Factorization and the Dressing Method for the Gel'fand-Dikii Hierarch

The isospectral flows of an $n^{th}$ order linear scalar differential operator $L$ under the hypothesis that it possess a Baker-Akhiezer function were originally investigated by Segal and Wilson from the point of view of infinite dimensional Grassmanians, and the reduction of the KP hierarchy to the Gel'fand-Dikii hierarchy. The associated first order systems and their formal asymptotic solutions have a rich Lie algebraic structure which was investigated by Drinfeld and Sokolov. We investigate the matrix Riemann-Hilbert factorizations for these systems, and show that different factorizations lead respectively to the potential, modified, and ordinary Gel'fand-Dikii flows. Lie algebra decompositions (the Adler-Kostant-Symes method) are obtained for the modified and potential flows. For $n>3$ the appropriate factorization for the Gel'fand-Dikii flows is not a group factorization, as would be expected; yet a modification of the dressing method still works. A direct proof, based on a Fredholm determinant associated with the factorization problem, is given that the potentials are meromorphic in $x$ and in the time variables. Potentials with Baker-Akhiezer functions include the multisoliton and rational solutions, as well as potentials in the scattering class with compactly supported scattering data. The latter are dense in the scattering class

Block Toeplitz determinants, constrained KP and Gelfand-Dickey hierarchies

We propose a method for computing any Gelfand-Dickey tau function living in Segal-Wilson Grassmannian as the asymptotics of block Toeplitz determinant associated to a certain class of symbols. Also truncated block Toeplitz determinants associated to the same symbols are shown to be tau function for rational reductions of KP. Connection with Riemann-Hilbert problems is investigated both from the point of view of integrable systems and block Toeplitz operator theory. Examples of applications to algebro-geometric solutions are given.Comment: 35 pages. Typos corrected, some changes in the introductio

The Solution Space of the Unitary Matrix Model String Equation and the Sato Grassmannian

The space of all solutions to the string equation of the symmetric unitary one-matrix model is determined. It is shown that the string equation is equivalent to simple conditions on points $V_1$ and $V_2$ in the big cell \Gr of the Sato Grassmannian $Gr$. This is a consequence of a well-defined continuum limit in which the string equation has the simple form \lb \cp ,\cq_- \rb =\hbox{\rm 1}, with \cp and \cq_- $2\times 2$ matrices of differential operators. These conditions on $V_1$ and $V_2$ yield a simple system of first order differential equations whose analysis determines the space of all solutions to the string equation. This geometric formulation leads directly to the Virasoro constraints \L_n\,(n\geq 0), where \L_n annihilate the two modified-KdV \t-functions whose product gives the partition function of the Unitary Matrix Model.Comment: 21 page

An axiomatization of cumulative prospect theory

This paper presents a method for axiomatizing a variety of models for decision making under uncertainty, including Expected Utility and Cumulative Prospect Theory. This method identifies, for each model, the situations that permit consistent inferences about the ordering of value differences. Examples of rankdependent and sign-dependent preference patterns are used to motivate the models and the tradeoff consistency axioms that characterize them. The major properties of the value function in Cumulative Prospect Theory—diminishing sensitivity and loss aversion—are contrasted with the principle of diminishing marginal utility that is commonly assumed in Expected Utility