一类缺项算子矩阵的四类点谱的扰动

有界线性算子的点谱可进一步细分为4类,分别为$\sigma_{p1}$, $\sigma_{p2}$, $\sigma_{p3}$ 和$\sigma_{p4}$.设 $H, K$为无穷维可分的Hilbert空间,用$M_C$表示$2\times 2$上三角算子矩阵$\left(\begin{array}{cc} A & C \\ 0 & B \\ \end{array} \right)$,对于给定的 $A\in B(H),~B\in B(K)$,描述了集合$\bigcap\limits_{C\in B(K,H)}\sigma_{p1}(M_C)$, $\bigcap\limits_{C\in B(K,H)}\sigma_{p2}(M_C)$, $\bigcap\limits_{C\in B(K,H)}\sigma_{p3}(M_C)$和$\bigcap\limits_{C\in B(K,H)}\sigma_{p4}(M_C)$.     

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.     

Cn中单位球上$mu$-Bloch函数的等价刻画

设$\mu$是$[0,1)$上的正规函数, 给出了${\bf C}^{\it n}$中单位球$B$上$\mu$-Bloch空间$\beta_{\mu}$中函数的几种刻画. 证明了下列条件是等价的: (1) $f\in \beta_{\mu}$; \ (2) $f\in H(B)$且函数$\mu(|z|)(1-|z|^{2})^{\gamma-1}R^{\alpha,\gamma}f(z)$ 在$B$上有界; (3) $f\in H(B)$ 且函数${\mu(|z|)(1-|z|^{2})^{M_{1}-1}\frac{\partial^{M_{1}} f}{\partial z^{m}}(z)}$ 在$B$上有界, 其中$|m|=M_{1}$; (4) $f\in H(B)$ 且函数${\mu(|z|)(1-|z|^{2})^{M_{2}-1}R^{(M_{2})}f(z)}$ 在$B$上有界.     

一列连续函数的遍历优化

设$T:X\rightarrow X$是紧度量空间$X$上的连续映射, $\mathcal{F}=\{f_n\}_{n\geq 1}$是$X$上的一族连续函数. 如果 $\mathcal{F}$是渐近次可加的, 那么$\sup\limits_{x\in \mathrm{Reg}(\mathcal{F},T)}\lim\limits_{n\rightarrow\infty}\frac 1 n f_n (x)=\sup\limits_{x\in X} \limsup\limits_{n\rightarrow\infty}\frac 1 n f_n (x) =\lim\limits_{n\rightarrow\infty}\frac 1 n \max\limits_{x\in X}f_n (x)=\sup\{\mathcal{F}^*(\mu):\mu\in\mathcal{M}_T\}$, 其中$\mathcal{M}_T$表示$T$-\!\!不变的Borel概率测度空间, $\mathrm{Reg}(\mathcal{F},T)$ 表示函数族$\mathcal{F}$的正规点集, $\mathcal{F}^*(\mu)=\lim\limits_{n\rightarrow\infty}\frac 1 n \int f_n \mathrm{d}\mu$. 这把Jenkinson, Schreiber 和 Sturman 等人的一些结果推广到渐近次可加势函数, 并且给出了次可加势函数从属原理成立的充分条件, 最后给出了 一些相关的应用.     

强不可约算子的K0群及其近似相似不变量

令 $\mathcal H$ 表示复可分的Hilbert空间, ${\mathcal L}({\mathcal H})$ 表示 $\mathcal H$上全体有界线性算子的集合. 算子 $T \in{\mathcal L}{(\mathcal H)}$称为是强不可约的, 如果不存在非平凡的幂等元与 T 可交换. 对强不可约算子的近似不变量给出比以往文献更精细的刻画. 主要结果如下: 对任意具有连通谱的有界线性算子 T 及 ε>0, 存在强不可约算子A, 使得 $\|A-T\|<\varepsilon$, $V({\mathcal A}^{\prime}(A))\cong{\mathbb{N}}$, $K_{0}({\mathcal A}^{\prime}(A))\cong{\mathbb{Z}}$, 且 ${{\mathcal A}^{\prime}(A)}/{\rm rad}{{\mathcal A}^{\prime}(A)}$ 可交换, 这里${\mathcal A}^{\prime}(A)$ 表示A 的换位代数, 且 ${\rm rad}{\mathcal A}^{\prime}(A)$ 表示${\mathcal A}^{\prime}(A)$的Jacobson根.     

