关于$\mathcal{F}$-S-可补子群

设$\mathcal{F}$是一个群类. 群$G$的子群$H$称为在$G$中$\mathcal{F}$-S-可补的，如果存在$G$的一个子群$K$,使得$G=HK$且$K/K\cap{H_G}\in\mathcal{F}$, 其中$H_G=\bigcap_{g\in G}H^g$是包含在$H$中的$G$的最大正规子群．本文利用子群的$\mathcal{F}$-S-可补性, 给出了有限群的可解性, 超可解性和幂零性的一些新的刻画. 应用这些结果, 我们可以得到一系列推论, 其中包括有关已知的著名结果.     

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。     

Function Spaces in Lipschitz Domains and Optimal Rates of Convergence for Sampling

Assume that we want to recover $f : \Omega \to {\bf C}$ in the $L_r$-quasi-norm ($0 < r \le \infty$) by a linear sampling method $$ S_n f = \sum_{j=1}^n f(x^j) h_j , $$ where $h_j \in L_r(\Omega )$ and $x^j \in \Omega$ and $\Omega \subset {\bf R}^d$ is an arbitrary bounded Lipschitz domain. We assume that $f$ is from the unit ball of a Besov space $B^s_{pq} (\Omega)$ or of a Triebel--Lizorkin space $F^s_{pq} (\Omega)$ with parameters such that the space is compactly embedded into $C(\overline{\Omega})$. We prove that the optimal rate of convergence of linear sampling methods is $$ n^{ -{s}/{d} + ({1}/{p}-{1}/{r})_+} , $$ nonlinear methods do not yield a better rate. To prove this we use a result from Wendland (2001) as well as results concerning the spaces $B^s_{pq} (\Omega) $ and $F^s_{pq}(\Omega)$. Actually, it is another aim of this paper to complement the existing literature about the function spaces $B^s_{pq} (\Omega)$ and $F^s_{pq} (\Omega)$ for bounded Lipschitz domains $\Omega \subset {\bf R}^d$. In this sense, the paper is also a continuation of a paper by Triebel (2002).     

An Identity Relating Moments of Functionals of Convex Hulls

Denote by $K_n$ the convex hull of $n$ independent random points distributed uniformly in a convex body $K$ in $\R^d$, by $V_n$ the volume of $K_n$, by $D_n$ the volume of $K\backslash K_n$, and by $N_n$ the number of vertices of $K_n$. A well-known identity due to Efron relates the expected volume ${\it ED}_n$---and thus ${\it EV}_n$---to the expected number ${\it EN}_{n+1}$. This identity is extended from expected values to higher moments. The planar case of the arising identity for the variances provides in a simple way the corrected version of a central limit theorem for $D_n$ by Cabo and Groeneboom ($K$ being a convex polygon) and an improvement of a central limit theorem for $D_n$ by Hsing ($K$ being a circular disk). Estimates of $\var D_n$ ($K$ being a two-dimensional smooth convex body) and $\var N_n$ ($K$ being a $d$-dimensional smooth convex body, $d\geq 4$) are obtained. The identity for moments of arbitrary order shows that the distribution of $N_n$ determines ${\it EV}_{n-1}, {\it EV}_{n-2}^2,\dots, {\it EV}_{d+1}^{n-d-1}$. Reversely it is proved that these $n-d-1$ moments determine the distribution of $N_n$ entirely. The resulting formula for the probability that $N_n=k\ (k=d+1,\dots , n)$ appears to be new for $k\geq d+2$ and yields an answer to a question raised by Baryshnikov. For $k=d+1$ the formula reduces to an identity which has been repeatedly pointed out.     

Tractability of Approximation for Weighted Korobov Spaces on Classical and Quantum Computers

