A necessary and sufficient condition is obtained for the incompleteness of a complex exponential system E(A, M) in Cα, where Cα is the weighted Banach space consisting of all complex continuous functions f on the re...A necessary and sufficient condition is obtained for the incompleteness of a complex exponential system E(A, M) in Cα, where Cα is the weighted Banach space consisting of all complex continuous functions f on the real axis R with f(t)exp(-α(t)) vanishing at infinity, in the uniform norm ‖f‖α = sup{|f(t)e-α(t)|: t ∈ R} with respect to the weight α(t). If the incompleteness holds, then the complex exponential system E(∧, M) is minimal and each function in the closure of the linear span of complex exponential system E(∧, M) can be extended to an entire function represented by a Taylor-Dirichlet series.展开更多
A topological dynamical system is n-sensitive,if there is a positive constant such that in each non-empty open subset there are n distinct points whose iterates will be apart from the constant at least for a same mome...A topological dynamical system is n-sensitive,if there is a positive constant such that in each non-empty open subset there are n distinct points whose iterates will be apart from the constant at least for a same moment.The properties of n-sensitivity in minimal systems are investigated.It turns out that a minimal system is n-sensitive if and only if the n-th regionally proximal relation Q_n contains a point whose coordinates are pairwise distinct.Moreover,the structure of a minimal system which is n-sensitive but not(n+1)-sensitive(n≥2)is determined.展开更多
1 Introduction A discrete dynamical system can be expressed as xn+1 ?f(xn), n = 0,1,2,... where X is a metric space and f : X →X is a continuous map. The study of it tells us how the points in the base space X moved....1 Introduction A discrete dynamical system can be expressed as xn+1 ?f(xn), n = 0,1,2,... where X is a metric space and f : X →X is a continuous map. The study of it tells us how the points in the base space X moved. Nevertheless, this is not enough for the researches of biological species, demography, numerical simulation and attractors (see [1], [2]). It is necessary to know how the subsets of X moved. In this direction, we consider the set-valued discrete system associated to f, An+1 = (f|-)(An), n = 0,1,2,... where (f|-) is the natural extension of f to K(X) (the class of all compact subsets of X).展开更多
Necessary and sufficient conditions are obtained for the incompleteness and the minimality of the exponential system E（Λ,M） = {z l e λ n z ： l = 0,1,...,m n-1;n = 1,2,...} in the Banach space E 2 [σ] consisting ...Necessary and sufficient conditions are obtained for the incompleteness and the minimality of the exponential system E（Λ,M） = {z l e λ n z ： l = 0,1,...,m n-1;n = 1,2,...} in the Banach space E 2 [σ] consisting of some analytic functions in a half strip.If the incompleteness holds,each function in the closure of the linear span of exponential system E（Λ,M） can be extended to an analytic function represented by a Taylor-Dirichlet series.Moreover,by the conformal mapping ζ = φ（z） = e z ,the similar results hold for the incompleteness and the minimality of the power function system F （Λ,M） = {（log ζ） l ζ λ n ： l = 0,1,...,m n-1;n = 1,2,...} in the Banach space F 2 [σ] consisting of some analytic functions in a sector.展开更多
The definition of nonlinear control sysms on fibre bundles proposed by Brockett and Willems is incomplete from the mathematical view geometric framework is proposed and a minimal realization theory is developed for no...The definition of nonlinear control sysms on fibre bundles proposed by Brockett and Willems is incomplete from the mathematical view geometric framework is proposed and a minimal realization theory is developed for nonlinear control systems on fibre bundles which is elaborated as a natural generalization of Sussmann's theory and differs essentially from Van der Schaft's approach. Limitations of realization theory given by Van der Schaft are also discussed.展开更多
A sufficient condition is obtained for the minimality of the complex exponential system E（A, M） = {z^le^λnz： l = 0, 1,,.., mn - 1; n = 1, 2,...} in the Banaeh space La^p consisting of all functions f such that f^...A sufficient condition is obtained for the minimality of the complex exponential system E（A, M） = {z^le^λnz： l = 0, 1,,.., mn - 1; n = 1, 2,...} in the Banaeh space La^p consisting of all functions f such that f^-a ∈ LP（N）. Moreover, if the incompleteness holds, each function in the closure of the linear span of exponential system E（A, M） can be extended to an analytic function represented by a Taylor-Dirichlet series.展开更多
Exponential systems of the form are considered, where is a degenerate coefficient, is a set of all integers and . The basis properties of these systems in , when, generally speaking, doesn’t satisfy the Muckenhoupt c...Exponential systems of the form are considered, where is a degenerate coefficient, is a set of all integers and . The basis properties of these systems in , when, generally speaking, doesn’t satisfy the Muckenhoupt condition are investigated.展开更多
The dynamical system on T^2 which is a group extension over an irrational rotation on T^1 is investigated. The criterion when the extension is minimal, a system of order 2 and when the maximal equicontinuous factor is...The dynamical system on T^2 which is a group extension over an irrational rotation on T^1 is investigated. The criterion when the extension is minimal, a system of order 2 and when the maximal equicontinuous factor is the irrational rotation is given. The topological complexity of the extension is computed, and a negative answer to the latter part of an open question raised by Host et al.(2014) is obtained.展开更多
We show that for every topological dynamical system with the approximate product property, zero topological entropy is equivalent to unique ergodicity. Equivalence of minimality is also proved under a slightly stronge...We show that for every topological dynamical system with the approximate product property, zero topological entropy is equivalent to unique ergodicity. Equivalence of minimality is also proved under a slightly stronger condition. Moreover, we show that unique ergodicity implies the approximate product property if the system has periodic points.展开更多
基金This work was partially supported by the Research Foundation for Doctor Programme(Grant No.20060027023)the National Natural Science Foundation of China(Grant No.10671022)
文摘A necessary and sufficient condition is obtained for the incompleteness of a complex exponential system E(A, M) in Cα, where Cα is the weighted Banach space consisting of all complex continuous functions f on the real axis R with f(t)exp(-α(t)) vanishing at infinity, in the uniform norm ‖f‖α = sup{|f(t)e-α(t)|: t ∈ R} with respect to the weight α(t). If the incompleteness holds, then the complex exponential system E(∧, M) is minimal and each function in the closure of the linear span of complex exponential system E(∧, M) can be extended to an entire function represented by a Taylor-Dirichlet series.
