In this paper,we give a complete characterization of all self-adjoint domains of odd order differential operators on two intervals.These two intervals with all four endpoints are singular(one endpoint of each interval...In this paper,we give a complete characterization of all self-adjoint domains of odd order differential operators on two intervals.These two intervals with all four endpoints are singular(one endpoint of each interval is singular or all four endpoints are regulars are the special cases).And these extensions yield"new"self-adjoint operators,which involve interactions between the two intervals.展开更多
This paper mainly discusses the(2+1)-dimensional modified dispersive water-wave(MDWW) system which will be proved nonlinear self-adjointness. This property is applied to construct conservation laws corresponding to th...This paper mainly discusses the(2+1)-dimensional modified dispersive water-wave(MDWW) system which will be proved nonlinear self-adjointness. This property is applied to construct conservation laws corresponding to the symmetries of the system. Moreover, via the truncated Painlev′e analysis and consistent tanh-function expansion(CTE)method, the soliton-cnoidal periodic wave interaction solutions and corresponding images will be eventually achieved.展开更多
Let Bs （7-/） be the real linear space of all self-adjoint operators on a complex Hilbert spae 7-/ with dimT/〉 2. It is proved that a linear surjective map on Bs（T/） preserves the nonzero projections of Jordan pro...Let Bs （7-/） be the real linear space of all self-adjoint operators on a complex Hilbert spae 7-/ with dimT/〉 2. It is proved that a linear surjective map on Bs（T/） preserves the nonzero projections of Jordan products of two operators if and only if there is a unitary or an anti-unitary operator U on T/such that ^（X） = AU＊XU, VX E Bs（TI） for some constant ~ with k E {1,-1}.展开更多
Assume that L is a non-negative self-adjoint operator on L^(2)(ℝ^(n))with its heat kernels satisfying the so-called Gaussian upper bound estimate and that X is a ball quasi-Banach function space onℝ^(n) satisfying som...Assume that L is a non-negative self-adjoint operator on L^(2)(ℝ^(n))with its heat kernels satisfying the so-called Gaussian upper bound estimate and that X is a ball quasi-Banach function space onℝ^(n) satisfying some mild assumptions.Let HX,L(ℝ^(n))be the Hardy space associated with both X and L,which is defined by the Lusin area function related to the semigroup generated by L.In this article,the authors establish various maximal function characterizations of the Hardy space HX,L(ℝ^(n))and then apply these characterizations to obtain the solvability of the related Cauchy problem.These results have a wide range of generality and,in particular,the specific spaces X to which these results can be applied include the weighted space,the variable space,the mixed-norm space,the Orlicz space,the Orlicz-slice space,and the Morrey space.Moreover,the obtained maximal function characterizations of the mixed-norm Hardy space,the Orlicz-slice Hardy space,and the Morrey-Hardy space associated with L are completely new.展开更多
In this paper, based on the idea of El-Mistikawy and Werle[1] we construct a difference scheme for a singularly perturbed self-adjoint ordinary differential equation in conservation form. We prove that it is a uniform...In this paper, based on the idea of El-Mistikawy and Werle[1] we construct a difference scheme for a singularly perturbed self-adjoint ordinary differential equation in conservation form. We prove that it is a uniformly convergent second order scheme.展开更多
We present a class of the second order optimal splines difference schemes derived from ex- ponential cubic splines for self-adjoint singularly perturbed 2-point boundary value problem. We prove an optimal error estima...We present a class of the second order optimal splines difference schemes derived from ex- ponential cubic splines for self-adjoint singularly perturbed 2-point boundary value problem. We prove an optimal error estimate and give illustrative numerical example.展开更多
