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Bisymmetric and quasitrivial operations: characterizations and enumerations Devillet, Jimmy in Aequationes Mathematicae (in press) We investigate the class of bisymmetric and quasitrivial binary operations on a given set and provide various characterizations of this class as well as the subclass of bisymmetric, quasitrivial, and ... [more ▼] We investigate the class of bisymmetric and quasitrivial binary operations on a given set and provide various characterizations of this class as well as the subclass of bisymmetric, quasitrivial, and order-preserving binary operations. We also determine explicitly the sizes of these classes when the set is finite. [less ▲] Detailed reference viewed: 80 (36 UL)A classification of polynomial functions satisfying the Jacobi identity over integral domains Marichal, Jean-Luc ; in Aequationes Mathematicae (2017), 91(4), 601-618 The Jacobi identity is one of the properties that are used to define the concept of Lie algebra and in this context is closely related to associativity. In this paper we provide a complete description of ... [more ▼] The Jacobi identity is one of the properties that are used to define the concept of Lie algebra and in this context is closely related to associativity. In this paper we provide a complete description of all bivariate polynomials that satisfy the Jacobi identity over infinite integral domains. Although this description depends on the characteristic of the domain, it turns out that all these polynomials are of degree at most one in each indeterminate. [less ▲] Detailed reference viewed: 88 (11 UL)On the generalized associativity equation Marichal, Jean-Luc ; Teheux, Bruno in Aequationes Mathematicae (2017), 91(2), 265-277 The so-called generalized associativity functional equation G(J(x,y),z) = H(x,K(y,z)) has been investigated under various assumptions, for instance when the unknown functions G, H, J, and K are real ... [more ▼] The so-called generalized associativity functional equation G(J(x,y),z) = H(x,K(y,z)) has been investigated under various assumptions, for instance when the unknown functions G, H, J, and K are real, continuous, and strictly monotonic in each variable. In this note we investigate the following related problem: given the functions J and K, find every function F that can be written in the form F(x,y,z) = G(J(x,y),z) = H(x,K(y,z)) for some functions G and H. We show how this problem can be solved when any of the inner functions J and K has the same range as one of its sections. [less ▲] Detailed reference viewed: 145 (26 UL)On spectral analysis in varieties containing the solutions of inhomogeneous linear functional equations Kiss, Gergely ; in Aequationes Mathematicae (2017) The aim of the paper is to investigate the solutions of special inhomogeneous linear functional equations using spectral analysis in a translation invariant closed linear subspace of additive ... [more ▼] The aim of the paper is to investigate the solutions of special inhomogeneous linear functional equations using spectral analysis in a translation invariant closed linear subspace of additive/multiadditive functions containing the restrictions of the solutions to finitely generated fields. The application of spectral analysis in some related varieties is a new and important trend in the theory of functional equations; especially they have successful applications in the case of homogeneous linear functional equations. The foundations of the theory can be found in Kiss and Varga (Aequat Math 88(1):151–162, 2014) and Kiss and Laczkovich (Aequat Math 89(2):301–328, 2015). We are going to adopt the main theoretical tools to solve some inhomogeneous problems due to Koclȩga-Kulpa and Szostok (Ann Math Sylesianae 22:27–40, 2008), see also Koclȩga-Kulpa and Szostok (Georgian Math J 16:725–736, 2009; Acta Math Hung 130(4):340–348, 2011). They are motivated by quadrature rules of approximate integration. [less ▲] Detailed reference viewed: 43 (1 UL)On spectral synthesis in varieties containing the solutions of inhomogeneous linear functional equations Kiss, Gergely ; in Aequationes Mathematicae (2017) As a continuation of our previous work [2] the aim of the recent paper is to investigate the solutions of special inhomogeneous linear functional equations by using spectral synthesis in translation ... [more ▼] As a continuation of our previous work [2] the aim of the recent paper is to investigate the solutions of special inhomogeneous linear functional equations by using spectral synthesis in translation invariant closed linear subspaces of additive/multiadditive functions containing the restrictions of the solutions to finitely generated fields. The idea is based on the fundamental work of [5]. Using spectral analysis in some related varieties we can prove the existence of special solutions (automorphisms) of the functional equation but spectral synthesis allows us to describe the entire space of solutions on a large class of finitely generated fields. It is spanned by the so-called exponential monomials which can be given in terms of automorphisms of CC and differential operators. We apply the general