let $g$ be a simple graph of order $n$‎. ‎we consider the‎ ‎independence polynomial and the domination polynomial of a graph‎ ‎$g$‎. ‎the value of a graph polynomial at a specific point can give‎ ‎sometimes a very surprising information about the structure of the‎ ‎graph‎. ‎in this paper we investigate independence and domination‎ ‎polynomial at $-1$ and $1$‎.

The Weyl relations, the harmonic oscillator, the hydrogen atom, the Dirac equation on the lattice are presented with the help of the difference equations and the orthogonal polynomials of discrete variable. This area of research is attracting more interest due to the lattice field theories and the hypothesis of a finite space. 1 Weyl group on finite space We defined the position space of dimens...

— This paper is a continuation of [18] where we studied a randomisation method based on the Laplacian with harmonic potential. Here we extend our previous results to the case of any polynomial and confining potential V on R. We construct measures, under concentration type assumptions, on the support of which we prove optimal weighted Sobolev estimates on R. This construction relies on accurate ...

We show that a polynomial Hˆ(N) of degree N harmonic oscillator hamiltonian allows us to devise fully solvable continuous quantum system for which the first discrete energy eigenvalues can be chosen at will. In general such choice leads re-ordering associated eigenfunctions Hˆ number their nodes does not increase monotonically with increasing level number. Systems have certain ‘universal’ featu...

We describe the structure QHO = QHON (dependent on the positive integer number N) on the universe L which is a finite cover, of order N, of the projective line P = P(F), F an algebraically closed field of characteristic 0. We prove that QHO is a complete irreducible Zariski geometry of dimension 1. We also prove that QHO is not classical in the sense that the structure is not interpretable in a...

The relativistic Hermite polynomials (RHP) were introduced in 1991 by Aldaya et al. [3] in a generalization of the theory of the quantum harmonic oscillator to the relativistic context. These polynomials were later related to the more classical Gegenbauer (or more generally Jacobi) polynomials in a study by Nagel [4]. For this reason, they do not deserve any special study since their properties...

The Kravchuk and Meixner polynomials of discrete variable are introduced for the discrete models of the harmonic oscillator and hydrogen atom. Starting from Rodrigues formula we construct raising and lowering operators, commutation and anticommutation relations. The physical properties of discrete models are figured out through the equivalence with the continuous models obtained by limit proces...

We prove polynomial and exponential decay at infinity of eigen-vectors of partial differential operators related to radiation problems for time-harmonic generalized Maxwell systems in an exterior domain Ω ⊂ RN , N ≥ 1, with non-smooth inhomogeneous, anisotropic coefficients converging near infinity with a rate r−τ , τ > 1, towards the identity. As a canonical application we show that the corres...