This paper discusses the order-preserving convergence for spectral approximation of the self-adjoint completely continuous operator T.Under the condition that the approximate operator Th converges to T in norm,it is p...This paper discusses the order-preserving convergence for spectral approximation of the self-adjoint completely continuous operator T.Under the condition that the approximate operator Th converges to T in norm,it is proven that the k-th eigenvalue of Th converges to the k-th eigenvalue of T.(We sorted the positive eigenvalues in decreasing order and negative eigenvalues in increasing order.) Then we apply this result to conforming elements,nonconforming elements and mixed elements of self-adjoint elliptic differential operators eigenvalue problems,and prove that the k-th approximate eigenvalue obtained by these methods converges to the k-th exact eigenvalue.展开更多
In this paper, the expression of the norm of a self-adjoint integral operator T ： L^2（0, ∞） → L^2 （0, ∞） is obtained. As applications, a new bilinear integral inequality with a best constant factor is establis...In this paper, the expression of the norm of a self-adjoint integral operator T ： L^2（0, ∞） → L^2 （0, ∞） is obtained. As applications, a new bilinear integral inequality with a best constant factor is established and some particular cases are considered.展开更多
In the present paper,the self-adjointness of the product of two ruth-order differential operators on [0,+∞)is studied.By means of the construction theory of self-adjoint operators and matrix computation,we obtain a s...In the present paper,the self-adjointness of the product of two ruth-order differential operators on [0,+∞)is studied.By means of the construction theory of self-adjoint operators and matrix computation,we obtain a sufficient and necessary condition to ensure that the product operator is self-adjoint,which extends the results in the second order case.展开更多
Symplectic self-adjointness of Hamiltonian operator matrices is studied, which is important to symplectic elasticity and optimal control. For the cases of diagonal domain and off-diagonal domain, necessary and suffici...Symplectic self-adjointness of Hamiltonian operator matrices is studied, which is important to symplectic elasticity and optimal control. For the cases of diagonal domain and off-diagonal domain, necessary and sufficient conditions are shown. The proofs use Frobenius-Schur factorizations of unbounded operator matrices.Under additional assumptions, sufficient conditions based on perturbation method are obtained. The theory is applied to a problem in symplectic elasticity.展开更多
Non-self-adjoint dynamical systems, e.g., nonholonomic systems, can admit an almost Poisson structure, which is formulated by a kind of Poisson bracket satisfying the usual properties except for the Jacobi identity. A...Non-self-adjoint dynamical systems, e.g., nonholonomic systems, can admit an almost Poisson structure, which is formulated by a kind of Poisson bracket satisfying the usual properties except for the Jacobi identity. A general theory of the almost Poisson structure is investigated based on a decompo- sition of the bracket into a sum of a Poisson one and an almost Poisson one. The corresponding rela- tion between Poisson structure and symplectic structure is proved, making use of Jacobiizer and symplecticizer. Based on analysis of pseudo-symplectic structure of constraint submanifold of Chaplygin’s nonholonomic systems, an almost Poisson bracket for the systems is constructed and decomposed into a sum of a canonical Poisson one and an almost Poisson one. Similarly, an almost Poisson structure, which can be decomposed into a sum of canonical one and an almost "Lie-Poisson" one, is also constructed on an affine space with torsion whose autoparallels are utilized to describe the free motion of some non-self-adjoint systems. The decomposition of the almost Poisson bracket di- rectly leads to a decomposition of a dynamical vector field into a sum of usual Hamiltionian vector field and an almost Hamiltonian one, which is useful to simplifying the integration of vector fields.展开更多
Self adjointness of the product of differential operators on is studiedhere, using the construction theory of self adjoint operators, we give a sufficient and necessary condition of self adjointness problem.
