关于S\''{a}rk\"{o}zy和S\''{o}s问题的一个注记

令$k,\ell \geq 2$是正整数.令$A$是无限非负整数的集合.对$n\in \mathbb{N}$, 令$r_{1,k,\ldots,k^{\ell-1}}(A, n)$表示方程$n=a_0+ka_1+\cdots +k^{\ell-1}a_{\ell-1}$, $a_0, \ldots, a_{\ell-1}\in A$解的个数. 在本文中, 我们证明了对所有$n\geq 0$, $r_{1,k,\ldots,k^{\ell-1}}(A, n)=1$当且仅当$A$是$k^\ell$进制展开中数位小于$k$的所有非负整数的集合. 这个结果部分回答了S\''{a}rk\"{o}zy and S\''{o}s关于多维线性型表示的一个问题.     

$C^*$-代数上可加Jordan左$*$-导子的刻画

设$\delta$是一个$*$-代数$\mathcal A$到其左$\mathcal A$-模$\mathcal M$的可加映射, 如果对任意$A\in\mathcal A$, 有$\delta(A^2)=A\delta(A)+A^*\delta(A)$, 则称$\delta$~是一个可加Jordan左$*$-导子. 在本文中, 我们证明了复的单位$C^*$- 代数到其Banach左模的每个可加Jordan左$*$-导子都恒等于零. 设$G\in\mathcal A$, 如果对任意$A,B\in \mathcal A$, 当$AB=G$时, 有$\delta(AB)=A\delta(B)+B^*\delta(A)$, 则称$\delta$在$G$处左$*$-可导. 我们证明了复的单位$C^*$-代数到其Banach左模的在单位点处左$*$-可导的连续可加映射恒等于零.     

本征平方函数在极大变指数Herz空间上的有界性

我们证明了本征平方函数及其交换子在Herz空间$\dot{K}_{q(\cdot)}^{\alpha(\cdot), p),\theta}({\Bbb{R}}^n)$空间上的有界性,其中$\alpha$, $q$均为变指数。当$\alpha(\cdot)\equiv \alpha$为常数时,所得结果也是新的.     

三角Banach代数的Lie弱顺从性

设$\mathcal {A,\ B}$ 是含单位元的Banach代数, $\mathcal M$ 是一个Banach $\mathcal {A,\ B}$-双模. $\mathcal {T}=\left ( \begin{array}{cc} \mathcal {A} & \mathcal M \\ & \mathcal {B} \\ \end{array} \right )$按照通常矩阵加法和乘法,范数定义为$\|\left( \begin{array}{cc} a & m \\ & b\\ \end{array} \right)\|=\|a\|_{\mathcal A}+\|m\|_{\mathcal M}+\|b\|_{\mathcal B}$,构成三角Banach 代数.如果从$\mathcal T$到其$n$次对偶空间$\mathcal T^{n}$上的Lie导子都是标准的,则称$\mathcal T$是Lie $n$弱顺从的.本文研究了三角Banach代数$\mathcal T$上的Lie $n$弱顺从性,证明了有限维套代数是Lie $n$弱顺从的.     

与群PSL2(\mathcal{R})相关的交叉积{\mathcal{R}(\mathcal{A}},α)的一点注记

在上半复平面$\mathbb{H}$上给定双曲测度$dxdy/y^{2}$, 群$G={\rm PSL}_{2}(\mathbb{R})$ 在$\mathbb{H}$上的分式线性作用导出了$G$在Hilbert空间$L^{2}(\mathbb{H}, dxdy/y^{2})$上的酉表示$\alpha$. 证明了交叉积 $\mathcal{R}(\mathcal{A}, \alpha)$是$\mathrm{I}$型von Neumann代数, 其中$\mathcal{A}= \{M_{f}:f\in L^{\infty}(\mathbb{H},dxdy/y^{2} )\}$. 具体地, 交叉积代数$\mathcal{R}(\mathcal{A}, \alpha)$与von Neumann代数$\mathcal{B}(L^{2}(P, \nu))\overline{\otimes}\mathcal{L}_{K}$是*-同构的, 其中$\mathcal{L}_{K}$是$G$中子群 $K$的左正则表示生成的群von Neumann代数.     

