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
In this paper, Leibniz' formula of generalized divided difference with respect to a class of differential operators whose basic sets of solutions have power form, is considered. The recurrence formula of Green fun...In this paper, Leibniz' formula of generalized divided difference with respect to a class of differential operators whose basic sets of solutions have power form, is considered. The recurrence formula of Green function about the operators is also given.展开更多
We consider the space of rapidly decreasing sequences s and the derivative operator D defined on it. The object of this article is to study the equivalence of a differential operator of infinite order; that is φ（D）...We consider the space of rapidly decreasing sequences s and the derivative operator D defined on it. The object of this article is to study the equivalence of a differential operator of infinite order; that is φ（D） =^∞∑k=0φkD^k.φk constant numbers an a power of D.Dn, meaning, is there a isomorphism X （from s onto s） such that Xφ（D） = D^nX？. We prove that if φ（D） is equivalent to Dn, then φ（D） is of finite order, in fact a polynomial of degree n. The question of the equivalence of two differential operators of finite order in the space s is addressed too and solved completely when n = 1.展开更多
When one function is defined as a differential operation on another function, it’s often desirable to invert the definition, to effectively “undo” the differentiation. A Green’s function approach is often used to ...When one function is defined as a differential operation on another function, it’s often desirable to invert the definition, to effectively “undo” the differentiation. A Green’s function approach is often used to accomplish this, but variations on this theme exist, and we examine a few such variations. The mathematical analysis of is sought in the form if such an inverse operator exists, but physics is defined by both mathematical formula and ontological formalism, as I show for an example based on the Dirac equation. Finally, I contrast these “standard” approaches with a novel exact inverse operator for field equations.展开更多
In this paper, we consider the general ordinary quasi-differential expression τ of order n with complex coefficients and its formal adjoint τ<sup>+</sup> on the interval [a,b). We shall show in the case ...In this paper, we consider the general ordinary quasi-differential expression τ of order n with complex coefficients and its formal adjoint τ<sup>+</sup> on the interval [a,b). We shall show in the case of one singular end-point and under suitable conditions that all solutions of a general ordinary quasi-differential equation are in the weighted Hilbert space provided that all solutions of the equations and its adjoint are in . Also, a number of results concerning the location of the point spectra and regularity fields of the operators generated by such expressions may be obtained. Some of these results are extensions or generalizations of those in the symmetric case, while the others are new.展开更多
In the cost function of three- or four-dimensional variational dataassimilation, each term is weighted by the inverse of its associated error covariance matrix and thebackground error covariance matrix is usually much...In the cost function of three- or four-dimensional variational dataassimilation, each term is weighted by the inverse of its associated error covariance matrix and thebackground error covariance matrix is usually much larger than the other covariance matrices.Although the background error covariances are traditionally normalized and parameterized by simplesmooth homogeneous correlation functions, the covariance matrices constructed from these correlationfunctions are often too large to be inverted or even manipulated. It is thus desirable to finddirect representations of the inverses of background error correlations. This problem is studied inthis paper. In particular, it is shown that the background term can be written into ∫ dx∣Dυ(x)∣~2, that is, a squared 1/2 norm of a vector differential operator D, called theD-operator, applied to the field of analysis increment υ(x). For autoregressive correlationfunctions, the D-operators are of finite orders. For Gaussian correlation functions, the D-operatorsare of infinite order. For practical applications, the Gaussian D-operators must be truncated tofinite orders. The truncation errors are found to be small even when the Gaussian D-operators aretruncated to low orders. With a truncated D-operator, the background term can be easily constructedwith neither inversion nor direct calculation of the covariance matrix. D-operators are also derivedfor non-Gaussian correlations and transformed into non-isotropic forms.展开更多
The spectrum of a class of fourth order left-definite differential operators is studied. By using the theory of indefinite differential operators in Krein space and the relationship between left-definite and right-def...The spectrum of a class of fourth order left-definite differential operators is studied. By using the theory of indefinite differential operators in Krein space and the relationship between left-definite and right-definite operators, the following conclusions are obtained： if a fourth order differential operator with a self-adjoint boundary condition that is left-definite and right-indefinite, then all its eigenvalues are real, and there exist countably infinitely many positive and negative eigenvalues which are unbounded from below and above, have no finite cluster point and can be indexed to satisfy the inequality …≤λ-2≤λ-1≤λ-0〈0〈λ0≤λ1≤λ2≤…展开更多
For a real valued function f defined on a finite interval I we consider the problem of approximating f from null spaces of differential operators of the form Ln（ψ） =∑k=0^n akψ（k） where the constant coefficients...For a real valued function f defined on a finite interval I we consider the problem of approximating f from null spaces of differential operators of the form Ln（ψ） =∑k=0^n akψ（k） where the constant coefficients ak C R may be adapted to f.展开更多
The quantum Euclidean space is a kind of noncommutative space that is obtained from ordinary Euclidean space by deformation with parameter q. When N is odd, the structure of this space is similar to . Motivated by r...The quantum Euclidean space is a kind of noncommutative space that is obtained from ordinary Euclidean space by deformation with parameter q. When N is odd, the structure of this space is similar to . Motivated by realization of by differential operators in , we give such realization for and cases and generalize our results to (N odd) in this paper, that is, we show that the algebra of can be realized by differential operators acting on C<SUP>∞</SUP> functions on undeformed space .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.
