关于渐近非扩张映象不动点迭代的一点注记

设E是一致凸Banach空间,C是E的非空闭凸子集,T:C→C是具有不动点的渐近非扩张映象．该文证明了在某些适当的条件下,由下列修改了的Ishikawa迭代程序所定义的序列{xn}=xn 1=rpn,pn=(1-an)xn anTmn ryn un,yn=(1-bn)xn bnTkn xn vn, (n≥1)弱收敛到t的不动点．     

ERGODIC THEOREMS FOR NON-LIPSCHITZIAN SEMIGROUPS WITHOUT CONVEXITY

§１．ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔＨｂｅａＨｉｌｂｅｒｔｓｐａｃｅｗｉｔｈｎｏｒｍ‖·‖ａｎｄｉｎｎｅｒｐｒｏｄｕｃｔ（·，·）ａｎｄｌｅｔＣｂｅａｎｏｎｅｍｐｔｙｓｕｂｓｅｔｏｆＨ．ＡｍａｐｐｉｎｇＴ：Ｃ｜→ＣｉｓｓａｉｄｔｏｂｅＬｉｐｓｃｈｉｔ...     

Banach空间中伪压缩映象不动点的迭代逼近

Let K be a nonempty closed convex subset of a real p-uniformly convex Banach space E and T be a Lipschitz pseudocontractive self-mapping of K with F（T） ：= {x ∈ K：Tx=x}≠φ. Let a sequence {xn} be generated from x1 ∈ K by xn＋1 = anxn,＋ bnTyn＋＋ cnun, yn= a′nxn~ ＋ b′nTx,＋ c′n,un, for all integers n ≥ 1. Then ‖xn - Txn,‖ → 0 as n→∞. Moreover, if T is completely continuous, then {xn} converges strongly to a fixed point of T.     

Solutions to the $\sigma_k$-Loewner-Nirenberg Problem on Annuli are Locally Lipschitz and Not Differentiable

We show for $k \geq 2$ that the locally Lipschitz viscosity solution to the $\sigma_k$-Loewner-Nirenberg problem on a given annulus $\{a < |x| < b\}$ is $C^{1,\frac{1}{k}}_{\rm loc}$ in each of $\{a < |x| \leq \sqrt{ab}\}$ and $\{\sqrt{ab} \leq |x| < b\}$ and has a jump in radial derivative across $|x| = \sqrt{ab}$. Furthermore, the solution is not $C^{1,\gamma}_{\rm loc}$ for any $\gamma > \frac{1}{k}$. Optimal regularity for solutions to the $\sigma_k$-Yamabe problem on annuli with finite constant boundary values is also established.     

关于伪压缩映象不动点迭代的一点注记

设$K$是自反的并且具有一致Gateaux可微范数的Banach空间$E$的非空有界闭凸子集.设$T:K\rightarrow K$是一致连续的伪压缩映象.假设$K$的每一非空有界闭凸子集对非扩张映象具有不动点性质.设$\{\lambda_n\}$是$(0,\frac{1}{2}]$中序列满足: (i) $\lim_{n\rightarrow \infty}\lambda_n=0$; (ii) $\sum_{n=0}^{\infty}\lambda_n=\infty$.任给$x_1\in K$,定义迭代序列$\{x_n\}$为:$x_{n+1}=(1-\lambda_n)x_n+\lambda_nTx_n-\lambda_n(x_n-x_1),n\geq 1.$若$\lim_{n\rightarrow \infty}\|x_n-Tx_n\|=0$, 则上述迭代产生的$\{x_n\}$强收敛到$T$的不动点.     

划分围长至少为6 的平面图为有界大小的分支

图$G$的$(\mathcal{O}_{k_1}, \mathcal{O}_{k_2})$-划分是将$V(G)$划分成两个非空子集$V_{1}$和$V_{2}$, 使得$G[V_{1}]$和$G[V_{2}]$分别是分支的阶数至多$k_1$和$k_2$的图.在本文中,我们考虑了有围长限制的平面图的点集划分问题,使得每个部分导出一个具有有界大小分支的图.我们证明了每一个围长至少为6并且$i$-圈不与$j$-圈相交的平面图允许$(\mathcal{O}_{2}$, $\mathcal{O}_{3})$-划分,其中$i\in\{6,7,8\}$和$j\in\{6,7,8,9\}$.     

On the construction of frames for Triebel-Lizorkin and Besov spaces

We present a general method for construction of frames for Triebel-Lizorkin and Besov spaces, whose nature can be prescribed. In particular, our method allows for constructing frames consisting of rational functions or more general functions which are linear combinations of a fixed (small) number of shifts and dilates of a single smooth and rapidly decaying function such as the Gaussian . We also study the boundedness and invertibility of the frame operator on Triebel-Lizorkin and Besov spaces and give necessary and sufficient conditions for the dual system to be a frame as well.

