References of "Mathonet, Pierre"
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See detailA classification of bisymmetric polynomial functions over integral domains of characteristic zero
Marichal, Jean-Luc; Mathonet, Pierre ULg

in Aequationes Mathematicae (2012), 84

We describe the class of n-variable polynomial functions that satisfy Aczel’s bisymmetry property over an arbitrary integral domain of characteristic zero with identity.

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See detailSymmetric approximations of pseudo-Boolean functions with applications to influence indexes
Marichal, Jean-Luc; Mathonet, Pierre ULg

in Applied Mathematics Letters (2012), 25(8), 1121-1126

We introduce an index for measuring the influence of the $k$th smallest variable on a pseudo-Boolean function. This index is defined from a weighted least squares approximation of the function by linear ... [more ▼]

We introduce an index for measuring the influence of the $k$th smallest variable on a pseudo-Boolean function. This index is defined from a weighted least squares approximation of the function by linear combinations of order statistic functions. We give explicit expressions for both the index and the approximation and discuss some properties of the index. Finally, we show that this index subsumes the concept of system signature in engineering reliability and that of cardinality index in decision making. [less ▲]

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See detailOn osp(p+1,q+1|2r)-equivariant quantizations
Leuther, Thomas ULg; Mathonet, Pierre ULg; Radoux, Fabian ULg

in Journal of Geometry & Physics (2012), 62

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See detailOn signature-based expressions of system reliability
Marichal, Jean-Luc; Mathonet, Pierre ULg; Waldhauser, Tamás

in Journal of Multivariate Analysis (2011), 102(10), 1410-1416

The concept of signature was introduced by Samaniego for systems whose components have i.i.d. lifetimes. This concept proved to be useful in the analysis of theoretical behaviors of systems. In particular ... [more ▼]

The concept of signature was introduced by Samaniego for systems whose components have i.i.d. lifetimes. This concept proved to be useful in the analysis of theoretical behaviors of systems. In particular, it provides an interesting signature-based representation of the system reliability in terms of reliabilities of k-out-of-n systems. In the non-i.i.d. case, we show that, at any time, this representation still holds true for every coherent system if and only if the component states are exchangeable. We also discuss conditions for obtaining an alternative representation of the system reliability in which the signature is replaced by its non-i.i.d. extension. Finally, we discuss conditions for the system reliability to have both representations. [less ▲]

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See detailA description of n-ary semigroups polynomial-derived from integral domains
Marichal, Jean-Luc; Mathonet, Pierre ULg

in Semigroup Forum (2011)

We provide a complete classification of the n-ary semigroup structures defined by polynomial functions over infinite commutative integral domains with identity, thus generalizing Glazek and Gleichgewicht ... [more ▼]

We provide a complete classification of the n-ary semigroup structures defined by polynomial functions over infinite commutative integral domains with identity, thus generalizing Glazek and Gleichgewicht's classification of ternary semigroups. [less ▲]

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See detailProjectively equivariant quantizations over the superspace R^{p|q}
Mathonet, Pierre ULg; Radoux, Fabian ULg

in Letters in Mathematical Physics (2011), 98(3), 311-331

We investigate the concept of projectively equivariant quantization in the framework of super projective geometry. When the projective superalgebra pgl(p+1|q) is simple, our result is similar to the ... [more ▼]

We investigate the concept of projectively equivariant quantization in the framework of super projective geometry. When the projective superalgebra pgl(p+1|q) is simple, our result is similar to the classical one in the purely even case: we prove the existence and uniqueness of the quantization except in some critical situations. When the projective superalgebra is not simple (i.e. in the case of pgl(n|n)\not\cong sl(n|n)), we show the existence of a one-parameter family of equivariant quantizations. We also provide explicit formulas in terms of a generalized divergence operator acting on supersymmetric tensor fields. [less ▲]

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See detailWeighted Banzhaf power and interaction indexes through weighted approximations of games
Marichal, Jean-Luc; Mathonet, Pierre ULg

in European Journal of Operational Research (2011), 211

The Banzhaf power index was introduced in cooperative game theory to measure the real power of players in a game. The Banzhaf interaction index was then proposed to measure the interaction degree inside ... [more ▼]

The Banzhaf power index was introduced in cooperative game theory to measure the real power of players in a game. The Banzhaf interaction index was then proposed to measure the interaction degree inside coalitions of players. It was shown that the power and interaction indexes can be obtained as solutions of a standard least squares approximation problem for pseudo-Boolean functions. Considering certain weighted versions of this approximation problem, we define a class of weighted interaction indexes that generalize the Banzhaf interaction index. We show that these indexes define a subclass of the family of probabilistic interaction indexes and study their most important properties. Finally, we give an interpretation of the Banzhaf and Shapley interaction indexes as centers of mass of this subclass of interaction indexes. [less ▲]

