关于算子的Orlicz-Hardy空间

设$L$为$L^2({{\mathbb R}^n})$上的线性算子且$L$生成的解析半群 $\{e^{-tL}\}_{t\ge 0}$的核满足Poisson型上界估计, 其衰减性由$\theta(L)\in(0,\infty)$刻画. 又设$\omega$为定义在$(0,\infty)$上的$1$-\!上型及临界 $\widetilde p_0(\omega)$-\!下型函数, 其中 $\widetilde p_0(\omega)\in (n/(n+\theta(L)), 1]$. 并记 $\rho(t)={t^{-1}}/\omega^{-1}(t^{-1})$, 其中$t\in (0,\infty).$ 本文引入了一类 Orlicz-Hardy空间 $H_{\omega,\,L}({\mathbb R}^n)$及 $\mathrm{BMO}$-\!型空间${\mathrm{BMO}_{\rho,\,L} ({\mathbb R}^n)}$, 并建立了关于${\mathrm{BMO}_{\rho,\,L}({\mathbb R}^n)}$函数的John-Nirenberg不等式及 $H_{\omega,\,L}({\mathbb R}^n)$与 $\mathrm{BMO}_{\rho,\,L^\ast}({\mathbb R}^n)$的对偶关系, 其中 $L^\ast$为$L$在$L^2({\mathbb R}^n)$中的共轭算子. 利用该对偶关系, 本文进一步获得了$\mathrm{BMO}_{\rho,\,L^\ast}(\rn)$的$\ro$-\!Carleson 测度特征及 $H_{\omega,\,L}({\mathbb R}^n)$的分子特征, 并通过后者建立了广义分数次积分算子 $L^{-\gamma}_\rho$从$H_{\omega,\,L}({\mathbb R}^n)$到 $H_L^1({\mathbb R}^n)$或$L^q({\mathbb R}^n)$的有界性, 其中$q>1$, $H_L^1({\mathbb R}^n)$为Auscher, Duong 和 McIntosh引入的Hardy空间. 如取$\omega(t)=t^p$,其中$t\in(0,\infty)$及$p\in(n/(n+\theta(L)), 1]$, 则所得结果推广了已有的结果.     

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).     

{C}^n$中单位球上Bergman型空间的一种积分算子

设$\omega_1,\omega_2$为正规函数, $\varphi$是$B_n$ 上的全纯自映射,$ g\in H(B_n)$ 满足 $g(0)=0$. 对所有的$0相似文献   

一类椭圆型随机偏微分方程弱解的存在性

设$D$是$R^N$ ($N>1$)中有界开集,$(\Omega, {\cal F}, P)$是一个完备的概率空间.该文研究了下列随机边值问题弱解的存在性问题\[\left\{\begin{array}{ll}-{\rm div} A(x,\omega,u, \nabla u)=f(x,\omega, u),\,\, &;(x,\omega)\in D\times \Omega,\\u=0, &;(x,\omega)\in \partial D\times \Omega,\end{array}\right.\]其中, div与 $\nabla $ 表示仅对 $x$求微分. 首先,作者引入了弱解的概念; 然后,作者转化随机问题为高维确定性问题;最后,作者证明了该问题弱解的存在性.     

Global Multi-Holder Estimate of Solutions to Elliptic Equations of Higher Order

In this paper the global multi-Holder estimate of solutions to general boundary value problem of elliptic equations of higher order is discussed. Let м be the solution of Pu=f of m-th order elliptic equation with Dirichlet conditions $D_n^iu=f_j,0\leq j \leq m/2-1$ where f\inC^r,\delta(\Omega),g_j\in C^{m-j+r,\delta}(\partial \Omega) with {0<\gamma =0,\delta>1} or {\gamma =1,\delta \leq 0}.Then u\inC^{m+[\tilde \gamma],[\tilde \delta]},where ([\tilde \gamma],[\tilde \delta])=(\gamma,\delta) if 0<\gamma <1 and \delta \in R^1,([\tilde \gamma],[\tilde \delta])=(\gamma,\delta -1) if \gamma=0,\delta >1 or \gamma =1,\delta \leq 0.Moreover,in the case \gamma =0 and 0\leq \delta <1,u\in C^(m-1)+1,\delta -1.     

