By extending both arithmetical operations into finite sets of natural numbers, from the entire set of natural numbers successively deleting some residue classes modulo a prime, we invented a recursive sieve method or ...By extending both arithmetical operations into finite sets of natural numbers, from the entire set of natural numbers successively deleting some residue classes modulo a prime, we invented a recursive sieve method or algorithm on natural numbers and their sets. The algorithm mechanically yields a sequence of sets, which converges to the set of all primes p such that 2p + 1 divides the Mersenne number Mp. The cardinal sequence corresponding to the sequence of sets is strictly increasing. So that we have captured enough usable structures, without any estimation, the existing theories of those structures allow us to prove an exact result: there are infinitely many Mersenne composite numbers with prime exponents Mp.展开更多
We generalize the Eulerian numbers ?to sets of numbers Eμ(k,l), (μ=0,1,2,···) where the Eulerian numbers appear as the special case μ=1. This can be used for the evaluation of generalizations Eμ(k,Z...We generalize the Eulerian numbers ?to sets of numbers Eμ(k,l), (μ=0,1,2,···) where the Eulerian numbers appear as the special case μ=1. This can be used for the evaluation of generalizations Eμ(k,Z) of the Geometric series G0(k;Z)=G1(0;Z) by splitting an essential part (1-Z)-(μK+1) where the numbers Eμ(k,l) are then the coefficients of the remainder polynomial. This can be extended for non-integer parameter k to the approximative evaluation of generalized Geometric series. The recurrence relations and for the Generalized Eulerian numbers E1(k,l) are derived. The Eulerian numbers are related to the Stirling numbers of second kind S(k,l) and we give proofs for the explicit relations of Eulerian to Stirling numbers of second kind in both directions. We discuss some ordering relations for differentiation and multiplication operators which play a role in our derivations and collect this in Appendices.展开更多
Premenstrual dysphoric disorder(PMDD) affects nearly 5% of women of reproductive age. Symptomatic heterogeneity, together with largely unknown genetics, has greatly hindered its effective treatment. In the present stu...Premenstrual dysphoric disorder(PMDD) affects nearly 5% of women of reproductive age. Symptomatic heterogeneity, together with largely unknown genetics, has greatly hindered its effective treatment. In the present study, analysis of genomic sequencing-based copy number variations(CNVs) called from 100 kb white blood cell DNA sequence windows by means of semisupervized clustering led to the segregation of patient genomes into the D and V groups, which correlated with the depression and invasion clinical types,respectively, with 89.0% consistency. Application of diagnostic CNV features selected using the correlation-based machine learning method enabled the classification of the CNVs obtained into the D group, V group, total patient group, and control group with an average accuracy of 83.0%. The power of the diagnostic CNV features was 0.98 on average, suggesting that these CNV features could be used for the molecular diagnosis of the major clinical types of PMDD. This demonstrated concordance between the CNV profiles and clinical types of PMDD supported the validity of symptom-based diagnosis of PMDD for differentiating between its two major clinical types, as well as the predominantly genetic nature of PMDD with a host of overlaps between multiple susceptibility genes/pathways and the diagnostic CNV features as indicators of involvement in PMDD etiology.展开更多
In this paper, we study, mathematically speaking, the problem-the number of admissible preference orderings for the transitivity of simple majority vote(SMV) derived from Arrow’s Impossibility Theorem. In our researc...In this paper, we study, mathematically speaking, the problem-the number of admissible preference orderings for the transitivity of simple majority vote(SMV) derived from Arrow’s Impossibility Theorem. In our research, we find, by computer enumerating, that some results given by Craven are not correct. By defining a set of constraints, we give the recurrence formula of the local maximal number of admissible preference orderings and some other useful results.展开更多
基金Supported by the National Natural Science Foundation of China under Grant No.60772023by the Slpported Project under Grant No.SKLSDE-2010ZX-07 of the State Key Laboratory of Software Development Environment,Beijing University of Aeronautics and As tronautics+2 种基金by the Specialized Research Fund for the Doctoral Program of Higher Educatioi under Grant No.200800130006Chinese Ministry of Education,and by the Innovation Foundation for Ph.D.Graduates under Grant Nos.30-0350 and 30-0366Beijing University of Aeronautics and Astronautics
文摘By extending both arithmetical operations into finite sets of natural numbers, from the entire set of natural numbers successively deleting some residue classes modulo a prime, we invented a recursive sieve method or algorithm on natural numbers and their sets. The algorithm mechanically yields a sequence of sets, which converges to the set of all primes p such that 2p + 1 divides the Mersenne number Mp. The cardinal sequence corresponding to the sequence of sets is strictly increasing. So that we have captured enough usable structures, without any estimation, the existing theories of those structures allow us to prove an exact result: there are infinitely many Mersenne composite numbers with prime exponents Mp.
