In this article, we will show that the super-bihamiltonian structures of the Kuper- KdV equation in [3], the Kuper-CH equation in [17, 18] and the super-HS equation in [11, 16, 19] can be obtained by applying a super-...In this article, we will show that the super-bihamiltonian structures of the Kuper- KdV equation in [3], the Kuper-CH equation in [17, 18] and the super-HS equation in [11, 16, 19] can be obtained by applying a super-bihamiltonian reduction of different super-Poisson pairs defined on the loop algebra of osp(1|2).展开更多
We prove a generalization of the classical Gauss-Bonnet formula for a conical metric on a compact Riemann surface provided that the Gaussian curvature is Lebesgue integrable with respect to the area form of the metric...We prove a generalization of the classical Gauss-Bonnet formula for a conical metric on a compact Riemann surface provided that the Gaussian curvature is Lebesgue integrable with respect to the area form of the metric.We also construct explicitly some conical metrics whose curvature is not integrable.展开更多
We give a summary on the recent development of chaos theory in topological dynamics, focusing on Li-Yorke chaos, Devaney chaos, distributional chaos, positive topological entropy, weakly mixing sets and so on, and the...We give a summary on the recent development of chaos theory in topological dynamics, focusing on Li-Yorke chaos, Devaney chaos, distributional chaos, positive topological entropy, weakly mixing sets and so on, and their relationships.展开更多
For the Monge-Amp`ere equation det D2u = 1, the authors find new auxiliary curvature functions which attain their respective maxima on the boundary. Moreover, the upper bounded estimates for the Gauss curvature and th...For the Monge-Amp`ere equation det D2u = 1, the authors find new auxiliary curvature functions which attain their respective maxima on the boundary. Moreover, the upper bounded estimates for the Gauss curvature and the mean curvature of the level sets for the solution to this equation are obtained.展开更多
The limiting behavior of stochastic evolution processes with small noise intensityεis investigated in distribution-based approaches.Letμεbe a stationary measure for stochastic process Xεwith smallεand X0 be a sem...The limiting behavior of stochastic evolution processes with small noise intensityεis investigated in distribution-based approaches.Letμεbe a stationary measure for stochastic process Xεwith smallεand X0 be a semiflow on a Polish space.Assume that{με:0<ε≤ε0}is tight.Then all their limits in the weak sense are X0-invariant and their supports are contained in the Birkhoff center of X0.Applications are made to various stochastic evolution systems,including stochastic ordinary differential equations,stochastic partial differential equations,and stochastic functional differential equations driven by Brownian motion or Levy processes.展开更多
Based on the Lax operator L and Orlov-Shulman's M operator,the string equations of the q-KP hierarchy are established from the special additional symmetry flows,and the negative Virasoro constraint generators {L-n...Based on the Lax operator L and Orlov-Shulman's M operator,the string equations of the q-KP hierarchy are established from the special additional symmetry flows,and the negative Virasoro constraint generators {L-n,n ≥ 1} of the 2-reduced q-KP hierarchy are also obtained.展开更多
In this paper,we use the 2-descent method to find a series of odd non-congruent numbers≡1(mod 8)whose prime factors are≡1(mod 4)such that the congruent elliptic curves have second lowest Selmer groups,which include ...In this paper,we use the 2-descent method to find a series of odd non-congruent numbers≡1(mod 8)whose prime factors are≡1(mod 4)such that the congruent elliptic curves have second lowest Selmer groups,which include Li and Tian’s result as special cases.展开更多
The q-deformation of W(2, 2) Lie algebra is well defined based on a realization of this Lie algebra by using the famous bosonic and fermionic oscillators in physics. Furthermore, the quantum group structures on the ...The q-deformation of W(2, 2) Lie algebra is well defined based on a realization of this Lie algebra by using the famous bosonic and fermionic oscillators in physics. Furthermore, the quantum group structures on the q-deformation of W(2, 2) Lie algebra are completely determined. Finally, the 1-dimensional central extension of the q-deformed W(2, 2) Lie algebra is studied, which turns out to be coincided with the conventional W(2, 2) Lie algebra in the q → 1 limit.展开更多