THE SUFFICIENT AND NECESSARY CONDITIONS FOR NOETHER’S SOLVABILITY OF SINGULAR INTEGRAL EQUATIONS WITH TWO CARLEMAN’S SHIFTS

In this paper we consider the problem of solvability of singular integral equtions with two Carleman's shifts \[\begin{gathered} (\mathcal{K}\varphi )(t) \equiv {a_0}(t)\varphi (t) + {a_1}(t)\varphi [\alpha (t)] + {a_2}(t)\varphi [\beta (t)] + {a_3}(t)\varphi [\gamma (t)] \hfill \ + \frac{{{b_0}(t)}}{{\pi i}}\int_\Gamma {\frac{{\varphi (\tau )}}{{\tau - t}}} d\tau + \frac{{{b_1}(t)}}{{\pi i}}\int_\Gamma {\frac{{\varphi (\tau )}}{{\tau - \alpha (t)}}} d\tau + \frac{{{b_2}(t)}}{{\pi i}}\int_\Gamma {\frac{{\varphi (\tau )}}{{\tau - \beta (t)}}d\tau } \hfill \ + \frac{{{b_s}(t)}}{{\pi i}}\int_\Gamma {\frac{{\varphi (\tau )}}{{\tau - \gamma (t)}}} d\tau + \int_\Gamma {K(t,\tau )\varphi (\tau )d\tau = g(t){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (1,1)} \hfill \\ \end{gathered} \] Suppose that Г is a closed simple Lyapunoff's curve and \[\alpha (t)\], \[\beta (t)\] which satisfy Carleman's. conditions and \[\alpha [\beta (t)] = \beta [\alpha (t)]\] are two different homeomorphisms of Г onto itself, and that \[{a_k}(t),{b_k}(t)\], k = 0, 1, 2, 3 belong to the,space \[{H_\mu }(\Gamma ),g(t)\] belongs to the space \[{L_p}(\Gamma ),p > 1\]), p>l and \[K(t,\tau )\] has only weak singularity. The following main results are obtained: 1. Singular integral eqution (1.1) is solvable if and only if the Noether's conditions \[det(p(t) \pm q(t)) \ne 0\] are satisfied. 2. Index of sigular integral eqution (1.1) is calculated by the formula \[Ind{\kern 1pt} {\kern 1pt} {\kern 1pt} \mathcal{K} = \frac{1}{{8\pi }}{\{ arg\frac{{\det (p(t) - q(t))}}{{\det (p(t) + q(t))}}\} _\Gamma }\] where p(t) and q(t) are matrices of coeffioents of so-called corresponding system of equtions. All these results have been generalized for systems of singular integral equtions with two Carleman's shifts and complex conjugate of unknown functions.     

On Applications of Vector Measure to the Optimal Control Theory for Distributed Parameter Systems

Let X and Z be two reflexive Banach spaces, U\in Z and b(\cdot,\cdot):[t_0,T]*U\rightarrow X continuous. Suppose $x(t)\equiv x(t,u(\cdot))$ is a function from [t_0, T] into X , satisfying the distrbnted parameter system $dx(t)\dt=A(t)x(t)+b(t,u(t)),t_0+\int_t_0^T {+r(t,u(t))dt}$. We have proved the following theorem. Theorem. Suppose u^*(\cdot) is the optimal control function, $x^*(t)=x(t,u^*(\cdot))$ and $\psi (t)=-U'(T,t)Q_1x^*(T)-\int_t^T{U'(\sigma,t)Q(\sigma)x^*(\sigma)d\sigma}$, then the maximum principle $<\psi(t),b(t,u^*(t))>-1/2r(t,u^*(t))=\mathop {\max }\limits_{u \in U} {\psi (t),b(t,u)>-1/2r(t,u)}$ (16) holds for almost all t on [t_0, T ].     

平面图和系列平行图的无圈边染色

图$G$的正常边染色称为无圈的, 如果图$G$中不含2-色圈, 图$G$的无圈边色数用$a''(G)$表示, 是使图$G$存在正常无圈边染色所需要的最少颜色数. Alon等人猜想: 对简单图$G$, 有$a''(G)\leq{\Delta(G)+2}$. 设图$G$是围长为$g(G)$的平面图, 本文证明了: 如果$g(G)\geq3$, 则$a''(G)\leq\max\{2\Delta(G)-2,\Delta(G)+22\}$; 如果 $g(G)\geq5$, 则$a''(G)\leq{\Delta(G)+2}$; 如果$g(G)\geq7$, 则$a''(G)\leq{\Delta(G)+1}$; 如果$g(G)\geq16$并且$\Delta(G)\geq3$, 则$a''(G)=\Delta(G)$; 对系列平行图$G$, 有$a''(G)\leq{\Delta(G)+1}$.     

