Zalcman引理在随机迭代函数族动力系统中的应用

本文根据Schwick的思想，利用Zalcman引理讨论了随机迭代函数族动力系统，指出了函数族随机迭代动力系统的Fatou集和函数族衍生半群动力系统的Fatou集定义差别明显但却等价.并获得了如下正规定则，设F={f_i|f_i为C（C）上的非线性解析函数，i ∈ M}，其中M为非空指标集，Σ_M={（j₁，j₂，…，j_n，…）|j_i ∈ M，i ∈ N}，若对任意的指标序列σ=（j₁，j₂，…，j_n，…）∈ Σ_M，迭代序列{W_σⁿ=f_jn º f_jn-1 º … ºf_j1（z）|n ∈ N}在点z处正规，则函数族F本身在点z处正规.     

带非光滑核的多线性Marcinkiewicz积分的加权界

我们引入了带非光滑核的多线性Marcinkiewicz积分算子.设p_1,…,p_m∈(1,∞)和p∈(0,+∞)满足1/p_1+…+1/p_m=1/p,记P=(p_1,…,p_m),又设向量权ω=(ω_1,…,ω_m)∈A_p和v_ω=Π_(k=1)~mω_k~(p/pk),得到了Marcinkiewicz积分算子从L~(p_1)(ω_1)×…×L~(p_m)(ω_m)到L~p(v_ω)的常数界.     

一类新的求解P_*(κ)阵线性互补问题的多项式内点算法

利用核函数及其性质,对P_*(k)阵线性互补问题提出了一种新的宽邻域不可行内点算法.对核函数作了一些适当的改进,所以是不同于Peng等人介绍的自正则障碍函数.最后证明了算法具有近似O((1+2k)n3/4log(nμ~0)/ε)多项式复杂性,是优于传统的基于对数障碍函数求解宽邻域内点算法的复杂性.     

Interior-point algorithms for P_{*}(\kappa )-LCP based on a new class of kernel functions

In this paper, we propose interior-point algorithms for $P_* (\kappa )$ -linear complementarity problem based on a new class of kernel functions. New search directions and proximity measures are defined based on these functions. We show that if a strictly feasible starting point is available, then the new algorithm has $\mathcal{O }\bigl ((1+2\kappa )\sqrt{n}\log n \log \frac{n\mu ^0}{\epsilon }\bigr )$ and $\mathcal{O }\bigl ((1+2\kappa )\sqrt{n} \log \frac{n\mu ^0}{\epsilon }\bigr )$ iteration complexity for large- and small-update methods, respectively. These are the best known complexity results for such methods.     

On Levine–O'Sullivan's Sequence and Its Conjecture

Let q_1=1,and q_n be the largest value of(k+1)(n-q_k) for all integers 1≤k≤n-1with n≥2.The sequence Q={q1,q2,q3,...} is called Levine–O'Sullivan sequence.In this paper,we use the combinational and analysis skill and the mathematical induction to study the asymptotic properties of q_n,and give an interesting asymptotic formula for it.This solved a conjecture proposed by Professor Chen Yonggao.     

Jacobian-determinant equation in the critical Besov spaces

Let Ω be a bounded domain in R~n with smooth boundary. Here we consider the following Jacobian-determinant equation det u(x)=f(x),x∈Ω;u(x)=x,x∈?Ω where f is a function on Ω with min_Ω f = δ 0 and Ωf(x)dx = |Ω|. We prove that if f ∈B_(p1)~(np)(Ω) for some p∈(n,∞), then there exists a solution u ∈ B_(p1)~(np+1)(Ω)C~1(Ω) to this equation. On the other hand, we give a simple example such that u ∈ C_0~1(R~2, R~2) while detu does not lie in B_(p1)~(2p)(R~2) for any p∞.     

Reinhardt域上一类推广的Roper-Suffridge算子

研究一类推广的Roper-Suffridge算子F(z)=(f(z_1)+f′(z_1)∑_(k=2)~nakz_k~pk,f′(z)1)(~1/p2)z_2,…,f′(z_1)~(1/pn)z_n)′,证明该算子在复欧氏空间中的Reinhardt域Ω_(n,p2,%…,pn)={z=(z_1,…,z_n)∈C~n:|z_|~2+∑_(k=2)~n|zk|~(pk)1,Pk∈N~+,k=2,…,n}上分别保持α次的殆β型螺形性,α次的β型螺形性及强β型螺形性.     

P_*(κ)线性互补问题基于一类新核函数的大步校正内点算法(英文)

本文采用一簇新的核函数设计原始-对偶内点算法用于解决P*(κ)线性互补问题.通过利用一些优良、简洁的分析工具,证明该算法具有O(q(2κ+1)n1/p(logn)1+1/qlog(n/ε))迭代复杂性.     