theory to some inhomogeneous problems motivated by quadrature rules of approximate integration [8], see also [7, 9]. [less ▲] Detailed reference viewed: 33 (3 UL)Power series solutions of Tarski's associativity law and of the cyclic associativity law Schölzel, Karsten ; Tomaschek, Jörg in Aequationes Mathematicae (2016), 90(2), 411425 Detailed reference viewed: 20 (3 UL)A classification of barycentrically associative polynomial functions Marichal, Jean-Luc ; ; in Aequationes Mathematicae (2015), 89(5), 1281-1291 We describe the class of polynomial functions which are barycentrically associative over an infinite commutative integral domain. Detailed reference viewed: 88 (14 UL)Linear functional equations, differential operators and spectral synthesis. Kiss, Gergely ; in Aequationes Mathematicae (2015), 89(2), 301328 Detailed reference viewed: 52 (10 UL)Existence of nontrivial solutions of linear functional equations. Kiss, Gergely ; in Aequationes Mathematicae (2014), 88(1-2), 151162 We investigate the existence of a solution of linear functional equations. Detailed reference viewed: 42 (5 UL)The equality problem in the class of conjugate means Dascal, Judita ; in Aequationes Mathematicae (2012), 84(1), 77-90 Detailed reference viewed: 30 (3 UL)A classification of bisymmetric polynomial functions over integral domains of characteristic zero Marichal, Jean-Luc ; Mathonet, Pierre in Aequationes Mathematicae (2012), 84(1-2), 125-136 We describe the class of n-variable polynomial functions that satisfy Aczél's bisymmetry property over an arbitrary integral domain of characteristic zero with identity. Detailed reference viewed: 85 (9 UL)Formal solutions of the generalized Dhombres functional equation with value one at zero Tomaschek, Jörg ; in Aequationes Mathematicae (2012), 83(1), 117-126 We study formal solutions f of the generalized Dhombres functional equation f(zf(z))=φ(f(z)). Unlike in the situation where f(0) = w 0 and w0∈C∖E where E denotes the complex roots of 1, which were already ... [more ▼] We study formal solutions f of the generalized Dhombres functional equation f(zf(z))=φ(f(z)). Unlike in the situation where f(0) = w 0 and w0∈C∖E where E denotes the complex roots of 1, which were already discussed, we investigate solutions f where f(0) = 1. To obtain solutions in this case we use new methods which differ from the already existing ones [less ▲] Detailed reference viewed: 30 (0 UL)Axiomatizations of quasi-Lovász extensions of pseudo-Boolean functions Couceiro, Miguel ; Marichal, Jean-Luc in Aequationes Mathematicae (2011), 82(3), 213-231 We introduce the concept of quasi-Lov\'asz extension as being a mapping $f\colon I^n\to\R$ defined on a nonempty real interval $I$ containing the origin and which can be factorized as $f(x_1,\ldots,x_n)=L ... [more ▼] We introduce the concept of quasi-Lov\'asz extension as being a mapping $f\colon I^n\to\R$ defined on a nonempty real interval $I$ containing the origin and which can be factorized as $f(x_1,\ldots,x_n)=L(\varphi(x_1),\ldots,\varphi(x_n))$, where $L$ is the Lov\'asz extension of a pseudo-Boolean function $\psi\colon\{0,1\}^n\to\R$ (i.e., the function $L\colon\R^n\to\R$ whose restriction to each simplex of the standard triangulation of $[0,1]^n$ is the unique affine function which agrees with $\psi$ at the vertices of this simplex) and $\varphi\colon I\to\R$ is a nondecreasing function vanishing at the origin. These functions appear naturally within the scope of decision making under uncertainty since they subsume overall preference functionals associated with discrete Choquet integrals whose variables are transformed by a given utility function. To axiomatize the class of quasi-Lov\'asz extensions, we propose generalizations of properties used to characterize the Lov\'asz extensions, including a comonotonic version of modularity and a natural relaxation of homogeneity. A variant of the latter property enables us to axiomatize also the class of symmetric quasi-Lov\'asz extensions, which are compositions of symmetric Lov\'asz extensions with $1$-place nondecreasing odd functions. [less ▲] Detailed reference viewed: 41 (2 UL)Self-commuting lattice polynomial functions on chains Couceiro, Miguel ; Lehtonen, Erkko in Aequationes Mathematicae (2011), 81(3), 263-278 We provide sufficient conditions for a lattice polynomial function to be self-commuting. We explicitly describe self-commuting polynomial functions on chains. Detailed reference viewed: 30 (0 UL)Quasi-polynomial functions over bounded distributive lattices Couceiro, Miguel ; Marichal, Jean-Luc in Aequationes Mathematicae (2010), 80(3), 319-334 In [6] the authors introduced the notion of quasi-polynomial function as being a mapping $f\colon X^n\to X$ defined and valued on a bounded chain $X$ and which can be factorized as $f(x_1,\ldots,x_n)=p ... [more ▼] In [6] the authors introduced the notion of quasi-polynomial function as being a mapping $f\colon X^n\to X$ defined and valued on a bounded chain $X$ and which can be factorized as $f(x_1,\ldots,x_n)=p(\varphi(x_1),\ldots,\varphi(x_n))$, where $p$ is a polynomial function (i.e., a combination of variables and constants using the chain operations $\wedge$ and $\vee$) and $\varphi$ is an order-preserving map. In the current paper we study this notion in the more general setting where the underlying domain and codomain sets are, possibly different, bounded distributive lattices, and where the inner function is not necessarily order-preserving. These functions appear naturally within the scope of decision making under uncertainty since, as shown in this paper, they subsume overall preference functionals associated with Sugeno integrals whose variables are transformed by a given utility function. To axiomatize the class of quasi-polynomial functions, we propose several generalizations of well-established properties in aggregation theory, as well as show that some of the characterizations given in [6] still hold in this general setting. Moreover, we investigate the so-called transformed polynomial functions (essentially, compositions of unary mappings with polynomial functions) and show that, under certain conditions, they reduce to quasi-polynomial functions. [less ▲] Detailed reference viewed: 57 (2 UL)Solving Chisini's functional equation Marichal, Jean-Luc in Aequationes Mathematicae (2010), 79(3), 237-260 We investigate the n-variable real functions G that are solutions of the Chisini functional equation F(x) = F(G(x),...,G(x)), where F is a given function of n real variables. We provide necessary and ... [more ▼] We investigate the n-variable real functions G that are solutions of the Chisini functional equation F(x) = F(G(x),...,G(x)), where F is a given function of n real variables. We provide necessary and sufficient conditions on F for the existence and uniqueness of solutions. When F is nondecreasing in each variable, we show in a constructive way that if a solution exists then a nondecreasing and idempotent solution always exists. We also provide necessary and sufficient conditions on F for the existence of continuous solutions and we show how to construct such a solution. We finally discuss a few applications of these results. [less ▲] Detailed reference viewed: 61 (10 UL)Axiomatizations of quasi-polynomial functions on bounded chains Couceiro, Miguel ; Marichal, Jean-Luc in Aequationes Mathematicae (2009), 78(1-2), 195-213 Two emergent properties in aggregation theory are investigated, namely horizontal maxitivity and comonotonic maxitivity (as well as their dual counterparts) which are commonly defined by means of certain ... [more ▼] Two emergent properties in aggregation theory are investigated, namely horizontal maxitivity and comonotonic maxitivity (as well as their dual counterparts) which are commonly defined by means of certain functional equations. We completely describe the function classes axiomatized by each of these properties, up to weak versions of monotonicity in the cases of horizontal maxitivity and minitivity. While studying the classes axiomatized by combinations of these properties, we introduce the concept of quasi-polynomial function which appears as a natural extension of the well-established notion of polynomial function. We give further axiomatizations for this class both in terms of functional equations and natural relaxations of homogeneity and median decomposability. As noteworthy particular cases, we investigate those subclasses of quasi-term functions and quasi-weighted maximum and minimum functions, and provide characterizations accordingly. [less ▲] Detailed reference viewed: 39 (4 UL)Meaningful aggregation functions mapping ordinal scales into an ordinal scale: a state of the art Marichal, Jean-Luc ; in Aequationes Mathematicae (2009), 77(3), 207-236 We present an overview of the meaningful aggregation functions mapping ordinal scales into an ordinal scale. Three main classes are discussed, namely order invariant functions, comparison meaningful ... [more ▼] We present an overview of the meaningful aggregation functions mapping ordinal scales into an ordinal scale. Three main classes are discussed, namely order invariant functions, comparison meaningful functions on a single ordinal scale, and comparison meaningful functions on independent ordinal scales. It appears that the most prominent meaningful aggregation functions are lattice polynomial functions, that is, functions built only on projections and minimum and maximum operations. [less ▲] Detailed reference viewed: 51 (1 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: 36 (4 UL)On an axiomatization of the quasi-arithmetic mean values without the symmetry axiom Marichal, Jean-Luc in Aequationes Mathematicae (2000), 59(1-2), 74-83 Kolmogoroff and Nagumo proved that the quasi-arithmetic means correspond exactly to the decomposable sequences of continuous, symmetric, strictly increasing in each variable and reflexive functions. We ... [more ▼] Kolmogoroff and Nagumo proved that the quasi-arithmetic means correspond exactly to the decomposable sequences of continuous, symmetric, strictly increasing in each variable and reflexive functions. We replace decomposability and symmetry in this characterization by a generalization of the decomposability. [less ▲] Detailed reference viewed: 56 (4 UL) |
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