In this paper, we give the definition of weak WT2-class of differential forms, and then obtain its weak reverse Holder inequality. As an application, we give an alternative proof of the higher integrability result of ...In this paper, we give the definition of weak WT2-class of differential forms, and then obtain its weak reverse Holder inequality. As an application, we give an alternative proof of the higher integrability result of weakly A-harmonic tensors due to B. Stroffolini.展开更多
Let(M,g_(0))be a compact Riemannian manifold-with-boundary.We present a new proof of the classical Gaffney inequality for differential forms in boundary value spaces over M,via a variational approach a la Kozono-Yanag...Let(M,g_(0))be a compact Riemannian manifold-with-boundary.We present a new proof of the classical Gaffney inequality for differential forms in boundary value spaces over M,via a variational approach a la Kozono-Yanagisawa[Lr-variational inequality for vector fields and the Helmholtz-Weyl decomposition in bounded domains,Indiana Univ.Math.J.58(2009),1853-1920],combined with global computations based on the Bochner technique.展开更多
In the article, we discuss basic concepts of the residue theory of logarithmic and multi-logarithmic differential forms, and describe some aspects of the theory, de-veloped by the author in the past few years. In part...In the article, we discuss basic concepts of the residue theory of logarithmic and multi-logarithmic differential forms, and describe some aspects of the theory, de-veloped by the author in the past few years. In particular, we introduce the notion of logarithmic differential forms with the use of the classical de Rham lemma and give an explicit description of regular meromorphic differential forms in terms of residues of logarithmic or multi-logarithmic differential forms with respect to hypersurfaces, com-plete intersections or pure-dimensional Cohen-Macaulay spaces. Among other things, several useful applications are considered, which are related with the theory of holo-nomic D-modules, the theory of Hodge structures, the theory of residual currents and others.展开更多
Effective Hamiltonians in periodically driven systems have received widespread attention for realization of novel quantum phases, non-equilibrium phase transition, and Majorana mode. Recently, the study of effective H...Effective Hamiltonians in periodically driven systems have received widespread attention for realization of novel quantum phases, non-equilibrium phase transition, and Majorana mode. Recently, the study of effective Hamiltonian using various methods has gained great interest. We consider a vector differential equation of motion to derive the effective Hamiltonian for any periodically driven two-level system, and the dynamics of the spin vector are an evolution under the Bloch sphere. Here, we investigate the properties of this equation and show that a sudden change of the effective Hamiltonian is expected. Furthermore, we present several exact relations, whose expressions are independent of the different starting points. Moreover, we deduce the effective Hamiltonian from the high-frequency limit, which approximately equals the results in previous studies. Our results show that the vector differential equation of motion is not affected by a convergence problem, and thus, can be used to numerically investigate the effective models in any periodic modulating system. Finally, we anticipate that the proposed method can be applied to experimental platforms that require time-periodic modulation, such as ultracold atoms and optical lattices.展开更多
We prove local weighted integral inequalities for differential forms. Then by using the local results, we prove global weighted integral inequalities for differential forms in L^s(μ)-averaging domains and in John dom...We prove local weighted integral inequalities for differential forms. Then by using the local results, we prove global weighted integral inequalities for differential forms in L^s(μ)-averaging domains and in John domains, respectively, which can be considered as generalizations of the classical Poincare-type inequality.展开更多
In this paper, we suggested exact-volume differential form (for short:EVDF) and proved four theorems correlative with them: 1. existence theorem, 2. cohomology theorem,3. constant multiple theorem, and 4. equal gauge ...In this paper, we suggested exact-volume differential form (for short:EVDF) and proved four theorems correlative with them: 1. existence theorem, 2. cohomology theorem,3. constant multiple theorem, and 4. equal gauge theorem. And their application were discussed also. For examlpe, we deduced the particle dynamic equation of the special theory of relativity. At the same time we analyzed and contrasted cohomology theory with Hamilton's variational principle. The contrast showed the superiority of cohomology theory. Moreover,we gave a more complete classification list of differential forms.展开更多
We present a new method to calculate the focal value of ordinary differential equation by applying the theorem defined the relationship between the normal form and focal value,with the help of a symbolic computation l...We present a new method to calculate the focal value of ordinary differential equation by applying the theorem defined the relationship between the normal form and focal value,with the help of a symbolic computation language M ATHEMATICA,and extending the matrix representation method.This method can be used to calculate the focal value of any high order terms.This method has been verified by an example.The advantage of this method is simple and more readily applicable.the result is directly obtained by substitution.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