We explore recurrence properties arising from dynamical approach to the van der Waerden theorem and similar combinatorial problems. We describe relations between these properties and study their consequences for dynam...We explore recurrence properties arising from dynamical approach to the van der Waerden theorem and similar combinatorial problems. We describe relations between these properties and study their consequences for dynamics. In particular, we present a measure-theoretical analog of a result of Glasner on multi-transitivity of topologically weakly mixing minimal maps. We also obtain a dynamical proof of the existence of a C-set with zero Banach density.展开更多
We focus on the complexity of a special flow built over an irrational rotation of the unit circle and under a roof function on the unit circle.We construct a weak mixing minimal special flow with bounded topological c...We focus on the complexity of a special flow built over an irrational rotation of the unit circle and under a roof function on the unit circle.We construct a weak mixing minimal special flow with bounded topological complexity.We also prove that if the roof function is C^(∞),then the special flow has sub-polynomial topological complexity and the time one map meets the condition of Sarnak’s conjecture.展开更多
The family of pairwise independently determined (PID) systems, i.e., those for which the independent joining is the only self joining with independent 2-marginals, is a class of systems for which the long standing o...The family of pairwise independently determined (PID) systems, i.e., those for which the independent joining is the only self joining with independent 2-marginals, is a class of systems for which the long standing open question by Rokhlin, of whether mixing implies mixing of all orders, has a positive answer. We show that in the class of weakly mixing PID one finds a positive answer for another long-standing open problem, whether the multiple ergodic averagesalmost surely converge.展开更多
Invariant symmetric bilinear forms and derivation algebras of a unitary Lie algebra L over R are characterized: (L) ≌ (R+/([R, R] A R+))* and Der(L) = Inn(L) + Der(L)0 = Inn(L) + SDer(R), which ...Invariant symmetric bilinear forms and derivation algebras of a unitary Lie algebra L over R are characterized: (L) ≌ (R+/([R, R] A R+))* and Der(L) = Inn(L) + Der(L)0 = Inn(L) + SDer(R), which recover what of the special linear Lie algebra and Steinberg Lie algebra over R, where R is a unital involutory associative algebra over a field F.展开更多
Let(X,T)be a weakly mixing minimal system,p_(1),...,p_(d) be integer-valued generalized polynomials and(p_(1),p_(2),...,p_(d))be non-degenerate.Then there exists a residual subset X_(0) of X such that for all x∈X0,{(...Let(X,T)be a weakly mixing minimal system,p_(1),...,p_(d) be integer-valued generalized polynomials and(p_(1),p_(2),...,p_(d))be non-degenerate.Then there exists a residual subset X_(0) of X such that for all x∈X0,{(T^(p1(n))x,...,T^(pd(n))x):n∈Z}is dense in X^(d).展开更多
It is known that a minimal prime system is either a subshift or with a connected phase space(Keynes and Newton TransAmMath Soc 217:237-255,1976).We show that a double minimal system is a subshift;this implies immediat...It is known that a minimal prime system is either a subshift or with a connected phase space(Keynes and Newton TransAmMath Soc 217:237-255,1976).We show that a double minimal system is a subshift;this implies immediately that no non-periodic map has 4-fold topological minimal self-joinings.We also prove that a POD system is either uniformly rigid or is a subshift.展开更多
In this paper we completely classify the homogeneous two-spheres,especially,the minimal homogeneous ones in the quaternionic projective space HPn.According to our classification,more minimal constant curved two-sphere...In this paper we completely classify the homogeneous two-spheres,especially,the minimal homogeneous ones in the quaternionic projective space HPn.According to our classification,more minimal constant curved two-spheres in HPnare obtained than what Ohnita conjectured in the paper"Homogeneous harmonic maps into complex projective spaces.Tokyo J Math,1990,13:87–116".展开更多
