Cohomology of the Lie Superalgebra of Contact Vector Fields on R1∣1\mathbb{R}^{1|1} and Deformations of the Superspace of Symbols

Following Feigin and Fuchs, we compute the first cohomology of the Lie superalgebra $\mathcal{K}(1)$ of contact vector fields on the (1,1)-dimensional real superspace with coefficients in the superspace of linear differential operators acting on the superspaces of weighted densities. We also compute the same, but $\mathfrak{osp}(1|2)$-relative, cohomology. We explicitly give 1-cocycles spanning these cohomology. We classify generic formal $\mathfrak{osp}(1|2)$-trivial deformations of the $\mathcal{K}(1)$-module structure on the superspaces of symbols of differential operators. We prove that any generic formal $\mathfrak{osp}(1|2)$-trivial deformation of this $\mathcal{K}(1)$-module is equivalent to a polynomial one of degree $\leq4$. This work is the simplest superization of a result by Bouarroudj [On $\mathfrak{sl}$(2)-relative cohomology of the Lie algebra of vector fields and differential operators, J. Nonlinear Math. Phys., no.1, (2007), 112--127]. Further superizations correspond to $\mathfrak{osp}(N|2)$-relative cohomology of the Lie superalgebras of contact vector fields on $1|N$-dimensional superspace

Rôle de la variation spatiale et temporelle des états de surface SUR la genèse des écoulements sur les sols en milieu cultive mediterraneen CAS du Bassin versant d’el gouazine (Tunisie centrale)

According to the analysis of Hillel and Gardner (1969) for steady infiltration, the hydraulic resistance of the soil surface crust, Rc (h) is determined by − hs,c / K (hs,c), hs,c (cm) being the water pressure at the interface soil/crust and K (hs,c) (cm/h), the soil hydraulic conductivity corresponding to hs,c. For steady infiltration, K (hs,c) is equal to the infiltration rate so that the knowledge of the hydraulic conductivity curve allow the determination of the crust resistance in this case. The steady infiltration rate was measured during rain simulation experiments conducted in Central Tunisia. The measurement plot was a square metallic frame driven 5 to 10 cm into the soil. Runoff from the plot was collected in a reservoir and continuously measured. Infiltration rate was determined as the difference between rain intensity and runoff intensity. Soil hydraulic properties were described by a combination of Van Genuchten and Brooks and Corey expressions for the water retention and hydraulic conductivity relations respectively. The parameters of these relations were determined from analysis of both grain size distribution and single ring infiltration test. This test was performed on decapped soil in the immediate vicinity of the rain simulation plot. The relevance of the soil hydraulic properties determined in this way was tested by numerically simulating the single ring infiltration test and by comparing measured and simulated results. The crust resistance was inferred from the steady infiltration flux measured during the rain simulation experiment and from the hydraulic conductivity curve. These informations were used to numerically reproduce the rain simulation infiltration experiment. The comparison with experimental data showed that this method is reliable at least for this type of soil