The relative cohomology Hdiff1(K(1|3), osp(2, 3);Dγ,µ(S1|3)) of the contact Lie superalgebra K(1|3) with coefficients in the space of differential operators Dγ,µ(S1|3) acting on tensor densities on S1|3, is calculated in N.Ben Fraj, I. Laraied, S. Omri (2013) and the generating 1-cocycles are expressed in terms of the infinitesimal super-Schwarzian derivative 1-cocycle s(Xf) = D1D2D3(f)α31/2, Xf \in K(1|3) which is invariant with respect to the conformal subsuperalgebra osp(2, 3) of K(1|3). In this work we study the supergroup case. We give an explicit construction of 1-cocycles of the group of contactomorphisms K(1|3) on the supercircle S1|3 generating the relative cohomology Hdiff1(K(1|3), PC(2, 3); Dγ,µ(S1|3) with coefficients in Dγ,µ(S1|3). We show that they possess properties similar to those of the super-Schwarzian derivative 1-cocycle S3(Φ) = EΦ-1 (D1(D2),D3)α31/2, Φ ∈ K(1|3) introduced by Radul which is invariant with respect to the conformal group PC(2, 3) of K(1|3). These cocycles are expressed in terms of S3(Φ) and possess its properties., Boujemaa Agrebaoui, Raja Hattab., and Obsahuje seznam literatury
This paper deals with integrability issues of the Euler-Lagrange equations associated to a variational problem, where the energy function depends on acceleration and drag. Although the motivation came from applications to path planning of underwater robot manipulators, the approach is rather theoretical and the main difficulties result from the fact that the power needed to push an object through a fluid increases as the cube of its speed.
Two new time-dependent versions of div-curl results in a bounded domain \domain⊂\RR3 are presented. We study a limit of the product \vectorvk\vectorwk, where the sequences \vectorvk and \vectorwk belong to \Lp2. In Theorem ??? we assume that \rotor\vectorvk is bounded in the Lp-norm and \diver\vectorwk is controlled in the Lr-norm. In Theorem ??? we suppose that \rotor\vectorwk is bounded in the Lp-norm and \diver\vectorwk is controlled in the Lr-norm. The time derivative of \vectorwk is bounded in both cases in the norm of \Hk−1. The convergence (in the sense of distributions) of \vectorvk\vectorwk to the product \vectorv\vectorw of weak limits of \vectorvk and \vectorwk is shown.
There exists a rich literature on systems of connections and systems of vector fields, stimulated by the irimportance in geometry and physis. In the previous papers [T1], [T2] we examined a simple type of systems of vector fields, called parameter dependent vector fields, and established their varionational equation.
In this paper we generalize the above equation to the projectable system of vector fields. The material is organized as follows: in the first section the geometry of the product bundle is presented. In the second we introduce the notion of derivative along a direction and prove Theorem 1. The third section is devoted to Theorem
2, which represents the main result of the paper. Some examples are presented in the last section. In a further paper we will apply the results in order to investigate some special systems as strong systems, “nice” systems and systems of connections generated by systems of vector fields.