On a compact Riemann surface with finite punctures P_(1),…P_(k),we define toric curves as multivalued,totallyunramified holomorphic maps to P^(n)with monodromy in a maximal torus of PSU(n+1).Toric solutions to SU(n+1...On a compact Riemann surface with finite punctures P_(1),…P_(k),we define toric curves as multivalued,totallyunramified holomorphic maps to P^(n)with monodromy in a maximal torus of PSU(n+1).Toric solutions to SU(n+1)Todasystems on X\{P_(1);…;P_(k)}are recognized by the associated toric curves in.We introduce character n-ensembles as-tuples of meromorphic one-forms with simple poles and purely imaginary periods,generating toric curves on minus finitelymany points.On X,we establish a correspondence between character-ensembles and toric solutions to the SU(n+1)system with finitely many cone singularities.Our approach not only broadens seminal solutions with two conesingularities on the Riemann sphere,as classified by Jost-Wang(Int.Math.Res.Not.,2002,(6):277-290)andLin-Wei-Ye(Invent.Math.,2012,190(1):169-207),but also advances beyond the limits of Lin-Yang-Zhong’s existencetheorems(J.Differential Geom.,2020,114(2):337-391)by introducing a new solution class.展开更多
In this note,we study the Yang-Mills bar connection,i.e.,the curvature of obeys,δ_(A)^(*)F_(A)^(0.2)on a principal G-bundle P over a compact complex manifold.According to the Koszul-Malgrange criterion,any holomorphi...In this note,we study the Yang-Mills bar connection,i.e.,the curvature of obeys,δ_(A)^(*)F_(A)^(0.2)on a principal G-bundle P over a compact complex manifold.According to the Koszul-Malgrange criterion,any holomorphic structure on can be seen as a solution to this equation.Suppose that G=SU(2)or SO(3)and X is a complex surface with H_(1)(X,Z_(2))=0.We then prove that the-part curvature of an irreducible Yang-Mills bar connection vanishes,i.e.,(P,δ_(A))is holomorphic.展开更多
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 introduce directional complexities forZ^(q)-measure preserving dynamical systems via a collection of new metrics along non-zero directions inR^(q).It turns out that aZ^(q)-measure preserving dynamical system is rig...We introduce directional complexities forZ^(q)-measure preserving dynamical systems via a collection of new metrics along non-zero directions inR^(q).It turns out that aZ^(q)-measure preserving dynamical system is rigid if and only if the invariant measure has bounded directional complexity.We also obtain ergodic decomposition formula for the measure-theoretic directional entropy of aZ^(q)-measure preserving dynamical system.展开更多
Sarnak’s Möbius disjointness conjecture states that Möbius function is disjoint to any zero entropy dynamics.We prove that Möbius disjointness conjecture holds for onefrequency analytic quasi-periodic ...Sarnak’s Möbius disjointness conjecture states that Möbius function is disjoint to any zero entropy dynamics.We prove that Möbius disjointness conjecture holds for onefrequency analytic quasi-periodic cocycles which are almost reducible,which extends(Liu and Sarnak in Duke Math J 164(7):1353–1399,2015;Wang in Invent Math 209:175–196,2017)to the noncommutative case.The proof relies on quantitative version of almost reducibility.展开更多
We prove that there exists an open and dense subset U in the space of C^(2) expanding self-maps of the circle T such that the Lyapunov minimizing measures of any T∈U are uniquely supported on a periodic orbit.This an...We prove that there exists an open and dense subset U in the space of C^(2) expanding self-maps of the circle T such that the Lyapunov minimizing measures of any T∈U are uniquely supported on a periodic orbit.This answers a conjecture of Jenkinson-Morris in the C^(2) topology.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
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
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
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
In this paper, we use the 2-descent method to find a series of odd non-congruent numbers = 1 （nmd 8） whose prime factors are = 1 （rood 4） such that the congruent elliptic curves have second lowest Selmer groups, w...In this paper, we use the 2-descent method to find a series of odd non-congruent numbers = 1 （nmd 8） whose prime factors are = 1 （rood 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
In this article,we prove that there exists a unique local smooth solution for the Cauchy problem of the Navier–Stokes–Schrodinger system.Our methods rely upon approximating the system with a sequence of perturbed sy...In this article,we prove that there exists a unique local smooth solution for the Cauchy problem of the Navier–Stokes–Schrodinger system.Our methods rely upon approximating the system with a sequence of perturbed system and parallel transport and are closer to the one in Ding and Wang(Sci China 44(11):1446–1464,2001)and McGahagan(Commun Partial Differ Equ 32(1–3):375–400,2007).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
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
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
基金supported by the National Natural Science Foundation of China(11931009,12271495,11971450,and 12071449)Anhui Initiative in Quantum Information Technologies(AHY150200)the Project of Stable Support for Youth Team in Basic Research Field,Chinese Academy of Sciences(YSBR-001).
