行分块矩阵值域上正交投影的展开式

We give in this note some expansion formulas for the orthogonal projectors onto the range of the row block matrix [ A, B ], and use the expansion formulas to examine relations among the orthogonal projectors onto the ranges of A, B and [A, B]. In particular, we present some identifying conditions for a pair of orthogonal projectors of the same size to commute.     

Path integral and effective Hamiltonian in loop quantum cosmology

We study the path integral formulation of Friedmann universe filled with a massless scalar field in loop quantum cosmology. All the isotropic models of $k=0,+1,-1$ are considered. To construct the path integrals in the timeless framework, a multiple group-averaging approach is proposed. Meanwhile, since the transition amplitude in the deparameterized framework can be expressed in terms of group-averaging, the path integrals can be formulated for both deparameterized and timeless frameworks. Their relation is clarified. It turns out that the effective Hamiltonian derived from the path integral in deparameterized framework is equivalent to the effective Hamiltonian constraint derived from the path integral in timeless framework, since they lead to same equations of motion. Moreover, the effective Hamiltonian constraints of above models derived in canonical theory are confirmed by the path integral formulation.     

Substituent effects on the conformational stability of allyl fluorides

By the B3P86/6‐311G(3d,2p) method, remote substituent effects on trans‐YCH?CHCH₂F were investigated by examining their conformational stabilities, molecular geometries, and stereoelectronic interactions in this paper. The cis conformer is favored for Y?H, Cl, Me, Vinyl, CF₃, CN, CHO, and NO₂, whereas the gauche is favored for Y?OMe, OH. A correlation of ΔH with the substituent constants σ⁺(Y) shows that the increasing electron‐withdrawing ability of the substituent Y increases the relative stability of the cis conformer. It was found that the substituent effect on the molecule stabilization energies (relative to CH₂?CHCH₂F) is more significant in the gauche conformers than in the cis conformers. In agreement, molecular structures of the gauche conformers were also observed to vary more significantly with the substitution than those of the cis conformers. By the second‐order perturbation energy (E(2)) in NBO analysis, it was found that the total C₂–C₃ vicinal hyperconjugation is determinant in the enthalpy difference and consequently controls the conformational stability. Further analysis shows that the substituent effect on the C₂–C₃ vicinal hyperconjugations is much higher in the gauche conformers than in the cis conformers. The highly sensitive π_C?C→σ*_{C? F} interaction to the substitution in the gauche conformers, is the leading factor in variation of molecular stability and geometry. Copyright © 2010 John Wiley & Sons, Ltd.     

HL－1M装置杂质VUV辐射及Te观测

本文介绍了ＨＬ－１Ｍ装置等离子体杂质真空紫外辐射观测的初步结果。用类Ｌｉ离子谱线强度比法估计出Ｔｅ≈４００ｅＶ。镀膜后的ＣＥＭ探测器的灵敏度提高。杂质对装置放电有重要影响     

On a family of matrix equalities that involve multiple products of generalized inverses

Aequationes mathematicae - One of the fundamental matrix equalities that involve multiple products of matrices and their generalized inverses is given by $$A_1B_1^{-}A_2B_2^{-} ldots...     

Separations by Random Oracles and “Almost” Classes for Generalized Reducibilities

Let ≤_r and ≤_sbe two binary relations on 2^ℕ which are meant as reducibilities. Let both relations be closed under finite variation (of their set arguments) and consider the uniform distribution on 2^ℕ, which is obtained by choosing elements of 2^ℕ by independent tosses of a fair coin.Then we might ask for the probability that the lower ≤_r‐cone of a randomly chosen set X, that is, the class of all sets A with A ≤_r X, differs from the lower ≤_s‐cone of X. By c osure under finite variation, the Kolmogorov 0‐1 aw yields immediately that this probability is either 0 or 1; in case it is 1, the relations are said to be separable by random oracles.Again by closure under finite variation, for every given set A, the probability that a randomly chosen set X is in the upper ≤_r‐cone of A is either 0 or 1; let Almost_r be the class of sets for which the upper ≤_r‐cone has measure 1. In the following, results about separations by random oracles and about Almost classes are obtained in the context of generalized reducibilities, that is, for binary relations on 2^ℕ which can be defined by a countable set of total continuous functionals on 2^ℕ in the same way as the usual resource‐bounded reducibilities are defined by an enumeration of appropriate oracle Turing machines. The concept of generalized reducibility comprises a natura resource‐bounded reducibilities, but is more general; in particular, it does not involve any kind of specific machine model or even effectivity. The results on generalized reducibilities yield corollaries about specific resource‐bounded reducibilities, including several results which have been shown previously in the setting of time or space bounded Turing machine computations.     

On constrained generalized inverses of matrices and their properties

A matrix G is called a generalized inverse (g-invserse) of matrix A if AGA = A and is denoted by G = A ⁻. Constrained g-inverses of A are defined through some matrix expressions like E(AE)⁻, (FA)⁻ F and E(FAE)⁻ F. In this paper, we derive a variety of properties of these constrained g-inverses by making use of the matrix rank method. As applications, we give some results on g-inverses of block matrices, and weighted least-squares estimators for the general linear model.     

The effects of indium segregation on the valence band structure and optical gain of GaInAs/GaAs quantum wells

The effects of indium segregation on the valence band structures and the optical gain in GaInAs/GaAs quantum wells are theoretically investigated using 4×4 Luttinger–Kohn Hamiltonian matrix. The method for the band structure calculation is based on the finite difference method, then the optical gain is calculated using the density matrix approach. For segregation coefficient R less than 0.7, indium segregation has little influence on optical gain, but for segregation coefficient R more than 0.7, it has a significant influence on optical gain, the gain spectra can be blue-shifted with the increase of segregation coefficient R, and the peak gains are decreased as segregation coefficient R increases, which is mainly due to the reduction of the carrier population inversion.     

Extremal ranks of submatrices in an Hermitian solution to the matrix equation AXA *=B with applications

Suppose that AXA ^*=B is a consistent matrix equation and partition its Hermitian solution X ^*=X into a 2-by-2 block form. In this paper, we give some formulas for the maximal and minimal ranks of the submatrices in an Hermitian solution X to AXA ^*=B. From these formulas we derive necessary and sufficient conditions for the submatrices to be zero or to be unique, respectively. As applications, we give some properties of Hermitian generalized inverses for an Hermitian matrix.