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Integer sequence #A127682 Marichal, Jean-Luc Diverse speeches and writings (2007) Number of non-isomorphic (i.e. defined up to a rotation and a reflection) maximal independent sets of the n-cycle graph having at least one symmetry axis. Detailed reference viewed: 40 (4 UL)Axiomatic characterizations of generalized values Marichal, Jean-Luc ; ; in Discrete Applied Mathematics (2007), 155(1), 26-43 In the framework of cooperative game theory, the concept of generalized value, which is an extension of that of value, has been recently proposed to measure the overall influence of coalitions in games ... [more ▼] In the framework of cooperative game theory, the concept of generalized value, which is an extension of that of value, has been recently proposed to measure the overall influence of coalitions in games. Axiomatizations of two classes of generalized values, namely probabilistic generalized values and generalized semivalues, which extend probabilistic values and semivalues, respectively, are first proposed. The axioms we utilize are based on natural extensions of axioms involved in the axiomatizations of values. In the second half of the paper, special instances of generalized semivalues are also axiomatized. [less ▲] Detailed reference viewed: 84 (2 UL)Counting non-isomorphic maximal independent sets of the n-cycle graph Bisdorff, Raymond ; Marichal, Jean-Luc Scientific Conference (2007, January) It is known that the number of maximal independent sets of the $n$-cycle graph $C_n$ is given by the $n$th term of the Perrin sequence. The action of the automorphism group of $C_n$ on the family of these ... [more ▼] It is known that the number of maximal independent sets of the $n$-cycle graph $C_n$ is given by the $n$th term of the Perrin sequence. The action of the automorphism group of $C_n$ on the family of these maximal independent sets partitions this family into disjoint orbits, which represent the non-isomorphic (i.e., defined up to a rotation and a reflection) maximal independent sets. We provide exact formulas for the total number of orbits and the number of orbits having a given number of isomorphic representatives. We also provide exact formulas for the total number of unlabelled (i.e., defined up to a rotation) maximal independent sets and the number of unlabelled maximal independent sets having a given number of isomorphic representatives. It turns out that these formulas involve both Perrin and Padovan sequences. [less ▲] Detailed reference viewed: 46 (9 UL)Derivative relationships between volume and surface area of compact regions in Rd Marichal, Jean-Luc ; in Rocky Mountain Journal of Mathematics (2007), 37(2), 551-571 We explore the idea that the derivative of the volume, V, of a region in R^d with respect to r equals its surface area, A, where r=d V/A. We show that the families of regions for which this formula for r ... [more ▼] We explore the idea that the derivative of the volume, V, of a region in R^d with respect to r equals its surface area, A, where r=d V/A. We show that the families of regions for which this formula for r is valid, which we call homogeneous families, include all the families of similar regions. We determine equivalent conditions for a family to be homogeneous, provide examples of homogeneous families made up of non-similar regions, and offer a geometric interpretation of r in a few cases. [less ▲] Detailed reference viewed: 24 (0 UL)Cumulative distribution functions and moments of lattice polynomials Marichal, Jean-Luc in Statistics & Probability Letters (2006), 76(12), 1273-1279 We give the cumulative distribution functions, the expected values, and the moments of lattice polynomials when regarded as real functions. Since lattice polynomial functions include order statistics, our ... [more ▼] We give the cumulative distribution functions, the expected values, and the moments of lattice polynomials when regarded as real functions. Since lattice polynomial functions include order statistics, our results encompass the corresponding formulas for order statistics. [less ▲] Detailed reference viewed: 100 (3 UL)Weighted lattice polynomials Marichal, Jean-Luc in Proc. 11th Int. Conf. on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU 2006), Paris, France, July 2-7, 2006 (2006, July) We define the concept of weighted lattice polynomials as lattice polynomials constructed from both variables and parameters. We provide equivalent forms of these functions in an arbitrary bounded ... [more ▼] We define the concept of weighted lattice polynomials as lattice polynomials constructed from both variables and parameters. We provide equivalent forms of these functions in an arbitrary bounded distributive lattice. We also show that these functions include the class of discrete Sugeno integrals and that they are characterized by a remarkable median based decomposition formula. [less ▲] Detailed reference viewed: 82 (0 UL)Cumulative distribution