振荡超奇性Hilbert变换的Sobolev有界性

主要研究R~n上沿曲线Γ(t)=(t~(p_1),t~(p_2),…,t~(p_n))的振荡超奇性Hilbert变换H_(n,α,β)=∫_0~1 f(x-Γ(t))e~(it-β)t~(-1-α),在Sobolev空间上的有界性,其中0p_1P_2…P_n,αβ0.证明了对于0γ(nα)/((n+1))(p_1+α),当|1/p-1/2|(β-(n+1)[α-(β+p_1)γ])/(2β)时,H_(n,α,β)是从L_γ~2(R~n))到L~2(R~n)的有界算子.特别地,当β≥(α-γp_1)/(γ+1/(n+1))等时,H_(n,α,β)是从L_γ~2(R~n)到L~2(R~n)的有界算子·     

von Neumann代数上的局部可乘Lie n-导子

A₁，…，A_n的（n-1）-换位子记为p_n（A₁，…，A_n）.令M是von Neumann代数，n ≥ 2是任意正整数，L：M → M是一个映射.本文证明了，若M不含I₁型中心直和项，且L满足L（p_n（A₁，…，A_n））=∑_k=1ⁿ p_n（A₁，…，A_k-1，L（A_k），A_k+1，…，A_n）对所有满足条件A₁A₂=0的A₁，A₂，…，A_n ∈ M成立，则L（A）=φ（A）+f（A）对所有A ∈ M成立，其中φ：M → M和f：M → Z（M）（M的中心）是两个映射，且满足φ在P_iMP_j上是可加导子，f（p_n（A₁，A₂，…，A_n））=0对所有满足A₁A₂=0的A₁，A₂，…，A_n ∈ P_iMP_j成立（1 ≤ i，j ≤ 2），P₁ ∈ M是core-free投影，P₂=I-P₁；若M还是因子且n ≥ 3，则L满足条件L（p_n（A₁，A₂，…，A_n））=∑_k=1ⁿ p_n（A₁，…，A_k-1，L（A_k），A_k+1，…，A_n）对所有满足A₁A₂A₁=0的A₁，A₂，…，A_n ∈ M成立当且仅当L（A）=φ（A）+h（A）I对所有A ∈ M成立，其中φ是M上的可加导子，h是M上的泛函且满足h（p_n（A₁，A₂，…，A_n））=0对所有满足条件A₁A₂A₁=0的A₁，A₂，…，A_n ∈ M成立.     

关于限制性m-位分拆函数(英文)

Let n,m,k be positive integers and b_m,k be the number of representations of n as n = m^a₀ + m^a₁ + ··· + m^a_j with 0 ≤ a₀ ≤ a₁ ≤···≤ a_j < k.In this note,we obtain some congruences and distribution properties of b_m,k.     

A large-update interior-point algorithm for convex quadratic semi-definite optimization based on a new kernel function

In this paper, we present a large-update interior-point algorithm for convex quadratic semi-definite optimization based on a new kernel function. The proposed function is strongly convex. It is not self-regular function and also the usual logarithmic function. The goal of this paper is to investigate such a kernel function and show that the algorithm has favorable complexity bound in terms of the elegant analytic properties of the kernel function. The complexity bound is shown to be $O\left( {\sqrt n \left( {\log n} \right)^2 \log \frac{n} {\varepsilon }} \right)$ . This bound is better than that by the classical primal-dual interior-point methods based on logarithmic barrier function and recent kernel functions introduced by some authors in optimization fields. Some computational results have been provided.     

球面稳定同伦群中非平凡元素γ_sξ_n

当p≥5, n≥0时,(i_1i_0)_*(h_n)∈Ext_■~(1,p~nq)(H~*K,Z_p)在Adams谱序列中是永久循环,并且收敛到π_(p~nq-1)K中的非零元.本文在此基础上,考虑了涉及第三希腊字母类乘积元素的收敛性,并且扩大了球面稳定同伦群中非平凡元素滤子s+1的取值范围,即当p+1 s+1 2p时,■_sh_n∈Ext_■~(s+1,t)(Z_p,Z_p)在Adams谱序列中是永久循环,并且收敛到π_(t-s-1)S中的非零元γ_sξ_n,其中p≥7, n≥3, t=p~nq+sp~2q+(s-1)pq+(s-2)q+s-3,q=2(p-1).     

Primal-Dual Interior-Point Algorithms for Semidefinite Optimization Based on a Simple Kernel Function

Interior-point methods (IPMs) for semidefinite optimization (SDO) have been studied intensively, due to their polynomial complexity and practical efficiency. Recently, J. Peng et al. introduced so-called self-regular kernel (and barrier) functions and designed primal-dual interior-point algorithms based on self-regular proximities for linear optimization (LO) problems. They also extended the approach for LO to SDO. In this paper we present a primal-dual interior-point algorithm for SDO problems based on a simple kernel function which was first presented at the Proceedings of Industrial Symposium and Optimization Day, Australia, November 2002; the function is not self-regular. We derive the complexity analysis for algorithms based on this kernel function, both with large- and small-updates. The complexity bounds are and , respectively, which are as good as those in the linear case. Mathematics Subject Classifications (2000) 90C22, 90C31.     

