The author considers the Feigenbaum's functional equation fP(λx) =λf(x) for each p ≥ 2.The existence of even unimodal C1 solutions to this equation is discussed and a feasible methodto construct such solutions ...The author considers the Feigenbaum's functional equation fP(λx) =λf(x) for each p ≥ 2.The existence of even unimodal C1 solutions to this equation is discussed and a feasible methodto construct such solutions is given.展开更多
We consider a class of generalized Fibonacci unimodal maps for which the central return times {Sn} satisfy that sn= sn-1 ＋ ksh-2 for some k≥ 1. We show that such a unimodal map admits a unique absolutely continuous...We consider a class of generalized Fibonacci unimodal maps for which the central return times {Sn} satisfy that sn= sn-1 ＋ ksh-2 for some k≥ 1. We show that such a unimodal map admits a unique absolutely continuous invariant probability with exactly stretched exponential decay of correlations if its critical order lies in （1, k ＋ 1）.展开更多
Replete unimodal function and other conceptions are defined. It is proved that for any continuous function φ, if φ has a unimodal orbit , then φ has all unimodal orbits whose types precede the orbit type of. They a...Replete unimodal function and other conceptions are defined. It is proved that for any continuous function φ, if φ has a unimodal orbit , then φ has all unimodal orbits whose types precede the orbit type of. They are altogether distributed on a compact subset X of I. The restriction φ|X of φ to X is a replete unimodal function. Moreover, an ordered classification φ of the function space C°(I,R) is given. It is a refinement of the famous Sarkovskii’s ordered classification F, and also a generalization of the conclusion obtained by Bhatia and Egerland not long ago.展开更多
In this work, we analyzed the impact of interventions on populations which exhibit unimodal dynamics. The six landmarks that characterize the “shape” of the unimodal reproduction curve f ( x ) of the difference equa...In this work, we analyzed the impact of interventions on populations which exhibit unimodal dynamics. The six landmarks that characterize the “shape” of the unimodal reproduction curve f ( x ) of the difference equation, X n+1 = f ( X n )?, are defined and used in order to examine and determine the behavior of dynamics of populations. By using the Li-Yorke criterion for determination of chaos we propose a qualitative intervention rule that can be applied without any explicit population equation. This proposed strategy for intervention brings out many interesting behaviors in population dynamics. A qualitative decision rule can be applied with a straight edge without any population equation and therefore offers a robust strategy for the management of populations.展开更多
For the so-called quadratic family, it is proved that, there exists a positive Lebesgue measure set E in the parameter space such that the corresponding map has a dense critical orbit in the support set of the invaria...For the so-called quadratic family, it is proved that, there exists a positive Lebesgue measure set E in the parameter space such that the corresponding map has a dense critical orbit in the support set of the invariant measure for almost all parameters in E; it is also proved that there is a dense set in E such that the critical orbit of the corresponding map enter the reversed fixed point.展开更多
We consider the topological entropy h(θ) of real unimodal maps as a function of the kneading parameter θ(equivalently, as a function of the external angle in the Mandelbrot set). We prove that this function is local...We consider the topological entropy h(θ) of real unimodal maps as a function of the kneading parameter θ(equivalently, as a function of the external angle in the Mandelbrot set). We prove that this function is locally Hlder continuous where h(θ) > 0, and more precisely for any θ which does not lie in a plateau the local Hlder exponent equals exactly, up to a factor log 2, the value of the function at that point.This confirms a conjecture of Isola and Politi(1990), and extends a similar result for the dimension of invariant subsets of the circle.展开更多
Let f（q） = arq^r ＋…＋asqs, with ar ≠ 0 and as ≠ 0, be a real polynomial. It is a palindromic polynomial of darga n if r＋s= n and ar＋i =as-i for all i. Polynomials of darga n form a linear subspace Pn（q） of R...Let f（q） = arq^r ＋…＋asqs, with ar ≠ 0 and as ≠ 0, be a real polynomial. It is a palindromic polynomial of darga n if r＋s= n and ar＋i =as-i for all i. Polynomials of darga n form a linear subspace Pn（q） of R（q）n＋l of dimension [n/2] ＋ 1. We give transition matrices between two bases {q^j（1 ＋ q ＋… q^n-2j）}, {q^j（1 ＋ q）^n-2j } and the standard basis {q^j（1 ＋ q^n-2j）} of Pn （q）. We present some characterizations and sufficient conditions for palindromic polynomials that can be expressed in terms of these two bases with nonnegative coefficients. We also point out the link between such polynomials and rank-generating functions of posets.展开更多