We study the approximation problem (or problem of optimal recovery in the $L_2$-norm) for weighted Korobov spaces with smoothness parameter $\a$. The weights $\gamma_j$ of the Korobov spaces moderate the behavior of periodic functions with respect to successive variables. The nonnegative smoothness parameter $\a$ measures the decay of Fourier coefficients. For $\a=0$, the Korobov space is the $L_2$ space, whereas for positive $\a$, the Korobov space is a space of periodic functions with some smoothness and the approximation problem corresponds to a compact operator. The periodic functions are defined on $[0,1]^d$ and our main interest is when the dimension $d$ varies and may be large. We consider algorithms using two different classes of information. The first class $\lall$ consists of arbitrary linear functionals. The second class $\lstd$ consists of only function values and this class is more realistic in practical computations. We want to know when the approximation problem is tractable. Tractability means that there exists an algorithm whose error is at most $\e$ and whose information cost is bounded by a polynomial in the dimension $d$ and in $\e^{-1}$. Strong tractability means that the bound does not depend on $d$ and is polynomial in $\e^{-1}$. In this paper we consider the worst case, randomized, and quantum settings. In each setting, the concepts of error and cost are defined differently and, therefore, tractability and strong tractability depend on the setting and on the class of information. In the worst case setting, we apply known results to prove that strong tractability and tractability in the class $\lall$ are equivalent. This holds if and only if $\a>0$ and the sum-exponent $s_{\g}$ of weights is finite, where $s_{\g}= \inf\{s>0 : \xxsum_{j=1}^\infty\g_j^s\,<\,\infty\}$. In the worst case setting for the class $\lstd$ we must assume that $\a>1$ to guarantee that functionals from $\lstd$ are continuous. The notions of strong tractability and tractability are not equivalent. In particular, strong tractability holds if and only if $\a>1$ and $\xxsum_{j=1}^\infty\g_j<\infty$. In the randomized setting, it is known that randomization does not help over the worst case setting in the class $\lall$. For the class $\lstd$, we prove that strong tractability and tractability are equivalent and this holds under the same assumption as for the class $\lall$ in the worst case setting, that is, if and only if $\a>0$ and $s_{\g} < \infty$. In the quantum setting, we consider only upper bounds for the class $\lstd$ with $\a>1$. We prove that $s_{\g}<\infty$ implies strong tractability. Hence for $s_{\g}>1$, the randomized and quantum settings both break worst case intractability of approximation for the class $\lstd$. We indicate cost bounds on algorithms with error at most $\e$. Let $\cc(d)$ denote the cost of computing $L(f)$ for $L\in \lall$ or $L\in \lstd$, and let the cost of one arithmetic operation be taken as unity. The information cost bound in the worst case setting for the class $\lall$ is of order $\cc (d) \cdot \e^{-p}$ with $p$ being roughly equal to $2\max(s_\g,\a^{-1})$. Then for the class $\lstd$ in the randomized setting, we present an algorithm with error at most $\e$ and whose total cost is of order $\cc(d)\e^{-p-2} + d\e^{-2p-2}$, which for small $\e$ is roughly $$ d\e^{-2p-2}. $$ In the quantum setting, we present a quantum algorithm with error at most $\e$ that uses about only $d + \log \e^{-1}$ qubits and whose total cost is of order $$ (\cc(d) +d) \e^{-1-3p/2}. $$ The ratio of the costs of the algorithms in the quantum setting and the randomized setting is of order $$ \frac{d}{\cc(d)+d}\,\left(\frac1{\e}\right)^{1+p/2}. $$ Hence, we have a polynomial speedup of order $\e^{-(1+p/2)}$. We stress that $p$ can be arbitrarily large, and in this case the speedup is huge.     

Noether’s problem for dihedral 2-groups

Let $K$ be any field and $G$ be a finite group. Let $G$ act on the rational function field $K(x_g: \, g \in G)$ by $K$-automorphisms defined by $g \cdot x_h= x _{gh}$ for any $g, \, h \in G$. Denote by $K(G)$ the fixed field $K(x_g: \, g \in G)^G$. Noethers problem asks whether $K(G)$ is rational (= purely transcendental) over $K$. We shall prove that $K(G)$ is rational over $K$ if $G$ is the dihedral group (resp. quasi-dihedral group, modular group) of order 16. Our result will imply the existence of the generic Galois extension and the existence of the generic polynomial of the corresponding group.     

ADMISSIBLE ESTIMATES IN THE IMPORTANT CLASS OF ESTIMATES OF THE COVARIANCE MATRIX

Let X1,…XN(where N&gt;m)be independent Nm(μ,∑）random vectors,and put X^-=1/N ∑i=1^N Xi and T‘T=A=∑i=1^N(Xi-X^-)(Xi-X^-)‘,where T is upper-triangular with positive diagonal elements.The author considers the problem of estimating ∑,and restricts his attention to the class of estimates D={T‘△^*T+Nb^*X^-X^‘&#183;△^* is any diagonal matrix and b^* is any nonnegative constant}because it has the following attractive features:(a)Its elements are all quadratic forms of the sufficient and complete statistics(X^-,T).(b)It contains all estimates of the form αA+NbX^-X^-‘(α≥0 and b≥0),which construct a complete subclass of the class of nonnegative quadratic estimates D^8={X‘BX:B≥0}(where X=(X1,…,XN)‘)for any strict convex loss function.(c)It contains all invariant estimates under the transformation group of upper-triangular matrices.The author obtains the characteristics for an estimate of the form.T‘△T+NbX^-X^-‘(△=diag{δ1,…，δm}≥0 and b≥0)of ∑ to be admissible in D when the loss function is chosen as tr(∑^-1∑-I)^2,and shows,by an example,that αA+NX^-X^-‘(α≥0 and b≥0)is admissible in D^* can not imply its admissibility in D.     

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.     

Distance Sets of Well-Distributed Planar Point Sets

We prove that a well-distributed subset of ${\Bbb R}^2$ can have a distance set $\Delta$ with $\#(\Delta\cap [0,N])\leq CN^{3/2-\epsilon}$ only if the distance is induced by a polygon $K$. Furthermore, if the above estimate holds with $\epsilon=\frac12$, then $K$ can have only finitely many sides.     

Hardy-Orlicz 空间到 Bloch-Orlicz 型空间上的复合算子

Let φ be an analytic self-map of D. The composition operator C_φ is the operator defined on H(D) by C_φ(f) = f ? φ. In this paper, we investigate the boundedness and compactness of the composition operator C_φ from Hardy-Orlicz spaces to Bloch-Orlicz type spaces.     

