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
Given an open bounded subset Ω of ℝ^(n) we consider the eigenvalue problem{Δu-(■u,■V)=-λvu,u>0inΩ,u=0 onδΩ,where V is a given function defined inΩandλV is the relevant eigenvalue.We determine sufficient c...Given an open bounded subset Ω of ℝ^(n) we consider the eigenvalue problem{Δu-(■u,■V)=-λvu,u>0inΩ,u=0 onδΩ,where V is a given function defined inΩandλV is the relevant eigenvalue.We determine sufficient conditions on V such that ifΩis convex,the solution u is log-concave.We also determine sufficient conditions ensuring that λ_(V),as a function of the setΩ,verifies a convexity inequality with respect to the Minkowski addition of sets.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
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
This paper is devoted to the Polynomial Preserving Recovery (PPR) based a posteriori error analysis for the second-order elliptic non-symmetric eigenvalue problem. An asymptotically exact a posteriori error estimator ...This paper is devoted to the Polynomial Preserving Recovery (PPR) based a posteriori error analysis for the second-order elliptic non-symmetric eigenvalue problem. An asymptotically exact a posteriori error estimator is proposed for solving the convection-dominated non-symmetric eigenvalue problem with non-smooth eigenfunctions or multiple eigenvalues. Numerical examples confirm our theoretical analysis.展开更多
Let G be a simple undirected graph.For any real numberα∈[0,1],Nikiforov defined the A_(α)-matrix of G as A_(α)(G)=αD(G)+(1-α)A(G),where A(G)and D(G)are the adjacency matrix and the degree diagonal matrix of G,re...Let G be a simple undirected graph.For any real numberα∈[0,1],Nikiforov defined the A_(α)-matrix of G as A_(α)(G)=αD(G)+(1-α)A(G),where A(G)and D(G)are the adjacency matrix and the degree diagonal matrix of G,respectively.In this paper,we investigate how the least eigenvalue of A_(α)(G)changes when the graph G is perturbed by deleting a vertex,subdividing edges or moving edges,respectively.展开更多
In this paper we investigate the least eigenvalue of a graph whose complement is connected,and present a lower bound for the least eigenvalue of such graph.We also characterize the unique graph whose least eigenvalue ...In this paper we investigate the least eigenvalue of a graph whose complement is connected,and present a lower bound for the least eigenvalue of such graph.We also characterize the unique graph whose least eigenvalue attains the second minimum among all graphs of fixed order.展开更多
The graphs which maximize and minimize respectively the largest eigenvalue over all unicyclic mixed graphs U on n vertices are determined. The unicyclic mixed graphs U with the largest eigenvalue λ 1(U)=n or λ 1(U...The graphs which maximize and minimize respectively the largest eigenvalue over all unicyclic mixed graphs U on n vertices are determined. The unicyclic mixed graphs U with the largest eigenvalue λ 1(U)=n or λ 1(U)∈(n,n+1] are characterized.展开更多
The problem of joint eigenvalue estimation for the non-defective commuting set of matrices A is addressed. A procedure revealing the joint eigenstructure by simultaneous diagonalization of. A with simultaneous Schur d...The problem of joint eigenvalue estimation for the non-defective commuting set of matrices A is addressed. A procedure revealing the joint eigenstructure by simultaneous diagonalization of. A with simultaneous Schur decomposition （SSD） and balance procedure alternately is proposed for performance considerations and also for overcoming the convergence difficulties of previous methods based only on simultaneous Schur form and unitary transformations, it is shown that the SSD procedure can be well incorporated with the balancing algorithm in a pingpong manner, i. e., each optimizes a cost function and at the same time serves as an acceleration procedure for the other. Under mild assumptions, the convergence of the two cost functions alternately optimized, i. e., the norm of A and the norm of the left-lower part of A is proved. Numerical experiments are conducted in a multi-dimensional harmonic retrieval application and suggest that the presented method converges considerably faster than the methods based on only unitary transformation for matrices which are not near to normality.展开更多
In present paper, using some methods of approximation theory, the trace formulas for eigenvalues of a eigenvalue problem are calculated under the periodic condition and the decaying condition at x∞.
