For integer n≥1 and real u,let Δ(n,u):=|{d:d] n,e^(u)<d≤e^(u+1)}|.The Erdos-Hooley Deltafunction is then defined by Δ(n):=Max_(u∈R)Δ(n,u).We improve the current upper bounds for the average and normal orders ...For integer n≥1 and real u,let Δ(n,u):=|{d:d] n,e^(u)<d≤e^(u+1)}|.The Erdos-Hooley Deltafunction is then defined by Δ(n):=Max_(u∈R)Δ(n,u).We improve the current upper bounds for the average and normal orders of this arithmetic function.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
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 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.
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).
In this paper,we study three types of Cantor sets.For any integer m≥4,we show that every real number in[0,k]is the sum of at most k m-th powers of elements in the Cantor ternary set C for some positive integer k,and ...In this paper,we study three types of Cantor sets.For any integer m≥4,we show that every real number in[0,k]is the sum of at most k m-th powers of elements in the Cantor ternary set C for some positive integer k,and the smallest such k is 2~m.Moreover,we generalize this result to the middle-1/αCantor set for1<α<2+√5 and m sufficiently large.For the naturally embedded image W of the Cantor dust C×C into the complex plane C,we prove that for any integer m≥3,every element in the closed unit disk in C can be written as the sum of at most 2^(m+8)m-th powers of elements in W.At last,some similar results on p-adic Cantor sets are also obtained.展开更多
针对类似文明寨油田的油藏条件,采用W aring B lender法,筛选出一种适合该类油藏特点的新型氮气泡沫体系:发泡剂0.5%HPAM,稳泡剂0.15%HDPM。通过室内流动实验,系统评价了该体系驱油性能,实验结果表明,筛选出的氮气泡沫体系能在水驱油的...针对类似文明寨油田的油藏条件,采用W aring B lender法,筛选出一种适合该类油藏特点的新型氮气泡沫体系:发泡剂0.5%HPAM,稳泡剂0.15%HDPM。通过室内流动实验,系统评价了该体系驱油性能,实验结果表明,筛选出的氮气泡沫体系能在水驱油的基础上提高采收率26.9%,提高采收率效果显著。展开更多
Liaoning is located in northeast China with a high level of seismic activity,and earthquake early warning is important for the mitigation of seismic hazard.In this work,we implement PRESTo,an open-source software plat...Liaoning is located in northeast China with a high level of seismic activity,and earthquake early warning is important for the mitigation of seismic hazard.In this work,we implement PRESTo,an open-source software platform for earthquake early warning based on regional seismic records,to the Liaoning seismic network.For the early warning of earthquakes in Liaoning,a travel-time table is created for event detection and location using an average crustal model,and the empirical relation is established between the earthquake magnitude and the initial P-wave amplitudes.Using archived seismic records of past earthquakes,we determine the optimal values for Liaoning using the core algorithms of PRESTo.Based on the optimal parameters,the uncertainty in event location is generally less than 5 km,and the lead time of the early warning is~15 s at 100-km epicentral distance.The implemented system can be directly put into routine earthquake early warning operation by linking it with the real-time data stream from the Liaoning seismic network.展开更多
In this paper, we prove the following estimate on exponential sums over primes: Let κ≥1,βκ=1/2+log κ/log2, x≥2 and α=a/q+ λ subject to (a, q) = 1, 1≤a≤q, and λ ∈ R. Then As an application, we prove that wi...In this paper, we prove the following estimate on exponential sums over primes: Let κ≥1,βκ=1/2+log κ/log2, x≥2 and α=a/q+ λ subject to (a, q) = 1, 1≤a≤q, and λ ∈ R. Then As an application, we prove that with at most O(N2/8+ε) exceptions, all positive integers up to N satisfying some necessary congruence conditions are the sum of three squares of primes. This result is as strong as what has previously been established under the generalized Riemann hypothesis.展开更多
In this paper we establish one new estimate on exponential sums over primes in short intervals. As an application of this result, we sharpen Hua's result by proving that each sufficiently large integer N congruent...In this paper we establish one new estimate on exponential sums over primes in short intervals. As an application of this result, we sharpen Hua's result by proving that each sufficiently large integer N congruent to 5 modulo 24 can be written as N = p12+p22+p32+p42+p52, with |pj-(N/5)^(1/2)|≤U = N1/2-1/20+ε, where pj are primes. This result is as good as what one can obtain from the generalized Riemann hypothesis.展开更多
It is proved that with at most O(N^(11/12+ε)) exceptions, all positiveintegers n ≤ N satisfying some necessary congruence conditions are the sum of three squares ofprimes. This improves substantially the previous re...It is proved that with at most O(N^(11/12+ε)) exceptions, all positiveintegers n ≤ N satisfying some necessary congruence conditions are the sum of three squares ofprimes. This improves substantially the previous results in this direction.展开更多
文摘For integer n≥1 and real u,let Δ(n,u):=|{d:d] n,e^(u)<d≤e^(u+1)}|.The Erdos-Hooley Deltafunction is then defined by Δ(n):=Max_(u∈R)Δ(n,u).We improve the current upper bounds for the average and normal orders of this arithmetic function.
