DAXUESHUXUEJIKAN ISSN0076 CN690/N 大学数学辑刊 犕犃犜犎犆犗犔犔犗犙犝犐犝犕犗犉犆犎犐犖犈犛犈犝犖犐犞犈犚犛犐犜犐犈犛 编委会主任委员赵荣侠 教授, 博士 ) 编委会副主任委员邓珠平 编审 ) 委员 以姓氏笔画为序 ) 主编赵荣侠 副主编 邓珠平 编审 ) 李海龙

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1 DAXUESHUXUEJIKAN ISSN0076 CN690/N 大学数学辑刊 犕犃犜犎犆犗犔犔犗犙犝犐犝犕犗犉犆犎犐犖犈犛犈犝犖犐犞犈犚犛犐犜犐犈犛 主办单位协办单位 宝鸡文理学院 杭州师范大学 徐州师范大学绍兴文理学院湖州师范学院 Math. Colloq. Chin. Univ., 0/) 中国宝鸡犅犃犗犑犐犆犎犐犖犃 / 0

2 DAXUESHUXUEJIKAN ISSN0076 CN690/N 大学数学辑刊 犕犃犜犎犆犗犔犔犗犙犝犐犝犕犗犉犆犎犐犖犈犛犈犝犖犐犞犈犚犛犐犜犐犈犛 编委会主任委员赵荣侠 教授, 博士 ) 编委会副主任委员邓珠平 编审 ) 委员 以姓氏笔画为序 ) 主编赵荣侠 副主编 邓珠平 编审 ) 李海龙 教授 ) 杨秀良 教授 ) 姚云飞 教授 ) 赵天绪 教授 ) 赵荣侠 倪仁兴 教授 ) 扈生彪 教授 ) 褚玉明 教授 ) 盛兴平 副教授 ) 邓珠平 主管单位 陕西省教育厅 主办单位 宝鸡文理学院 协办单位 杭州师范大学 徐州师范大学 Math. Colloq. Chin. Univ., 0/) 绍兴文理学院湖州师范学院 0 年第 / 期 0 年 5 月 5 日出版 本期责任编辑邓珠平校对方岩马加佳王晓癑陈杨军李哲峰张晓峰 英文 ) 等

3 大学数学辑刊 0 年 第 / 期 犕犪狋犺犆狅犾狅狇. 犆犺犻. 犝犻狏. 犖狅./ 0) 目次 犆狅狋犲狋狊 ) 广义 N 半正则环 英文 ) 殷晓斌, 王瑞 ) GeneralizedNsemiregularrings YIN Xiaobin,WANG Rui) 直观模糊赋范空间中双指标序列的收敛性 英文 ) 司海燕, 曹怀信, 杨萍 4) Convergenceofdoubleindexedsequencesinanintuitionisticfuzzynormedspace SIHaiyan,CAO Huaixin,YANGPing4) Sylow 塔群的新判别准则 黄建红 9) AnewcriterionofSylowtowergroups HUANGJianhong9) 正则化学习算法的误差分析 英文 ) 张际雄, 王建力, 盛宝怀 ) Erroranalysisonregularizedlearning ZHANGJixiong,WANGJianli,SHENGBaohuai) Wick 型随机广义 sinhsine)gordon 方程的白噪声泛函解 英文 ) 孙信秀, 刘海洋 8) Whitenoisefunctionalsolutionsfor Wicktypestochasticgeneralizedsinhsine)Gordon equations SUN Xinxiu,LIU Haiyang8) FCR 代数在 Cleft 扩张下的不变性 张棉棉 3) InvariantpropertiesofFCRalgebrasunderCleftextension ZHANG Mianmian3) Hilbert 空间中的平衡问题与不动点问题的复合迭代方法 李亚琼, 谷峰 6) CompositeiterativemethodsforequilibriumproblemsandfixedpointproblemsinHilbert spaces LIYaqiong,GU Feng6) Math. Colloq. Chin. Univ., 0/) 子环扩张的 morphic 性质 英文 ) 张丽婷 3) Themorphicpropertiesofsubringextension ZHANGLiting3) Weideman 公式的一种狇 模拟 英文 ) 郑德印, 陈广 36) A 狇 analogoftheweideman'sformula ZHENGDeyin,CHEN Guang36) 锥距离空间中 个映射的公共不动点定理 英文 ) 王云杰, 朱江 39) Commonfixedpointtheoremsfortwomappingsinconemetricspaces WANG Yunjie,ZHUJiang39) 涉及到 4 个自映象的一个新的公共不动点定理张贺军, 陆竞, 谷峰 43) Thenewcommonfixedpointtheoremforfourselfmappings ZHANGJunhe,LUJing,GU Feng43)

4 特定数字集的非谱问题仲明 47) Nonspectralproblemsofcertaindigitset ZHONG Ming47) 一类三阶非局部边值问题在共振条件下的可解性许圣梅 5) Solvabilityofathirdordernonlocalboundaryvalueproblematresonance XUShengmei5) 一类具有无限长区间摄动方程的渐近解陈升, 唐荣荣 56) Asymptoticsolutionforaclassoftheperturbationequationwithanunboundedinterval CHENSheng,TANG Rongrong56) 关于图及其主子图的自同构群的注记甘少君, 阿勇嘎 60) Anoteonautomorphismgroupsofagraphanditsprimarysubgraph GANShaojun,A Yongga60) 次线性积分方程组的正解郝喜国 68) Positivesolutionsofsublinearintegralequations HAO Xiguo68) 稳定律吸引场中两两 NQD 列的犔狉收敛性 沈建伟, 王文胜 7) 狉犔 convergenceofpairwisenqdsequencesindomainofatractionofstablelaw SHENJianwei,WANG Wensheng7) 可逆 Hom 余代数上的有理模性质 徐宁, 李金其, 张爱利 74) ThepropertyofrationalmodulesoninvertibleHomcoalgebras XU Ning,LIJinqi,ZHANG Aili74) 在线分类算法收敛性分析 田明党, 盛宝怀 76) Convergenceanalysisofonlineclassificationalgorithms TIAN Mingdang,SHENGBaohuai76) 纵向剪切问题中一类对偶积分方程组的封闭解及其应力场王文友 83) Theenclosedsolutionsofakindofdualintegralequationsandstressfieldofelasticlayer intheproblemoflongitudinalshear WANG Wenyou83) 几类冠图的第一类弱全色数杨随义, 包世堂, 文飞, 等 87) Math. Colloq. Chin. Univ., 0/) Onthefirstkindweaktotalchromaticnumberofseveralkindsofcoronagraphs YANGSuiyi,BAOShitang,WENFei,etal87) 具有特殊数量旗曲率的犿次根式 Finsler 度量祖豆蔻 89) MthrootFinslermetricswithspecialscalarflagcurvature ZU Doukou89) 七对角线性方程组的追赶法叶新涛, 张志 9) Catchupmethodofsevendiagonallinearequations YEXintao,ZHANGZhi9)

5 杭州师范大学学报 自然科学版 ), 第 0 卷, 第 期, 第 9700 页,0 年 3 月 JournalofHangzhouNormalUniversityNaturalScienceEdition),Vol.0,No.,pp.9700,Mar.0 DOI:0.3969/j.issn.6743X 犌犲犲狉犪犾犻狕犲犱犖 狊犲犿犻狉犲犵狌犾犪狉狉犻犵狊 YIN Xiaobin,WANG Rui ColegeofMathematicsandComputerScience,AnhuiNormalUniversity,Wuhu4000,Anhui,China) 犃犫狊狋狉犪犮狋 :A generalizationofapinjectiveringsgeneralized Nsemiregularringsisintroduced, andmainlyobtainsthat 犚 isastronglyregularringifandonlyif 犚 isareducedandgeneraliednsemi regularring.somepropertiesofgeneralized Nsemiregularringsarealsostudiedandsomeresults aboutapinjectiveringsareextended. 犓犲狔狑狅狉犱狊 :APinjectiverings;generalizedNsemiregularrings;stronglyregularrings 犆犔犆狌犿犫犲狉 :O53.3 犃狉狋犻犮犾犲犮犺犪狉犪犮狋犲狉 :A 犃狉狋犻犮犾犲犐犇 :6743X0) 犕犛犆 00:6E50 广义犖 半正则环 殷晓斌, 王 安徽师范大学数学与计算机科学学院, 安徽芜湖 4000) 摘要 : 介绍了 AP 内射环的推广 广义 N 半正则环, 主要得到了犚是强正则环当且仅当犚是约化 的广义 N 半正则环. 文章研究了广义 N 半正则环的性质且对 AP 内射环的某些结果进行了推广. 关键词 :AP 内射环 ; 广义 N 半正则环 ; 强正则环 中图分类号 O53.3 文献标志码 :A 文章编号 :6743X0) Throughoutthisarticle, 犚 denotesanasso ciativeringwithidentityandmodulesareunita ry.j,z resp.y),soc 犚犚 )Soc 犚犚 ))wilde noterespectivelythejacobsonradical,theleft singularidealresp.rightsingularideal)theleft socletherightsocle)of 犚. 犾 犡 ) 狉 犡 ))de notestheleftright)annihilatorof 犡 in 犚.If 犡 ={ 犪 },wewilwriteitfor 犾 犪 ) 狉 犪 )).Nil 犚 ) denotesthebiggestnilideal.recalthat 犚 is saidtobeleftright)nonsingularif 犣 =0 犢 = 0).Aring 犚 iscaledvonneumann)regular [] ifforany 犪 犚,thereexists 犫 犚 suchthat 犪 = 犪犫犪.Aring 犚 iscaledstronglyregular [] iffor any 犪 犚,thereexists 犫 犚 suchthat 犪 = 犪犫. 犇犲犳犻犻狋犻狅 Aring 犚 iscalednsemireg ular,ifforal 犪 犚,thereexistsanidempotent 瑞 犲 犪犚 suchthat- 犲 ) 犪 Nil 犚 ). 犇犲犳犻犻狋犻狅 犪 犚 iscalgeneralized N semiregularifthereexiststworightideals 犘, 犔 of 犚,suchthat 狉犾 犪 )= 犘 犔 with 犘 犪犚 and 犪犚 犔 isnil.aring 犚 iscaledgeneralized N semiregularifeachofitselementsisgeneralized Nsemiregular. 犘狉狅狆狅狊犻狋犻狅 If 犚 isaleftapinjectiveor Nsemiregularring,then 犚 isageneralized N Math. Colloq. Chin. Univ., 0/) semiregular. 犘狉狅狅犳 )Let 犚 bealeftapinjectivering. Then 狉犾 犪 )= 犪犚 犔犪 forany 犪 犚,where 犔犪 is arightidealof 犚.Notethat 犪犚 犔犪 isnil,thus 犚 isgeneralizednsemiregular. )Let 犚 beansemiregularringand 犪 犚. Thenthereexists 犲 = 犲 犪犚 suchthat- 犲 ) 犪 Receiveddate:00090 Foundationitem:SupportedbyNationalNaturalScienceFoundationofChina08706,09000)andNSFofAnhuiProvince EducationCommiteeKJ008A06). Biography:YIN Xiaobin97),male,borninZongyang,AnhuiProvince,China,associateprofessor,engagedinalgebra. xbyinzh@gmail.com

