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combinatorial structure of the Polytope of 2-Additive Measures

IEEE TRANSACTIONS ON FUZZY SYSTEMS 2020年第11期28卷 2864-2874页

作者： Miranda, Pedro Garcia-Segador, Pedro Univ Complutense Madrid Dept Stat & Operat Res Madrid 28040 Spain Natl Stat Inst Madrid 28071 Spain

In this article, we study the polytope of 2-additive measures, an important subpolytope of the polytope of fuzzy measures. For this polytope, we obtain its combinatorial structure, namely the adjacency structure and the structure of two-dimensional faces, three-dimensional faces, and so on. Based on this information, we build a triangulation of this polytope satisfying that all simplices in the triangulation have the same volume. As a consequence, this allows a very simple and appealing way to generate points in a random way in this polytope, an interesting problem arising in the practical identification of 2-additive measures. Finally, we also derive the volume and the centroid of this polytope.

关键词： 2-additive measures combinatorial structure fuzzy measures random generation triangulation

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combinatorial structure of rigid transformations in 2D digital images

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COMPUTER VISION AND IMAGE UNDERSTANDING 2013年第4期117卷 393-408页

作者： Ngo, Phuc Kenmochi, Yukiko Passat, Nicolas Talbot, Hugues Univ Paris Est LIGM UPEMLV ESIEE CNRS Paris France Univ Strasbourg LSIIT UMR CNRS 7005 Strasbourg France

Rigid transformations are involved in a wide range of digital image processing applications. When applied on discrete images, rigid transformations are usually performed in their associated continuous space, requiring a subsequent digitization of the result. In this article, we propose to study rigid transformations of digital images as fully discrete processes. In particular, we investigate a combinatorial structure modelling the whole space of digital rigid transformations on arbitrary subset of Z(2) of size N x N. We describe this combinatorial structure, which presents a space complexity O(N-9) and we propose an algorithm enabling to construct it in linear time with respect to its space complexity. This algorithm, which handles real (i.e., non-rational) values related to the continuous transformations associated to the discrete ones, is however defined in a fully discrete form, leading to exact computation. (C) 2012 Elsevier Inc. All rights reserved.

关键词： Digital rigid transformations combinatorial structure Discrete algorithm

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combinatorial structure and iconicity in artificial whistled languages 35

Combinatorial structure and iconicity in artificial whistled...

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35th Annual Meeting of the Cognitive Science Society - Cooperative Minds: Social Interaction and Group Dynamics, CogSci 2013

作者： Verhoef, Tessa Kirby, Simon de Boer, Bart ACLC University of Amsterdam Amsterdam1012 VT Netherlands LEC University of Edinburgh EdinburghEH8 9AD United Kingdom AI lab Vrije Universiteit Brussel Brussels1050 Belgium

ISBN: (纸本)9780976831891

This article reports on an experiment in which artificial languages with whistle words for novel objects are culturally transmitted in the laboratory. The aim of this study is to investigate the origins and evolution of combinatorial structure in speech. Participants learned the whistled language and reproduced the sounds with the use of a slide whistle. Their reproductions were used as input for the next participant. Cultural transmission caused the whistled systems to become more learnable and more structured. In addition, two conditions were studied: one in which the use of iconic form-meaning mappings was possible and one in which the use of iconic mappings was experimentally made impossible, so that we could investigate the influence of iconicity on the emergence of structure. © CogSci *** rights reserved.

