This paper deals with the Nash inequalities and the related ones for general symmetric forms which can be very much unbounded. Some sufficient conditions in terms of the isoperimetric inequalities and some necessary c...This paper deals with the Nash inequalities and the related ones for general symmetric forms which can be very much unbounded. Some sufficient conditions in terms of the isoperimetric inequalities and some necessary conditions for the inequalities are presented. The resulting conditions can be sharp qualitatively as illustrated by some examples. It turns out that for a probability measure, the Nash inequalities are much stronger than the Poincare and the logarithmic Sobolev inequalities in the present context.展开更多
Some estimates of logarithmic Sobolev constant for general symmetric forms are obtained in terms of new Cheeger’s constants. The estimates can be sharp in some sense.
Nakao＇s stochastic integrals for continuous additive functionals of zero energy are extended from the symmetric Dirichlet forms setting to the non-symmetric Dirichlet forms setting. ItS＇s formula in terms of the ext...Nakao＇s stochastic integrals for continuous additive functionals of zero energy are extended from the symmetric Dirichlet forms setting to the non-symmetric Dirichlet forms setting. ItS＇s formula in terms of the extended stochastic integrals is obtained.展开更多
Lp Poincare inequalities for general symmetric forms are established by new Cheeger＇s isoperimetric constants. Lp super-Poincare inequalities are introduced to describe the equivalent conditions for the Lp compact em...Lp Poincare inequalities for general symmetric forms are established by new Cheeger＇s isoperimetric constants. Lp super-Poincare inequalities are introduced to describe the equivalent conditions for the Lp compact embedding, and the criteria via the new Cheeger＇s constants for those inequalities are presented. Finally, the concentration or the volume growth of measures for these inequalities are studied.展开更多
Weak log-Sobolev and Lp weak Poincare inequalities for general symmetric forms are investigated by using newly defined Cheeger＇s isoperimetric constants. Some concrete examples of ergodic birth-death processes are al...Weak log-Sobolev and Lp weak Poincare inequalities for general symmetric forms are investigated by using newly defined Cheeger＇s isoperimetric constants. Some concrete examples of ergodic birth-death processes are also presented to illustrate the results.展开更多
Some sufficient conditions for the F-Sobolev inequality for symmetric forms are presented in terms of new Cheeger’s constants. Meanwhile, an estimate of the F-Sobolev constants is obtained.
In this paper we study the singularity at the origin with three–fold zeroeigenvalue for symmetric vector fields with nilpotent linear part and 3–jet C~∞–equivalent toy(partial deriv)/(partial deriv) + z(partial de...In this paper we study the singularity at the origin with three–fold zeroeigenvalue for symmetric vector fields with nilpotent linear part and 3–jet C~∞–equivalent toy(partial deriv)/(partial deriv) + z(partial deriv)/(partial deriv)y + ax^2y (partialderiv)/(partial deriv)/z with a ≠ 0. We first obtain several subfamilies of the symmetric versalunfoldings of this singularity by using the normal form and blow–up methods under some conditions,and derive the local and global bifurcation behavior, then prove analytically the existence of theSilnikov homoclinic bifurcation for some subfamilies of the symmetric versal unfoldings of thissingularity, by using the generalized Melnikov methods of a homoclinic orbit to a hyperbolic ornon–hyperbolic equilibrium in a highdimensional space.展开更多
We survey the recent development of the DeGiorgi-Nash-Moser-Aronson type theory for a class of symmetric jump processes(or equivalently,a class of symmetric integro-differential operators).We focus on the sharp two-si...We survey the recent development of the DeGiorgi-Nash-Moser-Aronson type theory for a class of symmetric jump processes(or equivalently,a class of symmetric integro-differential operators).We focus on the sharp two-sided estimates for the transition density functions(or heat kernels) of the processes,a priori Hlder estimate and parabolic Harnack inequalities for their parabolic functions.In contrast to the second order elliptic differential operator case,the methods to establish these properties for symmetric integro-differential operators are mainly probabilistic.展开更多
The authors establish a Cheeger-Müller type theorem for the complex valued analytic torsion introduced by Burghelea and Hailer for fiat vector bundles carrying nondegenerate symmetric bilinear forms. As a consequ...The authors establish a Cheeger-Müller type theorem for the complex valued analytic torsion introduced by Burghelea and Hailer for fiat vector bundles carrying nondegenerate symmetric bilinear forms. As a consequence, they prove the Burghelea-Haller conjecture in full generality, which gives an analytic interpretation of （the square of） the Turaev torsion.展开更多