Abstract A new function-valued partial Padé-type approximation was introduced in the polynomial space, and an explicit determinant formula was derived by means of some orthogonal polynomials. This method can be a...Abstract A new function-valued partial Padé-type approximation was introduced in the polynomial space, and an explicit determinant formula was derived by means of some orthogonal polynomials. This method can be applied to estimating surplus eigenvalues of the Fredholm integral equation of the second kind when its partial eigenvalues have been known, and at the same time, it can be applied to solving the approximating solution of the given equation.展开更多
This paper considers one computational method of the eigenvalues approximate value of the horizontal vibration problem of beam. The proof of our main result is based on the variational formula. First of all, Cauchy in...This paper considers one computational method of the eigenvalues approximate value of the horizontal vibration problem of beam. The proof of our main result is based on the variational formula. First of all, Cauchy inequality is used to obtain a basic inequality. Secondly, the functions of basis are made by Galerkin method, and the error estimates of eignevalues are obtained by Cauchy inequality. At last, the computational method of the approximate value of the eigenvalues turns out immediately, and acc...展开更多
In this article, we consider the eigenvalue problem for the bi-Kohn Laplacian and obtain universal bounds on the （k ＋ 1）-th eigenvalue in terms of the first k eigenvalues independent of the domains.
In this study, the boundary-value problem with eigenvalue parameter generated by the differential equation with discontinuous coefficients and boundary conditions which contains not only endpoints of the considered in...In this study, the boundary-value problem with eigenvalue parameter generated by the differential equation with discontinuous coefficients and boundary conditions which contains not only endpoints of the considered interval, but also point of discontinuity and linear functionals is investigated. So, the problem is not pure boundary-value. The authors single out a class of linear functionals and find simple algebraic conditions on coefficients, which garantee the existence of infinit number eigenvalues. Also the asymptotic formulas for eigenvalues are found.展开更多
Machine learning-based modeling of reactor physics problems has attracted increasing interest in recent years.Despite some progress in one-dimensional problems,there is still a paucity of benchmark studies that are ea...Machine learning-based modeling of reactor physics problems has attracted increasing interest in recent years.Despite some progress in one-dimensional problems,there is still a paucity of benchmark studies that are easy to solve using traditional numerical methods albeit still challenging using neural networks for a wide range of practical problems.We present two networks,namely the Generalized Inverse Power Method Neural Network(GIPMNN)and Physics-Constrained GIPMNN(PC-GIPIMNN)to solve K-eigenvalue problems in neutron diffusion theory.GIPMNN follows the main idea of the inverse power method and determines the lowest eigenvalue using an iterative method.The PC-GIPMNN additionally enforces conservative interface conditions for the neutron flux.Meanwhile,Deep Ritz Method(DRM)directly solves the smallest eigenvalue by minimizing the eigenvalue in Rayleigh quotient form.A comprehensive study was conducted using GIPMNN,PC-GIPMNN,and DRM to solve problems of complex spatial geometry with variant material domains from the fleld of nuclear reactor physics.The methods were compared with the standard flnite element method.The applicability and accuracy of the methods are reported and indicate that PC-GIPMNN outperforms GIPMNN and DRM.展开更多
Let A= (aij)∈Cn×n and ri = ∑ j≠i｜aij｜. Suppose thatforeach row of A there isatleastonenonzero off-diagonalentry. Itis proved thatalleigenvalues ofAarecontained in Ω～= ∪aij≠0,i≠j｛z∈C:｜z- aii｜｜z- ...Let A= (aij)∈Cn×n and ri = ∑ j≠i｜aij｜. Suppose thatforeach row of A there isatleastonenonzero off-diagonalentry. Itis proved thatalleigenvalues ofAarecontained in Ω～= ∪aij≠0,i≠j｛z∈C:｜z- aii｜｜z- ajj｜≤rirj｝. The resultre- duces the num berofovals in originalBrauer'stheorem in m any cases. Eigenval- ues(and associated eigenvectors) thatlocate in theboundary ofΩ～ arediscussed.展开更多
Let Ω be a connected bounded domain in R^n. Denote by λi the i-th eigenvalue of the Lapla^ian operator with any order p：{u=Эn→^-Эu=…=Эn→p-1^-Эp-1u=0 on ЭΩ （-△）pu=λu in Ω.In this article, we give som...Let Ω be a connected bounded domain in R^n. Denote by λi the i-th eigenvalue of the Lapla^ian operator with any order p：{u=Эn→^-Эu=…=Эn→p-1^-Эp-1u=0 on ЭΩ （-△）pu=λu in Ω.In this article, we give some expressions for upper bound of the （k ＋ 1）-th eigenvalue ）λk＋l in terms of the first k eigenvalues.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 project Disuguaglianze analitiche e geometriche,funded by the Gruppo per Analisi Matematica la Probabilitàe le loro Applicazioni.
