arXiv subjects

Olivier Sète

Publications and source records attributed to Olivier Sète.

23 records · Page 2Linked to original sources

arXiv2015

Creating images by adding masses to gravitational point lenses

A well-studied maximal gravitational point lens construction of S. H. Rhie produces $5n$ images of a light source using $n+1$ deflector masses. The construction arises from a circular, symmetric deflector configuration on $n$ masses (producing only $3n+1$ images) by adding a tiny mass in the center of the other mass positions (and reducing all the other masses a little bit). In a recent paper we studied this "image creating effect" from a purely mathematical point of view (Sète, Luce & Liesen, Comput. Methods Funct. Theory 15(1):9-35, 2015). Here we discuss a few consequences of our findings for gravitational microlensing models. We present a complete characterization of the effect of adding small masses to these point lens models, with respect to the number of images. In particular, we give several examples of maximal lensing models that are different from Rhie's construction and that do not share its highly symmetric appearance. We give generally applicable conditions that allow the construction of maximal point lenses on $n+1$ masses from maximal lenses on $n$ masses.

astro-ph.IM↗

arXiv2014

A Note on the Maximum Number of Zeros of $r(z) - \bar{z}$

An important theorem of Khavinson & Neumann (Proc. Amer. Math. Soc. 134(4), 2006) states that the complex harmonic function $r(z) - \bar{z}$, where $r$ is a rational function of degree $n \geq 2$, has at most $5 (n - 1)$ zeros. In this note we resolve a slight inaccuracy in their proof and in addition we show that for certain functions of the form $r(z) - \bar{z}$ no more than $5 (n - 1) - 1$ zeros can occur. Moreover, we show that $r(z) - \bar{z}$ is regular, if it has the maximal number of zeros.

math.CV↗

arXiv2014

Perturbing rational harmonic functions by poles

We study how adding certain poles to rational harmonic functions of the form $R(z)-\bar{z}$, with $R(z)$ rational and of degree $d\geq 2$, affects the number of zeros of the resulting functions. Our results are motivated by and generalize a construction of Rhie derived in the context of gravitational microlensing (ArXiv e-print 2003). Of particular interest is the construction and the behavior of rational functions $R(z)$ that are {\em extremal} in the sense that $R(z)-\bar{z}$ has the maximal possible number of $5(d-1)$ zeros.

math.CV↗

arXiv2014

Sharp parameter bounds for certain maximal point lenses

Starting from an $n$-point circular gravitational lens having $3n+1$ images, Rhie (2003) used a perturbation argument to construct an $(n+1)$-point lens producing $5n$ images. In this work we give a concise proof of Rhie's result, and we extend the range of parameters in Rhie's model for which maximal lensing occurs. We also study a slightly different construction given by Bayer and Dyer (2007) arising from the $(3n+1)$-point lens. In particular, we extend their results and give sharp parameter bounds for their lens model. By a substitution of variables and parameters we show that both models are equivalent in a certain sense.

astro-ph.IM↗

arXiv2013

Some properties of Faber-Walsh polynomials

Walsh introduced a generalisation of Faber polynomials to certain compact sets which need not be connected. We derive several equivalent representations of these Faber-Walsh polynomials, analogous to representations of Faber polynomials. Some simple asymptotic properties of the Faber-Walsh polynomials on the complement of the compact set are established. We further show that suitably normalised Faber-Walsh polynomials are asymptotically optimal polynomials in the sense of [Eiermann and Niethammer 1983].

math.CV↗