In this paper, the problem of self-adjointness of the product of two differential operators is considered. A number of results concerning self-adjointness of the product L<sub>2</sub>L<sub>1</sub&...In this paper, the problem of self-adjointness of the product of two differential operators is considered. A number of results concerning self-adjointness of the product L<sub>2</sub>L<sub>1</sub> of two second-order self-adjoint differential operators are obtained by using the general construction theory of self-adjoint extensions of ordinary differential operators.展开更多
In this paper, we first establish an abstract inequality for lower order eigenvalues of a self-adjoint operator on a Hilbert space which generalizes and extends the recent results of Cheng et al. （Calc. Var. Partial ...In this paper, we first establish an abstract inequality for lower order eigenvalues of a self-adjoint operator on a Hilbert space which generalizes and extends the recent results of Cheng et al. （Calc. Var. Partial Differential Equations, 38, 409-416 （2010））. Then, making use of it, we obtain some universal inequalities for lower order eigenvalues of the biharmonic operator on manifolds admitting some speciM functions. Moreover, we derive a universal inequality for lower order eigenvalues of the poly-Laplacian with any order on the Euclidean space.展开更多
The authors extend Hua’s fundamental theorem of the geometry of Hermitian matri- ces to the in?nite-dimensional case. An application to characterizing the corresponding Jordan ring automorphism is also presented.
Symplectic self-adjointness of infinite dimensional Hamiltonian operators is studied, the necessary and sufficient conditions are given. Using the relatively bounded perturbation, the sufficient conditions about sympl...Symplectic self-adjointness of infinite dimensional Hamiltonian operators is studied, the necessary and sufficient conditions are given. Using the relatively bounded perturbation, the sufficient conditions about symplectic self-adjointness are shown.展开更多
Let H be a complex Hilbert space with dimH ≥3, Bs（H） the （real） Jordan algebra of all self-adjoint operators on H. Every surjective map Ф ： Bs（H）→13s（H） preserving numerical radius of operator products （r...Let H be a complex Hilbert space with dimH ≥3, Bs（H） the （real） Jordan algebra of all self-adjoint operators on H. Every surjective map Ф ： Bs（H）→13s（H） preserving numerical radius of operator products （respectively, Jordan triple products） is characterized. A characterization of surjective maps on Bs （H） preserving a cross operator norm of operator products （resp. Jordan triple products of operators） is also given.展开更多
We introduce the variable integral and the smooth exponent Besov spaces associated to non-negative self-adjoint operators.Then we give the equivalent norms via the Peetre type maximal functions and atomic decompositio...We introduce the variable integral and the smooth exponent Besov spaces associated to non-negative self-adjoint operators.Then we give the equivalent norms via the Peetre type maximal functions and atomic decomposition of these spaces.展开更多
In this paper, we study a weakly over-penalized interior penalty method for non-self- adjoint and indefinite problems. An optimal a priori error estimate in the energy norm is derived. In addition, we introduce a resi...In this paper, we study a weakly over-penalized interior penalty method for non-self- adjoint and indefinite problems. An optimal a priori error estimate in the energy norm is derived. In addition, we introduce a residual-based a posteriori error estimator, which is proved to be both reliable and efficient in the energy norm. Some numerical testes are presented to validate our theoretical analysis.展开更多
A new technique is proposed for the analysis of shape optimization problems. The technique uses the asymptotic analysis of boundary value problems in singularly perturbed geometrical domains. The asymptotics of soluti...A new technique is proposed for the analysis of shape optimization problems. The technique uses the asymptotic analysis of boundary value problems in singularly perturbed geometrical domains. The asymptotics of solutions are derived in the framework of compound and matched asymptotics expansions. The analysis involves the so-called interior topology variations. The asymptotic expansions are derived for a model problem, however the technique applies to general elliptic boundary value problems. The self-adjoint extensions of elliptic operators and the weighted spaces with detached asymptotics are exploited for the modelling of problems with small defects in geometrical domains, The error estimates for proposed approximations of shape functionals are provided.展开更多
基金Supported by NSFC (No.12361027)NSF of Inner Mongolia (No.2018MS01021)+1 种基金NSF of Shandong Province (No.ZR2020QA009)Science and Technology Innovation Program for Higher Education Institutions of Shanxi Province (No.2024L533)。
文摘In this paper,we give a complete characterization of all self-adjoint domains of odd order differential operators on two intervals.These two intervals with all four endpoints are singular(one endpoint of each interval is singular or all four endpoints are regulars are the special cases).And these extensions yield"new"self-adjoint operators,which involve interactions between the two intervals.