6 大学数学辑刊 MathColoquiumofChineseUniversities) 0 年 Nil 犚 ).Thus 犚 = 犲犚 - 犲 ) 犚,where 犲犚 犪犚犪犱 - 犲 ) 犪犚 isnil.notethat 犪犚 狉犾 犪 ). Bymodularlaw,wehave 狉犾 犪 )= 狉犾 犪 ) 犚 = 狉犾 犪 ) 犲犚 - 犲 ) 犚 )= 犲犚 狉犾 犪 ) - 犲 ) 犚 ) and 犪犚 狉犾 犪 ) - 犲 ) 犚 )= 犪犚 - 犲 ) 犚 =- 犲 ) 犪犚 isnil.thus 犚 isgeneralizednsemiregular. 犔犲犿犿犪 If 犚 isageneralizednsemiregu larringandforevery 犪 犚 thereexists 犲 = 犲 犚 suchthat 犾 犪 )= 犾 犲 ),then 犚 isnsemiregular. 犘狉狅狅犳 Forany 犪 犚.Since 犚 isgeneral izednsemiregular,thereexiststworightideals 犘, 犔 of 犚 suchthat 狉犾 犪 )= 犘 犔,where 犘 犪犚 and 犪犚 犔 isnil.thus 犲犚 = 犘 犔 since 犾 犲 )= 犾 犪 ).As 犾 犲 )= 犾 犪 ),wehave 狉犾 犪 )= 狉犾 犲 )= 犲犚,thisgives 犪 = 犲犪.Take 犲 = 犵 + 狋, where 犵 = 犪狉 犘 犪犚 and 狋 犔.Then 犪 = 犲犪 = 犵犪 + 狋犪 = 犪狉犪 + 狋犪 and 犪狉 = 犪狉犪狉 + 狋犪狉 whichimplies that 犪狉 - 犪狉犪狉 = 狋犪狉 犘 犔 =0and 犪 - 犪狉犪 = 狋犪 犪犚 犔 Nil 犚 ).Thisshowsthat 犵 = 犵 犪犚 and - 犵 ) 犪 Nil 犚 ), 犪 is Nsemiregular. Hence 犚 isnsemiregular. 犆狅狉狅犾犪狉狔 Let 犚 beageneralizednsemi regularelement.if 犚犪 犚犲,where 犲 = 犲,then 犪 isnsemiregular. 犘狉狅狅犳 Forany 犪 犚,letφbetheisomor phismof 犚犪 onto 犚犲.By [3],Lemma.), thereexistsanidempotent 犳 of 犚 suchthat 犾 犪 ) = 犾 犳 ).ByLemma, 犪 isnsemiregular. If 狉犾 犪 )isadirectsummandof 犚,then thereexists 犲 = 犲 犚 suchthat 狉犾 犪 )= 犲犚.Thus 犾 犪 )= 犾狉犾 犪 )= 犾 犲 )andsothefolowingresults isimmediate. 犆狅狉狅犾犪狉狔 If 狉犾 犪 )isadirectsummandof 犚 forany 犪 犚 and 犚 isgeneralizednsemiregu lar,then 犚 isnsemiregular. Aring 犚 tobebaer [4] iftherightannihila torofeverynonemptysubsetof 犚 isgenerated, asarightideal,byanidempotent.thisdefini tionisleftrightsymmetric.aring 犚 iscaled right resp.left)pp [5] ifeveryprincipalright resp.left)idealof 犚 isprojective. 犚 isaright PPringifandonlyifforevery 犪 犚 thereexists anidempotent 犲 犚 suchthat 狉 犪 )= 犲犚.Clear ly,everybaerringisrightandleftppring. 犆狅狉狅犾犪狉狔 3 Let 犚 bealeftppring.if 犚 is generalizednsemiregular,then 犚 isnsemireg ular. From [3],Corolary.3),weseethatif 犚 isarightapinjectiveringthen 犑 = 犢. 犘狉狅狆狅狊犻狋犻狅 If 犚 isageneralizednsemi regularring,then 犣 犑. 犘狉狅狅犳 Let0 犪 犣.Forany 犫 犚,we havethat 犫犪 犣.Put 狌 =- 犫犪.Then 狌 0, and 犾 狌 )=0since 犾 犫犪 ) 犾 狌 )=0.Thus 犚 = 狉犾 狌 )= 犘 犔,where 犘 狌犚 and 狌犚 犔 isnil. Wehavethat 犘 = 犲犚 with 犲 = 犲 犚.Henceitis suficienttoprovethat 犲 =.Ifnot,thenthere exists0 狉 - 犲 ) 犚 - 犲 ) 犾 犫犪 )since 犾 犫犪 ) isessentialin 犚.Thisgives 狉 - 犲 ) 狌 = 狉 - 犲 ). Put 狌 = 犲狊 + 狋 forsome 狊 犚, 狋 犔.Then 狉 - 犲 ) 狌 = 狉 - 犲 ) 狋.Hence 狉 - 犲 )= 狉 - 犲 ) 狋 andtherefor 狉 - 犲 )- 狋 )=0.Notethat 狋 = 狌 - 犲狊 狌犚 犔 Nil 犚 ). Then- 狋 isunit,whichimplies 狉 - 犲 )=0,a contradiction.sowehavethat 犲 =and 犚 = 犘 = 狌犚.Thus 犪 犑. 犆狅狉狅犾犪狉狔 4 If 犚 isa Nsemiregularring, then 犣 犑. Aring 犚 iscaledleftmininjective,ifevery isomorphism betweensimpleleftidealsisgiven [6] bymultiplicationbyanelementof 犚.Equviv alently,if 狉犾 犓 )= 犓 foreveryrightideals 犓 = 犽犚 forwhich 犚犽 issimple. 犜犺犲狅狉犲犿 If 犚 isageneralizednsemireg ular,leftmininjectiveringandsoc 犚犚 )isan essentialsubmoduleof 犚犚,then 犑 = 犣. Math. Colloq. Chin. Univ., 0/) 犘狉狅狅犳 Since 犚 isleftmininjective,soc 犚犚 ) Soc 犚犚 )by [6].By [6],Proposition.8)), 犑 犣.Since 犚 isgeneralized Nsemiregular,By Proposition, 犣 犑.Thus 犑 = 犣. Anidempotentelement 犲 犚 isleft resp. right)semicentral [7] in 犚 if 犚犲 = 犲犚犲 resp. 犲犚 = 犲犚犲 ). 犘狉狅狆狅狊犻狋犻狅 3 Let 犚 beageneralized N semiregularring.ifanidempotent 犲 of 犚 isleft semicentral,then 犲犚犲 isgeneralizednsemiregular. 犘狉狅狅犳 Let 犪 犲犚犲,thenthereexiststwo rightideals 犘 and 犔 of 犚,suchthat 狉犾 犪 )= 犘

7 第 / 期 YIN Xiaobin,etal GeneralizedNsemiregularrings 3 犔,where 犘 犪犚 and 犪犚 犔 isnil.weclaim thatr 犲犚犲犾犲犚犲 犪 )= 犘犲 犔犲.Infact, 犘犲 犔犲 犘 犔 =0.Takeany 狔 犘犲 犲犘犲,where 狔 = 狔 犲, 狔 犘 狉犾 犪 ).Thenforany 狓 犾犲犚犲 犪 ) 犾 犪 ), 狓狔 =0,whichgives 狓狔 = 狓狔 犲 =0.Hence 狔 狉犲犚犲犾犲犚犲 犪 ), 犘犲 狉犲犚犲犾犲犚犲 犪 ).Similarly,wehave 犔犲 狉犲犚犲犾犲犚犲 犪 ),ontheotherhand,take 狓 狉犲犚犲犾犲犚犲 犪 ),thenforany 狔 犾 犪 ),wehave 犲狔犲犪 =0since 犪 犲犚犲.So 犲狔犲狓 =0,whichgives 狔狓 = 犲狔犲狓 =0since 狓 犲犚犲 and 犲 isleftsemicentral. Thus 狉犲犚犲犾犲犚犲 犪 ) 狉犾 犪 ).Take 狓 = 狊 + 狋,where 狊 犘, 狋 犔.Then 狓 = 狓犲 = 狊犲 + 狋犲 犘犲 + 犔犲.This showsthat 狉犲犚犲犾犲犚犲 犪 )= 犘犲 犔犲. Itremainstoprovethat 犪犲犚犲 犔犲 isnil. Since 犘犲 犪犚犲 = 犪犲犚犲.Since 犲 isleftsemicen tral,wehave 犪犲犚犲 犔犲 犲 犪犲犚犲 犔犲 ) 犲 犲犚犲. But 犪犲犚犲 犔犲 犪犚 犔 isnil.thus 犲犚犲 isgener alizednsemiregular. 犜犺犲狅狉犲犿 Let 犲 beanidempotentof 犚 suchthat 犚犲犚 = 犚.If 犚 isageneralizednsemi regularring, 犲犚犲 isgeneralizednsemiregular. 犘狉狅狅犳 Let 犪 犲犚犲,thenthereexiststwo rightideals 犘 and 犔 of 犚 suchthat 狉犾 犪 )= 犘 犔,where 犘 犪犚 and 犪犚 犔 isnil.weclaim that 狉犲犚犲犾犲犚犲 犪 )= 犲犘犲 犲犔犲.Since- 犲 犾 犪 ), weseethat- 犲 ) 狋 =0forany 狋 犔,whichim plies 犔 = 犲犔.Similarly, 犘 = 犲犘,Thus 犲犘犲 犲犔犲 =0.Clearly, 犲犘犲 狉犲犚犲犾犲犚犲 犪 )and 犲犔犲 狉犲犚犲犾犲犚犲 犪 )since 犲犘 = 犘 and 犲犔 = 犔.Ontheother hand,take 狓 狉犲犚犲犾犲犚犲 犪 ),and write = 犻犪犻犲犫犻 forsome 犪犻, 犫犻 in 犚.Thenforany 狔 = 犾 犪 ),weget 犲犫犻狔犲犪 = 犲犫犻狔犪 =0foreach 犻.This impliesthat 犲犫犻狔犲狓 =0foreach 犻,whichgive 狔狓 = 狔犲狓 = 犪犻犲犫犻狔犲狓 = 0since 狓 犲犚犲.Thus 犻 = 狉犲犚犲犾犲犚犲 犪 ) 狉犾 犪 ).Take 狓 = 狊 + 狋,where 狊 犘, 狋 犔.Hence 狓 = 犲狓犲 = 犲狊犲 + 犲狋犲 犲犘犲 + 犲犔犲.This showsthat 狉犲犚犲犾犲犚犲 犪 )= 犲犘犲 犲犔犲,andtheclaim iscomplete.itremainstoprovethat 犪犲犚犲 犲犔犲 isnilsince 犲犘犲 犪犲犚犲.Since 犔 = 犲犔,wehave 犪犲犚犲 犲犔犲 犪犚 犔 isnil.thus 犲犚犲 isgeneral izednsemiregular. 犘狉狅狆狅狊犻狋犻狅 4 IfNil 犚 )=0,then 犚 isa generalizednsemiregularringifandonlyif 犚 is aleftapinjectivering. 犘狉狅狅犳 Onedirectionisobvious.Converse ly,forany 犪 犚, 狉犾 犪 )= 犘 犔,where 犘 犪犚 and 犪犚 犔 isnil.byassumption,nil 犚 )=0, hence 犪犚 犔 Nil 犚 )=0.Itisclearlythat 狉犾 犪 )= 犪犚 + 犔.So 狉犾 犪 )= 犪犚 犔,andthisim pliesthat 犚 isaleftapinjectivering. 犘狉狅狆狅狊犻狋犻狅 5 犚 isareducedandgeneral ized Nsemiregularringifand onlyif 犚 isa stronglyregular. 犘狉狅狅犳 Onedirectionisobvious.Converse ly,since 犚 isreduced, 犣 =0.ByProposition4, 犚 isleftapinjectiveand 犣 = 犑 =0.Letany0 犪 犚,then 犪 0andthereexistsarightideal 犔犪 of 犚 suchthat 狉犾 犪 )= 犪犚 犔犪.But 犾 犪 ) = 犾 犪 )since 犚 isreduced,thus 犪 = 犪狉 + 狓 with 狉 犚 and 狓 犔犪,whichimplies 犪 - 犪狉犪 = 狓犪 犪犚 犔犪 =0,so 犪 = 犪狉犪.Then- 狉犪 狉 犪 ) = 狉 犪 ),so 犪 = 犪狉犪.Thisprovedthat 犚 isa stronglyregularring. 犔犲犿犿犪 [8] Let 犮 犆 犚 ),where 犆 犚 )is centerof 犚.If 犮 isvonneumannregularin 犚, thensois 犮 in 犆 犚 ). 犜犺犲狅狉犲犿 3 Let 犚 beasemiprimitiveand generalizednsemiregularring,thenthecenter 犆 犚 )of 犚 isvonneumannregular. 犘狉狅狅犳 ByProposition, 犚 isleftnonsin gular,thus 犚 hasvonneumannregularmaximal leftquotientringssee,e.g.corolary.3 [] ). Consequently,the center 犆 犛 )of 犛 is von Neummanregularby Lemma.Forany 犪 犆 犚 ) 犆 犛 ),thereexists 狊 犆 犛 )suchthat 犪 = 犪狊犪 = 犪狊 = 狊犪.Then 犆 犚 )isreduced.by Proposition4, 犚 isleftapinjective.thusthere Math. Colloq. Chin. Univ., 0/) existsarightideal 犔犪 of 犚 suchthat 犪 狉犾 犪 )= 狉犾 犪 )= 犪犚 犔犪.AsintheproofofProposi tion5, 犪 isavon Neumannregularelementin 犚.UsingLemmaagain,weconcludethat 犪 is vonneumannregularin 犆 犚 ). 犚犲犳犲狉犲犮犲狊 : []Goodear K R.Ringtheory:nonsingularringsand modules[j]. 犘狌狉犲牔犃狆狆犾犕犪狋犺,976,8:06. []Goodear K R.Von Neumann Regular Rings[M]. Florida:KriegerPublishingCompany,99. [3]PageSS,Zhou Yiqiang.Generalizationsofprincipaly injectiverings[j]. 犑犃犾犵犲犫狉犪,998,06):7067.

8 宝鸡文理学院学报 自然科学版 ), 第 3 卷, 第 期, 第 5 页,0 年 3 月 JournalofBaojiUniversityofArtsandSciencesNaturalScience),Vol.3,No.,pp.5,Mar.0 DOI:CNKI:6-90/N htp:// 犆狅狏犲狉犵犲犮犲狅犳犱狅狌犫犾犲 犻犱犲狓犲犱狊犲狇狌犲犮犲狊犻犪犻狋狌犻狋犻狅犻狊狋犻犮犳狌狕狕狔狅狉犿犲犱狊狆犪犮犲 SIHaiyan,CAO Huaixin,YANGPing ColegeofMathematicsandInformationScience,ShaanxiNormalUniversity,Xi an7006,shaanxi,china) 犃犫狊狋狉犪犮狋 : 犃犻犿 Todiscusstheconvergenceofdoubleindexedsequencesinanintuitionisticfuzzynormed spaceifns)andprovethattheaddition,scalarmultiplicationandintuitionisticfuzzynormofdoubleindexed sequenceinanifnsarealstatisticalycontinuous. 犕犲狋犺狅犱狊 Byintroducingtheboundednessandconver genceofdoubleindexedsequencesinanifns,theaforesaidaimisdiscussed. 犚犲狊狌犾狋狊 Theresultsareob tainedthattheaddition,scalarmultiplicationandintuitionisticfuzzynormofdoubleindexedsequenceinan IFNSarealstatisticalycontinuous. 犆狅犮犾狌狊犻狅 Theresultshowsthatthetopologicalandalgebraicstruc turesinanifnsarecompatible. 犓犲狔狑狅狉犱狊 :convergence;doubleindexedsequence;continuity;intuitionisticfuzzynormedspace 犆犔犆狌犿犫犲狉 :O89.3 犇狅犮狌犿犲狋犮狅犱犲 :A 犃狉狋犻犮犾犲犐犇 :00760)00005 犕犛犆 00:46S40;54E50 直观模糊赋范空间中双指标序列的收敛性 司海燕, 曹怀信, 杨 萍 陕西师范大学数学与信息科学学院, 陕西西安 7006) 摘要 : 目的研究直观模糊赋范空间中双指标序列的收敛性, 证明直观模糊赋范空间中的加法 数 乘及观模糊范数的连续性 方法定义直观模糊赋范空间中双指标序列的收敛性及有界性 结果证明 直观模糊赋范空间中的加法 数乘及直观模糊范数关于双指标序列的连续性 结论 观模糊赋范空间中的代数结构与拓扑结构是相容的 本文结果说明了直 檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶檶 [4]KaplanskyI.RingsofOperators[M].New York: W.A.Benjamin,968. [5]ArmendarizEP.AnoteonextensionsofBearandP. P.rings[J]. 犑犃狌狊狋狉犪犾犕犪狋犺犛狅犮,974,8: [6]Nicholson W K,YousifM F.Mininjectiverings[J]. 犑犃犾犵犲犫狉犪,997,87: [7]BirkenmeierGF.Idempotentsandcompletelysemiprime ideals[j]. 犆狅犿犿犃犾犵犲犫狉犪,983,: [8]Xiao Guangshi,Tong Wenting.Generalizationsof semiregularrings[j]. 犆狅犿犿犃犾犵犲犫狉犪,005,33: [9]Nicholson W K.Semiregularmodulesandrings[J]. 犆犪犑犕犪狋犺,976,85):050. [0]XiaoGuangshi,YinXiaobin,TongWenting.Anote on APinjectiverings[J]. 犑犕犪狋犺,003,4): Math. Colloq. Chin. Univ., 0/) 404. []AndersonF W,FulerK R.Ringsandcategoriesof modules[m].new York:SpringerVerlag,974. Editor:DENGZhuping Proofreader:FANG Yan) Receiveddate:000,Modificationdate:005,Webpublication:00365:58. Foundationitems:SupportedbytheNNSFofChinaNo.0573,0874) Biography:SIHaiyan985),female,Huayin,Shaanxi,master,majorinoperatortheoryandwaveletanalysis. sihaiyan53@63.com Correspondingauthor:CAO Huaixin958),male,professor,majorinoperatortheoryandwaveletanalysis. caohx@snnu.edu.cn

