In this paper we prove that perfect graphs are kernel solvable, as it was conjectured by Berge and Duchet (1983). The converse statement, i.e. that kernel solvable graphs are perfect, was also conjectured in the same paper, and is still open. In this direction we prove that it is always possible to substitute some of the vertices of a non-perfect graph by cliques so that the resulting graph is ...

The dynamical systems method (DSM), for solving operator equations, especially nonlinear and ill-posed, is developed in this paper. Consider an operator equation F (u) = 0 in a Hilbert space H and assume that this equation is solvable. Let us call the problem of solving this equation illposed if the operator F ′(u) is not boundedly invertible, and well-posed otherwise. The DSM for solving linea...

Abstract. This paper obtains all solvable 3-Lie algebras with the m-dimensional filiform 3-Lie algebra N (m ≥ 5) as a maximal hypo-nilpotent ideal, and proves that the m-dimensional filiform 3-Lie algebra N can’t be as the nilradical of solvable non-nilpotent 3-Lie algebras. By means of one dimensional extension of Lie algebras to the 3-Lie algebras, we get some classes of solvable Lie algebras...

The following structure theorems are proved: An orbit of a real solvable Lie group in projective space that is a complex submanifold is isomorphic to C x (C*)m x Í2 , where Í2 is an open orbit of a real solvable Lie group in a projective rational variety. Also, any homogeneous space of a complex Lie group that is isomorphic to C" can be realized as an orbit in some projective space. As a conseq...

By exploiting the hidden algebraic structure of the Schrödinger Hamiltonian, namely the sl(2), we propose a unified approach of generating both exactly solvable and quasi-exactly solvable potentials. We obtain, in this way, two new classes of quasi-exactly solvable systems one of which is of periodic type while the other hyperbolic. Tracking down solvable quantum potentials has always aroused i...

The exactly and quasi-exactly solvable problems for spin one-half in one dimension on the basis of a hidden dynamical symmetry algebra of Hamiltonian are discussed. We take the supergroup, OSP (2|1), as such a symmetry. A number of exactly solvable examples are considered and their spectrum are evaluated explicitly. Also, a class of quasi-exactly solvable problems on the basis of such a symmetr...

A lattice model of critical dense polymers is solved exactly for finite strips. The model is the first member of the principal series of the recently introduced logarithmic minimal models. The key to the solution is a functional equation in the form of an inversion identity satisfied by the commuting double-row transfer matrices. This is established directly in the planar Temperley-Lieb algebra...

S.P.Novikov: University of Maryland at College Park, Department of Mathematics and IPST, College Park MD 20742-2431 USA and Landau Institute for Theoretical Physics, Moscow 117940, Kosygin street 2, Russia. E-mail adress [email protected] A.P.Veselov: Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU, UK and Department of Mathematics and Mec...

We present a soluble variant of the replicator model well established in theoretical biology and game theory. By using methods of statistical physics we derive an analytical solution to our model which becomes exact in the long time limit. We apply our model to the iterated prisoner’s dilemma game and compare our results to numerical simulations. The replicator model describes the evolution of ...

Using the ideas of supersymmetry and shape invariance we show that the eigenvalues and eigenfunctions of a wide class of noncentral potentials can be obtained in a closed form by the operator method. This generalization considerably extends the list of exactly solvable potentials for which the solution can be obtained algebraically in a simple and elegant manner. As an illustration, we discuss ...