After having laid down the Axiom of Algebra, bringing the creation of the square root of -1 by Euler to the entire circle and thus authorizing a simple notation of the nth roots of unity, the author uses it to organiz...After having laid down the Axiom of Algebra, bringing the creation of the square root of -1 by Euler to the entire circle and thus authorizing a simple notation of the nth roots of unity, the author uses it to organize homogeneous divisions of the limited development of the exponential function, that is opening the way to the use of a whole bunch of new primary functions in Differential Calculus. He then shows how new supercomplex products in dimension 3 make it possible to calculate fractals whose connexity depends on the product considered. We recall the geometry of convex polygons and regular polygons.展开更多
This paper presents an analysis of the challenges in risk-based resource allocation in engineering projects.Sub-sequently,an alternative resource allocation evaluation method based on language information and informat...This paper presents an analysis of the challenges in risk-based resource allocation in engineering projects.Sub-sequently,an alternative resource allocation evaluation method based on language information and information axioms is proposed.Firstly,the evaluation team uses language information to give the evaluation information of the alternatives of risk resource allocation and provides the corresponding expected information for each resource.Secondly,according to the transformation formula of language information and fuzzy numbers,the above information is transformed into the evaluation information and expected information of the alternatives of risk-based resource allocation.Thirdly,according to the transformation formula of language information and fuzzy numbers,the above information is transformed into evalu-ation information and expectation information of alternative risk resource allocation.Finally,according to the information amount of each risk resource and the corresponding weight,the comprehensive information amount of the expected risk-based resource allocation alternatives is determined.展开更多
The Ti-axiom,the Ti-ordered axiom and Ti-pairwise axiom(i = 0,1,2,3,4) of topological ordered space are discussed and proved that they are equivalence under the certain conditions.
The facility layout problem belongs to typical multiple-attribute decision-making（MADM） problems. To make the information axiom fit to MADM problems, the original computation method for the information content is im...The facility layout problem belongs to typical multiple-attribute decision-making（MADM） problems. To make the information axiom fit to MADM problems, the original computation method for the information content is improved by increasing the satisfaction degree item. Attribute values are divided into precise type, uncertainty type and fuzzy type. For benefit, cost, fixation, and interval type attributes, the computation methods for the information content on the three types of attribute values are presented. The improved information content can reflect the system success probability and the decision-maker satisfaction degree simultaneously and evaluate the MADM problem including multiple type attribute values. Finally, as a case study, the facility layout alternatives of a welding assembly workshop are evaluated. The result verifies the validity and the feasibility of the improved information axiom on the MADM problems.展开更多
This paper reports the new progresses in the axiomatization of tensor anal- ysis, including the thought of axiomatization, the concept of generalized components, the axiom of covariant form invariability, the axiomati...This paper reports the new progresses in the axiomatization of tensor anal- ysis, including the thought of axiomatization, the concept of generalized components, the axiom of covariant form invariability, the axiomatized definition, the algebraic structure, the transformation group, and the simple calculation of generalized covariant differentia- tions. These progresses strengthen the tendency of the axiomatization of tensor analysis.展开更多
Whether a unified theory of everything(TOE)is possible or not is a philosophical question and yes or no can be chosen in a two-valued logic system.Currently the two schools are in conflict with each other.Based on the...Whether a unified theory of everything(TOE)is possible or not is a philosophical question and yes or no can be chosen in a two-valued logic system.Currently the two schools are in conflict with each other.Based on the relativity of simultaneity axiom proposed in this paper,the present author suggests to use a midway philosophy to replace the present materialist philosophy for modern sciences;then this conflict together with many other conflicts among different theories such as classical mechanics(CM),general relativity(GR),and quantum mechanics(QM)can be solved and a unified theory of everything for the world we can observe can be constructed.In this paper,the axiomatic foundation for a TOE is proposed which contains six fundamental axioms.Various problems related to these foundational issues are discussed.It is hoped that the present paper might show a new promise and a new direction for TOE which would be helpful for the further development of modern sciences.展开更多
