A Noetherian(Artinian)Lie algebra satisfies the maximal(minimal)condition for ideals.Generalisations include quasi-Noetherian and quasi-Artinian Lie algebras.We study conditions on prime ideals relating these properti...A Noetherian(Artinian)Lie algebra satisfies the maximal(minimal)condition for ideals.Generalisations include quasi-Noetherian and quasi-Artinian Lie algebras.We study conditions on prime ideals relating these properties.We prove that the radical of any ideal of a quasi-Artinian Lie algebra is the intersection of finitely many prime ideals,and an ideally finite Lie algebra is quasi-Noetherian if and only if it is quasi-Artinian.Both properties are equivalent to soluble-by-finite.We also prove a structure theorem for serially finite Artinian Lie algebras.展开更多
This paper is mainly concerned with the following nonlinear p-Laplacian equation-△pu(x)+(λa(x)+1)|u|^(p-2)(x)u(x)=f(x,u(x)),in V on a locally finite graph G=(V,E)with more general nonlinear term,whereΔp is the disc...This paper is mainly concerned with the following nonlinear p-Laplacian equation-△pu(x)+(λa(x)+1)|u|^(p-2)(x)u(x)=f(x,u(x)),in V on a locally finite graph G=(V,E)with more general nonlinear term,whereΔp is the discrete pLaplacian on graphs,p≥2.Under some suitable conditions on f and a(x),we can prove that the equation admits a positive solution by the Mountain Pass theorem and a ground state solution uλvia the method of Nehari manifold,for anyλ>1.In addition,asλ→+∞,we prove that the solution uλconverge to a solution of the following Dirichlet problem{-△pu(x)+|u|^(p-2)(x)u(x)=f(x,u(x)),inΩ,u(x)=0,onδΩwhereΩ={x∈V:a(x)=0}is the potential well and δΩ denotes the the boundary ofΩ.展开更多
A local bivariate C1 quasi-interpolating spline operator with a four directional mesh is considered and studied. Based on the above operator we present cubature formulas for 2-D singular integrals, defined in the Hada...A local bivariate C1 quasi-interpolating spline operator with a four directional mesh is considered and studied. Based on the above operator we present cubature formulas for 2-D singular integrals, defined in the Hadamard finite part sense. Convergence results are obtained for a wide class of functions. Moreover numerical tests are given.展开更多
In this paper, we first propose the maximum arc problem, normal rational curve conjecture, and extensions of normal rational curves over finite local rings, analogously to the finite geometry over finite fields. We th...In this paper, we first propose the maximum arc problem, normal rational curve conjecture, and extensions of normal rational curves over finite local rings, analogously to the finite geometry over finite fields. We then study the deep hole problem of generalized Reed-Solomon (RS) codes over finite local rings. Several different classes of deep holes are constructed. The relationship between finite geometry and deep holes of RS codes over finite local rings are also studied.展开更多
We study the algebraic structure of rings R whose zero-divisor graph T(R)has clique number four.Furthermore,we give complete characterizations of all the finite commutative local rings with clique number 4.
Let be a function on locally finite connect graph G=(V,E)andΩbe a bounded subset of V.We consider the nonlinear Dirichlet boundary condition problem{-△u=f(u),inΩ,u=0,onδΩ.Let f:R→R be a function satisfying certa...Let be a function on locally finite connect graph G=(V,E)andΩbe a bounded subset of V.We consider the nonlinear Dirichlet boundary condition problem{-△u=f(u),inΩ,u=0,onδΩ.Let f:R→R be a function satisfying certain assumptions.Then under the functional framework we use the three-solution theorem and the variational method to prove that the above equation has at least three solutions,of which one is trivial and the others are strictly positive.展开更多
By using the perpetual cutoff method,we prove two discrete versions of gradient estimates for bounded Laplacian on locally finite graphs with exception sets under the condition of CDE′(K,N).This generalizes a main re...By using the perpetual cutoff method,we prove two discrete versions of gradient estimates for bounded Laplacian on locally finite graphs with exception sets under the condition of CDE′(K,N).This generalizes a main result of F.Münch who considers the case of CD(K,∞)curvature.Hence,we answer a question raised by Münch.For that purpose,we characterize some basic properties of radical form of the perpetual cutoff semigroup and give a weak commutation relation between bounded LaplacianΔand perpetual cutoff semigroup P w t in our setting.展开更多
There are no simple singular Whittaker modules over most of important algebras,such as simple complex finite-dimensional Lie algebras,affine Kac-Moody Lie algebras,the Virasoro algebra,the Heisenberg-Virasoro algebra ...There are no simple singular Whittaker modules over most of important algebras,such as simple complex finite-dimensional Lie algebras,affine Kac-Moody Lie algebras,the Virasoro algebra,the Heisenberg-Virasoro algebra and the Schrödinger-Witt algebra.In this paper,however,we construct simple singular Whittaker modules over the Schrödinger algebra.Moreover,simple singular Whittaker modules over the Schrödinger algebra are classified.As a result,simple modules for the Schrödinger algebrawhich are locally finite over the positive part are completely classified.We also give characterizations of simple highestweight modules and simple singular Whittaker modules.展开更多
文摘A Noetherian(Artinian)Lie algebra satisfies the maximal(minimal)condition for ideals.Generalisations include quasi-Noetherian and quasi-Artinian Lie algebras.We study conditions on prime ideals relating these properties.We prove that the radical of any ideal of a quasi-Artinian Lie algebra is the intersection of finitely many prime ideals,and an ideally finite Lie algebra is quasi-Noetherian if and only if it is quasi-Artinian.Both properties are equivalent to soluble-by-finite.We also prove a structure theorem for serially finite Artinian Lie algebras.
