The restriction of the Ising model to a layer

We discuss the status of recent Gibbsian descriptions of the restriction (projection) of the Ising phases to a layer. We concentrate on the projection of the two-dimensional low temperature Ising phases for which we prove a variational principle.Comment: 38 page

Measures and dynamics of entangled states

We develop an original approach for the quantitative characterisation of the entanglement properties of, possibly mixed, bi- and multipartite quantum states of arbitrary finite dimension. Particular emphasis is given to the derivation of reliable estimates which allow for an efficient evaluation of a specific entanglement measure, concurrence, for further implementation in the monitoring of the time evolution of multipartite entanglement under incoherent environment coupling. The flexibility of the technical machinery established here is illustrated by its implementation for different, realistic experimental scenarios.Comment: Physics Reports, in pres

Efficient enumeration of solutions produced by closure operations

In this paper we address the problem of generating all elements obtained by the saturation of an initial set by some operations. More precisely, we prove that we can generate the closure of a boolean relation (a set of boolean vectors) by polymorphisms with a polynomial delay. Therefore we can compute with polynomial delay the closure of a family of sets by any set of "set operations": union, intersection, symmetric difference, subsets, supersets $\dots$). To do so, we study the $Membership_{\mathcal{F}}$ problem: for a set of operations $\mathcal{F}$, decide whether an element belongs to the closure by $\mathcal{F}$ of a family of elements. In the boolean case, we prove that $Membership_{\mathcal{F}}$ is in P for any set of boolean operations $\mathcal{F}$. When the input vectors are over a domain larger than two elements, we prove that the generic enumeration method fails, since $Membership_{\mathcal{F}}$ is NP-hard for some $\mathcal{F}$. We also study the problem of generating minimal or maximal elements of closures and prove that some of them are related to well known enumeration problems such as the enumeration of the circuits of a matroid or the enumeration of maximal independent sets of a hypergraph. This article improves on previous works of the same authors.Comment: 30 pages, 1 figure. Long version of the article arXiv:1509.05623 of the same name which appeared in STACS 2016. Final version for DMTCS journa

On association in regression: the coefficient of determination revisited

Universal coefficients of determination are investigated which quantify the strength of the relation between a vector of dependent variables Y and a vector of independent covariates X. They are defined as measures of dependence between Y and X through theta(x), with theta(x) parameterizing the conditional distribution of Y given X=x. If theta(x) involves unknown coefficients gamma the definition is conditional on gamma, and in practice gamma, respectively the coefficient of determination has to be estimated. The estimates of quantities we propose generalize R^2 in classical linear regression and are also related to other definitions previously suggested. Our definitions apply to generalized regression models with arbitrary link functions as well as multivariate and nonparametric regression. The definition and use of the proposed coefficients of determination is illustrated for several regression problems with simulated and real data sets

On non-monotonic Choquet integrals as aggregation functions

This paper deals with non-monotonic Choquet integral, a generalization of the regular Choquet integral. The discrete non-monotonic Choquet integral is considered under the viewpoint of aggregation. In particular we give an axiomatic characterization of the class of non-monotonic Choquet integrals.We show how the Shapley index, in contrast with the monotonic case, can assume positive values if the criterion is in average a benefit, depending on its effect in all the possible coalition coalitions, and negative values in the opposite case of a cost criterion.

Influence functions, followers and command games

We study and compare two frameworks: a model of influence, and command games. In the influence model, in which players are to make a certain acceptance/rejection decision, due to influence of other players, the decision of a player may be different from his inclination. We study a relation between two central concepts of this model: influence function, and follower function. We deliver sufficient and necessary conditions for a function to be a follower function, and we describe the structure of the set of all influence functions that lead to a given follower function. In the command structure introduced by Hu and Shapley, for each player a simple game called the command game is built. One of the central concepts of this model is the concept of command function. We deliver sufficient and necessary conditions for a function to be a command function,and describe the minimal sets generating a normal command game. We also study the relation between command games and influence functions. A sufficient and necessary condition for the equivalence between an influence function and a normal command game is delivered.influence function, follower function, lower and upper inverses, kernel, command game, command function, minimal sets generating a command game