代数图递归迭代函数系与开集条件（英文）

本文研究R上一类代数图递归迭代函数系的开集条件与代数参数β之间的关系.我们证明若图递归迭代函数系满足开集条件且递归图是几何型的,则βs必是一个代数整数,其中s为图递归迭代函数系不变集的最大的Hausdorff维数.     

约束优化问题的一类光滑罚算法的全局收敛特性

对约束优化问题给出了一类光滑罚算法.它是基于一类光滑逼近精确罚函数 l_p(p\in(0,1]) 的光滑函数 L_p 而提出的.在非常弱的条件下, 建立了算法的一个摄动定理, 导出了算法的全局收敛性.特别地, 在广义Mangasarian-Fromovitz约束规范假设下, 证明了当 p=1 时, 算法经过有限步迭代后, 所有迭代点都是原问题的可行解; p\in(0,1) 时,算法经过有限迭代后, 所有迭代点都是原问题可行解集的内点.     

广义有限型的注解

本文讨论了满足广义有限型的迭代函数系.首先构造了一个满足广义有限型条件的迭代函数系,证明了当且仅当不变集为(0,1)区间子集时它才是基本集.随后证明了当压缩比的指数是不可公度时,R~d上任何迭代函数系在指标套{A_k}_(k=0)~∞。下均不满足广义有限型条件.最后构造了一类具有广义有限型条件的自相似集,同时给出它们的Hausdorff维数.     

约束优化问题的一类光滑罚算法的全局收敛特性（英文）

对约束优化问题给出了一类光滑罚算法.它是基于一类光滑逼近精确罚函数l_p(p∈(0,1])的光滑函数L_p而提出的.在非常弱的条件下,建立了算法的一个摄动定理,导出了算法的全局收敛性.特别地,在广义Mangasarian-Fromovitz约束规范假设下,证明了当p=1时,算法经过有限步迭代后,所有迭代点都是原问题的可行解;当p∈(0,1)时,算法经过有限迭代后,所有迭代点都是原问题可行解集的内点.     

有重叠的自相似集的Hausdorff维数

对于一类满足一定条件的相似压缩迭代函数系生成的不变集,本文证明了一个计算其 Hausdorff 维数的简单公式.该公式是通过把满足所给条件的迭代函数系联系到一个非重叠的无穷迭代函数系,然后利用 Moran 的计算无穷迭代函数系生成的不变集的 Hausdorff维数的方法得到的. 该方法可以应用于一些不满足 Ngai 和 Wang引进的有限型条件的迭代函数系.     

基于拟相对内部集值优化问题弱有效元的最优性条件

本文研究了基于拟相对内部的非凸集值优化问题弱有效元的最优性条件.首先,讨论了弱有效元与线性子空间之间的关系,利用涉及拟相对内部的凸集分离定理,获得了弱有效元的最优性条件.其次,给出了基于拟相对内部弱有效元的Lagrange乘子定理.     

集值向量优化的二阶最优性条件

对集值映射引入了高阶Clarke导数,给出了判别集值向量优化所有效性的二阶Kuhn-Tucker条件,并且,借助于集值映射的强(弱)伪凸性给出了一个弱有效解的充分条件.     

求可测函数的总体极小值方法的收敛性分析

[1]中提出了求解连续函数f(x)总体极小值的均值算法，并证明了算法的全局收敛性．若假设f(t)是定义在某可测集G上的可测函数，本证明了均值算法产生的迭代序列全局收敛到f(t)的本质极小值，若进一步假设函数f(t)满足测度Lipschitz条件，还证明了求可测函数的均值算法是线性收敛的．     

一类锥约束变分不等式问题的最优性条件

利用像空间分析法,本文研究了带锥约束的变分不等式的最优性条件.利用Gerstewitz非线性标量化函数,给出了三个非线性弱分离函数、两个非线性正则弱分离函数和一个非线性强分离函数.然后,利用此分离函数,得到了带锥约束的变分不等式的弱或强的最优性条件.     

R上自相似迭代的表示的等价性

讨论自相似迭代系统的等价性,在满足开集条件与等缩条件的情况下,除了平凡情形之外,我们给出了迭代系统的等价性的一个十分简洁的充要条件。     

Absolute continuity of vector-valued self-affine measures

This paper is devoted to the absolute continuity of (scalar-valued or vector-valued) self-affine measures and their properties on the boundary of an invariant set. We first extend the definition of WSC to self-affine IFS, and then obtain a necessary and sufficient condition for the vector-valued self-affine measures to be absolutely continuous with respect to the Lebesgue measure. In addition, we prove that, for any IFS and any invariant open set V, the corresponding (scalar-valued or vector-valued) invariant measure is supported either in V or in ∂V.     

On the structures of generating iterated function systems of Cantor sets

A generating IFS of a Cantor set F is an IFS whose attractor is F. For a given Cantor set such as the middle-3rd Cantor set we consider the set of its generating IFSs. We examine the existence of a minimal generating IFS, i.e. every other generating IFS of F is an iterating of that IFS. We also study the structures of the semi-group of homogeneous generating IFSs of a Cantor set F in under the open set condition (OSC). If dim_HF<1 we prove that all generating IFSs of the set must have logarithmically commensurable contraction factors. From this Logarithmic Commensurability Theorem we derive a structure theorem for the semi-group of generating IFSs of F under the OSC. We also examine the impact of geometry on the structures of the semi-groups. Several examples will be given to illustrate the difficulty of the problem we study.     