In this paper,we prove the existence of general Cartesian vector solutions u=b(t)+A(t)x for the Ndimensional compressible Navier–Stokes equations with density-dependent viscosity,based on the matrix and curve integra...In this paper,we prove the existence of general Cartesian vector solutions u=b(t)+A(t)x for the Ndimensional compressible Navier–Stokes equations with density-dependent viscosity,based on the matrix and curve integration theory.Two exact solutions are obtained by solving the reduced systems.展开更多
In this paper,we study the Sil’nikov heteroclinic bifurcations,which display strange attractors,for the symmetric versal unfoldings of the singularity at the origin with a nilpotent linear part and 3-jet,using the no...In this paper,we study the Sil’nikov heteroclinic bifurcations,which display strange attractors,for the symmetric versal unfoldings of the singularity at the origin with a nilpotent linear part and 3-jet,using the normal form,the blow-up and the generalized Mel’nikov methods of heteroclinic orbits to two hyperbolic or nonhyperbolic equilibria in a high-dimensional space.展开更多
We prove the following vanishing theorem. Let M be an irreducible symmetric space of noncompact type whose dimension exceeds 2 and M ≠SO0（2, 2）/SO（2） × SO（2）. Let π ： E →＊ M be any vector bundle. Then ...We prove the following vanishing theorem. Let M be an irreducible symmetric space of noncompact type whose dimension exceeds 2 and M ≠SO0（2, 2）/SO（2） × SO（2）. Let π ： E →＊ M be any vector bundle. Then any E-valued L2 harmonic 1-form over M vanishes. In particular we get the vanishing theorem for harmonic maps from irreducible symmetric spaces of noncompact type.展开更多
In this paper the projective semi symmetric connection D is studied, which is projectively equivalent to the Levi_Civita connection . An intrinsic projective invariant is found out and a necessary and suffici...In this paper the projective semi symmetric connection D is studied, which is projectively equivalent to the Levi_Civita connection . An intrinsic projective invariant is found out and a necessary and sufficient condition is given. Furthermore, another condition is obtained when the convariant derivative of the projective invariant is kept.展开更多
The fact that infinite-dimensional algebra exists in a 2-dimensional Lax-pair system has caused keen interest.Using a variety of particular models, many explicit expressions have already been derived. Since the hidden...The fact that infinite-dimensional algebra exists in a 2-dimensional Lax-pair system has caused keen interest.Using a variety of particular models, many explicit expressions have already been derived. Since the hidden symmetry algebra was introduced in principal chiral model, the study of axially symmetric gravity with展开更多
The complete space-like hypersurfaces with constant normal saclar curvature is discussed in a locally symmetric Lorentz space. A classified theorem is obtained by the operator L1 introduced by S Y Cheng and S T Yau [3].
基金Research supported in part by NSFC (No. 19631060), Math. Tian Yuan Found., Qiu Shi Sci. & Tech. Found., RFDP and MCME
文摘This paper deals with the Nash inequalities and the related ones for general symmetric forms which can be very much unbounded. Some sufficient conditions in terms of the isoperimetric inequalities and some necessary conditions for the inequalities are presented. The resulting conditions can be sharp qualitatively as illustrated by some examples. It turns out that for a probability measure, the Nash inequalities are much stronger than the Poincare and the logarithmic Sobolev inequalities in the present context.
文摘Some estimates of logarithmic Sobolev constant for general symmetric forms are obtained in terms of new Cheeger’s constants. The estimates can be sharp in some sense.
基金supported by National Natural Science Foundation of China (Grant No.10961012)Natural Sciences and Engineering Research Council of Canada (Grant No. 311945-2008)
文摘Nakao＇s stochastic integrals for continuous additive functionals of zero energy are extended from the symmetric Dirichlet forms setting to the non-symmetric Dirichlet forms setting. ItS＇s formula in terms of the extended stochastic integrals is obtained.
基金Supported in part by Program for New Century Excellent Talents in University (NCET)973 Project (Grant No.2006CB805901)National Natural Science Foundation of China (Grant No.10721091)
文摘Lp Poincare inequalities for general symmetric forms are established by new Cheeger＇s isoperimetric constants. Lp super-Poincare inequalities are introduced to describe the equivalent conditions for the Lp compact embedding, and the criteria via the new Cheeger＇s constants for those inequalities are presented. Finally, the concentration or the volume growth of measures for these inequalities are studied.