9 第 / 期 SIHaiyan,etal Convergenceofdoubleindexedsequencesin 5 关键词 : 收敛性 ; 双指标序列 ; 连续性 ; 直观模糊赋范空间 中图分类号 :O89.3 文献标志码 :A 文章编号 :00760)00005 ThetheoryoffuzzysetswasintroducedbyZadehin965 [].AfterthepioneeringworkofZadeh, therehasbeenagreateforttoobtainfuzzyanaloguesofclassicaltheories.amongotherfields,pro gressivedevelopmentsaremadeinthefieldoffuzzytopology [0].Theconceptoffuzzytopologymay haveveryimportantapplicationsinquantum particlephysicsparticularlyinconnections withboth stringandε ) theorywhichweregivenandstudiedbyelnaschie [4] type.oneofthemostimportant problemsinfuzzytopologyistoobtainanappropriateconceptofintuitionisticfuzzymetricspace.this problemhasbeeninvestigatedbypark [5],whereanotionofintuitionisticfuzzymetricspacewasin troducedandstudied. TheaimofthispaperistodiscussconvergenceofdoubleindexedsequencesinanIFNS.Wewil provethattheaddition,scalar multiplicationandintuitionisticfuzzynorminanintuitionisticfuzzy normedspacearealstatisticalycontinuous. 犇犲犳犻犻狋犻狅 [6] The5tuple 犞, μ,ν,, )issaidtobeanifnsif 犞 isavectorspaceover 犉 犚, 犆 ), and arecontinuous 狋 normand 狋 conorm,respectively,andμ, νarefuzzysetson 犞 0, )suchthatforevery 狓, 狔 犞 and 狊, 狋 >0, a) μ 狓, 狋 )+ν 狓, 狋 ) ; b) μ 狓, 狋 ) >0 ; c) μ 0, 狋 )=forevery 狋 >0;ifthereexists 狋 >0suchthatμ 狓, 狋 )=,then 狓 =0; d) μ α 狓, 狋 )=μ 狓, 狋 )foreachα 0in 犉 ; 狘 α 狘 e) μ 狓, 狋 )μ 狔, 狊 ) μ 狓 + 狔, 狋 + 狊 ); f) μ 狓, ):0, ) [0,]iscontinuous; g)lim μ 狓, 狋 )=andlim μ 狓, 狋 )=0; 狋 狋 0 + h)ν 狓, 狋 ) < ; i)ν0, 狋 )=0forevery 狋 >0;ifthereexists 狋 >0suchthatν 狓, 狋 )=0,then 狓 =0; j)να 狓, 狋 )=ν 狓, 狋 )foreachα 0in 犉 ; 狘 α 狘 k)ν 狓, 狋 ) ν 狔, 狊 ) ν 狓 + 狔, 狋 + 狊 ); l)ν 狓, ):0, ) [0,]iscontinuous; m)lim ν 狓, 狋 )=0andlim ν 狓, 狋 )=; 狋 狋 0 + Inthiscase μ,ν)iscaledanintuitionisticfuzzynorm. 犈狓犪犿狆犾犲 [6] Let 犞, )beanormedspaceover 犉.Denote 犪 犫 = 犪犫 and 犪 犫 = min{ 犪 + 犫,} foral 犪, 犫 [0,]andletμ 0andν0befuzzysetson 犞 0, )definedasfolows μ 0 狓, 狋狋 )= 狋 + 狓, ν0 狓, 狋 )= 狓 狋 + 狓 foral 狋 犉 +.Then 犞, μ 0,ν0,, )isanifnsover 犉. 犇犲犳犻犻狋犻狅 [7] Math. Colloq. Chin. Univ., 0/) Let 犞, μ,ν,, )beanifns.{ 狓犼, 犽 } 犼, 犽 = 犞 issaidtobestatisticalyintu itionisticfuzzybounded 狊狋 IFB,forshort)if 狋 >0and0< 狉 <suchthat δ{ 犼, 犽 ): μ 狓犼, 犽, 狋 )- 狉 orν 狓犼, 犽, 狋 )} 狉 }=0. 犇犲犳犻犻狋犻狅 3 [7] Let 犞, μ,ν,, )beanifns.adoubleindexedsequence{ 狓犼, 犽 } 犼, 犽 =issaid tobestatisticalyintuitionisticfuzzyconvergentst IFC,forshort)to 狓 犞 withrespectedto μ,ν) ifforeveryε>0and 狋 >0,wehave δ{ 犼, 犽 ): μ 狓犼, 犽, 狋 ) -εorν 狓犼, 犽, 狋 )} ε }=0.

10 6 大学数学辑刊 MathColoquiumofChineseUniversities) 0 年 thatis, goesto0as 犿,.Inthiscasewewrite 狘 { 犼, 犽 ): μ 狓犼, 犽, 狋 ) -εorν 狓犼, 犽, 狋 )} ε } 狘犿 狊狋 μ,ν) - lim 狓犼, 犽 = 狓. 犼, 犽 犜犺犲狅狉犲犿 Let 犞, μ,ν,, )beanifnswith 犪 犫 >0whenever 犪, 犫 >0.If{ 狓犼, 犽 } 犼, 犽 =, { 狔犼, 犽 } 犼, 犽 = 犞 are 狊狋 IFB,thenforal 犮 犉 犮 0),{ 狓犼, 犽 + 狔犼, 犽 } 犼, 犽 =and{ 犮狓犼, 犽 } 犼, 犽 =arealso 狊狋 IFB. 犘狉狅狅犳 Since{ 狓犼, 犽 } 犼, 犽 =is 狊狋 IFB,wesee 狋 >0and0< 狉 <suchthatδ 犃 )=0,where 犃 = { 犼, 犽 ): μ 狓犼, 犽, 狋 ) - 狉 orν 狓犼, 犽, 狋 ) 狉 }. Using{ 狔犼, 犽 } 犼, 犽 = 狊狋 IFB,wehave 狋 >0and0< 狉 <suchthatδ 犅 )=0,where 犅 = { 犼, 犽 ): μ 狔犼, 犽, 狋 ) - 狉 orν 狔犼, 犽, 狋 ) 狉 }. Let 狋 = 狋 + 狋,choose0< 狉 <suchthat- 狉 )- 狉 ) >- 狉 and 狉 狉 < 狉.Now we showthat 犖 犖 \ 犃 犅 ) 犇,where 犇 isthesetofal 犼, 犽 )suchthatμ 狓犼, 犽 + 狔犼, 犽, 狋 ) >- 狉 andν 狓犼, 犽 + 狔犼, 犽, 狋 ) < 狉 }.Put Then 犼, 犽 ) 瓝犃 and 犼, 犽 ) 瓝犅,thatis, and Therefore, 犼, 犽 ) 犖 犖 \ 犃 犅 ). μ 狓犼, 犽, 狋 ) >- 狉 andν 狓犼, 犽, 狋 ) < 狉 μ 狔犼, 犽, 狋 ) >- 狉 andν 狔犼, 犽, 狋 ) < 狉. μ 狓犼, 犽 + 狔犼, 犽, 狋 )=μ 狓犼, 犽 + 狔犼, 犽, 狋 + 狋 ) μ 狓犼, 犽, 狋 )μ 狔犼, 犽, 狋 ) - 狉 )- 狉 ) >- 狉, and ν 狓犼, 犽 + 狔犼, 犽, 狋 )=ν 狓犼, 犽 + 狔犼, 犽, 狋 + 狋 ) ν 狓犼, 犽, 狋 ) ν 狔犼, 犽, 狋 ) 狉 狉 < 狉. Thisshowsthat 犼, 犽 ) 犇.So,δ 犇 )=.Hence,δ 犇犮 )= 0.By Definition,wesee { 狓犼, 犽 + 狔犼, 犽 } 犼, 犽 =is 狊狋 IFB. Similarly,let 狋 = 狘犮狘狋, 狉 = 狉,Now weshowthat 犖 犖 \ 犃 犓, 犓 isthesetofal 犼, 犽 )such thatμ 犮狓犼, 犽, 狋 ) >- 狉 andν 犮狓犼, 犽, 狋 ) < 狉. Let 犼, 犽 ) 犖 犖 \ 犃.Then 犼, 犽 ) 瓝犃,thatis, Therefore, and μ 狓犼, 犽, 狋 ) >- 狉 andν 狓犼, 犽, 狋 ) < 狉. μ 犮狓犼, 犽, 狋 )=μ 犮狓犼, 犽, 狘犮狘狋 )=μ 狓犼, 犽, 狋 ) >- 狉 =- 狉 ν 犮狓犼, 犽, 狋 )=ν 犮狓犼, 犽, 狘犮狘狋 )=ν 狓犼, 犽, 狋 )< 狉 = 狉. Thisshowsthat 犼, 犽 ) 犓.So,δ 犓 )=.Hence,δ 犓犮 )=0.ByusingDefinitionagain,we see{ 犮狓犼, 犽 } 犼, 犽 =isalso 狊狋 IFB. 犜犺犲狅狉犲犿 Math. Colloq. Chin. Univ., 0/) 犘狉狅狅犳 Let 狊狋 μ,ν) - lim InanIFNS,If 狊狋 μ,ν) - lim 狓犼, 犽 = 狓, 狊狋 μ,ν) - lim 狔犼, 犽 = 狔,then 犼, 犽 狊狋 μ,ν) 犼, 犽 - lim 犼, 犽 μ,ν) 犼, 犽 狓犼, 犽 ± 狔犼, 犽 )= 狓 ± 狔. 狓犼, 犽 = 狓 and 狊狋 - lim 狔犼, 犽 = 狔.Foragivenε>0,choose 狉 >0such 犼, 犽 that- 狉 )- 狉 )>-εand 狉 狉 <ε.then,forany 狋 >0,definethefolowingsets: 犓 μ, 狉, 狋 )= { 犼, 犽 ): μ 狓犼, 犽 - 狓, 狋 )- 狉 }, 犓 μ, 狉, 狋 )= { 犼, 犽 ): μ 狔犼, 犽 - 狔, 狋 )- 狉 }, 犓 ν, 狉, 狋 )= { 犼, 犽 ):ν 狓犼, 犽 - 狓, 狋 ) 狉 }, 犓 ν, 狉, 狋 )= { 犼, 犽 ):ν 狔犼, 犽 - 狔, 狋 ) 狉 }. Since 狊狋 μ,ν) - lim 狓犼, 犽 = 狓,wehaveδ 犓 μ, 狉, 狋 ))=0=δ 犓 ν, 狉, 狋 )) 狋 >0). 犼, 犽 μ,ν) Furthermore,using 狊狋 - lim 狔犼, 犽 = 狔,weget 犼, 犽

