On Solving Centrosymmetric Linear Systems

The current paper is mainly devoted for solving centrosymmetric linear systems of equations. Formulae for the determinants of tridiagonal centrosymmetric matrices are obtained explicitly. Two efficient computational algorithms are established for solving general centrosymmetric linear systems. Based on these algorithms, a MAPLE procedure is written. Some illustrative examples are given.

Flexo-photocatalysis in centrosymmetric semiconductors

The separation of photogenerated electron–hole pairs is vitally important for photocatalysis,which can be effectively promoted by polarization field.However,it only manifests in piezoelectric/pyroelectric/ferroelectric materials that have a non-centrosymmetric structure.Here,we demonstrate that the polarization enhanced photocatalysis(with wide spectra from ultraviolet(UV)light to visible light)can be achieved in centrosymmetric semiconductors,such asδ-MnO_(2) and TiO_(2) nanosheets integrated nanoflowers,by using the strain-gradient-induced flexoelectric polarization that is always overlooked in polarization-enhanced catalysis.Under ultrasonic and illumination excitation,the organic pollutants(methylene blue(MB),etc.)can be effectively degraded within 30 min with excellent stability and repeatability.Compared with photocatalysis,the flexo-photocatalytic performance of above centrosymmetric semiconductors is substantially increased by 85%.Moreover,the factors related to flexo-photocatalysis such as material morphology,mechanical stimuli source,and adsorption are explored to deeply understand the mechanism of flexo-photocatalysis.This work opens up a way for high-performance photocatalysis in centrosymmetric semiconductors.

Dual excited state deactivation pathways in TPZ2:A centrosymmetric dye with both high fluorescence and triplet state quantum yield

In this letter, excited state dynamics of TPZ2, a centrosymmetric PRODAN dye, has been studied by using several time-resolved spectroscopy techniques. Fluorescence quantum yield of TPZ2 is found to be 0.50 in both acetonitrile and ethanol solution. The radiative decay rate of the excited state of TPZ2 is determined to be 2.0×10^8 s^-1. Meanwhile, highly efficient triplet state and singlet oxygen generation have been observed in TPZ2 and the intersystem crossing（ISC） rate is determined to be 2.0×10^8s^-1. The almost identical ISC and non-radiative decay rates indicate that ISC is the only non-radiative decay pathway in TPZ2. Thus, dual excited state（S1） deactivation mechanism（50/50, fluorescence/ISC） of TPZ2 is proposed.Because of this unique property, TPZ2 has the potential to be used as biocompatible imaging and photodynamic therapy agent in the same time.

SOME PROPERTIES OF COMMUTING AND ANTI-COMMUTING m-INVOLUTIONS

We define an m-involution to be a matrix K ∈ Cn×n for which Km -= I. In this article, we investigate the class Sm （A） of m-involutions that commute with a diagonalizable matrix A E Cn×n. A number of basic properties of Sm （A） and its related subclass Sm （A, X） are given, where X is an eigenvector matrix of A. Among them, Sm （A） is shown to have a torsion group structure under matrix multiplication if A has distinct eigenvalues and has non-denumerable cardinality otherwise. The constructive definition of Sm （A, X） allows one to generate all m-involutions commuting with a matrix with distinct eigenvalues. Some related results are also given for the class S,, （A） of m-involutions that anti-commute with a matrix A ∈ Cnn×n.

The Crystal structure of Hexakisammonium Diacety loctamoly bdate Tetrahydrate

The title compound, 〔NH 4〕 6〔(CH 3CO) 2(Mo 8O 28 )〕·4H 2O, was produced from (NH 4) 6〔Mo 7O 24 〕·4H 2O aqueous solution in the presence of N 2H 4·H 2SO 4 and its structure has been determined by X ray single crystal analysis. The crystallographic data: C 4H 38 Mo 8N 6O 34 , M r =1481.92, triclinic, P1, a=8.127(2), b = 10\^373(2), c = 12.383(3) , α=99.51(3), β=105.47(3), γ=112.92(3)°, V=883.0(3) 3, Z=1, D c =2.787g/cm 3, F(000)= 712, μ =2.864 mm -1 , Final R =0.0368 and wR =0.0924 for 2926 reflections with I】2σ(I ). The X ray crystal structure analysis reveals that there is one independent molecule in the unit cell of the title compound which contains one centrosymmetric diacetyloctamolybdate anion, six ammonium cations and four water molecules.