基金Supported by National Natural Science Foundation of China under Grant Nos.11371293,11505090the Natural Science Foundation of Shaanxi Province under Grant No.2014JM2-1009+1 种基金Research Award Foundation for Outstanding Young Scientists of Shandong Province under Grant No.BS2015SF009the Science and Technology Innovation Foundation of Xi’an under Grant No.CYX1531WL41
文摘This paper mainly discusses the(2+1)-dimensional modified dispersive water-wave(MDWW) system which will be proved nonlinear self-adjointness. This property is applied to construct conservation laws corresponding to the symmetries of the system. Moreover, via the truncated Painlev′e analysis and consistent tanh-function expansion(CTE)method, the soliton-cnoidal periodic wave interaction solutions and corresponding images will be eventually achieved.
基金Supported by the National Natural Science Foundation of China (Grant No. 10971123)the Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20090202110001)
文摘Let Bs （7-/） be the real linear space of all self-adjoint operators on a complex Hilbert spae 7-/ with dimT/〉 2. It is proved that a linear surjective map on Bs（T/） preserves the nonzero projections of Jordan products of two operators if and only if there is a unitary or an anti-unitary operator U on T/such that ^（X） = AU＊XU, VX E Bs（TI） for some constant ~ with k E {1,-1}.
基金supported by the National Key Research and Development Program of China(2020YFA0712900)the National Natural Science Foundation of China(12371093,12071197,12122102 and 12071431)+2 种基金the Key Project of Gansu Provincial National Science Foundation(23JRRA1022)the Fundamental Research Funds for the Central Universities(2233300008 and lzujbky-2021-ey18)the Innovative Groups of Basic Research in Gansu Province(22JR5RA391).
文摘Assume that L is a non-negative self-adjoint operator on L^(2)(ℝ^(n))with its heat kernels satisfying the so-called Gaussian upper bound estimate and that X is a ball quasi-Banach function space onℝ^(n) satisfying some mild assumptions.Let HX,L(ℝ^(n))be the Hardy space associated with both X and L,which is defined by the Lusin area function related to the semigroup generated by L.In this article,the authors establish various maximal function characterizations of the Hardy space HX,L(ℝ^(n))and then apply these characterizations to obtain the solvability of the related Cauchy problem.These results have a wide range of generality and,in particular,the specific spaces X to which these results can be applied include the weighted space,the variable space,the mixed-norm space,the Orlicz space,the Orlicz-slice space,and the Morrey space.Moreover,the obtained maximal function characterizations of the mixed-norm Hardy space,the Orlicz-slice Hardy space,and the Morrey-Hardy space associated with L are completely new.
文摘In this paper, based on the idea of El-Mistikawy and Werle[1] we construct a difference scheme for a singularly perturbed self-adjoint ordinary differential equation in conservation form. We prove that it is a uniformly convergent second order scheme.
文摘We present a class of the second order optimal splines difference schemes derived from ex- ponential cubic splines for self-adjoint singularly perturbed 2-point boundary value problem. We prove an optimal error estimate and give illustrative numerical example.
基金supported by the National Natural Science Foundation of China (Grant No. 10761003)Guizhou Province Scientific Research for Senior Personnels
文摘This paper discusses the order-preserving convergence for spectral approximation of the self-adjoint completely continuous operator T.Under the condition that the approximate operator Th converges to T in norm,it is proven that the k-th eigenvalue of Th converges to the k-th eigenvalue of T.(We sorted the positive eigenvalues in decreasing order and negative eigenvalues in increasing order.) Then we apply this result to conforming elements,nonconforming elements and mixed elements of self-adjoint elliptic differential operators eigenvalue problems,and prove that the k-th approximate eigenvalue obtained by these methods converges to the k-th exact eigenvalue.