文摘On a compact Riemann surface with finite punctures P_(1),…P_(k),we define toric curves as multivalued,totallyunramified holomorphic maps to P^(n)with monodromy in a maximal torus of PSU(n+1).Toric solutions to SU(n+1)Todasystems on X\{P_(1);…;P_(k)}are recognized by the associated toric curves in.We introduce character n-ensembles as-tuples of meromorphic one-forms with simple poles and purely imaginary periods,generating toric curves on minus finitelymany points.On X,we establish a correspondence between character-ensembles and toric solutions to the SU(n+1)system with finitely many cone singularities.Our approach not only broadens seminal solutions with two conesingularities on the Riemann sphere,as classified by Jost-Wang(Int.Math.Res.Not.,2002,(6):277-290)andLin-Wei-Ye(Invent.Math.,2012,190(1):169-207),but also advances beyond the limits of Lin-Yang-Zhong’s existencetheorems(J.Differential Geom.,2020,114(2):337-391)by introducing a new solution class.
基金supported by the National Natural Science Foundation of China(12271496)the Youth Innovation Promotion Association CAS,the Fundamental Research Funds of the Central Universities,and the USTC Research Funds of the Double First-Class Initiative.
文摘In this note,we study the Yang-Mills bar connection,i.e.,the curvature of obeys,δ_(A)^(*)F_(A)^(0.2)on a principal G-bundle P over a compact complex manifold.According to the Koszul-Malgrange criterion,any holomorphic structure on can be seen as a solution to this equation.Suppose that G=SU(2)or SO(3)and X is a complex surface with H_(1)(X,Z_(2))=0.We then prove that the-part curvature of an irreducible Yang-Mills bar connection vanishes,i.e.,(P,δ_(A))is holomorphic.
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 Postdoctoral Fellowship Program of CPSF(Grant No.GZC20230363)NNSF of China(Grant No.12401228)+3 种基金supported by NNSF of China(Grant Nos.12031019,12371197)supported by NNSF of China(Grant No.12101582)USTC Research Funds of the Double First-Class Initiative and NSFC grant(Grant Nos.12090012,12090010)supported by NNSF of China(Grant Nos.11801193,12171175)。
文摘We introduce directional complexities forZ^(q)-measure preserving dynamical systems via a collection of new metrics along non-zero directions inR^(q).It turns out that aZ^(q)-measure preserving dynamical system is rigid if and only if the invariant measure has bounded directional complexity.We also obtain ergodic decomposition formula for the measure-theoretic directional entropy of aZ^(q)-measure preserving dynamical system.
基金Tianyuan Mathematical Center in Southwest(No.11826102)supported by NSFC grant(Nos.12090012,12031019,12090010)+8 种基金supported by National Key R&D Program of China(No.2021YFA1001600)NSFC grant(No.11971233)the Outstanding Youth Foundation of Jiangsu Province(No.BK20200074)Qing Lan Project of Jiangsu provincesupported by NSF grant(No.DMS-1753042)supported by National Key R&D Program of China(No.2020YFA0713300)NSFC grant(No.12071232)The Science Fund for Distinguished Young Scholars of Tianjin(No.19JCJQJC61300)Nankai Zhide Foundation.
文摘Sarnak’s Möbius disjointness conjecture states that Möbius function is disjoint to any zero entropy dynamics.We prove that Möbius disjointness conjecture holds for onefrequency analytic quasi-periodic cocycles which are almost reducible,which extends(Liu and Sarnak in Duke Math J 164(7):1353–1399,2015;Wang in Invent Math 209:175–196,2017)to the noncommutative case.The proof relies on quantitative version of almost reducibility.
基金partially supported by National Key R&D Program of China(Grant Nos.2022YFA1005801)NSFC(Grant Nos.12171348,12325106,ZXL2024386)+2 种基金partially supported by NSFC(Grant Nos.12090012,12031019,11731003)partially supported by NSFC(Grant Nos.12031019,11801538,11871188)Jiangsu Specially Appointed Professorship。
文摘We prove that there exists an open and dense subset U in the space of C^(2) expanding self-maps of the circle T such that the Lyapunov minimizing measures of any T∈U are uniquely supported on a periodic orbit.This answers a conjecture of Jenkinson-Morris in the C^(2) topology.
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 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 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 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 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 （nmd 8） whose prime factors are = 1 （rood 4） such that the congruent elliptic curves have second lowest Selmer groups, which include Li and Tian＇s result as special cases.
基金This work was supported by the National Natural Science Foundation of China(Grant No.11771415).
文摘In this article,we prove that there exists a unique local smooth solution for the Cauchy problem of the Navier–Stokes–Schrodinger system.Our methods rely upon approximating the system with a sequence of perturbed system and parallel transport and are closer to the one in Ding and Wang(Sci China 44(11):1446–1464,2001)and McGahagan(Commun Partial Differ Equ 32(1–3):375–400,2007).
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 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 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).
