The k-tuple conjecture of Hardy and Little wood predicts that there are infinitely many primes p such that p+2 and p+6 are primes simultaneously.In this paper,we prove that there are infinitely many primes p such that...The k-tuple conjecture of Hardy and Little wood predicts that there are infinitely many primes p such that p+2 and p+6 are primes simultaneously.In this paper,we prove that there are infinitely many primes p such that Ω(p+2)≤3 and Ω(p+6)≤6,where Ω(n) denotes the total number of prime divisors of an integer n.We also prove a better conditional result,with the above Ω(p+6)≤6 replaced by Ω(p+6)≤3,under the Elliott-Halberstam conjecture.展开更多
This special issue on Analytic Number Theory is dedicated to Jing-run Chen's Theorem(1+2)on the Goldbach Conjecture,the proof of which was first published in SCIENCE CHINA Mathematics exactly 50 years ago.Jing-run...This special issue on Analytic Number Theory is dedicated to Jing-run Chen's Theorem(1+2)on the Goldbach Conjecture,the proof of which was first published in SCIENCE CHINA Mathematics exactly 50 years ago.Jing-run Chen was born on May 22,1933 in Fujian province,China.In 1953,he graduated from Xiamen University with a B.Sc.degree in Mathematics.展开更多
It is proved that every large integer N≡5（mod24）can be written as N=p<sub>1</sub><sup>2</sup>+…+p<sub>5</sub><sup>2</sup>with each prime p<sub>j</sub> sa...It is proved that every large integer N≡5（mod24）can be written as N=p<sub>1</sub><sup>2</sup>+…+p<sub>5</sub><sup>2</sup>with each prime p<sub>j</sub> satisfying |p<sub>J</sub>-（N/5|）<sup>1/2</sup>≤N<sup>11/23</sup>.This gives a short interval version of Hua’stheorem on the quadratic Waring-Goldbach展开更多
Statistical machine learning models should be evaluated and validated before putting to work.Conventional k-fold Monte Carlo cross-validation(MCCV)procedure uses a pseudo-random sequence to partition instances into k ...Statistical machine learning models should be evaluated and validated before putting to work.Conventional k-fold Monte Carlo cross-validation(MCCV)procedure uses a pseudo-random sequence to partition instances into k subsets,which usually causes subsampling bias,inflates generalization errors and jeopardizes the reliability and effectiveness of cross-validation.Based on ordered systematic sampling theory in statistics and low-discrepancy sequence theory in number theory,we propose a new k-fold cross-validation procedure by replacing a pseudo-random sequence with a best-discrepancy sequence,which ensures low subsampling bias and leads to more precise expected-prediction-error(EPE)estimates.Experiments with 156 benchmark datasets and three classifiers(logistic regression,decision tree and na?ve bayes)show that in general,our cross-validation procedure can extrude subsampling bias in the MCCV by lowering the EPE around 7.18%and the variances around 26.73%.In comparison,the stratified MCCV can reduce the EPE and variances of the MCCV around 1.58%and 11.85%,respectively.The leave-one-out(LOO)can lower the EPE around 2.50%but its variances are much higher than the any other cross-validation(CV)procedure.The computational time of our cross-validation procedure is just 8.64%of the MCCV,8.67%of the stratified MCCV and 16.72%of the LOO.Experiments also show that our approach is more beneficial for datasets characterized by relatively small size and large aspect ratio.This makes our approach particularly pertinent when solving bioscience classification problems.Our proposed systematic subsampling technique could be generalized to other machine learning algorithms that involve random subsampling mechanism.展开更多
We consider the numberπ(x,y;q,a)of primes p≤such that p≡a(mod q)and(p-a)/q is free of prime factors greater than y.Assuming a suitable form of Elliott-Halberstam conjecture,it is proved thatπ(x,y:q,a)is asymptotic...We consider the numberπ(x,y;q,a)of primes p≤such that p≡a(mod q)and(p-a)/q is free of prime factors greater than y.Assuming a suitable form of Elliott-Halberstam conjecture,it is proved thatπ(x,y:q,a)is asymptotic to p(log(x/q)/log y)π(x)/φ(q)on average,subject to certain ranges of y and q,where p is the Dickman function.Moreover,unconditional upper bounds are also obtained via sieve methods.As a typical application,we may control more effectively the number of shifted primes with large prime factors.展开更多
This paper establishes an asymptotic formula with a power-saving error term for the number of rational points of bounded height on the singular cubic surface of P^3 Q given by the following equation X0(X^21+X^22)-X^33...This paper establishes an asymptotic formula with a power-saving error term for the number of rational points of bounded height on the singular cubic surface of P^3 Q given by the following equation X0(X^21+X^22)-X^33=0 in agreement with the Manin-Peyre conjectures.展开更多
Let f and g be holomorphic cusp forms of weights k1 and k2 for the congruence subgroups TO(N1)and Γ0(N2),respectively.In this paper the square moment of the Rankin-Selberg L-function for f and g in the aspect of both...Let f and g be holomorphic cusp forms of weights k1 and k2 for the congruence subgroups TO(N1)and Γ0(N2),respectively.In this paper the square moment of the Rankin-Selberg L-function for f and g in the aspect of both weights in short intervals is bounded,when k1^ε <<k^2<<k1^1-ε.These bounds are the mean Lindelof hypothesis in one case and subconvexity bounds on average in other cases.These square moment estimates also imply subconvexity bounds for individual L(1/2+it,f×g) for all g when f is chosen outside a small exceptional set.In the best case scenario the subconvexity bound obtained reaches the Weyl-type bound proved by Lau et al.(2006) in both the k1 and k2 aspects.展开更多
基金supported by the National Key Research and Development Program of China (Grant No. 2021YFA1000700)National Natural Science Foundation of China (Grant No. 12031008)。
文摘The k-tuple conjecture of Hardy and Little wood predicts that there are infinitely many primes p such that p+2 and p+6 are primes simultaneously.In this paper,we prove that there are infinitely many primes p such that Ω(p+2)≤3 and Ω(p+6)≤6,where Ω(n) denotes the total number of prime divisors of an integer n.We also prove a better conditional result,with the above Ω(p+6)≤6 replaced by Ω(p+6)≤3,under the Elliott-Halberstam conjecture.
