On different geometric formulations of Lagrangian formalism

We consider two geometric formulations of Lagrangian formalism on fibred manifolds: Krupka's theory of finite order variational sequences, and Vinogradov's infinite order variational sequence associated with the -spectral sequence. On one hand, we show that the direct limit of Krupka's variational bicomplex is a new infinite order variational bicomplex which yields a new infinite order variational sequence. On the other hand, by means of Vinogradov's -spectral sequence, we provide a new finite order variational sequence whose direct limit turns out to be the Vinogradov's infinite order variational sequence. Finally, we provide an equivalence of the two finite order and infinite order variational sequences modulo the space of Euler-Lagrange morphisms.     

Symmetries in finite order variational sequences

We refer to Krupka's variational sequence, i.e. the quotient of the de Rham sequence on a finite order jet space with respect to a variationally trivial subsequence. Among the morphisms of the variational sequence there are the Euler-Lagrange operator and the Helmholtz operator.In this note we show that the Lie derivative operator passes to the quotient in the variational sequence. Then we define the variational Lie derivative as an operator on the sheaves of the variational sequence. Explicit representations of this operator give us some abstract versions of Noether's theorems, which can be interpreted in terms of conserved currents for Lagrangians and Euler-Lagrange morphisms.     

May谱序列中的一个非平凡积

In this paper,we prove the non-triviality of the product h 0 k o δ s＋4 ∈ Ext s＋6,t（s） A （Z p ,Z p ） in the classical Adams spectral sequence,where p ≥ 11,0 ≤ s p-4,t（s） = （s ＋ 4）p 3 q ＋ （s ＋ 3）p 2 q ＋ （s ＋ 4）pq ＋ （s ＋ 3）q ＋ s with q = 2（p-1）.The elementary method of proof is by explicit combinatorial analysis of the （modified） May spectral sequence.     

Representation of the Variational Sequence by Differential Forms

In the paper the representation of the finite order variational sequence on fibered manifolds in field theory is studied. The representation problem is completely solved by a generalization of the integration by parts procedure using the concept of the Lie derivative of forms with respect to vector fields along canonical maps of prolongations of fibered manifolds. A close relationship between the obtained coordinate invariant representation of the variational sequence and some familiar objects of physics, such as Lagrangians, dynamical forms, Euler–Lagrange mapping and Helmholtz–Sonin mapping is pointed out and illustrated by examples.Mathematics Subject Classifications (2000) 58E99, 49F99.Jana Musilová: Research of both authors supported by grants MSM 0021622409 and 201/03/0512.     

Moore谱的Adams谱序列中的一个非平凡微分

令p是一个大于5的奇素数．本文证明了在收敛到Moore谱的稳定同伦群的Adams谱序列中,在相差一个非零系数下,h_2的2p阶Adams微分是a_1b_0~p．     

May谱序列的一些注记

令 p>5 是素数, A 表示模 p Steenrod代数, S 表示球谱的 p 局部化. 首先给出了有关May谱序列的一些重要定理, 然后作为应用, 利用May谱序列和Adams谱序列发觉了一族新的非零的球面稳定元素. 该新元素族次数为2(p-1)(pⁿ+sp²+sp+s)-7,在Adams谱序列中由 b_n-1g0γ_s∈ Ext_A^s+4,﹡( Z_pZ_p)所表示, 其中n≥4, 3≤s

     

Spectral gap on path spaces with infinite time-interval

Under certain curvature condition, the existence of spectral gap is proved on path spaces with infinite time-interval. Project supported in part by the National Natural Science Foundation of China (Grant No. 19631060), Beijing Normal University and the State Education Commission of China.     

到序列空间的乘子

这篇论文刻画了 (Ap,α,lq) (0

0 ,p ≤ q≤∞ )的特征 ,作为其推论 ,获得了 (Al;lq)、(Hp,lq)、(Gp,lq)和 (Bp,lq) (0

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