11 第 / 期 SIHaiyan,etal Convergenceofdoubleindexedsequencesin 7 δ 犓 μ, 狉, 狋 ))=0=δ 犓 ν, 狉, 狋 )) 狋 >0). Nowlet 犓 μ, ν 狉, 狋 )= 犓 μ, 狉, 狋 ) 犓 μ, 狉, 狋 )) 犓 ν, 狉, 狋 ) 犓 ν, 狉, 狋 )). Thenδ 犓 μ, ν 狉, 狋 ))=0,whichimpliesthatδ 犖 犖 \ 犓 μ, ν 狉, 狋 ))=.Now weshowthat 犖 犖 \ 犓 μ, ν 狉, 狋 )) 犅,where 犅 isthesetofal 犼, 犽 )suchthatμ 狓犼, 犽 ± 狔犼, 犽 )- 狓 ± 狔 ), 狋 ) >-εand ν 狓犼, 犽 ± 狔犼, 犽 )- 狓 ± 狔 ), 狋 ) <ε.let 犼, 犽 ) 犖 犖 \ 犓 μ, ν 狉, 狋 ).Then thatis 犼, 犽 ) 瓝犓 μ, 狉, 狋 ), 犼, 犽 ) 瓝犓 μ, 狉, 狋 ), 犼, 犽 ) 瓝犓 ν, 狉, 狋 )and 犼, 犽 ) 瓝犓 ν, 狉, 狋 ), μ 狓犼, 犽 - 狓, 狋 ) >- 狉 andν 狓犼, 犽 - 狓, 狋 ) < 狉, μ 狔犼, 犽 - 狔, 狋 ) >- 狉 andν 狔犼, 犽 - 狔, 狋 ) < 狉. Therefore, μ 狓犼, 犽 ± 狔犼, 犽 )- 狓 ± 狔 ), 狋 )=μ 狓犼, 犽 - 狓 )± 狔犼, 犽 - 狔 ), 狋 ) μ 狓犼, 犽 - 狓, 狋 )μ ± 狔犼, 犽 - 狔 ), 狋 ) - 狉 )- 狉 ) >-ε, and ν 狓犼, 犽 ± 狔犼, 犽 )- 狓 ± 狔 ), 狋 )=ν 狓犼, 犽 - 狓 )± 狔犼, 犽 - 狔 ), 狋 ) ν 狓犼, 犽 - 狓, 狋 ) ν± 狔犼, 犽 - 狔 ), 狋 ) 狉 狉 <ε. Thisshowsthat 犼, 犽 ) 犅.So,δ 犅 )=andδ 犅犮 )=0.Hence, 狊狋 μ,ν) - lim 狓犼, 犽 ± 狔犼, 犽 )= 狓 ± 狔. 犼, 犽 犜犺犲狅狉犲犿 3 Let 犞, μ,ν,, )beanifns.if 狊狋 μ,ν) - lim 狓犼, 犽 = 狓.Then foralα 犉. 狊狋 μ,ν) - lim 犼, 犽 α 狓犼, 犽 =α 狓 犼, 犽 犘狉狅狅犳 Thisisobviousforα=0.Nowletα 0.Thenforagivenε>0and 狋 >0,δ 犓 ε))=,δ 犓犮 ε))=0,where 犓 ε)denotesthesetofal 犼, 犽 )suchthatμ 狓犼, 犽 - 狓, 狋 ) >-εandν 狓犼, 犽 - 狓, 狋 ) <ε.foreveryε>0and 狋 >0,let 犃 bethesetofal 犼, 犽 )suchthatμ α 狓犼, 犽 - 狓, 狘 α 狘狋 ) >-εandνα 狓犼, 犽 - 狓, 狘 α 狘狋 ) <ε.itissuficienttoprovethat 犓 ε) 犃.Let 犼, 犽 ) 犓 ε).then wehaveμ 狓犼, 犽 - 狓, 狋 ) >-εandν 狓犼, 犽 - 狓, 狋 ) <ε.so,wehave and μ α 狓犼, 犽 - 狓, 狘 α 狘狋 )=μ 狓犼, 犽 - 狓, 狘 α 狘狋 ) 狘 α 狘 μ 狓犼, 犽 - 狓, 狋 )μ 0, 狋 )=μ 狓犼, 犽 - 狓, 狋 ) >-ε, να 狓犼, 犽 - 狓, 狘 α 狘狋 )=ν 狓犼, 犽 - 狓, 狘 α 狘狋 ) ν 狓犼, 犽 - 狓, 狋 ) ν0, 狋 )=ν 狓犼, 犽 - 狓, 狋 ) <ε. 狘 α 狘 Hence, 犼, 犽 ) 犃.So, 犓 ε) 犃.Usingδ 犓犮 ε)= 0,wehaveδ 犃犮 )= 0.Therefore, 狊狋 μ,ν) lim α 狓犼, 犽 =α 狓. 犼, 犽 犜犺犲狅狉犲犿 4 Let μ,ν)- lim 狓犼, 犽 = 狓 inanifns 犞, μ,ν,, ).Thenforevery 狋 >0, and 犘狉狅狅犳 Let 狓犼, 犽 μ,ν) 犼, 犽 狊狋 - lim μ 狓犼, 犽, 狋犼, )=μ 狓, 狋 ) 犽 狊狋 - lim ν 狓犼, 犽, 狋 )=ν 狓, 狋 ). 犼, 犽 - 狓 犼, 犽 )inanifnsv, μ,ν,, ).Thenforevery 狋 > 0,wehave μ 狓犼, 犽, 狋 ) μ 狓, 狋 ) 犼, 犽 )andν 狓犼, 犽, 狋 ) ν 狓, 狋 ) 犼, 犽 ). So,foreachε>0,and 狋 >0,thereexistsanumber 犽 0 犖 + suchthat 狘 μ 狓犼, 犽, 狋 )-μ 狓, 狋 ) 狘 <ε and 狘 ν 狓犼, 犽, 狋 )-ν 狓, 狋 ) 狘 <εforal 犼, 犽 犽 0.Hence,thesets and haveatmostfinitelymanyelements.thus, and Math. Colloq. Chin. Univ., 0/) { 犼, 犽 ): 狘 μ 狓犼, 犽, 狋 )-μ 狓, 狋 ) 狘 ε} { 犼, 犽 ): 狘 ν 狓犼, 犽, 狋 )-ν 狓, 狋 ) 狘 ε} δ { 犼, 犽 ): 狘 μ 狓犼, 犽, 狋 )-μ 狓, 狋 ) 狘 ε}=0 δ { 犼, 犽 ): 狘 ν 狓犼, 犽, 狋 )-ν 狓, 狋 ) 狘 ε}=0. Thatis, 狊狋 - lim μ 狓犼, 犽, 狋犼, )=μ 狓, 狋 )and 狊狋 - lim ν 狓犼, 犽, 狋 )=ν 狓, 狋 ). 犽 犼, 犽

12 8 大学数学辑刊 MathColoquiumofChineseUniversities) 0 年 犜犺犲狅狉犲犿 5 InanIFNS 犞, μ,ν,, ),let 狓犼, 犽, 狋犼, 犽 ) 狓, 狋 ) 犼, 犽 )in 犞 0, ).Then 狊狋 - lim μ 狓犼, 犽, 狋犼, 犽犼, )=μ 狓, 狋 ) 犽 and 狊狋 - lim ν 狓犼, 犽, 狋犼, 犽 )=ν 狓, 狋 ). 犼, 犽 犘狉狅狅犳 Let 狓犼, 犽, 狋犼, 犽 ) 狓, 狋 ) 犼, 犽 )in 犞 0, ).It is easy to get lim μ 狓犼, 犽, 狋犼, 犽犼, )=μ 狓, 狋 )andlim ν 狓犼, 犽, 狋犼, 犽 )=ν 狓, 狋 ). 犽 犼, 犽 So,foreachε>0and 狋 >0,thereexistsanumber 犽 0 犖 + suchthat 狘 μ 狓犼, 犽, 狋犼, 犽 )-μ 狓, 狋 ) 狘 < εand 狘 ν 狓犼, 犽, 狋犼, 犽 )-ν 狓, 狋 ) 狘 <εforal 犼, 犽 犽 0.Hence,thesets { 犼, 犽 ): 狘 μ 狓犼, 犽, 狋犼, 犽 )-μ 狓, 狋 ) 狘 ε} and haveatmostfinitelymanyelements.thus, and { 犼, 犽 ): 狘 ν 狓犼, 犽, 狋犼, 犽 )-ν 狓, 狋 ) 狘 ε} δ { 犼, 犽 ): 狘 μ 狓犼, 犽, 狋犼, 犽 )-μ 狓, 狋 ) 狘 ε}=0 δ { 犼, 犽 ): 狘 ν 狓犼, 犽, 狋犼, 犽 )-ν 狓, 狋 ) 狘 ε}=0. Thatis, 狊狋 - lim μ 狓犼, 犽, 狋犼, 犽犼, )=μ 狓, 狋 )and 狊狋 - lim ν 狓犼, 犽, 狋犼, 犽 )=ν 狓, 狋 ). 犽 犼, 犽 犚犲犳犲狉犲犮犲 : [] SCHWEIZERB,SKLAR A.Statisticalmetricspaces[J]. 犘犪犮犻犳犻犮犑犕犪狋犺,960,0): [] AMINIM,SAADATIR.Topicsinfuzzymetricspace[J]. 犉狌狕狕狔犕犪狋犺,003,4: [3] AMINIM,SAADATIR.SomepropertiesofContinuoustnormsandsnorms[J]. 犘狌狉犲犃狆狆犾犕犪狋犺,004,6: [4] HU C.CstructureofFTS,V:Fuzzymetricspaces[J]. 犑犉狌狕狕狔犕犪狋犺,995,3:77. [5] GEORGEA,VEERAMANIP.Onsomeresultofanalysisforfuzzymetricspaces[J]. 犉狌狕狕狔犛犲狋狊犛狔狊狋,997,90 3): [6] GREGORIV,ROMAGUERA S.Somepropertiesoffuzzy metricspaces[j]. 犉狌狕狕狔犛犲狋狊犛狔狊狋,000,53): [7] GREGORIV,ROMAGUERAS.Oncompletionoffuzzymetricspaces[J]. 犉狌狕狕狔犛犲狋狊犛狔狊狋,00,303): [8] GREGORIV,ROMAGUERAS.Characterizingcompletablefuzzymetricspaces[J]. 犉狌狕狕狔犛犲狋狊犛狔狊狋,004,44 3):440. [9] GREGORI V,ROMAGUERA S,VEERAMANIP.A noteonintuitionisticfuzzy metricspaces[j]. 犆犺犪狅狊, 犛狅犾犻狋狅狊犪犱犉狉犪犮狋犪犾狊,006,84): [0] LOWEN R.Fuzzysettheory[M].Dordrecht:KluwerAcademicPublishers,996. [] EINASCHIE M S.OntheuncertaintyofCantoriangeometryandtwoslitexperiment[J]. 犆犺犪狅狊, 犛狅犾犻狋狅狊犪犱 犉狉犪犮狋犪犾狊,998,93):5759. [] EINASCHIE M S.Onafuzzy Kahlerlike manifold whichisconsistent withthetwoslitexperiment[j]. 犐狋犲狉犪狋犻狅犪犾犑狅狌狉犪犾狅犳犖狅犾犻犲犪狉犛犮犻犲犮犲犪犱犖狌犿犲狉犻犮犪犾犛犻犿狌犾犪狋犻狅,005,6):9598. [3] ZADEH L A.Fuzzysets[J]. 犐犳狅狉犿犪狋犻狅犪犱犆狅狋狉狅犾,965,83): [4] EINASCJOE M S.AreviewofEinfinitytheoryandthemassspectrumofhighenergyparticlephysics[J]. 犆犺犪狅狊, 犛狅犾犻狋狅狊犉狉犪犮狋犪犾狊,004,9:0936. [5] MEGGINSON RE.AnIntroductiontoBanachSpaceTheory[M].New York:SpringerVerlag,998. [6] SAADATIR,PARKJH.Ontheintuitionisticfuzzytopologicalspaces[J]. 犆犺犪狅狊, 犛狅犾犻狋狅狊犪犱犉狉犪犮狋犪犾狊,006, 7): Math. Colloq. Chin. [7] MURSALEEN M,MOHIUDDINESA.Statisticalconvergenceofdoublesequencesinintuitionisticfuzzynormed spaces[j]. 犆犺犪狅狊, 犛狅犾犻狋狅狊犪犱犉狉犪犮狋犪犾狊,009,45):444. Univ., 0/) Editor:DENGZhuping Proofreader:LIZhefeng)

13 徐州师范大学学报 自然科学版 ), 第 9 卷, 第 期, 第 57 页,0 年 3 月 JournalofXuzhouNormalUniversityNaturalScienceEdition),Vol.9,No.,pp.57,Mar.0 犛狔犾狅狑塔群的新判别准则 黄建红 徐州师范大学数学科学学院, 江苏徐州 6) 摘要 : 设犉是一个群类. 群犌的子群犎称为在犌中犉犺正规, 如果犌有一个正规子群犜, 使得犎犜 犉是犌的正规 Hal 子群, 且 犎 犜 ) 犎犌 / 犎犌 犣 犌 / 犎犌 ). 利用犉犺正规子群的概念, 得到了关于 Sylow 塔群的一个新的判别准则. 关键词 : 有限群 ; 犉犺正规子群 ;Sylow 塔群 ;Sylow 子群 中图分类号 :O5 文献标志码 :A 文章编号 : ) 犃犲狑犮狉犻狋犲狉犻狅狅犳犛狔犾狅狑狋狅狑犲狉犵狉狅狌狆狊 HUANGJianhong SchoolofMathematicalScience,XuzhouNormalUniversity,Xuzhou6,Jiangsu,China) 犃犫狊狋狉犪犮狋 :Let 犌 beafinitegroupand 犉 aclassofgroups.asubgroup 犎 of 犌 issaidtobe 犉犺 nor malin 犌 ifthereexistsanormalsubgroup 犜 of 犌 suchthat 犎犜 isanormalhalsubgroupof 犌 and 犎 犜 ) 犎犌 / 犎犌 iscontainedinthe 犉 hypercenter 犣犉 犌 / 犎犌 )of 犌 / 犎犌.Byusingtheconceptof 犉犺 normalsubgroup,anewcriterionofsylowtowergroupisobtained. 犓犲狔狑狅狉犱狊 :finitegroup; 犉犺 normalsubgroup;sylowtowergroup;sylowsubgroup 犕犛犆 00:0D0 本文中, 所有群均是有限群. 未交待的概念和 符号参见文献 []. 群犌的子群犎称为在犌中可补, 如果犌有 一个子群犅, 使得犌 = 犎犅且犎 犅 =. 群犌的 子群犎称为在犌中犮正规, 如果存在犌的正规子 群犓, 使得犌 = 犎犓且犎 犓 犎犌, 其中犎犌是犌 的包含在犎中的最大正规子群 [3]. 进一步地, 杨 南迎和郭文彬教授引入了犉可补充子群的概 念 [4] : 设犉是一个群类, 群犌的子群犎称为在犌 中犉可补充, 如果犌有一个正规子群犜, 使得犌 = 犎犜且 犎 犜 ) 犎犌 / 犎犌含于犌 / 犎犌的犉超中心 犌 / 犎犌 ) 中. 利用以上这些子群, 人们已经得到 犉犣 了许多有趣的结论 [30]. 作为它们的推广, 冯秀仙 和郭文彬教授给出了犉犺正规子群的概念 []. 定义 [] 设犉是一个群类. 群犌的子群犎 称为在犌中犉犺正规, 如果犌有一个正规子群犜, 使得犎犜是犌的正规 Hal 子群, 且 犉 犎 犜 ) 犎犌 / 犎犌 犣 犌 / 犎犌 ). 显然, 对于任一非空群系犉, 所有的正规子 群 犮正规子群 犉可补充子群都是犉犺正规子 群. 但反之不成立 [, 例.]. 本文主要利用犉犺正规子群的概念, 得到了 关于 Sylow 塔群的一个新的判别准则. 有关概念和基本结果 设犌是一个群, 狆 > 狆 > > 狆狋是犌的不同 的素因子. 如果存在犌的 Sylow 狆犻子群犘犻 犻 =,,, 狋 ), 使得犘 犘 犘犽正规于犌 犽 =,,, 狋 ), 则称犌具有 Sylow 塔性 或称犌是一个 Sylow Math. Colloq. Chin. Univ., 0/) 塔群 ). 设犉是一个群类, 群犌的主因子犎 / 犓称为 犉中心的, 如果 [ 犎 / 犓 ] 犌 / 犆犌 犎 / 犓 )) 犉 [,]. 群犌的正规子群犖称为在犌中犉超中心的, 如 果犖的每个犌主因子在犌中是犉中心的. 用 犉犣 犌 ) 表示犌的所有非单位的犉超中心子群的 积, 并称为群犌的犉超中心. 群类犉称为一个群系, 如果犉是同态象封闭 且次直积封闭的. 群系犉称为饱和群系, 如果当 收稿日期 :009 基金项目 : 国家自然科学基金资助项目 079); 江苏省高校自然科学基金资助项目 0KJD0004) 作者简介 : 黄建红, 女, 讲师, 博士研究生, 主要从事有限群的研究, jhh30@6.com