Rough set axiomatization is one aspect of rough set study to characterize rough set theory using dependable and minimal axiom groups. Thus, rough set theory can be studied by logic and axiom system methods. The classi...Rough set axiomatization is one aspect of rough set study to characterize rough set theory using dependable and minimal axiom groups. Thus, rough set theory can be studied by logic and axiom system methods. The classic rough set theory is based on equivalent relation, but rough set theory based on reflexive and transitive relation (called quasi-ordering) has wide applications in the real world. To characterize topological rough set theory, an axiom group named RT, consisting of 4 axioms, is proposed. It is proved that the axiom group reliability in characterizing rough set theory based on similar relation is reasonable. Simultaneously, the minimization of the axiom group, which requires that each axiom is an equation and each is independent, is proved. The axiom group is helpful for researching rough set theory by logic and axiom system methods.展开更多
In this paper,a common characteristic of real number system and well ordered set is revealed and proved to be an equivalent form of the Dedekind Axiom or the Continuous Induction in R. Basing on it,we get the unified ...In this paper,a common characteristic of real number system and well ordered set is revealed and proved to be an equivalent form of the Dedekind Axiom or the Continuous Induction in R. Basing on it,we get the unified form of the mathematical induction,the transfinite induction and the continuous induction and generalize induction to totally ordered set that has the same characteristic.展开更多
In this paper,a common characteristic of real number system and well ordered set is revealed and proved to be an equivalent form of the Dedekind Axiom or the Continuous Induction in R. Basing on it,we get the unif...In this paper,a common characteristic of real number system and well ordered set is revealed and proved to be an equivalent form of the Dedekind Axiom or the Continuous Induction in R. Basing on it,we get the unified form of the mathematical induction,the transfinite induction and the continuous induction and generalize induction to totally ordered set that has the same characteristic.展开更多
Let T：X → X be an Axiom A diffeomorphism,m the Gibbs state for a Hlder continuous function ɡ. Assume that f：X → R^d is a Hlder continuous function with ∫_X^（fdm） = 0.If the components of f are cohomologously i...Let T：X → X be an Axiom A diffeomorphism,m the Gibbs state for a Hlder continuous function ɡ. Assume that f：X → R^d is a Hlder continuous function with ∫_X^（fdm） = 0.If the components of f are cohomologously independent, then there exists a positive definite symmetric matrix σ~2：=σ~2 （f ） such that S^fn √ n converges in distribution with respect to m to a Gaussian random variable with expectation 0 and covariance matrix σ~2 . Moreover, there exists a real number A 〉 0 such that, for any integer n ≥ 1,Π（ m＊（ 1√ nS f n ）,N （0,σ~2 ） ≤A√n, where m＊（1√ n S^fn）denotes the distribution of 1√ n S^fn with respect to m, and Π is the Prokhorov metric.展开更多
Rough set axiomatization is one aspect of rough set study, and the purpose is to characterize rough set theory using independable and minimal axiom groups. Thus, rough set theory can be studied by logic and axiom syst...Rough set axiomatization is one aspect of rough set study, and the purpose is to characterize rough set theory using independable and minimal axiom groups. Thus, rough set theory can be studied by logic and axiom system methods. To characterize rough set theory, an axiom group named H consisting of 4 axioms, is proposed. That validity of the axiom group in characterizing rough set theory is reasonable, is proved. Simultaneously, the minimization of the axiom group, which requires that each axiom is an inequality and each is independent, is proved. The axiom group is helpful for researching rough set theory by logic and axiom system methods. Key words rough set - lower approximation - axioms - minimization CLC number TP 18 Foundation item: Supported by the 973 National Basic Research Program of China (2002CB312106) and Science & Technology Program of Zhejiang Province (2004C31G101003)Biography: DAI Jian-hua (1977-), male, Ph. D, research direction: data mining, artificial intelligence, rough sets, evolutionary computation.展开更多
