511 research outputs found
Multidimensional Toda Lattices: Continuous and Discrete Time
In this paper we present multidimensional analogues of both the continuous-
and discrete-time Toda lattices. The integrable systems that we consider here
have two or more space coordinates. To construct the systems, we generalize the
orthogonal polynomial approach for the continuous and discrete Toda lattices to
the case of multiple orthogonal polynomials
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
On 2D discrete Schr\"odinger operators associated with multiple orthogonal polynomials
A class of cross-shaped difference operators on a two dimensional lattice is
introduced. The main feature of the operators in this class is that their
formal eigenvectors consist of multiple orthogonal polynomials. In other words,
this scheme generalizes the classical connection between Jacobi matrices and
orthogonal polynomials to the case of operators on lattices. Furthermore we
also show how to obtain 2D discrete Schr\"odinger operators out of this
construction and give a number of explicit examples based on known families of
multiple orthogonal polynomials.Comment: 15 page
Hermite-Padé Approximants for a Pair of Cauchy Transforms with Overlapping Symmetric Supports
Hermite-Padé approximants of type II are vectors of rational functions with a common denominator that interpolate a given vector of power series at infinity with maximal order. We are interested in the situation when the approximated vector is given by a pair of Cauchy transforms of smooth complex measures supported on the real line. The convergence properties of the approximants are rather well understood when the supports consist of two disjoint intervals (Angelesco systems) or two intervals that coincide under the condition that the ratio of the measures is a restriction of the Cauchy transform of a third measure (Nikishin systems). In this work we consider the case where the supports form two overlapping intervals (in a symmetric way) and the ratio of the measures extends to a holomorphic function in a region that depends on the size of the overlap. We derive Szegő-type formulae for the asymptotics of the approximants, identify the convergence and divergence domains (the divergence domains appear for Angelesco systems but are not present for Nikishin systems), and show the presence of overinterpolation (a feature peculiar for Nikishin systems but not for Angelesco systems). Our analysis is based on a Riemann-Hilbert problem for multiple orthogonal polynomials (the common denominator)
Continuous twin screw granulation: influence of process variables on granule and tablet quality
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
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