Asymptotic spectral analysis in semiconductor nanowire heterostructures

Mathematical settings in which heterogeneous structures affect electron transport through a tube-shaped quantum waveguide are studied, highlighting the interaction between material composition and geometric parameters like curvature and torsion. First, the macroscopic behaviour of a nanowire made of composite fibres with microscopic periodic texture is analysed, which amounts to determining the asymptotic behaviour of the spectrum of an elliptic Dirichlet eigenvalue problem with finely oscillating coefficients in a tube with shrinking cross-section. A suitable formal expansion suggests that the effective one-dimensional limit problem is of Sturm–Liouville type and yields the explicit formula for the underlying potential. In the torsion-free case, these findings are made rigorous by performing homogenization and 3d–1d dimension reduction for the two-scale problem in a variational framework by means of Γ-convergence. Second, waveguides with non-oscillating inhomogeneities in the cross-section are investigated. This leads to explicit criteria for propagation and localization of eigenmodes.     

Stochastic homogenization of reinforced polymer with very small carbon inclusions

In this study, we wish to determine a homogenized model of a material reinforced by spherical inclusion that is randomly distributed in space. The method used for the transition to the limit is Γ-convergence [1] in the stochastic case. In addition to the stochastic framework, the very small size compared to the characteristic size of the materials makes the homogenization procedure unconventional. In this study, we want to determine a homogenized model of a material reinforced by a spherical inclusion distributed randomly in space. The peculiarity here is that these particles are of very small size, this generating an energy due to the strong contrast of microstructure. The method used for the transition to the limit is Γ-convergence [1] in the stochastic case. The random distribution is taken into account during the transition of scales, so as to preserve the statistical information, and that in spite of the passage to the limit. In addition to the stochastic framework, the very small size compared to the characteristic size of the materials makes the homogenization procedure unconventional.     

一个具最佳常数的复合核积分不等式

引入Γ-函数等特殊函数,利用权函数方法和实分析技巧,建立一个具最佳常数的复合核积分不等式.考虑了它的等价式,证明了它们的常数因子是最佳的,并通过取特殊的参数值,得到一些有意义的结果.     

On Extension of 1-Lipschitz Mappings between Two Unit Spheres of lp(Γ) Type Spaces (1

FANG Xi Nian 《数学研究与评论》2009,29(4)

混合新几何体Γ_(-p，i)K的不等式

近来,Lutwak,Yang和Zhang提出了新几何体Γ_(-p)K的概念,获得一些结论.本文引入混合新几何体Γ_(-p,i)K的概念,即新几何体是它的特殊类,得到了混合新几何体的一些性质,并建立了相关的不等式.     

Recursive Determination of the Generalized Moore–Penrose <Emphasis Type="Italic">M</Emphasis>-Inverse of a Matrix

In this paper, we obtain recursive relations for the determination of the generalized Moore–Penrose M-inverse of a matrix. We develop separate relations for situations when a rectangular matrix is augmented by a row vector and when such a matrix is augmented by a column vector.     

Valuation Extensions of Filtered and Graded Algebras

In this note we relate the valuations of the algebras appearing in the noncommutative geometry of quantized algebras to properties of sublattices in some vector spaces. We consider the case of algebras with PBW-bases and prove that under some mild assumptions, the valuations of the ground field extend to a noncommutative valuation. Later we introduce the notion of F-reductor and graded reductor and reduce the problem of finding an extending noncommutative valuation to finding a reductor in an associated graded ring having a domain for its reduction.