This paper is to extend the Poincar’e Lemma for differential forms in a bounded, convex domain [1] in Rn to a more general domain that, we call, is deformable to every point in itself. Then we extend the homotopy ope...This paper is to extend the Poincar’e Lemma for differential forms in a bounded, convex domain [1] in Rn to a more general domain that, we call, is deformable to every point in itself. Then we extend the homotopy operator T in [1] to the domain defromed to every point of itself.展开更多
Recently, many new features of Sobolev spaces W k,p ？RN ？ were studied in [4-6, 32]. This paper is devoted to giving a brief review of some known characterizations of Sobolev spaces in Euclidean spaces and describin...Recently, many new features of Sobolev spaces W k,p ？RN ？ were studied in [4-6, 32]. This paper is devoted to giving a brief review of some known characterizations of Sobolev spaces in Euclidean spaces and describing our recent study of new characterizations of Sobolev spaces on both Heisenberg groups and Euclidean spaces obtained in [12] and [13] and outlining their proofs. Our results extend those characterizations of first order Sobolev spaces in [32] to the Heisenberg group setting. Moreover, our theorems also provide diff erent characterizations for the second order Sobolev spaces in Euclidean spaces from those in [4, 5].展开更多
We present the noncommutative differential calculus on the function space of the infinite set and construct a homotopy operator to prove the analogue of the Poincare lemma for the difference complex. Then the horizont...We present the noncommutative differential calculus on the function space of the infinite set and construct a homotopy operator to prove the analogue of the Poincare lemma for the difference complex. Then the horizontal and vertical complexes are introduced with the total differential map and vertical exterior derivative. As the application of the differential calculus, we derive the schemes with the conservation of symplecticity and energy for Hamiltonian system and a two-dimensional integral models with infinite sequence of conserved currents. Then an Euler-Lagrange cohomology with symplectic structure-preserving is given in the discrete classical mechanics.展开更多
By introducing the noncommutative differential calculus on the function space of the infinite/finite set and construct a homotopy operator, one prove the analogue of the Poincare lemma for the difference complex. As a...By introducing the noncommutative differential calculus on the function space of the infinite/finite set and construct a homotopy operator, one prove the analogue of the Poincare lemma for the difference complex. As an application of the differential calculus, a two dimensional integral model can be derived from the noncommutative differential calculus.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
The author proves the Poincard lemma on some （n ＋1）-dimensional corank 1 sub-Riemannian structures, formulating the （n-1）n（n2＋3n-2） necessarily and sufficient- s ly ＂curl-vanishing＂ compatibility conditions...The author proves the Poincard lemma on some （n ＋1）-dimensional corank 1 sub-Riemannian structures, formulating the （n-1）n（n2＋3n-2） necessarily and sufficient- s ly ＂curl-vanishing＂ compatibility conditions. In particular, this result solves partially an open problem formulated by Calin and Chang. The proof in this paper is based on a Poincard lemma stated on l：tiemannian manifolds and a suitable Ceskro-Volterra path in- tegral formula established in local coordinates. As a byproduct, a Saint-Venant lemma is also provided on generic Riemannian manifolds. Some examples are presented on the hyperbolic space and Carnot/Heisenberg groups.展开更多
Let ■Ω=Γ=Γ<sub>1</sub>+Γ<sub>2</sub> （see Fig.1）,meas（Γ<sub>1</sub>）】0,V={v|v∈H<sup>1</sup>（Ω）,v|Γ<sub>1</sub>=0},and V<sub>0</sub&g...Let ■Ω=Γ=Γ<sub>1</sub>+Γ<sub>2</sub> （see Fig.1）,meas（Γ<sub>1</sub>）】0,V={v|v∈H<sup>1</sup>（Ω）,v|Γ<sub>1</sub>=0},and V<sub>0</sub>={ω|Δω=h in Ω,ω|Γ=0,（?）h∈V}.Let V<sub>0</sub>′=thedual space of V<sub>0</sub>,a（u,v）=∫<sub>Ω</sub>▽u·▽Δvdx,and F（v）=∫<sub>Ω</sub> fvdx+∫<sub>Γ<sub>2</sub></sub>g1Δvds-∫<sub>Γ</sub>g2（?）ds,f∈V′<sub>0</sub>,g1∈H<sup>-（1/2）</sup>（Γ<sub>2</sub>）,g2∈H<sup>-（3/2）</sup>（Γ）.Consider the variational problem:find u ∈ V such thata（u,v）=F（v）,（?）v∈V<sub>0</sub>. （1）Using Tartar’s lemma,we prove that for problem （1） there exists a unique展开更多
文摘This paper is to extend the Poincar’e Lemma for differential forms in a bounded, convex domain [1] in Rn to a more general domain that, we call, is deformable to every point in itself. Then we extend the homotopy operator T in [1] to the domain defromed to every point of itself.