基金Supported by the Emphases Natural Science Foundation of Guangdong Institutions of Higher Learning,College and University(No.05Z026)
文摘In this paper, the expression of the norm of a self-adjoint integral operator T ： L^2（0, ∞） → L^2 （0, ∞） is obtained. As applications, a new bilinear integral inequality with a best constant factor is established and some particular cases are considered.
基金Supported by the National Natural Science Foundation of China(10261004)
文摘In the present paper,the self-adjointness of the product of two ruth-order differential operators on [0,+∞)is studied.By means of the construction theory of self-adjoint operators and matrix computation,we obtain a sufficient and necessary condition to ensure that the product operator is self-adjoint,which extends the results in the second order case.
基金supported by National Natural Science Foundation of China(Grant Nos.11371185,11101200 and 11361034)Specialized Research Fund for the Doctoral Program of Higher Education of China(Grant No.20111501110001)+1 种基金Major Subject of Natural Science Foundation of Inner Mongolia of China(Grant No.2013ZD01)Natural Science Foundation of Inner Mongolia of China(Grant No.2012MS0105)
文摘Symplectic self-adjointness of Hamiltonian operator matrices is studied, which is important to symplectic elasticity and optimal control. For the cases of diagonal domain and off-diagonal domain, necessary and sufficient conditions are shown. The proofs use Frobenius-Schur factorizations of unbounded operator matrices.Under additional assumptions, sufficient conditions based on perturbation method are obtained. The theory is applied to a problem in symplectic elasticity.
基金Supported by the National Natural Science Foundation of China (Grant Nos. 10872084, 10472040)the Outstanding Young Talents Training Fund of Liaoning Province of China (Grant No. 3040005)the Research Program of Higher Educa-tion of Liaoning Province of China (Grant No. 2008S098)
文摘Non-self-adjoint dynamical systems, e.g., nonholonomic systems, can admit an almost Poisson structure, which is formulated by a kind of Poisson bracket satisfying the usual properties except for the Jacobi identity. A general theory of the almost Poisson structure is investigated based on a decompo- sition of the bracket into a sum of a Poisson one and an almost Poisson one. The corresponding rela- tion between Poisson structure and symplectic structure is proved, making use of Jacobiizer and symplecticizer. Based on analysis of pseudo-symplectic structure of constraint submanifold of Chaplygin’s nonholonomic systems, an almost Poisson bracket for the systems is constructed and decomposed into a sum of a canonical Poisson one and an almost Poisson one. Similarly, an almost Poisson structure, which can be decomposed into a sum of canonical one and an almost "Lie-Poisson" one, is also constructed on an affine space with torsion whose autoparallels are utilized to describe the free motion of some non-self-adjoint systems. The decomposition of the almost Poisson bracket di- rectly leads to a decomposition of a dynamical vector field into a sum of usual Hamiltionian vector field and an almost Hamiltonian one, which is useful to simplifying the integration of vector fields.
文摘Self adjointness of the product of differential operators on is studiedhere, using the construction theory of self adjoint operators, we give a sufficient and necessary condition of self adjointness problem.
基金Supported by the Royal Society and the National Natural Science Foundation of Chinathe Regional Science Foundation of Inner Mongolia
文摘In this paper, the problem of self-adjointness of the product of two differential operators is considered. A number of results concerning self-adjointness of the product L<sub>2</sub>L<sub>1</sub> of two second-order self-adjoint differential operators are obtained by using the general construction theory of self-adjoint extensions of ordinary differential operators.