In this paper, combining the idea of difference method and finite element method, we construct a difference scheme for a self-adjoint problem in conservation form. Its solution uniformly converges to that of the origi...In this paper, combining the idea of difference method and finite element method, we construct a difference scheme for a self-adjoint problem in conservation form. Its solution uniformly converges to that of the original differential equation problem with order h3.展开更多
A new Koppelman-Leray-Norguet formula of (p,q) differential forms for a strictly pseudoconvex polyhedron with not necessarily smooth boundary on a Stein manifold is obtained, and an integral representation for the sol...A new Koppelman-Leray-Norguet formula of (p,q) differential forms for a strictly pseudoconvex polyhedron with not necessarily smooth boundary on a Stein manifold is obtained, and an integral representation for the solution of -equation on this domain which does not involve integrals on boundary is given, so one can avoid complex estimates of boundary integrals.展开更多
A brief survey of fractional calculus and fractional differential forms was firstly given.The fractional exterior transition to curvilinear coordinate at the origin were discussed and the two coordinate transformation...A brief survey of fractional calculus and fractional differential forms was firstly given.The fractional exterior transition to curvilinear coordinate at the origin were discussed and the two coordinate transformations for the fractional differentials for three-dimensional Cartesian coordinates to spherical and cylindrical coordinates are obtained, respectively. In particular, for v=m=1 ,the usual exterior transformations, between the spherical coordinate and Cartesian coordinate, as well as the cylindrical coordinate and Cartesian coordinate, are found respectively, from fractional exterior transformation.展开更多
We obtain the global weighted estimates for Jacobian J(x,f) and its subdeterminants, as well as the components of K-quasiconformal mappings in L^p (μ)-averaging domains. To develop these estimates, both local and...We obtain the global weighted estimates for Jacobian J(x,f) and its subdeterminants, as well as the components of K-quasiconformal mappings in L^p (μ)-averaging domains. To develop these estimates, both local and global weighted Ponincaré inequalities for differential forms are established.展开更多
Lie algorithm combined with differential form Wu's method is used to complete the symmetry classification of partial differential equations(PDEs)containing arbitrary parameter.This process can be reduced to solve ...Lie algorithm combined with differential form Wu's method is used to complete the symmetry classification of partial differential equations(PDEs)containing arbitrary parameter.This process can be reduced to solve a large system of determining equations,which seems rather difficult to solve,then the differential form Wu's method is used to decompose the determining equations into a series of equations,which are easy to solve.To illustrate the usefulness of this method,we apply it to some test problems,and the results show the performance of the present work.展开更多
We give a study on the general Moiler transformation and emphatically introduce its differential form. In this paper, a definition of acceleration is given in spacetime language and the inertial reference frame is als...We give a study on the general Moiler transformation and emphatically introduce its differential form. In this paper, a definition of acceleration is given in spacetime language and the inertial reference frame is also settled. With a discussion of the geodesic equations of motion, the differential form of the general MФller transformation at arbitrary direction is presented.展开更多
This is the first paper on symmetry classification for ordinary differential equations(ODEs)based on Wu’s method.We carry out symmetry classification of two ODEs,named the generalizations of the Kummer-Schwarz equati...This is the first paper on symmetry classification for ordinary differential equations(ODEs)based on Wu’s method.We carry out symmetry classification of two ODEs,named the generalizations of the Kummer-Schwarz equations which involving arbitrary function.First,Lie algorithm is used to give the determining equations of symmetry for the given equations,which involving arbitrary functions.Next,differential form Wu’s method is used to decompose determining equations into a union of a series of zero sets of differential characteristic sets,which are easy to be solved relatively.Each branch of the decomposition yields a class of symmetries and associated parameters.The algorithm makes the classification become direct and systematic.Yuri Dimitrov Bozhkov,and Pammela Ramos da Conceição have used the Lie algorithm to give the symmetry classifications of the equations talked in this paper in 2020.From this paper,we can find that the differential form Wu’s method for symmetry classification of ODEs with arbitrary function(parameter)is effective,and is an alternative method.展开更多
