In this paper,the modern geometrical structure of analytical mechanics,the exterior differential forms and the geometrical meaning of dynamic equations are briefly discussed.
As it is known, the closed inexact exterior form and associated closed dual form make up a differential-geometrical structure. Such a differential-geometrical structure describes a physical structure, namely, a pseudo...As it is known, the closed inexact exterior form and associated closed dual form make up a differential-geometrical structure. Such a differential-geometrical structure describes a physical structure, namely, a pseudostructure on which conservation laws are fulfilled (A closed dual form describes a pseudostructure. And a closed exterior form, as it is known, describes a conservative quantity, since the differential of closed form is equal to zero). It has been shown that closed inexact exterior forms, which describe physical structures, are obtained from the equations of mathematical physics. This process proceeds spontaneously under realization of any degrees of freedom of the material medium described. Such a process describes an emergence of physical structures and this is accompanied by an appearance of observed formations such as fluctuations, waves, turbulent pulsations and so on.展开更多
The authors prove that the total descendant potential functions of the theory of Fan-Jarvis-Ruan-Witten for D4 with symmetry group J and for D4T with symmetry group Gmax, respectively, are both tau-functions of the D4...The authors prove that the total descendant potential functions of the theory of Fan-Jarvis-Ruan-Witten for D4 with symmetry group J and for D4T with symmetry group Gmax, respectively, are both tau-functions of the D4 Kac-Wakimoto/Drinfeld-Sokolov hierarchy. This completes the proof, begun in the article by Fan-Jarvis-Ruan（2013）, of the Witten Integrable Hierarchies Conjecture for all simple（ADE） singularities.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
基金Work supported by NSF of Henan Education Commission
文摘In this paper,the modern geometrical structure of analytical mechanics,the exterior differential forms and the geometrical meaning of dynamic equations are briefly discussed.
文摘As it is known, the closed inexact exterior form and associated closed dual form make up a differential-geometrical structure. Such a differential-geometrical structure describes a physical structure, namely, a pseudostructure on which conservation laws are fulfilled (A closed dual form describes a pseudostructure. And a closed exterior form, as it is known, describes a conservative quantity, since the differential of closed form is equal to zero). It has been shown that closed inexact exterior forms, which describe physical structures, are obtained from the equations of mathematical physics. This process proceeds spontaneously under realization of any degrees of freedom of the material medium described. Such a process describes an emergence of physical structures and this is accompanied by an appearance of observed formations such as fluctuations, waves, turbulent pulsations and so on.
基金supported by the National Natural Science Foundation of China(Nos.1132510111271028)+1 种基金the National Security Agency of USA(No.H98230-10-1-0181)the Doctoral Fund of the Ministry of Education of China(No.20120001110060)
文摘The authors prove that the total descendant potential functions of the theory of Fan-Jarvis-Ruan-Witten for D4 with symmetry group J and for D4T with symmetry group Gmax, respectively, are both tau-functions of the D4 Kac-Wakimoto/Drinfeld-Sokolov hierarchy. This completes the proof, begun in the article by Fan-Jarvis-Ruan（2013）, of the Witten Integrable Hierarchies Conjecture for all simple（ADE） singularities.
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.
