A set E ? R^d whose indicator function 1_E has maximal Gowers norm, among all sets of equal measure, is an ellipsoid up to Lebesgue null sets. If 1_E has nearly maximal Gowers norm then E nearly coincides with some el...A set E ? R^d whose indicator function 1_E has maximal Gowers norm, among all sets of equal measure, is an ellipsoid up to Lebesgue null sets. If 1_E has nearly maximal Gowers norm then E nearly coincides with some ellipsoid, in the sense that their symmetric difference has small Lebesgue measure.展开更多
Let A ■ ■_(N),and f_(A)(s)={1-|A/N,-|A|/N,for s ∈A,for s■ A.We define the pseudorandom measure of order k of the subset A as follows,P _(k)(A,N)=max D|∑n∈■_(N)|f_(A)(n+c_(1))f_(A)(n+c_(2))…f_(A)(n+c_(k))|where...Let A ■ ■_(N),and f_(A)(s)={1-|A/N,-|A|/N,for s ∈A,for s■ A.We define the pseudorandom measure of order k of the subset A as follows,P _(k)(A,N)=max D|∑n∈■_(N)|f_(A)(n+c_(1))f_(A)(n+c_(2))…f_(A)(n+c_(k))|where the maximum is taken over all D=(c_(1),c_(2),…,C_(K))∈■^(k) with 0≤c_(1)<c_(2)<…ck≤N-1.The subset A ■ ■_(N) is considered as a pseudorandom subset of degree k if P_(k)(A,N)is“small”in terms of N.We establish a link be tween the Gowers norm and our pseudorandom measure,and show that“good”pseudorandom subsets must have“small”Gowers norm.We give an example to suggest that subsets with"small" Gowers norm may have large pseudorandom measure.Finally,we prove that the pseudorandom subset of degree L(k)contains an arithmetic progression of length k,where L(k)=2·lcm(2,4,…,2|k/2|),for k≥4,and lcm(a1,a2,…,al)denotes the least common multiple of a1,a2,…,al.展开更多
This paper investigates the global dynamics of a reaction-diffusion-advection Leslie-Gower predator-prey model in open advective environments.We find that there exist critical advection rates,intrinsic growth rates,di...This paper investigates the global dynamics of a reaction-diffusion-advection Leslie-Gower predator-prey model in open advective environments.We find that there exist critical advection rates,intrinsic growth rates,diffusion rates and length of the domain,which classify the global dynamics of the Leslie-Gower predator-prey system into three scenarios:coexistence,persistence of prey only and extinction of both species.The results reveal some significant differences with the classical specialist and generalist predator-prey systems.In particular,it is found that the critical advection rates of prey and predator are independent of each other and the parameters about predation rate have no influence on the dynamics of system.The theoretical results provide some interesting highlights in ecological protection in streams or rivers.展开更多
基金Research supported in part by NSF(Grant DMS-1363324)
文摘A set E ? R^d whose indicator function 1_E has maximal Gowers norm, among all sets of equal measure, is an ellipsoid up to Lebesgue null sets. If 1_E has nearly maximal Gowers norm then E nearly coincides with some ellipsoid, in the sense that their symmetric difference has small Lebesgue measure.
基金supported in part by the National Natural Science Foundation of China(Grant No.11571277).
文摘Let A ■ ■_(N),and f_(A)(s)={1-|A/N,-|A|/N,for s ∈A,for s■ A.We define the pseudorandom measure of order k of the subset A as follows,P _(k)(A,N)=max D|∑n∈■_(N)|f_(A)(n+c_(1))f_(A)(n+c_(2))…f_(A)(n+c_(k))|where the maximum is taken over all D=(c_(1),c_(2),…,C_(K))∈■^(k) with 0≤c_(1)<c_(2)<…ck≤N-1.The subset A ■ ■_(N) is considered as a pseudorandom subset of degree k if P_(k)(A,N)is“small”in terms of N.We establish a link be tween the Gowers norm and our pseudorandom measure,and show that“good”pseudorandom subsets must have“small”Gowers norm.We give an example to suggest that subsets with"small" Gowers norm may have large pseudorandom measure.Finally,we prove that the pseudorandom subset of degree L(k)contains an arithmetic progression of length k,where L(k)=2·lcm(2,4,…,2|k/2|),for k≥4,and lcm(a1,a2,…,al)denotes the least common multiple of a1,a2,…,al.
基金supported by the National Natural Science Foundation of China(11871403)Fundamental Research Funds for the Central Universities(XDJK2020B050).
文摘This paper investigates the global dynamics of a reaction-diffusion-advection Leslie-Gower predator-prey model in open advective environments.We find that there exist critical advection rates,intrinsic growth rates,diffusion rates and length of the domain,which classify the global dynamics of the Leslie-Gower predator-prey system into three scenarios:coexistence,persistence of prey only and extinction of both species.The results reveal some significant differences with the classical specialist and generalist predator-prey systems.In particular,it is found that the critical advection rates of prey and predator are independent of each other and the parameters about predation rate have no influence on the dynamics of system.The theoretical results provide some interesting highlights in ecological protection in streams or rivers.