The association between biodiversity and belowground biomass(BGB) remains a central debate in ecology.In this study, we compared the variations in species richness(SR) and BGB as well as their interaction in the top(0...The association between biodiversity and belowground biomass(BGB) remains a central debate in ecology.In this study, we compared the variations in species richness(SR) and BGB as well as their interaction in the top(0–20 cm), middle(20–50 cm) and deep(50–100 cm) soil depths among 8 grassland types(lowland meadow, temperate desert, temperate desert steppe, temperate steppe desert, temperate steppe, temperate meadow steppe, mountain meadow and alpine steppe) and along environmental gradients(elevation, energy condition(annual mean temperature(AMT) and potential evapotranspiration(PET)), and mean annual precipitation(MAP)) based on a 2011–2013 survey of 379 sites in Xinjiang, Northwest China.The SR and BGB varied among the grassland types.The alpine steppe had a medium level of SR but the highest BGB in the top soil depth, whereas the lowland meadow had the lowest SR but the highest BGB in the middle and deep soil depths.The SR and BGB in the different soil depths were tightly associated with elevation, MAP and energy condition;however, the particular forms of trends in SR and BGB depended on environmental factors and soil depths.The relationship between SR and BGB was unimodal in the top soil depth, but SR was positively related with BGB in the middle soil depth.Although elevation, MAP, energy condition and SR had significant effects on BGB, the variations in BGB in the top soil depth were mostly determined by elevation, and those in the middle and deep soil depths were mainly affected by energy condition.These findings highlight the importance of environmental factors in the regulations of SR and BGB as well as their interaction in the grasslands in Xinjiang.展开更多
In this paper we consider a parallel algorithm that detects the maximizer of unimodal function f(x) computable at every point on unbounded interval (0, ∞). The algorithm consists of two modes: scanning and detecting....In this paper we consider a parallel algorithm that detects the maximizer of unimodal function f(x) computable at every point on unbounded interval (0, ∞). The algorithm consists of two modes: scanning and detecting. Search diagrams are introduced as a way to describe parallel searching algorithms on unbounded intervals. Dynamic programming equations, combined with a series of liner programming problems, describe relations between results for every pair of successive evaluations of function f in parallel. Properties of optimal search strategies are derived from these equations. The worst-case complexity analysis shows that, if the maximizer is located on a priori unknown interval (n-1], then it can be detected after cp(n)=「2log「p/2」+1(n+1)」-1 parallel evaluations of f(x), where p is the number of processors.展开更多
In this paper we prove that Sands' topological condition for Collet-Eckmann maps implies Tsujii's metrical condition; on the other hand, if a Collet-Eckmann map satisfies Tsujii's metrical condition, then ...In this paper we prove that Sands' topological condition for Collet-Eckmann maps implies Tsujii's metrical condition; on the other hand, if a Collet-Eckmann map satisfies Tsujii's metrical condition, then it satisfies Sands' topological condition. Thus we obtain three different versions of Benedicks-Carleson Theorem by using topological conditions.展开更多
For any continuous function f on the interval I=[0, 1] and any m, n≥1, let N(n, f)denote the number of n-periodic orbits in f. Put N(n, m)=min{N(n, f):f is a continuousfunction on I, and N(m, f)≥1}. The famous Sarko...For any continuous function f on the interval I=[0, 1] and any m, n≥1, let N(n, f)denote the number of n-periodic orbits in f. Put N(n, m)=min{N(n, f):f is a continuousfunction on I, and N(m, f)≥1}. The famous Sarkovskii’s theorem can be stated as follows:If n?m, then N(n,m)≥1. In this paper, we further obtain analytic expressions of the precisevalue of N(n, m) for all positive integers m and n, which are convenient for computing.展开更多
The series of 2-（1-（2,4-dibenzhydrylnaphthylimino）ethyl）-6-（1-（arylimino）ethyl）pyridine derivatives and their cobalt chlorides （Col-Co5） were synthesized and fully characterized. The representative complexes...The series of 2-（1-（2,4-dibenzhydrylnaphthylimino）ethyl）-6-（1-（arylimino）ethyl）pyridine derivatives and their cobalt chlorides （Col-Co5） were synthesized and fully characterized. The representative complexes Co2 and Co3 were confirmed by single crystal X-ray diffraction analyses, indicating pseudo-square-pyramidal geometry around the cobalt centre. All cobalt complexes, activated with either methylaluminoxane （MAO） or modified methylaluminoxane （MMAO）, exhibited high activities towards ethylene polymerization, and produced polyethylenes with high molecular weight and highly linear feature as well as unimodal distribution.展开更多
文摘The author considers the Feigenbaum's functional equation fP(λx) =λf(x) for each p ≥ 2.The existence of even unimodal C1 solutions to this equation is discussed and a feasible methodto construct such solutions is given.