关于广义{\Phi}-增生算子的最速下降法的收敛定理

设$E$为一致光滑Banach空间,$A:E\to E$为有界次连续广义${\it \Phi} $-增生算子满足:对任意$x_0\in E$,选取$m\ge 1$,使得$\| x_0 - x^* \| \le m$且$\mathop {\underline {\lim } }\limits_{r \to \infty } {\it \Phi} (r) > m\left\| {Ax_0 } \right\|$.设$\{C_n\}$为$[0,1]$中数列满足控制条件: i)$C_n\to 0\,(n\to\infty)$; ii)$\sum\limits_{n = 0}^\infty {C_n } = \infty $.设$\{x_n\}_{n\ge0}$由下式产生x_{n + 1} = x_n - C_n Ax_n ,\q n \ge 0, \eqno{(@)}$$则存在常数$a>0$,当$C_n < a$时,$\{x_n\}$强收敛于$A$的唯一零点$x^{*}$.     

有界完全Reinhardt域上推广的Roper-Suffridge算子

在$\C^n$中的有界完全Reinhardt域$\Omega$上推广的Roper-Suffridge算子$\Phi(f)$定义为 \begin{eqnarray*} \Phi^r_{n,\beta_2, \gamma_2,\ldots, \beta_n, \gamma_n}(f)(z)\!=\!\Big(rf\Big(\frac{z_1}{r}\Big), \Big(\frac{rf(\frac{z_1}{r})}{z_1}\Big)^{\beta_2}\Big(f’\Big(\frac{z_1}{r}\Big)\Big)^{\gamma_2}z_2,\ldots, \Big(\frac{rf(\frac{z_1}{r})}{z_1}\Big)^{\beta_n}\Big(f’\Big(\frac{z_1}{r}\Big)\Big)^{\gamma_n}z_n \Big), \end{eqnarray*} 其中 $n\geq2$, $(z_1, z_2,\ldots, z_n)\in \Omega$, $r=r(\Omega)=\sup\{|z_1|: (z_1, z_2,\ldots, z_n)\in \Omega\}, 0\leq \gamma_j\leq 1-\beta_j, 0\leq \beta_j\leq 1$, 这里选取幂函数的单值解析分支, 使得 $(\frac{f(z_1)}{z_1})^{\beta_j}|_{z_1=0}= 1$ 和 $(f’(z_1))^{\gamma_j}|_{z_1=0}=1, j=2,\ldots, n$. 证明了 $\Omega$上的算子 $\Phi^r_{n,\beta_2, \gamma_2,\ldots, \beta_n, \gamma_n}(f)$ 是将 $S^*_\alpha(U)$ 的子集映入$S^*_\alpha\,(\Omega)\,(0\leq \alpha<1)$, 且对于一些合适的常数 $\beta_j, \gamma_j, p_j$, $D_p$上的这个算子 $\Phi^r_{n,\beta_2, \gamma_2,\ldots, \beta_n, \gamma_n}(f)$ 保持$\alpha$阶星形性或保持$\beta$ 型螺形性, 其中 $ D_p=\bigg\{(z_1, z_2,\ldots, z_n)\in \C^n: \he{j=1}{n}|z_j|^{p_j}<1\bigg\},\quad p_j>0, j=1, 2,\ldots, n, $ $U$是复平面$\C$上的单位圆, $S^*_\alpha(\Omega)$ 是 $\Omega$ 上所有正规化$\alpha$阶星形映射所成的类. 也得到: 对于某些合适的常数 $\beta_j, \gamma_j, p_j$ 和 在$\C^n$中的有界完全Reinhardt域$\Omega$上推广的Roper-Suffridge算子$\Phi(f)$定义为 \begin{eqnarray*} \Phi^r_{n,\beta_2, \gamma_2,\ldots, \beta_n, \gamma_n}(f)(z)\!=\!