The Pythagorean triples (a, b | c) of planar geometry which satisfy the equation a<sup>2</sup>+b<sup>2</sup>=c<sup>2</sup> with integers (a, b, c) are generalized to 3D-Pythagorean ...The Pythagorean triples (a, b | c) of planar geometry which satisfy the equation a<sup>2</sup>+b<sup>2</sup>=c<sup>2</sup> with integers (a, b, c) are generalized to 3D-Pythagorean quadruples (a, b, c | d) of spatial geometry which satisfy the equation a<sup>2</sup>+b<sup>2</sup>+c<sup>2</sup>=d<sup>2</sup> with integers (a, b, c, d). Rules for a parametrization of the numbers (a, b, c, d) are derived and a list of all possible nonequivalent cases without common divisors up to d<sup>2</sup> is established. The 3D-Pythagorean quadruples are then generalized to 4D-Pythagorean quintuples (a, b, c, d | e) which satisfy the equation a<sup>2</sup>+b<sup>2</sup>+c<sup>2</sup>+d<sup>2</sup>=e<sup>2</sup> and a parametrization is derived. Relations to the 4-square identity are discussed which leads also to the N-dimensional case. The initial 3D- and 4D-Pythagorean numbers are explicitly calculated up to d<sup>2</sup>, respectively, e<sup>2</sup>.展开更多
针对Pythagorean模糊信息的决策问题,构建广义Pythagorean模糊信息加权有序加权平均(PF-GWOWA)算子.首先,提出PF-GWOWA算子,并证明Pythagorean模糊广义加权平均(PF-GWA)算子、Pythagorean模糊加权有序加权平均(PF-WOWA)算子与Pythagorea...针对Pythagorean模糊信息的决策问题,构建广义Pythagorean模糊信息加权有序加权平均(PF-GWOWA)算子.首先,提出PF-GWOWA算子,并证明Pythagorean模糊广义加权平均(PF-GWA)算子、Pythagorean模糊加权有序加权平均(PF-WOWA)算子与Pythagorean模糊加权平均(PF-WA)算子均为PF-GWOWA算子的特例;其次,根据G WO WA算子属性综合权重计算模型,利用P F-G WO WA算子对信息进行集结;最后,通过算例分析和传统方法对比,说明本文提出方法的合理性与有效性.展开更多
Prime integers and their generalizations play important roles in protocols for secure transmission of information via open channels of telecommunication networks. Generation of multidigit large primes in the design st...Prime integers and their generalizations play important roles in protocols for secure transmission of information via open channels of telecommunication networks. Generation of multidigit large primes in the design stage of a cryptographic system is a formidable task. Fermat primality checking is one of the simplest of all tests. Unfortunately, there are composite integers (called Carmichael numbers) that are not detectable by the Fermat test. In this paper we consider modular arithmetic based on complex integers;and provide several tests that verify the primality of real integers. Although the new tests detect most Carmichael numbers, there are a small percentage of them that escape these tests.展开更多
本文考虑决策者在实际问题中存有的多种决策心理,针对毕达哥拉斯模糊偏好下考虑双边主体损失规避和参照依赖心理行为的双边匹配问题提出一种决策方法。对于双边主体给出的毕达哥拉斯模糊偏好信息,依据TODIM(Tomada de Decisao Interativ...本文考虑决策者在实际问题中存有的多种决策心理,针对毕达哥拉斯模糊偏好下考虑双边主体损失规避和参照依赖心理行为的双边匹配问题提出一种决策方法。对于双边主体给出的毕达哥拉斯模糊偏好信息,依据TODIM(Tomada de Decisao Interativa Multicritério)思想获得双边主体相较于另一边匹配主体的总体优势度,进而构建双边主体的满意度矩阵;而后,在考虑双边主体一对一的数量匹配约束下,以实现双边主体满意度最大化为决策目标,建立多目标双边匹配决策模型;最后,通过线性加权法进一步将其转化为单目标双边匹配模型,通过模型求解获得最优双边匹配方案;一个实际供应链管理系统软件的交易匹配算例验证本方法的可行性和有效性。展开更多
针对毕达哥拉斯模糊环境下的多指标评价问题,根据决策信息的实际分布,提出一种基于数值分布的毕达哥拉斯模糊数等级划分方法,根据不同等级区间发展度及等级规模确定指标权重和密度权重,结合密度算子的思想,提出基于数值分布的激励型毕...针对毕达哥拉斯模糊环境下的多指标评价问题,根据决策信息的实际分布,提出一种基于数值分布的毕达哥拉斯模糊数等级划分方法,根据不同等级区间发展度及等级规模确定指标权重和密度权重,结合密度算子的思想,提出基于数值分布的激励型毕达哥拉斯模糊密度加权平均算子(incentive Pythagorean fuzzy density weighted average operator,I-PFDWA).将基于I-PFDWA算子的多指标评价方法运用于航空服务评价问题的结果证实了该方法的可行性和有效性.展开更多
