循环群诸类问题探

主要探讨:(i)循环群的定义和性质;(ii)循环群及其子群特征;(iii)同构于Vn、(Z+0)及其推广.     

Weak Cayley Table Groups II: Alternating Groups and Finite Coxeter Groups

A weak Cayley table isomorphism is a bijection φ: G → H of groups such that φ(xy) ～ φ(x)φ(y) for all x, y ∈ G. Here ～denotes conjugacy. When G = H the set of all weak Cayley table isomorphisms φ: G → G forms a group 𝒲(G) that contains the automorphism group Aut(G) and the inverse map I: G → G, x → x ^?1. Let 𝒲₀(G) = ?Aut(G), I? ≤ 𝒲(G) and say that G has trivial weak Cayley table group if 𝒲(G) = 𝒲₀(G). We show that all finite irreducible Coxeter groups (except possibly E ₈) have trivial weak Cayley table group, as well as most alternating groups. We also consider some sporadic simple groups.     

Jordan isomorphisms of generalized structural matrix rings

We describe the sub-bimodules of matrix bimodules over two structural matrix rings. Structural matrix bimodules arise as particular such sub-bimodules, and we discuss when such a bimodule is faithful or indecomposable. As an application, we obtain a large class of rings whose Jordan isomorphisms are either ring isomorphisms or ring anti-isomorphisms. Complete upper block triangular matrix rings over 2-torsion-free indecomposable rings are elements of this class.     

Non-Commuting Graphs of Nilpotent Groups

Let G be a non-abelian group and Z(G) be the center of G. The non-commuting graph Γ _G associated to G is the graph whose vertex set is G?Z(G) and two distinct elements x, y are adjacent if and only if xy ≠ yx. We prove that if G and H are non-abelian nilpotent groups with irregular isomorphic non-commuting graphs, then |G| = |H|.     

MORPHIC MODULES

A module M is called morphic if M/M α ? ker(α) for all endomorphisms α in end(M), and a ring R is called a left morphic ring if _RR is a morphic module. We consider the open question when the matrix ring M_n(R) is left morphic by relating it to when Rⁿ is morphic as a left R-module. More generally, we investigate when M being morphic implies that end(M) is left morphic, and conversely. Finally, we relate the morphic condition to internal cancellation in the module.     

Star-shaped Differentiable Functions and Star-shaped Differentials

Based on the isomorphism between the space of star-shaped sets and the space of continuous positively homogeneous real-valued functions, the star-shaped differential of a directionally differentiable function is defined. Formulas for star-shaped differential of a pointwise maximum and a pointwise minimum of a finite number of directionally differentiable functions, and a composite of two directionaUy differentiable functions are derived. Furthermore, the mean-value theorem for a directionaUy differentiable function is demonstrated.     

Seeded graph matching via large neighborhood statistics

We study a noisy graph isomorphism problem, where the goal is to perfectly recover the vertex correspondence between two edge‐correlated graphs, with an initial seed set of correctly matched vertex pairs revealed as side information. We show that it is possible to achieve the information‐theoretic limit of graph sparsity in time polynomial in the number of vertices n. Moreover, we show the number of seeds needed for perfect recovery in polynomial‐time can be as low as in the sparse graph regime (with the average degree smaller than ) and in the dense graph regime, for a small positive constant . Unlike previous work on graph matching, which used small neighborhoods or small subgraphs with a logarithmic number of vertices in order to match vertices, our algorithms match vertices if their large neighborhoods have a significant overlap in the number of seeds.     

基于同构理论的三支概念格的构造方法与算法研究

三支概念分析理论目前已经发展成为数据分析与知识发现的有效工具。主要通过形式背景特征的研究, 讨论三支概念格与概念格的同构关系, 进一步研究了三支概念格的构造方法。首先给出了对偶属性、属性对偶背景的定义, 并证明了在对偶背景下三支概念格与概念格是同构的。其次, 推广了对偶属性、属性诱导的对偶背景, 给出对偶可交属性及属性对偶可交背景的定义, 同时, 证明了在对偶可交背景下，三支概念格与概念格是同构的。最后基于上述理论, 给出了判定属性对偶背景与属性对偶可交背景的2种算法以及三支概念格的构造方法。