关于两个新数论函数的Dirichlet级数

定义两个新的算术函数U(n)及V(n),研究由U(n)及V(n)组成的一类Dirichlet级数的收敛性.利用初等方法和解析方法,给出两个新数论函数Dirichlet级数的恒等式,得到两个有趣的均值定理.从而拓展了经典算术函数的相关研究.

Some Hybrid Mean Values of a New Number Theoretic Function

Let p and q be two distinct primes, epq（n） denotes the largest exponent of power pq which divides n. In this paper, we study the mean value properties of function epq（n）, and give some hybrid mean value formulas for epq（n） and Dirichlet divisor function d（n）. Key words： largest exponent; asymptotic formula; hybrid mean value; Dirichlet divisor function d（n）

The prescription rule analysis of the New Theory on Spleen Dampness Syndrome by Academician TONG Xiaolin

Objective To analyze the basic characteristics,drug features,prescription rules,and drug-symptom relationships of patients in the splenic deficiency and impairment stage,by data mining of medical records under the New Theory on Spleen Dampness Syndrome(Pi Dan Xin Lun,《脾瘅新论》).Methods Medical records listed in the“New Theory on Spleen Dampness Syndrome-Under-standing and Treatment of Metabolic Syndrome from the Perspective of Traditional Chinese Medicine”,and which were diagnosed with the spleen dampness syndrome at the splenic de-ficiency and impairment stage,during January 2004 and December 2016 were selected.These patients’data,including basic information,clinical symptoms,laboratory examination res-ults,traditional Chinese medicine(TCM)and western medicine diagnoses,treatment meth-ods,prescriptions,etc.,were collected.The collected data were subsequently compiled into a medical record database using the Epidata 3.1 data management software,followed by the use of Apriori algorithm provided in the SPSS Modeler 14.2 statistical software to investigate the association rules between drug-drug,drug-symptom,and drug-western medicine indices.Results(i)A total of 51 medical records were included,involving 17 types of syndromes.Among them,the top three with frequency≥3 included“Phlegm and blood stasis,and thoracic obstruction”“Deficiency-weakness of the spleen Qi,and static blood blocking collat-erals”,and“Deficiency-weakness of the spleen Qi,and static blood blocking collaterals”.Al-ternatively,of the 14 treatment methods,the top three treatments with frequency of≥3 in-cluded“Activating Yang and eliminating turbidity,and removing phlegm and dredging chan-nel blockage”“Strengthening the spleen and benefiting Qi,and eliminating phlegm to activ-ate the channels”,and“Warming Yang and benefiting Qi,and expelling cold to remove ob-structions”.Among the 15 prescriptions,the top three used with frequency≥3 included Huangqi Guizhi Wuwu Tang(黄芪桂枝五物汤),Gualou Xiebai Banxia Tang(瓜蒌薤白半夏汤),and Ganjiang Huangqin Huanglian Renshen Tang(干姜黄芩黄连人参汤).Lastly,of the 83 drugs used for a total of 476 times,those with frequency≥15 included Huanglian(Coptid-is Rhizoma),Huangqi(Astragali Radix),Jiudahuang(Wine-processed Rhei Radix et Rhizoma),Jixueteng(Spatholobi Caulis),Shengjiang(Zingiberis Rhizoma Recens),Huangqin(Scutellariae Radix),and Guizhi(Cinnamomi Ramulus).(ii)For the drug-drug associations,under the criteria of support≥15%and confidence=100%,seven second-order association rules,seven third-order rules,and six fourth-order roles were identified.The top-ranking rule of each was“Huangqin(Scutellariae Radix)→Huanglian(Coptidis Rhizoma)”“Ganjiang(Zingiberis Rhizoma)+Huangqin(Scutellariae Radix)→Huanglian(Coptidis Rhizoma)”,and“Baishao(Paeoniae Radix Alba)+Guizhi(Cinnamomi Ramulus)+Jixueteng(Spatho-lobi Caulis)→Huangqin(Scutellariae Radix)”,respectively.Alternatively,the drug-symptom associations were analyzed under the criteria of support≥5%and confidence=100%,which derived eight second-order association rules,31 third-order rules,and 30 fourth-order rules.The top-ranking association rule of each order was“Huangqi(Astragali Radix)→Limb ed-ema”“Guizhi(Cinnamomi Ramulus)+Jixueteng(Spatholobi Caulis)→Limb numbness and pain”,and“Guizhi(Cinnamomi Ramulus)+Jixueteng(Spatholobi Caulis)+Huangqi(As-tragali Radix)→Limb numbness and pain”,respectively.Similarly,the drug-western medi-cine index associations were investigated under the criteria of support≥5%and confidence=100%,and five second-order association rules,16 third-order rules,and 16 fourth-order rules were identified.In this category,the top-ranking association rule of each order was“Qinpi(Fraxini Cortex)→Uric acid”“Huanglian(Coptidis Rhizoma)+Ganjiang(Zingiberis Rhizoma)→Glycated hemoglobin”,and“Huanglian(Coptidis Rhizoma)+Ganjiang(Zing-iberis Rhizoma)+Huangqin(Scutellariae Radix)→Glycated hemoglobin”,respectively.Conclusion Through association rule mining,this study objectively and quantitatively demonstrated the drug-drug,drug-symptom,and drug-physicochemical index associations of patients with the spleen dampness syndrome at the splenic deficiency and impairment stage treated by Academician TONG Xiaolin.The results indicated that treatment for these patients adopted the“state-target”syndrome differentiation method.The drug combination was characterized by“small prescriptions”,targeting both the patient’s symptoms and signs(syndrome target)and western medicine indices(treatment target).This study could provide references for future research on the academic thoughts and medical experience of Academi-cian TONG Xiaolin.

A Series of Variable Separation Solutions and New Soliton Structures of （2＋1）-Dimensional Korteweg-de Vries Equation

Variable separation approach is introduced to solve the （2＋1）-dimensional KdV equation. A series of variable separation solutions is derived with arbitrary functions in system. We present a new soliton excitation model （24）. Based on this excitation, new soliton structures such as the multi-lump soliton and periodic soliton are revealed by selecting the arbitrary function appropriately.

A contribution to large deviations for heavy-tailed random sums

In this paper we consider the large deviations for random sums $S(t) = \sum _{i = t}^{N(t)} X_i ,t \geqslant 0$ , whereX n,n?1 are independent, identically distributed and non-negative random variables with a common heavy-tailed distribution function F, andN(t), t?0 is a process of non-negative integer-valued random variables, independent ofX n,n?1. Under the assumption that the tail of F is of Pareto’s type (regularly or extended regularly varying), we investigate what reasonable condition can be given onN(t), t?0 under which precise large deviation for S( t) holds. In particular, the condition we obtain is satisfied for renewal counting processes.