The inverse scattering transform for the defocusing Manakov system with nonzero boundary conditions at infinity is rigorously studied. Several new results are obtained: (i) The analyticity of the Jost eigenfunctions is investigated, and precise conditions on the potential that guarantee such analyticity are provided. (ii) The analyticity of the scattering coefficients is established. (iii) The ...

We slightly improve the lower bound of Báez-Duarte, Balazard, Landreau and Saias in the Nyman-Beurling formulation of the Riemann Hypothesis as an approximation problem. We construct Hilbert space vectors which could prove useful in the context of the so-called “Hilbert-Pólya idea”. Université de NiceSophia Antipolis Laboratoire J.-A. Dieudonné Parc Valrose F-06108 Nice Cedex 02 France burnol@m...

We present classical and generalized Riemann-Hilbert problem in several contexts arising from K-theory and bordism theory. The language of Fredholm pairs turns out to be useful and unavoidable. We propose an abstract formulation of a notion of bordism in the context of Hilbert spaces equipped with splittings. §

We have recently solved the inverse scattering problem for oneparameter families of vector fields, and used this result to construct the formal solution of the Cauchy problem for a class of integrable nonlinear partial differential equations in multidimensions, including the second heavenly equation of Plebanski and the dispersionless Kadomtsev Petviashvili (dKP) equation. We showed, in particu...

We have recently solved the inverse scattering problem for oneparameter families of vector fields, and used this result to construct the formal solution of the Cauchy problem for a class of integrable nonlinear partial differential equations in multidimensions, including the second heavenly equation of Plebanski and the dispersionless Kadomtsev Petviashvili (dKP) equation. We showed, in particu...

We derive semiclassical asymptotics for the orthogonal polynomials P n (z) on the line with respect to the exponential weight exp(−NV (z)), where V (z) is a double-well quartic polynomial, in the limit when n, N → ∞. We assume that ε ≤ (n/N) ≤ λ cr − ε for some ε > 0, where λ cr is the critical value which separates orthogonal polynomials with two cuts from the ones with one cut. Simultaneously...

by the method of weight coefficients, techniques of real analysis andhermite-hadamard's inequality, a half-discrete hardy-hilbert-type inequalityrelated to the kernel of the hyperbolic cosecant function with the best possibleconstant factor expressed in terms of the extended riemann-zeta function is proved.the more accurate equivalent forms, the operator expressions with the norm,the reverses a...

This paper, a continuation of [16], connects the analysis of the length of the longest weakly increasing subsequence of inhomogeneous random words to a Riemann-Hilbert problem and an associated system of integrable PDEs. That such a connection exists is not so surprising given the fundamental work of Baik, Deift and Johansson [3] connecting the related problem involving random permutations to a...

Solutions to the Riemann-Hilbert problems with irregular singularities naturally associated to semisimple Frobenius manifold structures on Hurwitz spaces (moduli spaces of meromorphic functions on Riemann surfaces) are constructed. The solutions are given in terms of meromorphic bidifferentials defined on the underlying Riemann surface. The relationship between different classes of Frobenius ma...