Ruelle operator with nonexpansive IFS on the line

Ruelle operator defined by weakly contractive iterated function systems (IFS) satisfying the open set condition was discussed in the paper [K.S. Lau, Y.L. Ye, Ruelle operator with nonexpansive IFS, Studia Math. 148 (2001) 143-169]. There, one of our theorems gave a sufficient condition for the possession of the Perron-Frobenius property. In this paper we consider Ruelle operator defined by nonexpansive IFS on the line instead of by weakly contractive one. And we prove, under the same condition, that the newly defined Ruelle operator has the Perron-Frobenius property. It extends the Ruelle-Perron-Frobenius theorem partially to the nonexpansive IFS.     

Graphs induced by iterated function systems

For an iterated function system (IFS) of similitudes, we define two graphs on the representing symbolic space. We show that if the self-similar set \(K\) has positive Lebesgue measure or the IFS satisfies the weak separation condition, then the graphs are hyperbolic; moreover the hyperbolic boundaries are homeomorphic to the self-similar sets.     

Vector-valued Ruelle operator with weakly contractive IFS

The vector-valued Ruelle operator defined by contractive iterated function systems (IFS) was discussed by the author [Y.L. Ye, Vector-valued Ruelle operator, J. Math. Anal. Appl. 299 (2004) 341-356]. In this paper we consider vector-valued Ruelle operators defined by weakly contractive IFS. And, a vector-valued analogue of the Ruelle-Perron-Frobenius theorem for the scalar Ruelle operator is set up. Our theorem gives a sufficient condition for the vector-valued Ruelle operator to be quasi-compact. Under this sufficient condition, we prove that the rate of convergence of the iterated operators is exponential.     

Invariant measures for parabolic IFS with overlaps and random continued fractions

We study parabolic iterated function systems (IFS) with overlaps on the real line. An ergodic shift-invariant measure with positive entropy on the symbolic space induces an invariant measure on the limit set of the IFS. The Hausdorff dimension of this measure equals the ratio of entropy over Lyapunov exponent if the IFS has no ``overlaps.' We focus on the overlapping case and consider parameterized families of IFS, satisfying a transversality condition. Our main result is that the invariant measure is absolutely continuous for a.e. parameter such that the entropy is greater than the Lyapunov exponent. If the entropy does not exceed the Lyapunov exponent, then their ratio gives the Hausdorff dimension of the invariant measure for a.e. parameter value, and moreover, the local dimension of the exceptional set of parameters can be estimated. These results are applied to a family of random continued fractions studied by R. Lyons. He proved singularity above a certain threshold; we show that this threshold is sharp and establish absolute continuity for a.e. parameter in some interval below the threshold.

     

Remarks on limit sets of infinite iterated function systems

If an iterated function system (IFS) is finite, it is well known that there is a unique non-empty compact invariant set K and that K?=???(I ^??), where ?? is the coding map. For an infinite IFS, there are two different sets generalising K, namely ??(I ^??) and its closure ${\overline{\pi(I^\infty)}}$ . In this paper we investigate the relations between these sets and their Hausdorff dimensions. In particular, we show how to construct an IFS for any pair of prescribed dimensions for ??(I ^??) and ${\overline{\pi(I^\infty)}\setminus \pi(I^\infty)}$ . Moreover, we investigate a set which depends only on the first iteration of an IFS, and characterise its relation to the abovementioned sets. This also extends and clarifies recent results by Mihail and Miculescu, who investigated the coding map for an infinite IFS and a condition for this map to be onto. Finally, we study the special case of one-dimensional IFS and show that in terms of the relations of the abovementioned sets these systems exhibit some very special features which do not generalise to higher dimensional situations.     

Self-similar sets satisfying the common point property

A self-similar set is a fixed point of iterated function system (IFS) whose maps are similarities. We say that a self-similar set satisfies the common point property if the intersection of images of the attractor under the maps of the IFS is a singleton and this point has a common pre-image, under the maps of the IFS, and the pre-image is in the attractor.Self-similar sets satisfying the common point property were introduced in Sirvent (2008) in the context of space-filling curves. In the present article we study some basic topological and dynamical properties of self-similar sets satisfying the common point property. We show examples of this family of sets.We consider attractors of a sub-IFS, an IFS formed from the original IFS by removing some maps. We put conditions on this attractors for having the common point property, when the original IFS have this property.     

直觉模糊集的扩展原理

基于文[1]给出的直觉模糊集的截集、分解定理和表现定理,利用模糊集的扩展原理,本文建立了直觉模糊集的扩展原理.首先,给出了直觉模糊集的扩展原理及其等价形式;其次,讨论了直党模糊集的扩展原理的有关性质;最后,研究了复合函数的直觉模糊集扩展原理及其性质.     

Separation conditions for conformal iterated function systems

We extend both the weak separation condition and the finite type condition to include finite iterated function systems (IFSs) of injective C ¹ conformal contractions on compact subsets of . For conformal IFSs satisfying the bounded distortion property, we prove that the finite type condition implies the weak separation condition. By assuming the weak separation condition, we prove that the Hausdorff and box dimensions of the attractor are equal and, if the dimension of the attractor is α, then its α-dimensional Hausdorff measure is positive and finite. We obtain a necessary and sufficient condition for the associated self-conformal measure μ to be singular. By using these we give a first example of a singular invariant measure μ that is associated with a non-linear IFS with overlaps. The authors are supported in part by an HKRGC grant.