Spaces of polynomials with roots of bounded multiplicity

We describe an alternative approach to some results of Vassiliev on spaces of polynomials, by using the scanning method which was used by Segal in his investigation of spaces of rational functions. We explain how these two approaches are related by the Smale-Hirsch Principle or the h-Principle of Gromov. We obtain several generalizations, which may be of interest in their own right.Comment: 29 pages, AMS-Te

Asymptotic expansion of a partition function related to the sinh-model

This paper develops a method to carry out the large-$N$ asymptotic analysis of a class of $N$-dimensional integrals arising in the context of the so-called quantum separation of variables method. We push further ideas developed in the context of random matrices of size $N$, but in the present problem, two scales $1/N^{\alpha}$ and $1/N$ naturally occur. In our case, the equilibrium measure is $N^{\alpha}$-dependent and characterised by means of the solution to a $2\times 2$ Riemann--Hilbert problem, whose large-$N$ behavior is analysed in detail. Combining these results with techniques of concentration of measures and an asymptotic analysis of the Schwinger-Dyson equations at the distributional level, we obtain the large-$N$ behavior of the free energy explicitly up to $o(1)$. The use of distributional Schwinger-Dyson is a novelty that allows us treating sufficiently differentiable interactions and the mixing of scales $1/N^{\alpha}$ and $1/N$, thus waiving the analyticity assumptions often used in random matrix theory.Comment: 158 pages, 4 figures (V2 introduction extended, missprints corrected, clarifications added to lemma 3.1.9 and corollary 3.1.10

Form factor approach to dynamical correlation functions in critical models

We develop a form factor approach to the study of dynamical correlation functions of quantum integrable models in the critical regime. As an example, we consider the quantum non-linear Schr\"odinger model. We derive long-distance/long-time asymptotic behavior of various two-point functions of this model. We also compute edge exponents and amplitudes characterizing the power-law behavior of dynamical response functions on the particle/hole excitation thresholds. These last results confirm predictions based on the non-linear Luttinger liquid method. Our results rely on a first principles derivation, based on the microscopic analysis of the model, without invoking, at any stage, some correspondence with a continuous field theory. Furthermore, our approach only makes use of certain general properties of the model, so that it should be applicable, with possibly minor modifications, to a wide class of (not necessarily integrable) gapless one dimensional Hamiltonians.Comment: 33 page

Dysplazja kregosłupowo-przynasadowa : Kozlowski

Background: Kozlowski type spondylo-metaphyseal dysplasia is the most frequent form among spondylometaphyseal dysplasias. Case report: The first cases of such dysplasia among Gypsies - a father and two sons - have been described. Conclusions: Radiographic examination is the only means by which to diagnose Kozlowski type spondylometaphyseal dysplasia with certainty. Correct diagnosis is not only prognostic for the course of the disease and its complications, but also excludes the necessity of further, often expensive, investigations

An ice giant exoplanet interpretation of the anomaly in microlensing event OGLE-2011-BLG-0173

We analyze the microlensing event OGLE-2011-BLG-0173, which shows a small perturbation at the end of the microlensing event caused by the primary lens. We consider both binary lens and binary source models and we explore their degeneracies, some of which have not previously been recognized. There are two families of binary lens solutions, one with a mass ratio $q\approx4\times10^{-4}$ and a separation s~4.6 and the other with q~0.015 and s~0.22, i.e, both have companions in the planetary regime. We search for solutions by using Bayesian analysis that includes planet frequency as a prior and find that the s~4.6 family is the preferred one with ~4 M_Uranus mass planet on an orbit of ~10 AU. The degeneracies arise from a paucity of information on the anomaly, demonstrating that high-cadence observations are essential for characterizing wide-orbit microlensing planets. Hence, we predict that the planned WFIRST microlensing survey will be less prone to these degeneracies than the ongoing ground-based surveys. We discuss the known low-mass, wide-orbit companions and we notice that for the largest projected separations the mass ratios are either high (consistent with brown dwarf companions) or low (consistent with Uranus analogs), but intermediate mass ratios (Jupiter analogs on wide orbits) have not been detected to date, despite the fact that the sensitivity to such planets should be higher than that of Uranus analogs. This is therefore tentative evidence of the existence of a massive ice giant desert at wide separations. On the other hand, given their low intrinsic detection sensitivity, Uranus analogs may be ubiquitous.Comment: AJ accepted, 6 figures, 4 table