A Symbolic Approach to Some Indentities for Bernoulli-Barnes Polynomials

A symbolic method is used to establish some properties of the Bernoulli-Barnes polynomials.Comment: 12 page

The Zagier modification of Bernoulli numbers and a polynomial extension. Part I

The modified B_{n}^{*} = \sum_{r=0}^{n} \binom{n+r}{2r} \frac{B_{r}}{n+r}, \quad n > 0 introduced by D. Zagier in 1998 are extended to the polynomial case by replacing $B_{r}$ by the Bernoulli polynomials $B_{r}(x)$. Properties of these new polynomials are established using the umbral method as well as classical techniques. The values of $x$ that yield periodic subsequences $B_{2n+1}^{*}(x)$ are classified. The strange 6-periodicity of $B_{2n+1}^{*}$, established by Zagier, is explained by exhibiting a decomposition of this sequence as the sum of two parts with periods 2 and 3, respectively. Similar results for modifications of Euler numbers are stated.Comment: 35 pages, Submitted for publicatio

The affine equivariant sign covariance matrix: asymptotic behavior and efficiencies.

We consider the affine equivariant sign covariance matrix (SCM) introduced by Visuri et al. (J. Statist. Plann. Inference 91 (2000) 557). The population SCM is shown to be proportional to the inverse of the regular covariance matrix. The eigenvectors and standardized eigenvalues of the covariance, matrix can thus be derived from the SCM. We also construct an estimate of the covariance and correlation matrix based on the SCM. The influence functions and limiting distributions of the SCM and its eigenvectors and eigenvalues are found. Limiting efficiencies are given in multivariate normal and t-distribution cases. The estimates are highly efficient in the multivariate normal case and perform better than estimates based on the sample covariance matrix for heavy-tailed distributions. Simulations confirmed these findings for finite-sample efficiencies. (C) 2003 Elsevier Science (USA). All rights reserved.affine equivariance; covariance and correlation matrices; efficiency; eigenvectors and eigenvalues; influence function; multivariate median; multivariate sign; robustness; multivariate location; discriminant-analysis; principal components; dispersion matrices; tests; estimators;

Influence function and asymptotic efficiency of the affine equivariant rank covariance matrix.

Visuri et al (2001) proposed and illustrated the use of the affine equivariant rank covariance matrix (RCM) in classical multivariate inference problems. The RCM was shown to be asymptotically multinormal but explicit formulas for the limiting variances and covariances were not given yet. In this paper the influence functions and the limiting variances and covariances of the RCM and the corresponding scatter estimate are derived in the multivariate elliptic case. Limiting efficiencies are given in the multivariate normal and t-distribution cases. The estimates based on the RCM are highly efficient in the multinormal case, and for heavy tailed distribution, perform better than those based on the regular covariance matrix.Efficiency;

The finite Fourier transform of classical polynomials

The finite Fourier transform of a family of orthogonal polynomials $A_{n}(x)$, is the usual transform of the polynomial extended by $0$ outside their natural domain. Explicit expressions are given for the Legendre, Jacobi, Gegenbauer and Chebyshev families

An analytical solution of Shallow Water system coupled to Exner equation

In this paper, an exact smooth solution for the equations modeling the bedload transport of sediment in Shallow Water is presented. This solution is valid for a large family of sedimentation laws which are widely used in erosion modeling such as the Grass model or those of Meyer-Peter & Muller. One of the main interest of this solution is the derivation of numerical benchmarks to valid the approximation methods

Recursion Rules for the Hypergeometric Zeta Functions

The hypergeometric zeta function is defined in terms of the zeros of the Kummer function M(a, a + b; z). It is established that this function is an entire function of order 1. The classical factorization theorem of Hadamard gives an expression as an infinite product. This provides linear and quadratic recurrences for the hypergeometric zeta function. A family of associated polynomials is characterized as Appell polynomials and the underlying distribution is given explicitly in terms of the zeros of the associated hypergeometric function. These properties are also given a probabilistic interpretation in the framework of Beta distributions

Spectroscopic direct detection of reflected light from extra-solar planets

At optical wavelengths, an exoplanet's signature is essentially reflected light from the host star - several orders of magnitude fainter. Since it is superimposed on the star spectrum its detection has been a difficult observational challenge. However, the development of a new generation of instruments like ESPRESSO and next generation telescopes like the E-ELT put us in a privileged position to detect these planets' reflected light as we will have access to extremely high signal-to-noise ratio spectra. With this work, we propose an alternative approach for the direct detection of the reflected light of an exoplanet. We simulated observations with ESPRESSO@VLT and HIRES@E-ELT of several star+planet systems, encompassing 10h of the most favourable orbital phases. To the simulated spectra we applied the Cross Correlation Function to operate in a much higher signal-to-noise ratio domain than when compared with the spectra. The use of the Cross-Correlation Function permitted us to recover the simulated the planet signals at a level above 3 \sigma_{noise} significance on several prototypical (e.g., Neptune type planet with a 2 days orbit with the VLT at 4.4 \sigma_{noise} significance) and real planetary systems (e.g., 55 Cnc e with the E-ELT at 4.9 \sigma_{noise} significance). Even by using a more pessimistic approach to the noise level estimation, where systematics in the spectra increase the noise 2-3 times, the detection of the reflected light from large close-orbit planets is possible. We have also shown that this kind of study is currently within reach of current instruments and telescopes (e.g., 51 Peg b with the VLT at 5.2 \sigma_{noise} significance), although at the limit of their capabilities.Comment: Accepted for Publication on MNRAS: 2013 August 29; Online Article: http://mnras.oxfordjournals.org/content/early/2013/09/27/mnras.stt1642; 5 Figures, 11 page