涉及分担超平面的正规定则

在本文中, 作者继续讨论涉及分担超平面的全纯曲线的正规性, 得到了如下结果:设$\mathcal F$是一族从区域$D\subset\mathbb C$到$\mathbb P^N(\mathbb C)$上的全纯曲线,$H_j=\{x\in\mathbb P^N(\mathbb C):\langle\bm{x},\alpha_j\rangle=0\}$是$\mathbb P^N(\mathbb C)$中处于一般位置的超平面, 这里$\alpha_j=(a_{j0},\cdots,a_{jN})^{\rm T}$且$a_{j0}\ne0$, $j=1,2,\cdots,2N+1$.若对于任意的$f\in\mathcal F$, 满足下列两个条件:(i) 如果$f(z)\in H_j$, 那么$\nabla f\in H_j$, 这里$j=1,2,\cdots,2N+1$;(ii) 如果$f(z)\in\bigcup\limits_{j=1}^{2N+1} H_j$, 那么$\frac{|\langle f(z),H_0\rangle|}{\|f\|\|H_0\|}\ge \delta$, 这里$0<\delta<1$是一个常数,而$H_0=\{w_0=0\}$,\noindent 则$\mathcal F$在$D$上正规.     

素*-代数上的非线性混合双斜Jordan三重导子

设$\mathcal{A}$是一个包含非平凡投影的单位素*-代数.本文证明了一个映射$\Phi:\mathcal{A}\rightarrow\mathcal{A}$满足对任意$A,B,C\in\mathcal{A}$有$\Phi([A,B]_{\diamond}\circC)=[\Phi(A),B]_{\diamond}\circC+[A,\Phi(B)]_{\diamond}\circC+[A,B]_{\diamond}\circ\Phi(C)$当且仅当$\Phi$是一个可加的*-导子, 其中$A\circ B=A^{*}B+B^{*}A$和$[A,B]_{\diamond}=A^{*}B-B^{*}A$.     

