Asymptotically finite dimensional pullback behaviour of non-autonomous PDEs

The asymptotic behaviour of general non-autonomous partial differential equations can be described using the concept of pullback attractor. This is, under suitable hypotheses, a time-dependent family of finite-dimensional compact sets. In this work we investigate how this finite-dimensional dynamics on the attractor determines the asymptotic behaviour of non-autonomous PDEs.     

MgNet: A unified framework of multigrid and convolutional neural network

We develop a unified model, known as MgNet, that simultaneously recovers some convolutional neural networks(CNN) for image classification and multigrid(MG) methods for solving discretized partial differential equations(PDEs). This model is based on close connections that we have observed and uncovered between the CNN and MG methodologies. For example, pooling operation and feature extraction in CNN correspond directly to restriction operation and iterative smoothers in MG, respectively. As the solution space is often the dual of the data space in PDEs, the analogous concept of feature space and data space(which are dual to each other) is introduced in CNN. With such connections and new concept in the unified model, the function of various convolution operations and pooling used in CNN can be better understood. As a result,modified CNN models(with fewer weights and hyperparameters) are developed that exhibit competitive and sometimes better performance in comparison with existing CNN models when applied to both CIFAR-10 and CIFAR-100 data sets.     

Global random attractors are uniquely determined by attracting deterministic compact sets

It is shown that for continuous dynamical systems an analogue of the Poincaré recurrence theorem holds for Ω-limit sets. A similar result is proved for Ω-limit sets of random dynamical systems (RDS) on Polish spaces. This is used to derive that a random set which attracts every (deterministic) compact set has full measure with respect to every invariant probability measure for theRDS. Then we show that a random attractor coincides with the Ω-limit set of a (nonrandom) compact set with probability arbitrarily close to one, and even almost surely in case the base flow is ergodic. This is used to derive uniqueness of attractors, even in case the base flow is not ergodic. Entrata in Redazione il 10 marzo 1997.     

Asymptotic regularity conditions for the strong convergence towards weak limit sets and weak attractors of the 3D Navier-Stokes equations

The asymptotic behavior of solutions of the three-dimensional Navier-Stokes equations is considered on bounded smooth domains with no-slip boundary conditions and on periodic domains. Asymptotic regularity conditions are presented to ensure that the convergence of a Leray-Hopf weak solution to its weak ω-limit set (weak in the sense of the weak topology of the space H of square-integrable divergence-free velocity fields with the appropriate boundary conditions) are achieved also in the strong topology. It is proved that the weak ω-limit set is strongly compact and strongly attracts the corresponding solution if and only if all the solutions in the weak ω-limit set are continuous in the strong topology of H. Corresponding results for the strong convergence towards the weak global attractor of Foias and Temam are also presented. In this case, it is proved that the weak global attractor is strongly compact and strongly attracts the weak solutions, uniformly with respect to uniformly bounded sets of weak solutions, if and only if all the global weak solutions in the weak global attractor are strongly continuous in H.     

Equi-attraction and backward compactness of pullback attractors for point-dissipative Ginzburg-Landau equations

A new concept of an equi-attractor is introduced, and defined by the minimal compact set that attracts bounded sets uniformly in the past, for a non-autonomous dynamical system. It is shown that the compact equi-attraction implies the backward compactness of a pullback attractor. Also, an eventually equi-continuous and strongly bounded process has an equi-attractor if and only if it is strongly point dissipative and strongly asymptotically compact. Those results primely strengthen the known existence result of a backward bounded pullback attractor in the literature. Finally, the theoretical criteria are applied to prove the existence of both equi-attractor and backward compact attractor for a Ginzburg-Landau equation with some varying coefficients and a backward tempered external force.     

Evidence-theory-based numerical algorithms of attribute reduction with neighborhood-covering rough sets

Covering rough sets generalize traditional rough sets by considering coverings of the universe instead of partitions, and neighborhood-covering rough sets have been demonstrated to be a reasonable selection for attribute reduction with covering rough sets. In this paper, numerical algorithms of attribute reduction with neighborhood-covering rough sets are developed by using evidence theory. We firstly employ belief and plausibility functions to measure lower and upper approximations in neighborhood-covering rough sets, and then, the attribute reductions of covering information systems and decision systems are characterized by these respective functions. The concepts of the significance and the relative significance of coverings are also developed to design algorithms for finding reducts. Based on these discussions, connections between neighborhood-covering rough sets and evidence theory are set up to establish a basic framework of numerical characterizations of attribute reduction with these sets.     

Attractors for stochastic lattice dynamical systems with a multiplicative noise

In this paper, we consider a stochastic lattice differential equation with diffusive nearest neighbor interaction, a dissipative nonlinear reaction term, and multiplicative white noise at each node. We prove the existence of a compact global random attractor which, pulled back, attracts tempered random bounded sets.      

On attractors and stability for semigroup actions and control systems

The present paper is dedicated to the study of various aspects of attraction and stability for semigroup actions on topological spaces. The main purpose is to present the connections among the distinct notions of attractors and stable sets. The concept of Conley attractor is also investigated and related to the other notions of attractors. All the results are applied to the theory of control systems.     