文摘Given an open bounded subset Ω of ℝ^(n) we consider the eigenvalue problem{Δu-(■u,■V)=-λvu,u>0inΩ,u=0 onδΩ,where V is a given function defined inΩandλV is the relevant eigenvalue.We determine sufficient conditions on V such that ifΩis convex,the solution u is log-concave.We also determine sufficient conditions ensuring that λ_(V),as a function of the setΩ,verifies a convexity inequality with respect to the Minkowski addition of sets.
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 the National Natural Science Foundation of China (Grant Nos.1236108412001130)。
文摘This paper is devoted to the Polynomial Preserving Recovery (PPR) based a posteriori error analysis for the second-order elliptic non-symmetric eigenvalue problem. An asymptotically exact a posteriori error estimator is proposed for solving the convection-dominated non-symmetric eigenvalue problem with non-smooth eigenfunctions or multiple eigenvalues. Numerical examples confirm our theoretical analysis.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1207141112171222)the Basic Research Program(Natural Science)of Yancheng(Grant No.YCBK2024043)。
文摘Let G be a simple undirected graph.For any real numberα∈[0,1],Nikiforov defined the A_(α)-matrix of G as A_(α)(G)=αD(G)+(1-α)A(G),where A(G)and D(G)are the adjacency matrix and the degree diagonal matrix of G,respectively.In this paper,we investigate how the least eigenvalue of A_(α)(G)changes when the graph G is perturbed by deleting a vertex,subdividing edges or moving edges,respectively.
基金Supported by the National Basic Research Program(973 Program)of China(2013CB329402)the National Natural Science Foundation of China(61473215,61472306,61271302,61272282,61272176)
基金Supported by National Natural Science Foundation of China(Grant No.11071002)Program for New Century Excellent Talents in University,Key Project of Chinese Ministry of Education(Grant No.210091)+8 种基金Specialized Research Fund for the Doctoral Program of Higher Education(Grant No.20103401110002)Science and Technological Fund of Anhui Province for Outstanding Youth(Grant No.10040606Y33)the Natural Science Foundation of Department of Education of Anhui Province(Grant Nos.KJ2011A195KJ2010B136)Project of Anhui Province for Excellent Young Talents in Universities(Grant No.2009SQRZ017ZD)Scientific Research Fund for Fostering Distinguished Young Scholars of Anhui University(Grant No.KJJQ1001)Project for Academic Innovation Team of Anhui University(Grant No.KJTD001B)Fund for Youth Scientific Research of Anhui University(Grant No.KJQN1003)Innovation Fund for Graduates of Anhui University
文摘In this paper we investigate the least eigenvalue of a graph whose complement is connected,and present a lower bound for the least eigenvalue of such graph.We also characterize the unique graph whose least eigenvalue attains the second minimum among all graphs of fixed order.
基金Supported by the project item for young teachers of colleges and universities of Anhui province( 2 0 0 3jq1 0 1 ) and the project item of Anhui University for talents group construction
文摘The graphs which maximize and minimize respectively the largest eigenvalue over all unicyclic mixed graphs U on n vertices are determined. The unicyclic mixed graphs U with the largest eigenvalue λ 1(U)=n or λ 1(U)∈(n,n+1] are characterized.
基金The National Natural Science Foundation of China(No.60572072,60496311),the National High Technology Researchand Development Program of China (863Program ) ( No.2003AA123310),the International Cooperation Project on Beyond 3G Mobile of China (No.2005DFA10360).
文摘The problem of joint eigenvalue estimation for the non-defective commuting set of matrices A is addressed. A procedure revealing the joint eigenstructure by simultaneous diagonalization of. A with simultaneous Schur decomposition （SSD） and balance procedure alternately is proposed for performance considerations and also for overcoming the convergence difficulties of previous methods based only on simultaneous Schur form and unitary transformations, it is shown that the SSD procedure can be well incorporated with the balancing algorithm in a pingpong manner, i. e., each optimizes a cost function and at the same time serves as an acceleration procedure for the other. Under mild assumptions, the convergence of the two cost functions alternately optimized, i. e., the norm of A and the norm of the left-lower part of A is proved. Numerical experiments are conducted in a multi-dimensional harmonic retrieval application and suggest that the presented method converges considerably faster than the methods based on only unitary transformation for matrices which are not near to normality.