With the help of skew-symmetric differential forms the hidden properties of the mathematical physics equations are revealed. It is shown that the equations of mathematical physics can describe the emergence of various...With the help of skew-symmetric differential forms the hidden properties of the mathematical physics equations are revealed. It is shown that the equations of mathematical physics can describe the emergence of various structures and formations such as waves, vortices, turbulent pulsations and others. Such properties of the mathematical physics equations, which are hidden (they appear only in the process of solving these equations), depend on the consistency of derivatives in partial differential equations and on the consistency of equations, if the equations of mathematical physics are a set of equations. This is due to the integrability of mathematical physics equations. It is shown that the equations of mathematical physics can have double solutions, namely, the solutions on the original coordinate space and the solutions on integrable structures that are realized discretely (due to any degrees of freedom). The transition from the solutions of the first type to one of the second type describes discrete transitions and the processes of origin of various structures and observable formations. Only mathematical physics equations, on what no additional conditions such as the integrability conditions are imposed, can possess such properties. The results of the present paper were obtained with the help of skew-symmetric differential forms.展开更多
The authors first give the definition of degenerate weakly (K1,K2)-quasiregular mappings using the technique of exterior power and exterior differential forms, and then, using the method of McShane extension, a usef...The authors first give the definition of degenerate weakly (K1,K2)-quasiregular mappings using the technique of exterior power and exterior differential forms, and then, using the method of McShane extension, a useful inequality is obtained, which can be used to derive the self-improving regularity.展开更多
The meshless local Petrov_Galerkin (MLPG) method for solving the bending problem of the thin plate were presented and discussed. The method used the moving least_squares approximation to interpolate the solution varia...The meshless local Petrov_Galerkin (MLPG) method for solving the bending problem of the thin plate were presented and discussed. The method used the moving least_squares approximation to interpolate the solution variables, and employed a local symmetric weak form. The present method was a truly meshless one as it did not need a finite element or boundary element mesh, either for purpose of interpolation of the solution, or for the integration of the energy. All integrals could be easily evaluated over regularly shaped domains (in general, spheres in three_dimensional problems) and their boundaries. The essential boundary conditions were enforced by the penalty method. Several numerical examples were presented to illustrate the implementation and performance of the present method. The numerical examples presented show that high accuracy can be achieved for arbitrary grid geometries for clamped and simply_supported edge conditions. No post processing procedure is required to computer the strain and stress, since the original solution from the present method, using the moving least squares approximation, is already smooth enough.展开更多
In this paper, in order to investigate whether the impact of different forms of interest rate differential may pass on to the flexible price monetary model, two flexible price monetary models, which are separately der...In this paper, in order to investigate whether the impact of different forms of interest rate differential may pass on to the flexible price monetary model, two flexible price monetary models, which are separately derived from the generalized monetary models with log-level interest rate differential and that with interest rate differential, are tested for China yuan to US dollar exchange rate. Through Johansen maximum likelihood method, we find that there is little support in the cointegrating coefllcient estimates for both flexible price monetary models for yuan/dollar exchange rate. However, the latter is generally better than the former in the light of sum of squared residual and log likelihood statistics. Therefore, we conclude that there is no transitive impact of different forms of interest rate differential on the flexible price monetary model.展开更多
基金Supported by the National Natural Science Foundation of China (10971224)the Hebei Natural ScienceFoundation (07M003)
文摘In this paper, we give the definition of weak WT2-class of differential forms, and then obtain its weak reverse Holder inequality. As an application, we give an alternative proof of the higher integrability result of weakly A-harmonic tensors due to B. Stroffolini.
文摘Let(M,g_(0))be a compact Riemannian manifold-with-boundary.We present a new proof of the classical Gaffney inequality for differential forms in boundary value spaces over M,via a variational approach a la Kozono-Yanagisawa[Lr-variational inequality for vector fields and the Helmholtz-Weyl decomposition in bounded domains,Indiana Univ.Math.J.58(2009),1853-1920],combined with global computations based on the Bochner technique.
