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
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
An absolute value equation is established for linear combinations of two operators.When the parameters take special values, the parallelogram law of operator type is given. In addition, the operator equation in litera...An absolute value equation is established for linear combinations of two operators.When the parameters take special values, the parallelogram law of operator type is given. In addition, the operator equation in literature [3] and its equivalent deformation are obtained.Based on the equivalent deformation of the operator equation and using the properties of conjugate number as well as the operator, an absolute value identity of multiple operators is given by means of mathematical induction. As Corollaries, Bohr inequalities are extended to multiple operators and some related inequalities are reduced to, such as inequalities in [2]and [3].展开更多
The development of algebraic and numerical algorithms is a kind of complicated creative work and it is difficult to guarantee the correctness of the algorithms. This paper introduces a systematic and unified formal de...The development of algebraic and numerical algorithms is a kind of complicated creative work and it is difficult to guarantee the correctness of the algorithms. This paper introduces a systematic and unified formal development method of algebraic and numerical algorithms. The method implements the complete refinement process from abstract specifications to a concrete executable program. It uses the core idea of partition and recursion for formal derivation and combines the mathematical induction based on strict mathematical logic with Hoare axiom for correctness verification. This development method converts creative work into non-creative work as much as possible while ensuring the correctness of the algorithm, which can not only verify the correctness of the existing algebraic and numerical algorithms but also guide the development of efficient unknown algorithms for such problems. This paper takes the non-recursive implementation of the Extended Euclidean Algorithm and Horner's method as examples. Therefore, the effectiveness and feasibility of this method are further verified.展开更多
Aims It is important to explore the underlying mechanisms that cause triphasic species–area relationship(triphasic SAR)across different scales in order to understand the spatial patterns of biodiversity.Methods Inste...Aims It is important to explore the underlying mechanisms that cause triphasic species–area relationship(triphasic SAR)across different scales in order to understand the spatial patterns of biodiversity.Methods Instead of theory establishment or field data derivation,I adopted a data simulation method that used the power function of SAR to fit log-normal distribution of species abundance.Important Findings The results showed that one-step sampling caused biphasic SAR and n-step sampling could cause 2n-phasic SAR.Practical two-step sampling produced triphasic SAR due to the Preston and Pan effects in large areas.Furthermore,before exploring biological or ecological mechanisms for the nature phenomenon,we should identify or exclude potential mathematical,statistical or sampling reasons.展开更多
In this paper we study a group testing model ФX, Y,, XY. In other words, we consider a n-item set containing exactly two defective ones. The purpose of this paper is to find out the two defective items with a worst-c...In this paper we study a group testing model ФX, Y,, XY. In other words, we consider a n-item set containing exactly two defective ones. The purpose of this paper is to find out the two defective items with a worst-case minimum number of tests, each of which will indicates whether the subset being tested contains all good （normal） items or not and in the latter case, it is not sure that the tested subset is of one defective （bad） item or two. Based on the M-sharp algorithm obtained in the previous paper and some combinatorial skills, we derive an almost optimal algorithm for the so called n-problem.展开更多
In this paper we study further on a group testing problem of identifying the defective from a n-coin set containing one defective coin with a balance without weight. The defective coin is not of the same weight as eac...In this paper we study further on a group testing problem of identifying the defective from a n-coin set containing one defective coin with a balance without weight. The defective coin is not of the same weight as each of the normal ones. We derive a new testing algorithm which can tell out the defective from the n-coin set with the worst-case minimum number of tests.展开更多
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 Key Scientific and Technological Innovation Team Project in Shaanxi Province(2014KCT-15)
文摘An absolute value equation is established for linear combinations of two operators.When the parameters take special values, the parallelogram law of operator type is given. In addition, the operator equation in literature [3] and its equivalent deformation are obtained.Based on the equivalent deformation of the operator equation and using the properties of conjugate number as well as the operator, an absolute value identity of multiple operators is given by means of mathematical induction. As Corollaries, Bohr inequalities are extended to multiple operators and some related inequalities are reduced to, such as inequalities in [2]and [3].
基金Supported by the National Natural Science Foundation of China (61862033, 61762049, 61902162)Jiangxi Provincial Natural Science Foundation (20202BABL202026, 20202BABL202025, 20202BAB202015)。
文摘The development of algebraic and numerical algorithms is a kind of complicated creative work and it is difficult to guarantee the correctness of the algorithms. This paper introduces a systematic and unified formal development method of algebraic and numerical algorithms. The method implements the complete refinement process from abstract specifications to a concrete executable program. It uses the core idea of partition and recursion for formal derivation and combines the mathematical induction based on strict mathematical logic with Hoare axiom for correctness verification. This development method converts creative work into non-creative work as much as possible while ensuring the correctness of the algorithm, which can not only verify the correctness of the existing algebraic and numerical algorithms but also guide the development of efficient unknown algorithms for such problems. This paper takes the non-recursive implementation of the Extended Euclidean Algorithm and Horner's method as examples. Therefore, the effectiveness and feasibility of this method are further verified.
基金The work was supported by the National Key R&D Program of China(2018YFF0214905 and 2016YFC1200802).
文摘Aims It is important to explore the underlying mechanisms that cause triphasic species–area relationship(triphasic SAR)across different scales in order to understand the spatial patterns of biodiversity.Methods Instead of theory establishment or field data derivation,I adopted a data simulation method that used the power function of SAR to fit log-normal distribution of species abundance.Important Findings The results showed that one-step sampling caused biphasic SAR and n-step sampling could cause 2n-phasic SAR.Practical two-step sampling produced triphasic SAR due to the Preston and Pan effects in large areas.Furthermore,before exploring biological or ecological mechanisms for the nature phenomenon,we should identify or exclude potential mathematical,statistical or sampling reasons.
基金This research is supported by Natural Science Foundation of Beiing （1052007）.
文摘In this paper we study a group testing model ФX, Y,, XY. In other words, we consider a n-item set containing exactly two defective ones. The purpose of this paper is to find out the two defective items with a worst-case minimum number of tests, each of which will indicates whether the subset being tested contains all good （normal） items or not and in the latter case, it is not sure that the tested subset is of one defective （bad） item or two. Based on the M-sharp algorithm obtained in the previous paper and some combinatorial skills, we derive an almost optimal algorithm for the so called n-problem.
基金This research is supported partially by Natural Science Foundation of Beijing （1052007,1042007）
文摘In this paper we study further on a group testing problem of identifying the defective from a n-coin set containing one defective coin with a balance without weight. The defective coin is not of the same weight as each of the normal ones. We derive a new testing algorithm which can tell out the defective from the n-coin set with the worst-case minimum number of tests.
