Quadratic Hermite-Pade approximation to the exponential function: a Riemann-Hilbert approach

We investigate the asymptotic behavior of the polynomials p, q, r of degrees n in type I Hermite-Pade approximation to the exponential function, defined by p(z)e^{-z}+q(z)+r(z)e^{z} = O(z^{3n+2}) as z -> 0. These polynomials are characterized by a Riemann-Hilbert problem for a 3x3 matrix valued function. We use the Deift-Zhou steepest descent method for Riemann-Hilbert problems to obtain strong uniform asymptotics for the scaled polynomials p(3nz), q(3nz), and r(3nz) in every domain in the complex plane. An important role is played by a three-sheeted Riemann surface and certain measures and functions derived from it. Our work complements recent results of Herbert Stahl.Comment: 60 pages, 13 figure

Asymptotics for Hermite-Pade rational approximants for two analytic functions with separated pairs of branch points (case of genus 0)

We investigate the asymptotic behavior for type II Hermite-Pade approximation to two functions, where each function has two branch points and the pairs of branch points are separated. We give a classification of the cases such that the limiting counting measures for the poles of the Hermite-Pade approximants are described by an algebraic function of order 3 and genus 0. This situation gives rise to a vector-potential equilibrium problem for three measures and the poles of the common denominator are asymptotically distributed like one of these measures. We also work out the strong asymptotics for the corresponding Hermite-Pade approximants by using a 3x3 Riemann-Hilbert problem that characterizes this Hermite-Pade approximation problem.Comment: 102 pages, 31 figure

Security of Quantum Key Distribution with Coherent States and Homodyne Detection

We assess the security of a quantum key distribution protocol relying on the transmission of Gaussian-modulated coherent states and homodyne detection. This protocol is shown to be equivalent to a squeezed state protocol based on a CSS code construction, and is thus provably secure against any eavesdropping strategy. We also briefly show how this protocol can be generalized in order to improve the net key rate.Comment: 7 page

Periinfarct rewiring supports recovery after primary motor cortex stroke.

After stroke restricted to the primary motor cortex (M1), it is uncertain whether network reorganization associated with recovery involves the periinfarct or more remote regions. We studied 16 patients with focal M1 stroke and hand paresis. Motor function and resting-state MRI functional connectivity (FC) were assessed at three time points: acute (<10 days), early subacute (3 weeks), and late subacute (3 months). FC correlates of recovery were investigated at three spatial scales, (i) ipsilesional non-infarcted M1, (ii) core motor network (M1, premotor cortex (PMC), supplementary motor area (SMA), and primary somatosensory cortex), and (iii) extended motor network including all regions structurally connected to the upper limb representation of M1. Hand dexterity was impaired only in the acute phase (P = 0.036). At a small spatial scale, clinical recovery was more frequently associated with connections involving ipsilesional non-infarcted M1 (Odds Ratio = 6.29; P = 0.036). At a larger scale, recovery correlated with increased FC strength in the core network compared to the extended motor network (rho = 0.71;P = 0.006). These results suggest that FC changes associated with motor improvement involve the perilesional M1 and do not extend beyond the core motor network. Core motor regions, and more specifically ipsilesional non-infarcted M1, could hence become primary targets for restorative therapies

Matrix interpretation of multiple orthogonality

In this work we give an interpretation of a (s(d + 1) + 1)-term recurrence relation in terms of type II multiple orthogonal polynomials.We rewrite this recurrence relation in matrix form and we obtain a three-term recurrence relation for vector polynomials with matrix coefficients. We present a matrix interpretation of the type II multi-orthogonality conditions.We state a Favard type theorem and the expression for the resolvent function associated to the vector of linear functionals. Finally a reinterpretation of the type II Hermite- Padé approximation in matrix form is given

Can education change the world? Education amplifies differences in liberalization values and innovation between developed and developing countries

The present study investigated the relationship between level of education and liberalization values in large, representative samples administered in 96 countries around the world (total N = 139,991). These countries show meaningful variation in terms of the Human Development Index (HDI), ranging from very poor, developing countries to prosperous, developed countries. We found evidence of cross-level interactions, consistently showing that individuals' level of education was associated with an increase in their liberalization values in higher HDI societies, whereas this relationship was curbed in lower HDI countries. This enhanced liberalization mindset of individuals in high HDI countries, in turn, was related to better scores on national indices of innovation. We conclude that this 'education amplification effect' widens the gap between lower and higher HDI countries in terms of liberalized mentality and economic growth potential. Policy implications for how low HDI countries can counter this gap are discussed

Critical behavior in Angelesco ensembles

We consider Angelesco ensembles with respect to two modified Jacobi weights on touching intervals [a,0] and [0,1], for a < 0. As a \to -1 the particles around 0 experience a phase transition. This transition is studied in a double scaling limit, where we let the number of particles of the ensemble tend to infinity while the parameter a tends to -1 at a rate of order n^{-1/2}. The correlation kernel converges, in this regime, to a new kind of universal kernel, the Angelesco kernel K^{Ang}. The result follows from the Deift/Zhou steepest descent analysis, applied to the Riemann-Hilbert problem for multiple orthogonal polynomials.Comment: 32 pages, 9 figure

Strong asymptotics for Cauchy biorthogonal polynomials with application to the Cauchy two--matrix model

We apply the nonlinear steepest descent method to a class of 3x3 Riemann-Hilbert problems introduced in connection with the Cauchy two-matrix random model. The general case of two equilibrium measures supported on an arbitrary number of intervals is considered. In this case, we solve the Riemann-Hilbert problem for the outer parametrix in terms of sections of a spinorial line bundle on a three-sheeted Riemann surface of arbitrary genus and establish strong asymptotic results for the Cauchy biorthogonal polynomials.Comment: 31 pages, 12 figures. V2; typos corrected, added reference

Ladder operators and differential equations for multiple orthogonal polynomials

In this paper, we obtain the ladder operators and associated compatibility conditions for the type I and the type II multiple orthogonal polynomials. These ladder equations extend known results for orthogonal polynomials and can be used to derive the differential equations satisfied by multiple orthogonal polynomials. Our approach is based on Riemann-Hilbert problems and the Christoffel-Darboux formula for multiple orthogonal polynomials, and the nearest-neighbor recurrence relations. As an illustration, we give several explicit examples involving multiple Hermite and Laguerre polynomials, and multiple orthogonal polynomials with exponential weights and cubic potentials.Comment: 28 page