基金the National Natural Science Foundation of China(Grant Nos.10501042,10531010)the Ministry of Education of China(Grant No.20050358053)
文摘A topological dynamical system is n-sensitive,if there is a positive constant such that in each non-empty open subset there are n distinct points whose iterates will be apart from the constant at least for a same moment.The properties of n-sensitivity in minimal systems are investigated.It turns out that a minimal system is n-sensitive if and only if the n-th regionally proximal relation Q_n contains a point whose coordinates are pairwise distinct.Moreover,the structure of a minimal system which is n-sensitive but not(n+1)-sensitive(n≥2)is determined.
文摘1 Introduction A discrete dynamical system can be expressed as xn+1 ?f(xn), n = 0,1,2,... where X is a metric space and f : X →X is a continuous map. The study of it tells us how the points in the base space X moved. Nevertheless, this is not enough for the researches of biological species, demography, numerical simulation and attractors (see [1], [2]). It is necessary to know how the subsets of X moved. In this direction, we consider the set-valued discrete system associated to f, An+1 = (f|-)(An), n = 0,1,2,... where (f|-) is the natural extension of f to K(X) (the class of all compact subsets of X).
基金Supported by the National Natural Science Foundation of China (Grant No. 11071020)the Research Foundation for Doctor Program (Grant No. 20100003110004)
文摘Necessary and sufficient conditions are obtained for the incompleteness and the minimality of the exponential system E（Λ,M） = {z l e λ n z ： l = 0,1,...,m n-1;n = 1,2,...} in the Banach space E 2 [σ] consisting of some analytic functions in a half strip.If the incompleteness holds,each function in the closure of the linear span of exponential system E（Λ,M） can be extended to an analytic function represented by a Taylor-Dirichlet series.Moreover,by the conformal mapping ζ = φ（z） = e z ,the similar results hold for the incompleteness and the minimality of the power function system F （Λ,M） = {（log ζ） l ζ λ n ： l = 0,1,...,m n-1;n = 1,2,...} in the Banach space F 2 [σ] consisting of some analytic functions in a sector.
文摘The definition of nonlinear control sysms on fibre bundles proposed by Brockett and Willems is incomplete from the mathematical view geometric framework is proposed and a minimal realization theory is developed for nonlinear control systems on fibre bundles which is elaborated as a natural generalization of Sussmann's theory and differs essentially from Van der Schaft's approach. Limitations of realization theory given by Van der Schaft are also discussed.
基金Supported by the National Natural Science Foundation of China (Grant No.10671022)the Research Foundation for Doctor Programme (Grant No.20060027023)
文摘A sufficient condition is obtained for the minimality of the complex exponential system E（A, M） = {z^le^λnz： l = 0, 1,,.., mn - 1; n = 1, 2,...} in the Banaeh space La^p consisting of all functions f such that f^-a ∈ LP（N）. Moreover, if the incompleteness holds, each function in the closure of the linear span of exponential system E（A, M） can be extended to an analytic function represented by a Taylor-Dirichlet series.
文摘Exponential systems of the form are considered, where is a degenerate coefficient, is a set of all integers and . The basis properties of these systems in , when, generally speaking, doesn’t satisfy the Muckenhoupt condition are investigated.
基金National Natural Science Foundation of China (Grant Nos. 11225105 and 11371339)
文摘The dynamical system on T^2 which is a group extension over an irrational rotation on T^1 is investigated. The criterion when the extension is minimal, a system of order 2 and when the maximal equicontinuous factor is the irrational rotation is given. The topological complexity of the extension is computed, and a negative answer to the latter part of an open question raised by Host et al.(2014) is obtained.
基金Supported by National Natural Science Foundation of China(Grant No.11571387)CUFE Young Elite Teacher Project(Grant No.QYP1902)。
文摘We show that for every topological dynamical system with the approximate product property, zero topological entropy is equivalent to unique ergodicity. Equivalence of minimality is also proved under a slightly stronger condition. Moreover, we show that unique ergodicity implies the approximate product property if the system has periodic points.
