Let R be a Noetherian unique factorization domain such that 2 and 3 are units,and let A=R[α]be a quartic extension over R by adding a rootαof an irreducible quartic polynomial p(z)=z4+az2+bz+c over R.We will compute...Let R be a Noetherian unique factorization domain such that 2 and 3 are units,and let A=R[α]be a quartic extension over R by adding a rootαof an irreducible quartic polynomial p(z)=z4+az2+bz+c over R.We will compute explicitly the integral closure of A in its fraction field,which is based on a proper factorization of the coefficients and the algebraic invariants of p(z).In fact,we get the factorization by resolving the singularities of a plane curve defined by z4+a(x)z2+b(x)z+c(x)=0.The integral closure is expressed as a syzygy module and the syzygy equations are given explicitly.We compute also the ramifications of the integral closure over R.展开更多
In this paper we first compute the out-of-time-order correlators （OTOC） for both a phenomenological model and a random-field XXZ model in the many-body localized phase. We show that the OTOC decreases in power law i...In this paper we first compute the out-of-time-order correlators （OTOC） for both a phenomenological model and a random-field XXZ model in the many-body localized phase. We show that the OTOC decreases in power law in a many-body localized system at the scrambling time. We also find that the OTOC can also be used to distinguish a many-body localized phase from an Anderson localized phase, while a normal correlator cannot. Furthermore, we prove an exact theorem that relates the growth of the second Renyi entropy in the quench dynamics to the decay of the OTOC in equilibrium. This theorem works for a generic quantum system. We discuss various implications of this theorem.展开更多
We study the existence and uniqueness of the solution to a forward-backward stochastic differential equation with subdifferential operator in the backward equation. This kind of equations includes, as a particular cas...We study the existence and uniqueness of the solution to a forward-backward stochastic differential equation with subdifferential operator in the backward equation. This kind of equations includes, as a particular case, multi-dimensional forward-backward stochastic differential equation where the backward equation is reflected on the boundary of a closed convex(time-independent) domain. Moreover, we give a probabilistic interpretation for the viscosity solution of a kind of quasilinear variational inequalities.展开更多
基金supported by National Natural Science Foundation of China(Grant No.11231003)the Science Foundation of Shanghai(Grant No.13DZ2260600)East China Normal University Reward for Excellent Doctors in Academics(Grant No.XRZZ2012014)
文摘Let R be a Noetherian unique factorization domain such that 2 and 3 are units,and let A=R[α]be a quartic extension over R by adding a rootαof an irreducible quartic polynomial p(z)=z4+az2+bz+c over R.We will compute explicitly the integral closure of A in its fraction field,which is based on a proper factorization of the coefficients and the algebraic invariants of p(z).In fact,we get the factorization by resolving the singularities of a plane curve defined by z4+a(x)z2+b(x)z+c(x)=0.The integral closure is expressed as a syzygy module and the syzygy equations are given explicitly.We compute also the ramifications of the integral closure over R.
基金supported by the National Key Research and Development Plan (2016YFA0301600)the National Natural Science Foundation of China (11325418)Tsinghua University Initiative Scientific Research Program
文摘In this paper we first compute the out-of-time-order correlators （OTOC） for both a phenomenological model and a random-field XXZ model in the many-body localized phase. We show that the OTOC decreases in power law in a many-body localized system at the scrambling time. We also find that the OTOC can also be used to distinguish a many-body localized phase from an Anderson localized phase, while a normal correlator cannot. Furthermore, we prove an exact theorem that relates the growth of the second Renyi entropy in the quench dynamics to the decay of the OTOC in equilibrium. This theorem works for a generic quantum system. We discuss various implications of this theorem.
基金supported by Australian Research Council’s Discovery Projects Funding Scheme(Grant No.DP120100895)
文摘We study the existence and uniqueness of the solution to a forward-backward stochastic differential equation with subdifferential operator in the backward equation. This kind of equations includes, as a particular case, multi-dimensional forward-backward stochastic differential equation where the backward equation is reflected on the boundary of a closed convex(time-independent) domain. Moreover, we give a probabilistic interpretation for the viscosity solution of a kind of quasilinear variational inequalities.
