图簇Y_(λ_kδ)∪β_kS_δ~*的伴随分解及其补图的色等价性

设P_n是具有n个顶点的路,令δ=rn+1,我们S_δ~*表示把rP_(n+1)的每个分支的一个1度点重迭在一起得到的图.用Y_(λ_1δ)~(S*)表示把r_1S_δ~*中每个分支的r度顶点与S_δ~*的r度顶点依次邻接后得到的图,Y_(λ_2δ)~(S*)表示把用r_2Y_(λ_1δ)~(S*)中每个分支的r+r1度顶点与S_δ~*的r度顶点依次邻接后得到的图,一般地,Y_(λ_kδ)~(S*)表示把用r_kY_(λ_(k-1)δ)~(S*)中每个分支的r+r_k-1度顶点与S_δ~*的r度顶点依次邻接后得到的图,运用图的伴随多项式的性质,证明了图Y_(λ_kδ)~(S*)∪β_kS_δ~*的伴随多项式的因式分解定理,进而得到了这类图的补图的色等价性.     

完全解决图$\overline{B_{n-8,1,4}}$的色等价类

Two graphs are defined to be adjointly equivalent if and only if their complements are chromatically equivalent.Using the properties of the adjoint polynomials and the fourth character R₄（G）,the adjoint equivalence class of graph B_n-8,1,4 is determined.According to the relations between adjoint polynomial and chromatic polynomial,we also simultaneously determine the chromatic equivalence class of B_n-8,1,4 that is the complement of B_n-8,1,4.     

A note on the Glauberman correspondence

In this paper, we will investigate the Glauberman-Watanabe correspondence between the characters and blocks. Motivated by the work of Puig and Zhou [10 Puig, L., Zhou, Y. (2012). Glauberman correspondents and extensions of nilpotent block algebras. J. London Math. Soc. (2) 85809–837.[Crossref] , [Google Scholar]] which is about the extension of a nilpotent block and its Glauberman correspondent, we will modify the basic Morita equivalence obtained in their work and get a new equivalence such that the correspondence between characters induced by this equivalence coincides with the Glauberman correspondence under the situation that the block is a p-extension of a nilpotent block.     

Asymptotic revenue equivalence of asymmetric auctions with interdependent values

We prove an asymptotic revenue equivalence among weakly asymmetric auctions with interdependent values, in which bidders have either asymmetric utility functions or asymmetric distributions of signals.     

Elementary equivalence of rings with finitely generated additive groups

We give algebraic characterizations of elementary equivalence between rings with finitely generated additive groups. They are similar to those previously obtained for finitely generated nilpotent groups. Here, the rings are not supposed associative, commutative or unitary.     

Invariant characterization of scalar third‐order ODEs that admit the maximal point symmetry Lie algebra

The Cartan equivalence method is used to deduce an invariant characterization of the scalar third‐order ordinary differential equation , which admits the maximal 7‐dimensional point symmetry Lie algebra. The method provides auxiliary functions that can be used to efficiently obtain the point transformation that does the reduction to the simplest linear equation . Moreover, examples are given to illustrate the method.     

Nonlinear Perturbations for Linear Nonautonomous Impulsive Differential Equations and Nonuniform $(h,k,\mu,\nu)$-dichotomy

We explore nonlinear perturbations of a flow generated by a linear nonautonomous impulsive differential equation $x''=A(t)x,t\neq\tau_i,\Delta x|_{t=\tau_i}=B_ix(\tau_i),~ i \in \Z$ in Banach spaces. Here we assume that the linear nonautonomous impulsive equation admits a more general dichotomy on $\R$ called the nonuniform $(h,k,\mu,\nu)$-dichotomy, which extends the existing uniform or nonuniform dichotomies and is related to the theory of nonuniform hyperbolicity. Under nonlinear perturbations, we establish a new version of the Grobman-Hartman theorem and construct stable and unstable invariant manifolds for nonlinear nonautonomous impulsive differential equations $x''=A(t)x+f(t,x)$, $t\neq \tau_i, \Delta x|_{t=\tau_i}=B_ix(\tau_i)+g_i(x(\tau_i)),i \in \Z$ with the help of nonuniform $(h,k,\mu,\nu)$-dichotomies.     

集值映射下的粗糙集

粗糙集是一种在信息系统中处理粗糙性和颗粒性的数据挖掘工具.本文从集值映射的角度研究并推广粗糙模型,使其能解决在论域和分类信息变化下对集合的近似问题.最后,讨论了集值映射下的粗糙集的性质.     

On quasi-reduced quadratic forms

With the help of continued fractions, we plan to list all the elements of the set Q△ = {aX2 ＋ bXY ＋ cY2 ： a,b, c ∈Z, b2 - 4ac = △ with 0 ≤ b 〈 √△}of quasi-reduced quadratic forms of fundamental discriminant △. As a matter of fact, we show that for each reduced quadratic form f = aX2 ＋ bXY ＋ cY2 = （a, b, c） of discriminant △〉0（and of sign σ（f） equal to the sign of a）, the quadratic forms associated with f and defined by {〈a＋bu＋cu2,b＋2cu.c〉},with 1≤σ（f）u≤b/2｜c｜ （whenever they exist）, 〈c,-b-2cu,a＋bu＋cu2〉 with b/2｜c｜≤σ（f）u≤[w（f）]=[b＋√△/2｜c｜], are all different from one another and build a set I（f） whose cardinality is #I（f）={1＋[ω（f）],when（2c）｜b,[ω（f）],when （2c）｜b. If f and g are two different reduced quadratic forms, we show that I（f） ∩ I（g） = Ф. Our main result is that the set Q△ is given by the disjoint union of all I（f） with f running through the set of reduced quadratic forms of discriminant △〉0. This allows us to deduce a formula for #（Q△） involving sums of partial quotients of certain continued fractions.