文摘Weak log-Sobolev and Lp weak Poincare inequalities for general symmetric forms are investigated by using newly defined Cheeger＇s isoperimetric constants. Some concrete examples of ergodic birth-death processes are also presented to illustrate the results.
文摘Some sufficient conditions for the F-Sobolev inequality for symmetric forms are presented in terms of new Cheeger’s constants. Meanwhile, an estimate of the F-Sobolev constants is obtained.
基金Project supported by the National Natural Science Foundation of China(No.10171044)the Foundation for University Key Teachers of the Ministry of Education 34C05,34C15,58F14,58F30
文摘In this paper we study the singularity at the origin with three–fold zeroeigenvalue for symmetric vector fields with nilpotent linear part and 3–jet C~∞–equivalent toy(partial deriv)/(partial deriv) + z(partial deriv)/(partial deriv)y + ax^2y (partialderiv)/(partial deriv)/z with a ≠ 0. We first obtain several subfamilies of the symmetric versalunfoldings of this singularity by using the normal form and blow–up methods under some conditions,and derive the local and global bifurcation behavior, then prove analytically the existence of theSilnikov homoclinic bifurcation for some subfamilies of the symmetric versal unfoldings of thissingularity, by using the generalized Melnikov methods of a homoclinic orbit to a hyperbolic ornon–hyperbolic equilibrium in a highdimensional space.
基金supported by National Science Foundation of USA(Grant No.DMS-0600206)
文摘We survey the recent development of the DeGiorgi-Nash-Moser-Aronson type theory for a class of symmetric jump processes(or equivalently,a class of symmetric integro-differential operators).We focus on the sharp two-sided estimates for the transition density functions(or heat kernels) of the processes,a priori Hlder estimate and parabolic Harnack inequalities for their parabolic functions.In contrast to the second order elliptic differential operator case,the methods to establish these properties for symmetric integro-differential operators are mainly probabilistic.
基金the Qiushi Foundationthe National Natural Science Foundation of China (Nos.10571088,10621101)
文摘The authors establish a Cheeger-Müller type theorem for the complex valued analytic torsion introduced by Burghelea and Hailer for fiat vector bundles carrying nondegenerate symmetric bilinear forms. As a consequence, they prove the Burghelea-Haller conjecture in full generality, which gives an analytic interpretation of （the square of） the Turaev torsion.
基金This research is partially supported by the National Science Foundation of China(Grant No.11271079,10671095)RG 11/2015-2016R from the Education University of Hong Kong。
文摘In this paper,we prove the existence of general Cartesian vector solutions u=b(t)+A(t)x for the Ndimensional compressible Navier–Stokes equations with density-dependent viscosity,based on the matrix and curve integration theory.Two exact solutions are obtained by solving the reduced systems.
文摘In this paper,we study the Sil’nikov heteroclinic bifurcations,which display strange attractors,for the symmetric versal unfoldings of the singularity at the origin with a nilpotent linear part and 3-jet,using the normal form,the blow-up and the generalized Mel’nikov methods of heteroclinic orbits to two hyperbolic or nonhyperbolic equilibria in a high-dimensional space.
文摘We prove the following vanishing theorem. Let M be an irreducible symmetric space of noncompact type whose dimension exceeds 2 and M ≠SO0（2, 2）/SO（2） × SO（2）. Let π ： E →＊ M be any vector bundle. Then any E-valued L2 harmonic 1-form over M vanishes. In particular we get the vanishing theorem for harmonic maps from irreducible symmetric spaces of noncompact type.
文摘In this paper the projective semi symmetric connection D is studied, which is projectively equivalent to the Levi_Civita connection . An intrinsic projective invariant is found out and a necessary and sufficient condition is given. Furthermore, another condition is obtained when the convariant derivative of the projective invariant is kept.
文摘The fact that infinite-dimensional algebra exists in a 2-dimensional Lax-pair system has caused keen interest.Using a variety of particular models, many explicit expressions have already been derived. Since the hidden symmetry algebra was introduced in principal chiral model, the study of axially symmetric gravity with
基金Supported the NSF of the Education Department of Jiangsu Province(04KJD110192)
文摘The complete space-like hypersurfaces with constant normal saclar curvature is discussed in a locally symmetric Lorentz space. A classified theorem is obtained by the operator L1 introduced by S Y Cheng and S T Yau [3].
