Let a, b, c be relatively prime positive integers such that a^2＋ b^2= c^2. Jesmanowicz＇conjecture on Pythagorean numbers states that for any positive integer N, the Diophantine equation（aN）x＋（b N）y=（cN）zhas n...Let a, b, c be relatively prime positive integers such that a^2＋ b^2= c^2. Jesmanowicz＇conjecture on Pythagorean numbers states that for any positive integer N, the Diophantine equation（aN）x＋（b N）y=（cN）zhas no positive solution（x, y, z） other than x = y = z = 2. In this paper, we prove this conjecture for the case that a or b is a power of 2.展开更多
In emergency decision making(EDM),it is necessary to generate an effective alternative quickly.Case-based reasoning(CBR)has been applied to EDM;however,choosing the most suitable case from a set of similar cases after...In emergency decision making(EDM),it is necessary to generate an effective alternative quickly.Case-based reasoning(CBR)has been applied to EDM;however,choosing the most suitable case from a set of similar cases after case retrieval remains challenging.This study proposes a dynamic method based on case retrieval and group decision making(GDM),called dynamic casebased reasoning group decision making(CBRGDM),for emergency alternative generation.In the proposed method,first,similar historical cases are identified through case similarity measurement.Then,evaluation information provided by group decision makers for similar cases is aggregated based on regret theory,and comprehensive perceived utilities for the similar cases are obtained.Finally,the most suitable historical case is obtained from the case similarities and the comprehensive perceived utilities for similar historical cases.The method is then applied to an example of a gas explosion in a coal company in China.The results show that the proposed method is feasible and effective in EDM.The advantages of the proposed method are verified based on comparisons with existing methods.In particular,dynamic CBRGDM can adjust the emergency alternative according to changing emergencies.The results of application of dynamic CBRGDM to a gas explosion and comparison with existing methods verify its feasibility and practicability.展开更多
This socialized environment among educated and developed people causes themto focusmore on their appearance and health,which turns them towards medical-related treatments,leading us to discuss anti-aging treatment met...This socialized environment among educated and developed people causes themto focusmore on their appearance and health,which turns them towards medical-related treatments,leading us to discuss anti-aging treatment methods for each age group,particularly for urban people who are interested in this.Some anti-aging therapies are used to address the alterations brought on by aging in human life without the need for surgery or negative effects.Five anti-aging therapies such as microdermabrasion or dermabrasion,laser resurfacing anti-aging skin treatments,chemical peels,dermal fillers for aged skin,and botox injections are considered in this study.Based on the criteria of safety risk,investment cost,customer happiness,and side effects,the optimal alternative is picked.As a result,a NormalWiggly Hesitant Pythagorean Fuzzy Set(NWHPFS)is constructed and used in Multi-Criteria Decision-Making(MCDM)using traditional wavy mathematical approaches.The entropy approach is utilized to determine weight values,and the Normal Wiggly Hesitant Pythagorean-VlseKriterijumska Optimizacija I Kompromisno Resenje(NWHPF-VIKOR)method is utilized to rank alternatives using MCDM methodologies.Sensitivity analysis and comparative analysis were performed to ensure the robustness and reliability of the proposed method.The smart final choice will undoubtedly assist Decision Makers(DM)in making the right judgments,and the MCDM approach will undoubtedly assist individuals in understanding the medicine.展开更多
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
18世纪的法国传教士若瑟·玛利·阿米奥(Joseph Maric Amiot),是民族音乐学历史中真正的先驱,是第一位全面写作有关中国音乐的权威性文章的欧洲人。他不仅是一位天才的音乐家,在北京居住的那些年代里,他对中国音乐文献研究之深...18世纪的法国传教士若瑟·玛利·阿米奥(Joseph Maric Amiot),是民族音乐学历史中真正的先驱,是第一位全面写作有关中国音乐的权威性文章的欧洲人。他不仅是一位天才的音乐家,在北京居住的那些年代里,他对中国音乐文献研究之深,在外国人士中也是史无前例的。他的著作曾立即产生持久的影响,这在以后研究中国音乐的学术界中是显而易见的。 阿米奥研究中国音乐的主要著作《论中国古代和现代音乐》(Mémoire sar la musique des Chinois, tant anciens quc modernes),1779年在巴黎出版,1780年重印。他的一些其它短小文章也付印过,但还有许多并没有出版,其中就包括一篇对上述著作的补遗。在这篇补遗中,他提供了新的材料,并且答复了法国评论家们对他的评论。 本文旨在重点论述阿米奥在古代文化中关于毕达哥拉斯定律法(Pythagorean tuning)起源的争论中的作用。这场争论是由于让——菲力普·拉莫(Jean—Philippe Rameau)在《实用音乐辞典》(Code de musique practique)中,以阿米奥早期手稿为依据,错误地描绘了中国古代定律法所引起的。后来,其他18世纪认为毕达哥拉斯定律法起源于希腊、埃 及和中国的理论家让——亚克·卢梭(Jean—Jacques Rousseau)、德尼斯·狄德罗(DenisDiderot)。展开更多
