HILBERTIAN INVARIANCE PRINCIPLE FOR EMPIRICAL PROCESS ASSOCIATED WITH A MARKOV PROCESS

The authors establish the Hilbertian invariance principle for the empirical process of a stationary Markov process, by extending the forward-backward martingale decomposition of Lyons-Meyer-Zheng to the Hilbert space valued additive functionals associated with general non-reversible Markov processes.     

THE STRUCTURE AND APPROXIMATION OF A.S. SELF-SIMILAR SET

The structure of any a.s. self-similar set K(w) generated by a class of random elements {gn,wσ} taking values in the space of contractive operators is given and the approximation of K(w) by the fixed points {Pn,wσ} of {gn,ow} is obtained. It is useful to generate the fractal in computer.     

ASYMPTOTIC NORMALITY OF KERNEL ESTIMATES OF A DENSITY FUNCTION UNDER ASSOCIATION DEPENDENCE

Let {X_n, n≥1} be a strictly stationary sequence of random variables, whichare either associated or negatively associated, f(·) be their common density. In this paper,the author shows a central limit theorem for a kernel estimate of f(·) under certain regularconditions.     

THE LIMIT SET OF A CONTINUOUS SELF-MAP OF THE TREE

Let f denote a continuous map of a tree T to itself. A point x ∈ T is called a 7-limit point of f if it is both an ω-limit point and an α-limit point. In the present paper, we show that (1) Ω-Γ is countable, (2) A -Γ and P - Γ are either empty or countably infinite, where P denotes the closure of the set of periodic points P.     

DISTRIBUTIONS OF LIMIT CYCLES FOR POLYNOMIAL VECTOR FIELDS (P_2, Q_3)

The distributions of limit cycles of cubic vector fields (P2, Q3) are considered in this paper, where P2 and Q3 are polynomials of x and y of order two and three, respectively. It is possibly seven different distributions of limit cycles given in [1]. We now prove that in which three kinds of distributions are impossible and other four kinds all can be realized by concrete vector fields of (P2,Q3). Some related results are also given.     

A NECESSARY AND SUFFICIENT CONDITION FOR THE COEXISTENCE OF A CLASS OF CUBIC CURVE SEPARATRIX CYCLES AND LIMIT CYCLES TO THE CUBIC SYSTEM

In this paper, we give the necessary and sufficient condition for the coexistence of a class of cubic curve separatrix cycles and limit cycles to the cubic system, and study their topological structures.     

THE BOUNDARY VALUE PROBLEMS OF A CLASS OF ELLIPTIC EQUATION WITH A CURVE OF TURNING POINT

The boundary value problems of a class of elliptic equation with a curveof turning point are considered. Using the method of multiple scales and thecomparison theorem, the asymptotic behavior of solution for the boundary valueproblem is studied.     

A localization argument for characters of reductive Lie groups

This article provides a geometric bridge between two entirely different character formulas for reductive Lie groups and answers the question posed by Schmid (in: Deformation Theory and Symplectic Geometry, Mathematical Physics Studies, Vol. 20, Kluwer Academic Publishers, Dordrecht, 1997, pp. 259-270).A corresponding problem in the compact group setting was solved by Berline et al. (Heat Kernels and Dirac Operators, Springer, Berlin, 1992) by an application of the theory of equivariant forms and particularly the fixed point integral localization formula. This article (besides its representation-theoretical significance) provides a whole family of examples where it is possible to localize integrals to fixed points with respect to an action of a noncompact group. Moreover, a localization argument given here is not specific to the particular setting considered in this article and can be extended to a more general situation.There is a broadly accessible article (Libine, A Localization Argument for Characters of Reductive Lie Groups: An Introduction and Examples, 2002, math.RT/0208024) which explains how the argument works in the case, where the key ideas are not obstructed by technical details and where it becomes clear how it extends to the general case.     

Travelling waves for reaction-diffusion equations with time depending nonlinearities

The existence of travelling wave with given end points for parabolic system of nonlinear equations is proven. The nonlinear term depends also on a·x−ct where x is the multidimensional space variable, t—time, c—the speed of the wave and a—the direction of travel.     

Uniform asymptotic regularity for Mann iterates

In Numer. Funct. Anal. Optim. 22 (2001) 641-656, we obtained an effective quantitative analysis of a theorem due to Borwein, Reich, and Shafrir on the asymptotic behavior of general Krasnoselski-Mann iterations for nonexpansive self-mappings of convex sets C in arbitrary normed spaces. We used this result to obtain a new strong uniform version of Ishikawa's theorem for bounded C. In this paper we give a qualitative improvement of our result in the unbounded case and prove the uniformity result for the bounded case under the weaker assumption that C contains a point x whose Krasnoselski-Mann iteration (x_k) is bounded. We also consider more general iterations for which asymptotic regularity is known only for uniformly convex spaces (Groetsch). We give uniform effective bounds for (an extension of) Groetsch's theorem which generalize previous results by Kirk, Martinez-Yanez, and the author.