基金supported by AcRF-Tier 1 grant from MOE,Singapore(Grant No.R-146-000-199-112)
文摘We consider a class of generalized Fibonacci unimodal maps for which the central return times {Sn} satisfy that sn= sn-1 ＋ ksh-2 for some k≥ 1. We show that such a unimodal map admits a unique absolutely continuous invariant probability with exactly stretched exponential decay of correlations if its critical order lies in （1, k ＋ 1）.
基金Project supported by the National Natural Science Foundation of China.
文摘Replete unimodal function and other conceptions are defined. It is proved that for any continuous function φ, if φ has a unimodal orbit , then φ has all unimodal orbits whose types precede the orbit type of. They are altogether distributed on a compact subset X of I. The restriction φ|X of φ to X is a replete unimodal function. Moreover, an ordered classification φ of the function space C°(I,R) is given. It is a refinement of the famous Sarkovskii’s ordered classification F, and also a generalization of the conclusion obtained by Bhatia and Egerland not long ago.
文摘In this work, we analyzed the impact of interventions on populations which exhibit unimodal dynamics. The six landmarks that characterize the “shape” of the unimodal reproduction curve f ( x ) of the difference equation, X n+1 = f ( X n )?, are defined and used in order to examine and determine the behavior of dynamics of populations. By using the Li-Yorke criterion for determination of chaos we propose a qualitative intervention rule that can be applied without any explicit population equation. This proposed strategy for intervention brings out many interesting behaviors in population dynamics. A qualitative decision rule can be applied with a straight edge without any population equation and therefore offers a robust strategy for the management of populations.
文摘For the so-called quadratic family, it is proved that, there exists a positive Lebesgue measure set E in the parameter space such that the corresponding map has a dense critical orbit in the support set of the invariant measure for almost all parameters in E; it is also proved that there is a dense set in E such that the critical orbit of the corresponding map enter the reversed fixed point.
文摘We consider the topological entropy h(θ) of real unimodal maps as a function of the kneading parameter θ(equivalently, as a function of the external angle in the Mandelbrot set). We prove that this function is locally Hlder continuous where h(θ) > 0, and more precisely for any θ which does not lie in a plateau the local Hlder exponent equals exactly, up to a factor log 2, the value of the function at that point.This confirms a conjecture of Isola and Politi(1990), and extends a similar result for the dimension of invariant subsets of the circle.
基金Supported by National Natural Science Foundation of China(Grant Nos.11071030,11371078)the Specialized Research Fund for the Doctoral Program of Higher Education of China(Grant No.20110041110039)
文摘Let f（q） = arq^r ＋…＋asqs, with ar ≠ 0 and as ≠ 0, be a real polynomial. It is a palindromic polynomial of darga n if r＋s= n and ar＋i =as-i for all i. Polynomials of darga n form a linear subspace Pn（q） of R（q）n＋l of dimension [n/2] ＋ 1. We give transition matrices between two bases {q^j（1 ＋ q ＋… q^n-2j）}, {q^j（1 ＋ q）^n-2j } and the standard basis {q^j（1 ＋ q^n-2j）} of Pn （q）. We present some characterizations and sufficient conditions for palindromic polynomials that can be expressed in terms of these two bases with nonnegative coefficients. We also point out the link between such polynomials and rank-generating functions of posets.
