Jack superpolynomials: physical and combinatorial definitions

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 superpolynomials, enforcing orthogonality with respect to a one-parameter deformation of the combinatorial scalar product. Both constructions turns out to be equivalent. This provides strong support for the correctness of the various underlying constructions and for the pivotal role of Jack superpolynomials in the theory of symmetric superpolynomials.Comment: 6 pages. To appear in the proceedings of the {\it XIII International Colloquium on Integrable Systems and Quantum Groups}, Czech. J . Phys., June 17-19 2004, Doppler Institute, Czech Technical Universit

Probability Distributions on Partially Ordered Sets and Network Interdiction Games

This article poses the following problem: Does there exist a probability distribution over subsets of a finite partially ordered set (poset), such that a set of constraints involving marginal probabilities of the poset's elements and maximal chains is satisfied? We present a combinatorial algorithm to positively resolve this question. The algorithm can be implemented in polynomial time in the special case where maximal chain probabilities are affine functions of their elements. This existence problem is relevant for the equilibrium characterization of a generic strategic interdiction game on a capacitated flow network. The game involves a routing entity that sends its flow through the network while facing path transportation costs, and an interdictor who simultaneously interdicts one or more edges while facing edge interdiction costs. Using our existence result on posets and strict complementary slackness in linear programming, we show that the Nash equilibria of this game can be fully described using primal and dual solutions of a minimum-cost circulation problem. Our analysis provides a new characterization of the critical components in the interdiction game. It also leads to a polynomial-time approach for equilibrium computation

Evaluation and normalization of Jack superpolynomials

Two evaluation formulas are derived for the Jack superpolynomials. The evaluation formulas are expressed in terms of products of fillings of skew diagrams. One of these formulas is nothing but the evaluation formula of the Jack polynomials with prescribed symmetry, which thereby receives here a remarkably simple formulation. Among the auxiliary results required to establish the evaluation formulas, the determination of the conditions ensuring the non-vanishing coefficients in a Pieri-type rule for Jack superpolynomials is worth pointing out. An important application of the evaluation formulas is a new derivation of the combinatorial norm of the Jack superpolynomials. We finally mention that the introduction of a simpler version of the dominance ordering on superpartitions is fundamental to establish our results.Comment: 42 pages, 10 figures. v2: minor corrections in Eqs (16) and (17

The supersymmetric Ruijsenaars-Schneider model

An integrable supersymmetric generalization of the trigonometric Ruijsenaars-Schneider model is presented whose symmetry algebra includes the super Poincar\'e algebra. Moreover, its Hamiltonian is showed to be diagonalized by the recently introduced Macdonald superpolynomials. Somewhat surprisingly, the consistency of the scalar product forces the discreteness of the Hilbert space.Comment: v1: 11 pages, 1 figure. v2: new format, 5 pages, short section added at the end of the article addressing the problem of consistency of the scalar product (e.g., positivity of the weight functions and the normalization of the ground state wave function). To appear in Physical Review Letter

How is water context impacting the results of a role-playing game: an experimental study

Role-playing games (RPG) are decision-making supports, developed in the field, and are used to help stakeholders' "point of view wording" in resources management situations. A particular RPG (KatAware) was built in the Kat River watershed (in South Africa), according to the ComMod approach, in order to help people in designing a shared water management plan. Only two sessions of KatAware were played, in which the players seemed to behave in a cooperative way. Unfortunately, the results obtained after having run only two sessions are ambiguous, as stationary replications of these sessions are difficult to implement. The experimental method allows such a replication by controlling all the parameters. The objective of the experimental method put in place is to assess game sessions outcomes and to reflect about general design rules and dynamics followed when building role-playing games. Among all the parameters, the context of the game influences the outcomes. After having simplified KatAware, our study shows how the context, defined as the level of information carried by the game, could be decomposed and re-composed according to its main dimensions, i.e.: illustration of the instructions; communication; repetition of periods; players' experience. The impact on players' choices of different levels of these dimensions were experimentally tested with students (at the "Experimental Economics Lab in Montpellier", France). We assessed the impact of the two first dimensions: "illustration" and "communication", by comparing the outcomes obtained after having varied their levels with the results obtained with the referential treatment, in which neither illustration nor communication were introduced in the protocol. We showed that the addition of watery illustrative elements within the instructions, first by only introducing the sentence "this experience is based on a water management situation", and then by describing such a situation through a story telling, increases noise in behaviours. However, this watery context does not impact players' decisions in the same way as another context chosen - in our protocol: employees in a firm -, which provided more familiar cultural references to the players. Finally, players' argumentation of their choices - through an ad-hoc controlled communication treatment - improved learning outcomes: while the observed choices converge towards theoretical equilibrium faster when communication among players is allowed, the final equilibriums are not significantly different in presence or in absence of communication. (Résumé d'auteur