文摘This special issue on Analytic Number Theory is dedicated to Jing-run Chen's Theorem(1+2)on the Goldbach Conjecture,the proof of which was first published in SCIENCE CHINA Mathematics exactly 50 years ago.Jing-run Chen was born on May 22,1933 in Fujian province,China.In 1953,he graduated from Xiamen University with a B.Sc.degree in Mathematics.
基金Supported by MCSEC and the National Natural Science Foundation (Grant No. 19701019) Supported by MCSFC and the National Natural Science Foundation
文摘It is proved that every large integer N≡5（mod24）can be written as N=p<sub>1</sub><sup>2</sup>+…+p<sub>5</sub><sup>2</sup>with each prime p<sub>j</sub> satisfying |p<sub>J</sub>-（N/5|）<sup>1/2</sup>≤N<sup>11/23</sup>.This gives a short interval version of Hua’stheorem on the quadratic Waring-Goldbach
基金supported by the Qilu Youth Scholar Project of Shandong Universitysupported by National Natural Science Foundation of China(Grant No.11531008)+1 种基金the Ministry of Education of China(Grant No.IRT16R43)the Taishan Scholar Project of Shandong Province。
文摘Statistical machine learning models should be evaluated and validated before putting to work.Conventional k-fold Monte Carlo cross-validation(MCCV)procedure uses a pseudo-random sequence to partition instances into k subsets,which usually causes subsampling bias,inflates generalization errors and jeopardizes the reliability and effectiveness of cross-validation.Based on ordered systematic sampling theory in statistics and low-discrepancy sequence theory in number theory,we propose a new k-fold cross-validation procedure by replacing a pseudo-random sequence with a best-discrepancy sequence,which ensures low subsampling bias and leads to more precise expected-prediction-error(EPE)estimates.Experiments with 156 benchmark datasets and three classifiers(logistic regression,decision tree and na?ve bayes)show that in general,our cross-validation procedure can extrude subsampling bias in the MCCV by lowering the EPE around 7.18%and the variances around 26.73%.In comparison,the stratified MCCV can reduce the EPE and variances of the MCCV around 1.58%and 11.85%,respectively.The leave-one-out(LOO)can lower the EPE around 2.50%but its variances are much higher than the any other cross-validation(CV)procedure.The computational time of our cross-validation procedure is just 8.64%of the MCCV,8.67%of the stratified MCCV and 16.72%of the LOO.Experiments also show that our approach is more beneficial for datasets characterized by relatively small size and large aspect ratio.This makes our approach particularly pertinent when solving bioscience classification problems.Our proposed systematic subsampling technique could be generalized to other machine learning algorithms that involve random subsampling mechanism.
基金supported by the Programme de Recherche Conjoint CNRS-NSFC(Grant No.1457)supported by National Natural Science Foundation of China(Grant No.11531008)+3 种基金the Ministry of Education of China(Grant No.IRT16R43)the Taishan Scholar Project of Shandong Provincesupported by National Natural Science Foundation of China(Grant No.11601413)NSBRP of Shaanxi Province(Grant No.2017JQ1016)
文摘We consider the numberπ(x,y;q,a)of primes p≤such that p≡a(mod q)and(p-a)/q is free of prime factors greater than y.Assuming a suitable form of Elliott-Halberstam conjecture,it is proved thatπ(x,y:q,a)is asymptotic to p(log(x/q)/log y)π(x)/φ(q)on average,subject to certain ranges of y and q,where p is the Dickman function.Moreover,unconditional upper bounds are also obtained via sieve methods.As a typical application,we may control more effectively the number of shifted primes with large prime factors.
基金supported by the program PRC 1457-Au For Di P(CNRS-NSFC)supported by National Natural Science Foundation of China(Grant No.11531008)+1 种基金the Ministry of Education of China(Grant No.IRT16R43)the Taishan Scholar Project of Shandong Province
文摘This paper establishes an asymptotic formula with a power-saving error term for the number of rational points of bounded height on the singular cubic surface of P^3 Q given by the following equation X0(X^21+X^22)-X^33=0 in agreement with the Manin-Peyre conjectures.
基金supported by National Natural Science Foundation of China(Grant No.11531008)Ministry of Education of China(Grant No.IRT16R43)+3 种基金Taishan Scholar Project of Shandong Provincesupported by National Natural Science Foundation of China(Grant No.11601271)China Postdoctoral Science Foundation(Grant No.2016M602125)China Scholarship Council(Grant No.201706225004)。
文摘Let f and g be holomorphic cusp forms of weights k1 and k2 for the congruence subgroups TO(N1)and Γ0(N2),respectively.In this paper the square moment of the Rankin-Selberg L-function for f and g in the aspect of both weights in short intervals is bounded,when k1^ε <<k^2<<k1^1-ε.These bounds are the mean Lindelof hypothesis in one case and subconvexity bounds on average in other cases.These square moment estimates also imply subconvexity bounds for individual L(1/2+it,f×g) for all g when f is chosen outside a small exceptional set.In the best case scenario the subconvexity bound obtained reaches the Weyl-type bound proved by Lau et al.(2006) in both the k1 and k2 aspects.
