On some applications of a symbolic representation of non-centered L\'evy processes

By using a symbolic technique known in the literature as the classical umbral calculus, we characterize two classes of polynomials related to L\'evy processes: the Kailath-Segall and the time-space harmonic polynomials. We provide the Kailath-Segall formula in terms of cumulants and we recover simple closed-forms for several families of polynomials with respect to not centered L\'evy processes, such as the Hermite polynomials with the Brownian motion, the Poisson-Charlier polynomials with the Poisson processes, the actuarial polynomials with the Gamma processes, the first kind Meixner polynomials with the Pascal processes, the Bernoulli, Euler and Krawtchuk polynomials with suitable random walks

Multivariate time-space harmonic polynomials: a symbolic approach

By means of a symbolic method, in this paper we introduce a new family of multivariate polynomials such that multivariate L\'evy processes can be dealt with as they were martingales. In the univariate case, this family of polynomials is known as time-space harmonic polynomials. Then, simple closed-form expressions of some multivariate classical families of polynomials are given. The main advantage of this symbolic representation is the plainness of the setting which reduces to few fundamental statements but also of its implementation in any symbolic software. The role played by cumulants is emphasized within the generalized Hermite polynomials. The new class of multivariate L\'evy-Sheffer systems is introduced.Comment: In pres

On the computation of classical, boolean and free cumulants

This paper introduces a simple and computationally efficient algorithm for conversion formulae between moments and cumulants. The algorithm provides just one formula for classical, boolean and free cumulants. This is realized by using a suitable polynomial representation of Abel polynomials. The algorithm relies on the classical umbral calculus, a symbolic language introduced by Rota and Taylor in 1994, that is particularly suited to be implemented by using software for symbolic computations. Here we give a MAPLE procedure. Comparisons with existing procedures, especially for conversions between moments and free cumulants, as well as examples of applications to some well-known distributions (classical and free) end the paper.Comment: 14 pages. in press, Applied Mathematics and Computatio

Dust in active nuclei. II. Powder or gravel?

In a companion paper, Maiolino et al. (2000) presented various observational evidences for "anomalous" dust properties in the circumnuclear region of AGNs and, in particular, the reduced E(B-V)/N_H and Av/N_H ratios, the absence of the silicate absorption feature in mid-IR spectra of Sy2s and the absence of the carbon dip in UV spectra of reddened Sy1s. In this paper we discuss various explanations for these facts. The observational constraints favor a scenario where coagulation, catalyzed by the high densities in the circumnuclear region, yields to the formation of large grains. The resulting extinction curve is featureless, flatter than Galactic and the E(B-V)/N_H and Av/N_H ratios are significantly reduced. These results should warn about an unappropriate use of the standard Galactic extinction curve and Av/N_H ratio when dealing with the extreme gas conditions typical of the circumnuclear clouds of AGNs. We also investigated alternative scenarios for the observed anomalous properties of dust in AGNs. Some of these scenarios might explain some of the observed properties for a few objects, but they generally fail to account for all of the observational constraints obtained for the large sample of AGNs studied in these works.Comment: 13 pages, 7 figures, accepted for publication in A&

Early star formation traced by the highest redshift quasars

The iron abundance relative to alpha-elements in the circumnuclear region of quasars is regarded as a clock of the star formation history and, more specifically, of the enrichment by SNIa. We investigate the iron abundance in a sample of 22 quasars in the redshift range 3.0<z<6.4 by measuring their rest frame UV FeII bump, which is shifted into the near-IR, and by comparing it with the MgII 2798 flux. The observations were performed with a device that can obtain near-IR spectra in the range 0.8-2.4 um in one shot, thereby enabling an optimal removal of the continuum underlying the FeII bump. We detect iron in all quasars including the highest redshift (z=6.4) quasar currently known. The uniform observational technique and the wide redshift range allows a reliable study of the trend of the FeII/MgII ratio with redshift. We find the FeII/MgII ratio is nearly constant at all redshifts, although there is marginal evidence for a higher FeII/MgII ratio in the quasars at z~6. If the FeII/MgII ratio reflects the Fe/alpha abundance, this result suggests that the z~6 quasars have already undergone a major episode of iron enrichment. We discuss the possible implications of this finding for the star formation history at z>6. We also detect a population of weak iron emitters at z~4.5, which are possibly hosted in systems that evolved more slowly. Alternatively, the trend of the FeII/MgII ratio at high redshift may reflect significantly different physical conditions of the circumnuclear gas in such high redshift quasars.Comment: Replaced to match the accepted version (ApJL in press), 5 page

Near-infrared K-band Spectroscopic Investigation of Seyfert 2 Nuclei in the CfA and 12 Micron Samples

We present near-infrared K-band slit spectra of the nuclei of 25 Seyfert 2 galaxies in the CfA and 12 micron samples. The strength of the CO absorption features at 2.3-2.4 micron produced by stars is measured in terms of a spectroscopic CO index. A clear anti-correlation between the observed CO index and the nuclear K-L color is present, suggesting that a featureless hot dust continuum heated by an AGN contributes significantly to the observed K-band fluxes in the nuclei of Seyfert 2 galaxies. After correction for this AGN contribution, we estimate nuclear stellar K-band luminosities for all sources, and CO indices for sources with modestly large observed CO indices. The corrected CO indices for 10 (=40%) Seyfert 2 nuclei are found to be as high as those observed in star-forming or elliptical (=spheroidal) galaxies. We combine the K-band data with measurements of the L-band 3.3 micron polycyclic aromatic hydrocarbon (PAH) emission feature, another powerful indicator for star-formation, and find that the 3.3 micron PAH to K-band stellar luminosity ratios are substantially smaller than those of starburst galaxies. Our results suggest that the 3.3 micron PAH emission originates in the putative nuclear starbursts in the dusty tori surrounding the AGNs, because of its high surface brightness, whereas the K-band CO absorption features detected at the nuclei are dominated by old bulge (=spheroid) stars, and thus may not be a powerful indicator for the nuclear starbursts. We see no clear difference in the strength of the CO absorption and PAH emission features between the CfA and 12 micron Seyfert 2s.Comment: 28 pages, 6 figures, accepted for publication in ApJ (10 October 2004, v614 issue

Static solutions with nontrivial boundaries for the Einstein-Gauss-Bonnet theory in vacuum

The classification of certain class of static solutions for the Einstein-Gauss-Bonnet theory in vacuum is performed in $d\geq5$ dimensions. The class of metrics under consideration is such that the spacelike section is a warped product of the real line and an arbitrary base manifold. It is shown that for a generic value of the Gauss-Bonnet coupling, the base manifold must be necessarily Einstein, with an additional restriction on its Weyl tensor for $d>5$. The boundary admits a wider class of geometries only in the special case when the Gauss-Bonnet coupling is such that the theory admits a unique maximally symmetric solution. The additional freedom in the boundary metric enlarges the class of allowed geometries in the bulk, which are classified within three main branches, containing new black holes and wormholes in vacuum