用模糊Galois联络对L-模糊粗糙集进行公理化刻画

粗糙集理论是由Pawlak提出的一种表示与处理数据表中信息的形式化工具.作为粗糙集概念的推广,一种基于完备剩余格的L-模糊粗糙集已由Radzikowska与Kerre提出,在本文中,我们第一次借助于L-模糊Galois联络对L-模糊粗糙集进行了公理化刻画.由于L-模糊粗糙集及L-模糊Galois联络均为相应经典情形的推广,故本文的结论对于经典粗糙集来说也是成立的,这就意味着通过Galois联络可将经典粗糙集乃至L-模糊粗糙集的公理化统一起来.     

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.     

Lauwerier映射的奇怪吸引子的遍历性及其结构

本又考虑Ｌａｕｗｅｒｉｅｒ映射Ｆａ,ｂ（ｘ,ｙ）＝（ｂｘ（１－２ｙ）＋ｙ,ａｙ（１－ｙ））．我们证明对于参数α在一个正测度集合中,对应的映射有非平凡的拓扑可迁的吸引子,其中是某个双曲不动点的不稳定集的闭包．周期点是双曲的且在中稠密,而且中任两个周期点异宿相关（稳定与不稳定集的横截相交）．同时也构造支撑在吸引子上的Ｓｉｎａｌ－Ｂｏｗｅｎ－Ｒｕｅｌｌｅ测度,并研究其性质．     

Bounds for a new chaotic system and its application in chaos synchronization

This paper has investigated the localization problem of compact invariant sets of a new chaotic system with the help of the iteration theorem and the first order extremum theorem. If there are more iterations, then the estimation for the bound of the system will be more accurate, because the shape of the chaotic attractor is irregular. We establish that all compact invariant sets of this system are located in the intersection of a ball with two frusta and we also compute its parameters. It is a great advantage that we can attain a smaller bound of the chaotic attractor compared with the classical method. One numerical example illustrating a localization of a chaotic attractor is presented as well.     

Coexistence for species sharing a predator

A class of equations describing the dynamics of two prey sharing a common predator are considered. Even though the boundary and internal dynamics can exhibit oscillatory behavior, it is shown these equations are permanent if only if they admit a positive equilibrium. Going beyond permanence, a subclass of equations are constructed that are almost surely permanent but not permanent; there exists an attractor in the positive orthant that attracts Lebesgue almost every (but not every) initial condition.     

具有粘弹性和热粘弹性方程组的全局周期吸引子

证明了具有粘弹性和热粘弹性方程组在Dirichlet边界条件下,对于任意的非自治时间周期受迫力,均具有唯一的指数吸引任何有界集的周期解,即全局周期吸引子.并且如果受迫力是自治的,则全局周期吸引子恰是系统唯一的指数吸引有界集的平衡解.     

依零测度的Fuzzy集运算

给出了Fuzzy集合依零测度的包含,依零测度的相等。依零测度的并．交,补运算等。并补充地研究了这些基本运算,获得了若干好的性质。Fuzzy集依零测度的基本运算比标准fuzzy集合论的相应运算更有人情味,更为接近人的智能机理。有望在智能信息处理和软计算领域取得好的应用。     

Random attractors for the stochastic Benjamin-Bona-Mahony equation on unbounded domains

We prove the existence of a compact random attractor for the stochastic Benjamin-Bona-Mahony equation defined on an unbounded domain. This random attractor is invariant and attracts every pulled-back tempered random set under the forward flow. The asymptotic compactness of the random dynamical system is established by a tail-estimates method, which shows that the solutions are uniformly asymptotically small when space and time variables approach infinity.     

The liapunov dimension of strange attractors

Many papers have been published recently on studies of dynamical processes in which the attracting sets appear quite strange. In this paper the question of estimating the dimension of the attractor is addressed. While more general conjectures are made here, particular attention is paid to the idea that if the Jacobian determinant of a map is greater than one and a ball is mapped into itself, then generically, the attractor will have positive two-dimensional measure, and most of this paper is devoted to presenting cases with such Jacobians for which the attractors are proved to have non-empty interior.     

General random perturbations of hyperbolic and expanding transformations

Small random perturbations of a general form of diffeomorphisms having hyperbolic invariant sets and expanding maps are considered. The convergence of invariant measures of perturbations to the Sinaî-Bowen-Ruelle measure in the case of a hyperbolic attractor and to the smooth invariant measure in the expanding case are proved. The convergence of corresponding entropy characteristics and the approximation of the topological pressure by means of perturbations is considered as well.     

On the structures and dimensions of Moran sets

The Moran sets and the Moran class are defined by geometric fashion that distinguishes the classical self_similar sets from the following points:(i) The placements of the basic sets at each step of the constructions can be arbitrary.(ii) The contraction ratios may be different at each step.(iii) The lower limit of the contraction ratios permits zero.The properties of the Moran sets and Moran class are studied, and the Hausdorff, packing and upper Box_counting dimensions of the Moran sets are determined by net measure techniques. It is shown that some important properties of the self_similar sets no longer hold for Moran sets.