functions and moments of weighted lattice polynomials Marichal, Jean-Luc Presentation (2006, June 05) In the first part of this presentation we define the concept of weighted lattice polynomials as lattice polynomials constructed from both variables and parameters. We then show that, in any bounded ... [more ▼] In the first part of this presentation we define the concept of weighted lattice polynomials as lattice polynomials constructed from both variables and parameters. We then show that, in any bounded distributive lattice, these functions can always be written in conjunctive and disjunctive normal forms. We also show that these functions include the class of discrete Sugeno integrals and that they are characterized by a remarkable median based decomposition formula. In the second part we give the cumulative distribution functions, the expected values, and the moments of weighted lattice polynomials when regarded as real functions. Since weighted lattice polynomial functions include Sugeno integrals, lattice polynomial functions, and order statistics, our results encompass the corresponding formulas for these particular functions. We then conclude with some applications of our results to the reliability analysis of coherent systems. [less ▲] Detailed reference viewed: 34 (0 UL)Axiomatic characterizations of probabilistic and cardinal-probabilistic interaction indices ; ; Marichal, Jean-Luc in Games and Economic Behavior (2006), 55(1), 72-99 In the framework of cooperative game theory, the concept of interaction index, which can be regarded as an extension of that of value, has been recently proposed to measure the interaction phenomena among ... [more ▼] In the framework of cooperative game theory, the concept of interaction index, which can be regarded as an extension of that of value, has been recently proposed to measure the interaction phenomena among players. Axiomatizations of two classes of interaction indices, namely probabilistic interaction indices and cardinal-probabilistic interaction indices, generalizing probabilistic values and semivalues, respectively, are first proposed. The axioms we utilize are based on natural generalizations of axioms involved in the axiomatizations of values. In the second half of the paper, existing instances of cardinal-probabilistic interaction indices encountered thus far in the literature are also axiomatized. [less ▲] Detailed reference viewed: 90 (1 UL)The weighted lattice polynomials as aggregation functions Marichal, Jean-Luc in Proc. 63rd Meeting of the Eur. Working Group "Multiple Criteria Decision Aiding" (MCDA 63) (2006, March) We define the concept of weighted lattice polynomials as lattice polynomials constructed from both variables and parameters. We provide equivalent forms of these functions in an arbitrary bounded ... [more ▼] We define the concept of weighted lattice polynomials as lattice polynomials constructed from both variables and parameters. We provide equivalent forms of these functions in an arbitrary bounded distributive lattice. We also show that these functions include the class of discrete Sugeno integrals and that they are characterized by a remarkable median based decomposition formula. [less ▲] Detailed reference viewed: 30 (1 UL)Cumulative distribution functions and moments of weighted lattice polynomials Marichal, Jean-Luc in Fodor, János; Klement, Erich Peter; Roubens, Marc (Eds.) Proc. 27th Linz Seminar on Fuzzy Set Theory (LINZ 2006): Preferences, Games and Decisions (2006, February) We give the cumulative distribution functions, the expected values, and the moments of weighted lattice polynomials when regarded as real functions. Since weighted lattice polynomial functions include ... [more ▼] We give the cumulative distribution functions, the expected values, and the moments of weighted lattice polynomials when regarded as real functions. Since weighted lattice polynomial functions include Sugeno integrals, lattice polynomial functions, and order statistics, our results encompass the corresponding formulas for these particular functions. [less ▲] Detailed reference viewed: 39 (1 UL)Fonctions d'agrégation pour la décision Marichal, Jean-Luc in Bouyssou, Denis; Dubois, Didier; Pirlot, Marc (Eds.) et al Concepts et Méthodes pour l'Aide à la Décision Vol. 3 (2006) Detailed reference viewed: 50 (4 UL)A complete description of comparison meaningful functions Marichal, Jean-Luc ; ; in Montseny, Eduard; Sobrevilla, Pilar (Eds.) Proceedings of the Joint 4th Conference of the European Society for Fuzzy Logic and Technology and the 11th Rencontres Francophones sur la Logique Floue et ses Applications, Barcelona, Spain, September 7-9, 2005. (2005, September) Comparison meaningful functions acting on some real interval E are completely described as transformed coordinate projections on minimal invariant subsets. The case of monotone comparison meaningful ... [more ▼] Comparison meaningful functions acting on some real interval E are completely