Due to the numerous variables to take into account as well as the inherent ambiguity and uncertainty,evaluating educational institutions can be difficult.The concept of a possibility Pythagorean fuzzy hypersoft set(pP...Due to the numerous variables to take into account as well as the inherent ambiguity and uncertainty,evaluating educational institutions can be difficult.The concept of a possibility Pythagorean fuzzy hypersoft set(pPyFHSS)is more flexible in this regard than other theoretical fuzzy set-like models,even though some attempts have been made in the literature to address such uncertainties.This study investigates the elementary notions of pPyFHSS including its set-theoretic operations union,intersection,complement,OR-and AND-operations.Some results related to these operations are also modified for pPyFHSS.Additionally,the similarity measures between pPyFHSSs are formulated with the assistance of numerical examples and results.Lastly,an intelligent decision-assisted mechanism is developed with the proposal of a robust algorithm based on similarity measures for solving multi-attribute decision-making(MADM)problems.A case study that helps the decision-makers assess the best educational institution is discussed to validate the suggested system.The algorithmic results are compared with the most pertinent model to evaluate the adaptability of pPyFHSS,as it generalizes the classical possibility fuzzy set-like theoretical models.Similarly,while considering significant evaluating factors,the flexibility of pPyFHSS is observed through structural comparison.展开更多
The Pythagorean Bézier hodograph curves are Bézier curves {x(t), y(t)}, whose hodograph (first-order parametric derivative) components satisfy the Phythagorean condition x’2(t) +y’2(t) = σ’(t) for some p...The Pythagorean Bézier hodograph curves are Bézier curves {x(t), y(t)}, whose hodograph (first-order parametric derivative) components satisfy the Phythagorean condition x’2(t) +y’2(t) = σ’(t) for some polynomial σ(t). For nth degree PB curve (n is odd), its offset curve is represented by rational Bézier curve with (2n - 1)th degree and arc length by polynomial. Specially, the properties of cubic PB curve are studied, its geometric features are discussed and its quasi-Hermite interpolating curve and GC1 composite cubic PB curve are also constructed.展开更多
Pythagorean fuzzy set(PFS) can provide more flexibility than intuitionistic fuzzy set(IFS) for handling uncertain information, and PFS has been increasingly used in multi-attribute decision making problems. This paper...Pythagorean fuzzy set(PFS) can provide more flexibility than intuitionistic fuzzy set(IFS) for handling uncertain information, and PFS has been increasingly used in multi-attribute decision making problems. This paper proposes a new multiattribute group decision making method based on Pythagorean uncertain linguistic variable Hamy mean(PULVHM) operator and VIKOR method. Firstly, we define operation rules and a new aggregation operator of Pythagorean uncertain linguistic variable(PULV) and explore some properties of the operator.Secondly, taking the decision makers' hesitation degree into account, a new score function is defined, and we further develop a new group decision making approach integrated with VIKOR method. Finally, an investment example is demonstrated to elaborate the validity of the proposed method. Sensibility analysis and comprehensive comparisons with another two methods are performed to show the stability and advantage of our method.展开更多
The primary goal of this research is to determine the optimal agricultural field selection that would most effectively support manufacturing producers in manufacturing production while accounting for unpredictability ...The primary goal of this research is to determine the optimal agricultural field selection that would most effectively support manufacturing producers in manufacturing production while accounting for unpredictability and reliability in their decision-making.The PFS is known to address the levels of participation and non-participation.To begin,we introduce the novel concept of a PFZN,which is a hybrid structure of Pythagorean fuzzy sets and the ZN.The PFZN is graded in terms of membership and non-membership,as well as reliability,which provides a strong advice in real-world decision support concerns.The PFZN is a useful tool for dealing with uncertainty in decision-aid problems.The PFZN is a practical way for dealing with such uncertainties in decision-aid problems.The list of aggregation operators:PFZN Einstein weighted averaging and PFZN Einstein weighted geometric,is established