14 0 大学数学辑刊 MathColoquiumofChineseUniversities) 0 年 犌 /Φ 犌 ) 犉时, 总有犌 犉. 群系犉称为犛闭的, 如果犌 犉时, 犌的每个子群都属于犉. 众所周 知, 所有 Sylow 塔群构成的群类是一个犛闭的饱 和群系. 引理 [] 犓 犌, 则下列结论成立 : 设犉是一个非空饱和群系, 犎 ) 犎在犌中犉犺正规, 当且仅当犌有一个正 规子群犜, 使得犎犜是犌的正规 Hal 子群, 犎犌 犉犜且犎 / 犎犌 犜 / 犎犌 犣 犌 / 犎犌 ); ) 设犎犌, 若犓在犌中犉犺正规, 则犓 / 犎 在犌 / 犎中犉犺正规 ; 3) 设犎犌, 若犈在犌中犉犺正规且 犈, 犎 )=, 则犈犎 / 犎在犌 / 犎中犉犺正规 ; 4) 若犎在犌中犉犺正规且犉是犛闭的, 则 犎在犓中犉犺正规 ; 5) 若犌 犉, 则犌的每个子群在犌中犉犺正规. 引理 [3] 假设犉是包含所有超可解群的 饱和群系 ; 群犌有一个正规子群犈, 使得犌 / 犈 犉. 若犈循环, 则犌 犉. 引理 3 [4] 设犌是一个有限群. 若犌的子群 犎在犌中狊置换, 则犎 / 犎犌是幂零的. 主要结论 定理 设犉是一个具有 Sylow 塔性的群 类, 则群犌 犉, 当且仅当犌的每个非循环 Sylow 子群的极大子群在犌中犉犺正规. 证必要性显然, 只需证明充分性. 假设命题不成立, 设犌为极小阶反例, 下面 分 3 步证明. ) 犌可解. 只需证犌有一个交换的正规子群. 若不成 立, 则对犌的任意 Sylow 子群犛, 有犛犌 = 且 犉犣 犌 )=. 设狆是犌的最小素因子, 犘是犌的 Sylow 狆子群, 则犘犌 =. 由 [5,0..9)] 知, 犘 非循环且 犘 > 狆. 令犘 是犘的极大子群. 由条 件知, 犘 在犌中犉犺正规, 即犌有一个正规子群 犜, 使得犘 犜是犌的正规 Hal 子群, 且犘 犜 犉犣 犌 )=. 这说明 犜狆 = 狆, 其中犜狆是犜的 Sy low 狆子群. 再由 [5,0..9)] 知, 犜为狆幂零 群, 从而犜有正规 Hal 狆 子群犜狆. 根据 Feit Thompson 定理, 犜狆 可解. 因为犜狆 char 犜犌, 所以犜狆 犌. 又因犘 犜 / 犜狆 是狆群, 故犘 犜可解. 显然, 狆 犌 / 犘 犜. 于是, 犌 / 犘 犜可解, 从而犌可 解, 矛盾. 因此, 犌可解. ) 犌有唯一极小正规子群犎且犎 Φ 犌 ). 令犎是犌的一个极小正规子群. 由 ) 知, 对 某一素数狆, 犎是交换狆群. 由引理 之 ),3) 知, 犌 / 犎满足条件. 由犌的选择知, 犌 / 犎 犉. 因为 具有 Sylow 塔性的群类是一个饱和群系, 所以犌 有唯一极小正规子群犎, 且犎 Φ 犌 ). 3) 导出矛盾. 若犎循环, 则由引理 知犌 犉. 故对某一素 数狆, 犎是初等交换狆群. 若犣犉 犌 ), 则由犎 的唯一性知犎 犣犉 犌 ), 从而犌 犉, 矛盾. 因此, 犉犣 犌 )=. 因为犎 Φ 犌 ), 所以存在犌的极大子群犕, 使得犌 =[ 犎 ] 犕. 设犕狆是犕的 Sylow 狆子群, 使得犌狆 = 犎犕狆是犌的 Sylow 狆子群. 令 犘 是犌狆的包含犕狆的极大子群. 显然, 犘 ) 犌 =. 由条件知, 犘 在犌中犉犺正规, 即存在犌的正 规子群犜, 使得犘 犜是犌的正规 Hal 子群, 且 犉犘 犜 犣 犌 )=. 这表明犜的 Sylow 狆子群 犜狆是狆阶的, 从而犜. 又因为犜 犌且犎是 犌的唯一极小正规子群, 所以犎为狆阶群, 矛盾. 由此, 定理得证. 定理 设犉是一个具有 Sylow 塔性的群 类, 则群犌 犉, 当且仅当犌 = 犃犅, 其中犃在犌中 狊置换, 犅 犉, 且犃的每个非循环 Sylow 子群的 极大子群在犌中犉犺正规. 证必要性显然成立, 只需证明充分性. 假设断言不成立, 设犌是使得 犌 犃 最小 的反例, 下面分 9 步进行证明. ) 犌的每个包含犃的真子群具有 Sylow 塔性. 假设犃 犕 < 犌, 则犕 = 犕 犃犅 = 犃 犕 犅 ), 且犕 犅 犉. 由引理 之 4) 知, 犃的每个非循环 Sylow 子群的极大子群在犕中犉犺正规. 对于犕的 任一 Sylow 子群犘, 存在犌的 Sylow 子群犌狆, 使得 犘 犌狆. 由于犃在犌中狊置换, 故 犃犘 = 犃 犕 犌 ) = 犕 犃犌狆 = 狆 犕 犌狆犃 = 犕 犌狆 ) 犃 = 犘犃, Math. Colloq. Chin. Univ., 0/) 即犃在犕中狊置换. 因此, 条件对犕成立. 由犌 的选择知, 犕 犉. ) 假设对某一素数狆, 犎是犌的非单位正规 狆子群. 若犎 犃, 或犃的 Sylow 狆子群循环, 或 犃的一个 Sylow 狆子群含于犎, 或 狆, 犃 )=, 则犌 / 犎 犉. 如果犃 犎, 那么犌 / 犎 = 犅犎 / 犎 犅 / 犅 犎 犉. 假设犃 犎. 因为 犌 / 犎 < 犌, 所以只需验证条 件对犌 / 犎成立. 由于犌 / 犎 = 犃犎 / 犎 ) 犅犎 / 犎 ), 犃犎 / 犎在犌 / 犎中狊置换且犅犎 / 犅 犅 / 犅 犎 犉, 故只需验证犃犎 / 犎的每个非循环 Sylow 子群的极 大子群在犌 / 犎中犉犺正规.

15 第 / 期 黄建红 Sylow 塔群的新判别准则 设犙 / 犎是犃犎 / 犎的 Sylow 狇子群, 犕 / 犎是犙 / 犎的极大子群, 则存在犃的 Sylow 狇子群犙, 使得犙 = 犙 犎. 假设犙 / 犎不循环, 则犙 也不循环. 若犎 犃, 则由引理 之 ),3) 知, 犕 / 犎在犌 / 犎中犉犺正规. 假设犃的 Sylow 狆子群循环, 或犃的一个 Sylow 狆子群含于犎, 或 狆, 犃 )=, 则狆 狇. 我们断言, 犕 犙 是犙 的极大子群. 事实上, 因为犕 犙且犙 犎 = 犙, 所以犕 犙 犙. 假设犌有一个子群犜, 使得犕 犙 犜 犙, 且犕 犙 犜 犙, 则犕 = 犎 犕 犙 ) 犎犜 犎犙 = 犙. 因此, 犕 = 犎犜或犎犜 = 犎犙. 若犕 = 犎犜, 则犜 犕 犙, 矛盾. 若犎犜 = 犎犙, 则犙 = 犙 犎犜 = 犙 犎 ) 犜 犙 犕 ) 犜 = 犜, 矛盾. 因此, 犕 犙 是犙 的极大子群. 由条件知, 犕 犙 在犌中犉犺正规. 故由引理 之 3) 知, 犕 / 犎 = 犕 犙 ) 犎 / 犎在犌 / 犎中犉犺正规. 这样, 犌 / 犎满足条件. 由犌的选择知, 犌 / 犎 犉. 3) 犃可解. 若犃 < 犌, 则由 ) 及犌的选择知犃可解. 若犃 = 犌, 则由定理 知犌为 Sylow 塔群, 从而犌可解. 4) 犌可解. 由条件知犃在犌中狊置换, 故犃在犌中次正 规, 从而由 3) 知, 犃含于犌的某个可解正规子群 犈中. 因为犌 / 犈 犈犅 / 犈 犅 / 犅 犈具有 Sylow 塔 性, 所以犌可解. 5) 犃 犉且犃 犌. 若犃 = 犌, 则由定理 知, 犌 犉, 矛盾. 因此犃 犌. 由 ) 及犌的选择知犃 犉. 6) 若 狆, 犃 )=, 其中狆为某一素数, 则犗狆 犌 )=. 令犔 = 犗狆 犌 ) 且犎 犔, 其中犎是犌的极小正规子群, 则由 ) 知犌 / 犎 犉. 令 π=π 犃 ). 因为犃次正规于犌, 所以犃 犗 π 犌 ), 从而犌 / 犗 π 犌 ) 犅 / 犅 犗 π 犌 ) 犉. 于是, 犌 犌 / 犎 犗 π 犌 )) 犉, 矛盾. 7) 犃不是幂零群. 假设犃幂零. 令犘是犃的 Sylow 狆子群. 因为犃是犌的次正规子群, 所以犘也是犌的次正规子群, 从而犘 犗狆 犌 ). 由 ) 知, 犌 / 犗狆 犌 ) 犉. 因为犉是群系, 所以犃 = 犘. 令狇是犌 / 犗狆 犌 ) 的最小素因子, 则存在犌的包含犘的极大子群犕, 使 犌 : 犕 = 狇. 令狉是犌的最大素因子, 犚是犕的 Sylow 狉子群, 则由 ) 知犚犕, 从而犚犌. 由 6) 得狉 = 狆, 从而犌 犉, 矛盾. 8) 犃犌. 由假设, 犃在犌中狊置换, 所以由引理 3, 犃 / 犃犌为幂零群. 由 7) 知犃犌. 9) 导出矛盾. 因为犃犌, 所以可令犎是犌的含于犃的极 小正规子群. 由 4) 知, 犌为可解群, 从而对某一素数狆, 犎是交换狆群. 令犘是犃的 Sylow 狆子群, 狉是 犃的最大素因子, 且犚是犃的 Sylow 狉子群. 由 5) 知犚犃, 从而犚次正规于犌. 因此, 犚 犗狉 犌 ). 由 ) 知犌 / 犗狉 犌 ) 满足条件, 故由犌的选择知, 犌 / 犗狉 犌 ) 犉. 再由 ) 知犌 / 犎 犉. 因此, 狉 = 狆, 从而犘犃. 显然, 犘在犌中狊置换. 又因为犉是一个饱和群系, 所以犎 Φ 犌 ). 于是, 存在犌的一个极大子群犕, 使得犌 = 犎犕且犕 犌 / 犎 犉. 考虑犌 = 犘犕. 由 犌 犃 的极小性知犘 = 犃, 与 7) 矛盾. 综上可知, 定理得证. 参考文献 : [] DoerkK,Hawkes T.Finite Soluble Group[M]. Berlin:WalterdeGruyter,99. [] Guo Wenbin.TheTheoryofClassofGroups[M]. Beijing:SciPress& KluwerAcadPubl,000. [3] WangYanming. 犮 Normality of groups and its properties[j]. 犑犃犾犵犲犫狉犪,996,803):954. [4] YangNanying,Guo Wenbin.On 犉 supplemented subgroupsoffinitegroups[j]. 犃狊犻犪 犈狌狉狅犑犕犪狋犺, 008,4):69. [5] GuoXiuyun,Shum KP.On 犮 normalmaximaland minimalsubgroupsofsylow 狆 subgroupsoffinite groups[j]. 犃狉犮犺犕犪狋犺,003,806):56. [6] 胡滨. 关于有限群的犛 犆置换嵌入子群的新探讨 [J]. 徐州师范大学学报 : 自然科学版,00,8):5. [7] LiDeyu,GuoXiuyun.Theinfluenceof 犮 normality ofsubgroupsonthestructureoffinitegroups[j]. 犑犘狌狉犲犃狆狆犾犃犾犵犲犫狉犪,000,50):53. [8] WeiHuaquan.On 犮 normal maximaland minimal subgroupsofsylowsubgroupsoffinitegroups[j]. 犆狅犿犿犃犾犵犲犫狉犪,00,95):93. [9] AlsheikA Y,JaradenJJ,SkibaA N.On 犝犮 nor mal subgroups of finite groups [J]. 犃犾犵犲犫狉犪犆狅犾狅狇狌犻狌犿,007,4):5. [0] Guo Wenbin,Skiba A N.On 犡 permutablesub groupsoffinitegroups[j]. 徐州师范大学学报 : 自然科学版,009,7):6. [] FengXiuxian,Guo Wenbin.On 犉犺 normalsub groupsoffinitegroups[j]. 犉狉狅狋犕犪狋犺犆犺犻犪, 00,54):653. Math. Colloq. Chin. Univ., 0/)

