ofamedicalinformationsystemisconsideredandisshowntobepreservedunder

Size: px
Start display at page:

Download "ofamedicalinformationsystemisconsideredandisshowntobepreservedunder"

Transcription

1 StepwiseDevelopmentofHigh-LevelPetriNets CompatibilityofNetInvariantsand RenementversusVerication: J.Padberg,M.Gajewsky,C.Ermel TechnischeUniversitatBerlin asanintegrationofalgebraicspecicationsandpetrinets.inalargecasestudy powerfulconceptforverticalstucturingofpetrinets.thisincludeslow-leveland high-levelpetrinets,especiallyalgebraichigh-levelnetswhichcanbeconsidered rule-basedmodicationofalgebraichigh-levelnetshasbeenappliedsuccessfully transformationsandhigh-levelreplacementsystemshasrecentlyshowntobea fortherequirementsanalysisofamedicalinformationsystem.themainnew Theconceptofrule-basedmodicationdevelopedintheareaofalgebraicgraph Abstract ofamedicalinformationsystemisconsideredandisshowntobepreservedunder asaverticaldevelopmentstrategythisextensionisanimportantnewtechnique.it resultinthispaperextendsrule-basedmodicationofalgebraichigh-levelnets suchthatitpreservessafetypropertiesformulatedintermsoftemporallogic.for iscalledrule-basedrenement.asarunningexampleanimportantsafetyproperty softwaredevelopmentbasedonrule-basedmodicationofalgebraichigh-levelnets (Humboldt-UniversitatzuBerlin),supportedbytheGermanResearchCouncil(DFG). betweenh.weber(coordinator),h.ehrig(bothfromthetechnischeuniversitatberlin)andw.reisig Thisworkispartofthejointresearchproject\DFG-ForschergruppePetrinetz-Technologie" 1

2 Contents 4CompatibilityofRule-BasedModication 3PreservingInvariantswithAHLNetMorphisms 1Introduction 2VericationandRenementinHDMS andinvariants Conclusion 39 2

3 1Introduction Petrinetsarewell-knownasabasicmodelforthegeneraltheoryofconcurrencyand asaformalspecicationtechniquefordistributedandconcurrentsystems.high-level netscanbeconsideredastheintegrationofprocessanddatatypedescription,most [GL81,Gen91]andalgebraichigh-levelnets[Vau87,Rei91,PER95].Thepractical relevanceofhigh-levelpetrinetsisconsideredtobeveryhigh,astherearemanyhighlevelpetrinettoolsusedinrealsoftwareproduction(e.g.leu[sm97],design/cpn abstractdatatypes(seee.g.[em85])weusealgebraichigh-levelnets,butthereisno [JCHH91],INCOME[OSS94]).Sincealgebraicspecicationsarewelldevelopedfor prominentclassesarecolouredpetrinets[jen92,jen95],predicate/transitionnets veriedproperties. problemoftranferringresultstootherhigh-levelnetclassesastheseclassescanbe IntheareaofPetrinetstherearemanycontributionsconcerningvericationwithtemporallogic[DDGJ90,BS90,HRH91]andrenement[BGV90,DM90,GG90,BDH92, Onemainproblemofvericationinformalsoftwareengineeringcanbedescribed restrictedandanentirelynewvericationateachstepisusuallyconsideredtobetoo expensiveandtimeconsuming.thus,verticalstructuringtechniquesshouldpreserve tionduringallphasesofthesoftwaredevelopmentprocess.nevertheless,resourcesare bythefollowingdemand:rigoroussoftwaredevelopmentrequirescontinuousverica- conceivedasdierentinstancesofageneraltheoryofabstractpetrinets(see[pad96]). safetyproperties.thetheoryofrule-basedmodicationisaninstanceofthetheory Peu97].Theyaremainlyintheareaoflow-levelnets.Intheareaofhigh-levelnets, sideoftherule)andwhichnewpartsaretobeadded(rightsideoftherule).this levelnets(developedin[per95])andextendittorule-basedrenementpreserving ofhigh-levelreplacementsystems[ehkp91],ageneralizationofgraphtransformation [Ehr79]inacategoricalway.Rulesdescribewhichpartsofanetaretobedeleted(left systempropertieswithrenement. verication[jen95,sch96]ismuchmoredicultandevenmorethecompatibilityof transformationofnetsyieldsaresultingnetwhichiswell-denedandnounspecied changeshavebeenmade.theadvantageofthisapproachisthelocaldescriptionof Inthisreportweconsiderournotionofrule-basedmodicationofalgebraichighphisms,calledplacepreservingmorphisms,allowtransferringspecictemporallogiphismsin[PER95]{preservesafetyproperties,inthesenseof[MP92].Thesemor- formulasexpressingnetpropertiesfromthesourcetothetargetnet.thisfactiscapturedbyourrstmaintheorem3.17thatstatesthefactthatplacepreservingmorphisms obtainsafetypropertypreservingalgebraichigh-levelnetmorphisms. Moreover,wecombinetheseplacepreservingmorphismswithrule-basedmodication. ofthenewconcept4.1thatistheextensionofrule-basedmodicationtorule-based videdwithsuchasafetypropertypreservingmorphism.thisallowstheformulation Thesecondmainresultofthisreportisformulatedintheorem4.15.Itstatesthe preservationofsafetypropertiesundertransformationofnetsviasomerulethatispro- renement,aformaltechniqueforverticalstructuringinsoftwaredevelopment. change. Inordertoextendrule-basedmodicationofalgebraichigh-levelnetsweintroduce morphismsforalgebraichigh-levelnets,that{incontrasttotransitionpreservingmor- preserveinvariantformulas.asinvariantformulasdescribesafetypropertieswehereby 3