\Big(rf\Big(\frac{z_1}{r}\Big), \Big(\frac{rf(\frac{z_1}{r})}{z_1}\Big)^{\beta_2}\Big(f’\Big(\frac{z_1}{r}\Big)\Big)^{\gamma_2}z_2,\ldots, \Big(\frac{rf(\frac{z_1}{r})}{z_1}\Big)^{\beta_n}\Big(f’\Big(\frac{z_1}{r}\Big)\Big)^{\gamma_n}z_n \Big), \end{eqnarray*} 其中 $n\geq2$, $(z_1, z_2,\ldots, z_n)\in \Omega$, $r=r(\Omega)=\sup\{|z_1|: (z_1, z_2,\ldots, z_n)\in \Omega\}, 0\leq \gamma_j\leq 1-\beta_j, 0\leq \beta_j\leq 1$, 这里选取幂函数的单值解析分支, 使得 $(\frac{f(z_1)}{z_1})^{\beta_j}|_{z_1=0}= 1$ 和 $(f’(z_1))^{\gamma_j}|_{z_1=0}=1, j=2,\ldots, n$. 证明了 $\Omega$上的算子 $\Phi^r_{n,\beta_2, \gamma_2,\ldots, \beta_n, \gamma_n}(f)$ 是将 $S^*_\alpha(U)$ 的子集映入$S^*_\alpha\,(\Omega)\,(0\leq \alpha<1)$, 且对于一些合适的常数 $\beta_j, \gamma_j, p_j$, $D_p$上的这个算子 $\Phi^r_{n,\beta_2, \gamma_2,\ldots, \beta_n, \gamma_n}(f)$ 保持$\alpha$阶星形性或保持$\beta$ 型螺形性, 其中 $ D_p=\bigg\{(z_1, z_2,\ldots, z_n)\in \C^n: \he{j=1}{n}|z_j|^{p_j}<1\bigg\},\quad p_j>0, j=1, 2,\ldots, n, $ $U$是复平面$\C$上的单位圆, $S^*_\alpha(\Omega)$ 是 $\Omega$ 上所有正规化$\alpha$阶星形映射所成的类. 也得到: 对于某些合适的常数 $\beta_j, \gamma_j, p_j$ 和 在C~n中的有界完全Reinhardt域Ω上推广的Roper-Suffridge算子Φ(f)定义为Φ_(n,β_2,γ_2,…,β_n,γ_n)~r(f)(z)=(rf(z_1/r),((rf(z_1/r))/z_1)~(β_2)(f′(z_1/r))~γ_2_(z_2,…,)((rf(z_1/r))/z_1)~(β_n)(f′(z_1/r))~(γ_n)_(z_n),其中n≥2,(z_1,z_2,…,z_n)∈Ω,r=r(Ω)=sup{|z_1|:(z_1,z_2,…,z_n)∈Ω},0≤γ_j≤1-β_j,0≤β_j≤1,这里选取幂函数的单值解析分支,使得((f(z_1))/z_1)~(β_j)|_(z_1=0)=1和(f′(z_1))~(γ_j)|_(z_1=0)=1,j= 2,…,n.证明了Ω上的算子Φ_(n,β_2,γ_2,…,β_n,γ_n)~r(f)是将S_α~*(U)的子集映入S_α~*(Ω)(0≤α<1),且对于一些合适的常数β_j,γ_j,p_j,D_p上的这个算子Φ_(n,β_2,γ_2,…,β_n,γ_n)~r(f)保持α阶星形性或保持β型螺形性,其中(?) U是复平面C上的单位圆,S_α~*(Ω)是Ω上所有正规化α阶星形映射所成的类.也得到:对于某些合适的常数β_j,γ_j,p_j和0≤α<1,Φ_(n,β_2,γ_2,…,β_n,γ_n)~r(f)∈S_α~*(D_p)当且仅当f∈S_α~*(U).     

Hom-$\omega$-smash 积Hopf 代数的拟三角结构

设 $(A,\alpha)$和$(H,\beta)$ 是 Hom-\!\!双代数, $\omega:H\otimes A\rightarrow A\otimes H$ 是线性映射, 定义了 Hom-$\omega$-smash 积$(A\sharp_{\omega} H,\gamma)$,并给出了 $(A\bowtie_{\omega}H,\gamma)$ 是 Hom-bialgebra 的充要条件. 最后,研究了$(A\bowtie_{\omega} H,\gamma)$上的拟三角结构, 并给出了它是拟 三角 Hom-Hopf 代数的充要条件.     