缺项算子矩阵的本性谱

Let =(A C X B)be a 2×2 operator matrix acting on the Hilbert space н( )κ.For given A ∈B (H),B ∈B(K)and C ∈B(K,H)the set Ux∈B(H,к)σe(Mx)is determined,where σe(T)denotes the essential spectrum.     

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.     

A Remark on the Norm of Integer Order Favard Spaces

For a generator $A$ of a $C_0$-semigroup $T(\cdot)$ on a Banach space $X$ we consider the semi-norm $M^{k}_x:=\limsup_{t\to 0+}\|t^{-1}(T(t)-I)A^{k-1}x\|$ on the Favard space ${\cal F}_{k}$ of order $k$ associated with $A$. The use of this semi-norm is motivated by the functional analytic treatment of time-discretization methods of linear evolution equations. We show that sharp inequalities for bounded linear operators on ${\cal D}(A^k)$ can be extended to the larger space ${\cal F}_{k}$ by using the semi-norm $M^{k}_{(\cdot)}$. We also show that $M^{k}_{(\cdot)}$ is a norm equivalent to the norms that are usually considered in the literature if A has a bounded inverse.     

Infinitely many low- and high-energy solutions for a class of elliptic equations with variable exponent

This paper is concerned with the $p(x)$-Laplacian equation of the form $$ \left\{\begin{array}{ll} -\Delta_{p(x)} u=Q(x)|u|^{r(x)-2}u, &\mbox{in}\ \Omega,\u=0, &\mbox{on}\ \partial \Omega, \end{array}\right. \eqno{0.1} $$ where $\Omega\subset\R^N$ is a smooth bounded domain, $1p^+$ and $Q: \overline{\Omega}\to\R$ is a nonnegative continuous function. We prove that (0.1) has infinitely many small solutions and infinitely many large solutions by using the Clark''s theorem and the symmetric mountain pass lemma.     

从$A^{p}_{\alpha}$到$A^{\infty}(\varphi)$的加权复合算子

设$u \in H(D), \ \phi$为$D$上的解析自映射,定义$H(D)$上的加权复合算子为$u C_{\phi}(f)=$$uf\circ\phi$, \ $f\in H(D)$.本文得到了从$A^{p}_{\alpha}$到$A^{\infty}(\varphi)\ (A_{0}^{\infty}(\varphi))$的加权复合算子$u C_{\phi}$的有界性和紧性的充要条件.     

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$\bigcap\limits_{C\in B(K,H)}\sigma_{p4}(M_C)$.     

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.     

Hilbert空间中的Bessel列构成的 $C^*$-代数 $B_H(I)$

Let H be a separable Hilbert space, B H(I), B(H) and K(H) the sets of all Bessel sequences {f i}i∈I in H, bounded linear operators on H and compact operators on H, respectively. Two kinds of multiplications and involutions are introduced in light of two isometric linear isomorphisms αH : B H(I) → B(?2), β : B H(I) → B(H), respectively, so that B H(I) becomes a unital C*-algebra under each kind of multiplication and involution. It is proved that the two C*-algebras(B H(I), ?, ?) and(B H(I), ·, *) are *-isomorphic. It is also proved that the set F H(I) of all frames for H is a unital multiplicative semi-group and the set R H(I) of all Riesz bases for H is a self-adjoint multiplicative group, as well as the set K H(I) := β-1(K(H)) is the unique proper closed self-adjoint ideal of the C*-algebra B H(I).     

广义超特殊p-群的自同构群Ⅲ

确定了广义超特殊p-群G的自同构群的结构.设|G|=p~(2n+m),|■G|=p~m,其中n≥1,m≥2,Aut_fG是AutG中平凡地作用在Frat G上的元素形成的正规子群,则(1)当G的幂指数是p~m时,(i)如果p是奇素数,那么AutG/AutfG≌Z_((p-1)p~(m-2)),并且AutfG/InnG≌Sp(2n,p)×Zp.(ii)如果p=2,那么AutG=Aut_fG(若m=2)或者AutG/AutfG≌Z_(2~(m-3))×Z_2(若m≥3),并且AutfG/InnG≌Sp(2n,2)×Z_2.(2)当G的幂指数是p~(m+1)时,(i)如果p是奇素数,那么AutG=〈θ〉■Aut_fG,其中θ的阶是(p-1)p~(m-1),且Aut_f G/Inn G≌K■Sp(2n-2,p),其中K是p~(2n-1)阶超特殊p-群.(ii)如果p=2,那么AutG=〈θ_1,θ_2〉■Aut_fG,其中〈θ_1,θ_2〉=〈θ_1〉×〈θ_2〉≌Z_(2~(m-2))×Z_2,并且Aut_fG/Inn G≌K×Sp(2n-2,2),其中K是2~(2n-1)阶初等Abel 2-群.特别地,当n=1时...     

RANDOM ITERAtION OF HOLOMORPHIC SELF-MAPS OVER BOUNDED DOMAINS IN C~N

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