连续函数之集在上半连续函数之集中的一种拓扑位置

设$(X,\rho)$是一个度量空间. 用$\dd {\rm USCC}(X)$和$\dd {\rm CC}(X)$ 分别表示从$X$ 到 $\I=[0,1]$的紧支撑的上半连续函数和紧支撑的连续函数下方图形全体. 赋予 Hausdorff 度量后, 它们是拓扑空间. 文中证明了, 如果 $X$ 是一个无限的且孤立点集稠密的紧度量空间, 则 $(\dd {\rm USCC}(X),\dd {\rm CC}(X))\approx(Q,c_0\cup (Q\setminus \Sigma))$, 即存在一个同胚 $h:~\dd {\rm USCC}(X)\to Q$, 使得 $h(\dd {\rm CC}(X))=c_0\cup (Q\setminus \Sigma)$, 这里 $Q=[-1,1]^{\omega},\,\Sigma=\{(x_n)_{n}\in Q: {\rm sup}|x_n|<1\},\, c_0=\Big\{(x_n)_{n}\in \Sigma: \lim\limits_{n\to +\infty}x_n=0\Big\}.$ 结合这个论断和另一篇文章的结果, 可以得到: 如果 $X$ 是一个无限的紧度量空间, 则 $(\uscc(X), \cc(X))\approx \left\{ \begin{array}{ll} (Q,c_0\cup (Q\setminus \Sigma)), &;\quad \text{如 果 孤 立 点 集 在} X \text{中稠密},\\ (Q, c_0), &;\quad \text{ 其他}. \end{array} \right.$ 还证明了, 对一个度量空间$X$, $(\dd {\rm USCC}(X),\dd {\rm CC}(X))\approx (\Sigma,c_0)$ 当且仅当 $X$是一个非紧的、局部紧的、非离散的可分空间.     

一类推广的随机分形的 Hausdorff维数

该文研究一类推广的${\bf R}^{d}$中具有有限记忆的随机递归模型,引入了一个与该结构有关的函数$\Psi(\beta),\beta\geq 0$,构造了一个随机测度$\mu_\omega$,证明了由该结构产生的随机集 $K(\omega)$的Hausdorff维数是$\alpha:=\inf\{\beta:\Psi(\beta)\leq1\}$.     

ON TAYLOR'S CONJECTURE ABOUT THE PACKING MEASURES OF CARTESIAN PRODUCT SETS

1.IntroductionInthegeometryoffractals,Hausdorffmeasurealiddimensionplayaveryimportantrole.Olltheotherhand,therecelltilltroductionofpackingmeasureshasledtoagreaterunderstandillgofthegeometrictheoryoffractals,aspackingmeasuresbehaveillawnythatis'dual'toHausdoofmeasure8inmanyrespectsl2].Forexample,denotingHausdorffdimellsionandpackingdimensionbydimandDimrespectively,wehavedim(ExF)2dimE dimF,whileDim(ExF)5DimE DimF.Itiswell-kllowenthatifECRm,FCR",thenH(ExFW1T2)2b'H((E,W1)H(FW2)forsome…     

有循环极大子群的素数幂阶群的作用是边传递的图(II)

假定Γ是一个有限的、单的、无向的且无孤立点的图,G是Aut(Γ)的一个子群.如果G在Γ的边集合上传递,则称Γ是G-边传递图.我们完全分类了当G为一个有循环的极大子群的素数幂阶群时的G-边传递图.结果为:设图Γ含有一个阶为pn(p是素数,n≥2)的自同构群,且G有一个极大子群循环,则Γ是G-边传递的,当且仅当Γ同构于下列图之一1)pmK1,pn-1-m,0≤m≤n-1;2)pmK1,pn-m,0≤m≤n;3)pmKp,pn-m-1,0≤m≤n-2;4)pn-mCpm,pm≥3,m＜n;5)2n-2K1,1;6)pn-1-mCpm,pm≥3,m≤n-1;7)2pn-mCpm,pm≥3,m≤n-1;8)2pn-mK1,pm,0≤m≤n;9)pn-mK1,2pm,0≤m≤n;10)pn-mK2,pm,0＜m≤n;11)C(2pn-m,1,pm);12)pkC(2pm-k,1,pn-m),0＜k＜m,0＜m≤n;13)(t-s,2m)C(2m 1/(t-s,2m),1,2n-1-m),其中0≤m≤n-1,2n-2(s-1)≡0(mod 2m),t≡1(mod 2),s(≠)t(mod 2m),1≤s≤2m,1≤t≤2n-1;14)∪p i=1 Ci p n-1,其中Ci p n-1=Ca1a1 [1 (i-1)pn-2]a 1 2[1 (i--1)p n-2]…a 1 (pn-1-1)[1 (i-1)p n-2]≌Cp n-1,i=1,2,…,p;15)∪2 i=1 Ci 2n-1,其中Ci 2n-1=Ca1a 1 [1 (i-1)(2n-2-1)]a1 2[1 (i-1)(2n-2-1)]…a1 (2n-1-1)[1 (i-1)(2n-2-1)]≌C2n-1,i=1,2.     