We prove weighted q-variation inequalities with 2<q<∞for sharp truncations of singular integral operators in higher dimensions.The vector-valued extensions of these inequalities are also given.Parallel results ...We prove weighted q-variation inequalities with 2<q<∞for sharp truncations of singular integral operators in higher dimensions.The vector-valued extensions of these inequalities are also given.Parallel results are proven for differential operators.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
The author considers the embedding problem of weighted Sobolev spaces H<sup>n</sup><sub>p</sub> in weighted L<sub>s</sub> spaces L<sub>s,r</sub>,and some sufficient cond...The author considers the embedding problem of weighted Sobolev spaces H<sup>n</sup><sub>p</sub> in weighted L<sub>s</sub> spaces L<sub>s,r</sub>,and some sufficient conditions and necessary conditions are given, when weight functions satisfy certain conditions.The author uses the results obtained to the qualitative analysis of the spectrum of 2n-order weighted differential operator,and gives some sufficient conditions and necessary conditions to ensure that the spectrum is discrete.展开更多
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
Insa and Pauer presented a basic theory of Grobner basis for differential operators with coefficients in a commutative ring in 1998, and a criterion was proposed to determine if a set of differential operators is a Gr...Insa and Pauer presented a basic theory of Grobner basis for differential operators with coefficients in a commutative ring in 1998, and a criterion was proposed to determine if a set of differential operators is a GrSbner basis. In this paper, we will give a new criterion such that Insa and Pauer＇s criterion could be concluded as a special case and one could compute the Grobner basis more efficiently by this new criterion.展开更多
The first aim of the paper is to study the Hermitizability of secondorder differential operators,and then the corresponding isospectral operators.The explicit criteria for the Hermitizable or isospectral properties ar...The first aim of the paper is to study the Hermitizability of secondorder differential operators,and then the corresponding isospectral operators.The explicit criteria for the Hermitizable or isospectral properties are presented.The second aim of the paper is to study a non-Hermitian model,which is now well known.In a regular sense,the model does not belong to the class of Hermitizable operators studied in this paper,but we will use the theory developed in the past years,to present an alternative and illustrated proof of the discreteness of its spectrum.The harmonic function plays a critical role in the study of spectrum.Two constructions of the function are presented.The required conclusion for the discrete spectrum is proved by some comparison technique.展开更多
In the paper under review,we consider the generation of fractional resolvent families by abstract differential operators.Our results can be simply incorporated in the study of corresponding abstract time-fractional eq...In the paper under review,we consider the generation of fractional resolvent families by abstract differential operators.Our results can be simply incorporated in the study of corresponding abstract time-fractional equations with Caputo fractional derivatives.展开更多
This paper discusses some problems on the cardinal spline interpolation correspond- ing to infinite order differential operators.The remainder formulas and a dual theorem are es- tablished for some convolution classes...This paper discusses some problems on the cardinal spline interpolation correspond- ing to infinite order differential operators.The remainder formulas and a dual theorem are es- tablished for some convolution classes,where the kernels are PF densities.Moreover,the exact error of approximation of a convolution class with interpolation cardinal splines is determined. The exact values of average n-Kolmogorov widths are obtained for the convolution class.展开更多
This study focuses on vector-valued anisotropic Sobolev-Lions spaces associated with Banach spaces E0, E. Several conditions are found that ensure the continuity and compactness of embedding operators that are optimal...This study focuses on vector-valued anisotropic Sobolev-Lions spaces associated with Banach spaces E0, E. Several conditions are found that ensure the continuity and compactness of embedding operators that are optimal regular in these spaces in terms of interpolations of spaces E0 and E. In particular, the most regular class of interpolation spaces Eα between E0, E depending on α and the order of space are found and the boundedness of differential operators D^α from this space to Eα-valued Lp,γ spaces is proved. These results are applied to partial differential-operator equations with parameters to obtain conditions that guarantee the maximal Lp,γ regularity and R-positivity uniformly with respect to these parameters.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.
