一个集值映射的不动点定理

得到了从度量空间X(不必是完备的)到X的非空有界子集族B(X)的集值映射的一些不动点,其结果推广了一些已有的结论.     

局部Lipschitz方程解的叠代逼近

设X=L_p或l_p(p≥2),T:D(T)→X是局部Lipschitz和严格增殖算子.本文给出非线性方程T_x=y的解的叠代逼近并讨论了局部Lipschitz和严格伪收缩映射不动点的叠代逼近.     

Lipschitz局部强增殖算子的非线性方程的解的迭代构造

本文研究p一致光滑Banach空间X中Ishikawa迭代法.设T:X→K是Lipschitz局部强增殖算子,方程T_x=f的解集sol(T)非空.我们证明了sol(T)是一个单点集且Ishikawa序列强收敛到方程T_x=f的唯一解.另行,当T是从X的非空凸子集K到X的Lipschitz局部伪压缩映像且T的不动点集F(T)非空时,我们证明了F(T)是一个单点集且Ishikawa序列强收敛到T的唯一不动点.我们的结果改进和推广了[4]与[5]的结果.     

Banach空间中含强增生算子的非线性方程的迭代解

设X为实Banach空间,X*为其一致凸的共轭空间.设T:X→X为Lipschitzian强增生映象,L≥1为其Lipschitzian常数,k∈(0,1)为其强增生常数.设{α_n},{β_n}为[0,1]中的两个实数列满足:(ⅰ)α_n→0(n→∞);(ⅱ)β_n<L(1+L)/k(1-k)(n≥0);(ⅲ).假设为X中两序列满足:=o(β_n)与μ_n→0(n→∞).任取x₀∈X,则由(IS)1x_n+1=(1-α_n)x_n+α_nSy_n+u_ny_n=(1-β_n)x_n+β_nSx_n+μ_n(_n≥0){所定义的迭代序列{x_n强收敛于方程T     

二维振荡流动中污染云团的收缩*

当垂向扩散时间尺度与流动的周期相当时,在转流过程中,污染云团将会出现收缩.这时水平剪切分散导数将会出现负值奇性.本文根据作者两维延迟扩散方程[7]: 其中u(t),v (t)为深度平均水平速度.导出X(t,τ),Y(t,τ)坐标位移,D_ij(t,τ)为剪切扩散导数的方程.一般情况下,?D_ij(t,τ)/?τ是正的.不存在奇异性.但在转流的初期.记忆函数D_ij(t,τ)就有可能是负的.本文给出了D_ij和X、Y的解析表示式.     

高速扩展平面应力裂纹尖端的各向异性塑性场

在裂纹尖端的应力分量都只是θ的函数的条件下,利用定常运动方程,Hill各向异性屈服条件及应力应变关系,我们得到高速扩展平面应力裂纹尖端的各向异性塑性场的一般解.将这个一般解用于四种各向异性特殊情形,我们就导出这四种特殊情形的一般解.最后,本文给出X=Y=Z情形的高速扩展平面应力Ⅰ型裂纹尖端的各向异性塑性场.     

吸引子的逼近

本文考虑耦合振子系统及对其按时间离散化(时间长为h)得到的离散动力系统.证明在一定条件下,这两个系统都有一维整体吸引子l和l_h.进一步证明当时间步长h→0时,l_h→l.     

广义符号动力系统的混沌性

本文给出了符号动力系统的一般数学模型,它是离散时空系统的一种特殊情形.在现有离散时空系统的混沌概念和研究方法的基础上.研究了这类广义符号动力系统的混沌性,得到了一类在Devaney意义下新的广义符号混沌动力系统,从而推广了现有符号动力系统混沌性的研究范围.     

乘积拓扑动力系统的转移不变集

研究一般拓扑动力系统的复杂性是很困难的,拓扑共轭、拓扑半共轭、嵌入映射和转移不变集都可以不同程度保持动力系统的复杂性.通过研究拓扑动力系统与符号动力系统拓扑共轭,找到了拓扑动力(子)系统存在转移不变集的条件,同时证明了拓扑动力系统存在转移不变集与乘积拓扑动力系统存在转移不变集的关系,通过研究相对简单、直观的符号动力系统,间接的反应一般拓扑动力(子)系统与其乘积拓扑动力系统的动力性状.     

离散变量结构优化设计的组合算法*

本文首先给出了离散变量优化设计局部最优解的定义,然后提出了一种综合的组合算法.该算法采用分级优化的方法,第一级优化首先采用计算效率很高且经过随机抽样性能实验表明性能较高的启发式算法─—相对差商法,求解离散变量结构优化设计问题近似最优解 X ;第二级采用组合算法,在 X 的离散邻集内建立离散变量结构优化设计问题的(-1,0.1)规划模型,再进一步将其化为(0,1)规划模型,应用定界组合算法或相对差商法求解该(0,1)规划模型,求得局部最优解.解决了采用启发式算法无法判断近似最优解是否为局部最优解这一长期未得到解决的问题,提高了计算精度,同时,由于相对差商法的高效率与高精度,以上综合的组合算法的计算效率也还是较高的.     