16 杭州师范大学学报 自然科学版 ), 第 0 卷, 第 期, 第 733 页,0 年 月 JournalofHangzhouNormalUniversityNaturalScienceEdition),Vol.0,No.,pp.733,Jan.0 DOI:0.3969/j.issn.6743X 犈狉狉狅狉犪犪犾狔狊犻狊狅狉犲犵狌犾犪狉犻狕犲犱犾犲犪狉犻犵 ZHANGJixiong,WANGJianli,SHENGBaohuai.ColegeofScience,HangzhouNormalUniversity,Hangzhou30036,Zhejiang,China;.DepartmentofMathematics,ShaoxingUniversity,Shaoxing3000,Zhejiang,China) 犃犫狊狋狉犪犮狋 :Theerroranalysisonakindofregularizationlearningalgorithmisproposed,andthe sampleerrorisoferedbythelargenumberlaw,andtheestimationforapproximationerrorisgiven witha 犓 functional. 犓犲狔狑狅狉犱狊 :learningtheory;regularizationframework;function reconstruction;reproducing kernelhilbertspaces 犆犔犆狌犿犫犲狉 :O74.4 犃狉狋犻犮犾犲犮犺犪狉犪犮狋犲狉 :A 犃狉狋犻犮犾犲犐犇 :6743X0) 犕犛犆 00:68Q3;68T05 摘 正则化学习算法的误差分析 张际雄, 王建力, 盛宝怀. 杭州师范大学理学院, 浙江杭州 30036;. 绍兴文理学院数学系, 浙江绍兴 3000) 要 : 给出了一类正则化样本学习算法的误差分析. 借助于大数定律给出了样本误差, 用一种 犓 泛函给出了逼近误差的估计. 关键词 : 学习理论 ; 正则化模型 ; 函数重构 ; 再生核 Hilbert 空间 中图分类号 :O74.4 文献标志码 :A 文章编号 :6743X0) 犐狋狉狅犱狌犮狋犻狅 ItisknownthattheShannonsamplingtheoremisthecoreoffunctionreconstructionwhichisthetheory foundationofsignalprocessandmanyotherrelatedfieldsseee.g.[4]).todealwiththenoiseinthesam plingdata,s.smaleandd.x.zhouconsidered,fromtheviewoflearningtheoryandregressionanalysis, theregularizationlearningalgorithmforrandomsamplingandprovidedaneroranalysisinprobabilitysee [4]).Inthepresentpaper,weshalgivesomefurtherinvestigationsontheestimateoftheeroranalysisfor thealgorithm.tothisend,weneedtostatethelearning framework. 檵檵檵檵檵檵檵檵檵檵檵檵檵檵檵檵檵檵檵檵檵檵檵檵檵檵檵檵檵檵檵檵檵檵檵檵檵檵檵檵檵檵檵檵檵檵檵 [] ShemetkovL A,SkibaA N.FormationsofAlge braicsystems[m].moscow:nauka,989. [3] Guo Wenbin.On 犉 supplementedsubgroupsoffi nitegroups[j]. 犕犪狌狊犮狉犻狆狋犪犕犪狋犺,008,7 ):39. Math. Colloq. Chin. [4] ItN,Szép J. Uber die quasinormalteiler von Univ., 0/) Endlichengruppen[J]. 犃犮狋犛犮犻犕犪狋犺,96,3: 68. [5] RobinsonDJS. A Course in the Theory of Groups[M].New York:SpringerVerlag,98. 编辑 : 邓珠平校对 : 刘春林 ) Receiveddate:00090 Foundationitem:SupportedbytheNationalNSF 0876)ofPRC. Biography:ZHANGJixiong985),male,borninJiujiang,JiangxiProvince,master,engagedinlearningtheory. Correspondingauthor:SHENGBaohuai96),male,borninBaoji,ShaanxiProvince,China,Ph.D.,professor,engagedin learningtheory. bhsheng@usx.edu.cn

17 第 / 期 ZHANGJixiong,etal Erroranalysisonregularizedlearning 3 Let 狇 beapositiveinteger, 犡 beacompactsubsetofthe 狇 dimensionaleuclideanspace 犚狇, 狋 犡 beagivendiscretesubset.foracontinuoussymmetricmap 犓 狓, 狔 )= 犓狓 狔 ): 犡 犡 犚 wedefinea matrixpossiblyinfinite) 犓狋, 狋 : 犾 狋 ) 犾 狋 )correspondingto 狋 by 犓狋, 狋犪 ) 狊 = 狊, 狋 ) 犪狋, 狊 狋, 犪 犾 狋 ). 狋 狋犓 Here 犾 狋 )isthesetofsequence 犪 = 犪狋 ) 狋 狋 withtheinnerproductdefinedas 犪, 犫 犾 狋 ) = for 犪狋 狋犪狋犫狋 = 犪狋 ) 狋 狋 and 犫 = 犫狋 ) 狋 狋.Weassume 犓狋, 狋 isweldefined,bounded,andpositivesemidefinite,i.e., foranygivenpositiveintegers 犿 andanyfinitesetofdistinctpoints 狋 ={ 狓, 狓,, 狓犿 } 犡,theMer cerkernelmatrices 犓狋, 狋 = 犓 狓犻, 狓犼 )) 犿犻, 犼 =arepositivesemidefinite.thenorm 犓狋, 狋犾 Suchmapsarecaled Mercerkernels. 狋 ) = sup α 0,α 犾 狋 ) 犓狋, 狋 α 犾 狋 ) α 犾 狋 ). 犓狋, 狋犾 狋 )isdefinedas Let 犘 beanonnegativeborelmeasureon 犡 犚, 犘犡 bethemarginalmeasureinducedby 犘 on 犡, i.e.,themeasureon 犡 definedby 犘犡 犛 )= 犘 π - 犛 )),whereπ: 犡 犚 犡 istheprojection.notice that 犘 狓, 狔 ), 犘 狔 狓 )and 犘犡 狓 ), 狓 犡,satisfythefolowingrelationseee.g.[],[5]) φ 狓, 狔 )d 犘 = 犡 犚 犡 φ 狓, 狔 )d 犘 狔狘狓 ))d 犘犡 狓 ). 犚 This breaking of 犘 intothemeasure 犘 狔 狓 )and 犘犡 狓 )correspondstolookingat 犡 犚 asthe productof 犡 and 犚. Let 狕 ={ 狕狓 } 狓 犡 ={ 狓, 狔狓 )} 狓 犡 bearandomsampledrawnindependentaccordingtotheunknown 犘 狓, 狔 ), 狑 ={ 狑狓 } 狓 犡 beaweightsequencewith 狑狓 >0.Then,S.SmaleandD.X.Zhouintroduced in [4]thefolowingregularizedsamplinglearningframeworks 犳狕,γ 狓 ):= 犳狕,γ, 犓, 狓 )= 狋犪狉犵 min 狑狓 犳 犳 狓 )- 狔狓 ) +γ 犳犓 ), ) 犓, 狋狓 犡 where>λ>0aregivenregularizationparameterswhicharecommonlyusedtoovercometheilpos edness, 狋 isagivendiscretesubsetof 犡, 犓, 狋 isahilbertspacedefinedasthecompleteofthelinear spanoffinitelinearcombinationsof 犓狋, 狋 狋,i.e., 狋 狋犪狋犓狋 狓 )withonlyafinitenumberof 犪狋 arenon zero.aninnerproducton 犓, 狋 isdefinedbylinearextensionfrom 犓狋, 犓狊 犓 = 犓 狋, 狊 )andhasthefolow ingreproducingproperty 犳, 犓狓 犓 = 犳 狓 ), 狓 犡, 犳 犓, 狋.Denotedby 犔 犘犡 )= { 犳 狓 ): 犳, 犘犡 = 狘犳 狓 ) 狘 d 犘犡 ) <+ }.If 犓 狓, 狔 )satisfies 犓狓 犔 犘犡 )forevery 狓 犡 and 犡 狘犓 狓, 狔 ) 狘 犡犡 d 犘犡 狓 )d 犘犡 狔 ) <+, ) then,by[6]or[7]weknow 犓, 狋 犔 犘犡 )isacompactembeddingandthereisaconstant 犽 >0suchthat Math. Colloq. Chin. Univ., 0/) 犳, 犘 犡槡犽 犳, 犓, 狋犳 犓, 狋. 3) Foragiven Mercerkernelon 犡 犡 wedefinethelinearoperator 犓犡, 狋 : 犾 狋 ) 犾 狓 )anditsad joint 犓狋, 犡 : 犾 犡 ) 犾 狋 )by 犓犡, 狋犪 ) 狓 = 犓 狓, 狋 ) 犪狋, 狓 犡,andassume 犓犡, 狋 andhence 犓狋, 犡 )iswel 狋 狋 definedandbounded.further,denotedby 犇狑 : 犾 犡 ) 犾 犡 )thediagonalmatrixmultiplicationoper atoron 犾 犡 ))withmaindiagonalentries{ 狑狓 } 狓 犡.Then,[4]showedthat)hasauniquesolution ofthetype 犳狕,γ 狓 ):= 狋 狋 犔狔 ) 狋犓狋 狓 ), 狓 犡, 4) where 狔 ={ 狔狓 } 狓 犡 isthesampleset,and 犔 = 犓狋, 犡犇狑犓犡, 狋 +γ 犓狋, 狋 ) - 犓狋, 犡犇狑.Ontheotherhand,by [8]weknowthatthemostidealregressionfunctionw.r.t.withrespectto)to 犘 is 犳 狓 ):= 犪狉犵 min 犳 犘 犳 )= 犚狔 d 犘 狔狘狓 ), 5) where 犘 犳 )= 犣 狔 - 犳 狓 )) d 犘 and 犣 := 犡 犚,theminimumistakenoveralthemeasurablefunc

18 4 大学数学辑刊 MathColoquiumofChineseUniversities) 0 年 狋犻狅狊犳 狓 )w.r.t. 犘犡.Hence,theerroranalysisbetween 犳狕,γand 犳 isneeded.whenthesample numberofthesample 狕 ={ 狓, 狔狓 )} 狓 犡 isafinitenumber 犿 andtheweights 狑狓 =,theerroranalysis 犿 hasbeendiscussedfulyby many mathematicians see [],[5]).Inthepresentpaper,weshal showarateforerror 犘 犳狕,γ ) - 犘 犳 ).4)remindsustorewrite)as α : 狕,γ = 犪狉犵 min ) 狑狓 狔狓 α 犾 - 犳 α 狓 )) +γ 犳 α 犓 ), 6) 狋狓 犡 where 犳 α 狓 )= α 狋犓狋 狓 ),α= α 狋 ) 狋 狋 犾 狋 ).Thecorespondingintegralregularizedriskminimizationis 狋 狋 α 犘 ) :=α 犘 ) λ,γ = 犪狉犵 min 犘 犳 ) α +γ 犳 α 犓 ). 7) α 犾 狋 ) Fromnowonweshalassumethatthedistributions 犘 狓, 狔 )satisfy 狘犘狘 = 犣狔 d 犘 <+,there arerealnumbersλ 狋 >0andaconstant 犆 >0suchthatthekernels 犓 狓, 狔 )satisfy λ 狋狘犓 狓, 狋 ) 狘 狋 狋 犆 <+ uniformlyholdson 犡 anddefineby 犙狕 狓, 狔 )theempiricalmeasureon 犣 as 犙 [ 狕犳 狓, 狔 )]= 犳 狓, 狔 )d 犙狕 = 犣 狓, 狔狓 ) 8) 狓 犡狑狓犳 foranybounded 犘 measurablefunction 犳 狓, 狔 )on 犣 andbyδ 犻, 犼 wedenotetheδfunctionwhosevalues isif 犻 = 犼 and0if 犻 犼. Withabovenotationsinhand,wegivethemainresultofthepresentpaper. 犜犺犲狅狉犲犿 Let 犓 狓, 狔 )beamercerkernelon 犡 犡 satisfying ).α 狕,γisthesolutionof6), 狑狓 =.Then,forany0<δ<,withconfidence-δ,thereholds 狓 犡 8 犆犽狘犘狘 犘 犳 ) α - 犘 + 狕犳 ),γ 槡 ) 烄 狓 γ 烌 狓狓 犡狑 狓 犡狑 + γγ 狋槡 δ 槡 δ + 犓 犳,γ), 9) 烆烎 where 狘犘狘 = 犣狔 d 犘,Γ 狋 = inf α 犾 狋 ) α 犓狋, 狋 α Τ 犓 犳,γ)= inf α 犘 犳 α 犾 狋 )and ) - 犘 犳 )+γ 犳 α 犓 ). α 犾 狋 ) Theestimateforthe 犓 functional 犓 犳,γ)dependsupontheuniversalityoftheMercerkernels seee.g.[9]).itisrelatedtotherkhsapproximationproblemsee[],[0],[]).γ 犘 ) 犓, 狋 andγ 狋 maybeestimatedaccordingtothewayusedin[].inarticular,ifthesamplenumberofthesample 狕 ={ 狓, 狔狓 )} 狓 犡 is 犿 andweights 狑狓 =,wehave 犿 8 犆犽狘犘狘 犘 犳 ) α - 犘 + 狕犳 ),γ 槡 ) γ γγ + ) 狋 δ 犿槡 δ + 犓 犳,γ). 犜犺犲犚狅犫狌狊狋犲狊 BeforeshowingTheoremwegivetherobustnessofthesolutions. 犘狉狅狆狅狊犻狋犻狅 Let 犘, 犙 bedistributionson 犣 with 犘 <+, 犙 <+. 犓 狓, 狔 )aremercer kernelson 犡 犡 satisfying).α 犘 ) andα 犙 )aresolutionsofscheme7)fordistributions 犘 and 犙 re spectively, 犓狋 狓 )= 犓 狓, 狋 )) 狋 狋.Then,thereholds α 犘 ) -α 犙 ) 犾 狋 ) 狔 - 犳 α γγ 狋 犘 ) 狓 )) 犓狋 狓 )d 犘 - 狔 - 犳 α 犣 犘 ) 狓 )) 犓狋 狓 )d 犙, 0) 犣犾 狋 ) whereforthevectorfunction 犳 狓, 狔 )={ 犳狋 狓, 狔 )} 狋 狋 andafunctionα 狓 )wedefine 犳 狓, 狔 )α 狓 )= 犳狋 狓, 狔 )α 狓 )) 狋 狋 and 犳 狓, 狔 )α 狓 )d 犘 = 狓, 狔 )α 狓 )d 犘 ) 狋 狋. 犣 犣犳狋 犔犲犿犿犪 Let 犘 beadistributionon 犣 = 犡 犚 suchthat 犘 < +. 犓 狓, 狔 )arekernelson 犡 犡 satisfying ).Then, Math. Colloq. Chin. Univ., 0/) 犿槡

