In this paper, we construct a symplectic bigraded Toda hierarchy which contains an symplectic deformation of the original Toda lattice hierarchy. In particular, we give the rational solutions which are expressed by th...In this paper, we construct a symplectic bigraded Toda hierarchy which contains an symplectic deformation of the original Toda lattice hierarchy. In particular, we give the rational solutions which are expressed by the products of the symplectic Schur polynomials.展开更多
This paper studies Morita duality of semigroup-graded rings, and discusses an equiva, lence between duality functors of graded module category and bigraded bimodules. An important result is obtained： A semigroup bigr...This paper studies Morita duality of semigroup-graded rings, and discusses an equiva, lence between duality functors of graded module category and bigraded bimodules. An important result is obtained： A semigroup bigraded R-A-bimodule Q defines a semigroup graded Morita duality if and only if Q is gr-faithfully balanced and Ref（RQ）, Ref（QA） is closed under graded submodules and graded quotients.展开更多
Let S = K[x1,... ,xn] be the polynomial ring over a field K, and let I C S be a graded ideal. It is shown that the higher iterated Hilbert coefficients of the graded S-modules Tori^S（M,Ik） and Exts^i（M,Ik） are pol...Let S = K[x1,... ,xn] be the polynomial ring over a field K, and let I C S be a graded ideal. It is shown that the higher iterated Hilbert coefficients of the graded S-modules Tori^S（M,Ik） and Exts^i（M,Ik） are polynomial functions in k, and an upper bound for their degree is given. These results are derived by considering suitable bigraded modules.展开更多
This paper analyzes the reduction of the well known Kadomtsev-Petviashvili hierarchy. The reduction yields a previously unknown dispersion counterpart of the dispersionless hierarchy which has a Lax function of the fo...This paper analyzes the reduction of the well known Kadomtsev-Petviashvili hierarchy. The reduction yields a previously unknown dispersion counterpart of the dispersionless hierarchy which has a Lax function of the form p＋u（x）（p-φ）^-1＋v（x）（p-φ）^-2. This paper also describes the bihamiltonian structure of the reduced hierarchy using Dirac reduction and proves that the approximation for the reduced hierarchy up to the second order of the dispersion parameter coincides with the hierarchy of integrable systems constructed from a particular twodimensional Frobenius manifold using the approach of Dubrovin and Zhang.展开更多
The authors give finite dimensional exponential solutions of the bigraded Toda hierarchy （BTH）. As a specific example of exponential solutions of the BTH, the authors consider a regular solution for the （1, 2）-BTH...The authors give finite dimensional exponential solutions of the bigraded Toda hierarchy （BTH）. As a specific example of exponential solutions of the BTH, the authors consider a regular solution for the （1, 2）-BTH with a 3 × 3-sized Lax matrix, and discuss some geometric structures of the solution from which the difference between the （1, 2）- BTH and the original Toda hierarchy is shown. After this, the authors construct another kind of Lax representation of （N, 1）-BTH which does not use the fractional operator of Lax operator. Then the authors introduce the lattice Miura transformation of （N, 1）-BTH which leads to equations depending on one field, and meanwhile some specific examples which contain the Volterra lattice equation （a useful ecological competition model） are given.展开更多
基金Supported by National Natural Science Foundation of China (Grant No. 12071237)。
文摘In this paper, we construct a symplectic bigraded Toda hierarchy which contains an symplectic deformation of the original Toda lattice hierarchy. In particular, we give the rational solutions which are expressed by the products of the symplectic Schur polynomials.
基金Foundation item： the National Natural Science Foundation of China （No. 10571043 10671053） the Natural Science Foundation of Hebei Province （No. 102132） and the Fundation of the Education Department of Hebei Province （No. 2004108）.
文摘This paper studies Morita duality of semigroup-graded rings, and discusses an equiva, lence between duality functors of graded module category and bigraded bimodules. An important result is obtained： A semigroup bigraded R-A-bimodule Q defines a semigroup graded Morita duality if and only if Q is gr-faithfully balanced and Ref（RQ）, Ref（QA） is closed under graded submodules and graded quotients.
文摘Let S = K[x1,... ,xn] be the polynomial ring over a field K, and let I C S be a graded ideal. It is shown that the higher iterated Hilbert coefficients of the graded S-modules Tori^S（M,Ik） and Exts^i（M,Ik） are polynomial functions in k, and an upper bound for their degree is given. These results are derived by considering suitable bigraded modules.
文摘This paper analyzes the reduction of the well known Kadomtsev-Petviashvili hierarchy. The reduction yields a previously unknown dispersion counterpart of the dispersionless hierarchy which has a Lax function of the form p＋u（x）（p-φ）^-1＋v（x）（p-φ）^-2. This paper also describes the bihamiltonian structure of the reduced hierarchy using Dirac reduction and proves that the approximation for the reduced hierarchy up to the second order of the dispersion parameter coincides with the hierarchy of integrable systems constructed from a particular twodimensional Frobenius manifold using the approach of Dubrovin and Zhang.
基金supported by the National Natural Science Foundation of China(Nos.11201251,10971109)the Natural Science Foundation of Zhejiang Province(No.LY12A01007)the K.C.Wong Magna Fundin Ningbo University
文摘The authors give finite dimensional exponential solutions of the bigraded Toda hierarchy （BTH）. As a specific example of exponential solutions of the BTH, the authors consider a regular solution for the （1, 2）-BTH with a 3 × 3-sized Lax matrix, and discuss some geometric structures of the solution from which the difference between the （1, 2）- BTH and the original Toda hierarchy is shown. After this, the authors construct another kind of Lax representation of （N, 1）-BTH which does not use the fractional operator of Lax operator. Then the authors introduce the lattice Miura transformation of （N, 1）-BTH which leads to equations depending on one field, and meanwhile some specific examples which contain the Volterra lattice equation （a useful ecological competition model） are given.