DMk OPERATORS AND SPECTRAL OPERATORS

1谱位于平面上的有界\[{\mathcal{D}_{ < {M_k} > }}\]型算子 记号与[1,2]相同，不再一一赘述.设序列 {Mk}满足（M.1)，(M.2)，(M.3)即.对数凸性、非拟解析性、可微性[1]. 由{M（k）}我们可以 定义二元相关函数\[M({t_1},{t_2})\](详见[7])以及二元\[{\mathcal{D}_{ < {M_k} > }}\]空间 \[{\mathcal{D}_{ < {M_k} > }} = \{ \varphi |\varphi \in \mathcal{D};\exists \nu ,st{\left\| \varphi \right\|_\nu } = \mathop {\sup }\limits_\begin{subarray}{l} s \in {R^2} \\ {k_i} \geqslant 0 \\ (i = 1,2) \end{subarray} |\frac{{{\partial ^{{k_1} + {k_2}}}}}{{{\partial ^{{k_1}}}{s_1}\partial _{{s_2}}^{{k_2}}}}\varphi (s)|/{\nu ^k}{M_k} < + \infty \} \] 其中\[s = ({s_1},{s_2})k = {k_1} + {k_2}\].关于谱位于复平面上的有界\[{\mathcal{D}_{ < {M_k} > }}\]型算子的定义及性质可 参看[3,4].设X为Banach空间，B(X)为X上有界线性算子的全体组成的环.当 \[T \in B(X)\]为\[{\mathcal{D}_{ < {M_k} > }}\]型算子时，有\[T = {T_1} + i{T_2};{T_1} = {U_{Ret}}{T_2}{\text{ = }}{U_{\operatorname{Im} {\kern 1pt} t}}\] ,此处U为T的谱超广义函数，t为复变量.由于supp(U)为紧集，故可将U延拓到\[{\varepsilon _{ < {M_k} > }}\]上且保持连续性. 经过简单的计算，若\[T \in B(X)\]为谱位于平面上的一个\[{\mathcal{D}_{ < {M_k} > }}\]型算子，则T的一个谱 超广义函数（1）U可表成 \[{U_\varphi } = \int_{ - \infty }^{ + \infty } {\int_{ - \infty }^{ + \infty } {{e^{i({t_1}{T_1} + {t_2}{T_2})}}\hat \varphi } } ({t_1},{t_2})d{t_1}d{t_2}\] 设\[T \in B(X)\]为谱算子，S、N、E（.)分别为T的标量部分、根部、谱测度.下面的定理给出了谱算子成为\[{\mathcal{D}_{ < {M_k} > }}\]型算子的一个充分条件： 定理1设T为谱算子适合下面的条件 \[\mathop {\sup }\limits_{k > 0} \mathop {\sup }\limits_\begin{subarray}{l} |{\mu _j}| < 1 \\ {\delta _j} \in \mathcal{B} \\ j = 1,2,...,k \end{subarray} {(\left\| {\frac{{{N^n}}}{{n!}}\sum\limits_{j = 1}^k {{\mu _j}E({\delta _j})} } \right\|{M_n})^{\frac{1}{n}}} \to 0(n \to \infty )\] 其中\[\mathcal{B}\]为平面本的Borel集类.则T为\[{\mathcal{D}_{ < {M_k} > }}\]型算子且它的一个谱广义函数可表为 \[{U_\varphi } = \sum\limits_{n = 0}^\infty {\frac{{{N^n}}}{{n!}}} \int {{\partial ^n}} \varphi (s)dE(s)\] 推论1设E（?），N满足 \[{(\frac{{{M_n}}}{{n!}} \vee ({N^n}E))^{\frac{1}{n}}} \to 0\] 则T为\[{\mathcal{D}_{ < {M_k} > }}\]型算子. 推论2设N为广义幂零算子，则对于任何与N可换的标量算子S,S+N为\[{\mathcal{D}_{ < {M_k} > }}\]型算子的充分必要条件是 \[{(\frac{{\left\| {{N^n}} \right\|}}{{n!}}{M_n})^{\frac{1}{n}}} \to 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (n \to \infty )\] 在[4]中称满足上式的算子为\[\{ {M_k}\} \]广义幂零算子.显然\[\{ {M_k}\} \]广义幂零算子必为通 常的广义幂零算子.下面的命题给出了\[\{ {M_k}\} \] 广义幂零算子的一些性质. 命题 设N为广义幂零算子，则下列事实等价： (i ) N为\[\{ {M_k}\} \]广义幂零算子； (ii)对于任给的\[\lambda > 0\],存在\[{B_\lambda } > 0\]使(1) \[\left\| {R(\xi ,N)} \right\| \leqslant {B_\lambda }{e^{{M^*}(\frac{\lambda }{{|\xi |}})}}\](\[{|\xi |}\]充分小)； (iii)对于任给的\[\mu > 0\],存在\[{A_\mu } > 0\]使 \[\left\| {{e^{izN}}} \right\| \leqslant {A_\mu }{e^{M(\mu |z|)}}\] 2谱位于实轴上的有界\[{\mathcal{D}_{ < {M_k} > }}\]型算子本节讨论有界\[{\mathcal{D}_{ < {M_k} > }}\]型算子T成为谱算子 的条件，这里假定\[{\mathcal{D}_{ < {M_k} > }}\]中的函数是一元的，于是Т的谱位于实轴上.X^*表示X的共轭 空间. 设\[f \in {\mathcal{D}^'}_{ < {M_k} > }\]，由[8, 9]，存在测度\[{\mu _n}(n \geqslant 0)\]使得对任何h>0,存在A>0适合 \[\sum\limits_{n = 0}^\infty {\frac{{{h^n}}}{{n!}}} {M_n}\int {|d{\mu _n}| \leqslant A} \]且 \[ < f,\varphi > = \sum\limits_{n = 0}^\infty {\frac{1}{{n!}}} \int {{\varphi ^{(n)}}} (t)d{\mu _n}(t)\] 一般说，上述\[{\mu _n}(n \geqslant 0)\]不是唯一的，为此我们引入 定义设\[{n_0}\]为正整，如果对一切\[n \geqslant {n_0}\]，存在测度\[{{\mu _n}}\]，它们的支集均包含在某一L 零测度闭集内，则称f是\[{n_0}\]奇异的，若\[{n_0}\] = 1，则称f是奇异的.设\[T \in B(X)\]为\[{\mathcal{D}_{ < {M_k} > }}\]型 算子，U为其谱超广义函数，如果对于任何\[x \in X{x^*} \in {X^*},{x^*}U\].x是\[{n_0}\]奇异的（奇异 的)，则称T是\[{n_0}\]奇异的(奇异的）\[{\mathcal{D}_{ < {M_k} > }}\]型算子. 经过若干准备，可以证明下面的 定理2 设X为自反的Banach空间，则\[T \in B(X)\]为奇异\[{\mathcal{D}_{ < {M_k} > }}\]型算子的充分必要 条件是T为满足下列条件的谱算子： (i)对每个\[x \in X\]及\[{x^*} \in X\]，\[\sup p({x^*}{N^n}E()x)\]包含在一个与\[n \geqslant 1\]无关的L零测 度闭集F内(F可以依赖于\[x{x^*}\])，此处E(?)、N分别是T的谱测度与根部； (ii)算子N是\[\{ {M_k}\} \]广义幂零算子. 推论 设X为自反的banach空间，\[T \in B(X)\]为奇异\[{\mathcal{D}_{ < {M_k} > }}\]型算子且\[\sigma (T)\]的测度 为零的充分必要条件是T为满足下列条件的谱算子： (i) E(?）的支集为L零测度集； (ii) 算子N是\[\{ {M_k}\} \]广义幂零算子.；     

