黎曼流形上不变集的分形维数估计

本文利用整体Lyapunov指数给出了光滑黎曼流形上C1-映射不变集的分形维数的上界.     

分形插值函数图象的Hausdorff维数

本文给出了插值点为的分形插值函数图象的Hausdorff维数的下界估计     

稠密的s维分形集

对任意给定的0≤s≤1,本文构造Cantor型集E_s,使dim_H E_s=s,且E_s在[0,1]内稠密。     

A New Type of Reduced Dimension Path Following Methods

To solve $F(x)=0$ numerically, we first prove that there exists a tube-like neighborhood around the curve in $R^n$ defined by the Newton homotopy in which $F(x)$ possesses some good properties. Then in this neighborhood, we set up an algorithm which is numerically stable and convergent. Since we can ensure that the iterative points are not far from the homotopy curve while computing, we need not apply the predictor-corrector which is often used in path following methods.     

Construction and Dimension Analysis for a Class of Fractal Functions

In this paper, we construct a class of nowhere differentiable continuous functions by means of the Cantor series expression of real numbers. The constructed functions include some known nondifferentiable functions, such as Bush type functions. These functions are fractal functions since their graphs are in general fractal sets. Under certain conditions, we investigate the fractal dimensions of the graphs of these functions, compute the precise values of Box and Packing dimensions, and evaluate the Hausdorff dimension. Meanwhile, the Holder continuity of such functions is also discussed.     

Fractal Dimension of Random Attractors for Non-autonomous Fractional Stochastic Reaction-diffusion Equations

This paper dealswith non-autonomous fractional stochastic reaction-diffusion equations driven by multiplicative noise with s ∈ (0,1). We first present some conditions for estimating the boundedness of fractal dimension of a random invariant set. Then we establish the existence and uniqueness of tempered pullback random attractors. Finally, the finiteness of fractal dimension of the random attractors is proved.     

Probabilistic Methods for Decomposition Dimension of Graphs

In a graph G, the distance from an edge e to a set FE(G) is the vertex distance from e to F in the line graph L(G). For a decomposition of E(G) into k sets, the distance vector of e is the k-tuple of distances from e to these sets. The decomposition dimension dec(G) of G is the smallest k such that G has a decomposition into k sets so that the distance vectors of the edges are distinct. For the complete graph K _n and the k-dimensional hypercube Q _k , we prove that (2–o(1))lgndec(K _n )(3.2+o(1))lgn and k/lgk dec(Q _k ) (3.17+o(1))k/lgk. The upper bounds use probabilistic methods directly or indirectly. We also prove that random graphs with edge probability p such that p n ^1– for some positive constant have decomposition dimension (lnn) with high probability. Acknowledgments.The authors thank Noga Alon for clarifying and strengthening the results in Sections 3 and 4. Thanks also go to a referee for repeated careful readings and suggestions.AMS classifications: 05C12, 05C35, 05D05, 05D40     

Weierstrass型函数图象的分形维数

考虑函数f(x)=sum from i=1 to ∞(?)~(-1)φ((?) θ_n)和w(x)=sum from n=1 to ∞(?)φ_(?)((?)x θ_(?)),式中0<α<(?)是任意实数,在一定条件下,估计了函数f图象的Hausdorff维数的下界,并求得了w函数图象的Box维数和Packing维数。     

A Comparison of Methods for Estimating the Extremal Index

The extremal index, (01), is the key parameter when extending discussions of the limiting behavior of the extreme values from independent and identically distributed sequences to stationary sequences. As measures the limiting dependence of exceedances over a threshold u, as u tends to the upper endpoint of the distribution, it may not always be informative about the extremal dependence at levels of practical interest. Therefore we also consider a threshold-based extremal index, (u). We compare the performance of a range of different estimators for and (u) covering processes with < 1 and = 1. We find that the established methods for estimating actually estimate (u), so perform well only when (u) . For Markov processes, we introduce an estimator which is as good as the established methods when (u) but provides an improvement when (u) < = 1. We illustrate our methods using simulated data and daily rainfall measurements.     

