Equalities for orthogonal projectors and their operations

A complex square matrix A is called an orthogonal projector if A ² = A = A*, where A* denotes the conjugate transpose of A. In this paper, we give a comprehensive investigation to matrix expressions consisting of orthogonal projectors and their properties through ranks of matrices. We first collect some well-known rank formulas for orthogonal projectors and their operations, and then establish various new rank formulas for matrix expressions composed by orthogonal projectors. As applications, we derive necessary and sufficient conditions for various equalities for orthogonal projectors and their operations to hold.     

On the equivalence of estimations under a general linear model and its transformed models

Some necessary and sufficient conditions are given for two equalities of ordinary least-squares estimators and best linear unbiased estimators of an estimable vector of parametric functions under a general linear model and its transformed linear model to hold     

New vanadium phosphate ([(phen)VIVO]2(VV2O5)(HPO4))n with chiral layer architecture

A new layered vanadium phosphate ([(phen)V(IV)O](2)(V(V)(2)O(5))(HPO(4))n has been hydrothermally synthesized and structurally characterized by elemental analysis, IR, and single-crystal X-ray diffraction. This compound crystallized in the monoclinic space group P2/c with a = 9.979(2) A, b = 9.886(2) A, c = 15.298(3) A, beta = 102.86(3) degrees, V = 1471.3(5) A(3), and Z = 4. In the structure of this compound, two nitrogen atoms of 1,10-phen are directly coordinated to the vanadium skeleton. Interestingly, the crystal structure consists of two types of chiral layers, one left-handed and the other right-handed, which lead to a racemic solid-state compound. In each layer, there coexist three distinct rings, described as a 8-membered ring [V(2)(oct)P(2)O(4)], 12-membered ring [V(2)(omicron)(ct)V(2)(tet)P(2)O(6)], and 16-membered ring [V(2)(omicron)(ct)V(4)(tet)P(2)O(8)].     

A kinetically controlled trans bifunctionalized organoimido derivative of the Lindqvist-type hexamolybdate: synthesis, spectroscopic characterization, and crystal structure of (n-Bu4N)2{trans-[Mo6O17(NAr)2]} (Ar = 2,6-dimethylphenyl)

A kinetically controlled trans bifunctionalized organoimido derivative of hexamolybdate, (n-Bu(4)N)(2){trans-[Mo(6)O(17)(NAr)(2)]} (Ar = 2,6-dimethylphenyl) 1, in which the two 2,6-dimethylphenylimido groups are bonded to hexamolybdate at the trans positions, has been successfully synthesized in ca. 60% yield under mild reaction conditions. Its trans structure has been confirmed by a single-crystal X-ray diffraction study. In the crystals, cluster anions of 1 self-assemble into a 3D netlike structure via two different kinds of C-H...O hydrogen bondings, in which 1D supramolecular rectangular channels containing tetrabutylammonium cations form along the a axis. Compound 1 has also been characterized by (1)H NMR, IR, and UV-vis spectroscopic studies. UV-vis-near-IR reflectance spectroscopy measurements reveal the compound's nature of semiconductivity with an optical energy gap of 2.55 eV.     

HL—1M托卡马克脉冲超声分子束注入等离子体及杂质特性研究

文内叙述了在ＨＬ－１Ｍ托卡马克脉冲分子束注入实验中观测到的等离子体和杂质行为。分子束注入减小了杂质辐射,有效地提高了电子密度,改善了等离子体的约束性能。     

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.     

毛细管气相色谱测定多菌种发酵无醇饮料中的氨基酸

多菌种发酵饮料是含有氨基酸、维生素等营养丰富的高级饮料。本文叙述采用毛细管气相色谱和化学衍生化技术相结合的方法测定其中的多种氨基酸。该法具有分析速度快、灵敏度高、选择性好等特点,定性、定量分析结果令人满意。     

The crystal structure of hexaammonium diacetyl‐octa‐molybdate tetrahydrate

The crystal structure of hexaammonium diacetyl‐octa‐molybdate tetrahydrate, (NH₄)₄₂[Mo₁₃₂O₃₇₂ (CH₃COO)₃₀(H₂O)₇₂] is documented. The crystals are triclinic, space group P‐1, with a=8.1018(16) Å, b=10.334(2) Å, c=12.238(2) Å, α=68.20(3)°, β=74.98(3)°, γ=67.25(3)°, V=869.3(3) ų, Z=1. The structure was solved by direct methods and refined by least squares methods to a Final R1 = 0.0374 and wR2 = 0.1074 for 3805 observed reflections with I > 2σ(I). The structure contains ammonium cations and isolated acetyl octamolybdate(6‐) anions, [Mo₈O₂₈(CH₃CO)₂]^6‐. The crystallographic data of the structure was deposited with the Cambridge Data Center as No. CCDC 249565. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)