In this work we have to deal with the axiomatization of cosmology, but it is only recently that we have hit upon a new mathematical approach to capitalize on our new set identities for the basic laws of cosmology. So ...In this work we have to deal with the axiomatization of cosmology, but it is only recently that we have hit upon a new mathematical approach to capitalize on our new set identities for the basic laws of cosmology. So our proposal of settlement is that we will propose some new laws (e.g., formation of the black hole). We introduce the concept of axiom cosmology. This principle describes the cosmology which can get freedom from the notion of the induction. We present a large-scale structure model of the universe, and this leads to successfully explanation of problem of closed universe or open universe (because from the outset it is theorem and its succinct proof). In this paper we prove that the non-singular point theorem means that a singularity cannot be mathematically defined nor physical. It allows us to overcome the mysterious, physical singularity conundrum and explain meaningful antimatter annihilations for general configurations.展开更多
This article presents four (4) additions to a book on the brain’s OS published by SciRP in 2015 [1]. It is a kind of appendix to the book. Some familiarity with the earlier book is presupposed. The book itself propos...This article presents four (4) additions to a book on the brain’s OS published by SciRP in 2015 [1]. It is a kind of appendix to the book. Some familiarity with the earlier book is presupposed. The book itself proposes a complete physical and mathematical blueprint of the brain’s OS. A first addition to the book (see Chapters 5 to 10 below) concerns the relation between the afore-mentioned blueprint and the more than 2000-year-old so-called fundamental laws of thought of logic and philosophy, which came to be viewed as being three (3) in number, namely the laws of 1) Identity, 2) Contradiction, and 3) the Excluded Middle. The blueprint and the laws cannot both be the final foundation of the brain’s OS. The design of the present paper is to interpret the laws in strictly mathematical terms in light of the blueprint. This addition constitutes the bulk of the present article. Chapters 5 to 8 set the stage. Chapters 9 and 10 present a detailed mathematical analysis of the laws. A second addition to the book (Chapter 11) concerns the distinction between the laws and the axioms of the brain’s OS. Laws are part of physics. Axioms are part of mathematics. Since the theory of the brain’s OS involves both physics and mathematics, it exhibits both laws and axioms. A third addition (Chapter 12) to the book involves an additional flavor of digitality in the brain’s OS. In the book, there are five (5). But brain chemistry requires a sixth. It will be called Existence Digitality. A fourth addition (Chapter 13) concerns reflections on the role of imagination in theories of physics in light of the ignorance of deeper causes. Chapters 1 to 4 present preliminary matter, for the most part a brief survey of general concepts derived from what is in the book [1]. Some historical notes are gathered at the end in Chapter 14.展开更多
In the present paper we introduce and study pre-T0-,pre-R0-,pre-R1-,pre-T2(pre-Hausdorff)-,pre-T3(pre-regularity)-,pre-T4(pre-normality)-,pre-strong T3- and pre-strong T4-separation axioms in fuzzifying topology and g...In the present paper we introduce and study pre-T0-,pre-R0-,pre-R1-,pre-T2(pre-Hausdorff)-,pre-T3(pre-regularity)-,pre-T4(pre-normality)-,pre-strong T3- and pre-strong T4-separation axioms in fuzzifying topology and give some of their characterisations as well as the relations of these axioms and other separation axioms in fuzzifying topology introduced by Shen[7].展开更多
文摘After having laid down the Axiom of Algebra, bringing the creation of the square root of -1 by Euler to the entire circle and thus authorizing a simple notation of the nth roots of unity, the author uses it to organize homogeneous divisions of the limited development of the exponential function, that is opening the way to the use of a whole bunch of new primary functions in Differential Calculus. He then shows how new supercomplex products in dimension 3 make it possible to calculate fractals whose connexity depends on the product considered. We recall the geometry of convex polygons and regular polygons.
文摘This paper presents an analysis of the challenges in risk-based resource allocation in engineering projects.Sub-sequently,an alternative resource allocation evaluation method based on language information and information axioms is proposed.Firstly,the evaluation team uses language information to give the evaluation information of the alternatives of risk resource allocation and provides the corresponding expected information for each resource.Secondly,according to the transformation formula of language information and fuzzy numbers,the above information is transformed into the evaluation information and expected information of the alternatives of risk-based resource allocation.Thirdly,according to the transformation formula of language information and fuzzy numbers,the above information is transformed into evalu-ation information and expectation information of alternative risk resource allocation.Finally,according to the information amount of each risk resource and the corresponding weight,the comprehensive information amount of the expected risk-based resource allocation alternatives is determined.
