Polynomials with a common composite

We develop a conditional entropy theory for infinite measure preserving actions of countable discrete amenable groups with respect to a σ-finite factor. This includes ‘infinite’ analogues of relative Kolmogorov-Sinai, Rokhlin and Krieger theorems on generating partitions, Pinsker theorem on disjointness, Furstenberg decomposition and disjointness theorems, etc. In case of ℤ-action, our concept of relative entropy matches well the ‘absolute’ entropy h _Kr introduced by Krengel. Answering in part his question and a question of Silva and Thieullen, we show that for any non-distal transformation S there exists an infinite measure preserving transformation T with h _Kr(T × S) = ∞ but h _Kr(T) = 0. This project was supported in part by a CRDF grant UM1-2546-KH-03.     

Acoustic Scattering and the Extended Korteweg– de Vries Hierarchy

The acoustic scattering operator on the real line is mapped to a Schrödinger operator under the Liouville transformation. The potentials in the image are characterized precisely in terms of their scattering data, and the inverse transfor- mation is obtained as a simple, linear quadrature. An existence theorem for the associated Harry Dym flows is proved, using the scattering method. The scattering problem associated with the Camassa–Holm flows on the real line is solved explicitly for a special case, which is used to reduce a general class of such problems to scattering problems on finite intervals.     

Action-angle variables for the Gel'fand-Dikii flows

Using the scattering transform fornth order linear scalar operators, the Poisson bracket found by Gel'fand and Dikii, which generalizes the Gardner Poisson bracket for the KdV hierarchy, is computed on the scattering side. Action-angle variables are then constructed. Using this, complete integrability is demonstrated in the strong sense. Real action-angle variables are constructed in the self-adjoint case.Dedicated to Professor Klaus Kirchgässner on the occasion of his sixtieth birthdayResearch supported in part by N.S.F. Grants DMS-8916968 and DMS-8901607.We would like to thank Boris Konopelchenko for helpful discussions during the preparation of this paper.     

Constructively Factoring Linear Partial Differential Operators in Two Variables

We study conditions under which a partial differential operator of arbitrary order n in two variables or an ordinary linear differential operator admits a factorization with a first-order factor on the left.The process of factoring consists of recursively solving systems of linear equations subject to certain differential compatibility conditions.In the general case of partial differential operators, it is not necessary to solve a differential equation. In special degenerate cases, such as an ordinary differential operator, the problem eventually reduces to solving some Riccati equation(s). We give the factorization conditions explicitly for the second and third orders and in outline form for higher orders. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 145, No. 2, pp. 165–180, November, 2005.     

Approximate identities from Laguerre functions and singular integrals on the Heisenberg group

Suitably scaled Laguerre functions are an approximate identity for multiplicative convolution with test functions on the half line. As an application, we derive a precise connection between the Mikhlin-type expansion of a singular integral operator on a Heisenberg groupH _n and its natural restriction toH _n modulo the center. Research supported by NSF Grant DMS-9800605 and by NSERC Grant OGP0003017.