Let e_λ(x) be a Neumann eigenfunction with respect to the positive Laplacian A on a compact Riemannian manifold M with boundary such that △e_λ=λ~2e_λ in the interior of M and the normal derivative of e\ vanishes ...Let e_λ(x) be a Neumann eigenfunction with respect to the positive Laplacian A on a compact Riemannian manifold M with boundary such that △e_λ=λ~2e_λ in the interior of M and the normal derivative of e\ vanishes on the boundary of M.Let χλ be the unit band spectral projection operator associated with the Neumann Laplacian and f be a square integrable function on M.The authors show the following gradient estimate for χλf as λ≥1:‖▽χλ f‖∞≤C(λ‖χλ f‖∞+λ^(-1)‖△χλf‖∞),where C is a positive constant depending only on M,As a corollary,the authors obtain the gradient estimate of e_λ:For every λ≥1,it holds that ‖▽e_λ‖∞≤Cλ‖e_λ‖∞.展开更多
This paper is devoted to studying the asymptotic behavior of the solution to nonlocal Fisher-KPP type reaction diffusion equations in heterogeneous media.The kernel K is assumed to depend on the media.First,we give an...This paper is devoted to studying the asymptotic behavior of the solution to nonlocal Fisher-KPP type reaction diffusion equations in heterogeneous media.The kernel K is assumed to depend on the media.First,we give an estimate of the upper and lower spreading speeds by generalized principal eigenvalues.Second,we prove the existence of spreading speeds in the case where the media is periodic or almost periodic by showing that the upper and lower generalized principal eigenvalues are equal.展开更多
For any C;diffeomorphism on a smooth compact Riemannian manifold that admits an ergodic measure with positive entropy,a lower bound of the Hausdorff dimension for the local stable and unstable sets is given in terms o...For any C;diffeomorphism on a smooth compact Riemannian manifold that admits an ergodic measure with positive entropy,a lower bound of the Hausdorff dimension for the local stable and unstable sets is given in terms of the measure-theoretic entropy and the maximal Lyapunov exponent.The main line of our approach to this result is under the setting of topological dynamical systems,which is also applicable to infinite-dimensional C;dynamical systems.展开更多
In this paper,we consider weak horseshoe with bounded-gap-hitting times.For a flow(M,Ф),it is shown that if the time one map(M,Ф_(1))has weak horseshoe with boundedgap-hitting times,so is(M,Ф_(τ))for all τ≠0.In ...In this paper,we consider weak horseshoe with bounded-gap-hitting times.For a flow(M,Ф),it is shown that if the time one map(M,Ф_(1))has weak horseshoe with boundedgap-hitting times,so is(M,Ф_(τ))for all τ≠0.In addition,we prove that for an affine homeomorphism of a compact metric abelian group,positive topological entropy is equivalent to weak horseshoe with bounded-gap-hitting times.展开更多
基金partially supported by"PCSIRT"the Fundamental Research Funds for the Central Universities(WK0010000024)+3 种基金NCET-13-0550SRF for ROCS,SEM and OATF,USTCNSFC(11271345,11371138)Natural Science Foundation of Anhui Province and Outstanding Young Talent Funds of Anhui Province(2013SQRL092ZD)
文摘In this article, we will show that the super-bihamiltonian structures of the Kuper- KdV equation in [3], the Kuper-CH equation in [17, 18] and the super-HS equation in [11, 16, 19] can be obtained by applying a super-bihamiltonian reduction of different super-Poisson pairs defined on the loop algebra of osp(1|2).
基金Support by the Project of Stable Support for Youth Team in Basic Research Field,CAS(Grant No.YSBR-001)NSFC(Grant Nos.12271495,11971450 and 12071449).
文摘We prove a generalization of the classical Gauss-Bonnet formula for a conical metric on a compact Riemann surface provided that the Gaussian curvature is Lebesgue integrable with respect to the area form of the metric.We also construct explicitly some conical metrics whose curvature is not integrable.
基金Supported by NNSF of China(Grant Nos.11371339,11431012,11401362,11471125)NSF of Guangdong province(Grant No.S2013040014084)
文摘We give a summary on the recent development of chaos theory in topological dynamics, focusing on Li-Yorke chaos, Devaney chaos, distributional chaos, positive topological entropy, weakly mixing sets and so on, and their relationships.
基金supported by the Chinese Universities Scientific Fund(No.WK0010000028)supported by the National Science Fund for Distinguished Young Scholars of China and Wu Wen-Tsun Key Laboratory of Mathematics+1 种基金partially supported by the National Natural Science Foundation of China(Nos.11101110,11326144)the Foundation of Harbin Normal University(No.KGB201224)
文摘For the Monge-Amp`ere equation det D2u = 1, the authors find new auxiliary curvature functions which attain their respective maxima on the boundary. Moreover, the upper bounded estimates for the Gauss curvature and the mean curvature of the level sets for the solution to this equation are obtained.