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.
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→∞
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.
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).
文摘In this paper,we study three types of Cantor sets.For any integer m≥4,we show that every real number in[0,k]is the sum of at most k m-th powers of elements in the Cantor ternary set C for some positive integer k,and the smallest such k is 2~m.Moreover,we generalize this result to the middle-1/αCantor set for1<α<2+√5 and m sufficiently large.For the naturally embedded image W of the Cantor dust C×C into the complex plane C,we prove that for any integer m≥3,every element in the closed unit disk in C can be written as the sum of at most 2^(m+8)m-th powers of elements in W.At last,some similar results on p-adic Cantor sets are also obtained.
文摘针对类似文明寨油田的油藏条件,采用W aring B lender法,筛选出一种适合该类油藏特点的新型氮气泡沫体系:发泡剂0.5%HPAM,稳泡剂0.15%HDPM。通过室内流动实验,系统评价了该体系驱油性能,实验结果表明,筛选出的氮气泡沫体系能在水驱油的基础上提高采收率26.9%,提高采收率效果显著。
文摘Liaoning is located in northeast China with a high level of seismic activity,and earthquake early warning is important for the mitigation of seismic hazard.In this work,we implement PRESTo,an open-source software platform for earthquake early warning based on regional seismic records,to the Liaoning seismic network.For the early warning of earthquakes in Liaoning,a travel-time table is created for event detection and location using an average crustal model,and the empirical relation is established between the earthquake magnitude and the initial P-wave amplitudes.Using archived seismic records of past earthquakes,we determine the optimal values for Liaoning using the core algorithms of PRESTo.Based on the optimal parameters,the uncertainty in event location is generally less than 5 km,and the lead time of the early warning is~15 s at 100-km epicentral distance.The implemented system can be directly put into routine earthquake early warning operation by linking it with the real-time data stream from the Liaoning seismic network.
基金The author is supported by Post-Doctoral Fellowsbip of The University of Hong Kong.
文摘In this paper, we prove the following estimate on exponential sums over primes: Let κ≥1,βκ=1/2+log κ/log2, x≥2 and α=a/q+ λ subject to (a, q) = 1, 1≤a≤q, and λ ∈ R. Then As an application, we prove that with at most O(N2/8+ε) exceptions, all positive integers up to N satisfying some necessary congruence conditions are the sum of three squares of primes. This result is as strong as what has previously been established under the generalized Riemann hypothesis.
基金supported by the National Natural Science Foundation of China(Grant Nos.10125101&10531060)a Major Grant Program in Science and Technology by the Ministry of EducationTianyuan Mathematics Foundation(Grant No.10526028).
文摘In this paper we establish one new estimate on exponential sums over primes in short intervals. As an application of this result, we sharpen Hua's result by proving that each sufficiently large integer N congruent to 5 modulo 24 can be written as N = p12+p22+p32+p42+p52, with |pj-(N/5)^(1/2)|≤U = N1/2-1/20+ε, where pj are primes. This result is as good as what one can obtain from the generalized Riemann hypothesis.
基金Supported by The National Science Foundation(Grants #10125101 and #10131010)by a Ministry of Education Major Grant Program in Sciences and Technology
文摘It is proved that with at most O(N^(11/12+ε)) exceptions, all positiveintegers n ≤ N satisfying some necessary congruence conditions are the sum of three squares ofprimes. This improves substantially the previous results in this direction.
