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
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
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
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
For a number field F and a prime number p, the Z_(p)-torsion module of the Galois group of the maximal abelian pro-p extension of F unramified outside p over F, denoted by T_(p)(F), is an important subject in the abel...For a number field F and a prime number p, the Z_(p)-torsion module of the Galois group of the maximal abelian pro-p extension of F unramified outside p over F, denoted by T_(p)(F), is an important subject in the abelian p-ramification theory. In this paper, we study the group T_(2)(F) = T_(2)(m) of the quadratic field F = Q(m_(1/2)). Firstly, assuming m > 0, we prove an explicit 4-rank formula for quadratic fields that rk4(T_(2)(-m))= rk2(T_(2)(-m))-rank(R), where R is a certain explicitly described Rédei matrix over F_(2). Furthermore, using this formula, we obtain the 4-rank density formula of T_(2)-groups of imaginary quadratic fields. Secondly, for l an odd prime, we obtain the results about the 2-power divisibility of orders of T_(2)(±l) and T_(2)(±2l), both of which are cyclic 2-groups. In particular, we find that #T_(2)(l) ≡ 2#T_(2)(2l) ≡ h_(2)(-2l)(mod 16) if l ≡ 7(mod 8),where h_(2)(-2l) is the 2-class number of Q((-2l)_(1/2)). We then obtain the density results for T_(2)(±l) and T_(2)(±2l)when the orders are small. Finally, based on our density results and numerical data, we propose distribution conjectures about T_(p)(F) when F varies over real or imaginary quadratic fields for any prime p, and about T_(2)(±l)and T_(2)(±2l) when l varies, in the spirit of Cohen-Lenstra heuristics. Our conjecture in the T_(2)(l) case is closely connected to Shanks-Sime-Washington’s speculation on the distributions of the zeros of 2-adic L-functions and to the distributions of the fundamental units.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
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
The regionally proximal relation of order d along arithmetic progressions,namely AP[d]for d 2 N,is introduced and investigated.It turns out that if(X;T)is a topological dynamical system with AP[d]=Δ,then each ergodic...The regionally proximal relation of order d along arithmetic progressions,namely AP[d]for d 2 N,is introduced and investigated.It turns out that if(X;T)is a topological dynamical system with AP[d]=Δ,then each ergodic measure of(X;T)is isomorphic to a d-step pro-nilsystem,and thus(X;T)has zero entropy.Moreover,it is shown that if(X;T)is a strictly ergodic distal system with the property that the maximal topological and measurable d-step pro-nilsystems are isomorphic,then AP[d]=RP[d]for each d 2 N.It follows that for a minimal 1-pro-nilsystem,AP[d]=RP[d]for each d 2 N.An example which is a strictly ergodic distal system with discrete spectrum whose maximal equicontinuous factor is not isomorphic to the Kronecker factor is constructed.展开更多
Area-preserving parameterization is now widely applied,such as for remeshing and medical image processing.We propose an efficient and stable approach to compute area-preserving parameterization on simply connected ope...Area-preserving parameterization is now widely applied,such as for remeshing and medical image processing.We propose an efficient and stable approach to compute area-preserving parameterization on simply connected open surfaces.From an initial parameterization,we construct an objective function of energy.This consists of an area distortion measure and a new regularization,termed as the Tutte regularization,combined into an optimization problem with sliding boundary constraints.The original area-preserving problem is decomposed into a series of subproblems to linearize the boundary constraints.We design an iteration framework based on the augmented Lagrange method to solve each linear constrained subproblem.Our method generates a high-quality parameterization with area-preserving on facets.The experimental results demonstrate the efficacy of the designed framework and the Tutte regularization for achieving a fine parameterization.展开更多
In this paper we introduce two metrics:the max metric d_(n,q)and the mean metric d_(n,q).We give an equivalent characterization of rigid measure preserving systems by the two metrics.It turns out that an invariant mea...In this paper we introduce two metrics:the max metric d_(n,q)and the mean metric d_(n,q).We give an equivalent characterization of rigid measure preserving systems by the two metrics.It turns out that an invariant measureμon a topological dynamical system(X,T)has bounded complexity with respect to d_(n,q)if and only ifμhas bounded complexity with respect to d_(n,q)if and only if(X,B_X,μ,T)is rigid.We also obtain computation formulas of the measure-theoretic entropy of an ergodic measure preserving system(resp.the topological entropy of a topological dynamical system)by the two metrics dn,q and dn,q.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(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supported by Anhui Initiative in Quantum Information Technologies(Grant No.AHY150200)。