关键词： combinatorial structure cultural evolution duality of patterning iconicity iterated learning phonology

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Iconicity and the Emergence of combinatorial structure in Language

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COGNITIVE SCIENCE 2016年第8期40卷 1969-1994页

作者： Verhoef, Tessa Kirby, Simon de Boer, Bart Univ Calif San Diego Ctr Res Language 9500 Gilman dr La Jolla CA 92093 USA Univ Edinburgh Sch Philosophy Psychol & Language Sci Edinburgh Midlothian Scotland Vrije Univ Brussel Artificial Intelligence Lab Brussels Belgium

In language, recombination of a discrete set of meaningless building blocks forms an unlimited set of possible utterances. How such combinatorial structure emerged in the evolution of human language is increasingly being studied. It has been shown that it can emerge when languages culturally evolve and adapt to human cognitive biases. How the emergence of combinatorial structure interacts with the existence of holistic iconic form-meaning mappings in a language is still unknown. The experiment presented in this paper studies the role of iconicity and human cognitive learning biases in the emergence of combinatorial structure in artificial whistled languages. Participants learned and reproduced whistled words for novel objects with the use of a slide whistle. Their reproductions were used as input for the next participant, to create transmission chains and simulate cultural transmission. Two conditions were studied: one in which the persistence of iconic form-meaning mappings was possible and one in which this was experimentally made impossible. In both conditions, cultural transmission caused the whistled languages to become more learnable and more structured, but this process was slightly delayed in the first condition. Our findings help to gain insight into when and how words may lose their iconic origins when they become part of an organized linguistic system.

关键词： Cultural evolution Cognitive biases Iterated learning combinatorial structure Iconicity Language evolution

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combinatorial structures associated with Lie algebras of finite dimension

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LINEAR ALGEBRA AND ITS APPLICATIONS 2004年第1-3期389卷 43-61页

作者： Carriazo, A Fernández, LM Núñez, J Univ Seville Fac Math Dept Geometry & Topol Seville 41080 Spain

Given a Lie algebra of finite dimension, with a selected basis of it, we show in this paper that it is possible to associate it with a combinatorial structure, of dimension 2, in general. In some particular cases, this structure is reduced to a weighted graph. We characterize such graphs, according to they have 3-cycles or not. (C) 2004 Elsevier Inc. All rights reserved.

关键词： combinatorial structure weighted graph oriented digraph bracket product Lie algebra

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combinatorial structures of three vertices and Lie algebras

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INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 2012年第13-14期89卷 1879-1900页

作者： Caceres, J. Ceballos, M. Nunez, J. Puertas, M. L. Tenorio, A. F. Univ Seville Fac Matemat Dept Geometria & Topol E-41080 Seville Spain Univ Almeria Dept Estadist & Matemat Aplicada La Canada De San Urbano 04120 Almeria Spain Univ Pablo de Olavide Escuela Politecn Super Dept Econ Metodos Cuantitativos & Hist Econ Seville 41013 Spain

This paper shows a characterization of digraphs of three vertices associated with Lie algebras, as well as determining the list of isomorphism classes for Lie algebras associated with these digraphs. Additionally, we introduce and implement two algorithmic procedures related to this study: the first is devoted to draw, if exists, the digraph associated with a given Lie algebra;whereas the other corresponds to the converse problem and allows us to test if a given digraph is associated or not with a Lie algebra. Finally, we give the complete list of all non-isomorphic combinatorial structures of three vertices associated with Lie algebras and we study the type of Lie algebra associated with each configuration.

关键词： digraph combinatorial structure Lie algebra isomorphism class algorithm

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combinatorial structures of max-type functions characterizing the optimal solution of the equivalence problem

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Electronic Notes in Discrete Mathematics 2001年 8卷 1-4页

作者： Pickl, S. Institute of Mathematics Center for Applied Computer Science University of Cologne Germany

This paper presents a solution to the so called equivalence problem which was introduced in [2]. The equivalence problem consists of a semismooth system of equations which has the following form: (Formula Presented) This system can be formulated in a more simplified representation by ũ = M̃(x, y, z, w)(x, y, z, w) ^T = M̃(x̃)x̃ ^T The vector ũ can be interpreted as possible control parameters of a time-discrete system, whereas the vector x̃ can be seen as bargaining solution of a cooperative game. In [2] the bargaining solution is identical to the τ-value, which was introduced in [3]. For the special case of three actors the τ-value lies always in the core, if we assume 1-convexity. Under these assumptions, all properties are expressed by the general formulation of (2) where the functions f, g and h are of the form ξ(u) = min{u ₁, . . . , u _n} = - max{-u ₁, . . . , -u _n}. By exploiting the combinatorial structure of max-type functions we can show that a solution of (2) exists and may be found via Newton type methods which are treated in [1].