一类非线性分数阶薛定谔-泊松系统非零解的存在性

本文研究了分数阶薛定谔-泊松系统$$\left\{\begin{array}{l}(-\Delta)^su+u+\phi u=\lambda f(u)\ \text {in} \ \mathbb {R}^3, \\ (-\Delta)^{\alpha}\phi =u^2\ \text {in} \ \mathbb {R}^3\emph{},\end{array}\right. $$ 非零解的存在性, 其中$s\in (\frac{3}{4},1), \alpha\in(0,1),\lambda$ 是正参数, $(-\Delta)^s,(-\Delta)^{\alpha}$是分数阶拉普拉斯算子. 在一定的假设条件下, 利用扰动法和Morse迭代法, 得到了系统至少一个非平凡解.     

EXISTENCE OF DISCONTINUOUS SOLUTIONS FOR A DOUBLY DEGENERATE ELLIPTIC EQUATIONS ON $R^N$

It is demonstrated that under the hypotheses I—III the problem $\[\left\{ {\begin{array}{*{20}{c}} {div((k(U) + \varepsilon )|DU{|^{M - 1}}DU) = f(|x|,U) + \varepsilon U{\text{ }}in{\text{ }}{R^N},N > 1,{\text{ (1}}{\text{.1}}{{\text{)}}_\varepsilon }} \ {U(0) > 0,U(x) \geqslant 0{\text{ on }}{R^N},U(x) \to 0{\text{ as }}|x| \to + \infty {\text{ }}(1.2)} \end{array}} \right.\]$ for each fixed $\epsilon >0$ has infinitely many distinct radially symmetric solutions $U_\epsilon=V_\epsilon(|x|)$ such that $V_\epsilon(s),s^{N-1}(k(V_\epsilon(s))+\epsilon)|V''(s)|^{M-1}V''_\epsilon(s)\in C[0,+\infinity)\capC^1(0,+\infinity)$, $\[\left\{ {\begin{array}{*{20}{c}} {({s^{N - 1}}(k({V_\varepsilon }(s)) + \varepsilon )|V''(s){|^{M - 1}}V''(s)) = {\varepsilon ^{N - 1}}(f(s,{V_\varepsilon }(s)) + \varepsilon {V_\varepsilon }(s))for{\text{ }}s > 0,{{(1.3)}_\varepsilon }} \ {{V_\varepsilon }(0) = B > 0,{V_\varepsilon }(s) \geqslant 0{\text{ for }}s > 0,and{\text{ }}{V_\varepsilon }( + \infty ) = 0,(1.4)} \end{array}} \right.\]$ where B is a positive number chosen arbitrarily, which extends the result in [3]. In particular, the author proves that $U_0(x)=V_0(|x|)$ is a weak solution of the problem $(l.l)_0-(1.2)$.     

