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简介:AnewgeometricmethodtoprovethetotalpositivityofUEsplinebasiswasproposed.UEsplinebasisisakindofbasisdefinedoveralgebraic-trigonometricunifiedspace.UEsplinebasissharesmostpropertiesoftheusualpolynomialB-Splines.Totalpositivityisanimportantpropertyforsplinebasis,itishighlyrelatedwithshapepreservingandvariationdiminishingproperties.Knotinsertedalgorithmisthemostusefulalgorithmforsplinecurvessincemanyotherusefulpropertiesarebasedonit.ItisnecessarytoprovethetotalpositivityofUEsplinebasisusingknotinsertedalgorithmintuitively,notonlyenrichtheUEsplinebasistheory,butalsocanbetreatedassupplementtothetotalpositivityinalgebraicsense.Thisapproachalsocanbeextendedtootheranalogicalbases.
简介:Coronaryarterychronictotalocclusion(CTO)isdefinedasanoccludedcoronaryarterysegmentwithoutanterogradeflowforatleastthreemonths.Itcanbeclassifiedasa“true”or“functional”CTObasedonflowcharacteristics.In“true”CTO,thereisnoanterogradeflow.In“functional”CTO,thereisminimalanterogradeflowthroughtheoccludedsegmentofthecoronaryartery.CTOisacommonfindingduringcoronaryangiographyanditsprevalencemayvarydependingonthereportedliterature.Amongpatientswithoutpreviouscoronaryarterybypassgrafting(CABG),CTOisfoundinabout20–30%ofthepatients.CTOmaydevelopinsidiouslyoveraperiodoftimeandinvolveacomplexinterplaybetweenintracellularandextracellularfactors,smoothmuscleandfoamcells,calcification,andneovascularization.ThereisagrowingbodyofevidencetosupportthatCTOrevascularizationmayimproveclinicaloutcomewhencomparedtomedicalmanagement.BoththeEuropeanandAmericancardiovascularsocietiessupportCTOrevascularizationwithaclass2arecommendation(levelofevidenceB).Historically,duetolowproceduralsuccessrate,apparentinefficientresourceutilization,potentialincreaseincomplicationratesanduncertainclinicalbenefits,onlyabout10–20%ofpatientswithCTOaretreatedwithpercutaneouscoronaryintervention(PCI).RecentadvancesusingnovelandinnovativetechniqueswithdedicatedequipmenthavesignificantlyimprovedtheproceduralsuccessrateforCTOPCItoabout90%inthehandsofexperiencedoperators.WithincreasinginterestinCTOPCIcoupledwithincreasededucationaleffort,CTOPCIlikelywillbecomemoreaccessibletopatientsinneedofCTOrevascularization.OngoingadvancementininnovativetechniquesandequipmentwillcontinuetoimproveproceduralsuccessratesandreduceproceduralcomplicationrateforCTOPCI.Furthermore,thereareanumberofprospectiveclinicaltrialsonthehorizonwhichshouldhelpdefinetheclinicalbenefitsandlimitationsofCTOPCIinthenear
简介:Itisprovedthatisa(Δ+1)-colorablegraph,soarethegraphsG×PnandC×Cn,wherePnandCnarerespectivelythepathandcyclewithnvertices,andΔthemaximumedgedegreeofthegraph.TheexactchromaticnumbersoftheproductgraphsPr1×Pr2×…×PrnandC3k×C2m1×C2m2×…×C2mnarealsopresented.Thusthetltalcoloringconjectureisprovedtobetrueformanyothergraphs.