Effective inverse spectral problem for rational Lax matrices and applications

We reconstruct a rational Lax matrix of size R+1 from its spectral curve (the desingularization of the characteristic polynomial) and some additional data. Using a twisted Cauchy--like kernel (a bi-differential of bi-weight (1-nu,nu)) we provide a residue-formula for the entries of the Lax matrix in terms of bases of dual differentials of weights nu and 1-nu respectively. All objects are described in the most explicit terms using Theta functions. Via a sequence of ``elementary twists'', we construct sequences of Lax matrices sharing the same spectral curve and polar structure and related by conjugations by rational matrices. Particular choices of elementary twists lead to construction of sequences of Lax matrices related to finite--band recurrence relations (i.e. difference operators) sharing the same shape. Recurrences of this kind are satisfied by several types of orthogonal and biorthogonal polynomials. The relevance of formulae obtained to the study of the large degree asymptotics for these polynomials is indicated.Comment: 33 pages. Version 2 with added references suggested by the refere

Algebro-Geometric Solutions of the Boussinesq Hierarchy

We continue a recently developed systematic approach to the Bousinesq (Bsq) hierarchy and its algebro-geometric solutions. Our formalism includes a recursive construction of Lax pairs and establishes associated Burchnall-Chaundy curves, Baker-Akhiezer functions and Dubrovin-type equations for analogs of Dirichlet and Neumann divisors. The principal aim of this paper is a detailed theta function representation of all algebro-geometric quasi-periodic solutions and related quantities of the Bsq hierarchy.Comment: LaTeX, 48 page

Normal Families and Complex Dynamics

The schedule comprised more than 25 ordinary and problem session talks from a broad range of areas in function theory, including but not limited to: Nevanlinna theory, iteration of rational functions, dynamics of transcendental entire and meromorphic functions, function algebras, Riemann surfaces, each of them in close connection to the main topic Normal Families and Complex Dynamics

Generalizations on the results of Cao and Zhang

summary:We establish some uniqueness results for meromorphic functions when two nonlinear differential polynomials $P(f)\prod _{i=1}^{k}(f^{(i)})^{n_{i}}$ and $P(g)\prod _{i=1}^{k}(g^{(i)})^{n_{i}}$ share a nonzero polynomial with certain degree and our results improve and generalize some recent results in Y.-H. Cao, X.-B. Zhang (2012). Also we exhibit two examples to show that the conditions used in the results are sharp