In this paper, a closed-loop supply chain composed of dual-channel retailers and manufacturers, a dynamic game model under the direct recovery, and an entrusted third-party recycling mode of the manufacturer is constructed. The impact of horizontal fairness concern behavior is introduced on the pricing strategies and utility of decision makers under different recycling models. The equilibrium strategy at fair neutrality is used as a reference to compare offline retails sales. Research shows that in the closed-loop supply chain of dual-channel sales, whether in the case of fair neutrality or horizontal fairness concerns, the manufacturer’s direct recycling model is superior to the entrusted third-party recycling, and the third-party recycling model is transferred by the manufacturer. In the direct recycling model, the horizontal fairness concern of offline retailers makes two retailers in the positive supply chain compete to lower the retail price in order to increase market share. Manufacturers will lower the wholesale price to encourage competition, and the price will be the horizontal fairness concern coefficient, which is negatively correlated. In the reverse supply chain, manufacturers increase the recycling rate of used products. This pricing strategy increases the utility of manufacturers and the entire supply chain system compared to fair neutral conditions, while two retailers receive diminished returns. Manufacturers, as channel managers to encourage retailers to compete for price cuts, can be coordinated through a three-way revenue sharing contract to achieve Pareto optimality.

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We prove some improvements of the classical Weil bound for one variable additive and multiplicative character sums associated to a polynomial over a finite field k = Fq for two classes of polynomials which are invariant under a large abelian group of automorphisms of the affine line A 1 k : those invariant under translation by elements of k and those invariant under homotheties with ratios in a large subgroup of the multiplicative group of k. In both cases, we are able to improve the bound by a factor of √ q over an extension of k of cardinality sufficiently large compared to the degree of f .Partially supported by P08-FQM-03894 (Junta de Andalucía), MTM2007-66929 and FEDER.

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ABSTRACT. I propose here a new concept of information based on two relevant aspects of its expression. The first related to the undeniable fact that the expression of information modifies the physical state of its receiver. The second to the arbitrariness of such physical changes. In fact, these changes are not deducible from physical laws but from a code established arbitrarily. Thus, physical information is proposed here as the capacity of producing arbitrary changes. Once defined physical information from this physical point of view, I deduce some basic properties of informed systems. These properties (renewal, self-reproducing, evolution, diversification) are immediately recognized as the attributes most characteristic of living beings, the only natural informed systems we know. I also propose here a double evaluation of information. The former is an absolute measure of the physical effects of its expression based on Einstein's probability. The second is a functional measure based on the probability that an informed systems attain a given objective as consequence of the expression of its information.