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See detailMeasuring the interactions among variables of functions over the unit hypercube
Marichal, Jean-Luc; Mathonet, Pierre ULg

in Journal of Mathematical Analysis & Applications (2011), 380

By considering a least squares approximation of a given square integrable function f: [0,1]^n\to\R by a multilinear polynomial of a specified degree, we define an index which measures the overall ... [more ▼]

By considering a least squares approximation of a given square integrable function f: [0,1]^n\to\R by a multilinear polynomial of a specified degree, we define an index which measures the overall interaction among variables of f. This definition extends the concept of Banzhaf interaction index introduced in cooperative game theory. Our approach is partly inspired from multilinear regression analysis, where interactions among the independent variables are taken into consideration. We show that this interaction index has appealing properties which naturally generalize several properties of the Banzhaf interaction index. In particular, we interpret this index as an expected value of the difference quotients of f or, under certain natural conditions on f, as an expected value of the derivatives of f. Finally, we discuss a few applications of the interaction index in aggregation function theory. [less ▲]

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See detailExtensions of system signatures to dependent lifetimes: Explicit expressions and interpretations
Marichal, Jean-Luc; Mathonet, Pierre ULg

in Journal of Multivariate Analysis (2011), 102

The concept of system signature was introduced by Samaniego for systems whose components have i.i.d. lifetimes. We consider its extension to the continuous dependent case and give an explicit expression ... [more ▼]

The concept of system signature was introduced by Samaniego for systems whose components have i.i.d. lifetimes. We consider its extension to the continuous dependent case and give an explicit expression for this extension as a difference of weighted means of the structure function values. We then derive a formula for the computation of the coefficients of these weighted means in the special case of independent continuous lifetimes. Finally, we interpret this extended concept of signature through a natural least squares approximation problem. [less ▲]

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See detailMeasuring the Interactions among Variables of Functions over the Unit hypercube
Marichal, Jean-Luc; Mathonet, Pierre ULg

in Modeling Decisions for Artificial Intelligence 7th International Conference, MDAI 2010 (2010)

By considering a least squares approximation of a given square integrable function f:[0,1]^n to R by a multilinear polynomial of a specified degree, we define an index which measures the overall ... [more ▼]

By considering a least squares approximation of a given square integrable function f:[0,1]^n to R by a multilinear polynomial of a specified degree, we define an index which measures the overall interaction among variables of f. This definition extends the concept of Banzhaf interaction index introduced in cooperative game theory. Our approach is partly inspired from multilinear regression analysis, where interactions among the independent variables are taken into consideration. We show that this interaction index has appealing properties which naturally generalize the properties of the Banzhaf interaction index. In particular, we interpret this index as an expected value of the difference quotients of f or, under certain natural conditions on f, as an expected value of the derivatives of f. These interpretations show a strong analogy between the introduced interaction index and the overall importance index defined by Grabisch and Labreuche. Finally, we discuss a few applications of the interaction index. [less ▲]

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See detailMeasuring the Influence of the kth Largest Variable on Functions over the Unit Hypercube
Marichal, Jean-Luc; Mathonet, Pierre ULg

in Modeling Decisions for Artificial Intelligence 7th International Conference, MDAI 2010 (2010)

By considering a least squares approximation of a given square integrable function f: [0,1]^n\to R by a shifted L-statistic function (a shifted linear combination of order statistics), we define an index ... [more ▼]

By considering a least squares approximation of a given square integrable function f: [0,1]^n\to R by a shifted L-statistic function (a shifted linear combination of order statistics), we define an index which measures the global influence of the k-th largest variable on f. We show that this influence index has appealing properties and we interpret it as an average value of the difference quotient of f in the direction of the k-th largest variable or, under certain natural conditions on f, as an average value of the derivative of f in the direction of the k-th largest variable. We also discuss a few applications of this index in statistics and aggregation theory. [less ▲]

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See detailExistence of natural and conformally invariant quantizations of arbitrary symbols
Mathonet, Pierre ULg; Radoux, Fabian ULg

in Journal of Nonlinear Mathematical Physics (2010), 17

A quantization can be seen as a way to construct a differential operator with prescribed principal symbol. The map from the space of symbols to the space of differential operators is moreover required to be ... [more ▼]

A quantization can be seen as a way to construct a differential operator with prescribed principal symbol. The map from the space of symbols to the space of differential operators is moreover required to be a linear bijection. In general, there is no natural quantization procedure, that is, spaces of symbols and of differential operators are not equivalent, if the action of local diffeomorphisms is taken into account. However, considering manifolds endowed with additional structures, one can seek for quantizations that depend on this additional structure and that are natural if the dependence with respect to the structure is taken into account. The existence of such a quantization was proved recently in a series of papers in the context of projective geometry. Here, we show that the construction of the quantization based on Cartan connections can be adapted from projective to pseudo-conformal geometry to yield the natural and conformally invariant quantization for arbitrary symbols, outside some critical situations. [less ▲]