文摘In this paper, Leibniz' formula of generalized divided difference with respect to a class of differential operators whose basic sets of solutions have power form, is considered. The recurrence formula of Green function about the operators is also given.
文摘We consider the space of rapidly decreasing sequences s and the derivative operator D defined on it. The object of this article is to study the equivalence of a differential operator of infinite order; that is φ（D） =^∞∑k=0φkD^k.φk constant numbers an a power of D.Dn, meaning, is there a isomorphism X （from s onto s） such that Xφ（D） = D^nX？. We prove that if φ（D） is equivalent to Dn, then φ（D） is of finite order, in fact a polynomial of degree n. The question of the equivalence of two differential operators of finite order in the space s is addressed too and solved completely when n = 1.
文摘When one function is defined as a differential operation on another function, it’s often desirable to invert the definition, to effectively “undo” the differentiation. A Green’s function approach is often used to accomplish this, but variations on this theme exist, and we examine a few such variations. The mathematical analysis of is sought in the form if such an inverse operator exists, but physics is defined by both mathematical formula and ontological formalism, as I show for an example based on the Dirac equation. Finally, I contrast these “standard” approaches with a novel exact inverse operator for field equations.
文摘In this paper, we consider the general ordinary quasi-differential expression τ of order n with complex coefficients and its formal adjoint τ<sup>+</sup> on the interval [a,b). We shall show in the case of one singular end-point and under suitable conditions that all solutions of a general ordinary quasi-differential equation are in the weighted Hilbert space provided that all solutions of the equations and its adjoint are in . Also, a number of results concerning the location of the point spectra and regularity fields of the operators generated by such expressions may be obtained. Some of these results are extensions or generalizations of those in the symmetric case, while the others are new.
文摘In the cost function of three- or four-dimensional variational dataassimilation, each term is weighted by the inverse of its associated error covariance matrix and thebackground error covariance matrix is usually much larger than the other covariance matrices.Although the background error covariances are traditionally normalized and parameterized by simplesmooth homogeneous correlation functions, the covariance matrices constructed from these correlationfunctions are often too large to be inverted or even manipulated. It is thus desirable to finddirect representations of the inverses of background error correlations. This problem is studied inthis paper. In particular, it is shown that the background term can be written into ∫ dx∣Dυ(x)∣~2, that is, a squared 1/2 norm of a vector differential operator D, called theD-operator, applied to the field of analysis increment υ(x). For autoregressive correlationfunctions, the D-operators are of finite orders. For Gaussian correlation functions, the D-operatorsare of infinite order. For practical applications, the Gaussian D-operators must be truncated tofinite orders. The truncation errors are found to be small even when the Gaussian D-operators aretruncated to low orders. With a truncated D-operator, the background term can be easily constructedwith neither inversion nor direct calculation of the covariance matrix. D-operators are also derivedfor non-Gaussian correlations and transformed into non-isotropic forms.
基金Supported by the National Natural Science Foundation of China(10561005)the Doctor's Discipline Fund of the Ministry of Education of China(20040126008)
文摘The spectrum of a class of fourth order left-definite differential operators is studied. By using the theory of indefinite differential operators in Krein space and the relationship between left-definite and right-definite operators, the following conclusions are obtained： if a fourth order differential operator with a self-adjoint boundary condition that is left-definite and right-indefinite, then all its eigenvalues are real, and there exist countably infinitely many positive and negative eigenvalues which are unbounded from below and above, have no finite cluster point and can be indexed to satisfy the inequality …≤λ-2≤λ-1≤λ-0〈0〈λ0≤λ1≤λ2≤…
文摘For a real valued function f defined on a finite interval I we consider the problem of approximating f from null spaces of differential operators of the form Ln（ψ） =∑k=0^n akψ（k） where the constant coefficients ak C R may be adapted to f.