基金supported by National Natural Science Foundation of China(Grant Nos.11771295,11271356,11371041,11431014 and 11401557)Key Laboratory of Random Complex Structures and Data Science,Academy of Mathematics and Systems Science,Chinese Academy of Sciences,and the Fundamental Research Funds for the Central Universities(Grant No.WK0010000048)。
文摘The limiting behavior of stochastic evolution processes with small noise intensityεis investigated in distribution-based approaches.Letμεbe a stationary measure for stochastic process Xεwith smallεand X0 be a semiflow on a Polish space.Assume that{με:0<ε≤ε0}is tight.Then all their limits in the weak sense are X0-invariant and their supports are contained in the Birkhoff center of X0.Applications are made to various stochastic evolution systems,including stochastic ordinary differential equations,stochastic partial differential equations,and stochastic functional differential equations driven by Brownian motion or Levy processes.
基金supported by the National Natural Science Foundation of China (Nos.10971109,10971209,10825101)
文摘Based on the Lax operator L and Orlov-Shulman's M operator,the string equations of the q-KP hierarchy are established from the special additional symmetry flows,and the negative Virasoro constraint generators {L-n,n ≥ 1} of the 2-reduced q-KP hierarchy are also obtained.
基金supported by National Natural Science Foundation of China(Grant No.11171317)National Key Basic Research Program of China(Grant No.2013CB834202)
文摘In this paper,we use the 2-descent method to find a series of odd non-congruent numbers≡1(mod 8)whose prime factors are≡1(mod 4)such that the congruent elliptic curves have second lowest Selmer groups,which include Li and Tian’s result as special cases.
基金Supported by National Natural Science Foundation of China (Grant No. 10825101)
文摘The q-deformation of W(2, 2) Lie algebra is well defined based on a realization of this Lie algebra by using the famous bosonic and fermionic oscillators in physics. Furthermore, the quantum group structures on the q-deformation of W(2, 2) Lie algebra are completely determined. Finally, the 1-dimensional central extension of the q-deformed W(2, 2) Lie algebra is studied, which turns out to be coincided with the conventional W(2, 2) Lie algebra in the q → 1 limit.
基金supported by the National Science Centre (Grant No. DEC-2012/07/E/ ST1/00185)National Natural Science Foundation of China (Grant Nos. 11401362, 11471125, 11326135, 11371339 and 11431012)+1 种基金Shantou University Scientific Research Foundation for Talents (Grant No. NTF12021)the Project of LQ1602 IT4Innovations Excellence in Science
文摘We explore recurrence properties arising from dynamical approach to the van der Waerden theorem and similar combinatorial problems. We describe relations between these properties and study their consequences for dynamics. In particular, we present a measure-theoretical analog of a result of Glasner on multi-transitivity of topologically weakly mixing minimal maps. We also obtain a dynamical proof of the existence of a C-set with zero Banach density.
基金supported by NNSF of China(11431012,11731003)supported by NNSF of China(11801538,11871188).
文摘We focus on the complexity of a special flow built over an irrational rotation of the unit circle and under a roof function on the unit circle.We construct a weak mixing minimal special flow with bounded topological complexity.We also prove that if the roof function is C^(∞),then the special flow has sub-polynomial topological complexity and the time one map meets the condition of Sarnak’s conjecture.
基金supported by the National Science Center(Poland)grant 2013/08/A/ST1/00275 the the National Science Center(Poland)grant 2016/22/E/ST1/00448+1 种基金supported by NNSF of China(Grant Nos.11371339,11431012,11571335 and 11225105)"the Fundamental Research Funds for the Central Universities"
文摘The family of pairwise independently determined (PID) systems, i.e., those for which the independent joining is the only self joining with independent 2-marginals, is a class of systems for which the long standing open question by Rokhlin, of whether mixing implies mixing of all orders, has a positive answer. We show that in the class of weakly mixing PID one finds a positive answer for another long-standing open problem, whether the multiple ergodic averagesalmost surely converge.
文摘Invariant symmetric bilinear forms and derivation algebras of a unitary Lie algebra L over R are characterized: (L) ≌ (R+/([R, R] A R+))* and Der(L) = Inn(L) + Der(L)0 = Inn(L) + SDer(R), which recover what of the special linear Lie algebra and Steinberg Lie algebra over R, where R is a unital involutory associative algebra over a field F.