under the novel Pythagorean fuzzy ZNs.It is a more precise mathematical instrument for dealing with precision and uncertainty.The core of this research is to develop a numerical algorithmto tackle the uncertainty in real-life problems using PFZNs.To show the applicability and effectiveness of the proposed algorithm,we illustrate the numerical case study related to determining the optimal agricultural field.The main purpose of this work is to describe the extended EDAS approach,then compare the proposed methodology with many other methodologies now in use,and then demonstrate how the suggested methodology may be applied to real-world problems.In addition,the final ranking results that were obtained by the devised techniques weremore efficient and dependable in comparison to the results provided by other methods presented in the literature.展开更多
Hypersoft set theory is a most advanced form of soft set theory and an innovative mathematical tool for dealing with unclear complications.Pythagorean fuzzy hypersoft set(PFHSS)is the most influential and capable leew...Hypersoft set theory is a most advanced form of soft set theory and an innovative mathematical tool for dealing with unclear complications.Pythagorean fuzzy hypersoft set(PFHSS)is the most influential and capable leeway of the hypersoft set(HSS)and Pythagorean fuzzy soft set(PFSS).It is also a general form of the intuitionistic fuzzy hypersoft set(IFHSS),which provides a better and more perfect assessment of the decision-making(DM)process.The fundamental objective of this work is to enrich the precision of decision-making.A novel mixed aggregation operator called Pythagorean fuzzy hypersoft Einstein weighted geometric(PFHSEWG)based on Einstein’s operational laws has been developed.Some necessary properties,such as idempotency,boundedness,and homogeneity,have been presented for the anticipated PFHSEWG operator.Multi-criteria decision-making(MCDM)plays an active role in dealing with the complications of manufacturing design for material selection.However,conventional methods of MCDM usually produce inconsistent results.Based on the proposed PFHSEWG operator,a robust MCDM procedure for material selection in manufacturing design is planned to address these inconveniences.The expected MCDM method for material selection(MS)of cryogenic storing vessels has been established in the real world.Significantly,the planned model for handling inaccurate data based on PFHSS is more operative and consistent.展开更多
基金Supported by Grant in Aid for JSPS Fellows(Grant No.25484)
文摘Let a, b, c be relatively prime positive integers such that a^2＋ b^2= c^2. Jesmanowicz＇conjecture on Pythagorean numbers states that for any positive integer N, the Diophantine equation（aN）x＋（b N）y=（cN）zhas no positive solution（x, y, z） other than x = y = z = 2. In this paper, we prove this conjecture for the case that a or b is a power of 2.
基金partly supported by the National Natural Science Foundation of China under the Grant Nos.71371053 and 71902034Humanities and Social Sciences Foundation of Chinese Ministry of Education,No.20YJC630229+1 种基金Humanities and Social Science Foundation of Fujian Province,No.FJ2019B079Science and Technology Development Center of Chinese Ministry of Education.No.2018A0I019.
文摘In emergency decision making(EDM),it is necessary to generate an effective alternative quickly.Case-based reasoning(CBR)has been applied to EDM;however,choosing the most suitable case from a set of similar cases after case retrieval remains challenging.This study proposes a dynamic method based on case retrieval and group decision making(GDM),called dynamic casebased reasoning group decision making(CBRGDM),for emergency alternative generation.In the proposed method,first,similar historical cases are identified through case similarity measurement.Then,evaluation information provided by group decision makers for similar cases is aggregated based on regret theory,and comprehensive perceived utilities for the similar cases are obtained.Finally,the most suitable historical case is obtained from the case similarities and the comprehensive perceived utilities for similar historical cases.The method is then applied to an example of a gas explosion in a coal company in China.The results show that the proposed method is feasible and effective in EDM.The advantages of the proposed method are verified based on comparisons with existing methods.In particular,dynamic CBRGDM can adjust the emergency alternative according to changing emergencies.The results of application of dynamic CBRGDM to a gas explosion and comparison with existing methods verify its feasibility and practicability.