基金The project supported by National Natural Science Foundation of China under Grant Nos.10075042 and 10375056
文摘The quantum Euclidean space is a kind of noncommutative space that is obtained from ordinary Euclidean space by deformation with parameter q. When N is odd, the structure of this space is similar to . Motivated by realization of by differential operators in , we give such realization for and cases and generalize our results to (N odd) in this paper, that is, we show that the algebra of can be realized by differential operators acting on C<SUP>∞</SUP> functions on undeformed space .
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 National Natural Science Foundation of China(Grant Nos.11671308 and 11431011)Ministerio de Economia y Competitividad/al Fondo Europeo de Desarrollo Regional(Grant No.MTM2015-66157-C2-1-P)
文摘We prove weighted q-variation inequalities with 2<q<∞for sharp truncations of singular integral operators in higher dimensions.The vector-valued extensions of these inequalities are also given.Parallel results are proven for differential operators.
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 the National Natural Science Fundation of Chinathe Natural Science Foundation of Inner Mongolia.
文摘The author considers the embedding problem of weighted Sobolev spaces H<sup>n</sup><sub>p</sub> in weighted L<sub>s</sub> spaces L<sub>s,r</sub>,and some sufficient conditions and necessary conditions are given, when weight functions satisfy certain conditions.The author uses the results obtained to the qualitative analysis of the spectrum of 2n-order weighted differential operator,and gives some sufficient conditions and necessary conditions to ensure that the spectrum is discrete.
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 Nos.10971217, 60821002/F02)NC Special Project
文摘Insa and Pauer presented a basic theory of Grobner basis for differential operators with coefficients in a commutative ring in 1998, and a criterion was proposed to determine if a set of differential operators is a GrSbner basis. In this paper, we will give a new criterion such that Insa and Pauer＇s criterion could be concluded as a special case and one could compute the Grobner basis more efficiently by this new criterion.
基金supported in part by the National Natural Science Foundation of China(Grant No.11771046)the project from the Ministry of Education in China,and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.
文摘The first aim of the paper is to study the Hermitizability of secondorder differential operators,and then the corresponding isospectral operators.The explicit criteria for the Hermitizable or isospectral properties are presented.The second aim of the paper is to study a non-Hermitian model,which is now well known.In a regular sense,the model does not belong to the class of Hermitizable operators studied in this paper,but we will use the theory developed in the past years,to present an alternative and illustrated proof of the discreteness of its spectrum.The harmonic function plays a critical role in the study of spectrum.Two constructions of the function are presented.The required conclusion for the discrete spectrum is proved by some comparison technique.
基金Supported by Ministry of Science and Technological Development,Republic of Serbia(Grant No.174024)
文摘In the paper under review,we consider the generation of fractional resolvent families by abstract differential operators.Our results can be simply incorporated in the study of corresponding abstract time-fractional equations with Caputo fractional derivatives.
文摘This paper discusses some problems on the cardinal spline interpolation correspond- ing to infinite order differential operators.The remainder formulas and a dual theorem are es- tablished for some convolution classes,where the kernels are PF densities.Moreover,the exact error of approximation of a convolution class with interpolation cardinal splines is determined. The exact values of average n-Kolmogorov widths are obtained for the convolution class.
基金This work is supported by the grant of Istanbul University (Project UDP-227/18022004)
文摘This study focuses on vector-valued anisotropic Sobolev-Lions spaces associated with Banach spaces E0, E. Several conditions are found that ensure the continuity and compactness of embedding operators that are optimal regular in these spaces in terms of interpolations of spaces E0 and E. In particular, the most regular class of interpolation spaces Eα between E0, E depending on α and the order of space are found and the boundedness of differential operators D^α from this space to Eα-valued Lp,γ spaces is proved. These results are applied to partial differential-operator equations with parameters to obtain conditions that guarantee the maximal Lp,γ regularity and R-positivity uniformly with respect to these parameters.