${R}_{0}$代数中的滤子格

Abstract In the present paper, some basic properties of MP filters of Ro algebra M are investigated. It is proved that（FMP（M）,包含,′∧^-∨^-,{1},M）is a bounded distributive lattice by introducing the negation operator ′, the meet operator ∧^-, the join operator ∨^- and the implicati on operator → on the set FMP（M） of all MP filters of M. Moreover, some conditions under which （FMP（M）,包含,′∨^-,→{1},M）is an Ro algebra are given. And the relationship between prime elements of FMP （M） and prime filters of M is studied. Finally, some equivalent characterizations of prime elements of .FMP （M） are obtained.     

Refined asymptotic profiles for a semilinear heat equation

We study the large time behavior of the solutions of the Cauchy problem for a semilinear heat equation,

The behaviour of microstructures with small shears of the austenite-martensite interface in martensitic phase transformations

Let $\Omega \subset \Bbb{R}^2$ denote a bounded domain whose boundary $\partial \Omega$ is Lipschitz and contains a segment $\Gamma_0$ representing the austenite-twinned martensite interface. We prove $$\displaystyle{\inf_{{u\in \cal W}(\Omega)} \int_\Omega \varphi(\nabla u(x,y))dxdy=0}$$ for any elastic energy density $\varphi : \Bbb{R}^2 \rightarrow [0,\infty)$ such that $\varphi(0,\pm 1)=0$. Here ${\cal W}(\Omega)$ consists of all Lipschitz functions $u$ with $u=0$ on $\Gamma_0$ and $|u_y|=1$ a.e. Apart from the trivial case $\Gamma_0 \subset \reel \times \{a\},~a\in \Bbb{R}$, this result is obtained through the construction of suitable minimizing sequences which differ substantially for vertical and non-vertical segments.     

Universal approach to the Takesaki-Takai $\gamma $-duality for crossed products

In this article, we generalize and simplify the proof of the Takesaki-Takai $\gamma $-duality theorem. Assume a morphism \textbf{\textit{$\omega \; :\; G\to Aut\left({\rm A}\right)$}} is a projective representation of the locally compact Abel group \textbf{\textit{$G$}} in \textbf{\textit{$Aut\left({\rm A}\right)$}}, mapping $\gamma \; :\; G\to G$ is continuous, and $\left({\rm A},\; G,\; \omega \right)$ is a dynamic system then there exists isomorphism \[\Upsilon \; :\; Env_{\hat{\omega }} {}^{\gamma } \left(L^{1} \left(\hat{G},\; Env_{\omega } {}^{\gamma } \left(L^{1} \left(G,\; {\rm A}\right)\right)\right)\right)\to {\rm A}\otimes LK\left(L^{2} \left(G\right)\right) \] which is the equivariant for the double dual action \[\hat{\hat{\omega }}\; :\; G\to Aut\left(Env_{\hat{\omega }} {}^{\gamma } \left(L^{1} \left(\hat{G},\; Env_{\omega } {}^{\gamma } \left(L^{1} \left(G,\; {\rm A}\right)\right)\right)\right)\right).\] These results deepen our understanding of the representation theory and are especially interesting given their possible applications to problems of the quantum theory.     