From Experience to Experiments in South African Water Management: Defining the Framework

A role-playing game (RPG), KatAware, was developed in the Kat River catchment of South Africa to support the negotiation process among water users on the allocation rules of the resource. Playing the RPG with local stakeholders exhibited some regularity in the behaviour of players, particularly on their attitude of defining binding agreements. These regularities were first formalized through a model of cooperative game theory (CGT), and then, to confirm the results of the model, tested by an experimental protocol. Both the model and the protocol were based and calibrated on the results of the RPG. The progressive simplification (decontextualization) required to bring the RPG into the laboratory suggested to explore the role of context (in our case water related issues) on players’ behaviour. The objective of this paper is to illustrate the process that conducted the research team from the experience in the Kat River to the first experiments to test the hypotheses exhibited in the experience and then to analyze the influence of context on players’ behaviour. Terms and concepts are clarified in order to provide a clear research framework in this new field at the border between experiences and experiments in social sciences for commons management.

From "tools to tell" to "tools to test"

The emphasis in this paper will be on the illustration and explanation of the concepts and terminology defining the research framework within which the research programme was developed. Having crossed a somehow wide field, which ranges from two extremes which could be identified as "experience" and "experiment", has prompted us to a reflection on the similarities and distinctions between these two terms and also on the dimensions among which this diversity is expressed. Research trajectory that started with the construction of a Role-Playing Game (RPG) to support local participatory decision-making about water management, continued with the development of an experimental protocol to test economic hypotheses exhibited by the RPG and developed into the analysis of the influence of context on players' behaviour. RPG decomposition started from the definition of the concept of "context" that groups several informational dimensions: the game instructions; communication among players; and their involvement. In this presentation, only the two first dimensions are treated. They open perspectives to study the third one and the relations between these three attributes of the game context. (Résumé d'auteur

Two-flux and multiflux matrix models for colored surfaces

International audienceThis paper presents various extensions of the so-called two-flux models for prediction of reflectance and transmittance of diffusing media, i.e. the ubelka-Munk model, and the extension of Kubelka-Munk for stacks of diffusing layers. A first matrix formulation of the Kubelka-Munk differential equations leads to a matrix framework based on transfer matrices, which can be extended to stacks of diffusing layers, stacks of nonscattering films, and stacks of scattering and non-scatterings films as a generalization of the Williams-Clapper model for prediction of the reflectance of paper photographs, each of these configurations being illustrated through various examples. This paper also exposes the limitsof the two flux approach and shows that the matrix formalism extends in a straightforward manner to multiflux models, where the size of the matrices is increased

Geometric, Variational Discretization of Continuum Theories

This study derives geometric, variational discretizations of continuum theories arising in fluid dynamics, magnetohydrodynamics (MHD), and the dynamics of complex fluids. A central role in these discretizations is played by the geometric formulation of fluid dynamics, which views solutions to the governing equations for perfect fluid flow as geodesics on the group of volume-preserving diffeomorphisms of the fluid domain. Inspired by this framework, we construct a finite-dimensional approximation to the diffeomorphism group and its Lie algebra, thereby permitting a variational temporal discretization of geodesics on the spatially discretized diffeomorphism group. The extension to MHD and complex fluid flow is then made through an appeal to the theory of Euler-Poincar\'{e} systems with advection, which provides a generalization of the variational formulation of ideal fluid flow to fluids with one or more advected parameters. Upon deriving a family of structured integrators for these systems, we test their performance via a numerical implementation of the update schemes on a cartesian grid. Among the hallmarks of these new numerical methods are exact preservation of momenta arising from symmetries, automatic satisfaction of solenoidal constraints on vector fields, good long-term energy behavior, robustness with respect to the spatial and temporal resolution of the discretization, and applicability to irregular meshes