Let(M,g)be a Kähler surface andΣbe aβ-symplectic critical surface in M.If L_(q)(Σ)is bounded for some q>3,then we give a uniform upper bound for the Kähler angle onΣ.This bound only depends on M,q,βa...Let(M,g)be a Kähler surface andΣbe aβ-symplectic critical surface in M.If L_(q)(Σ)is bounded for some q>3,then we give a uniform upper bound for the Kähler angle onΣ.This bound only depends on M,q,βand the Lq functional ofΣ.For q>4,this estimate is known and we extend the scope of q.展开更多
In this paper,the authors propose a neural network architecture designed specifically for a class of Birkhoffian systems—The Newtonian system.The proposed model utilizes recurrent neural networks(RNNs)and is based on...In this paper,the authors propose a neural network architecture designed specifically for a class of Birkhoffian systems—The Newtonian system.The proposed model utilizes recurrent neural networks(RNNs)and is based on a mathematical framework that ensures the preservation of the Birkhoffian structure.The authors demonstrate the effectiveness of the proposed model on a variety of problems for which preserving the Birkhoffian structure is important,including the linear damped oscillator,the Van der Pol equation,and a high-dimensional example.Compared with the unstructured baseline models,the Newtonian neural network(NNN)is more data efficient,and exhibits superior generalization ability.展开更多
In this paper,we start to study the gradient flow of the functional L_(β) introduced by Han-Li-Sun in[8].As a first step,we show that if the initial surface is symplectic in a Kähler surface,then the symplectic ...In this paper,we start to study the gradient flow of the functional L_(β) introduced by Han-Li-Sun in[8].As a first step,we show that if the initial surface is symplectic in a Kähler surface,then the symplectic property is preserved along the gradient flow.Then we show that the singularity of the flow is characterized by the maximal norm of the second fundamental form.When β=1,we derive a monotonicity formula for the flow.As applications,we show that the l-tangent cone of the flow consists of the finite flat planes.展开更多
基金supported by the National Natural Science Foundation of China(Grant No.11871436).
文摘Let(M,g)be a Kähler surface andΣbe aβ-symplectic critical surface in M.If L_(q)(Σ)is bounded for some q>3,then we give a uniform upper bound for the Kähler angle onΣ.This bound only depends on M,q,βand the Lq functional ofΣ.For q>4,this estimate is known and we extend the scope of q.
基金supported by the National Natural Science Foundation of China under Grant Nos.12171466 and 12271025.
文摘In this paper,the authors propose a neural network architecture designed specifically for a class of Birkhoffian systems—The Newtonian system.The proposed model utilizes recurrent neural networks(RNNs)and is based on a mathematical framework that ensures the preservation of the Birkhoffian structure.The authors demonstrate the effectiveness of the proposed model on a variety of problems for which preserving the Birkhoffian structure is important,including the linear damped oscillator,the Van der Pol equation,and a high-dimensional example.Compared with the unstructured baseline models,the Newtonian neural network(NNN)is more data efficient,and exhibits superior generalization ability.
基金supported by theNationalNatural Science Foundation of China,Nos.11721101,12071352,12031017。
文摘In this paper,we start to study the gradient flow of the functional L_(β) introduced by Han-Li-Sun in[8].As a first step,we show that if the initial surface is symplectic in a Kähler surface,then the symplectic property is preserved along the gradient flow.Then we show that the singularity of the flow is characterized by the maximal norm of the second fundamental form.When β=1,we derive a monotonicity formula for the flow.As applications,we show that the l-tangent cone of the flow consists of the finite flat planes.
