Let G be a group and assume that ( A p ) p∈G is a family of algebras with identity. We have a Hopf G-coalgebra (in the sense of Turaev) if, for each pair p, q ∈ G, there is given a unital homomorphism p,q : A pq → A p ⊗ A q satisfying certain properties. Consider now the direct sum A of these algebras. It is an algebra, without identity, except when G is a finite group, but the product is non-degenerate. The maps p,q can be used to define a coproduct on A and the conditions imposed on these maps give that ( A, ) is a multiplier Hopf algebra. It is G-cograded as explained in this paper. We study these so-called group-cograded multiplier Hopf algebras. They are, as explained above, more general than the Hopf group-coalgebras as introduced by Turaev. Moreover, our point of view makes it possible to use results and techniques from the theory of multiplier Hopf algebras in the study of Hopf group-coalgebras (and generalizations). In a separate paper, we treat the quantum double in this context and we recover, in a simple and natural way (and generalize) results obtained by Zunino. In this paper, we study integrals, in general and in the case where the components are finite-dimensional. Using these ideas, we obtain most of the results of Virelizier on this subject and consider them in the framework of multiplier Hopf algebras.
Multiobjective oligopoly models are constructed. The objective of the first two models are to maximize profits and to maximize sales. In the third model, the objectives are to maximize profits and to minimize risk. Giving more weight to risk minimization decreased the profits. In all the three models, we found that the weight of the profit maximization has to be higher than a given threshold. Sufficient conditions for persistence of some multiobjective oligopolies are derived. Again, they require that the weight of profit maximization to be higher than certain value.
Multiobjective oligopoly models are constructed. The objectives of the first two models are to maximize profits and to maximize sales. In the third model the objectives are to maximize profits and to minimize risk. Giving more weight to risk minimization decreased the profits. In all three models, we found that the weight of profit maximization has to be higher than a given threshold. Again they require that the weight of profit maximization has to be higher than a certain value.
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