Weakly tight functions, their Jordan type decomposition and total variation in effect algebras

In the present paper, we have studied envelopes of a function m defined on a subfamily E (containing 0 and 1) of an effect algebra L. The notion of a weakly tight function is introduced and its relation to tight functions is investigated; examples and counterexamples are constructed for illustration. A Jordan type decomposition theorem for a locally bounded real-valued weakly tight function m defined on E is established. The notions of total variation |m| on the subfamily E and m-atoms on a sub-effect algebra E (along with a few examples of m-atoms for null-additive as well as non null-additive functions) are introduced and studied. Finally, it is proved for a real-valued additive function m on a sub-effect algebra E that, m is non-atomic if and only if its total variation |m| is non-atomic.     

基于MRA的梅花状伸缩矩阵的二元箱小波紧框架

本文针对梅花状的伸缩矩阵，给出从任何紧支撑的箱样条函数构造紧支撑箱小波紧框架的具体算法，最后给出若干构造算例。     

Pseudo 1-homogeneous distance-regular graphs

Let Γ be a distance-regular graph of diameter d≥2 and a ₁≠0. Let θ be a real number. A pseudo cosine sequence for θ is a sequence of real numbers σ ₀,…,σ _d such that σ ₀=1 and c _i σ _i−1+a _i σ _i +b _i σ _i+1=θ σ _i for all i∈{0,…,d−1}. Furthermore, a pseudo primitive idempotent for θ is E _θ =s ∑ _i=0 ^d σ _i A _i , where s is any nonzero scalar. Let be the characteristic vector of a vertex v∈VΓ. For an edge xy of Γ and the characteristic vector w of the set of common neighbours of x and y, we say that the edge xy is tight with respect to θ whenever θ≠k and a nontrivial linear combination of vectors , and Ew is contained in . When an edge of Γ is tight with respect to two distinct real numbers, a parameterization with d+1 parameters of the members of the intersection array of Γ is given (using the pseudo cosines σ ₁,…,σ _d , and an auxiliary parameter ε). Let S be the set of all the vertices of Γ that are not at distance d from both vertices x and y that are adjacent. The graph Γ is pseudo 1-homogeneous with respect to xy whenever the distance partition of S corresponding to the distances from x and y is equitable in the subgraph induced on S. We show Γ is pseudo 1-homogeneous with respect to the edge xy if and only if the edge xy is tight with respect to two distinct real numbers. Finally, let us fix a vertex x of Γ. Then the graph Γ is pseudo 1-homogeneous with respect to any edge xy, and the local graph of x is connected if and only if there is the above parameterization with d+1 parameters σ ₁,…,σ _d ,ε and the local graph of x is strongly regular with nontrivial eigenvalues a ₁ σ/(1+σ) and (σ ₂−1)/(σ−σ ₂).     

新团簇形成的动力学研究

利用紧束缚模型,通过对不同初始条件下的碱金属团簇的碰撞研究,得到形成大分子团簇的可能性.结果表明,对同一种团簇,具有空间对称性的团簇间的碰撞更易形成大团簇,而平面结构的团簇碰撞则反之,并且空间结构的团簇碰撞形成的新团簇复合体更加稳定.另一方面,团簇碰撞过程是可逆的.     

Tight Monomials for Type G 2 and A 3

The global crystal basis or canonical basis plays an important role in the theory of the quantum groups and their representations. The tight monomials are the simplest elements in the canonical basis. Based on Reineke, Deng, and Du's work, the tight monomials in quantized enveloping algebra of type G₂ and A₃ are completely determined in this article.     

Dynamics of a family of piecewise-linear area-preserving plane maps III. Cantor set spectra

This paper studies the behavior under iteration of the maps T _ab (x,y) = (F _ab (x) ? y, x) of the plane ?², in which F _ab (x) = ax if x ≥ 0 and bx if x < 0. These maps are area-preserving homeomorphisms of ?² that map rays from the origin to rays from the origin. Orbits of the map correspond to solutions of the nonlinear difference equation x _n+2 = 1/2(a ? b)|x _n+1|+1/2(a+b)x _n+1 ? x _n . This difference equation can be rewritten in an eigenvalue form for a nonlinear difference operator of Schrödinger type ? x _n+2+2x _n+1 ? x _n +V _μ(x _n+1)x _n+1 = Ex _n+1, in which μ = (1/2)(a ? b) is fixed, and V _μ(x) = μ(sgn(x)) is an antisymmetric step function potential, and the energy E = 2 ? 1/2(a+b). We study the set Ω _SB of parameter values where the map T _ab has at least one bounded orbit, which correspond to l _∞-eigenfunctions of this difference operator. The paper shows that for transcendental μ the set Spec_∞[μ] of energy values E having a bounded solution is a Cantor set. Numerical simulations suggest the possibility that these Cantor sets have positive (one-dimensional) measure for all real values of μ.     

Spectrum of Andreev bound states in Josephson junctions with a ferromagnetic insulator

Ferromagnetic-insulator (FI) based Josephson junctions are promising candidates for a coherent superconducting quantum bit as well as a classical superconducting logic circuit. Recently the appearance of an intriguing atomic-scale 0–π$0–\pi$ transition has been theoretically predicted. In order to uncover the mechanism of this phenomena, we numerically calculate the spectrum of Andreev bound states in a FI barrier by diagonalizing the Bogoliubov–de Gennes equation. We show that Andreev spectrum drastically depends on the parity of the FI-layer number L   and accordingly the π(0)$\pi (0)$ state is always more stable than the 0 (π$\pi$) state if L is odd (even).