Synthesis and Crystal Structure of Tetraimidazolium Dinicotinatooctamolybdate

The title compound, [C3H5N2]4[(C6H5NO2)2(Mo8O26)] 1, was synthesized by the reaction of H2MoO4, nicotinic acid and imidazole in aqueous solution. Single-crystal X-ray analysis has revealed that compound 1 (C24H30Mo8N10O30) crystallizes in the monoclinic system, space group P21/n, Mr = 1706.10, a = 12.529(1), b = 12.796(1), c = 14.038(1) , b = 100.2(1), V = 2215.3(2) 3, Z = 2, Dc = 2.558 g/cm3, F(000) = 1640, = 2.299 mm-1, the final R = 0.0527 and wR = 0.0964 for 3064 observed reflections with I > 2s(I). X-ray crystal structure analysis suggests that compound 1 consists of centrosymmetric dinicotinatooctamolybdate anion and four protonated imidazole cations, which are linked into a three-dimensional framework through hydrogen bonding.

Synthesis and Crystal Structure of a New Imidazole Coordinated Octamolybdate Compound

The title compound (Himi)4[Mo8O26(imi)2]4H2O (imi = imidazole) 1 was synthe- sized by the reaction of H2MoO4 and imidazole in aqueous solution. Single-crystal X-ray analysis reveals that compound 1 is crystallized in the triclinic system, space group P with a = 10.6219(8), b = 10.7260(8), c = 11.3220(9) , a = 92.842(2), b =117.364(1), g = 101.655(1), C18H36Mo8N12- O30, Mr = 1668.11, V = 1106.8(2) 3, Z = 1, Dc = 2.503 g/cm3, F(000) = 804, m = 2.298 mm-1, the final R = 0.0714 and wR = 0.1651 for 3121 observed reflections with I > 2s(I). The X-ray crystal structure analysis suggests that compound 1 is built up by two imidazole ligands coordinated by the centrosymmetric octamolybdate anions, protonated imidazole cations and crystallization water molecules.

Block Decompositions and Applications of Generalized Reflexive Matrices

Generalize reflexive matrices are a special class of matrices ?that have the relation where? and ?are some generalized reflection matrices. The nontrivial cases ( or ) of this class of matrices occur very often in many scientific and engineering applications. They are also a generalization of centrosymmetric matrices and reflexive matrices. The main purpose of this paper is to present block decomposition schemes for generalized reflexive matrices of various types and to obtain their decomposed explicit block-diagonal structures. The decompositions make use of unitary equivalence transformations and, therefore, preserve the singular values of the matrices. They lead to more efficient sequential computations and at the same time induce large-grain parallelism as a by-product, making themselves computationally attractive for large-scale applications. A numerical example is employed to show the usefulness of the developed explicit decompositions for decoupling linear least-square problems whose coefficient matrices are of this class into smaller and independent subproblems.

A System of Four Matrix Equations over von Neumann Regular Rings and Its Applications

We consider the system of four linear matrix equations A_1 X = C_1,XB_2 =C_2,A_3,XB_3, = C3 and A_4XB_4 = C_4 over R, an arbitrary von Neumann regular ring with identity. Anecessary and sufficient condition for the existence and the expression of the general solution tothe system are derived. As applications, necessary and sufficient conditions are given for thesystem of matrix equations A_1X = C_1 and A_3X = C_3 to have a bisymmetric solution, the system ofmatrix equations A_1X = C_1 and A_3XB_3 = C_3 to have a perselfconjugate solution over R with aninvolution and char R≠2, respectively. The representations of such solutions are also presented.Moreover, some auxiliary results on other systems over R are obtained. The previous known results onsome systems of matrix equations are special cases of the new results.