3$\times$3阶上三角型算子矩阵的可能谱

Let H1, H2 and H3 be infinite dimensional separable complex Hilbert spaces. We denote by M（D,V,F） a 3×3 upper triangular operator matrix acting on Hi ＋H2＋ H3 of theform M（D,E,F）=（A D F 0 B F 0 0 C）.For given A ∈ B（H1）, B ∈ B（H2） and C ∈ B（H3）, the sets ∪D,E,F^σp（M（D,E,F））,∪D,E,F ^σr（M（D,E,F））,∪D,E,F ^σc（M（D,E,F）） and ∪D,E,F σ（M（D,E,F）） are characterized, where D ∈ B（H2,H1）, E ∈B（H3, H1）, F ∈ B（H3,H2） and σ（·）, σp（·）, σr（·）, σc（·） denote the spectrum, the point spectrum, the residual spectrum and the continuous spectrum, respectively.     

Inequalities and Separation for a Biharmonic Laplace-Beltrami Differential Operator in a Hilbert Space Associated with the Existence and Uniqueness Theorem

In this paper, we have studied the separation for the biharmonic Laplace-Beltrami differential operator\begin{equation*}Au(x)=-\Delta \Delta u(x)+V(x)u(x),\end{equation*}for all $x\in R^{n}$, in the Hilbert space $H=L_{2}(R^{n},H_{1})$ with the operator potential $V(x)\in C^{1}(R^{n},L(H_{1}))$, where $L(H_{1})$ is the space of all bounded linear operators on the Hilbert space $H_{1}$, while $\Delta \Delta u$\ is the biharmonic differential operator and\begin{equation*}\Delta u{=-}\sum_{i,j=1}^{n}\frac{1}{\sqrt{\det g}}\frac{\partial }{{\partial x_{i}}}\left[ \sqrt{\det g}g^{-1}(x)\frac{\partial u}{{\partial x}_{j}}\right]\end{equation*}is the Laplace-Beltrami differential operator in $R^{n}$. Here $g(x)=(g_{ij}(x))$ is the Riemannian matrix, while $g^{-1}(x)$ is the inverse of the matrix $g(x)$. Moreover, we have studied the existence and uniqueness Theorem for the solution of the non-homogeneous biharmonic Laplace-Beltrami differential equation $Au=-\Delta \Delta u+V(x)u(x)=f(x)$ in the Hilbert space $H$ where $f(x)\in H$ as an application of the separation approach.     

