11,624 research outputs found
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 by the Bernoulli polynomials . Properties of
these new polynomials are established using the umbral method as well as
classical techniques. The values of that yield periodic subsequences
are classified. The strange 6-periodicity of ,
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
, is the usual transform of the polynomial extended by 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
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