19 第 / 期 ZHANGJixiong,etal Erroranalysisonregularizedlearning 5 i)thereexistsuniquelyaminimizerofα 犘 ) ofproblem 7)and 犳 α 犘 ) 犽 犘, 犘犡槡 γ. ) i)thereholdsthefolowingrelation 犓狋, 狋 α 犘 ) = γ 犣 狔 - 犳 α 犘 ) 狓 )) 犓狋 狓 )d 犘. ) 犘狉狅狅犳 Theuniquenessoftheminimizercanbeobtainedbythefactthatγ>0.Sinceα 犘 ) isthe minimizerof7),wehave 犘 犳 α 犘 ) ) +γ 犳 α 犘 ) 犓 犘 犳 0)= 犘 whichmakesγ 犳 α 犘 ) 犓 犘, combingwhichwith 3)yields). Since 犳 α 犓 =α Τ 犓狋, 狋 α,bythedefinitionofα 犘 ) andthefermattheorem weknowthegradient α 犘 犳 α ) +γ 犳 α 犓 )aboutαatα 犘 ) is0,i.e., 0= α 狔 - 犳 α 狓 )) d 犘 ) 犣 α=α 犘 ) +γ 犓狋, 狋 α 犘 ) =- 狔 - 犳 α 犘 ) 狓 )) 犓狋 狓 )d 犘 +γ 犓狋, 狋 α 犘 ). 犣 )thenholds. Proofof0).Theequality 狓 - 狔 狔 狓 - 狔 ), 狓, 狔 犚,yields Therefore, Also,theequalitysince 狔 - 犳 α 犙 ) 狓 )) - 狔 - 犳 α 犘 ) 狓 )) - 狔 - 犳 α 犘 ) 狓 )) 犳 α 犙 ) 狓 )- 犳 α 犘 ) 狓 )). 狔 - 犳 α 犙 ) 狓 )) d 犙 - 狔 - 犳 α 犣 犘 ) 狓 )) d 犙 犣 α 犙 ) -α 犘 ),- 狔 - 犳 α 犘 ) 狓 )) 犓狋 狓 )d 犙 ) 犾 狋 ). 3) 犣 犓, 狋 arehilbertspaces) 犳 α 犙 ) 犓 - 犳 α 犘 ) 犓 = 犳 α 犘 ), 犳 α 犙 ) - 犳 α 犘 ) ) + 犳 α 犘 ) - 犳 α 犙 ) 犓 4) and )yields 犙 犳 α 犙 ) ) +γ 犳 α 犙 ) 犓 )- 犙 犳 α 犘 ) ) +γ 犳 α 犘 ) 犓 ) α 犙 ) -α 犘 ), 狔 - 犳 α 犘 ) 狓 )) 犓狋 狓 )d 犘 - 狔 - 犳 α 犣 犘 ) 狓 )) 犓狋 狓 )d 犙 ) 犾 狋 )+γ 犳 α 犘 ) 犙 ) -α 犓. 犣 Thedefinitionsofα 犘 ) andα 犙 ) yields 犙 犳 α 犙 ) ) +γ 犳 α 犙 ) 犓 )- 犙 犳 α 犘 ) ) +γ 犳 α 犘 ) 犓 ) 0.Therefore, γ 犳 α 犘 ) 犙 ) -α 犓 α 犘 ) -α 犙 ), 狔 - 犳 α 犘 ) 狓 )) 犓狋 狓 )d 犘 - 狔 - 犳 α 犣 犘 ) 狓 )) 犓狋 狓 )d 犙 ) 犾 狋 ). 犣 Since 犳 α 犘 ) 犙 ) -α 犓 =α 犘 ) -α 犙 ) ) 犓狋, 狋 α 犘 ) -α 犙 ) ) T Γ 狋 α 犘 ) -α 犙 ) γγ 狋 α 犘 ) -α 犙 ) 犾 狋 ) α 犘 ) 犾 狋 ),wehave -α 犙 ), 狔 - 犳 α 犘 ) 狓 )) 犓狋 狓 )d 犘 - 狔 - 犳 α 犣 犘 ) 狓 )) 犓狋 狓 )d 犙 ) 犣 狔 - 犳 α 犘 ) 狓 )) 犓狋 狓 )d 犘 - 狔 - 犳 α 犣 犘 ) 狓 )) 犓狋 狓 )d 犙 犣犾 狋 ) α 犘 ) -α 犙 ) 犾 狋 ). 5) 5)yields0).Replacingthedistributions 犙 with 犙狕 in 0)wehavethefolowingcorolary. 犆狅狉狅犾犪狉狔 Let 犓 狓, 狔 )satisfytheconditionsoftheorem, 犘 beadistributionon 犣 with 犘 <+,α 犘 )andα 狕,γbethesolutionsof7)and 6)respectively.Then, α 犘 )-α 狕,γ 犾 狋 ) 狔 - 犳 α γγ 狋 犘 ) 狓 )) 犓狋 狓 )d 犘 - 狑狓 狔狓犣 - 犳 α 犘 ) 狓 )) 犓狋 狓 ). 6) 狓 犡犾 狋 ) 3 犜犺犲犛犪犿狆犾犲犈狉狉狅狉 Thedefinitionsofα 狕,γ, α 犘 )and 犳 yields 犘 犳 α 狕,γ ) - 犘 犳 )= 狘犘 犳 α 狕,γ ) - 犘 犳 ) 狘 狘犘 犳 α 犘 ) ) - 犘 犳 α 狕,γ ) 狘 + 犘 犳 α 犘 ) ) - 犘 犳 )= 犃 + 犅, where 犃 = 犘 犳 α 犘 ) ) - 犘 犳 α 狕,γ ) isusualycaledthesampleerror. 犅 = 犘 犳 α 犘 ) ) - 犘 犳 )isoften caledapproximationerrorwhoseestimateisoftensumuptoa 犓 functional.inthissectionweshal sumupthesampleerrorbetween 犳 α 犘 ) and 犳 α 狕,γ tothesampleerrorbetweenα 犘 ) andα 狕,γ. 犘狉狅狆狅狊犻狋犻狅 Let 犓 狓, 狔 )beamercerkernelon 犡 犡 satisfying ). 犘 isadistributionon 犣 with 犘 <+,α 犘 )andα 狕,γaredefinedasin7)and6)respectively.Then,forany0<δ<,with confidence-δ,thereholds Math. Colloq. Chin. Univ., 0/)

20 6 大学数学辑刊 MathColoquiumofChineseUniversities) 0 年 ) 犆犽槡狘犘狘 + 烄狘犘 犳 α 犘 ) ) - 犘 槡犳 ) γ α 狘狕,γ min λ + 狋槡狋 狋烆 Toshow 7),weneedalargenumberlaw. 烌狓 狓 犡狑 α 犘 )-α 狕,γ. 7) δ 烎 犔犲犿犿犪 Let 犎, 犎 )beahilbertspace, ξ bearandomvariableon 犣, ρ )andwithvalues in 犎,{ 狓 } 狓 犡 beindependentrandomdrawersofρ, 狑狓 >0and 狓 犡狑狓 =.Then,forany0<δ<, withconfidenceatmost-δ,thereholds 狑狓 ξ 狓 )- 犈 ξ ) 犎 狓 犡 槡 σ ξ ) 狓 狓 犡狑, 8) 槡 δ where 犈 ξ )isthemeanoftherandomvariableξandσ ξ )= 犣 ξ- 犈 ξ ) 犎 dρ. 犘狉狅狅犳 Takeη= 狓 犡狑狓 ξ 狓 )- 犈 ξ ) 犎,then, η isarandomvariableaboutthe 狓 of 犡.Bythe [ ] Markovsinequalitywehave 犘 犈 η >ε) η ) = 犈 ε ε 狑狓 ξ 狓 )- 犈 ξ ) 狓 犡犈 [ 狑狓 ξ 狓 )- 犈 ξ ) ] = 犈犎 狓 犡狓 犡 犎,where ) = 狑狓 ξ 狓 )- 犈 ξ ), 狓 犡狑狓 ξ 狓 )- 犈 ξ ) 犈 ξ 狓 )- 犈 ξ ), ξ 狓 )- 犈 ξ ))=σ 狓, 狓 犡狑狓狑狓 ξ ) 狑狓 犡 σ ξ ) 狑狓 σ ξ ) 狓 σ 狓 犡狓 Hence, 犘 η >ε) and 犘 ε η ε) - 犡狑 ξ ) 狓 狓 犡狑.Take = δ,then, ε ε σ ξ ) 狓 狓 犡狑 ε= and 8)holds. 槡 δ Proofof7).Equality 4)yields 狘犘 犳 α 犘 ) ) - 犘 犳 ) α 狘狕,γ { 犘 犳 ) α +γ 狕犳,γ α 狕犓 )-,γ 犘 犳 α 犘 ) ) +γ 犳 α 犘 ) 犓 )}+γ 狘 犳 α 犘 ) 犓 - 犳 α 狕,γ 犓狘 犃 +γ α 狕,γ -α 犘 ), 犓狋, 狋 α ) 狕,γ 犾 γ 犳 α 犘 ) - 犳 α 狕犓,γ 犃 +γ α 狕,γ -α 犘 ) Then,theinequality 狔 - 狓 狔 狔 - 狓 ), 狓, 狔 犚,andtheequality 犳 α 狕,γ gives 犃 = 犘 犳 α 狕,γ ) +γ 犳 α 狕,γ 狓. 犾 狋 ) 犓狋, 狋 α 狕,γ 犾 狋 )+γ 犳 α 犘 ) - 犳 α 狕,γ 犓 - 犳 α 犘 ) 犓 = 犳, α 狕犳 -,γ α 狕犳,γ α 犘 ) ) 犓 - 犳 α 犘 ) - 犳 α 狕,γ 犓 )- 犘 犳 α 犘 ) ) +γ 犳 α 犘 ) 犓 ) - 狔 - 犳 α 狓 )) 狕犓狋 狓 )d 犘 +γ 犓狋, 狋 α, 狕,γ,γ α 狕,γ -α 犘 ) ) 犾 狋 )-γ 犳 α - 狕犳,γ α 犘 ) 犣 狔 - 犳 α 狓 )) 犓狋 狓 )d 犘 - 狔狓 狕,γ 犣 - 犳 α 狓 )) 犓狋 狓 ) 狕,γ 狓 犡狑狓 α 狕,γ -α 犘 ) 犾 狋 ) 犓 狋 ) + 犓. 9) 犓 犾 狋 )-γ 犳 α - 犳狕,γ α 犘 ) 犓. 0) 9)and 0)yields 狘犘 犳 α 犘 ) ) - 犘 犳 ) α 狘狕,γ 犓狋 狓 ) 狔 - 犳 α 狓 ))d 犘 - 狕狔狓,γ 犣 - 犳 α 狓 )) 狕犓狋 狓 ),γ 狓 犡狑狓 + 犾 狋 ) 犓狋 狓 ) 狔 - 犳 α 狓 ))d 犘狕,γ 犣犾 ) 狋 ) α 狕,γ -α 犘 ) 犾 狋 ). ) Math. Colloq. Chin. Univ., 0/) Ononehand,by )weknow 犣犓狋 狓 ) 狔 - 犳 α 狕,γ 狓 ))d 犘 犾 狋 ) 狘狔狘 + 狘犳 α 狓 ) 狘 ) 狕犓狋 狓 ) 犾 狋 )d 犘,γ 槡狘犘狘 + 犳 α ) 狕,γ, 犘犡犣 min λ狋 ) 狋 狋犽 ) min ) 狋 λ 狋狘犓 狓, 狋 ) 狘 ) 狋 狋 犽狘犘狘 槡狘犘狘 +槡 γ ) min λ ) 狋 λ 狋狘犓 狓, 狋 ) 狘 ) 犆槡狘犘狘 + 狋 狋槡. ) γ λ 狋 狋狋 狋 Ontheotherhand,takeξ 狓, 狔 )= 狔 - 犳 α 狓 )) 狕犓狋 狓 ),then,by 8)weknow,withconfidence,γ -δ,thereholds 犅 = 犓狋 狓 ) 狔 - 犳 α 狓 ))d 犘 - 狕狔狓,γ 犣 - 犳 α 狓 )) 狕犓狋 狓 ),γ 狓 犡狑狓 犾 狋 )

