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P L equations are classical approximations to the neutron transport equations, which are obtained expanding the angular neutron flux in terms of spherical harmonics. These approximations are useful to study the behavior of reactor cores with complex fuel assemblies, for the homogenization of nuclear cross sections, etc., and most of these applications are in three-dimensional (3D) geometries. In this work, we review the multi-dimensional P L equations and describe a nodal collocation method for the spatial discretization of these equations for arbitrary odd order L, which is based on the expansion of the spatial dependence of the fields in terms of orthonormal Legendre polynomials. The performance of the nodal collocation method is studied by means of obtaining the k eff and the stationary power distribution of several 3D benchmark problems. The solutions are obtained are compared with a finite element method and a Monte Carlo method.

We study finite-dimensional extra symmetries of generic 2D dilaton gravity models. Using a non-linear sigma model formulation we show that the unique theories admitting an extra (conformal) symmetry are the models with an ex-which include the model of Callan, Giddings, Harvey and Strominger (CGHS) as a particular though limiting (β = 0) case. These models give rise to black hole solutions with * Work partially supported by the Comisión Interministerial de Ciencia y Tecnología and DGICYT. † cruz@lie.uv.es ‡ jnavarro@lie.uv.es § mnavarro@ugr.es ¶ talavera@lie.uv.es a mass-dependent temperature. The underlying extra symmetry can be maintained in a natural way in the one-loop effective action, thus implying the exact solubility of the semiclassical theory including back-reaction. Moreover, we also introduce three different classes of (non-conformal) transformations which are extra symmetries for generic 2D dilaton gravity models. Special linear combinations of these transformations turn out to be the (conformal) symmetries of the CGHS and V ∝ e βφ models. We show that one of the non-conformal extra symmetries can be converted into a conformal one by means of adequate field redefinitions involving the metric and the derivatives of the dilaton. Finally, by expressing the Polyakov-Liouville effective action in terms of an invariant metric, we are able to provide semiclassical models which are also invariant. This generalizes the solvable semiclassical model of Bose, Parker and Peleg (BPP)for a generic 2D dilaton gravity model.

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