     

自反Banach空间中非扩张非自映射的粘滞迭代逼近方法

主要在自反和严格凸的且具有一致G(a)teaux可微范数的Banach空间中研究了非扩张非自映射的粘滞迭代逼近过程,证明了此映射的隐格式与显格式粘滞迭代序列均强收敛到它的某个不动点.     

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 $\[{\ln _2}x = \ln (\ln x)\]$.     

Subspace Arrangements of Type B n and D n

Let ${\mathcal{D}}_{n,k} $ be the family of linear subspaces of ?ⁿ given by all equations of the form $\varepsilon _1 x_{i_1 } = \varepsilon _2 x_{i_2 } = \cdot \cdot \cdot \varepsilon _k x_{i_k } ,$ for 1 ≤ < ? ? ? < i _k ≤ i and $\left( {\varepsilon _1 ,...,\varepsilon _k } \right)\varepsilon \left\{ { + 1, - 1} \right\}^k $ Also let ${\mathcal{B}}_{n,k,h} $ be ${\mathcal{D}}_{n,k} $ enlarged by the subspaces $x_{j_1 } = x_{j_2 } = \cdot \cdot \cdot x_{j_h } = 0,$ for 1 ≤. The special cases ${\mathcal{B}}_{n,2,1} $ and ${\mathcal{D}}_{n,2} $ are well known as the reflection hyperplane arrangements corresponding to the Coxeter groups of type B _nand D _n respectively. In this paper we study combinatorial and topological properties of the intersection lattices of these subspace arrangements. Expressions for their Möbius functions and characteristic polynomials are derived. Lexicographic shellability is established in the case of ${\mathcal{B}}_{n,k,h,} 1 \leqslant h < k$ , which allows computation of the homology of its intersection lattice and the cohomology groups of the manifold $\begin{gathered} {\mathcal{D}}_{n,2} \\ M_{n,k,h,} = {\mathbb{R}}^n \backslash \bigcup {{\mathcal{B}}_{n,k,h,} } \\ \end{gathered} $ . For instance, it is shown that $H^d \left( {M_{n,k,k - 1} } \right)$ is torsion-free and is nonzero if and only if d = t(k ? 2) for some $t,0 \leqslant t \leqslant \left[ {{n \mathord{\left/ {\vphantom {n k}} \right. \kern-0em} k}} \right]$ . Torsion-free cohomology follows also for the complement in ?ⁿof the complexification ${\mathcal{B}}_{n,k,h}^C ,1 \leqslant h < k$ .     

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$x^{*}$.     

具误差隐格式迭代逼近严格伪压缩映像族公共不动点 (英)

设$K$是实Banach空间$E$中非空闭凸集, $\{T_i\}_i=1^{N}$是$N$个具公共不动点集$F$的严格伪压缩映像, $\{\alpha_n\}\subset [0,1]$是实数列, $\{u_n\}\subset K$是序列, 且满足下面条件 (i)\ 设$K$是实Banach空间$E$中非空闭凸集, $\{T_i\}_i=1^{N}$是$N$个具公共不动点集$F$的严格伪压缩映像, $\{\alpha_n\}\subset [0,1]$是实数列, $\{u_n\}\subset K$是序列, 且满足下面条件 (i)\ 设$K$是实Banach空间$E$中非空闭凸集, $\{T_i\}_i=1^{N}$是$N$个具公共不动点集$F$的严格伪压缩映像, $\{\alpha_n\}\subset [0,1]$是实数列, $\{u_n\}\subset K$是序列, 且满足下面条件 (i)\ 设K是实Banach空间E中非空闭凸集，{Ti}i=1^N是N个具公共不动点集F的严格伪压缩映像，{αn}包括于[0,1]是实数例，{un}包括于K是序列，且满足下面条件（i）0〈α≤αn≤1;（ii）∑n=1∞（1-αn）=＋∞.（iii）∑n=1∞ ‖un‖〈＋∞.设x0∈K,{xn}由正式定义xn=αnxn-1＋（1-αn）Tnxn＋un-1,n≥1,其中Tn=Tnmodn，则下面结论（i）limn→∞‖xn-p‖存在，对所有p∈F；（ii）limn→∞d（xn,F）存在，当d（xn,F）=infp∈F‖xn-p‖；（iii）lim infn→∞‖xn-Tnxn‖=0.文中另一个结果是，如果{xn}包括于[1-2^-n,1]，则{xn}收敛，文中结果改进与扩展了Osilike（2004）最近的结果，证明方法也不同。     

Approximate Representation of Bergman Submodules

In the present paper, the author shows that if a homogeneous submodule M of the Bergman module L_a~2(B_d) satisfies P_M-sum from i to ( M_(zi)P_MM*_(zi))≤c/(N + 1)P_M for some number c 0, then there is a sequence {f_j } of multipliers and a positive number c such that c'P_M ≤sum from j to ( M_(fj)M*_(fj))≤ P_M, i.e., M is approximately representable. The author also proves that approximately representable homogeneous submodules are p-essentially normal for p d.     

