The Waring distribution is an important two-parameter discrete distribution,commonly used in fields such as ecology,linguistics,and information science,where heavy tails are often observed.In this paper,we propose a n...The Waring distribution is an important two-parameter discrete distribution,commonly used in fields such as ecology,linguistics,and information science,where heavy tails are often observed.In this paper,we propose a new goodness-of-fit test for the Waring distribution,which is established through the hazard rate and a linear equivalent definition of the Waring distribution.We establish an asymptotic Chi-square null distribution for the proposed test and show that it is more powerful than classical methods in simulation studies.Finally,we apply the test to analyze the authorships of published papers on computer science.展开更多
Let G (k) denote the smallest s such that every sufficiently large natural number is the sum of at most s fcth powers of natural numbers. It is proved that G (16)<111. This improves the result of T. D. Wooley's.
It has been proved that every positive integer is expressible as a sum of no more than eight pyramidal numbers P(m)=(m-1)m(m+1)/6.This paper reports on a computer calculation of the partition of integers from n=1 to 1...It has been proved that every positive integer is expressible as a sum of no more than eight pyramidal numbers P(m)=(m-1)m(m+1)/6.This paper reports on a computer calculation of the partition of integers from n=1 to 109 into pyramidal numbers.We find that no integer≤10°needs more than five pyramidal numbers for its partition,and only 241 of them do need five.We define J(n) as the least number of pyramidal numbers to partition n,and Nk(n) as the number of positive integers l less than or equal to n for which J(l)=k.Based on our numerical results we make conjectures about the asymptotic form of Nk(n) for n→∞展开更多
Let G(k) denote the smallest number s such that every sufficiently large natural number is the sum of, at most, s k-th powers of natural numbers. In this paper, we give new bounds for G(12), G(13) and G(19).
Let k≥1 be an integer.Assume that RH holds.In this paper we prove that a suitable asymptotic formula for the average number of representations of integers n=p^(k)_(1)+p^(3)_(2)+p^(3)_(3)+p^(3)_(4)+p^(3)_(5),where p_(...Let k≥1 be an integer.Assume that RH holds.In this paper we prove that a suitable asymptotic formula for the average number of representations of integers n=p^(k)_(1)+p^(3)_(2)+p^(3)_(3)+p^(3)_(4)+p^(3)_(5),where p_(1),p_(2),p_(3),p_(4),p_(5)are prime numbers.This expands the previous results.展开更多
The two-parameter Waring is an important heavy-tailed discrete distribution,which extends the famous Yule Simon distribution and provides more flexibility when modelling the data.The commonly used EFF(Expectation-Firs...The two-parameter Waring is an important heavy-tailed discrete distribution,which extends the famous Yule Simon distribution and provides more flexibility when modelling the data.The commonly used EFF(Expectation-First Frequency)for parameter estimation can only be applied when the first moment exists,and it only uses the information of the expectation and the first frequency,which is not as efficient as the maximum likelihood estimator(MLE).However,the MLE may not exist for some sample data.We apply the profle method to the log-likelihood function and derive the necessary and sufficient Conditions for the existence of the MLE of the Waring parameters.We use extensive simulation studies to compare the MLE and EFF methods,and the goodness-of-fit comparison with the Yule Simon distribution.We also apply the Waring distribution to fit an insurance data.展开更多
In the early morning,the constant jingling and crackling echoed through the deep valley of Aima Township,Namling County,Xigaze City,where a group of apprentices at the Tsedong Gold-Copper Buddha Statue Center for Anci...In the early morning,the constant jingling and crackling echoed through the deep valley of Aima Township,Namling County,Xigaze City,where a group of apprentices at the Tsedong Gold-Copper Buddha Statue Center for Ancient Craftsmanship and Technique were diligently creating their handcrafted works.展开更多
In contemporary China,some domestic films are deeply rooted in traditional Chinese culture,contributing to more and more excellent film&TV programs,among which,the styling inspired by animal patterns on Late Shang...In contemporary China,some domestic films are deeply rooted in traditional Chinese culture,contributing to more and more excellent film&TV programs,among which,the styling inspired by animal patterns on Late Shang Dynasty bronze ware has given strong vitality to the film&TV programs.This paper is intended to compile and understand the cultural value and artistic features of the patterns on Late Shang Dynasty bronze ware as well as its value in the traditional pattern application.It provides certain references regarding the role of film&TV styling in the inheritance and application of traditional patterns,if we analyze the cognition and status quo of the Late Shang Dynasty bronze ware patterns in the contemporary film&TV styling,and if we go deep into its artistic value and its application within such makeup design and identify the problems and laws in current design cases.Such analysis also provides a way of thinking for the application of traditional patterns in current film&TV costume design.展开更多
基金supported by the National Natural Science Foundation of China 12371265Natural Science Foundation of Shanghai 21ZR1420700Fundamental Research Funds for the Central Universities(2022QKT001).