Two results on arithmetical functions

Generalizing two results of Rieger [8] and Selberg [10] we give asymptotic formulas for sums of type $${\matrix {\sum \limits_{n\leq x}\cr n\equiv l({\rm mod}k)\cr f_{\kappa}(n)\equiv s_{\kappa}({\rm mod}p_{\kappa})\cr (\kappa=1,\dots,r)\cr}}\qquad \chi(n)\qquad {\rm and} {\matrix {\sum \limits_{n\leq x}\cr n\equiv l({\rm mod}k)\cr f_{\kappa}(n)\equiv s_{\kappa}({\rm mod}p_{\kappa})\cr (\kappa=1,\dots,r)\cr}}\qquad \chi(n),$$ where χ is a suitable multiplicative function, f₁,…, f _r are “small” additive, prime-independent arithmetical functions and k, l are coprime. The proofs are based on an analytic method which consists of considering the Dirichlet series generated by $ \chi(n)z_{1}^{f_{1}(n)}\cdot... \cdot z_{r}^{f_{r}(n)},z_{1}\dots z_{r} $ complex.     

On Certain Finite Semigroups of Order-decreasing Transformations I

Let $D_n $ (${\cal O}_n$) be the semigroup of all finite order-decreasing (order-preserving) full transformations of an $n$-element chain, and let $D(n,r) = \{\alpha\in D_n: |\mbox{Im}\alpha| \leq r\}$ (${\cal C}(n,r) = D(n,r)\cap {\cal O}_n)$ be the two-sided ideal of $D_n $ ($D_n \cap {\cal O}_n$). Then it is shown that for $r \geq 2$, the Rees quotient semigroup $DP_r(n)= D(n,r) / D(n,r-1)$ (${\cal C}P_r(n)= {\cal C}(n,r)/{\cal C} (n,r-1)$) is an ${\cal R}$-trivial (${\cal J}$-trivial) idempotent-generated 0*-bisimple primitive abundant semigroup. The order of ${\cal C}P_r(n)$ is shown to be $1+ \left(\begin{array}{c} n-1 \\ r-1 \end{array} \right) \left(\begin{array}{c} n \\ r \end{array} \right)/(n-r+1)$. Finally, the rank and idempotent ranks of ${\cal C}P_r(n)\,(r<n)$ are both shown to be equal to $\left(\begin{array}{c} n-1 \\ r-1 \end{array} \right)$.     

A Note on Randomly Weighted Sums of Dependent Subexponential Random Variables

The author obtains that the asymptotic relations■hold as x→∞,where the random weightsθ_1,···,θ_(n )are bounded away both from 0 and from∞with no dependency assumptions,independent of the primary random variables X_1,···,X_(n )which have a certain kind of dependence structure and follow non-identically subexponential distributions.In particular,the asymptotic relations remain true whenX_1,···,X_(n )jointly follow a pairwise Sarmanov distribution.     

Extremal Functions for Adams Inequalities in Dimension Four

Let $\Omega\subset \mathbb{R}^4$ be a smooth bounded domain, $W_0^{2,2}(\Omega)$ be the usual Sobolev space. For any positive integer $\ell$, $\lambda_{\ell}(\Omega)$ is the $\ell$-th eigenvalue of the bi-Laplacian operator. Define $E_{\ell}=E_{\lambda_1(\Omega)}\oplus E_{\lambda_2(\Omega)}\oplus\cdots\oplus E_{\lambda_{\ell}(\Omega)}$, where $E_{\lambda_i(\Omega)}$ is eigenfunction space associated with $\lambda_i(\Omega)$. $E^{\bot}_{\ell}$ denotes the orthogonal complement of $E_\ell$ in $W_0^{2,2}(\Omega)$. For $0\leq\alpha<\lambda_{\ell+1}(\Omega)$, we define a norm by $\|u\|_{2,\alpha}^{2}=\|\Delta u\|^2_2-\alpha \|u\|^2_2$ for $u\in E^\bot_{\ell}$. In this paper, using the blow-up analysis, we prove the following Adams inequalities$$\sup_{u\in E_{\ell}^{\bot},\,\| u\|_{2,\alpha}\leq 1}\int_{\Omega}e^{32\pi^2u^2}{\rm d}x<+\infty;$$moreover, the above supremum can be attained by a function $u_0\in E_{\ell}^{\bot}\cap C^4(\overline{\Omega})$ with $\|u_0\|_{2,\alpha}=1$. This result extends that of Yang (J. Differential Equations, 2015), and complements that of Lu and Yang (Adv. Math. 2009) and Nguyen (arXiv: 1701.08249, 2017).