High order three-term recursions, Riemann-Hilbert minors and Nikishin systems on star-like sets

We study monic polynomials $Q_n(x)$ generated by a high order three-term recursion $xQ_n(x)=Q_{n+1}(x)+a_{n-p} Q_{n-p}(x)$ with arbitrary $p\geq 1$ and $a_n>0$ for all $n$. The recursion is encoded by a two-diagonal Hessenberg operator $H$. One of our main results is that, for periodic coefficients $a_n$ and under certain conditions, the $Q_n$ are multiple orthogonal polynomials with respect to a Nikishin system of orthogonality measures supported on star-like sets in the complex plane. This improves a recent result of Aptekarev-Kalyagin-Saff where a formal connection with Nikishin systems was obtained in the case when $\sum_{n=0}^{\infty}|a_n-a|0$. An important tool in this paper is the study of "Riemann-Hilbert minors", or equivalently, the "generalized eigenvalues" of the Hessenberg matrix $H$. We prove interlacing relations for the generalized eigenvalues by using totally positive matrices. In the case of asymptotically periodic coefficients $a_n$, we find weak and ratio asymptotics for the Riemann-Hilbert minors and we obtain a connection with a vector equilibrium problem. We anticipate that in the future, the study of Riemann-Hilbert minors may prove useful for more general classes of multiple orthogonal polynomials.Comment: 59 pages, 3 figure

Asymptotics of multiple orthogonal polynomials for a system of two measures supported on a starlike set

For a system of two measures supported on a starlike set in the complex plane, we study asymptotic properties of associated multiple orthogonal polynomials $Q_{n}$ and their recurrence coefficients. These measures are assumed to form a Nikishin-type system, and the polynomials $Q_{n}$ satisfy a three-term recurrence relation of order three with positive coefficients. Under certain assumptions on the orthogonality measures, we prove that the sequence of ratios $\{Q_{n+1}/Q_{n}\}$ has four different periodic limits, and we describe these limits in terms of a conformal representation of a compact Riemann surface. Several relations are found involving these limiting functions and the limiting values of the recurrence coefficients. We also study the $n$th root asymptotic behavior and zero asymptotic distribution of $Q_{n}$.Comment: 31 page

Nikishin systems on star-like sets: Ratio asymptotics of the associated multiple orthogonal polynomials, II

In this paper we continue the investigations initiated in \cite{LopLopstar} on ratio asymptotics of multiple orthogonal polynomials and functions of the second kind associated with Nikishin systems on star-like sets. We describe in detail the limiting functions found in \cite{LopLopstar}, expressing them in terms of certain conformal mappings defined on a compact Riemann surface of genus zero. We also express the limiting values of the recurrence coefficients, which are shown to be strictly positive, in terms of certain values of the conformal mappings. As a consequence, the limits depend exclusively on the location of the intervals determined by the supports of the measures that generate the Nikishin system.Comment: Change in title, corrections have been made. 27 page

Integrability, quantization and moduli spaces of curves

This paper has the purpose of presenting in an organic way a new approach to integrable (1+1)-dimensional field systems and their systematic quantization emerging from intersection theory of the moduli space of stable algebraic curves and, in particular, cohomological field theories, Hodge classes and double ramification cycles. This methods are alternative to the traditional Witten-Kontsevich framework and its generalizations by Dubrovin and Zhang and, among other advantages, have the merit of encompassing quantum integrable systems. Most of this material originates from an ongoing collaboration with A. Buryak, B. Dubrovin and J. Gu\'er\'e

Spectral Methods for Numerical Relativity

Version published online by Living Reviews in Relativity.International audienceEquations arising in General Relativity are usually too complicated to be solved analytically and one has to rely on numerical methods to solve sets of coupled partial differential equations. Among the possible choices, this paper focuses on a class called spectral methods where, typically, the various functions are expanded onto sets of orthogonal polynomials or functions. A theoretical introduction on spectral expansion is first given and a particular emphasis is put on the fast convergence of the spectral approximation. We present then different approaches to solve partial differential equations, first limiting ourselves to the one-dimensional case, with one or several domains. Generalization to more dimensions is then discussed. In particular, the case of time evolutions is carefully studied and the stability of such evolutions investigated. One then turns to results obtained by various groups in the field of General Relativity by means of spectral methods. First, works which do not involve explicit time-evolutions are discussed, going from rapidly rotating strange stars to the computation of binary black holes initial data. Finally, the evolutions of various systems of astrophysical interest are presented, from supernovae core collapse to binary black hole mergers

Spectral analysis of gravitational waves from binary neutron star merger remnants

In this work we analyze the gravitational wave signal from hypermassive neutron stars formed after the merger of binary neutron star systems, focusing on its spectral features. The gravitational wave signals are extracted from numerical relativity simulations of models already considered by De Pietri et al. [Phys. Rev. D 93, 064047 (2016)], Maione et al. [Classical Quantum Gravity 33, 175009 (2016)], and Feo et al. [Classical Quantum Gravity 34, 034001 (2017)], and allow us to study the effect of the total baryonic mass of such systems (from $2.4 M_{\odot}$ to $3 M_{\odot}$), the mass ratio (up to $q = 0.77$), and the neutron star equation of state, both in equal and highly unequal mass binaries. We use the peaks we find in the gravitational spectrum as an independent test of already published hypotheses of their physical origin and empirical relations linking them with the characteristics of the merging neutron stars. In particular, we highlight the effects of the mass ratio, which in the past was often neglected. We also analyze the temporal evolution of the emission frequencies. Finally, we introduce a modern variant of Prony's method to analyze the gravitational wave postmerger emission as a sum of complex exponentials, trying to overcome some drawbacks of both Fourier spectra and least-squares fitting. Overall, the spectral properties of the postmerger signal observed in our simulation are in agreement with those proposed by other groups. More specifically, we find that the analysis of Bauswein and Stergioulas [Phys. Rev. D 91, 124056 (2015)] is particularly effective for binaries with very low masses or with a small mass ratio and that the mechanical toy model of Takami et al. [Phys. Rev. D 91, 064001 (2015)] provides a comprehensive and accurate description of the early stages of the postmerger.Comment: 19 pages, 6 figure