文摘The Waring distribution is an important two-parameter discrete distribution,commonly used in fields such as ecology,linguistics,and information science,where heavy tails are often observed.In this paper,we propose a new goodness-of-fit test for the Waring distribution,which is established through the hazard rate and a linear equivalent definition of the Waring distribution.We establish an asymptotic Chi-square null distribution for the proposed test and show that it is more powerful than classical methods in simulation studies.Finally,we apply the test to analyze the authorships of published papers on computer science.
基金Project supported by the National Natural Science Foundation of China.
文摘Let G (k) denote the smallest s such that every sufficiently large natural number is the sum of at most s fcth powers of natural numbers. It is proved that G (16)<111. This improves the result of T. D. Wooley's.
基金Project partially supported by the Applied Mathematics Subprogram of the U.S. Department of Energy DE-FG02-90ER25084partially supported by NSF PHY CAP 91-08054
文摘It has been proved that every positive integer is expressible as a sum of no more than eight pyramidal numbers P(m)=(m-1)m(m+1)/6.This paper reports on a computer calculation of the partition of integers from n=1 to 109 into pyramidal numbers.We find that no integer≤10°needs more than five pyramidal numbers for its partition,and only 241 of them do need five.We define J(n) as the least number of pyramidal numbers to partition n,and Nk(n) as the number of positive integers l less than or equal to n for which J(l)=k.Based on our numerical results we make conjectures about the asymptotic form of Nk(n) for n→∞
文摘Let G(k) denote the smallest number s such that every sufficiently large natural number is the sum of, at most, s k-th powers of natural numbers. In this paper, we give new bounds for G(12), G(13) and G(19).
基金the National Natural Science Foundation of China(Grant No.11761048)the Natural Science Foundation of Jiangxi Province for Distinguished Young Scholars(Grant No.20212ACB211007)Natural Science Foundation of Jiangxi Province(Grant No.20224BAB201001).
文摘Let k≥1 be an integer.Assume that RH holds.In this paper we prove that a suitable asymptotic formula for the average number of representations of integers n=p^(k)_(1)+p^(3)_(2)+p^(3)_(3)+p^(3)_(4)+p^(3)_(5),where p_(1),p_(2),p_(3),p_(4),p_(5)are prime numbers.This expands the previous results.
基金This work is partially supported by National Natural Science Foundation of China[Grant Numbers 11671096,11690013,11731011,11871376]Natural Science Foundation of Shanghai[Grant Number 21ZR1420700].
文摘The two-parameter Waring is an important heavy-tailed discrete distribution,which extends the famous Yule Simon distribution and provides more flexibility when modelling the data.The commonly used EFF(Expectation-First Frequency)for parameter estimation can only be applied when the first moment exists,and it only uses the information of the expectation and the first frequency,which is not as efficient as the maximum likelihood estimator(MLE).However,the MLE may not exist for some sample data.We apply the profle method to the log-likelihood function and derive the necessary and sufficient Conditions for the existence of the MLE of the Waring parameters.We use extensive simulation studies to compare the MLE and EFF methods,and the goodness-of-fit comparison with the Yule Simon distribution.We also apply the Waring distribution to fit an insurance data.
文摘In the early morning,the constant jingling and crackling echoed through the deep valley of Aima Township,Namling County,Xigaze City,where a group of apprentices at the Tsedong Gold-Copper Buddha Statue Center for Ancient Craftsmanship and Technique were diligently creating their handcrafted works.
基金supported by the Education and Teaching Reform Program,Beijing Institute of Fashion Technology.
文摘In contemporary China,some domestic films are deeply rooted in traditional Chinese culture,contributing to more and more excellent film&TV programs,among which,the styling inspired by animal patterns on Late Shang Dynasty bronze ware has given strong vitality to the film&TV programs.This paper is intended to compile and understand the cultural value and artistic features of the patterns on Late Shang Dynasty bronze ware as well as its value in the traditional pattern application.It provides certain references regarding the role of film&TV styling in the inheritance and application of traditional patterns,if we analyze the cognition and status quo of the Late Shang Dynasty bronze ware patterns in the contemporary film&TV styling,and if we go deep into its artistic value and its application within such makeup design and identify the problems and laws in current design cases.Such analysis also provides a way of thinking for the application of traditional patterns in current film&TV costume design.