21 第 / 期 ZHANGJixiong,etal Erroranalysisonregularizedlearning 7 狓 ) 犈 ξ 狓 犡狑 犾 狋 ))+ 犈 ξ ) 犾 狋 )), δ 槡 whereby) 犈 ξ 犾 狋 ))= - 犳 α 狕,γ 犣狘狔 狓 ) 狘 犓狋 狓 ) 犾 狋 )d 犘 4 犣狔 d 犘 + 狘犳 α 狓 ) 狘 d 犘 ) 狕,γ 犣 λ 狋狘犓 狓, 狋 ) 狘狋 狋 min λ 狋狋 狋 犽狘犘狘 4 狘犘狘 + γ ) 犆 min λ 狋狋 狋, 3) and 犈 ξ ) 犾 狋 ) 狓 ) 狔 - 犳 α 狓 ))d 犘狕,γ 犣犓狋 犾 狘狔狘 + 狘犳狋 ) α 狓 ) 狘 ) 狕犓狋 狓 ) 犾 狋 )d 犘 ),γ 犣 狘犘狘槡 + 犳 α 狕,γ, 犘犡 Therefore, )-5)gives α 狕,γ -α 犘 ) 6)leadsto7). 4 犘狉狅狅犳狅犳犜犺犲狅狉犲犿 犾 狋 [ ) 犆 By 6)and 5)wehave 7)and 7)makes9). 犚犲犳犲狉犲犮犲狊 : min 狋 狋 λ 狋 犅 犆 4 犆 min 狋 狋 λ 狋 狘犘狘 + 犳 α 狕,γ, 犘 狓 ) 犽 狘犘狘 + 狓 犡狑 γ δmin λ狋狋 狋 槡 犡 ) 狘犘 犳 α 犘 ) ) - 犘 犳 α 狕,γ ) 狘 犽犆 ) 槡狘犘狘 + ) +4 犆 γ min λ狋 槡 α 犘 )-α 狕,γ 犾 狋 ) 4 犆 γγ 狋 []CuckerF,SmaleS.Onthemathematicalfoundations 槡 oflearningtheory[j]. 犅狌犾犃犿犲狉犕犪狋犺犛狅犮,00, 39):49. []CuckerF,ZhouDingxuan.Learningtheory:anap proximationtheoryviewpoint[m].new York:Cam bridgeuniversitypress,007. [3]HanDeguang,ZayedAI.Samplingexpansiosforfunc tionshavingvaluesinabanachspace[j]. 犘狉狅犮犃犿犲狉 犕犪狋犺犛狅犮,005,33): [4]SmaleS,ZhouDingxuan.Shannonsamplingandfunction reconstructionfrompointvalues[j]. 犅狌犾 犖犲狑犛犲狉犻犲狊 ) 犃犿犲狉犕犪狋犺犛狅犮,004,43): [5]CuckerF,SmaleS.Bestchoicesforregularizationpa rametersinlearningtheory:onthebiasvarianceproblem [J]. 犉狅狌犱犆狅犿狆狋犕犪狋犺,00,3):4348. [6]CarmeliC,DeVitoE,Toigo A.Vectorvaluedre producingkernel Hilbertspacesofintegrablefunc tionsand Mercertheorem[J]. 犃犪犾犃狆狆犾,006,4 4): [7]SunHongwei.MercertheoremforRKHSonnoncompact 狋 狋 ) 4 犆狘犘狘 犽 + min γ. 4) λ 狋狋 狋 ). 5) ) 狓 ) 犽 狘犘狘 + 狓 犡狑 γ δmin λ狋 槡 狋 狋 ]. 6) 犽狘犘狘 狓 狘犘狘 + ) 狓 犡狑 γ. 7) δmin λ狋 槡 狋 狋 sets[j]. 犑犆狅犿狆犾犲狓犻狋狔,005,): [8]DevroyeL,GyrfiL,LugosiG.Aprobabilistictheo ryofpaternrecognition[m].new York:Springer Verlag,997. [9]MiccheliCA,XuYuesheng,ZhangHaizhang.Uni versalkernels[j]. 犑犕犪犮犺犔犲犪狉犚犲狊,006,7: [0]SmaleS,Zhou Dingxuan.Estimatingtheapproxi mationerrorinlearningtheory[j]. 犃犪犾犪犱犃狆狆犾, Math. Colloq. Chin. Univ., 0/) 003,):74. []Sheng Baihuai,Xiang Daohong.Theconvergence ratefora 犓 functionalinlearningtheory[j]. 犑 犐犲狇狌犪犾犻狋狔犪犱犃狆狆犾,00,doi:0.55/00/ []ShengBaohuai.EstimatesofthenormoftheMercer kernelmatriceswithdiscreteorthogonaltransforms [J]. 犃犮狋犪犕犪狋犺犎狌犵犪狉,009,4): Editor:DENGZhuping Proofreader:MAJiajia)

22 徐州师范大学学报 自然科学版 ), 第 9 卷, 第 期, 第 4348 页,0 年 3 月 JournalofXuzhouNormalUniversityNaturalScienceEdition),Vol.9,No.,pp.4348,Mar.0 犠犺犻狋犲狅犻狊犲犳狌犮狋犻狅犪犾狊狅犾狌狋犻狅狊犳狅狉犠犻犮犽 狋狔狆犲 狊狋狅犮犺犪狊狋犻犮犵犲犲狉犪犾犻狕犲犱狊犻犺 狊犻犲 ) 犌狅狉犱狅犲狇狌犪狋犻狅狊 SUN Xinxiu,LIU Haiyang.DepartmentofBasicCourses,SuzhouVocationalUniversity,Suzhou504,Jiangsu,China;.DepartmentofBasicCourses,XuzhouAirForceColege,Xuzhou006,Jiangsu,China) 犃犫狊狋狉犪犮狋 :Anumberofhyperbolicandelipticwhitenoisefunctionalsolutions,hyperbolicfunc tionandjacobielipticfunctionsolutionsfor WicktypestochasticgeneralizedsinhandsineGordon equationswithvariablecoeficientareobtainedbyusingwhitenoiseanalysis,hermitetransform,the homogeneousbalanceprincipleand 犉 expansionmethod. 犓犲狔狑狅狉犱狊 :WicktypestochasticgeneralizedsinhandsineGordonequations;whitenoisefunc tionalanalysis;hermitetransform; 犉 expansionmethod 犆犔犆狌犿犫犲狉 :O.63 犇狅犮狌犿犲狋犮狅犱犲 :A 犃狉狋犻犮犾犲犐犇 : ) 犕犛犆 00:60H5;35R60;60H40 犠犻犮犽型随机广义狊犻犺 狊犻犲 ) 犌狅狉犱狅方程的白噪声泛函解 孙信秀, 刘海洋. 苏州市职业大学基础部, 江苏苏州 504;. 徐州空军学院基础部, 江苏徐州 006) 摘要 : 利用白噪声泛函分析理论 Hermite 变换 齐次平衡原理和犉扩张法, 分别得到了 Wick 型随 机广义 sinhsine)gordon 方程的双曲 椭圆白噪声函数解和变系数广义 sinhsine)gordon 方程的双 曲 Jacobi 椭圆函数解. 关键词 :Wick 型随机广义 sinhsine)gordon 方程 ; 白噪声泛函分析 ;Hermite 变换 ; 犉扩张法 中图分类号 :O.63 文献标志码 :A 文章编号 : ) 犐狋狉狅犱狌犮狋犻狅 ThefamoussinhandsineGordonequations 狌狓狋 狓, 狋 )=sinh 狌 狓, 狋 )), ) 狌狓狋 狓, 狋 )=sin 狌 狓, 狋 )), ) appearinmanybranchesofnonlinearscience [] andplayanimportantroleinphysics.sotheywere studiedbysomeauthors. Math. Colloq. Chin. Univ., 0/) Inthispaper,wewildeduceexactsolutionsforthegeneralizedsinhandsineGordonequations withvariablecoeficientasthefolowingforms: where 犵 狋 )isanarbitraryfunctionofindicatedvariable. 狌狓狋 狓, 狋 )+ 犵 狋 )sinh 狌 狓, 狋 ))=0, 3) 狌狓狋 狓, 狋 )+ 犵 狋 )sin 狌 狓, 狋 ))=0, 4) Inthepastfewdecades,avastvarietyofnewandpowerfulapproachestoconstructsolutionsof nonlineardiferentequationshavebeenproposed,suchasthehomogeneousbalancemethod [],theja Receiveddate:00099 Funditem:ResearchsupportedbytheNationalNaturalScienceFoundationofChina09780);and theyouthfoundationofsuzhouvocationaluniversity00szdq). Biography:SUN Xinxiu,female,associateprofessor.

23 第 / 期 SUN Xinxiu,etal WhitenoisefunctionalsolutionsforWicktypestochasticgeneralized 9 cobielipticfunctionexpansionmethod [3],thetanhfunctionmethod [4],andsoon. However,asfarasweknown,mostofaforementionedmethodsarerelatedtotheconstantcoefi cientmodels.recently,theinvestigationsofvariablecoeficientnonlinearequationshavebeenpaid atentiontobymanyresearchers.iftheseproblemsareconsideredinrandomenvironment,wecanget randomgeneralizedsinhandsinegordonequations.so,inthepaper,wealsoconsiderthewickver sionsinwhitenoiseenvironment,namely, 犝狓狋 狓, 狋 )+ 犌 狋 )sinh 犝 狓, 狋 ))=0, 5) 犝狓狋 狓, 狋 )+ 犌 狋 ) sin 犝 狓, 狋 ))=0, 6) where isthewickproductonthekondratievdistributionspace 犛 ) definedin [5], 犌 狋 ) 0is 犛 ) valuedfunctions. Random wavesareanimportantsubjectofstochasticpdestohavebeenstudiedby manyau thors,suchasxie,chen,wangandli,sunetal [58].Inthispaper,we lusewhitenoiseanalysis, Hermitetransformand 犉 expansionmethodtogethyperbolicfunctionandjacobielipticfunctionso lutionsand WickversionsofhyperbolicandelipticwhitenoisefunctionalsolutionsforEqs.3)6), respectively. 犠犺犻狋犲狅犻狊犲犳狌犮狋犻狅犪犾狊狅犾狌狋犻狅狊犳狅狉犈狇.5) Inthissection,we lusetheorem.in[5]with 犱 =toobtainwhitenoisefunctionalsolutions of5)andexactsolutionsof3). TakingtheHermitetransformofEq.5),wecangetthenextequation: 槇犝狓狋 狋, 狓, 狕 )+ 槇犌 狋, 狕 )sinh 槇犝 狋, 狓, 狕 ))=0, 7) 犖 where 狕 = 狕, 狕, ) 犆 isaparameter.thecondition 犌 狋 ) 0yields 槇犌 狋, 狕 ) 0.Now,wefirstly solvetheequation 7).Let 狌 狋, 狓, 狕 )= 槇犝 狋, 狓, 狕 ), 犵 狋, 狕 )= 槇犌 狋, 狕 ),Eq.7)yields 狌狓狋 狋, 狓, 狕 )+ 犵 狋, 狕 )sinh 狌 狋, 狓, 狕 ))=0. 8) Letξ= 狆 狋, 狕 ) 狓 + 狇 狋, 狕 )+ 狉,where 狆 狋, 狕 )and 狇 狋, 狕 )arenonzerofunctionstobedeterminedlater, 狉 isanarbitraryconstant,thereis 狌 狋, 狓, 狕 )= 狌 ξ ). Therefore,Eq.8)canberewritenas 狆狋 狋, 狕 ) 狌 ξ )+ 狆 狋, 狕 ) 狆狋 狋, 狕 ) 狓 + 狇狋 狋, 狕 )) 狌 ξ )+ 犵 狋, 狕 )sinh 狌 ξ ))=0. 9) InordertosolveEq.9),weintroducetransformation 狌 狏 ξ )=e ξ), 0) sothat - - sinh 狌 狏 ξ ))= ξ )- 狏 ξ ), cosh 狌 狏 ξ ))= ξ )+ 狏 ξ ), ) thatalsogives 狌 ξ )= - 狏 arccosh ξ )+ 狏 ξ ). ) Math. Colloq. Chin. Univ., 0/) SubstitutingEqs.0)and )intoeq.9)yields 狆 狋, 狕 ) 狆狋 狋, 狕 ) 狓 + 狇狋 狋, 狕 )) 狏 ξ ) 狏 ξ )- 狏 ξ ) 狏 ξ ) + 狆狋 狋, 狕 ) 狏 ξ ) 狏 ξ ) + 犵 狋, 狕 ) 狏 ξ )- 狏 ξ ) =0. 3) From Eq.3),wehave 犵 狋, 狕 ) 狏 3 ξ )- 犵 狋, 狕 ) 狏 ξ )+ 狆 狋, 狕 ) 狆狋 狋, 狕 ) 狓 + 狇狋 狋, 狕 )) 狏 ξ ) 狏 ξ )- 狆 狋, 狕 ) 狆狋 狋, 狕 ) 狓 + 狇狋 狋, 狕 )) 狏 ξ )+ 狆狋 狋, 狕 ) 狏 ξ ) 狏 ξ )=0. 4) WesupposethatthesolutionsofEq.4)canbeexpressedintheform 犕狏 ξ )= 犪犻 狋 ) 犉犻 ξ ), 5) 犻 =0 犻 where 犪犻 狋 )and 犉 ξ ) 犻 =0,,, 犕 )arefunctionstobedeterminedlater. 犕 isapositiveintegerthatcanbe determinedbybalancingthelineartermofthehighestorderwiththenonlineartermineq.4). Supposethatthefunction 犉 ξ )ineq.5)satisfiestheelipticequation6),i.e.,thefunction 犉 ξ )ineq.5)isthesolutionofelipticequation 6): 犉 ξ )= 犃 + 犃 犉 ξ )+ 犃 3 犉 4 ξ ), 6)