基金supported by the National Natural Science Foundation of China (U1603235, 31660127)the Tianshan Innovation Team Plan of Xinjiang (2017D14009)
文摘The association between biodiversity and belowground biomass(BGB) remains a central debate in ecology.In this study, we compared the variations in species richness(SR) and BGB as well as their interaction in the top(0–20 cm), middle(20–50 cm) and deep(50–100 cm) soil depths among 8 grassland types(lowland meadow, temperate desert, temperate desert steppe, temperate steppe desert, temperate steppe, temperate meadow steppe, mountain meadow and alpine steppe) and along environmental gradients(elevation, energy condition(annual mean temperature(AMT) and potential evapotranspiration(PET)), and mean annual precipitation(MAP)) based on a 2011–2013 survey of 379 sites in Xinjiang, Northwest China.The SR and BGB varied among the grassland types.The alpine steppe had a medium level of SR but the highest BGB in the top soil depth, whereas the lowland meadow had the lowest SR but the highest BGB in the middle and deep soil depths.The SR and BGB in the different soil depths were tightly associated with elevation, MAP and energy condition;however, the particular forms of trends in SR and BGB depended on environmental factors and soil depths.The relationship between SR and BGB was unimodal in the top soil depth, but SR was positively related with BGB in the middle soil depth.Although elevation, MAP, energy condition and SR had significant effects on BGB, the variations in BGB in the top soil depth were mostly determined by elevation, and those in the middle and deep soil depths were mainly affected by energy condition.These findings highlight the importance of environmental factors in the regulations of SR and BGB as well as their interaction in the grasslands in Xinjiang.
文摘In this paper we consider a parallel algorithm that detects the maximizer of unimodal function f(x) computable at every point on unbounded interval (0, ∞). The algorithm consists of two modes: scanning and detecting. Search diagrams are introduced as a way to describe parallel searching algorithms on unbounded intervals. Dynamic programming equations, combined with a series of liner programming problems, describe relations between results for every pair of successive evaluations of function f in parallel. Properties of optimal search strategies are derived from these equations. The worst-case complexity analysis shows that, if the maximizer is located on a priori unknown interval (n-1], then it can be detected after cp(n)=「2log「p/2」+1(n+1)」-1 parallel evaluations of f(x), where p is the number of processors.
基金the National Natural Science Foundation of China (No.19901035) andTWAS/CNPq associate fellowship.
文摘In this paper we prove that Sands' topological condition for Collet-Eckmann maps implies Tsujii's metrical condition; on the other hand, if a Collet-Eckmann map satisfies Tsujii's metrical condition, then it satisfies Sands' topological condition. Thus we obtain three different versions of Benedicks-Carleson Theorem by using topological conditions.
基金Project supported by the National Natural Science Foundation of China.
文摘For any continuous function f on the interval I=[0, 1] and any m, n≥1, let N(n, f)denote the number of n-periodic orbits in f. Put N(n, m)=min{N(n, f):f is a continuousfunction on I, and N(m, f)≥1}. The famous Sarkovskii’s theorem can be stated as follows:If n?m, then N(n,m)≥1. In this paper, we further obtain analytic expressions of the precisevalue of N(n, m) for all positive integers m and n, which are convenient for computing.
基金supported by the National Natural Science Foundation of China (U1362204, 51473170, 21572085)
文摘The series of 2-（1-（2,4-dibenzhydrylnaphthylimino）ethyl）-6-（1-（arylimino）ethyl）pyridine derivatives and their cobalt chlorides （Col-Co5） were synthesized and fully characterized. The representative complexes Co2 and Co3 were confirmed by single crystal X-ray diffraction analyses, indicating pseudo-square-pyramidal geometry around the cobalt centre. All cobalt complexes, activated with either methylaluminoxane （MAO） or modified methylaluminoxane （MMAO）, exhibited high activities towards ethylene polymerization, and produced polyethylenes with high molecular weight and highly linear feature as well as unimodal distribution.