分形维数排序农业生产结构的影响因素

自然资源、科学技术生产力、人口数量、市场需求、政府宏观调控5个因素是影响农业生产结构变动的主要因素,使用分形维数方法排序它们的影响力,以期为农业生产结构调整方案的制定,提供依据和参考.     

无穷格点上长波-短波共振方程组核截面的分形维数估计

该文证明了无穷格点上长波-短波共振方程组核截面的分形维数估计.     

Fractal Dimension of Random Attractors for Non-autonomous Fractional Stochastic Ginzburg–Landau Equations

This paper considers the dynamical behavior of solutions for non-autonomous stochastic fractional Ginzburg–Landau equations driven by additive noise with α∈(0, 1). First, we give some conditions for bounding the fractal dimension of a random invariant set of non-autonomous random dynamical system. Second, we derive uniform estimates of solutions and establish the existence and uniqueness of tempered pullback random attractors for the equation in H. At last, we prove the finiteness of fractal dimension of random attractors.     

Klein-Gordon-Schr(o)dinger格点系统全局吸引子的分形维数

本文利用估计Hilbert空间中紧子集的分形维数的准则,得到了Klein-Gordon-Schr(o)dinger格点系统的全局吸引子的分形维数的一个上界.     

An Anthology on the Homology of Simulation with Artificial Intelligence

Simulation and artificial intelligence (AI) can be related in the following ways: the methodological similarities, the actual and potential uses of AI concepts in simulation, expert systems in simulation environments, and the gains for AI when applying ideas from simulation. This paper highlights the personal experiences of the author by discussing, for each of the four above-mentioned areas, a topic or a computer system in which he is directly involved. To highlight the similarities, it will be shown that the three-phase simulation model can be considered as a production-system model. A natural-language understanding system that was used in the definition stage of a simulation project will provide the basis for discussing AI concepts in simulation environments. An expert system used in the development and verification stages will provide an insight into the application of expert systems in simulation modelling. An expert-system development tool which is partially based on simulation techniques will be used to outline the gains for AI.     

Dimension of the Fractal Curve in Plane and Its Derivative of the Fractional Order

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拟蒙特卡罗方法用于估计多元回归函数

拟蒙特卡罗（QMC）方法被广泛用于解决数值分析和统计学中的各种问题，比如数值积分，最优化，试验设计，随机过程的模拟等．本文研究该方法在估计多元回归函数中的应用．证明了，在相当一般的条件下，均匀设计（或者，“代表点设计”）与回归函数傅里叶系数的QMC估计（对应地，使用拟随机重要性抽样的QMC估计）一起，构成一个回归函数的渐近最优投影估计．     

A New Upper Bound for the Dimension of Trace Codes

We shall derive a new non-trivial upper bound for the dimensionof trace codes connected to algebraic-geometric codes. Furthermore,we shall deduce their true dimension if certain conditions aresatisfied.     

大数据背景下人工智能发展对大学数学教学的启示

大数据时代,以数据为载体的人工智能技术渗透到社会的各行各业,而人工智能技术的发展越来越依赖应用数学学科的进步.在此背景下,大学数学教学改革成为必需,以适应社会新时代的发展.结合自身的教学改革实践,探讨如何从基本的大学数学内容出发,进行拓展,向学生介绍前沿的人工智能技术.从而,增加数学课程教学的趣味性和实用性,加深学生对数学思想方法在人工智能领域应用理解.     

Fractal Dimension of Graph of Weierstrass Function and Its Derivative of the Fractional Order

In this paper,we obtain the fractal dimension of the graph of the Weierstrass function, its derivative of the fractional order and the relation between the dimension and the order of the fractional derivative.     

基于人工智能的黄金价格非线性组合预测

黄金价格时间序列数据具有较强的非线性,在采用单模型对其预测时较多采用线性模型、非线性模型以及加入外生变量的非线性模型.但是,单一模型较难全面体现黄金数据的非线性特征,因此,预测效果不很理想.在利用线性方法进行模型组合时若被组合模型与原数据序列无线性关系,此时采用线性组合预测效果较差,甚至组合后的模型预测精度低于被组合的单模型精度.为充分发挥单一模型的优势,采用人工智能的方法建立非线性组合预测模型,模型可有效利用各模型特点,预测精度优于采用的各单一模型.