R-黎曼-可积的若干等价命题及其特征

第20卷第3期湖南工程学院学报Vo1.20.No.320109JournalofHunanInstituteofEngineeringSept.2010:2010-04-02:(1981-),,,:.R()张明会,高婷婷(,742500):积分学是微积分的两大组成之一,数学分析中的积分包括不定积分和定积分(黎曼积分)两类,黎曼积分可以说是整个积分学的基础.本文这种讨论了R(黎曼)可积的若干等价命题及其特征.:黎曼可积;等价命题;特征:O175.7:A:1671-119X(2010)03-0049-03f(x)a,bR,11f(x)a,bR!baf=!baf.:f,,0,0,T,∀T∀!baf-#(f,i,T)!baf+.!baf-∃s(T)∃#(f,i,T)∃S(T)∃!baf+.lim∀T∀%0S(T)=lim∀T∀%0s(T)=!baf.,!baf=!baf=!baf.!baf=!baf=J,lim∀T∀%0s(T)=lim∀T∀%0S(T)=J,T,s(T)∃#(f,i,T)∃S(T),∀T∀%0,,,lim∀T∀%0#(f,i,T)=J,f!baf=!baf=!baf.2fa,bR0,T,#!i∀xi=#(Mi-mi)∀xi,S(T)-s(T).0,TS(T)-s(T)s(T)∃!baf∃!baf∃S(T),0∃!baf-!baf∃S(T)-s(T),!baf=!baf1,f(x)a,bR.a,bR,1.3fa,bR0,#0,T,!i&∋#∀i&#∀xi&.fa,bR,2,1=#,T,#!i∀xi1=#.!i#∀i&,##∀xi&∃#!i∀xi&∃#!i∀xi#,#∀xi&∋#/#=.,#=2(b-a),&=2(M-m),!i∋#=2(b-a)∀i&#∀xi2(M-m)=&.!i#∀i(,!i&∃M-m,!i(#,#!i∀xi=#!i&∀xi&+#!i(∀xi((M-m)#∀xi&+##∀xi(=(M-m)#∀xi&+2(b-a)#∀xi((M-m))2(M-m)+2(b-a))(b-a).2+2=2,f(x)[a,b].4f(x)[a,b]R0#0,!(x)∋#f(x)Jk,#|Jk|.:T,∀i!(x)∋#,!xi∋#,!xi#∀xi(!(x)∋#,!(x)∋#!i∋#∀i&T.f(x)[a,b],3,#,T,!xi∋#∀i&#∀xi&2,(xi&,xi&+1)Ji&.T,2.m,m2m(Jk&),2.!(x)∋#(∗Ji&)∗(∗Jk&),#|Ji&|+#|Jk&|2+2=.,f[∃,%]!(x)#T,!i#(,).Jk!(x)∋#.#|Jk|[a,b]-∗Jk[∃i,%i]!i(j)#.,[∃i,%i],f!(x)#,[∃i,%i],Tj!i(j)#,Tj[a,b]T,!i∋#∀i&Jk,#∀xi&=#|Jk|3,f[a,b].5f[a,b]RfJi(i=1,2,+,n,+),.:!(x)0x,f(x)=∗,&=1{x|!(x)∋1n},n=1,2,+.f,4,n=1,!(x)∋1J1i1(i1=1,2,+,n1)#|J1i1|2,n=2,!(x)∋12,J2i2(i2=1,2,+,n2)#|J2i2|22,n=k,!(x)∋1k,Jkik(ik=1,2,+,nk)#|Jkik|2k){Jkik}k=1,2,+,n,+ik=1,2,+,nk,,{Jkik}!(x)∋1k(k=1,2,+),!(x)0,:#|J1i1|+#|J2i2|+++#|Jkik|++2+22+23+++2k++=!(x)∋#,{Ji},,Jin=(n=1.2.+,k)!(x)∋#.#,i=1|Ji|,#kn=1|Jin|4,f(x)[a,b].2,.,:12;345.50201012,,,.,:1f[a,b],f[a,b].2[a,b]f,[a,b]R.3f[a,b],f[a,b].(1,:[a,b]f,,f[a,b])..345,R.,0E,(),L.,5:fRfL.fR,f(L).,5.,,fn,.参考文献[1].()[M].:,2001.[2].[M].:,2006.[3],.[M].:,2005.NumberofEquivalentPropositionsandCharacteristicforIntegrableintheSenseofRiemannZHANGMing!hui,GAOTing!ting(Datp.ofMath.,LongnanTeachersCollege,Chengxian742500,China)Abstract:Calculusisoneofthetwocomponentsfordifferentialandintegralcalculus.Integrationinmathematicalanalysisincludesindefiniteintegralanddefiniteintegral.Riemannintegralisthebaseofcalculus.Thispaperdis!cussedsnumberofequivalentpropositionsandcharacteristicsforintegrableinthesenseofRiemann.Keywords:integrableinthesenseofRiemann;equivalentproposition;characteristic(上接第32页)TeachingExperimentalDevicewithAutomationInstrumentationControlTechnologyLUHe!ping(GanxiScicence&TechonlogyInstitution,Xinyu338000,China)Abstract:Tohelpstudentsmastertheactualoperationandtheautomationinstrumentationcontroltechnologydur!ingtheschool.anexampleofatypicalindustrialair!conditioningisgiventoanalyzethephysicalparameterssuchastemperature,pressure,flow,level,humidity,windspeed,airpressure,speedandetc.,Acommonlyusedinstru!mentandsophisticatedformsofelectronicdevicesareselectedtoformavarietyofdetectiondevicesandautomaticcontrolsystem.Throughpracticeonthesedevices,studentcanquicklyandcomprehensivelymastertheuseofthethermalinstruments,instrumentcontrolsystemandindustrialair!conditioning.Keywords:teachingexperimentaldevice;automationinstrumentation;industrialair!conditioning513:R()