ON THE GENERALIZED POLYNOMIALS OF L. V. KANTOROVITCH AND THEIR ASYMPTOTIC BEHAVIORS

In this article we generahze the polynomials of Kantorovitch \({P_n}(f)\) . Let \({B_n}\) be a sequence of linear operators from C[a,b] into \({H_n}\), if \[f(t) \in L[a,b],F(u) = \int_a^u {f(t)dt} ,{A_n}(f(t),x) = \frac{d}{{dx}}{B_{n + 1}}(F(u),x)\], here \({B_n}\)satisfy\[\begin{array}{l} (a):{B_n}(1,x) \equiv 1,{B_n}(u,x) \equiv x;\(b):for{\kern 1pt} {\kern 1pt} g(u) \in C[a,b]{\kern 1pt} {\kern 1pt} we{\kern 1pt} {\kern 1pt} have{\kern 1pt} {\kern 1pt} {B_n}(g(u),b) = g(b). \end{array}\]. we call such \({A_n}(f)\) generalized polynomials of Kantorovitch (denoted by \({A_n}(f) \in K\) ). Let \[\begin{array}{l} {\varepsilon _n}({W^2};x)\mathop = \limits^{def} \mathop {\sup }\limits_{f \in {W^2}} \left| {{A_n}(f(t),x) - f(x) - f'(x)({A_n}(t,x) - x)} \right|,\{\varepsilon _n}{({W^2}{L^p})_{{L^p}}}\mathop = \limits^{def} \mathop {\sup }\limits_{f \in {W^2}{L^p}} {\left\| {{A_n}(f(t),x) - f(x) - f'(x)({A_n}(t,x) - x)} \right\|_p}. \end{array}\] We have proved the following results: Let An he a sequence of linear continuous operators of type \[C[a,b] \Rightarrow C[a,b],{D_n}(x,z)\mathop = \limits^{def} {A_n}(\left| {t - z} \right|,x) - \left| {x - z} \right| - ({A_n}(t,x) - x)Sgn(x - z),{A_n}(1,x) = 1\] then (1):\({\varepsilon _n}({W^2};x) = \frac{1}{2}\int_a^b {\left| {{D_n}(x,z)} \right|} dz\), (2): Moreover, if \({A_n}\) be a sequence of linear positive operators, then for \(\left[ {\begin{array}{*{20}{c}} {a \le x \le b}\{a \le z \le b} \end{array}} \right]\) ,we have \({D_n}(x,z) \ge 0\), and \({\varepsilon _n}({W^2};x) = \frac{1}{2}{A_n}({(t - x)^2},x)\). Let \({A_n}(f) \in K\) be a sequence of linear positive operators,\[{R_n}{(z)_L} = \frac{1}{2}\int_a^b {\left| {{D_n}(x,z)} \right|} dx\],then \[{R_n}{(z)_L} = \frac{1}{2}\left[ {{B_{n + 1}}({u^2},z) - {z^2}} \right]\] and \[{\varepsilon _n}{({W^2}L)_L}{\rm{ = }}\frac{1}{2}\left\| {{B_{n + 1}}({u^2},z) - {z^2}} \right\|\]. Let \[{g_n} = \frac{1}{2}\mathop {\max }\limits_{a \le x \le b} {A_n}({(t - x)^2},x),{h_n} = \frac{1}{2}\mathop {\max }\limits_{a \le z \le b} \left[ {{B_{n + 1}}({u^2},z) - {z^2}} \right],\] then \[{\varepsilon _n}{({W^2}{L^p})_{{L^p}}} \le {g_n}^{1 - \frac{1}{p}}{h_n}^{\frac{1}{p}}(1 < p < \infty ).\]     

On Proximal Relations in Transformation Semigroups Arising from Generalized Shifts

For a finite discrete topological space $X$ with at least two elements, a nonempty set $\Gamma$, and a map $\varphi:\Gamma \to \Gamma$, $\sigma_{\varphi}:X^{\Gamma} \to X^{\Gamma}$with $\sigma_{\varphi}((x_{\alpha})_{\alpha \in \Gamma})=(x_{\varphi(\alpha)})_{\alpha \in \Gamma}$ (for $(x_{\alpha})_{\alpha \in \Gamma} \in X^{\Gamma}$) is a generalized shift. In this text for $\mathcal{S} = \{\sigma_{\varphi}:\varphi \in \Gamma^{\Gamma}\}$ and $\mathcal{H}=\{\sigma_{\varphi}:\Gamma \xrightarrow{\varphi} \Gamma$ is bijective$\}$ we study proximal relations of transformation semigroups $(\mathcal{S}, X^{\Gamma})$ and $(\mathcal{H}, X^{\Gamma})$. Regarding proximal relation we prove: $$P(\mathcal{S}, X^{\Gamma}) = \{((x_{\alpha})_{\alpha \in \Gamma},(y_{\alpha})_{\alpha \in \Gamma}) \in X^{\Gamma} \times X^{\Gamma} : \exists \beta \in \Gamma (x_{\beta} = y_{\beta})\}$$and $P(\mathcal{H}, X^{\Gamma} ) \subseteq \{((x_{\alpha})_{\alpha \in \Gamma},(y_{\alpha})_{\alpha \in \Gamma}) \in X^{\Gamma} \times X^{\Gamma} : \{\beta \in \Gamma : x_{\beta} = y_{\beta}\}$ is infinite$\}$ $\cup\{($ $x,x) : x \in \mathcal{X}\}$. Moreover, for infinite $\Gamma$, both transformation semigroups $(\mathcal{S}, X^{\Gamma})$ and $(\mathcal{H}, X^{\Gamma})$ are regionally proximal, i.e., $Q(\mathcal{S}, X^{\Gamma}) = Q(\mathcal{H}, X^{\Gamma} ) = X^{\Gamma} \times X^{\Gamma}$, also for sydetically proximal relation we have $L(\mathcal{H}, X^{\Gamma}) = \{((x_{\alpha})_{\alpha \in \Gamma},(y_{\alpha})_{\alpha \in \Gamma}) \in X^{\Gamma} \times X^{\Gamma} : \{\gamma ∈ \Gamma :$ $x_{\gamma} \neq y_{\gamma}\}$ is finite$\}$.     