Magnetic and dielectric properties of Fe_(3)BO_(6) nanoplates prepared through self-combustion method

In the present investigation,a facile synthesis method is explored involving a self-combustion of a solid precursor mixture of iron oxide Fe_(2)O_(3) and boric acid(H_(3)BO_(3))using camphor(C_(10)H_(16)O)as fuel in ambient air in order to form a single phase Fe_(3)BO_(6) crystallites.X-ray diffraction(XRD),Field emission electron microscopy(FESEM),magnetic,and dielectric properties of as prepared sample are studied.From XRD pattern,a single phase compound is observed with an orthorhombic crystal structure(Pnma space group),with average crystallite size of 42 nm.A reasonably uniform size distribution of the plates and self-assemblies is retained in the sample.A magnetic transition is observed in dielectric permittivity(at-445 K)and power loss(at-435 K)when plotted against temperature.A weak peak occurs near 330 K due to the charge reordering in the sample.For temperatures above the transition temperature,a sharp increase of the dielectric loss is observed which occurs due to the presence of thermally activated charge carriers.A canted antiferromagnetic Fe_(3)þordering in a Fe_(3)BO_(6) lattice with a localized charge surface layer is an apparent source of exhibiting a ferroelectric feature in this unique example of a centrosymmetric compound.An induced spin current over the Fe sites thus could give rise to a polarization hysteresis loop.Due to the presence of both ferromagnetic as well as polarization ordering,Fe_(3)BO_(6) behaves like a single phase multiferroic ceramics.

Ni_(2)M^(Ⅱ)Te^(Ⅳ)_(2)O_(2)(PO_(4))_(2)(OH)_(4)(M^(Ⅱ)=Ni,Zn):Synthesis,Crystal Structure,and Magnetic Properties

Two isostructural tellurite phosphates Ni_(2)M^(Ⅱ)Te^(Ⅳ)_(2)O_(2)(PO_(4))_(2)(OH)_(4)(M^(Ⅱ)=Ni,Zn)have been synthesized via hydrothermal method.Both of them crystalize in monoclinic space group C2/m(No.12).For Ni_(2)ZnTe^(Ⅳ)_(2)O_(2)(PO_(4))_(2)(OH)_(4),a=19.3247(10),b=5.9697(3),c=4.7737(2)Å,β=103.637(5)°,V=535.18(5)Å^(3),Z=2,M_(r)=727.94,S=1.103,D_(c)=4.517 g·cm^(-3),μ(Mo Kα)=11.434 mm^(-1) and F(000)=672,the final R=0.0369and wR=0.1086 for 639 observed reflections with I>2σ(I).The crystal structure of Ni_(2)M^(Ⅱ)Te^(Ⅳ)_(2)O_(2)(PO_(4))_(2)(OH)_(4)(M^(Ⅱ)=Ni,Zn)features a 3D framework composed of[Ni_(2)O_(2)(PO_(4))_(2)]^(6-)layers interconnected by[MTe_(2)O_(2)(OH)_(4)]^(2+)single chains.Different magnetic susceptibility results at low temperature of the two title compounds confirm that Zn(1)completely occupies the Ni(2)position but not partially substitutes both Ni(1)and Ni(2)position atoms.Additionally,the acentric TeO_(3)(OH)_(2) tetragonal pyramids are aligned in an antiparallel manner,resulting in the crystallization of centrosymmetric(C2/m)crystal structure instead of the non-centrosymmetric crystal structure.The investigation of the origin of centrosymmetric crystal structure with strong dipole moment units provides deeper understanding for future rational design for non-centrosymmetric crystal structure.