This paper contains a general description of the theory of invariants under the adjoint action of a given finite-dimensional complex Lie algebra G. with special emphasis on polynomial and rational invariants. The familiar "Casimir" invariants are identified with the polynomial invariants in the enveloping algebra 21( G). More general structures (quotient fields) are required in order to investigate rational invariants. Some useful criteria for G having only polynomial or rational invariants are given. Moreover. in most of the physically relevant Lie algebras the exact computation of the maximal number of algebraically independent invariants turns out to be very easy. It reduces to finding the rank of a finite matrix. We apply the general method to some typical examples.

No abstract

The dispersionless limit of the scalar nonlocal ∂-problem is derived. It is given by a special class of nonlinear first-order equations. A quasi-classical version of the ∂-dressing method is presented. It is shown that the algebraic formulation of dispersionless hierarchies can be expressed in terms of properties of Beltrami tupe equations. The universal Whitham hierarchy and, in particular, the dispersionless KP hierarchy turn out to be rings of symmetries for the quasi-classical ∂-problem.

A note on reductions of the dispersionless Toda hierarchy J. Math. Phys. 51, 122704 (2010) The quasiclassical limit of the scalar nonlocal ‫ץ‬ -problem is derived and a quasiclassical version of the ‫ץ‬ -dressing method is presented. Dispersionless KadomtsevPetviashvili ͑KP͒, modified KP, and dispersionless two-dimensional Toda lattice ͑2DTL͒ hierarchies are discussed as illustrative examples. It is shown that the universal Whitham hierarchy is nothing but the ring of symmetries for the quasiclassical ‫ץ‬ -problem. The reduction problem is discussed and, in particular, the d2DTL equation of B type is derived.

This paper deals with the determination of the S-curves in the theory of non-hermitian orthogonal polynomials with respect to exponential weights along suitable paths in the complex plane. It is known that the corresponding complex equilibrium potential can be written as a combination of Abelian integrals on a suitable Riemann surface whose branch points can be taken as the main parameters of the problem. Equations for these branch points can be written in terms of periods of Abelian differentials and are known in several equivalent forms. We select one of these forms and use a combination of analytic an numerical methods to investigate the phase structure of asymptotic zero densities of orthogonal polynomials and of asymptotic eigenvalue densities of random matrix models. As an application we give a complete description of the phases and critical processes of the standard cubic model.

Some individuals can present important increases and decreases in corneal thickness values after anesthetic eye drops. This effect of anesthetic eye drops must be considered by refractive surgeons when carrying out preoperative laser in situ keratomileusis corneal thickness measurements.

Abstract. We present an implementation of the method of orthogonal polynomials which is particularly suitable to study the partition functions of Penner random matrix models, to obtain their explicit forms in the exactly solvable cases, and to determine the coefficients of their perturbative expansions in the continuum limit. The method relies on identities satisfied by the resolvent of the Jacobi matrix in the three-term recursion relation of the associated families of orthogonal polynomials. These identities lead to a convenient formulation of the string equations. As an application, we show that in the continuum limit the free energy of certain exactly solvable models like the linear and double Penner models can be written as a sum of gaussian contributions plus linear terms. To illustrate the one-cut case we discuss the linear, double and cubic Penner models, and for the two-cut case we discuss theoretically and numerically the existence of a double-branch structure of the free energy for the gaussian Penner model.

A general scheme for analyzing reductions of Whitham hierarchies is presented. It is based on a method for determining the S-function by means of a system of first order partial differential equations. Compatibility systems of differential equations characterizing both reductions and hodograph solutions of Whitham hierarchies are obtained. The method is illustrated by exhibiting solutions of integrable models such as the dispersionless Toda equation (heavenly equation) and the generalized Benney system.