4 resultsofthispaperinthecontextofacasestudy[erm96,epe96]concerningthe developmentofamedicalinformationsystem.asketchofthiscasestudyaswellas areviewofthebasicnotionsofalgebraichigh-levelnetsandrule-basedmodication isgiveninthenextsection.insection3weintroducethenotionofplacepreserving morphisms.ourrstmainresultstatesthatthesemorphismspreservesafetyproperties. Insection4,rule-basedmodicationisintegratedwiththesemorphisms.Wepresentour secondmaintheorem,showingthatrule-basedrenementpreservessafetyproperties. Throughoutthewholereportwegiveanongoingexamplewhichillustratesthe Moreover,wediscusstherelevanceofourresultsforsoftwareengineering,especiallythe combinationofhorizontalstructuringandrenement. aswellasampleillustrationofthetechnicalnotionsbymeansofourrunningmedical informationssystemexample.acompactversionofthisreportbythesameauthors discussedinfulldepth. of[pge98]areelaboratedinthisreport,thetechnicalbasicsandtheexamplesare ([PGE98])hasbeenpresentedatETAPS-FASE98inLisbon.Thesketchedproofideas Inthisreport,acompanionreportto[PGE98],wegivethefullproofsofourtheorems ReisigfromHumboldt-UniversitatzuBerlinwithinthe\DFG-ForschergruppePetrinetz- Acknowledgements areforexample[kin97,kp98]. ThisworkwasdevelopedonthebasisofacooperationwiththegroupofProf.W. vativeextensionsofpetrinetsviamorphismstogetherwiths.peukerandw.reisig Technologie"(seealsothefootnoteonpage1).ThecooperationonthesubjectConser- ofthenets.otherworksofthereisiggrouponcompositionalapproachesofverication forlow-levelpetrinetsandshowingthatthesemorphismspreserveinvariantproperties resultedinthecontribution[peu97]deningthenotionofplace-preservingmorphisms 4