文摘This paper is mainly concerned with the following nonlinear p-Laplacian equation-△pu(x)+(λa(x)+1)|u|^(p-2)(x)u(x)=f(x,u(x)),in V on a locally finite graph G=(V,E)with more general nonlinear term,whereΔp is the discrete pLaplacian on graphs,p≥2.Under some suitable conditions on f and a(x),we can prove that the equation admits a positive solution by the Mountain Pass theorem and a ground state solution uλvia the method of Nehari manifold,for anyλ>1.In addition,asλ→+∞,we prove that the solution uλconverge to a solution of the following Dirichlet problem{-△pu(x)+|u|^(p-2)(x)u(x)=f(x,u(x)),inΩ,u(x)=0,onδΩwhereΩ={x∈V:a(x)=0}is the potential well and δΩ denotes the the boundary ofΩ.
文摘A local bivariate C1 quasi-interpolating spline operator with a four directional mesh is considered and studied. Based on the above operator we present cubature formulas for 2-D singular integrals, defined in the Hadamard finite part sense. Convergence results are obtained for a wide class of functions. Moreover numerical tests are given.
基金The research of Jun Zhang was supported by the National Natural Science Foundation of China(Grant No.11971321)by National Key Research and Development Program of China(Grant No.2018YFA0704703)The research of Haiyan Zhou was supported by the National Natural Science Foundation of China(Grant No.12071221).
文摘In this paper, we first propose the maximum arc problem, normal rational curve conjecture, and extensions of normal rational curves over finite local rings, analogously to the finite geometry over finite fields. We then study the deep hole problem of generalized Reed-Solomon (RS) codes over finite local rings. Several different classes of deep holes are constructed. The relationship between finite geometry and deep holes of RS codes over finite local rings are also studied.
基金This research was supported by the National Natural Science Foundation of China(No.11801356,No.11401368,No.11971338)by the Natural Science Foundation of Shanghai(No.19ZR1424100).
文摘We study the algebraic structure of rings R whose zero-divisor graph T(R)has clique number four.Furthermore,we give complete characterizations of all the finite commutative local rings with clique number 4.
文摘Let be a function on locally finite connect graph G=(V,E)andΩbe a bounded subset of V.We consider the nonlinear Dirichlet boundary condition problem{-△u=f(u),inΩ,u=0,onδΩ.Let f:R→R be a function satisfying certain assumptions.Then under the functional framework we use the three-solution theorem and the variational method to prove that the above equation has at least three solutions,of which one is trivial and the others are strictly positive.
文摘By using the perpetual cutoff method,we prove two discrete versions of gradient estimates for bounded Laplacian on locally finite graphs with exception sets under the condition of CDE′(K,N).This generalizes a main result of F.Münch who considers the case of CD(K,∞)curvature.Hence,we answer a question raised by Münch.For that purpose,we characterize some basic properties of radical form of the perpetual cutoff semigroup and give a weak commutation relation between bounded LaplacianΔand perpetual cutoff semigroup P w t in our setting.
文摘There are no simple singular Whittaker modules over most of important algebras,such as simple complex finite-dimensional Lie algebras,affine Kac-Moody Lie algebras,the Virasoro algebra,the Heisenberg-Virasoro algebra and the Schrödinger-Witt algebra.In this paper,however,we construct simple singular Whittaker modules over the Schrödinger algebra.Moreover,simple singular Whittaker modules over the Schrödinger algebra are classified.As a result,simple modules for the Schrödinger algebrawhich are locally finite over the positive part are completely classified.We also give characterizations of simple highestweight modules and simple singular Whittaker modules.
