Let f,g and h be three distinct primitive holomorphic cusp forms of even integral weights k_(1),k_(2)and k_(3)for the full modular groupΓ=SL(2,Z),and denote byλ_(f)(n),λ_(g)(n),λ_(h)(n)the corresponding normalized...Let f,g and h be three distinct primitive holomorphic cusp forms of even integral weights k_(1),k_(2)and k_(3)for the full modular groupΓ=SL(2,Z),and denote byλ_(f)(n),λ_(g)(n),λ_(h)(n)the corresponding normalized Fourier coefficients,respectively.In this paper,we investigate the correlations of triple sums associated to these Fourier coefficientsλ_(f)(n),λ_(g)(n),λ_(h)(n)over certain polynomials,and obtain some power-saving asymptotic estimates which beat the trivial bounds.展开更多
In this paper,we study the eigenvalue problem of the Markov diffusion operator L^(2),and give generalized inequalities for eigenvalues of the operator L^(2)on a Markov diffusion triple.By applying these inequalities,w...In this paper,we study the eigenvalue problem of the Markov diffusion operator L^(2),and give generalized inequalities for eigenvalues of the operator L^(2)on a Markov diffusion triple.By applying these inequalities,we then get some new universal bounds for eigenvalues of a special Markov diffusion operator L^(2)on bounded domains in an Euclidean space.Moreover,our results can reveal the relationship between the(k+1)-th eigenvalue and the first k eigenvalues in a relatively straightforward manner.展开更多
In this paper,we investigate the weighted Dirichlet eigenvalue problem of polynomial operator of the drifting Laplacian on the cigar soliton■as follows■where is a positive continuous function on,denotes the outward ...In this paper,we investigate the weighted Dirichlet eigenvalue problem of polynomial operator of the drifting Laplacian on the cigar soliton■as follows■where is a positive continuous function on,denotes the outward unit normal to the boundary,and are two nonnegative constants.We establish some universal inequalities for eigenvalues of this problem.展开更多
Let Un be the set of connected unicyclic graphs of order n and girth g.Let C(T_(1),T_(2),...,T_(g))Un be obtained from a cycle v_(1)v_(2)…v_(g)v_(1)(in the anticlockwise direction)by identifying vi with the root of a...Let Un be the set of connected unicyclic graphs of order n and girth g.Let C(T_(1),T_(2),...,T_(g))Un be obtained from a cycle v_(1)v_(2)…v_(g)v_(1)(in the anticlockwise direction)by identifying vi with the root of a rooted tree Ti of order ni for each i=1,2,...,g,where ni≥1 and∑^(g)_(i=1)n_(i)=n.Let S(n_(1),n_(2),...,n_(g))be obtained from C(T_(1),T_(2),..,T_(g))by replacing each Ti by a rooted star Sni with the center as its root.Let U(n_(1),n_(2),...,ng)be the set of unicyclic graphs which differ from the unicyclic graph S(n_(1),n_(2),...,n_(g))only up to a permutation of ni's.In this paper,the graph with the minimal least signless Laplacian eigenvalue(respectively,the graph with maximum signless Laplacian spread)in U(n_(1),n_(2),...,n_(g))is determined.展开更多
This paper is concerned with the following nonlinear Steklov problemΔu=0 in D,∂vu=λf(u)on∂D,where D is the unit disk in the plane,∂v denotes the unit outward normal derivative.For each k∈N,under some natural condit...This paper is concerned with the following nonlinear Steklov problemΔu=0 in D,∂vu=λf(u)on∂D,where D is the unit disk in the plane,∂v denotes the unit outward normal derivative.For each k∈N,under some natural conditions on f,using the Crandall-Rabinowitz bifurcation theorem,we obtain a bifurcation curve emanating from(k,0).Furthermore,we also analyze the local structure of bifurcation curves and stability of solutions on them.Specifically,our results indicate the bifurcation is critical for each k and is subcritical(supercritical)if f'''(0)>0(f'''(0)<0).展开更多
基金Supported in part by NSFC(Nos.12401011,12201214)National Key Research and Development Program of China(No.2021YFA1000700)+3 种基金Shaanxi Fundamental Science Research Project for Mathematics and Physics(No.23JSQ053)Science and Technology Program for Youth New Star of Shaanxi Province(No.2025ZC-KJXX-29)Natural Science Basic Research Program of Shaanxi Province(No.2025JC-YBQN-091)Scientific Research Foundation for Young Talents of WNU(No.2024XJ-QNRC-01)。
文摘Let f,g and h be three distinct primitive holomorphic cusp forms of even integral weights k_(1),k_(2)and k_(3)for the full modular groupΓ=SL(2,Z),and denote byλ_(f)(n),λ_(g)(n),λ_(h)(n)the corresponding normalized Fourier coefficients,respectively.In this paper,we investigate the correlations of triple sums associated to these Fourier coefficientsλ_(f)(n),λ_(g)(n),λ_(h)(n)over certain polynomials,and obtain some power-saving asymptotic estimates which beat the trivial bounds.