亚BCI-代数的$(\alpha,\beta)$-软理想

首先将软集的参数集赋予亚BCI-代数, 给出了亚BCI-代数的$(\alpha,\beta)$-软理想的概念.当$U=[0,1], \alpha=U, \beta=\phi$时,相应地就得到了亚BCI-代数的犹豫模糊理想的概念.研究了亚BCI-代数的$(\alpha,\beta)$-软理想的一些重要性质.最后讨论了亚BCI-代数的$(\alpha,\beta)$-软理想的同态像和原像的性质.     

从$L^{\infty}$空间到Bloch型空间一类奇异积分算子的范数

本文研究了单位圆盘上从$L^{\infty}(\mathbb{D})$空间到Bloch型空间 $\mathcal{B}_\alpha$ 一类奇异积分算子$Q_\alpha, \alpha>0$的范数, 该算子可以看成投影算子$P$ 的推广,定义如下$$Q_\alpha f(z)=\alpha \int_{\mathbb{D}}\frac{f(w)}{(1-z\bar{w})^{\alpha+1}}\d A(w),$$ 同时我们也得到了该算子从 $C(\overline{\mathbb{D}})$空间到小Bloch型空间$\mathcal{B}_{\alpha,0}$上的范数.     

具有对称自同态的环及其扩张

Let R be a ring with an endomorphism α and an α-derivation δ. We introduce the notions of symmetric α-rings and weak symmetric α-rings which are generalizations of symmetric rings and weak symmetric rings, respectively, discuss the relations between symmetricα-rings and related rings and investigate their extensions. We prove that if R is a reduced ring and α(1) = 1, then R is a symmetric α-ring if and only if R[x]/(x n) is a symmetric ˉα-ring for any positive integer n. Moreover, it is proven that if R is a right Ore ring, α an automorphism of R and Q(R) the classical right quotient ring of R, then R is a symmetric α-ring if and only if Q(R) is a symmetric ˉα-ring. Among others we also show that if a ring R is weakly 2-primal and(α, δ)-compatible, then R is a weak symmetric α-ring if and only if the Ore extension R[x; α, δ] of R is a weak symmetric ˉα-ring.     

剩余有限Minimax可解群的4阶正则自同构

设G是剩余有限minimax可解群,α是G的4阶正则自同构,则下面结果成立:(1)如果映射φ:G→G (g→[g,α])是满射,那么G是中心子群被亚Abel群的扩张.(2)C_G(α~2)和[G,n-1α~2]/[G,nα~2](n∈Z~+)都是Abel群的有限扩张.     

A Hilbert Module Approach to the Haagerup Property

We develop a Hilbert module version of the Haagerup property for general C*-algebras ${{\mathcal{A} \subseteq \mathcal{B}}}$ . We show that if ${\alpha : \Gamma \curvearrowright \mathcal{A}}$ is an action of a countable discrete group Γ on a unital C*-algebra ${\mathcal{A}}$ , then the reduced C*-algebra crossed product ${\Gamma \ltimes _{\alpha, r} \mathcal{A}}$ has the Hilbert ${\mathcal{A}}$ -module Haagerup property if and only if the action α has the Haagerup property. We are particularly interested in the case when ${\mathcal{A} = C(X)}$ is a unital commutative C*-algebra. We compare the Haagerup property of such an action ${\alpha: \Gamma \curvearrowright C(X)}$ with the two special cases when (1) Γ has the Haagerup property and (2) Γ is coarsely embeddable into a Hilbert space. We also prove a contractive Schur mutiplier characterization for groups coarsely embeddable into a Hilbert space, and a uniformly bounded Schur multiplier characterization for exact groups.     