基金supported by National Natural Science Foundation of China(Grant No.11001130)
文摘In this paper, we first establish an abstract inequality for lower order eigenvalues of a self-adjoint operator on a Hilbert space which generalizes and extends the recent results of Cheng et al. （Calc. Var. Partial Differential Equations, 38, 409-416 （2010））. Then, making use of it, we obtain some universal inequalities for lower order eigenvalues of the biharmonic operator on manifolds admitting some speciM functions. Moreover, we derive a universal inequality for lower order eigenvalues of the poly-Laplacian with any order on the Euclidean space.
基金Project supported by the National Natural Science Foundation of China (No.10471082) and the ShanxiProvincial Natural Science Foundation of China.
文摘The authors extend Hua’s fundamental theorem of the geometry of Hermitian matri- ces to the in?nite-dimensional case. An application to characterizing the corresponding Jordan ring automorphism is also presented.
基金Supported by NNSF of China(Grant Nos.11761029 and 11561048)NSF of Inner Mongolia(Grant No.2015MS0116)Natural Science Foundation of Hetao College(Grant No.HYZY201702)
文摘Symplectic self-adjointness of infinite dimensional Hamiltonian operators is studied, the necessary and sufficient conditions are given. Using the relatively bounded perturbation, the sufficient conditions about symplectic self-adjointness are shown.
基金Supported by National Science Foundation of China (Grant Nos. 10771157, 10871111)the Provincial Science Foundation of Shanxi (Grant No. 2007011016)the Research Fund of Shanxi for Returned Scholars (Grant No. 2007-38)
文摘Let H be a complex Hilbert space with dimH ≥3, Bs（H） the （real） Jordan algebra of all self-adjoint operators on H. Every surjective map Ф ： Bs（H）→13s（H） preserving numerical radius of operator products （respectively, Jordan triple products） is characterized. A characterization of surjective maps on Bs （H） preserving a cross operator norm of operator products （resp. Jordan triple products of operators） is also given.
基金supported in part by the National Natural Science Foundation of China(Grant No.11761026,11761027)Natural Science Foundation of Guangxi(Grant No.2020GXNSFAA159085).
文摘We introduce the variable integral and the smooth exponent Besov spaces associated to non-negative self-adjoint operators.Then we give the equivalent norms via the Peetre type maximal functions and atomic decomposition of these spaces.
基金We thank the anonymous referees for their valuable comments and suggestions which lead to an improved presentation of this paper. This work was supported by NSFC under the grant 11371199, 11226334 and 11301275, the Jiangsu Provincial 2011 Program （Collaborative Innovation Center of Climate Change）, the Program of Natural Science Research of Jiangsu Higher Education Institutions of China （Grant No. 12KJB110013）, Natural Science Foundation of Guangdong Province of China （Grant No. S2012040007993） and Educational Commission of Guangdong Province of China （Grant No. 2012LYM0122）.
文摘In this paper, we study a weakly over-penalized interior penalty method for non-self- adjoint and indefinite problems. An optimal a priori error estimate in the energy norm is derived. In addition, we introduce a residual-based a posteriori error estimator, which is proved to be both reliable and efficient in the energy norm. Some numerical testes are presented to validate our theoretical analysis.
基金Partially supported by INRIA in the framework of the grant 00-01 from Institrt france-russe A.M.Liapunov d'informatique et de mathmatiques appliquesby the grant 4 TIIA 01524 of the State Committee for the Scientific Research of the Republic of Po
文摘A new technique is proposed for the analysis of shape optimization problems. The technique uses the asymptotic analysis of boundary value problems in singularly perturbed geometrical domains. The asymptotics of solutions are derived in the framework of compound and matched asymptotics expansions. The analysis involves the so-called interior topology variations. The asymptotic expansions are derived for a model problem, however the technique applies to general elliptic boundary value problems. The self-adjoint extensions of elliptic operators and the weighted spaces with detached asymptotics are exploited for the modelling of problems with small defects in geometrical domains, The error estimates for proposed approximations of shape functionals are provided.
