As applications, the additive decomposition of Hermitian generalized inverse C ⁻ = A ⁻ + B ⁻ for three Hermitian matrices A, B and C is also considered.