距离测度是毕达哥拉斯(Pythagorean)模糊决策的一个基本概念,其在用权重向量进行有序集成运算、有序加权距离和相似度构造时具有重要作用。然而,早期发表于International Journal of Intelligent Systems中的文章Extension of TOPSIS to...距离测度是毕达哥拉斯(Pythagorean)模糊决策的一个基本概念,其在用权重向量进行有序集成运算、有序加权距离和相似度构造时具有重要作用。然而,早期发表于International Journal of Intelligent Systems中的文章Extension of TOPSIS to multiple criteria decision making with Pythagorean fuzzy sets,在首次引入海明距离测度时,对距离公理化条件的证明存在不妥。为此,通过反例和分析指出了其在证明公理化条件的有界性和三点不等式时存在的错误,并统筹考虑犹豫度的绝对值项和其他绝对值项,通过分情况讨论给出了海明距离测度的严格证明,进一步,证明了其推广形式(欧几里得距离测度)的公理化条件。展开更多
文摘The Pythagorean triples (a, b | c) of planar geometry which satisfy the equation a<sup>2</sup>+b<sup>2</sup>=c<sup>2</sup> with integers (a, b, c) are generalized to 3D-Pythagorean quadruples (a, b, c | d) of spatial geometry which satisfy the equation a<sup>2</sup>+b<sup>2</sup>+c<sup>2</sup>=d<sup>2</sup> with integers (a, b, c, d). Rules for a parametrization of the numbers (a, b, c, d) are derived and a list of all possible nonequivalent cases without common divisors up to d<sup>2</sup> is established. The 3D-Pythagorean quadruples are then generalized to 4D-Pythagorean quintuples (a, b, c, d | e) which satisfy the equation a<sup>2</sup>+b<sup>2</sup>+c<sup>2</sup>+d<sup>2</sup>=e<sup>2</sup> and a parametrization is derived. Relations to the 4-square identity are discussed which leads also to the N-dimensional case. The initial 3D- and 4D-Pythagorean numbers are explicitly calculated up to d<sup>2</sup>, respectively, e<sup>2</sup>.
文摘针对Pythagorean模糊信息的决策问题,构建广义Pythagorean模糊信息加权有序加权平均(PF-GWOWA)算子.首先,提出PF-GWOWA算子,并证明Pythagorean模糊广义加权平均(PF-GWA)算子、Pythagorean模糊加权有序加权平均(PF-WOWA)算子与Pythagorean模糊加权平均(PF-WA)算子均为PF-GWOWA算子的特例;其次,根据G WO WA算子属性综合权重计算模型,利用P F-G WO WA算子对信息进行集结;最后,通过算例分析和传统方法对比,说明本文提出方法的合理性与有效性.
文摘Prime integers and their generalizations play important roles in protocols for secure transmission of information via open channels of telecommunication networks. Generation of multidigit large primes in the design stage of a cryptographic system is a formidable task. Fermat primality checking is one of the simplest of all tests. Unfortunately, there are composite integers (called Carmichael numbers) that are not detectable by the Fermat test. In this paper we consider modular arithmetic based on complex integers;and provide several tests that verify the primality of real integers. Although the new tests detect most Carmichael numbers, there are a small percentage of them that escape these tests.
文摘本文考虑决策者在实际问题中存有的多种决策心理,针对毕达哥拉斯模糊偏好下考虑双边主体损失规避和参照依赖心理行为的双边匹配问题提出一种决策方法。对于双边主体给出的毕达哥拉斯模糊偏好信息,依据TODIM(Tomada de Decisao Interativa Multicritério)思想获得双边主体相较于另一边匹配主体的总体优势度,进而构建双边主体的满意度矩阵;而后,在考虑双边主体一对一的数量匹配约束下,以实现双边主体满意度最大化为决策目标,建立多目标双边匹配决策模型;最后,通过线性加权法进一步将其转化为单目标双边匹配模型,通过模型求解获得最优双边匹配方案;一个实际供应链管理系统软件的交易匹配算例验证本方法的可行性和有效性。
文摘针对毕达哥拉斯模糊环境下的多指标评价问题,根据决策信息的实际分布,提出一种基于数值分布的毕达哥拉斯模糊数等级划分方法,根据不同等级区间发展度及等级规模确定指标权重和密度权重,结合密度算子的思想,提出基于数值分布的激励型毕达哥拉斯模糊密度加权平均算子(incentive Pythagorean fuzzy density weighted average operator,I-PFDWA).将基于I-PFDWA算子的多指标评价方法运用于航空服务评价问题的结果证实了该方法的可行性和有效性.
文摘距离测度是毕达哥拉斯(Pythagorean)模糊决策的一个基本概念,其在用权重向量进行有序集成运算、有序加权距离和相似度构造时具有重要作用。然而,早期发表于International Journal of Intelligent Systems中的文章Extension of TOPSIS to multiple criteria decision making with Pythagorean fuzzy sets,在首次引入海明距离测度时,对距离公理化条件的证明存在不妥。为此,通过反例和分析指出了其在证明公理化条件的有界性和三点不等式时存在的错误,并统筹考虑犹豫度的绝对值项和其他绝对值项,通过分情况讨论给出了海明距离测度的严格证明,进一步,证明了其推广形式(欧几里得距离测度)的公理化条件。
