The PDEs of biorthogonal polynomials arising in the two-matrix model

The two-matrix model can be solved by introducing bi-orthogonal polynomials. In the case the potentials in the measure are polynomials, finite sequences of bi-orthogonal polynomials (called "windows") satisfy polynomial ODEs as well as deformation equations (PDEs) and finite difference equations (Delta-E) which are all Frobenius compatible and define discrete and continuous isomonodromic deformations for the irregular ODE, as shown in previous works of ours. In the one matrix model an explicit and concise expression for the coefficients of these systems is known and it allows to relate the partition function with the isomonodromic tau-function of the overdetermined system. Here, we provide the generalization of those expressions to the case of bi-orthogonal polynomials, which enables us to compute the determinant of the fundamental solution of the overdetermined system of ODE+PDEs+Delta-E.Comment: 20 pages v1 18 Nov 2003; v2 9 Jan 2004: trivial Latex mistake correcte

On the location of poles for the Ablowitz-Segur family of solutions to the second Painlev\'e equation

Using a simple operator-norm estimate we show that the solution to the second Painlev\'e equation within the Ablowitz-Segur family is pole-free in a well defined region of the complex plane of the independent variable. The result is illustrated with several numerical examples.Comment: 8 pages, to appear in Nonlinearit

Mixed Correlation Functions of the Two-Matrix Model

We compute the correlation functions mixing the powers of two non-commuting random matrices within the same trace. The angular part of the integration was partially known in the literature: we pursue the calculation and carry out the eigenvalue integration reducing the problem to the construction of the associated biorthogonal polynomials. The generating function of these correlations becomes then a determinant involving the recursion coefficients of the biorthogonal polynomials.Comment: 16 page

Drop impact on spherical soft surfaces

The impact of water drops on spherical soft surfaces is investigated experimentally through high-speed imaging. The effect of a convex compliant surface on the dynamics of impacting drops is relevant to various applications, such as 3D ink-jet printing, where drops of fresh material impact on partially cured soft substrates with arbitrary shape. Several quantities which characterize the morphology of impacting drops are measured through image-processing, including the maximum and minimum spreading angles, length of the wetted curve, and dynamic contact angle. In particular, the dynamic contact angle is measured using a novel digital image-processing scheme based on a goniometric mask, which does not require edge fitting. It is shown that the surface with a higher curvature enhances the retraction of the spreading drop; this effect may be due to the difference of energy dissipation induced by the curvature of the surface. In addition, the impact parameters (elastic modulus, diameter ratio, and Weber number) are observed to significantly affect the dynamic contact angle during impact. A quantitative estimation of the deformation energy shows that it is significantly smaller than viscous dissipation

The Supernova Impostor Impostor SN 1961V: Spitzer Shows That Zwicky Was Right (Again)

SN 1961V, one of Zwicky's defining Type V supernovae (SN), was a peculiar transient in NGC 1058 that has variously been categorized as either a true core collapse SN leaving a black hole (BH) or neutron star (NS) remnant, or an eruption of a luminous blue variable (LBV) star. The former case is suggested by its association with a decaying non-thermal radio source, while the latter is suggested by its peculiar transient light curve and its low initial expansion velocities. The crucial difference is that the star survives a transient eruption but not an SN. All stars identified as possible survivors are significantly fainter, L_opt ~ 10^5 Lsun, than the L_opt ~ 3 10^6 Lsun progenitor star at optical wavelengths. While this can be explained by dust absorption in a shell of material ejected during the transient, the survivor must then be present as a L_IR ~ 3 10^6 Lsun mid-infrared source. Using archival Spitzer observations of the region, we show that such a luminous mid-IR source is not present. The brightest source of dust emission is only L_IR ~ 10^5 Lsun and does not correspond to the previously identified candidates for the surviving star. The dust cannot be made sufficiently distant and cold to avoid detection unless the ejection energy, mass and velocity scales are those of a SN or greater. We conclude that SN 1961V was a peculiar, but real, supernova. Its peculiarities are probably due to enhanced mass loss just prior to the SN, followed by the interactions of the SN blast wave with this ejecta. This adds to the evidence that there is a population of SN progenitors that have major mass loss episodes shortly before core collapse. The progenitor is a low metallicity, ~1/3 solar, high mass, M_ZAMS > 80 Msun, star, which means either that BH formation can be accompanied by an SN or that surprisingly high mass stars can form a NS.Comment: 26 pages, 7 figures, submitted to Ap

