A Unified Lorenz-Like System and Its Tracking Control

This paper introduces the finding of a unified Lorenz-like system.By gradually tuning the only parameter d,the reported system belongs to Lorenz-type system in the sense defined by Clikovsky.Meanwhile,this system belongs to Lorenz-type system,Lu-type system,Chen-type system with d less than,equivalent to and greater than 1.5,respectively,according to the classification defined by Yang.However,this system can only generate a succession of Lorenz-like attractors.Some basic dynamical properties of the system are investigated theoretically and numerically.Moreover,the tracking control of the system with exponential convergence rate is studied.Theoretical analysis and computer simulation show that the proposed scheme can allow us to drive the output variable x\ to arbitrary reference signals exponentially,and the guaranteed exponential convergence rate can be estimated accurately.     

Gevrey Regularity for the Attractor of the 3D Navier–Stokes–Voight Equations

Recently, the Navier–Stokes–Voight (NSV) model of viscoelastic incompressible fluid has been proposed as a regularization of the 3D Navier–Stokes equations for the purpose of direct numerical simulations. In this work, we prove that the global attractor of the 3D NSV equations, driven by an analytic forcing, consists of analytic functions. A consequence of this result is that the spectrum of the solutions of the 3D NSV system, lying on the global attractor, have exponentially decaying tail, despite the fact that the equations behave like a damped hyperbolic system, rather than the parabolic one. This result provides additional evidence that the 3D NSV with the small regularization parameter enjoys similar statistical properties as the 3D Navier–Stokes equations. Finally, we calculate a lower bound for the exponential decaying scale—the scale at which the spectrum of the solution start to decay exponentially, and establish a similar bound for the steady state solutions of the 3D NSV and 3D Navier–Stokes equations. Our estimate coincides with the known bounds for the smallest length scale of the solutions of the 3D Navier–Stokes equations, established earlier by Doering and Titi.      

非线性发展方程的H(o)lder连续惯性流形

引入非线性发展方程的H\"older连续惯性流形的概念,为原来惯性流形概念的推广和修正.惯性流形是有限维不变的Lipschiz流形,是研究发展方程解的长时间性态的合适工具,其缺点是需要谱间隙条件.提出H\"older连续惯性流形也是有限维不变的,但光滑性减弱为H\"older连续,不需要谱间隙条件.该流形与指数吸引子交集具有指数吸引性,无穷维动力系统可在H\"older连续惯性流形上约化为有限维常微分方程组.     

有限可解群的Fuzzy子群的阶数及其等价类数

定义Fuzzy子群的一种等价关系,即:如果两个Fuzzy子群的水平集的阶构成的集合相等, 就称这两个Fuzzy子群等价,并且通过研究群的子群列以及群阶的因数列和商数列找出了有限可解群的极大Fuzzy子群和Fuzzy子群的阶和等价类数的求解公式,并给出了二者之间的关系式.     

二值命题逻辑中的极大命题集与完备命题集

系统讨论了二值命题逻辑系统中极大命题集与完备命题集,给出了两种命题集的等价描述和表示定理,揭示了极大命题集和完备命题集的深刻内涵和联系.     

非紧L-凸度量空间中的GLsKKM定理及其对不动点问题的应用

建立了非紧完备L-凸度量空间中的GLsKKM定理.作为应用,获得了Ky Fan 截口定理,极大元定理和Fan-Browder型不动点定理.     

一类非自治时滞非局部模型的一致吸引子

本文研究了一类具有非局部反应项的非自治时滞偏微分方程的长时间性态,利用平移紧函数的性质,将相空间扩展使得非自治系统生成的过程提升为半群,再利用半群理论证明了一致吸引子的存在性.     

The Maximal Operator Associated to a Nonsymmetric Ornstein–Uhlenbeck Semigroup

Let (ℋ _t ) _t≥0 be the Ornstein–Uhlenbeck semigroup on ℝ ^d with covariance matrix I and drift matrix λ(R−I), where λ>0 and R is a skew-adjoint matrix, and denote by γ _∞ the invariant measure for (ℋ _t ) _t≥0. Semigroups of this form are the basic building blocks of Ornstein–Uhlenbeck semigroups which are normal on L ²(γ _∞). We prove that if the matrix R generates a one-parameter group of periodic rotations, then the maximal operator ℋ_* f(x)=sup  _t≥o |ℋ _t f(x)| is of weak type 1 with respect to the invariant measure γ _∞. We also prove that the maximal operator associated to an arbitrary normal Ornstein–Uhlenbeck semigroup is bounded on L ^p (γ _∞) if and only if 1<p≤∞.      

Random maximal independent sets and the unfriendly theater seating arrangement problem

People arrive one at a time to a theater consisting of m rows of length n. Being unfriendly they choose seats at random so that no one is in front of them, behind them or to either side. What is the expected number of people in the theater when it becomes full, i.e., it cannot accommodate any more unfriendly people? This is equivalent to the random process of generating a maximal independent set of an m×n grid by randomly choosing a node, removing it and its neighbors, and repeating until there are no nodes remaining. The case of m=1 was posed by Freedman and Shepp [D. Freedman, L. Shepp, An unfriendly seating arrangement (problem 62-3), SIAM Rev. 4 (2) (1962) 150] and solved independently by Friedman, Rothman and MacKenzie [H.D. Friedman, D. Rothman, Solution to: An unfriendly seating arrangement (problem 62-3), SIAM Rev. 6 (2) (1964) 180-182; J.K. MacKenzie, Sequential filling of a line by intervals placed at random and its application to linear adsorption, J. Chem. Phys. 37 (4) (1962) 723-728] by proving the asymptotic limit . In this paper we solve the case m=2 and prove the asymptotic limit . In addition, we consider the more general case of m×n grids, m≥1, and prove the existence of asymptotic limits in this general setting. We also make several conjectures based upon Monte Carlo simulations.