5 Anylargeandcomplexsystemcanonlybedevelopedusinghorizontalandvertical ofourcasestudy.themotivationaddressesgeneralproblemsinsoftwareengineering. Inthissectionwemotivatethenotionsandresultsofthesubsequentsectioninterms structuringthatis,stepwisedevelopmentofsubsystem.thisimpliesthattheentire 2VericationandRenementinHDMS Thenwebrieydenealgebraichigh-level(AHL)nets,theirbehaviourandAHLnet engineering.last,wegiveanexampleofverifyingasafetyproperty,rstinanetand systemisgivenonlyimplicitly.thus,vericationhastobeachievedaccordingto modicationviatransformationrulesusedtodescribedevelopmentstepsinsoftware thenforonedevelopmentstep.note,thisismerelyasmallexamplefromthelarger horizontalandverticalstructuring. contextofthemedicalinformationsystem. Wewillrstsketchtheaimandscopeofourmedicalinformationsystemcasestudy. ThemedicalinformationsystemHDMS Themedicalinformationsystem,calledHeterogeneousDistributedInformationManagementSystem(HDMS),hasbeenalargeprojectincludingthewholereorganisatioducedduringthetreatmentofDHZBpatients,whichisabletocommunicatethesedata Medizin/InformatikattheDHZBandtheTechnicalUniversityBerlin(foranoverview Herz-ZentrumBerlin(DHZB).ThisprojecthasbeendevelopedbytheProjektgruppe kindsofcardiacdiseases.theaimoftheprojecthasbeenthe\developmentofasupport andinformationsystemforallactivitiesofthemedicalandthenon-medicalpersonnel atthedhzb,whichisabletodigitallyrecordandstoreallmedicaldatawhicharepro- see[bj97])1thedhzbisaclinicalcenterwhichisdedicatedtothetreatmentofall ofthemedicalandmanagementdataofthegermancardiaccenterberlin,deutsches furtherhumanprocessing"([fhmo91]). withinthewholesystemandtopresenttheseinauniqueformattheuserinterfacefor functionalessencecomprisesabout100rulesandusesinasignicantwaycompatibility wepresentonerenementstepofthewholerule-basedrenementgivenexplicitlyin resultsform[per95]betweenhorizontalstructuringandrule-basedrenement.our ofhdms,theelectronicpatientdatarecord,usingalgebraichigh-levelnets.here techniquesdevelopedin[per95].thetransformationsequencefromactualstateto [Erm96].Infact,theAHLnetsmodellingtheactualstatein[Erm96]containabout 130placesand50transitionsandaremodelledusingsuitablehorizontalstructuring Inourcasestudy[Erm96]weprovideaformalrequirementanalysisforapart kindofahlnetmorphismsforthenettransformation. thatitispossibletoshowthatinvariantpropertiesarepreservedwhenusingaspecial goalinthischapterafterhavingintroducedthebasictechnicalnotions,istodemonstrate 1991and1994[CHL95]. KORSO,(KORrekteSOftware),nancedbytheMinisterofResearchandTechnology(BMFT)between 1ThecasestudyHDMS?A,thebasisforourwork,hasbeenapartoftheGermanBMFT-project 5

6 formaldescriptions.oneofthemainissuesforthepracticaluseofcategoricalspecicationformalismsisthepossibilityofhorizontalandverticalstructuring.fortheconcept AlgebraicHigh-LevelNetsandRenementTechniques InthecontextofPetrinets,categorytheoryhasbeenusedinliteraturetoformulate propertiesofspecicnetclasses,tostudycompositionalityandtherelationtoother withintheframeofhigh-levelreplacementsystems[el93].resultsfromthetheoryof AHLnetabstraction/renementstepsinthesenseofsoftwareengineering. AHLnets[PER95]comprisehorizontalstructuringtechniqueslikeunion(composition oftwonetswithrespecttoacommoninterfaceineachofthecomponents)andfusion ofalgebraichigh-levelnetsasusedinthispaper,structuringtechniquesareformulated viamorphismsandrulesasgivenin[per95].furtherinformationaboutdierentkinds likelocalconuenceandparallelism,andcompatibilityofhorizontalstructuringwith (thegluingofsubnetswithinagivennet),concurrencypropertiesoftransformations thatservesasabasistodenepreandpostdomainsoftransitions,ringsequencesand markingsofahlnets. ofalgebraicpetrinetscanbefoundforinstancein[vau86,ks91,rei91,epr94,lil94]. ThissectioncontainsbasicdenitionsofAHLnets,theirbehaviourandmodication Denition2.1(FreeCommutativeMonoid) LetPbeaset.ThenP=def(P;;)iscalledthefreemonoidgeneratedbyP, wherepisthesetofallwordsoverp,suchthatforallu;v;w2pthefollowing equationshold: BeforewedeneAHLnetsweintroducefreemonoids,analgebraicconstruction thatistosay,isanassociative,commutativebinaryoperationinpwithidentity. u(vw)=(uv)w, v=v=v, \2ab3cd".Inthiscase,wewillsticktothenotationPinordertodenotethe thenabcacdc=aabcccd2pcanberepresentedas freecommutativemonoidgeneratedbyp. AnywordofPcanberepresentedasalinearsum.Forinstance,ifP=deffa;b;c;dg, wv=vw Remark: 1.Ingeneral,wehavethefollowingformaldenitionforanyw2Pexpressedin linearsumform: w=defnxi=1kiai;ki2n;ai2p: 4 6