基金The project is supported by the NNSF of China(No.10971185,10971186)Fujian Province support college research plan project(No.JK2011031)
文摘The Ti-axiom,the Ti-ordered axiom and Ti-pairwise axiom(i = 0,1,2,3,4) of topological ordered space are discussed and proved that they are equivalence under the certain conditions.
基金Supported by the National Natural Science Foundation of China(50505017,50775111)the Qing Lan Project of China~~
文摘The facility layout problem belongs to typical multiple-attribute decision-making（MADM） problems. To make the information axiom fit to MADM problems, the original computation method for the information content is improved by increasing the satisfaction degree item. Attribute values are divided into precise type, uncertainty type and fuzzy type. For benefit, cost, fixation, and interval type attributes, the computation methods for the information content on the three types of attribute values are presented. The improved information content can reflect the system success probability and the decision-maker satisfaction degree simultaneously and evaluate the MADM problem including multiple type attribute values. Finally, as a case study, the facility layout alternatives of a welding assembly workshop are evaluated. The result verifies the validity and the feasibility of the improved information axiom on the MADM problems.
基金supported by the National Natural Science Foundation of China(Nos.11072125 and11272175)the Natural Science Foundation of Jiangsu Province(No.SBK201140044)the Specialized Research Fund for Doctoral Program of Higher Education(No.20130002110044)
文摘This paper reports the new progresses in the axiomatization of tensor anal- ysis, including the thought of axiomatization, the concept of generalized components, the axiom of covariant form invariability, the axiomatized definition, the algebraic structure, the transformation group, and the simple calculation of generalized covariant differentia- tions. These progresses strengthen the tendency of the axiomatization of tensor analysis.
文摘Whether a unified theory of everything(TOE)is possible or not is a philosophical question and yes or no can be chosen in a two-valued logic system.Currently the two schools are in conflict with each other.Based on the relativity of simultaneity axiom proposed in this paper,the present author suggests to use a midway philosophy to replace the present materialist philosophy for modern sciences;then this conflict together with many other conflicts among different theories such as classical mechanics(CM),general relativity(GR),and quantum mechanics(QM)can be solved and a unified theory of everything for the world we can observe can be constructed.In this paper,the axiomatic foundation for a TOE is proposed which contains six fundamental axioms.Various problems related to these foundational issues are discussed.It is hoped that the present paper might show a new promise and a new direction for TOE which would be helpful for the further development of modern sciences.
文摘Rough set axiomatization is one aspect of rough set study to characterize rough set theory using dependable and minimal axiom groups. Thus, rough set theory can be studied by logic and axiom system methods. The classic rough set theory is based on equivalent relation, but rough set theory based on reflexive and transitive relation (called quasi-ordering) has wide applications in the real world. To characterize topological rough set theory, an axiom group named RT, consisting of 4 axioms, is proposed. It is proved that the axiom group reliability in characterizing rough set theory based on similar relation is reasonable. Simultaneously, the minimization of the axiom group, which requires that each axiom is an equation and each is independent, is proved. The axiom group is helpful for researching rough set theory by logic and axiom system methods.
文摘In this paper,a common characteristic of real number system and well ordered set is revealed and proved to be an equivalent form of the Dedekind Axiom or the Continuous Induction in R. Basing on it,we get the unified form of the mathematical induction,the transfinite induction and the continuous induction and generalize induction to totally ordered set that has the same characteristic.
文摘In this paper,a common characteristic of real number system and well ordered set is revealed and proved to be an equivalent form of the Dedekind Axiom or the Continuous Induction in R. Basing on it,we get the unified form of the mathematical induction,the transfinite induction and the continuous induction and generalize induction to totally ordered set that has the same characteristic.