基金Supported by the National Natural Science Foundation of China(1146103211401267)+2 种基金the Foundation of Jiangxi University of Science and Technology(NSFJ2015-G25)the Youth Foundation of Jiangxi Provincial Education Department of China(GJJ150646GJJ151356)
文摘Recently, many new features of Sobolev spaces W k,p ？RN ？ were studied in [4-6, 32]. This paper is devoted to giving a brief review of some known characterizations of Sobolev spaces in Euclidean spaces and describing our recent study of new characterizations of Sobolev spaces on both Heisenberg groups and Euclidean spaces obtained in [12] and [13] and outlining their proofs. Our results extend those characterizations of first order Sobolev spaces in [32] to the Heisenberg group setting. Moreover, our theorems also provide diff erent characterizations for the second order Sobolev spaces in Euclidean spaces from those in [4, 5].
基金The project supported by National Natural Science Foundation of China under Grant No.10626016China Postdoctor Science Foundation of Henan University under Grant No.05YBZR014
文摘We present the noncommutative differential calculus on the function space of the infinite set and construct a homotopy operator to prove the analogue of the Poincare lemma for the difference complex. Then the horizontal and vertical complexes are introduced with the total differential map and vertical exterior derivative. As the application of the differential calculus, we derive the schemes with the conservation of symplecticity and energy for Hamiltonian system and a two-dimensional integral models with infinite sequence of conserved currents. Then an Euler-Lagrange cohomology with symplectic structure-preserving is given in the discrete classical mechanics.
基金Supported by the China Pcetdoctoral Science Foundation by a grant from Henan University(05YBZR014)Supported by the Tianyuan Foundation for Mathematics of National Natural Science Foundation of China(10626016)
文摘By introducing the noncommutative differential calculus on the function space of the infinite/finite set and construct a homotopy operator, one prove the analogue of the Poincare lemma for the difference complex. As an application of the differential calculus, a two dimensional integral model can be derived from the noncommutative differential calculus.
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.
文摘The author proves the Poincard lemma on some （n ＋1）-dimensional corank 1 sub-Riemannian structures, formulating the （n-1）n（n2＋3n-2） necessarily and sufficient- s ly ＂curl-vanishing＂ compatibility conditions. In particular, this result solves partially an open problem formulated by Calin and Chang. The proof in this paper is based on a Poincard lemma stated on l：tiemannian manifolds and a suitable Ceskro-Volterra path in- tegral formula established in local coordinates. As a byproduct, a Saint-Venant lemma is also provided on generic Riemannian manifolds. Some examples are presented on the hyperbolic space and Carnot/Heisenberg groups.
基金supported by the National Natural Science Foundation of China(Grant No.12071076)the Program for Education and Scientific Research Project of Young and Middle-Aged Teachers in Fujian Province(Grant Nos.JAT191128,JT180818).
基金This research was supported by the National Natural Science Foundation of China
文摘Let ■Ω=Γ=Γ<sub>1</sub>+Γ<sub>2</sub> （see Fig.1）,meas（Γ<sub>1</sub>）】0,V={v|v∈H<sup>1</sup>（Ω）,v|Γ<sub>1</sub>=0},and V<sub>0</sub>={ω|Δω=h in Ω,ω|Γ=0,（?）h∈V}.Let V<sub>0</sub>′=thedual space of V<sub>0</sub>,a（u,v）=∫<sub>Ω</sub>▽u·▽Δvdx,and F（v）=∫<sub>Ω</sub> fvdx+∫<sub>Γ<sub>2</sub></sub>g1Δvds-∫<sub>Γ</sub>g2（?）ds,f∈V′<sub>0</sub>,g1∈H<sup>-（1/2）</sup>（Γ<sub>2</sub>）,g2∈H<sup>-（3/2）</sup>（Γ）.Consider the variational problem:find u ∈ V such thata（u,v）=F（v）,（?）v∈V<sub>0</sub>. （1）Using Tartar’s lemma,we prove that for problem （1） there exists a unique
