THE PROXIMAL RELATION,REGIONALLY PROXIMAL RELATION AND BANACH PROXIMAL RELATION FOR AMENABLE GROUP ACTIONS

In this paper,we study the proximal relation,regionally proximal relation and Banach proximal relation of a topological dynamical system for amenable group actions.A useful tool is the support of a topological dynamical system which is used to study the structure of the Banach proximal relation,and we prove that above three relations all coincide on a Banach mean equicontinuous system generated by an amenable group action.

The Multi-sensitivity and Topological Sequence Entropy of Dynamical System with Group Action

In this paper,we study the relationship between the multi-sensitivity and the topological maximal sequence entropy of dynamical systems for general group action.Furthermore,we also discuss the consistency of multi-sensitivity of a dynamical system(G■X)and its hyperspace dynamical system G■K(X).Moreover,we research the relationship between the multi-sensitivity of two dynamical systems and the multi-sensitivity of their product space dynamical system.Finally,we prove that if the topological sequence entropy of G■X vanishes,then so does that of its induced system G■M(X);if the topological sequence entropy of G■X is positive,then that of its induced system G■M(X)is infinity.

THE SYSTEMS WITH ALMOST BANACH-MEAN EQUICONTINUITY FOR ABELIAN GROUP ACTIONS

In this paper,we present the concept of Banach-mean equicontinuity and prove that the Banach-,Weyl-and Besicovitch-mean equicontinuities of a dynamic system of Abelian group action are equivalent.Furthermore,we obtain that the topological entropy of a transitive,almost Banach-mean equicontinuous dynamical system of Abelian group action is zero.As an application of our main result,we show that the topological entropy of the Banach-mean equicontinuous system under the action of an Abelian groups is zero.

Eight predictive powers with historical and interim data for futility and efficacy analysis

When the historical data of the early phase trial and the interim data of the Phase II trial are avail-able,we should use them to give a more accurate prediction in both futility and efficacy analysis.The predictive power is an important measure of the practical utility of a proposed trial,and it is better than the classical statistical power in giving a good indication of the probability that the trial will demonstrate a positive or statistically significant outcome.In addition to the four predic-tive powers with historical and interim data available in the literature and summarized in Table 1,we discover and calculate another four predictive powers also summarized in Table 1,for one-sided hypotheses.Moreover,we calculate eight predictive powers summarized in Table 2,for the reversed hypotheses.The combination of the two tables gives us a complete picture of the pre-dictive powers with historical and interim data for futility and efficacy analysis.Furthermore,the eight predictive powers with historical and interim data are utilized to guide the futility analysis in the tamoxifen example.Finally,extensive simulations have been conducted to investigate the sensitivity analysis of priors,sample sizes,interim result and interim time on different predictive powers.

The area minimizing problem in conformal cones,Ⅱ

In this paper we continue to study the connection among the area minimizing problem,certain area functional and the Dirichlet problem of minimal surface equations in a class of conformal cones with a similar motivation from[15].These cones are certain generalizations of hyperbolic spaces.We describe the structure of area minimizing n-integer multiplicity currents in bounded C^2 conformal cones with prescribed C^1 graphical boundary via a minimizing problem of these area functionals.As an application we solve the corresponding Dirichlet problem of minimal surface equations under a mean convex type assumption.We also extend the existence and uniqueness of a local area minimizing integer multiplicity current with star-shaped infinity boundary in hyperbolic spaces into a large class of complete conformal manifolds.