基金supported by the National Natural Science Foundation of China(10571174)the Scientific Research Foundation of Ministry of Education for Returned Overseas Chinese ScholarsScientific Research Foundation of Ministry of Human Resources and Social Security for Returned Overseas Chinese Scholars
文摘Let T：X → X be an Axiom A diffeomorphism,m the Gibbs state for a Hlder continuous function ɡ. Assume that f：X → R^d is a Hlder continuous function with ∫_X^（fdm） = 0.If the components of f are cohomologously independent, then there exists a positive definite symmetric matrix σ~2：=σ~2 （f ） such that S^fn √ n converges in distribution with respect to m to a Gaussian random variable with expectation 0 and covariance matrix σ~2 . Moreover, there exists a real number A 〉 0 such that, for any integer n ≥ 1,Π（ m＊（ 1√ nS f n ）,N （0,σ~2 ） ≤A√n, where m＊（1√ n S^fn）denotes the distribution of 1√ n S^fn with respect to m, and Π is the Prokhorov metric.
文摘Rough set axiomatization is one aspect of rough set study, and the purpose is to characterize rough set theory using independable and minimal axiom groups. Thus, rough set theory can be studied by logic and axiom system methods. To characterize rough set theory, an axiom group named H consisting of 4 axioms, is proposed. That validity of the axiom group in characterizing rough set theory is reasonable, is proved. Simultaneously, the minimization of the axiom group, which requires that each axiom is an inequality and each is independent, is proved. The axiom group is helpful for researching rough set theory by logic and axiom system methods. Key words rough set - lower approximation - axioms - minimization CLC number TP 18 Foundation item: Supported by the 973 National Basic Research Program of China (2002CB312106) and Science & Technology Program of Zhejiang Province (2004C31G101003)Biography: DAI Jian-hua (1977-), male, Ph. D, research direction: data mining, artificial intelligence, rough sets, evolutionary computation.
文摘In this work we have to deal with the axiomatization of cosmology, but it is only recently that we have hit upon a new mathematical approach to capitalize on our new set identities for the basic laws of cosmology. So our proposal of settlement is that we will propose some new laws (e.g., formation of the black hole). We introduce the concept of axiom cosmology. This principle describes the cosmology which can get freedom from the notion of the induction. We present a large-scale structure model of the universe, and this leads to successfully explanation of problem of closed universe or open universe (because from the outset it is theorem and its succinct proof). In this paper we prove that the non-singular point theorem means that a singularity cannot be mathematically defined nor physical. It allows us to overcome the mysterious, physical singularity conundrum and explain meaningful antimatter annihilations for general configurations.
文摘This article presents four (4) additions to a book on the brain’s OS published by SciRP in 2015 [1]. It is a kind of appendix to the book. Some familiarity with the earlier book is presupposed. The book itself proposes a complete physical and mathematical blueprint of the brain’s OS. A first addition to the book (see Chapters 5 to 10 below) concerns the relation between the afore-mentioned blueprint and the more than 2000-year-old so-called fundamental laws of thought of logic and philosophy, which came to be viewed as being three (3) in number, namely the laws of 1) Identity, 2) Contradiction, and 3) the Excluded Middle. The blueprint and the laws cannot both be the final foundation of the brain’s OS. The design of the present paper is to interpret the laws in strictly mathematical terms in light of the blueprint. This addition constitutes the bulk of the present article. Chapters 5 to 8 set the stage. Chapters 9 and 10 present a detailed mathematical analysis of the laws. A second addition to the book (Chapter 11) concerns the distinction between the laws and the axioms of the brain’s OS. Laws are part of physics. Axioms are part of mathematics. Since the theory of the brain’s OS involves both physics and mathematics, it exhibits both laws and axioms. A third addition (Chapter 12) to the book involves an additional flavor of digitality in the brain’s OS. In the book, there are five (5). But brain chemistry requires a sixth. It will be called Existence Digitality. A fourth addition (Chapter 13) concerns reflections on the role of imagination in theories of physics in light of the ignorance of deeper causes. Chapters 1 to 4 present preliminary matter, for the most part a brief survey of general concepts derived from what is in the book [1]. Some historical notes are gathered at the end in Chapter 14.
文摘In the present paper we introduce and study pre-T0-,pre-R0-,pre-R1-,pre-T2(pre-Hausdorff)-,pre-T3(pre-regularity)-,pre-T4(pre-normality)-,pre-strong T3- and pre-strong T4-separation axioms in fuzzifying topology and give some of their characterisations as well as the relations of these axioms and other separation axioms in fuzzifying topology introduced by Shen[7].