基金Supported by NNSF of China(Grant Nos.11871188,12031019)。
文摘Let(X,T)be a weakly mixing minimal system,p_(1),...,p_(d) be integer-valued generalized polynomials and(p_(1),p_(2),...,p_(d))be non-degenerate.Then there exists a residual subset X_(0) of X such that for all x∈X0,{(T^(p1(n))x,...,T^(pd(n))x):n∈Z}is dense in X^(d).
基金Authors are supported by NNSF of China 11225105,11371339 and 11431012.
文摘It is known that a minimal prime system is either a subshift or with a connected phase space(Keynes and Newton TransAmMath Soc 217:237-255,1976).We show that a double minimal system is a subshift;this implies immediately that no non-periodic map has 4-fold topological minimal self-joinings.We also prove that a POD system is either uniformly rigid or is a subshift.
基金supported by National Natural Science Foundation of China(Grant Nos.11471299,11401481 and 11331002)。
文摘In this paper we completely classify the homogeneous two-spheres,especially,the minimal homogeneous ones in the quaternionic projective space HPn.According to our classification,more minimal constant curved two-spheres in HPnare obtained than what Ohnita conjectured in the paper"Homogeneous harmonic maps into complex projective spaces.Tokyo J Math,1990,13:87–116".
基金supported by the National Natural Science Foundation of China(Nos.10971104,11271343,11101387)the Anhui Provincial Natural Science Foundation(No.1208085MA01)the Fundamental Research Funds for the Central Universities(Nos.WK0010000020,WK0010000023,WK3470000003)
文摘Let e_λ(x) be a Neumann eigenfunction with respect to the positive Laplacian A on a compact Riemannian manifold M with boundary such that △e_λ=λ~2e_λ in the interior of M and the normal derivative of e\ vanishes on the boundary of M.Let χλ be the unit band spectral projection operator associated with the Neumann Laplacian and f be a square integrable function on M.The authors show the following gradient estimate for χλf as λ≥1:‖▽χλ f‖∞≤C(λ‖χλ f‖∞+λ^(-1)‖△χλf‖∞),where C is a positive constant depending only on M,As a corollary,the authors obtain the gradient estimate of e_λ:For every λ≥1,it holds that ‖▽e_λ‖∞≤Cλ‖e_λ‖∞.
基金Supported by the National Natural Science Foundation of China(Grant Nos.11971454 and 12001514)the Fundamental Research Funds for the Central Universities and the Japan Society for the Promotion of Science。
文摘This paper is devoted to studying the asymptotic behavior of the solution to nonlocal Fisher-KPP type reaction diffusion equations in heterogeneous media.The kernel K is assumed to depend on the media.First,we give an estimate of the upper and lower spreading speeds by generalized principal eigenvalues.Second,we prove the existence of spreading speeds in the case where the media is periodic or almost periodic by showing that the upper and lower generalized principal eigenvalues are equal.
基金supported by National Natural Science Foundation of China(Grant No.11701394)supported by National Natural Science Foundation of China(Grant Nos.11971455 and 11731003)supported by National Natural Science Foundation of China(Grant Nos.11671279 and 11541003)。
文摘For any C;diffeomorphism on a smooth compact Riemannian manifold that admits an ergodic measure with positive entropy,a lower bound of the Hausdorff dimension for the local stable and unstable sets is given in terms of the measure-theoretic entropy and the maximal Lyapunov exponent.The main line of our approach to this result is under the setting of topological dynamical systems,which is also applicable to infinite-dimensional C;dynamical systems.
基金Leiye Xu is partially supported by NNSF of China(11801538,11871188)the USTC Research Funds of the Double First-Class Initiative.Junren Zheng is partially supported by NNSF of China(11971455).
文摘In this paper,we consider weak horseshoe with bounded-gap-hitting times.For a flow(M,Ф),it is shown that if the time one map(M,Ф_(1))has weak horseshoe with boundedgap-hitting times,so is(M,Ф_(τ))for all τ≠0.In addition,we prove that for an affine homeomorphism of a compact metric abelian group,positive topological entropy is equivalent to weak horseshoe with bounded-gap-hitting times.