文摘In the article, we discuss basic concepts of the residue theory of logarithmic and multi-logarithmic differential forms, and describe some aspects of the theory, de-veloped by the author in the past few years. In particular, we introduce the notion of logarithmic differential forms with the use of the classical de Rham lemma and give an explicit description of regular meromorphic differential forms in terms of residues of logarithmic or multi-logarithmic differential forms with respect to hypersurfaces, com-plete intersections or pure-dimensional Cohen-Macaulay spaces. Among other things, several useful applications are considered, which are related with the theory of holo-nomic D-modules, the theory of Hodge structures, the theory of residual currents and others.
基金supported by the National Natural Science Foundation of China (Grant No. 11774328)。
文摘Effective Hamiltonians in periodically driven systems have received widespread attention for realization of novel quantum phases, non-equilibrium phase transition, and Majorana mode. Recently, the study of effective Hamiltonian using various methods has gained great interest. We consider a vector differential equation of motion to derive the effective Hamiltonian for any periodically driven two-level system, and the dynamics of the spin vector are an evolution under the Bloch sphere. Here, we investigate the properties of this equation and show that a sudden change of the effective Hamiltonian is expected. Furthermore, we present several exact relations, whose expressions are independent of the different starting points. Moreover, we deduce the effective Hamiltonian from the high-frequency limit, which approximately equals the results in previous studies. Our results show that the vector differential equation of motion is not affected by a convergence problem, and thus, can be used to numerically investigate the effective models in any periodic modulating system. Finally, we anticipate that the proposed method can be applied to experimental platforms that require time-periodic modulation, such as ultracold atoms and optical lattices.
文摘We prove local weighted integral inequalities for differential forms. Then by using the local results, we prove global weighted integral inequalities for differential forms in L^s(μ)-averaging domains and in John domains, respectively, which can be considered as generalizations of the classical Poincare-type inequality.
文摘In this paper, we suggested exact-volume differential form (for short:EVDF) and proved four theorems correlative with them: 1. existence theorem, 2. cohomology theorem,3. constant multiple theorem, and 4. equal gauge theorem. And their application were discussed also. For examlpe, we deduced the particle dynamic equation of the special theory of relativity. At the same time we analyzed and contrasted cohomology theory with Hamilton's variational principle. The contrast showed the superiority of cohomology theory. Moreover,we gave a more complete classification list of differential forms.
文摘We present a new method to calculate the focal value of ordinary differential equation by applying the theorem defined the relationship between the normal form and focal value,with the help of a symbolic computation language M ATHEMATICA,and extending the matrix representation method.This method can be used to calculate the focal value of any high order terms.This method has been verified by an example.The advantage of this method is simple and more readily applicable.the result is directly obtained by substitution.
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In this paper, combining the idea of difference method and finite element method, we construct a difference scheme for a self-adjoint problem in conservation form. Its solution uniformly converges to that of the original differential equation problem with order h3.
基金Supported by the National Natural Science Foundation and Mathematical "Tian Yuan" Foundation of China and the Natural Science Foundation of Fujian (Grant No. 10271097, TY10126033, F0110012)
文摘A new Koppelman-Leray-Norguet formula of (p,q) differential forms for a strictly pseudoconvex polyhedron with not necessarily smooth boundary on a Stein manifold is obtained, and an integral representation for the solution of -equation on this domain which does not involve integrals on boundary is given, so one can avoid complex estimates of boundary integrals.
文摘A brief survey of fractional calculus and fractional differential forms was firstly given.The fractional exterior transition to curvilinear coordinate at the origin were discussed and the two coordinate transformations for the fractional differentials for three-dimensional Cartesian coordinates to spherical and cylindrical coordinates are obtained, respectively. In particular, for v=m=1 ,the usual exterior transformations, between the spherical coordinate and Cartesian coordinate, as well as the cylindrical coordinate and Cartesian coordinate, are found respectively, from fractional exterior transformation.
文摘We obtain the global weighted estimates for Jacobian J(x,f) and its subdeterminants, as well as the components of K-quasiconformal mappings in L^p (μ)-averaging domains. To develop these estimates, both local and global weighted Ponincaré inequalities for differential forms are established.