Sur les ensembles représentés par les partitions d'un entier n

Let n = n₁ + n₂ + … + n_j a partition Π of n. One will say that this partition represents the integer a if there exists a subsum n_il + n_i2 + … + n_il equal to a. The set (Π) is defined as the set of all integers a represented by Π. Let be a subset of the set of positive integers. We denote by p( ,n) the number of partitions of n with parts in , and by (( ,n) the number of distinct sets represented by these partitions. Various estimates for ( ,n) are given. Two cases are more specially studied, when is the set {1, 2, 4, 8, 16, …} of powers of 2, and when is the set of all positive integers. Two partitions of n are said to be equivalent if they represent the same integers. We give some estimations for the minimal number of parts of a partition equivalent to a given partition.     

基于全牛顿求解P*(κ)阵水平线性互补问题的内点算法

提出一种求解P_*(k)阵水平线性互补问题的全牛顿内点算法,全牛顿算法的优势在于每次迭代中不需要线性搜寻.当给定适当的中心路径邻域的阈值和更新势垒参数,证明算法中心邻域的全牛顿是局部二次收敛的,最后给出算法迭代复杂性O(√n)log(n+1+k)/εμ₀.     

Horizontal Growth of Harmonic Functions

We show that positive harmonic functions in the upper halfplane grow at most quadratically in horizontal bands. This bound is sharp in a sense to be specified, which, at least implies that there are examples growing as fast as any power under 2. These results are extended to positive harmonic functions in a half-space of R ^{n +1}, with points represented by ( x , y ), where x ∈R ⁿ , and y ∈R, the sharp maximum rate of growth being now ¦ x ¦ ^{n +1}. The case of Poisson integrals of functions in L_p ( dx /(1+(¦ x ¦)² )^{( n +1)/2}) is also taken up; the bound condition is then O (¦ x ¦^{( n +1)/ p} ).     

Justification of a Perturbation Approximation of the Klein–Gordon Equation

Consider the nonlinear wave equation
u_tt − γ ² u_xx + f(u) = 0
with the initial conditions
u ( x ,0) = εφ ( x ), u _t( x ,0) = εψ ( x ),
where f ( u ) is either of the form f ( u )= c ² u −σ u ^{2 s +1}, s =1, 2,…, or an odd smooth function with f '(0)>0 and | f '( u )|≤ C ₀².The initial data φ( x )∈ C ² and ψ( x )∈ C ¹ are odd periodic functions that have the same period. We establish the global existence and uniqueness of the solution u ( x ,  t ; ɛ), and prove its boundedness in x ∈ R and t >0 for all sufficiently small ɛ>0. Furthermore, we show that the error between the solution u ( x ,  t ; ɛ) and the leading term approximation obtained by the multiple scale method is of the order ɛ³ uniformly for x ∈ R and 0≤ t ≤ T /ɛ², as long as ɛ is sufficiently small, T being an arbitrary positive number.     

New parameterized kernel functions for linear optimization

Recent studies on the kernel function-based primal-dual interior-point algorithms indicate that a kernel function not only represents a measure of the distance between the iteration and the central path, but also plays a critical role in improving the computational complexity of an interior-point algorithm. In this paper, we propose a new class of parameterized kernel functions for the development of primal-dual interior-point algorithms for solving linear programming problems. The properties of the proposed kernel functions and corresponding parameters are investigated. The results lead to a complexity bounds of ${O\left(\sqrt{n}\,{\rm log}\,n\,{\rm log}\,\frac{n}{\epsilon}\right)}$ for the large-update primal-dual interior point methods. To the best of our knowledge, this is the best known bound achieved.     

A primal-dual large-update interior-point algorithm for <Emphasis Type="Italic">P</Emphasis><Subscript>*</Subscript>(<Emphasis Type="Italic">κ</Emphasis>)-LCP based on a new class of kernel functions

In this paper, we propose a large-update primal-dual interior point algorithm for P_*(κ)-linear complementarity problem. The method is based on a new class of kernel functions which is neither classical logarithmic function nor self-regular functions. It is determines both search directions and the proximity measure between the iterate and the center path. We show that if a strictly feasible starting point is available, then the new algorithm has \(o\left( {(1 + 2k)p\sqrt n {{\left( {\frac{1}{p}\log n + 1} \right)}^2}\log \frac{n}{\varepsilon }} \right)\)iteration complexity which becomes \(o\left( {(1 + 2k)\sqrt n log{\kern 1pt} {\kern 1pt} n\log \frac{n}{\varepsilon }} \right)\)with special choice of the parameter p. It is matches the currently best known iteration bound for P_*(κ)-linear complementarity problem. Some computational results have been provided.