Computation of the Marcum Q-function

Methods and an algorithm for computing the generalized Marcum $Q-$function ($Q_{\mu}(x,y)$) and the complementary function ($P_{\mu}(x,y)$) are described. These functions appear in problems of different technical and scientific areas such as, for example, radar detection and communications, statistics and probability theory, where they are called the non-central chi-square or the non central gamma cumulative distribution functions. The algorithm for computing the Marcum functions combines different methods of evaluation in different regions: series expansions, integral representations, asymptotic expansions, and use of three-term homogeneous recurrence relations. A relative accuracy close to $10^{-12}$ can be obtained in the parameter region $(x,y,\mu) \in [0,\,A]\times [0,\,A]\times [1,\,A]$, $A=200$, while for larger parameters the accuracy decreases (close to $10^{-11}$ for $A=1000$ and close to $5\times 10^{-11}$ for $A=10000$).Comment: Accepted for publication in ACM Trans. Math. Soft

Asymptotic approximations to the nodes and weights of Gauss-Hermite and Gauss-Laguerre quadratures

Asymptotic approximations to the zeros of Hermite and Laguerre polynomials are given, together with methods for obtaining the coefficients in the expansions. These approximations can be used as a standalone method of computation of Gaussian quadratures for high enough degrees, with Gaussian weights computed from asymptotic approximations for the orthogonal polynomials. We provide numerical evidence showing that for degrees greater than $100$ the asymptotic methods are enough for a double precision accuracy computation ($15$-$16$ digits) of the nodes and weights of the Gauss--Hermite and Gauss--Laguerre quadratures.Comment: Submitted to Studies in Applied Mathematic

Conical: an extended module for computing a numerically satisfactory pair of solutions of the differential equation for conical functions

Conical functions appear in a large number of applications in physics and engineering. In this paper we describe an extension of our module CONICAL for the computation of conical functions. Specifically, the module includes now a routine for computing the function ${{\rm R}}^{m}_{-\frac{1}{2}+i\tau}(x)$, a real-valued numerically satisfactory companion of the function ${\rm P}^m_{-\tfrac12+i\tau}(x)$ for $x>1$. In this way, a natural basis for solving Dirichlet problems bounded by conical domains is provided.Comment: To appear in Computer Physics Communication

Ranking efficient DMUs using cooperative game theory

The problem of ranking Decision Making Units (DMUs) in Data Envelopment Analysis (DEA) has been widely studied in the literature. Some of the proposed approaches use cooperative game theory as a tool to perform the ranking. In this paper, we use the Shapley value of two different cooperative games in which the players are the efficient DMUs and the characteristic function represents the increase in the discriminant power of DEA contributed by each efficient DMU. The idea is that if the efficient DMUs are not included in the modified reference sample then the efficiency score of some inefficient DMUs would be higher. The characteristic function represents, therefore, the change in the efficiency scores of the inefficient DMUs that occurs when a given coalition of efficient units is dropped from the sample. Alternatively, the characteristic function of the cooperative game can be defined as the change in the efficiency scores of the inefficient DMUs that occurs when a given coalition of efficient DMUs are the only efficient DMUs that are included in the sample. Since the two cooperative games proposed are dual games, their corresponding Shapley value coincide and thus lead to the same ranking. The more an ef- ficient DMU impacts the shape of the efficient frontier, the higher the increase in the efficiency scores of the inefficient DMUs its removal brings about and, hence, the higher its contribution to the overall discriminant power of the method. The proposed approach is illustrated on a number of datasets from the literature and compared with existing methods

Jets as diagnostics of the circumstellar medium and the explosion energetics of supernovae: the case of Cas A

We present hydrodynamical models for the Cassiopeia A (Cas A) supernova remnant and its observed jet / counter-jet system. We include the evolution of the progenitor's circumstellar medium, which is shaped by a slow red supergiant wind that is followed by a fast Wolf-Rayet (WR) wind. The main parameters of the simulations are the duration of the WR phase and the jet energy. We find that the jet is destroyed if the WR phase is sufficiently long and a massive circumstellar shell has formed. We therefore conclude that the WR phase must have been short (a few thousand yr), if present at all. Since the actual jet length of Cas A is not known we derive a lower limit for the jet energy, which is ~10^{48} erg. We discuss the implications for the progenitor of Cas A and the nature of its explosion.Comment: 9 pages, 5 figures, ApJ accepted. Version with high resolution figures available at http://www.phys.uu.nl/~schure/CasA_jet.pd

La transgresión cenomanense en el sector septentrional de la "Serranía de Cuenca" (provincias de Cuenca y Guadalajara, Cordillera Ibérica)

[ES] Los materiales del Albense-Cenomanense que afloran al N. de la Serrania de Cuenca se encuentran delimitados entre dos discordancias, una previa al depósito de las arenas en Facies Utrillas y otra de edad Cenomanense superior. Estos materiales forman una sucesión con diez tramos litológicos diferentes, entre 10s que se indentifican, bastante reducidos de potencia, alguna de las unidades litoestratigráficas recientemente definidas en la Cordillera Ibérica. Estos tramos pueden agruparse en tres unidades litoestratigraficas con rango de Formación: Arenas de Utrillas, Dolomias y margas (sin denominación formal) y Calizas dolomiticas de Nuévalos-Dolomias Tableadas de Villa de ves. Desde el punto de vista evolutivo, estos materiales forman una transgresión compleja, con cinco secuencias deposicionales separadas por discontinuidades. Por ultimo, se define la existencia de un elemento paleogeografico en la región de Peñalen-Taravilla, un escalón que delimita la extensión hacia el N. de la unidad secuencia deposicional basal.[EN] The Albian-Cenomanian material outcropping in the Northern part of the Serrania de Cuenca are bounded by angular unconformities. The one at the base was prior to the ({Facies Utrillas)) (sandstones) deposition, and the unconformity which constitutes the upper boundary is considered to be Upper Cenomanian. Ten lithological units form the entire succession, being identified from among them some of the recently defined stratigraphical Units of the Iberian Ranges, although thicknesses are always very reduced. Th beds can be brouped into three lithoestratigraphical Formations, named: Utrillas Sandstones, Dolomites and marls (unformal denomination) and Dolomitic limestones of ~uévalos-Laminated dolomites of Villa de Ves. From the evolutionary point of view, the succession represents a complex transgression with several impulses, specifically, five stratigraphical cycles, separates by unconformities (non-depositional unconformities) are distinguished. Finally, it is defined the existence of a paleogeographical feature in the Peíialen- Taravilla region, a step that conditioned the extension towards the North of the basal cycle.Peer reviewe