基金Supported by the Open Research Fund of Key Laboratory of Nonlinear Analysis and Applications(Central China Normal University),Ministry of Education,P.R.China(Grant No.NAA2025ORG011)Science and Technology Plan Project of Jingmen(Grant No.2024YFZD076)+3 种基金Research Team Project of Jingchu University of Technology(Grant No.TD202006)Research Project of Jingchu University of Technology(Grant Nos.HX20240049HX20240200)the Teaching Reform Research Project of Hubei Province(Grant No.2024496)。
文摘In this paper,we study the eigenvalue problem of the Markov diffusion operator L^(2),and give generalized inequalities for eigenvalues of the operator L^(2)on a Markov diffusion triple.By applying these inequalities,we then get some new universal bounds for eigenvalues of a special Markov diffusion operator L^(2)on bounded domains in an Euclidean space.Moreover,our results can reveal the relationship between the(k+1)-th eigenvalue and the first k eigenvalues in a relatively straightforward manner.
基金Supported by National Natural Science Foundation of China(11001130,12272062)Fundamental Research Funds for the Central Universities(30917011335).
文摘In this paper,we investigate the weighted Dirichlet eigenvalue problem of polynomial operator of the drifting Laplacian on the cigar soliton■as follows■where is a positive continuous function on,denotes the outward unit normal to the boundary,and are two nonnegative constants.We establish some universal inequalities for eigenvalues of this problem.
基金This research is supported by NSFC(Nos.12171154,12301438)the Chenguang Program of Shanghai Education Development Foundation and Shanghai Municipal Education Commission(No.23CGA37)。
文摘Let Un be the set of connected unicyclic graphs of order n and girth g.Let C(T_(1),T_(2),...,T_(g))Un be obtained from a cycle v_(1)v_(2)…v_(g)v_(1)(in the anticlockwise direction)by identifying vi with the root of a rooted tree Ti of order ni for each i=1,2,...,g,where ni≥1 and∑^(g)_(i=1)n_(i)=n.Let S(n_(1),n_(2),...,n_(g))be obtained from C(T_(1),T_(2),..,T_(g))by replacing each Ti by a rooted star Sni with the center as its root.Let U(n_(1),n_(2),...,ng)be the set of unicyclic graphs which differ from the unicyclic graph S(n_(1),n_(2),...,n_(g))only up to a permutation of ni's.In this paper,the graph with the minimal least signless Laplacian eigenvalue(respectively,the graph with maximum signless Laplacian spread)in U(n_(1),n_(2),...,n_(g))is determined.
基金Supported by the National Natural Science Foundation of China(Grant No.12371110).
文摘This paper is concerned with the following nonlinear Steklov problemΔu=0 in D,∂vu=λf(u)on∂D,where D is the unit disk in the plane,∂v denotes the unit outward normal derivative.For each k∈N,under some natural conditions on f,using the Crandall-Rabinowitz bifurcation theorem,we obtain a bifurcation curve emanating from(k,0).Furthermore,we also analyze the local structure of bifurcation curves and stability of solutions on them.Specifically,our results indicate the bifurcation is critical for each k and is subcritical(supercritical)if f'''(0)>0(f'''(0)<0).
