Chebyshev, Legendre, Hermite and other orthonormal polynomials in D-dimensions

We propose a general method to construct symmetric tensor polynomials in the D-dimensional Euclidean space which are orthonormal under a general weight. The D-dimensional Hermite polynomials are a particular case of the present ones for the case of a gaussian weight. Hence we obtain generalizations of the Legendre and of the Chebyshev polynomials in D dimensions that reduce to the respective well-known orthonormal polynomials in D=1 dimensions. We also obtain new D-dimensional polynomials orthonormal under other weights, such as the Fermi-Dirac, Bose-Einstein, Graphene equilibrium distribution functions and the Yukawa potential. We calculate the series expansion of an arbitrary function in terms of the new polynomials up to the fourth order and define orthonormal multipoles. The explicit orthonormalization of the polynomials up to the fifth order (N from 0 to 4) reveals an increasing number of orthonormalization equations that matches exactly the number of polynomial coefficients indication the correctness of the present procedure.Comment: 20 page

Investigation of the existence of hybrid stars using Nambu-Jona-Lasinio models

We investigate the hadron-quark phase transition inside neutron stars and obtain mass-radius relations for hybrid stars. The equation of state for the quark phase using the standard NJL model is too soft leading to an unstable star and suggesting a modification of the NJL model by introducing a momentum cutoff dependent on the chemical potential. However, even in this approach, the instability remains. In order to remedy the instability we suggest the introduction of a vector coupling in the NJL model, which makes the EoS stiffer, reducing the instability. We conclude that the possible existence of quark matter inside the stars require high densities, leading to very compact stars.Comment: 4 pages, 2 figures; prepared for IV International Workshop on Astronomy and Relativistic Astrophysics (IWARA 2009), Maresias, 4-8 Oct 200

Fully dissipative relativistic lattice Boltzmann method in two dimensions

In this paper, we develop and characterize the fully dissipative Lattice Boltzmann method for ultra-relativistic fluids in two dimensions using three equilibrium distribution functions: Maxwell-J\"uttner, Fermi-Dirac and Bose-Einstein. Our results stem from the expansion of these distribution functions up to fifth order in relativistic polynomials. We also obtain new Gaussian quadratures for square lattices that preserve the spatial resolution. Our models are validated with the Riemann problem and the limitations of lower order expansions to calculate higher order moments are shown. The kinematic viscosity and the thermal conductivity are numerically obtained using the Taylor-Green vortex and the Fourier flow respectively and these transport coefficients are compared with the theoretical prediction from Grad's theory. In order to compare different expansion orders, we analyze the temperature and heat flux fields on the time evolution of a hot spot

Integrated Methodology for Physical and Economic Assessment of Coastal Interventions Impacts

Due to economic, environmental, and social interest of coastal areas, together with their erosion problems, different coastal management strategies can be considered, with different physical (shoreline evolution) and economic (net present value, ratio benefit-cost, break-even point) consequences and impacts. Therefore, this work presents an integrated methodology that aims to compare and discuss the most promising coastal intervention scenarios to mitigate erosion problems and climate change effects, considering costs and benefits related to each intervention. The proposed methodology takes a step forward in assessing the coastal erosion mitigation strategies, incorporating three well-defined and sequential stages: shoreline evolution in a medium-term perspective; structures pre-design; and a cost-benefit assessment. To show the relevance of the methodology, a hypothetic case study and several intervention scenarios were assessed. In order to mitigate costal erosion two different situations were analyzed: the reference scenario and the intervention scenarios. 34 intervention scenarios were proposed and evaluated to mitigate the erosion verified. Depending on the parameter considered (reduce erosion areas, protect the full extension of urban waterfronts, improve the economic performance of the intervention by increasing the net present value, the benefit-cost ratio or decreasing the break-even time), best results are obtained for different scenarios. The definition of the best option for coastal erosion mitigation is complex and depends on the main goal defined for the intervention. In conclusion, costs and benefits analysis are demanded and it is considered that the proposed methodology allows choosing better physical and economic options for future coastal interventions, helping decision-making processes related to coastal management