ON THE EXPECTED SAMPLE SIZES OF SOME POWER ONE TESTS FOR NORMAL MEAN WITH UNKNOWN VARIANCE

Suppose that $\[{x_1},{x_2}, \cdots \]$ are i i d. random variables on a probability space $\[(\Omega ,F,P)\]$ and $\[{x_1}\]$ is normally distributed with mean $\[\theta \]$ and variance $\[{\sigma ^2}\]$, both of which are unknown. Given $\[{\theta _0}\]$ and $\[0 < \alpha < 1\]$, we propose a concrete stopping rule T w. r. e.the $\[\{ {x_n},n \ge 1\} \]$ such that $$\[{P_{\theta \sigma }}(T < \infty ) \le \alpha \begin{array}{*{20}{c}} {for}&{\begin{array}{*{20}{c}} {all}&{\theta \le {\theta _0},\sigma > 0,} \end{array}} \end{array}\]$$ $$\[{P_{\theta \sigma }}(T < \infty ) = 1\begin{array}{*{20}{c}} {for}&{\begin{array}{*{20}{c}} {all}&{\theta > {\theta _0},\sigma > 0,} \end{array}} \end{array}\]$$ $$\[\mathop {\lim }\limits_{\theta \downarrow {\theta _0}} {(\theta - {\theta _0})^2}{({\ln _2}\frac{1}{{\theta - {\theta _0}}})^{ - 1}}{E_{\theta \sigma }}T = 2{\sigma ^2}{P_{{\theta _0}\sigma }}(T = \infty )\]$$ where $\[{\ln _2}x = \ln (\ln x)\]$.     

关于三圈图的拉普拉斯谱半径的一些结果

边数等于点数加二的连通图称为三圈图.~设 ~$\Delta(G)$~和~$\mu(G)$~
分别表示图~$G$~的最大度和其拉普拉斯谱半径,设${\mathcal
T}(n)$~表示所有~$n$~阶三圈图的集合,证明了对于~${\mathcal
T}(n)$~的两个图~$H_{1}$~和~$H_{2}$~,~若~$\Delta(H_{1})>
\Delta(H_{2})$ ~且 ~$\Delta(H_{1})\geq \frac{n+7}{2}$,~则~$\mu
(H_{1})> \mu (H_{2}).$ 作为该结论的应用,~确定了~${\mathcal
T}(n)(n\geq9)$~中图的第七大至第十九大的拉普拉斯谱半径及其相应的极图.     

Haddad和Helou定理的另一结论

Let K be a finite field of characteristic ≠ 2 and G the additive group of K × K. Let k_1, k_2 be integers not divisible by the characteristic p of K with(k_1, k_2) = 1. In 2004, Haddad and Helou constructed an additive basis B of G for which the number of representations of g ∈ G as a sum b_1+ b_2(b_1, b_2 ∈ B) is bounded by 18. For g ∈ G and B■G, let σk_1,k_2(B, g)be the number of solutions of g = k_1b_1 + k_2b_2, where b_1, b_2 ∈ B. In this paper, we show that there exists a set B ? G such that k_1 B + k2 B = G and σk_1,k_2(B, g)≤16.