文摘For a number field F and a prime number p, the Z_(p)-torsion module of the Galois group of the maximal abelian pro-p extension of F unramified outside p over F, denoted by T_(p)(F), is an important subject in the abelian p-ramification theory. In this paper, we study the group T_(2)(F) = T_(2)(m) of the quadratic field F = Q(m_(1/2)). Firstly, assuming m > 0, we prove an explicit 4-rank formula for quadratic fields that rk4(T_(2)(-m))= rk2(T_(2)(-m))-rank(R), where R is a certain explicitly described Rédei matrix over F_(2). Furthermore, using this formula, we obtain the 4-rank density formula of T_(2)-groups of imaginary quadratic fields. Secondly, for l an odd prime, we obtain the results about the 2-power divisibility of orders of T_(2)(±l) and T_(2)(±2l), both of which are cyclic 2-groups. In particular, we find that #T_(2)(l) ≡ 2#T_(2)(2l) ≡ h_(2)(-2l)(mod 16) if l ≡ 7(mod 8),where h_(2)(-2l) is the 2-class number of Q((-2l)_(1/2)). We then obtain the density results for T_(2)(±l) and T_(2)(±2l)when the orders are small. Finally, based on our density results and numerical data, we propose distribution conjectures about T_(p)(F) when F varies over real or imaginary quadratic fields for any prime p, and about T_(2)(±l)and T_(2)(±2l) when l varies, in the spirit of Cohen-Lenstra heuristics. Our conjecture in the T_(2)(l) case is closely connected to Shanks-Sime-Washington’s speculation on the distributions of the zeros of 2-adic L-functions and to the distributions of the fundamental units.
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This work was supported by National Natural Science Foundation of China(Grant Nos.11431012,11971455,11571335 and 11371339).
文摘The regionally proximal relation of order d along arithmetic progressions,namely AP[d]for d 2 N,is introduced and investigated.It turns out that if(X;T)is a topological dynamical system with AP[d]=Δ,then each ergodic measure of(X;T)is isomorphic to a d-step pro-nilsystem,and thus(X;T)has zero entropy.Moreover,it is shown that if(X;T)is a strictly ergodic distal system with the property that the maximal topological and measurable d-step pro-nilsystems are isomorphic,then AP[d]=RP[d]for each d 2 N.It follows that for a minimal 1-pro-nilsystem,AP[d]=RP[d]for each d 2 N.An example which is a strictly ergodic distal system with discrete spectrum whose maximal equicontinuous factor is not isomorphic to the Kronecker factor is constructed.
基金supported by Anhui Center for Applied Mathematics,the NSF of China (No.11871447)the special project of strategic leading science and technology of CAS (No.XDC08010100)the National Key Research and Development Program of MOST of China (No.2018AAA0101001).
文摘Area-preserving parameterization is now widely applied,such as for remeshing and medical image processing.We propose an efficient and stable approach to compute area-preserving parameterization on simply connected open surfaces.From an initial parameterization,we construct an objective function of energy.This consists of an area distortion measure and a new regularization,termed as the Tutte regularization,combined into an optimization problem with sliding boundary constraints.The original area-preserving problem is decomposed into a series of subproblems to linearize the boundary constraints.We design an iteration framework based on the augmented Lagrange method to solve each linear constrained subproblem.Our method generates a high-quality parameterization with area-preserving on facets.The experimental results demonstrate the efficacy of the designed framework and the Tutte regularization for achieving a fine parameterization.
基金Supported by NNSF of China(Grant Nos.11971455,11801538,11801193,11871188,11731003 and 12090012)supported by STU Scientific Research Foundation for Talents(Grant No.NTF19047)。
文摘In this paper we introduce two metrics:the max metric d_(n,q)and the mean metric d_(n,q).We give an equivalent characterization of rigid measure preserving systems by the two metrics.It turns out that an invariant measureμon a topological dynamical system(X,T)has bounded complexity with respect to d_(n,q)if and only ifμhas bounded complexity with respect to d_(n,q)if and only if(X,B_X,μ,T)is rigid.We also obtain computation formulas of the measure-theoretic entropy of an ergodic measure preserving system(resp.the topological entropy of a topological dynamical system)by the two metrics dn,q and dn,q.