described as transformed coordinate projections on minimal invariant subsets. The case of monotone comparison meaningful functions is further specified. Several already known results for comparison meaningful functions and invariant functions are [less ▲] Detailed reference viewed: 81 (0 UL)Entropy of bi-capacities ; Marichal, Jean-Luc in Montseny, Eduard; Sobrevilla, Pilar (Eds.) Proceedings of the Joint 4th Conference of the European Society for Fuzzy Logic and Technology and the 11th Rencontres Francophones sur la Logique Floue et ses Applications, Barcelona, Spain, September 7-9, 2005. (2005, September) The notion of Shannon entropy, recently generalized to capacities, is extended to bi-capacities and its main properties are studied. Detailed reference viewed: 58 (3 UL)An extension of the Shannon entropy to bi-capacities ; Marichal, Jean-Luc in Proc. 3rd Int. Summer School on Aggregation Operators and their Applications (AGOP 2005), Lugano, Switzerland, July 10-15, 2005 (2005, July) The notion of Shannon entropy, recently generalized to capacities, is extended to bi-capacities and its main properties are studied. Detailed reference viewed: 49 (1 UL)A complete description of comparison meaningful functions Marichal, Jean-Luc ; ; in Proc. 3rd Int. Summer School on Aggregation Operators and their Applications (AGOP 2005), Lugano, Switzerland (2005, July) Comparison meaningful functions acting on some real interval E are completely described as transformed coordinate projections on minimal invariant subsets. The case of monotone comparison meaningful ... [more ▼] Comparison meaningful functions acting on some real interval E are completely described as transformed coordinate projections on minimal invariant subsets. The case of monotone comparison meaningful functions is further specified. Several already known results for comparison meaningful functions and invariant functions are obtained as consequences of our description. [less ▲] Detailed reference viewed: 76 (1 UL)An axiomatic approach to the definition of the entropy of a discrete Choquet capacity ; Marichal, Jean-Luc ; in Information Sciences (2005), 172(1-2), 131-153 To extend the classical Shannon entropy to non-additive measures, Marichal recently introduced the concept of generalized entropy for discrete Choquet capacities. We provide a first axiomatization of this ... [more ▼] To extend the classical Shannon entropy to non-additive measures, Marichal recently introduced the concept of generalized entropy for discrete Choquet capacities. We provide a first axiomatization of this new concept on the basis of three axioms: the symmetry property, a boundary condition for which the entropy reduces to the Shannon entropy, and a generalized version of the well-known recursivity property. We also show that this generalized entropy fulfills several properties considered as requisites for defining an entropy-like measure. Lastly, we provide an interpretation of it in the framework of aggregation by the discrete Choquet integral. [less ▲] Detailed reference viewed: 230 (5 UL)A complete description of comparison meaningful functions Marichal, Jean-Luc ; ; in Aequationes Mathematicae (2005), 69(3), 309-320 Comparison meaningful functions acting on some real interval E are completely described as transformed coordinate projections on minimal invariant subsets. The case of monotone comparison meaningful ... [more ▼] Comparison meaningful functions acting on some real interval E are completely described as transformed coordinate projections on minimal invariant subsets. The case of monotone comparison meaningful functions is further specified. Several already known results for comparison meaningful functions and invariant functions are obtained as consequences of our description. [less ▲] Detailed reference viewed: 73 (4 UL)Derivative relationships between volume and surface area of compact regions in Rd Marichal, Jean-Luc Presentation (2005, April 08) Detailed reference viewed: 18 (1 UL)Sorting multiattribute alternatives: The TOMASO method Marichal, Jean-Luc ; ; in Computers and Operations Research (2005), 32(4), 861-877 We analyze a recently proposed ordinal sorting procedure (Tomaso) for the assignment of alternatives to graded classes and we present a freeware constructed from this procedure. We illustrate it by two ... [more ▼] We analyze a recently proposed ordinal sorting procedure (Tomaso) for the assignment of alternatives to graded classes and we present a freeware constructed from this procedure. We illustrate it by two examples, and do some testing in order to show its usefulness. [less ▲] Detailed reference viewed: 124 (4 UL)Proc. of the 61st Meeting of the Eur. Working Group "Multiple Criteria Decision Aiding" (Luxembourg, March 10-12, 2005) - Theme: Preference Modelling Bisdorff, Raymond ; Marichal, Jean-Luc ; Book published by SMA - University of Luxembourg (2005) Detailed reference viewed: 48 (6 UL) |
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