We provide the geometric actions for most general N = 1 supergravity in two spacetime dimensions. Our construction implies an extension to arbitrary N. This provides a supersymmetrization of any generalized dilaton gravity theory or of any theory with an action being an (essentially) arbitrary function of curvature and torsion. Technically we proceed as follows: The bosonic part of any of these theories may be characterized by a generically nonlinear Poisson bracket on a three-dimensional target space. In analogy to a given ordinary Lie algebra, we derive all possible N = 1 extensions of any of the given Poisson (or W -) algebras. Using the concept of graded Poisson sigma models, any extension of the algebra yields a possible supergravity extension of the original theory, local Lorentz and super-diffeomorphism invariance follow by construction. Our procedure automatically restricts the fermionic extension to the minimal one; thus local supersymmetry is realized on-shell. By avoiding a superfield approach we are also able to circumvent in this way the introduction of constraints and their solution. For many well-known dilaton theories different supergravity extensions are derived. In generic cases their field equations are solved explicitly.
I n this paper, a method developed by Loeb to correct the Poisson-Boltzmann equation for the self-atmosphere effect of the ions is applied to the case of a charged plane surface immersed in a binary symmetrical electrolyte. The two limiting cases of a perfectly conducting and a perfectly insulating wall are treated. Numerical values of the correction to the average potential for various distances from the wall are obtained. This correction term is 5% near the wall for a conducting wall and 3% for an insulating one when the wall potential is 1OOmv and is larger for higher potentials.
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