凸函数的等价定义及几何特征

5320097沈阳工程学院学报(自然科学版)JournalofShenyangInstituteofEngineering(NaturalScience)Vol5No3Ju.l2009:2008-09-21:(1968-),(),,,.岳贵鑫(辽宁省交通高等专科学校人事处,沈阳110122):从凸函数的基本概念出发,在此基础上引申出了凸函数的等价定义,并通过例题说明了凸函数的定义及等价定义在证明不等式中的应用.讨论了凸函数几何特征,并以实际算例介绍了凸函数的几何性质的应用,对凸函数研究有一定的应用价值.:凸函数;等价定义;几何特征:O1743:A:1673-1603(2009)03-0298-03,.,,.11()[1]y=f(x)I,I2x1,x2(0,1)f(x1+(1-)x2)f(x1)+(1-)f(x2)(1)f(x)I.f(x1+(1-)x2)f(x1)+(1-)f(x2)(2)f(x)I.(1)(2)f(x1+(1-)x2)f(x1)+(1-)f(x2)f(x1+(1-)x2)f(x1)+(1-)f(x2).1ea+b212(ea+eb)y=ex,1,y=exx(-,+)y(-,+).1,x1=a,x2=b,=12.y(a+b2)=y(12a+(1-12)b)12y(a)+(1-12)y(b)ea+b212(ea+eb)..,()y=f(x),1y=exx(-,+),y(-,+),,x1x2,1x1=a,x2=b,=12,,x1x2.22()[2]f(x)I,x1x2x3,f(x2)-f(x1)x2-x1f(x3)-f(x1)x3-x1f(x3)-f(x2)x3-x2,f(x).3()[2]f(x)I,x1x2x3,1x1f(x1)1x2f(x2)1x3f(x3)0,f(x).2ea+b212(ea+eb)1,y=exx(-,+)y(-,+).3岳贵鑫:凸函数的等价定义及几何特征299x1=a,x2=a+b2,x3=by=exx(-,+)y(-,+),2:f(x2)-f(x1)x2-x1f(x3)-f(x1)x3-x1ea+b2-eaa+b2-aeb-eab-aea+b2-ea12(eb-ea)ea+b212(eb+ea).3.x1=a,x2=a+b2,x3=b3,f(x)1x1f(x1)1x2f(x2)1x3f(x3)0,1aea1a+b2ea+b21beb0123,:1aea1b-a2ea+b2-ea1b-aeb-ea0b-a2(eb-ea)-(b-a)(ea+b2-ea)0ea+b212(ea+eb).,.3,1,1=,2=1,1+2=1,(1,2(0,1)),1:f(1x1+2x2)1f(x1)+2(x2)1,A1A2y=f(x)2,x1x2,x(x1,x2),1,20,1+2=1,x=1x1+2x2,xoxA,A1A2B,x=1x1+2x2A,B.,:2A1A2A1A2.1,[3].y=f(x),A1A2f(x),A1(x1,f(x1)),A2(x2,f(x2))x1x2,A1A2y=f(x1)+f(x2)-f(x1)x2-x1(x-x1),x(x1,x2),f(x1)+f(x2)-f(x1)x2-x1(x-x1)f(x),x(x1,x2).1f(x1)+f(x2)-f(x1)x2-x1(x-x1)f(x)x(x1,x2)3(a+b2)nan+bn2,a,b0.y=f(x)=xn,x0,f(x),300沈阳工程学院学报(自然科学版)5f(x)f(x1)+f(x2)-f(x1)x2-x1(x-x1)x1=aa+b,x2=ba+b,x=12(x1+x2)=12(12)n(aa+b)n+(ba+b)n+(aa+b)nba+b-aa+b(12-aa+b)(a+b2)n12(an+bn),,,x1x2,,.[1].[M].3.:,2003.[2].[J].,2003,17(2).[3].[M].:,2003.TheequivalentdefinitionandgeometriccharacteristicsofconvexfunctionYUEGuixin(LiaoningProvincialCollegeofCommunications,Shenyang110122,China)Abstract:Startingfromthebasicdefinitionofconvexfunction,thisarticleextendstheequivalentdefinitionofconvexfunction.Theapplicationofconvexfunctiondefinitionanditsequivalentdefinitioninprovinginequalityareexplainedbyexamples.Atthesametime,thegeometriccharacteristicsofconvexfunctionarediscussed,theapplicationareintroducedthroughexamples.Ithassomevalueinconvexfunctionresearch.Keywords:convexfunction;equivalentdefinitionofconvexfunction;geometricfeatures(218)DiscussiononthefourfansofultrasupercriticalboilerYINJun1a,YINMinquan1b,GONGYanjun2(1a.OperationDepartmen;t1b.ProductionTechnologyOffice,ShandongZouxianPowerPlan,tZoucheng273522,China;2.DadongBranchBureauofShenyangEnvironmentalProtectionBureau,Shenyang110042,China)Abstract:Now,thepowerindustryisrapidlydeveloping,newunitscontinuouslygointoproduction,andmoreandmoreplantsselectboilerswithlargecapacityandhighparameter.Withtheaddingofauxiliarymechanicalparametersmatchingwithboiler,thetechnologyrequireshigherandhigher.Furthermore,newnationalenvironmentalprotectionispublishedandimproved.Notonlythenewlybuiltunitsarerequiredtoputintoproduceatthesametime,butalsotheoldunitsarerequiredtooadddesulfurizationanddenitrificationequipmen.tSothenumberofboilerfansbecomesfourfromtwo.Becausethestructure,principleandfunctionsoffansaredifferen,ttheefficienciesaredifferen.tAimingatallkindsoffansinstalledinzouxianpowerplan,tsomeproblemsofoperation,maintenance,manufacturingandinstallingareanalyzedanddiscussed.Suggestionsareproposed,thisprovidesomereferenceforfansselectionandequipmentsmanagemen.tKeywords:boiler;kindsoffans;efficiencyanalysis;equipmentmanagement