The notion of preordering, which is a generalization of the notion of ordering, has been introduced by Serre. On the other hand, the notion of round quadratic forms has been introduced by Witt. Based on these ideas, i...The notion of preordering, which is a generalization of the notion of ordering, has been introduced by Serre. On the other hand, the notion of round quadratic forms has been introduced by Witt. Based on these ideas, it is here shown that 1) a field F is formally real n-pythagorean iff the nth radical, RnF is a preordering (Theorem 2), and 2) a field F is n-pythagorean iff for any n-fold Pfister form ρ. There exists an odd integer l(>1) such that l×ρ is a round quadratic form (Theorem 8). By considering upper bounds for the number of squares on Pfister’s interpretation, these results finally lead to the main result (Theorem 10) such that the generalization of pythagorean fields coincides with the generalization of Hilbert’s 17th Problem.展开更多
In this paper, we prove the main result: Let both (K, S) and (K*, S*) be preordered fields, and let (K*, S*) be a finitely generated extension of (K, S). If K* is transcendental over K, then (K*, S*) has the weak Hilb...In this paper, we prove the main result: Let both (K, S) and (K*, S*) be preordered fields, and let (K*, S*) be a finitely generated extension of (K, S). If K* is transcendental over K, then (K*, S*) has the weak Hilbert property. This result answers negatively an open problem posed by the author in reference[1]. Moreover, some results on the weak Hilbert property are established. In this paper, we prove the main result: Let both (K, S) and (K*, S*) be preordered fields, and let (K*, S*) be a finitely generated extension of (K, S). If K* is transcendental over K, then (K*, S*) has the weak Hilbert property. This result answers negatively an open problem posed by the author in reference [1]. Moreover, some results on the weak Hilbert property are established.展开更多
This article presents a connection between fuzzy preordered structures and hyperstructures. Firstly, we introduce the notion of fuzzy preordered semigroup and then, we construct a semihypergroup associated with it, gi...This article presents a connection between fuzzy preordered structures and hyperstructures. Firstly, we introduce the notion of fuzzy preordered semigroup and then, we construct a semihypergroup associated with it, giving some properties of the associated hyperstructure. Secondly, we define the notion of fuzzy preordered ring in order to construct a fuzzy hyperring.展开更多
In this paper,the criteria set related to the priority preorders of water resources projects is introduced.A fuzzy multiple criteria group decision-making model is established,which incorporates quantitative analysis,...In this paper,the criteria set related to the priority preorders of water resources projects is introduced.A fuzzy multiple criteria group decision-making model is established,which incorporates quantitative analysis,judgments,experience and preferences of decision-makers.The model is used in practice to determine the priority preorders of five water resources projects,and the results show that the best choice can supply more new employment,domestic water and irrigation water,and has better quality.展开更多
文摘The notion of preordering, which is a generalization of the notion of ordering, has been introduced by Serre. On the other hand, the notion of round quadratic forms has been introduced by Witt. Based on these ideas, it is here shown that 1) a field F is formally real n-pythagorean iff the nth radical, RnF is a preordering (Theorem 2), and 2) a field F is n-pythagorean iff for any n-fold Pfister form ρ. There exists an odd integer l(>1) such that l×ρ is a round quadratic form (Theorem 8). By considering upper bounds for the number of squares on Pfister’s interpretation, these results finally lead to the main result (Theorem 10) such that the generalization of pythagorean fields coincides with the generalization of Hilbert’s 17th Problem.
基金Project supported by National Natural Science Foundation of China
文摘In this paper, we prove the main result: Let both (K, S) and (K*, S*) be preordered fields, and let (K*, S*) be a finitely generated extension of (K, S). If K* is transcendental over K, then (K*, S*) has the weak Hilbert property. This result answers negatively an open problem posed by the author in reference[1]. Moreover, some results on the weak Hilbert property are established. In this paper, we prove the main result: Let both (K, S) and (K*, S*) be preordered fields, and let (K*, S*) be a finitely generated extension of (K, S). If K* is transcendental over K, then (K*, S*) has the weak Hilbert property. This result answers negatively an open problem posed by the author in reference [1]. Moreover, some results on the weak Hilbert property are established.
文摘This article presents a connection between fuzzy preordered structures and hyperstructures. Firstly, we introduce the notion of fuzzy preordered semigroup and then, we construct a semihypergroup associated with it, giving some properties of the associated hyperstructure. Secondly, we define the notion of fuzzy preordered ring in order to construct a fuzzy hyperring.
文摘In this paper,the criteria set related to the priority preorders of water resources projects is introduced.A fuzzy multiple criteria group decision-making model is established,which incorporates quantitative analysis,judgments,experience and preferences of decision-makers.The model is used in practice to determine the priority preorders of five water resources projects,and the results show that the best choice can supply more new employment,domestic water and irrigation water,and has better quality.
