Painlev\'e V and time dependent Jacobi polynomials

In this paper we study the simplest deformation on a sequence of orthogonal polynomials, namely, replacing the original (or reference) weight $w_0(x)$ defined on an interval by $w_0(x)e^{-tx}.$ It is a well-known fact that under such a deformation the recurrence coefficients denoted as $\alpha_n$ and $\beta_n$ evolve in $t$ according to the Toda equations, giving rise to the time dependent orthogonal polynomials, using Sogo's terminology. The resulting "time-dependent" Jacobi polynomials satisfy a linear second order ode. We will show that the coefficients of this ode are intimately related to a particular Painlev\'e V. In addition, we show that the coefficient of $z^{n-1}$ of the monic orthogonal polynomials associated with the "time-dependent" Jacobi weight, satisfies, up to a translation in $t,$ the Jimbo-Miwa $\sigma$-form of the same $P_{V};$ while a recurrence coefficient $\alpha_n(t),$ is up to a translation in $t$ and a linear fractional transformation $P_{V}(\alpha^2/2,-\beta^2/2, 2n+1+\alpha+\beta,-1/2).$ These results are found from combining a pair of non-linear difference equations and a pair of Toda equations. This will in turn allow us to show that a certain Fredholm determinant related to a class of Toeplitz plus Hankel operators has a connection to a Painlev\'e equation

Prediction of the bias voltage dependent magnetic contrast in spin-polarized scanning tunneling microscopy

This work is concerned with the theoretical description of the contrast, i.e., the apparent height difference between two lateral surface positions on constant current spin-polarized scanning tunneling microscopy (SP-STM) images. We propose a method to predict the bias voltage dependent magnetic contrast from single point tunneling current or differential conductance measurements, without the need of scanning large areas of the surface. Depending on the number of single point measurements, the bias positions of magnetic contrast reversals and of the maximally achievable magnetic contrast can be determined. We validate this proposal by simulating SP-STM images on a complex magnetic surface employing a recently developed approach based on atomic superposition. Furthermore, we show evidence that the tip electronic structure and magnetic orientation have a major effect on the magnetic contrast. Our theoretical prediction is expected to inspire experimentalists to considerably reduce measurement efforts for determining the bias dependent magnetic contrast on magnetic surfaces.Comment: 28 pages manuscript, 1 table, 6 figure

Determination of Frequency and Distribution of Hessian Fly (Diptera: Cecidomyiidae) Biotypes in the Northeastern Soft Wheat Region

Fifteen collections of Hessian flies from the northern soft winter wheat region of the United States were used to determine the composition and frequency of biotypes. The wheat cultivars \u27Seneca\u27 (H7Hs), \u27Monon\u27 (H3), \u27Knox 62\u27 (~, H7Hg), and \u27Abe\u27 (Hs) were used as differentials. Biotypes J and L replaced biotype B as the prevalent biotype in Indiana, since wheat cultivars having the Hs and the H6 genes have been grown. Biotype GP, the least virulent of any Hessian fly biotypes, was still present in New York indicating that wheat cuItivars with no genes for resistance are still being grown there. The genetic variability of Hessian fly biotypes that enables them to overcome the resistance in wheat cultivars is discussed

The GTC exoplanet transit spectroscopy survey X. Stellar spots versus Rayleigh scattering: the case of HAT-P-11b

Rayleigh scattering in a hydrogen-dominated exoplanet atmosphere can be detected from ground or space based telescopes, however, stellar activity in the form of spots can mimic Rayleigh scattering in the observed transmission spectrum. Quantifying this phenomena is key to our correct interpretation of exoplanet atmospheric properties. We obtained long-slit optical spectroscopy of two transits of HAT-P-11b with the Optical System for Imaging and low-Intermediate-Resolution Integrated Spectroscopy (OSIRIS) at Gran Telescopio Canarias (GTC) on August 30 2016 and September 25 2017. We integrated the spectrum of HAT-P-11 and one reference star in several spectroscopic channels across the $\lambda\sim$ 400-785 nm region, creating numerous light curves of the transits. We fit analytic transit curves to the data taking into account the systematic effects and red noise present in the time series in an effort to measure the change of the planet-to-star radius ratio ($R_\mathrm{p}/R_\mathrm{s}$) across wavelength. By fitting both transits together, we find a slope in the transmission spectrum showing an increase of the planetary radius towards blue wavelengths. A closer inspection to the transmission spectrum of the individual data sets reveals that the first transit presents this slope while the transmission spectrum of the second data set is flat. Additionally we detect hints of Na absorption in the first night, but not in the second. We conclude that the transmission spectrum slope and Na absorption excess found in the first transit observation are caused by unocculted stellar spots. Modeling the contribution of unocculted spots to reproduce the results of the first night we find a spot filling factor of $\delta=0.62^{+0.20}_{-0.17}$ and a spot-to-photosphere temperature difference of $\Delta T = 429^{+184}_{-299}$ K.Comment: Accepted for publication in Astronomy & Astrophysics, 13 page

Rate of Convergence Towards Semi-Relativistic Hartree Dynamics

We consider the semi-relativistic system of $N$ gravitating Bosons with gravitation constant $G$. The time evolution of the system is described by the relativistic dispersion law, and we assume the mean-field scaling of the interaction where $N \to \infty$ and $G \to 0$ while $GN = \lambda$ fixed. In the super-critical regime of large $\lambda$, we introduce the regularized interaction where the cutoff vanishes as $N \to \infty$. We show that the difference between the many-body semi-relativistic Schr\"{o}dinger dynamics and the corresponding semi-relativistic Hartree dynamics is at most of order $N^{-1}$ for all $\lambda$, i.e., the result covers the sub-critical regime and the super-critical regime. The $N$ dependence of the bound is optimal.Comment: 29 page