WHEN CAN THE STABLE ALGEBRA DETERMINE THE STRUCTURE OF A C^＊-ALGEBRA ？

ＷＨＥＮＣＡＮＴＨＥＳＴＡＢＬＥＡＬＧＥＢＲＡＤＥＴＥＲＭＩＮＥＴＨＥＳＴＲＵＣＴＵＲＥＯＦＡＣ－ＡＬＧＥＢＲＡ？￥ＷＵＬＩＡＮＧＳＥＮＡｂｓｔｒａｃｔ：ＬｅｔＡａｎｄＢｂｅＣ－ａｌｇｅｂｒａｓ．ＳｕｐｐｏｓｅｔｈａｔＫｉｓｔｈｅａｌｇｅｂｒａｏｆａ...     

$\alpha$-可分解4-圈系统

An m-cycle system of order v and index λ, denoted by m-CS（v,λ）, is a collection of cycles of length m whose edges partition the edges of λKv. An m-CS（v,λ） is α-resolvable if its cycles can be partitioned into classes such that each point of the design occurs in precisely α cycles in each class. The necessary conditions for the existence of such a design are m｜λv（v-1）/2,2｜λ（v -1）,m｜αv,α｜λ（v-1）/2. It is shown in this paper that these conditions are also sufficient when m = 4.     

Abdellatif Ghendir Aoun

In this paper, we study a fractional differential equation $$^{c}D^{\alpha}_{0^{+}}u(t)+f(t,u(t))=0,\quad t\in(0, +\infty)$$ satisfying the boundary conditions: $$u^{\prime}(0)=0,\quad \lim_{t\rightarrow +\infty}\,^{c}D^{\alpha-1}_{0^{+}}u(t)=g(u),$$ where $1<\alpha\leqslant2$, $^{c}D^{\alpha}_{0^{+}}$ is the standard Caputo fractional derivative of order $\alpha$. The main tools used in the paper is contraction principle in the Banach space and the fixed point theorem due to D. O''Regan. Some the compactness criterion and existence of solutions are established.     

NASH EQUILIBRIUM OF TWO GAME MODELS ON THE RANDOM SOCIAL NETWORK

A graph $G$ is $k$-triangular if each of its edge is contained in at least $k$ triangles. It is conjectured that every 4-edge-connected triangular graph admits a nowhere-zero 3-flow. A triangle-path in a graph $G$ is a sequence of distinct triangles $T_1 T_2\cdots T_k$ in $G$ such that for $1\leq i\leq k-1, |E(T_i )\cap E(T_{i+1})|=1$ and $E(T_i)\cap E(T_j)=\emptyset$ if $j>i+1$. Two edges $e,e''\in E(G)$ are triangularly connected if there is a triangle-path $T_1,T_2,\cdots, T_k$ in $G$ such that $e\in E(T_1)$ and $e''\in E(T_k)$. Two edges $e,e''\in E(G)$ are equivalent if they are the same, parallel or triangularly connected. It is easy to see that this is an equivalent relation. Each equivalent class is called a triangularly connected component. In this paper, we prove that every 4-edge-connected triangular graph $G$ is ${\mathbb Z}_3$-connected, unless it has a triangularly connected component which is not ${\mathbb Z}_3$-connected but admits a nowhere-zero 3-flow.     

THE INVARIANT CONTINUOUS-TRACE C^＊-ALGEBRAS BY THE ACTIONS OF COMPACT ABELIAN GROUPS

1.IntroductionLet(A,G,a)beaC*-dynamicsystem.TherelationbetweenAxosGandAhasbeenstudiedforalongtime,andconsiderableprogresshasbeenmade.SpeciallyifAisacontinuous-traceC*-algebra,I.Raeburnandhiscollaboratorsgotrichresultsseveralyearsago.Forexample,[17]and119]saythatifAisparacompact,Gisabelianandaislocallyunitary,thenAiscontinuous-traceiffAxosGiscontinuous-trace.Inotherdirection,therelationbetweenAxosGandA"withGcompacthasalsobeenstudiedformanyyears.Themovementofthispaperistoinvestigatethec…     

一类新的具有标量旗曲率Finsler度量

In this paper,we study a new class of general(α,β)-metrics F defined by a Riemannian metric α,a 1-form β and C ∞ function φ(b2,s).We provide the projective factor of a class of general(α,β)-metrics F=αφ(b2,s),and apply these formulae to compute its flag curvature.