64 research outputs found
Finite term relations for the exponential orthogonal polynomials
The exponential orthogonal polynomials encode via the theory of hyponormal
operators a shade function supported by a bounded planar shape. We prove
under natural regularity assumptions that these complex polynomials satisfy a
three term relation if and only if the underlying shape is an ellipse carrying
uniform black on white. More generally, we show that a finite term relation
among these orthogonal polynomials holds if and only if the first row in the
associated Hessenberg matrix has finite support. This rigidity phenomenon is in
sharp contrast with the theory of classical complex orthogonal polynomials. On
function theory side, we offer an effective way based on the Cauchy transforms
of , to decide whether a
-term relation among the exponential orthogonal polynomials exists; in
that case we indicate how the shade function can be reconstructed from a
resulting polynomial of degree and the Cauchy transform of . A
discussion of the relevance of the main concepts in Hele-Shaw dynamics
completes the article.Comment: 33 page
The solution to the q-KdV equation
Let KdV stand for the Nth Gelfand-Dickey reduction of the KP hierarchy. The
purpose of this paper is to show that any KdV solution leads effectively to a
solution of the q-approximation of KdV. Two different q-KdV approximations were
proposed, one by Frenkel and a variation by Khesin et al. We show there is a
dictionary between the solutions of q-KP and the 1-Toda lattice equations,
obeying some special requirement; this is based on an algebra isomorphism
between difference operators and D-operators, where . Therefore,
every notion about the 1-Toda lattice can be transcribed into q-language.Comment: 18 pages, LaTe
Integrability and Seiberg-Witten Theory: Curves and Periods
Interpretation of exact results on the low-energy limit of SUSY YM
in the language of integrability theory is reviewed. The case of elliptic
Calogero system, associated with the flow between and SUSY in ,
is considered in some detail.Comment: 26 page
Limiting Cases for Spectrum Closure Results
The spectrum of a first-order sentence is the set of cardinalities of its finite models. Given a spectrum S and a function f, it is not always clear whether or not the image of S under f is also a spectrum. In this paper, we consider questions of this form for functions that increase very quickly and for functions that increase very slowly. Roughly speaking, we prove that the class of all spectra is closed under functions that increase arbitrarily quickly, but it is not closed under some natural slowly increasing functions
Multivariate orthogonal polynomials and integrable systems
Multivariate orthogonal polynomials in D real dimensions are considered from the perspective of the Cholesky factorization of a moment matrix. The approach allows for the construction of corresponding multivariate orthogonal polynomials, associated second kind functions, Jacobi type matrices and associated three term relations and also Christoffel-Darboux formulae. The multivariate orthogonal polynomials, their second kind functions and the corresponding Christoffel-Darboux kernels are shown to be quasi-determinants as well as Schur complements of bordered truncations of the moment matrix; quasi-tau functions are introduced. It is proven that the second kind functions are multivariate Cauchy transforms of the multivariate orthogonal polynomials. Discrete and continuous deformations of the measure lead to Toda type integrable hierarchy, being the corresponding flows described through Lax and Zakharov-Shabat equations; bilinear equations are found. Varying size matrix nonlinear partial difference and differential equations of the 2D Toda lattice type are shown to be solved by matrix coefficients of the multivariate orthogonal polynomials. The discrete flows, which are shown to be connected with a Gauss-Borel factorization of the Jacobi type matrices and its quasi-determinants, lead to expressions for the multivariate orthogonal polynomials and their second kind functions in terms of shifted quasi-tau matrices, which generalize to the multidimensional realm, those that relate the Baker and adjoint Baker functions to ratios of Miwa shifted tau-functions in the 1D scenario. In this context, the multivariate extension of the elementary Darboux transformation is given in terms of quasi-determinants of matrices built up by the evaluation, at a poised set of nodes lying in an appropriate hyperplane in R^D, of the multivariate orthogonal polynomials. The multivariate Christoffel formula for the iteration of m elementary Darboux transformations is given as a quasi-determinant. It is shown, using congruences in the space of semi-infinite matrices, that the discrete and continuous flows are intimately connected and determine nonlinear partial difference-differential equations that involve only one site in the integrable lattice behaving as a Kadomstev-Petviashvili type system. Finally, a brief discussion of measures with a particular linear isometry invariance and some of its consequences for the corresponding multivariate polynomials is given. In particular, it is shown that the Toda times that preserve the invariance condition lay in a secant variety of the Veronese variety of the fixed point set of the linear isometry
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