文摘In present paper, using some methods of approximation theory, the trace formulas for eigenvalues of a eigenvalue problem are calculated under the periodic condition and the decaying condition at x∞.
基金Project supported by the National Natural Science Foundation of China(Grant No.10271074)
文摘Abstract A new function-valued partial Padé-type approximation was introduced in the polynomial space, and an explicit determinant formula was derived by means of some orthogonal polynomials. This method can be applied to estimating surplus eigenvalues of the Fredholm integral equation of the second kind when its partial eigenvalues have been known, and at the same time, it can be applied to solving the approximating solution of the given equation.
文摘This paper considers one computational method of the eigenvalues approximate value of the horizontal vibration problem of beam. The proof of our main result is based on the variational formula. First of all, Cauchy inequality is used to obtain a basic inequality. Secondly, the functions of basis are made by Galerkin method, and the error estimates of eignevalues are obtained by Cauchy inequality. At last, the computational method of the approximate value of the eigenvalues turns out immediately, and acc...
文摘In this article, we consider the eigenvalue problem for the bi-Kohn Laplacian and obtain universal bounds on the （k ＋ 1）-th eigenvalue in terms of the first k eigenvalues independent of the domains.
文摘In this study, the boundary-value problem with eigenvalue parameter generated by the differential equation with discontinuous coefficients and boundary conditions which contains not only endpoints of the considered interval, but also point of discontinuity and linear functionals is investigated. So, the problem is not pure boundary-value. The authors single out a class of linear functionals and find simple algebraic conditions on coefficients, which garantee the existence of infinit number eigenvalues. Also the asymptotic formulas for eigenvalues are found.
基金partially supported by the National Natural Science Foundation of China(No.11971020)Natural Science Foundation of Shanghai(No.23ZR1429300)Innovation Funds of CNNC(Lingchuang Fund)。
文摘Machine learning-based modeling of reactor physics problems has attracted increasing interest in recent years.Despite some progress in one-dimensional problems,there is still a paucity of benchmark studies that are easy to solve using traditional numerical methods albeit still challenging using neural networks for a wide range of practical problems.We present two networks,namely the Generalized Inverse Power Method Neural Network(GIPMNN)and Physics-Constrained GIPMNN(PC-GIPIMNN)to solve K-eigenvalue problems in neutron diffusion theory.GIPMNN follows the main idea of the inverse power method and determines the lowest eigenvalue using an iterative method.The PC-GIPMNN additionally enforces conservative interface conditions for the neutron flux.Meanwhile,Deep Ritz Method(DRM)directly solves the smallest eigenvalue by minimizing the eigenvalue in Rayleigh quotient form.A comprehensive study was conducted using GIPMNN,PC-GIPMNN,and DRM to solve problems of complex spatial geometry with variant material domains from the fleld of nuclear reactor physics.The methods were compared with the standard flnite element method.The applicability and accuracy of the methods are reported and indicate that PC-GIPMNN outperforms GIPMNN and DRM.
文摘Let A= (aij)∈Cn×n and ri = ∑ j≠i｜aij｜. Suppose thatforeach row of A there isatleastonenonzero off-diagonalentry. Itis proved thatalleigenvalues ofAarecontained in Ω～= ∪aij≠0,i≠j｛z∈C:｜z- aii｜｜z- ajj｜≤rirj｝. The resultre- duces the num berofovals in originalBrauer'stheorem in m any cases. Eigenval- ues(and associated eigenvectors) thatlocate in theboundary ofΩ～ arediscussed.
基金supported by NSFC (10471108,10631020) of ChinaNSF of Henan Provincial Education Department (2010A110008)
文摘Let Ω be a connected bounded domain in R^n. Denote by λi the i-th eigenvalue of the Lapla^ian operator with any order p：{u=Эn→^-Эu=…=Эn→p-1^-Эp-1u=0 on ЭΩ （-△）pu=λu in Ω.In this article, we give some expressions for upper bound of the （k ＋ 1）-th eigenvalue ）λk＋l in terms of the first k eigenvalues.