涉及无限族非扩张映象$\{T_n\}_{n=1}^\infty$的迭代算法$x_{n 1}=\alpha_{n 1}f(x_n) (1-\alpha_{n 1})T_{n 1}x_n$

Under the framework of uniformly smooth Banach spaces, Chang proved in 2006 that the sequence {xn} generated by the iteration xn＋1 =αn＋1f（xn） ＋ （1 - αn＋1）Tn＋1xn converges strongly to a common fixed point of a finite family of nonexpansive maps {Tn}, where f ： C → C is a contraction. However, in this paper, the author considers the iteration in more general case that {Tn} is an infinite family of nonexpansive maps, and proves that Chang＇s result holds still in the setting of reflexive Banach spaces with the weakly sequentially continuous duality mapping.     

一类带齐次核的奇异重积分算子的范数及其应用

设核函数K(u,v)具有对称性和齐次性,对如下定义的奇异重积分算子T:(Tf)(y)=∫R_+~n K(‖x‖α,‖y‖α)f(x)dx,y∈R_+~n,其中‖x‖α=(x_1~α+…+x_n~α)~1/α(α>0),研究了T的范数及其应用.     

图$(\bigcup\limits_{i{\in}A}P_i){\bigcup}(\bigcup\limits_{j{\in}B}U_j)$伴随唯一的充要条件

设$h(G; x) =h(G)$和$[G]_h$分别表示图$G$的伴随多项式和伴随等价类. 文中给出了$[G]_h$的一个新应用. 利用$[G]_h$, 给出了图$H{\;}(H \cong G)$伴随唯一的充要条件, 其中$H=(\bigcup_{i{\in}A}P_i){\bigcup}(\bigcup_{j{\in}B}U_j)$, $A \subseteq A^{'}=\{1,2,3,5\} \bigcup \{2n|n \in N, n \geq 3\}$, $B \subseteq B^{'}     

自伴算子空间上保持Jordan积投影的线性映射

Let Bs(H) be the real linear space of all self-adjoint operators on a complex Hilbert space H with dim H ≥ 2.It is proved that a linear surjective map on Bs (H) preserves the nonzero projections of Jordan products of two operators if and only if there is a unitary or an anti-unitary operator U on H such that (X)=λU XU,X∈Bs(H) for some constant λ with λ∈{1,1}.     

r-log-concavity of partition functions

The Ramanujan Journal - Let $$\hat{\mathscr {L}}$$ be the operator given by $$\hat{\mathscr {L}} \{a_n\}_{n \ge 0} = \{a_{n+1}^2 - a_{n} a_{n+2} \}_{n \ge 0}$$ . A sequence $$\{ a_n \}_{n \ge 0}$$...     

Pointwise estimates for Lagrange interpolation polynomials

Let f ∈ C[?1, 1]. Let the approximation rate of Lagrange interpolation polynomial of f based on the nodes $ \left\{ {\cos \frac{{2k - 1}} {{2n}}\pi } \right\} \cup \{ - 1,1\} $ be Δ _{n + 2}(f, x). In this paper we study the estimate of Δ _{n + 2}(f,x), that keeps the interpolation property. As a result we prove that $$ \Delta _{n + 2} (f,x) = \mathcal{O}(1)\left\{ {\omega \left( {f,\frac{{\sqrt {1 - x^2 } }} {n}} \right)\left| {T_n (x)} \right|\ln (n + 1) + \omega \left( {f,\frac{{\sqrt {1 - x^2 } }} {n}\left| {T_n (x)} \right|} \right)} \right\}, $$ where T _n (x) = cos (n arccos x) is the Chebeyshev polynomial of first kind. Also, if f ∈ C ^r [?1, 1] with r ≧ 1, then $$ \Delta _{n + 2} (f,x) = \mathcal{O}(1)\left\{ {\frac{{\sqrt {1 - x^2 } }} {{n^r }}\left| {T_n (x)} \right|\omega \left( {f^{(r)} ,\frac{{\sqrt {1 - x^2 } }} {n}} \right)\left( {\left( {\sqrt {1 - x^2 } + \frac{1} {n}} \right)^{r - 1} \ln (n + 1) + 1} \right)} \right\}. $$