基金funded by the Korean Government(MSIT)Grant NRF-2022R1C1C1006671.
文摘This socialized environment among educated and developed people causes themto focusmore on their appearance and health,which turns them towards medical-related treatments,leading us to discuss anti-aging treatment methods for each age group,particularly for urban people who are interested in this.Some anti-aging therapies are used to address the alterations brought on by aging in human life without the need for surgery or negative effects.Five anti-aging therapies such as microdermabrasion or dermabrasion,laser resurfacing anti-aging skin treatments,chemical peels,dermal fillers for aged skin,and botox injections are considered in this study.Based on the criteria of safety risk,investment cost,customer happiness,and side effects,the optimal alternative is picked.As a result,a NormalWiggly Hesitant Pythagorean Fuzzy Set(NWHPFS)is constructed and used in Multi-Criteria Decision-Making(MCDM)using traditional wavy mathematical approaches.The entropy approach is utilized to determine weight values,and the Normal Wiggly Hesitant Pythagorean-VlseKriterijumska Optimizacija I Kompromisno Resenje(NWHPF-VIKOR)method is utilized to rank alternatives using MCDM methodologies.Sensitivity analysis and comparative analysis were performed to ensure the robustness and reliability of the proposed method.The smart final choice will undoubtedly assist Decision Makers(DM)in making the right judgments,and the MCDM approach will undoubtedly assist individuals in understanding the medicine.
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18世纪的法国传教士若瑟·玛利·阿米奥(Joseph Maric Amiot),是民族音乐学历史中真正的先驱,是第一位全面写作有关中国音乐的权威性文章的欧洲人。他不仅是一位天才的音乐家,在北京居住的那些年代里,他对中国音乐文献研究之深,在外国人士中也是史无前例的。他的著作曾立即产生持久的影响,这在以后研究中国音乐的学术界中是显而易见的。 阿米奥研究中国音乐的主要著作《论中国古代和现代音乐》(Mémoire sar la musique des Chinois, tant anciens quc modernes),1779年在巴黎出版,1780年重印。他的一些其它短小文章也付印过,但还有许多并没有出版,其中就包括一篇对上述著作的补遗。在这篇补遗中,他提供了新的材料,并且答复了法国评论家们对他的评论。 本文旨在重点论述阿米奥在古代文化中关于毕达哥拉斯定律法(Pythagorean tuning)起源的争论中的作用。这场争论是由于让——菲力普·拉莫(Jean—Philippe Rameau)在《实用音乐辞典》(Code de musique practique)中,以阿米奥早期手稿为依据,错误地描绘了中国古代定律法所引起的。后来,其他18世纪认为毕达哥拉斯定律法起源于希腊、埃 及和中国的理论家让——亚克·卢梭(Jean—Jacques Rousseau)、德尼斯·狄德罗(DenisDiderot)。
基金supported by the Deanship of Graduate Studies and Scientific Research at Qassim University(QU-APC-2024-9/1).
文摘Due to the numerous variables to take into account as well as the inherent ambiguity and uncertainty,evaluating educational institutions can be difficult.The concept of a possibility Pythagorean fuzzy hypersoft set(pPyFHSS)is more flexible in this regard than other theoretical fuzzy set-like models,even though some attempts have been made in the literature to address such uncertainties.This study investigates the elementary notions of pPyFHSS including its set-theoretic operations union,intersection,complement,OR-and AND-operations.Some results related to these operations are also modified for pPyFHSS.Additionally,the similarity measures between pPyFHSSs are formulated with the assistance of numerical examples and results.Lastly,an intelligent decision-assisted mechanism is developed with the proposal of a robust algorithm based on similarity measures for solving multi-attribute decision-making(MADM)problems.A case study that helps the decision-makers assess the best educational institution is discussed to validate the suggested system.The algorithmic results are compared with the most pertinent model to evaluate the adaptability of pPyFHSS,as it generalizes the classical possibility fuzzy set-like theoretical models.Similarly,while considering significant evaluating factors,the flexibility of pPyFHSS is observed through structural comparison.