The impact of viscoplastic drops on a heated surface in the Leidenfrost regime

The impact morphology of viscoplastic drops on a heated surface in the Leidenfrost regime is investigated experimentally by high-speed imaging. In particular several important parameters which characterize the impact morphology (such as maximum spreading diameter, minimum retracting diameter and maximum bouncing height etc.) are measured by analysing the impact process, recorded using a high-speed camera. It is shown that as the yield stress grows, surface forces are no longer able to minimize the free surface of the drop, and the inertial deformation upon impact becomes permanent. For small values of the yield stress, the impact morphology of viscoplastic Leidenfrost drops is similar to that of Newtonian drops. These effects can be interpreted in terms of the Bingham–Capillary number, which compares the yield stress magnitude and the capillary (Laplace) pressure. These results suggest that the main contribution to drop rebound is due to surface forces, and not to the intrinsic elasticity of the vapour cushion between the drop and the surface, which is a major assumption in one of the existing models

Moment determinants as isomonodromic tau functions

We consider a wide class of determinants whose entries are moments of the so-called semiclassical functionals and we show that they are tau functions for an appropriate isomonodromic family which depends on the parameters of the symbols for the functionals. This shows that the vanishing of the tau-function for those systems is the obstruction to the solvability of a Riemann-Hilbert problem associated to certain classes of (multiple) orthogonal polynomials. The determinants include Haenkel, Toeplitz and shifted-Toeplitz determinants as well as determinants of bimoment functionals and the determinants arising in the study of multiple orthogonality. Some of these determinants appear also as partition functions of random matrix models, including an instance of a two-matrix model.Comment: 24 page

Red Radiolarite Availability in Western Liguria? A Challenging Enigma from Ortovero (Savona, Liguria, Northern Italy)

The authors present a new prehistoric site recently discovered at Ortovero, near Albenga (Savona Province, Italy) in Western Liguria that can be attributed on typological grounds to the Final Epigravettian or to the Sauveterrian. The raw-material characteristics of the chipped stone artefacts resemble those stemming from the red radiolarite outcrops of Eastern Liguria, although the large number of pieces manufactured from this material and the distance from the supposed sources challenge this hypothesis. As a result of field surveys carried out over the last few years, the authors discuss the possibility of a formerly unknown source of vitreous red-coloured radiolarite outcropping in the surroundings of the site that may correspond to the Arnasco radiolarite formation outcropping near OrtoveroLes auteurs pr\ue9sentent la r\ue9cente d\ue9couverte d'un nouveau site pr\ue9historique \ue0 Ortovero, pr\ue8s d\u2019Albenga (province de Savone, Italie) en Ligurie occidentale qui peut \ueatre dat\ue9e sur la base d\u2019observations typologiques \ue0 la fin de l\u2019\uc9pigravettien ou au Sauveterrien. La mati\ue8re premi\ue8re de l\u2019ensemble lithique ressemble \ue0 celles des affleurements de radiolarite rouge de Ligurie orientale, bien que le nombre \ue9lev\ue9 de pi\ue8ces r\ue9alis\ue9es sur ce mat\ue9riau et la distance aux sources suppos\ue9es nous am\ue8ne \ue0 consid\ue9rer cette hypoth\ue8se avec prudence. \uc0 la lumi\ue8re des prospections de terrain conduites durant ces derni\ue8res ann\ue9es, les auteurs signalent l\u2019existence d\u2019un affleurement inconnu de radiolarites vitreuses rouges dans les environs du site, correspondant \ue0 la formation des \uab radiolarites d\u2019Arnasco \ubb qui affleurent non loin d\u2019Ortovero