关键词： τ-Value combinatorial structure Cooperative Dynamic Games Equivalence Problem Lipschitz Property Max-Type Function

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Interaction indices for games on combinatorial structures with forbidden coalitions

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2011年第1期214卷 99-108页

作者： Labreuche, Christophe Thales Res & Technol F-91767 Palaiseau France

The notion of interaction among a set of players has been defined on the Boolean lattice and Cartesian products of lattices. The aim of this paper is to extend this concept to combinatorial structures with forbidden coalitions. The set of feasible coalitions is supposed to fulfil some general conditions. This general representation encompasses convex geometries, antimatroids, augmenting systems and distributive lattices. Two axiomatic characterizations are obtained. They both assume that the Shapley value is already defined on the combinatorial structures. The first one is restricted to pairs of players and is based on a generalization of a recursivity axiom that uniquely specifies the interaction index from the Shapley value when all coalitions are permitted. This unique correspondence cannot be maintained when some coalitions are forbidden. From this, a weak recursivity axiom is defined. We show that this axiom together with linearity and dummy player are sufficient to specify the interaction index. The second axiomatic characterization is obtained from the linearity, dummy player and partnership axioms. An interpretation of the interaction index in the context of surplus sharing is also proposed. Finally, our interaction index is instantiated to the case of games under precedence constraints. (C) 2011 Elsevier B.V. All rights reserved.

关键词： Game theory Cooperative games Interaction index combinatorial structure Shapley value

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Algorithmic method to obtain combinatorial structures associated with Leibniz algebras

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MATHEMATICS AND COMPUTERS IN SIMULATION 2016年 125卷 126-138页

作者： Ceballos, Manuel Nunez, Juan Tenorio, Angel F. Univ Seville Fac Matemat Dept Geometria & Topol Apdo 1160 E-41080 Seville Spain Univ Pablo de Olavide Escuela Politecn Super Dept Econ Metodos Cuantitat & Ha Econ Ctra Utrera Km 1 Seville 41013 Spain

In this paper, we introduce an algorithmic process to associate Leibniz algebras with combinatorial structures. More concretely, we have designed an algorithm to automatize this method and to obtain the restrictions over the structure coefficients for the law of the Leibniz algebra and so determine its associated combinatorial structure. This algorithm has been implemented with the symbolic computation package Maple. Moreover, we also present another algorithm (and its implementation) to draw the combinatorial structure associated with a given Leibniz algebra, when such a structure is a (pseudo)digraph. As application of these algorithms, we have studied what (pseudo)digraphs are associated with low-dimensional Leibniz algebras by determination of the restrictions over edge weights (i.e. structure coefficients) in the corresponding combinatorial structures. (C) 2014 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.

关键词： Pseudodigraph combinatorial structure Leibniz algebra structure theory Algorithm

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Homomorphic Image Orders on combinatorial structures

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ORDER-A JOURNAL ON THE THEORY OF ORDERED SETS AND ITS APPLICATIONS 2015年第2期32卷 205-226页

作者： Huczynska, Sophie Ruskuc, Nik Univ St Andrews Sch Math & Stat St Andrews KY16 9SS Fife Scotland

combinatorial structures have been considered under various orders, including substructure order and homomorphism order. In this paper, we investigate the homomorphic image order, corresponding to the existence of a surjective homomorphism between two structures. We distinguish between strong and induced forms of the order and explore how they behave in the context of different common combinatorial structures. We focus on three aspects: antichains and partial well-order, the joint preimage property and the dual amalgamation property. The two latter properties are natural analogues of the well-known joint embedding property and amalgamation property, and are investigated here for the first time.

关键词： Partial order Homomorphism combinatorial structure

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