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See detailEntanglement equivalence of N-qubit symmetric states
Mathonet, Pierre ULg; Krins, Stéphanie ULg; Godefroid, M. et al

in Physical Review. A (2010), 81

We study the interconversion of multipartite symmetric N-qubit states under stochastic local operations and classical communication (SLOCC). We demonstrate that if two symmetric states can be connected ... [more ▼]

We study the interconversion of multipartite symmetric N-qubit states under stochastic local operations and classical communication (SLOCC). We demonstrate that if two symmetric states can be connected with a nonsymmetric invertible local operation (ILO), then they belong necessarily to the separable, W, or Greenberger-Horne-Zeilinger (GHZ) entanglement class, establishing a practical method of discriminating subsets of entanglement classes. Furthermore, we prove that there always exists a symmetric ILO connecting any pair of symmetric N-qubit states equivalent under SLOCC, simplifying the requirements for experimental implementations of local interconversion of those states. [less ▲]

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See detailOn natural and conformally invariant quantizations
Mathonet, Pierre ULg; Radoux, Fabian ULg

in Journal of the London Mathematical Society (2009), 80(1), 256-272

The concept of conformally equivariant quantization was introduced by Duval, Lecomte and Ovsienko for manifolds endowed with flat conformal structures. They obtained results of existence and uniqueness ... [more ▼]

The concept of conformally equivariant quantization was introduced by Duval, Lecomte and Ovsienko for manifolds endowed with flat conformal structures. They obtained results of existence and uniqueness (up to normalization) of such a quantization procedure. A natural generalization of this concept is to seek for a quantization procedure, over a manifold M, that depends on a pseudo-Riemannian metric, is natural, and is invariant with respect to a conformal change of the metric. The existence of such a procedure was conjectured by P. Lecomte and proved by C. Duval and V. Ovsienko for symbols of degree at most 2 and by S. Loubon Djounga for symbols of degree 3. In two recent papers, we investigated the question of existence of projectively equivariant quantizations using the framework of Cartan connections. Here we will show how the formalism developed in these works adapts in order to deal with the conformally equivariant quantization for symbols of degree at most 3. This will allow us to easily recover earlier results on the subject. We will then show how it can be modified in order to prove the existence of conformally equivariant quantizations for symbols of degree 4. [less ▲]

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See detailOperational Families of Entanglement Classes for Symmetric N-Qubit States
Bastin, Thierry ULg; Krins, Stéphanie ULg; Mathonet, Pierre ULg et al

in Physical Review Letters (2009), 103

We solve the entanglement classification under stochastic local operations and classical communication (SLOCC) for all multipartite symmetric states in the general N-qubit case. For this purpose, we ... [more ▼]

We solve the entanglement classification under stochastic local operations and classical communication (SLOCC) for all multipartite symmetric states in the general N-qubit case. For this purpose, we introduce 2 parameters playing a crucial role, namely, the diversity degree and the degeneracy configuration of a symmetric state. Those parameters give rise to a simple method of identifying operational families of SLOCC entanglement classes of all symmetric N-qubit states, where the number of families grows as the partition function of the number of qubits. [less ▲]

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See detailApproximations of Lovász extensions and their induced interaction index
Marichal, Jean-Luc; Mathonet, Pierre ULg

in Discrete Applied Mathematics (2008), 156(1), 11-24

The Lovasz extension of a pseudo-Boolean function f : (0, 1)(n) -> R is defined on each simplex of the standard triangulation of [0, 1](n) as the unique affine function (f) over cap : [0, 1](n) -> R that ... [more ▼]

The Lovasz extension of a pseudo-Boolean function f : (0, 1)(n) -> R is defined on each simplex of the standard triangulation of [0, 1](n) as the unique affine function (f) over cap : [0, 1](n) -> R that interpolates f at the n + 1 vertices of the simplex. Its degree is that of the unique multilinear polynomial that expresses f. In this paper we investigate the least squares approximation problem of an arbitrary Lovasz extension (f) over cap by Lovasz extensions of (at most) a specified degree. We derive explicit expressions of these approximations. The corresponding approximation problem for pseudo-Boolean functions was investigated by Hammer and Holzman [Approximations of pseudo-Boolean functions; applications to game theory, Z. Oper. Res. 36(1) (1992) 3-21] and then solved explicitly by Grabisch et al. [Equivalent representations of set functions, Math. Oper. Res. 25(2) (2000) 157-178], giving rise to an alternative definition of Banzhaf interaction index. Similarly we introduce a new interaction index from approximations of (f) over cap and we present some of its properties. It turns out that its corresponding power index identifies with the power index introduced by Grabisch and Labreuche [How to improve acts: an alternative representation of the importance of criteria in MCDM, Internat. J. Uncertain. Fuzziness Knowledge-Based Syst. 9(2) (2001) 145-157]. (c) 2007 Elsevier B.V. All rights reserved. [less ▲]

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