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.     

Expansion formulas for the inertias of Hermitian matrix polynomials and matrix pencils of orthogonal projectors

This paper gives a group of expansion formulas for the inertias of Hermitian matrix polynomials A−A², I−A² and A−A³ through some congruence transformations for block matrices, where A is a Hermitian matrix. Then, the paper derives various expansion formulas for the ranks and inertias of some matrix pencils generated from two or three orthogonal projectors and Hermitian unitary matrices. As applications, the paper establishes necessary and sufficient conditions for many matrix equalities to hold, as well as many inequalities in the Löwner partial ordering to hold.     

χ-Octamolybdate [Mo(V)4Mo(VI)4O24](4-): an unusual small polyoxometalate in partially reduced form from nonaqueous solvent reduction

The first mixed-valent octamolybdate anion [Mo(V)(4)Mo(VI)(4)O(24)](4-), termed χ-octamolybdate due to its shape, was obtained by partial reduction of α-(Bu(4)N)(4)-[Mo(8)O(26)] in dry acetonitrile. Single-crystal X-ray structure analysis revealed that the anion includes an unusual Mo(V)(4)O(8) cubane-like cluster core, whose four side faces are capped by four MoO(4) units to form a crosslike cluster. X-ray photoelectron spectroscopy (XPS) and bond valence sum (BVS) calculations were carried out to validate the presence of mixed-valent Mo centers.     

A simultaneous decomposition of a matrix triplet with applications

Researches on ranks of matrix expressions have posed a number of challenging questions, one of which is concerned with simultaneous decompositions of several given matrices. In this paper, we construct a simultaneous decomposition to a matrix triplet (A, B, C), where A=±A*. Through the simultaneous matrix decomposition, we derive a canonical form for the matrix expressions A?BXB*?CYC* and then solve two conjectures on the maximal and minimal possible ranks of A?BXB*?CYC* with respect to X=±X* and Y=±Y*. As an application, we derive a sufficient and necessary condition for the matrix equation BXB* + CYC*=A to have a pair of Hermitian solutions, and then give the general Hermitian solutions to the matrix equation. Copyright © 2010 John Wiley & Sons, Ltd.     

Trifluoromethylation of Tetraphenylborate Counterions in Cationic Iridium(III) Complexes: Enhanced Electrochemical Stabilities,Charge‐Transport Abilities,and Device Performance

Trifluoromethylation of tetraphenlyborate counterions was successfully used to improve the electrochemical stabilities and device performance of cationic iridium(III) complexes. Melioration of the thermal, photoluminescent, electrochemical, and electrophosphorescent characteristics was achieved. Interionic hydrogen bonds were first found between the aromatic hydrogen atoms in the ancillary ligands of cations and the fluorine atoms in the trifluoromethyl groups of the anions. The strong impact of the counterions on the charge transport in the devices was investigated. A compound with two trifluoromethyl groups in the tetraphenlyborate ion shows the highest photoluminescent efficiency, the best electrochemical stability, and the greatest performance in green‐blue‐emitting devices, with a high current efficiency of 12.4 cd A^?1 and an emission peak at λ=480 nm. The efficiencies achieved are the highest reported for OLEDs with ionic complexes emitting in the blue‐green region.     

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.