Trajectory Segmentation and Symbolic Representation of Dynamics of Delayed Recurrent Inhibitory Neural Loops

We develop a general symbolic dynamics framework to examine the dynamics of an analogue of the integrate-and-fire neuron model of recurrent inhibitory loops with delayed feedback, which incorporates the firing procedure and absolute refractoriness. We first show that the interaction of the delay, the inhibitory feedback and the absolute refractoriness can generate three basic types of oscillations, and these oscillations can be pinned together to form interesting coexisting periodic patterns in the case of short feedback duration. We then develop a natural symbolic dynamics formulation for the segmentation of a typical trajectory in terms of the basic oscillatory patterns, and use this to derive general principles that determine whether a periodic pattern can and should occur.     

On the expressiveness and decidability of o-minimal hybrid systems

This paper is driven by a general motto: bisimulate a hybrid system by a finite symbolic dynamical system. In the case of o-minimal hybrid systems, the continuous and discrete components can be decoupled, and hence, the problem reduces in building a finite symbolic dynamical system for the continuous dynamics of each location. We show that this can be done for a quite general class of hybrid systems defined on o-minimal structures. In particular, we recover the main result of a paper by G. Lafferriere, G.J. Pappas, and S. Sastry, on o-minimal hybrid systems. We also provide an analysis and extension of results on decidability and complexity of problems and constructions related to o-minimal hybrid systems.     

Level sets of multiple ergodic averages

We propose to study multiple ergodic averages from multifractal analysis point of view. In some special cases in the symbolic dynamics, the Hausdorff dimensions of the level sets for the limit of these multiple ergodic averages are determined by using Riesz products.     

Application of Gray codes to the study of the theory of symbolic dynamics of unimodal maps

In this paper we provide a closed mathematical formulation of our previous results in the field of symbolic dynamics of unimodal maps. This being the case, we discuss the classical theory of applied symbolic dynamics for unimodal maps and its reinterpretation using Gray codes. This connection was previously emphasized but no explicit mathematical proof was provided. The work described in this paper not only contributes to the integration of the different interpretations of symbolic dynamics of unimodal maps, it also points out some inaccuracies that exist in previous works.     

Transverse disks, symbolic dynamics, homology direction vectors, and Thurston-Nielson theory

We review some properties of transverse disks and use symbolic dynamics to determine rotation vectors from the return map to a transverse disk. We also prove connections between the symbolic dynamics and Nielson equivalence of orbits.     

Computation of symbolic dynamics for one-dimensional maps

In this paper we design and implement rigorous algorithms for computing symbolic dynamics for piecewise-monotone-continuous maps of the interval. The algorithms are based on computing forwards and backwards approximations of the boundary, discontinuity and critical points. We explain how to handle the discontinuities in the symbolic dynamics which occur when the computed partition element boundaries are not disjoint. The method is applied to compute the symbolic dynamics and entropy bounds for the return map of the singular limit of a switching system with hysteresis and the forced Van der Pol equation.     

传递子集的性质及其在符号动力系统中的应用

讨论了传递子集的一些性质,并且应用这些性质研究符号动力系统的弱混合子集和传递子集之间的关系,给出了符号动力系统的传递子集是弱混合子集的一个充分条件.     

Intrinsic ergodicity beyond specification: β-shifts, S-gap shifts, and their factors

We give sufficient conditions for a shift space (Σ, σ) to be intrinsically ergodic, along with sufficient conditions for every subshift factor of Σ to be intrinsically ergodic. As an application, we show that every subshift factor of a β-shift is intrinsically ergodic, which answers an open question included in Mike Boyle’s article “Open problems in symbolic dynamics”. We obtain the same result for S-gap shifts, and describe an application of our conditions to more general coded systems. One novelty of our approach is the introduction of a new version of the specification property that is well adapted to the study of symbolic spaces with a non-uniform structure.     

Computation of the normal forms for general M-DOF systems using multiple time scales. Part I: autonomous systems

This paper is concerned with the symbolic computation of the normal forms of general multiple-degree-of-freedom oscillating systems. A perturbation technique based on the method of multiple time scales, without the application of center manifold theory, is generalized to develop efficient algorithms for systematically computing normal forms up to any high order. The equivalence between the perturbation technique and Poincaré normal form theory is proved, and general solution forms are established for solving ordered perturbation equations. A number of cases are considered, including the non-resonance, general resonance, resonant case containing 1:1 primary resonance, and combination of resonance with non-resonance. “Automatic” Maple programs have been developed which can be executed by a user without knowing computer algebra and Maple. Examples are presented to show the efficiency of the perturbation technique and the convenience of symbolic computation. This paper is focused on autonomous systems, and non-autonomous systems are considered in a companion paper.     

Inversion of general tridiagonal matrices

In the current work, the authors present a symbolic algorithm for finding the inverse of any general nonsingular tridiagonal matrix. The algorithm is mainly based on the work presented in [Y. Huang, W.F. McColl, Analytic inversion of general tridiagonal matrices, J. Phys. A 30 (1997) 7919–7933] and [M.E.A. El-Mikkawy, A fast algorithm for evaluating nth order tridiagonal determinants, J. Comput. Appl. Math. 166 (2004) 581–584]. It removes all cases where the numeric algorithm in [Y. Huang, W.F. McColl, Analytic inversion of general tridiagonal matrices, J. Phys. A 30 (1997) 7919–7933] fails. The symbolic algorithm is suited for implementation using Computer Algebra Systems (CAS) such as MACSYMA, MAPLE and MATHEMATICA. An illustrative example is given.