Geometry and Singularities of the Prony mapping

Prony mapping provides the global solution of the Prony system of equations $$\Sigma_{i=1}^{n}A_{i}x_{i}^{k}=m_{k},\ k=0,1,...,2n-1.$$ This system appears in numerous theoretical and applied problems arising in Signal Reconstruction. The simplest example is the problem of reconstruction of linear combination of $\delta$-functions of the form $g(x)=\sum_{i=1}^{n}a_{i}\delta(x-x_{i})$, with the unknown parameters $a_{i},\ x_{i},\ i=1,...,n,$from the "moment measurements"$m_{k}=\int x^{k}g(x)dx.$Global solution of the Prony system, i.e. inversion of the Prony mapping, encounters several types of singularities. One of the most important ones is a collision of some of the points$x_{i}.$ The investigation of this type of singularities has been started in \cite{yom2009Singularities} where the role of finite differences was demonstrated. In the present paper we study this and other types of singularities of the Prony mapping, and describe its global geometry. We show, in particular, close connections of the Prony mapping with the "Vieta mapping" expressing the coefficients of a polynomial through its roots, and with hyperbolic polynomials and "Vandermonde mapping" studied by V. Arnold.Comment: arXiv admin note: text overlap with arXiv:1301.118

Automatic detection and branch switching methods for steady bifurcation in fluid mechanics

International audienceThis paper deals with the computation of steady bifurcations in the framework of 2D incompressible Navier-Stokes flow. We first propose a numerical method to accurately detect the critical Reynolds number where this kind of bifurcation appears. From this singular value, we introduce a numerical tool to compute all the steady bifurcated branches. All these algorithms are based on the Asymptotic Numerical Method. The critical values are determined by using a bifurcation indicator and the bifurcated branches are computed by using an augmented system which was first introduced in solid mechanics. Several numerical examples from 2D Navier-Stokes show the reliability and the efficiency of the proposed methods

A numerical algorithm coupling a bifurcating indicator and a direct method for the computation of Hopf bifurcation points in fluid mechanics

International audienceThis paper deals with the computation of Hopf bifurcation points in fluid mechanics. This computation is done by coupling a bifurcation indicator proposed recently and a direct method which consists in solving an augmented system whose solutions are Hopf bifurcation points. The bifurcation indicator gives initial critical values (Reynolds number, Strouhal frequency) for the direct method iterations. Some classical numerical examples from fluid mechanics, in two dimensions, are studied to demonstrate the efficiency and the reliability of such an algorithm

First Forcer results on deep-inelastic scattering and related quantities

We present results on the fourth-order splitting functions and coefficient functions obtained using Forcer, a four-loop generalization of the Mincer program for the parametric reduction of self-energy integrals. We have computed the respective lowest three even-N and odd-N moments for the non-singlet splitting functions and the non-singlet coefficient functions in electromagnetic and nu+nu(bar) charged-current deep-inelastic scattering, and the N=2 and N=4 results for the corresponding flavour-singlet quantities. Enough moments have been obtained for an LLL-based determination of the analytic N-dependence of the nf^3 and nf^2 parts, respectively, of the singlet and non-singlet splitting functions. The large-N limit of the latter provides the complete nf^2 contributions to the four-loop cusp anomalous dimension. Our results also provide additional evidence of a non-vanishing contribution of quartic group invariants to the cusp anomalous dimension.Comment: 11 pages, LaTeX (PoS style), 4 eps-figures. To appear in the proceedings of `Loops & Legs 2016', Leipzig (Germany), April 201

Continued fractions

[ES]Las fracciones continuas han tenido un papel fundamental en el desarrollo de numerosas teorías matemáticas y actualmente siguen siendo un tema de investigación muy activo. El estudio de las fracciones continuas con coeficientes en los complejos se asienta en el análisis de las regiones donde la fracción continua converge. Esta teoría de convergencia nos permite definir funciones meromorfas como fracciones continuas a partir de su serie de potencias mediante una sucesión de aproximaciones racionales conocidas como aproximaciones de Padé. Además, se puede establecer una equivalencia entre los números reales y las fracciones continuas simples (un tipo especial de fracciones continuas con coeficientes enteros) y, mediante esta equivalencia, también se pueden estudiar numerosos problemas de teoría de números como la aproximación de números irracionales por aproximaciones racionales o la resolución de la ecuación de Pell, una ecuación diofántica[EN]Continued fractions have played a central role in the development of manu mathematical theories and, even today, they are still a very active line of research. The study of continued fractions with complex coefficients relies o the analysis fo the regions where the continued fraction converges. This convergence theory allows us to define meromorphic functions as continued fractions fron their formal power series with the help of a sequence of rational approximations known as Padé approximants. Furthermore, there is an equivalence between real numbers and simple cotinued fractions (a special case of continued fractions with integer coefficients) and, based on this equivalence, one can study problems of number theory such as how well irrational numbers can be approximated by rational numbers or how to solve Pell's equation, a kind of Diophantine equatio

Parton densities and structure functions at next-to-next-to-leading order and beyond

We summarize recent results on the evolution of unpolarized parton densities and deep-inelastic structure functions in massless perturbative QCD. Due to last year's extension of the integer-moment calculations of the three-loop splitting functions, the NNLO evolution of the parton distributions can now be performed reliably at momentum fractions x >= 10^-4, facilitating a considerably improved theoretical accuracy of their extraction from data on deep-inelastic scattering. The NNLO corrections are not dominated, at relevant values of x, by their leading small-x terms. At large x the splitting-function series converges very rapidly, hence, employing results on the three-loop coefficient functions, the structure functions can be analysed at N^3LO for x > 10^-2. The resulting values for alpha_s do not significantly change beyond NNLO, their renormalization scale dependence reaches about +-1% at N^3LO.Comment: 10 pages, LaTeX, 6 figures. Talk presented at the workshops `New Trends in HERA Physics 2001', Ringberg Castle (Germany), June 2001 and `DIS 2001', Bologna (Italy), April 2001. To appear, slightly shortened in the latter case, in the proceeding