文摘We generalize the Eulerian numbers ?to sets of numbers Eμ(k,l), (μ=0,1,2,···) where the Eulerian numbers appear as the special case μ=1. This can be used for the evaluation of generalizations Eμ(k,Z) of the Geometric series G0(k;Z)=G1(0;Z) by splitting an essential part (1-Z)-(μK+1) where the numbers Eμ(k,l) are then the coefficients of the remainder polynomial. This can be extended for non-integer parameter k to the approximative evaluation of generalized Geometric series. The recurrence relations and for the Generalized Eulerian numbers E1(k,l) are derived. The Eulerian numbers are related to the Stirling numbers of second kind S(k,l) and we give proofs for the explicit relations of Eulerian to Stirling numbers of second kind in both directions. We discuss some ordering relations for differentiation and multiplication operators which play a role in our derivations and collect this in Appendices.
基金supported by grants to HX from University Grants Council(SRF116SC01UROP18SC06+10 种基金UROP20SC07)Innovation and Technology Commission(ITS/085/10ITS113/15FPITCPD/17-9ITT/023/17GPITT/026/18GP)of Hong Kong SARShenzhen Municipal Council of Science and Technology,Guangdong(JCYJ20170818113656988)Guangdong Province Basic and Applied Basic Research Fund(2021A1515011169)Shandong Province First Class Disciple Development Grant and Tai-Shan Scholar Program,Shandongand Ministry of Science and Technology(National Science and Technology Major Project,No.2017ZX09301064,2017ZX09301064004)People’s Republic of China,as well as grants from National Natural Science Foundation of China to M.Q.(8157151623)and J.W.(81603510)。
文摘Premenstrual dysphoric disorder(PMDD) affects nearly 5% of women of reproductive age. Symptomatic heterogeneity, together with largely unknown genetics, has greatly hindered its effective treatment. In the present study, analysis of genomic sequencing-based copy number variations(CNVs) called from 100 kb white blood cell DNA sequence windows by means of semisupervized clustering led to the segregation of patient genomes into the D and V groups, which correlated with the depression and invasion clinical types,respectively, with 89.0% consistency. Application of diagnostic CNV features selected using the correlation-based machine learning method enabled the classification of the CNVs obtained into the D group, V group, total patient group, and control group with an average accuracy of 83.0%. The power of the diagnostic CNV features was 0.98 on average, suggesting that these CNV features could be used for the molecular diagnosis of the major clinical types of PMDD. This demonstrated concordance between the CNV profiles and clinical types of PMDD supported the validity of symptom-based diagnosis of PMDD for differentiating between its two major clinical types, as well as the predominantly genetic nature of PMDD with a host of overlaps between multiple susceptibility genes/pathways and the diagnostic CNV features as indicators of involvement in PMDD etiology.
基金This project is supported by Doctoral Fundation of National Educational Committee.
文摘In this paper, we study, mathematically speaking, the problem-the number of admissible preference orderings for the transitivity of simple majority vote(SMV) derived from Arrow’s Impossibility Theorem. In our research, we find, by computer enumerating, that some results given by Craven are not correct. By defining a set of constraints, we give the recurrence formula of the local maximal number of admissible preference orderings and some other useful results.