文摘The Pythagorean Bézier hodograph curves are Bézier curves {x(t), y(t)}, whose hodograph (first-order parametric derivative) components satisfy the Phythagorean condition x’2(t) +y’2(t) = σ’(t) for some polynomial σ(t). For nth degree PB curve (n is odd), its offset curve is represented by rational Bézier curve with (2n - 1)th degree and arc length by polynomial. Specially, the properties of cubic PB curve are studied, its geometric features are discussed and its quasi-Hermite interpolating curve and GC1 composite cubic PB curve are also constructed.
基金supported by the National Natural Science Foundation of China(61402260,61473176)Taishan Scholar Project of Shandong Province(TSQN201812092)
文摘Pythagorean fuzzy set(PFS) can provide more flexibility than intuitionistic fuzzy set(IFS) for handling uncertain information, and PFS has been increasingly used in multi-attribute decision making problems. This paper proposes a new multiattribute group decision making method based on Pythagorean uncertain linguistic variable Hamy mean(PULVHM) operator and VIKOR method. Firstly, we define operation rules and a new aggregation operator of Pythagorean uncertain linguistic variable(PULV) and explore some properties of the operator.Secondly, taking the decision makers' hesitation degree into account, a new score function is defined, and we further develop a new group decision making approach integrated with VIKOR method. Finally, an investment example is demonstrated to elaborate the validity of the proposed method. Sensibility analysis and comprehensive comparisons with another two methods are performed to show the stability and advantage of our method.
文摘The primary goal of this research is to determine the optimal agricultural field selection that would most effectively support manufacturing producers in manufacturing production while accounting for unpredictability and reliability in their decision-making.The PFS is known to address the levels of participation and non-participation.To begin,we introduce the novel concept of a PFZN,which is a hybrid structure of Pythagorean fuzzy sets and the ZN.The PFZN is graded in terms of membership and non-membership,as well as reliability,which provides a strong advice in real-world decision support concerns.The PFZN is a useful tool for dealing with uncertainty in decision-aid problems.The PFZN is a practical way for dealing with such uncertainties in decision-aid problems.The list of aggregation operators:PFZN Einstein weighted averaging and PFZN Einstein weighted geometric,is established under the novel Pythagorean fuzzy ZNs.It is a more precise mathematical instrument for dealing with precision and uncertainty.The core of this research is to develop a numerical algorithmto tackle the uncertainty in real-life problems using PFZNs.To show the applicability and effectiveness of the proposed algorithm,we illustrate the numerical case study related to determining the optimal agricultural field.The main purpose of this work is to describe the extended EDAS approach,then compare the proposed methodology with many other methodologies now in use,and then demonstrate how the suggested methodology may be applied to real-world problems.In addition,the final ranking results that were obtained by the devised techniques weremore efficient and dependable in comparison to the results provided by other methods presented in the literature.
基金funding this work through General Research Project under Grant No.GRP/93/43.
文摘Hypersoft set theory is a most advanced form of soft set theory and an innovative mathematical tool for dealing with unclear complications.Pythagorean fuzzy hypersoft set(PFHSS)is the most influential and capable leeway of the hypersoft set(HSS)and Pythagorean fuzzy soft set(PFSS).It is also a general form of the intuitionistic fuzzy hypersoft set(IFHSS),which provides a better and more perfect assessment of the decision-making(DM)process.The fundamental objective of this work is to enrich the precision of decision-making.A novel mixed aggregation operator called Pythagorean fuzzy hypersoft Einstein weighted geometric(PFHSEWG)based on Einstein’s operational laws has been developed.Some necessary properties,such as idempotency,boundedness,and homogeneity,have been presented for the anticipated PFHSEWG operator.Multi-criteria decision-making(MCDM)plays an active role in dealing with the complications of manufacturing design for material selection.However,conventional methods of MCDM usually produce inconsistent results.Based on the proposed PFHSEWG operator,a robust MCDM procedure for material selection in manufacturing design is planned to address these inconveniences.The expected MCDM method for material selection(MS)of cryogenic storing vessels has been established in the real world.Significantly,the planned model for handling inaccurate data based on PFHSS is more operative and consistent.