基金National Natural Science Foundation of China(No.61862048)。
文摘Lie algorithm combined with differential form Wu's method is used to complete the symmetry classification of partial differential equations(PDEs)containing arbitrary parameter.This process can be reduced to solve a large system of determining equations,which seems rather difficult to solve,then the differential form Wu's method is used to decompose the determining equations into a series of equations,which are easy to solve.To illustrate the usefulness of this method,we apply it to some test problems,and the results show the performance of the present work.
文摘We give a study on the general Moiler transformation and emphatically introduce its differential form. In this paper, a definition of acceleration is given in spacetime language and the inertial reference frame is also settled. With a discussion of the geodesic equations of motion, the differential form of the general MФller transformation at arbitrary direction is presented.
文摘This is the first paper on symmetry classification for ordinary differential equations(ODEs)based on Wu’s method.We carry out symmetry classification of two ODEs,named the generalizations of the Kummer-Schwarz equations which involving arbitrary function.First,Lie algorithm is used to give the determining equations of symmetry for the given equations,which involving arbitrary functions.Next,differential form Wu’s method is used to decompose determining equations into a union of a series of zero sets of differential characteristic sets,which are easy to be solved relatively.Each branch of the decomposition yields a class of symmetries and associated parameters.The algorithm makes the classification become direct and systematic.Yuri Dimitrov Bozhkov,and Pammela Ramos da Conceição have used the Lie algorithm to give the symmetry classifications of the equations talked in this paper in 2020.From this paper,we can find that the differential form Wu’s method for symmetry classification of ODEs with arbitrary function(parameter)is effective,and is an alternative method.
文摘With the help of skew-symmetric differential forms the hidden properties of the mathematical physics equations are revealed. It is shown that the equations of mathematical physics can describe the emergence of various structures and formations such as waves, vortices, turbulent pulsations and others. Such properties of the mathematical physics equations, which are hidden (they appear only in the process of solving these equations), depend on the consistency of derivatives in partial differential equations and on the consistency of equations, if the equations of mathematical physics are a set of equations. This is due to the integrability of mathematical physics equations. It is shown that the equations of mathematical physics can have double solutions, namely, the solutions on the original coordinate space and the solutions on integrable structures that are realized discretely (due to any degrees of freedom). The transition from the solutions of the first type to one of the second type describes discrete transitions and the processes of origin of various structures and observable formations. Only mathematical physics equations, on what no additional conditions such as the integrability conditions are imposed, can possess such properties. The results of the present paper were obtained with the help of skew-symmetric differential forms.
基金Research supported by NSFC (10471149)Special Fund of Mathematics Research of Natural Science Foundation of Hebei Province (07M003)Doctoral Foundation of Hebei Province Ministry of Education (B2004103).
文摘The authors first give the definition of degenerate weakly (K1,K2)-quasiregular mappings using the technique of exterior power and exterior differential forms, and then, using the method of McShane extension, a useful inequality is obtained, which can be used to derive the self-improving regularity.
文摘The meshless local Petrov_Galerkin (MLPG) method for solving the bending problem of the thin plate were presented and discussed. The method used the moving least_squares approximation to interpolate the solution variables, and employed a local symmetric weak form. The present method was a truly meshless one as it did not need a finite element or boundary element mesh, either for purpose of interpolation of the solution, or for the integration of the energy. All integrals could be easily evaluated over regularly shaped domains (in general, spheres in three_dimensional problems) and their boundaries. The essential boundary conditions were enforced by the penalty method. Several numerical examples were presented to illustrate the implementation and performance of the present method. The numerical examples presented show that high accuracy can be achieved for arbitrary grid geometries for clamped and simply_supported edge conditions. No post processing procedure is required to computer the strain and stress, since the original solution from the present method, using the moving least squares approximation, is already smooth enough.
基金This project is supported by National Natural Science Foundation of China (70371055)
文摘In this paper, in order to investigate whether the impact of different forms of interest rate differential may pass on to the flexible price monetary model, two flexible price monetary models, which are separately derived from the generalized monetary models with log-level interest rate differential and that with interest rate differential, are tested for China yuan to US dollar exchange rate. Through Johansen maximum likelihood method, we find that there is little support in the cointegrating coefllcient estimates for both flexible price monetary models for yuan/dollar exchange rate. However, the latter is generally better than the former in the light of sum of squared residual and log likelihood statistics. Therefore, we conclude that there is no transitive impact of different forms of interest rate differential on the flexible price monetary model.