A power series is introduced that is an extension to three sets of variables of the Cauchy sum for Jack symmetric functions in the Jack parameter α. We conjecture that the coefficients of this series with respect to the power sum basis are nonnegative integer polynomials in b, the Jack parameter shifted by 1. More strongly, we make the Matchings-Jack Conjecture, that the coefficients are counti...

We investigate the link between rectangular Jack polynomials and Hankel hyperdeterminants. As an application we give an expression of the even power of the Vandermonde in term of Jack polynomials.

It is shown that the deformed Calogero-Moser-Sutherland (CMS) operators can be described as the restrictions on certain affine subvarieties of the usual CMS operators for infinite number of particles. The ideals of these varieties are shown to be generated by the Jack symmetric functions related to the Young diagrams with special geometry. Combinatorial formulas for the related super-Jack and s...

We give new proofs of the rationality of the N = 1 superconformal minimal model vertex operator superalgebras and of the classification of their modules in both the Neveu-Schwarz and Ramond sectors. For this, we combine the standard free field realisation with the theory of Jack symmetric functions. A key role is played by Jack symmetric polynomials with a certain negative parameter that are la...

In Quebec, jack pine (Pinus banksiana Lamb.) is an important commercial species, used principally for timber wood and pulp. The natural ecology of jack pine is related to fire, which allow opening of the cones and dispersal of seeds (Burns and Honkala 1990). The exploitation of jack pine stands disturbs this cycle and the regeneration remains low after harvesting and plantation must be done to ...

Jack superpolynomials are eigenfunctions of the supersymmetric extension of the quantum trigonometric Calogero-Moser-Sutherland. They are orthogonal with respect to the scalar product, dubbed physical, that is naturally induced by this quantum-mechanical problem. But Jack superpolynomials can also be defined more combinatorially, starting from the multiplicative bases of symmetric superpolynomi...

Abstract. The Calogero model is a one-dimensional quantum integrable system with inverse-square long-range interactions confined in an external harmonic well. It shares the same algebraic structure with the Sutherland model, which is also a one-dimensional quantum integrable system with inverse-sine-square interactions. Inspired by the Rodrigues formula for the Jack polynomials, which form the ...

A recent novel derivation of the representation of Virasoro singular vectors in terms of Jack polynomials is extended to the supersymmetric case. The resulting expression of a generic super-Virasoro singular vector is given in terms of a simple di erential operator (whose form is characteristic of the sector, Neveu-Schwarz or Ramond) acting on a Jack superpolynomial. The latter is indexed by a ...

On a jack-up structure acts static loads (own weight, machines, installations and materials weight) and dynamic loads (waves, wind, marine currents, earthquake, operating installations). The authors led their analysis on the determining the dynamic loads produced by waves, wind and marine currents, in the Black Sea, because these are the main source in collapse of the structure. The structural ...