7 2.Theoperations, 3.CommutativemonoidstogetherwithmonoidhomomorphismsdeneasubcategoryCMonofMon,thelatteronebeingthecategoryofmonoidsandmonoid homomorphisms.,andaretheobviousaddition,inverseandpreorder operationsonlinearsums:forexample,vwiforallcoecientsviwiholds thenumberofelementsinthelinearsum. (within).aj2wikj>0andjwjdenotesthecardinalityofw2pthatis, 4.Thenotionofcommutativemonoidscorrespondstomultisets. SPEC=(S;OP;E)inthesenseof[EM85],setsPandT(placesandtransitions), namelyactualization,renamingandinclusionasdenedin[cew93]. AnAHLnetN=(SPEC;P;T;pre;post;cond;A)consistsofanalgebraicspecication Denition2.2(AlgebraicHigh-LevelNet) ofthedatapartisachievedbytheusualstructuringtechniquesofalgebraicspecication, tionsofanalgebraicspecicationdeningthedatatypepartofthenet.thestructuring Analgebraichigh-levelnetconsists-roughlyspeaking-ofaPetrinetwithinscrip- functionspre;post:t!(top(x)p)assigningtoeachtransitiontanelementof thecommutativemonoidoverthecartesianproductoftermstop(x)withvariablesin XandthesetofplacesPandafunctioncond:T!Pfin(EQNS(SIG))assigningto eachtransitiontanitesetofequationsoversig=(s;op),thesignatureofspec, Denition2.3(AHLNetMarking) andaspec-algebraa.ncanberepresentedbythediagram LetN=(SPEC;P;T;pre;post;A;cond)beanAHLnet.ThenamarkingMofNis anelementofthecommutativemonoid(ap).hereaisthedisjointunionofall carriersetsofthealgebraa,thatisa=u Pfin(EQNS(SIG))T ocond post(top(x)p) pre/ / s2sas hospital:thepatientislocatedattheward.hisbloodpressureistaken,forexample, TheideaofthenetVVMingure1istomodelthefollowingsituationattheDHZB ifthishasbeendemandedintheprescriptionsheet.thevalueistakendownintothe Figure1showsanexamplenetfromourcasestudy,theVitalValueMeasurement.The Example2.4(TheAHLNetVitalValueMeasurement) netisinscribedwithtermsoverthespecicationvvm-specwhichissketchedbelow. 4 Note,thatwerestrictourexampletothesmallsubsystemvitalvaluemeasurement. chartbelongstothepatientrecordthatiskeptattheward.alltheseactivitiesare representedastransitionsinthenetvvmingure1. pulse,...arealsomeasured,ifdemandedintheprescriptionsheet.thetemperature temperaturechart.othervitalvalues,asmediumarterialbloodpressure,temperature, 7

8 Taking blood pressure getpat(patient)=patid Prescr=get_Prescr(get_Treats(PatRecord)) bl_pressure_wantd(prescr)=true v(bpd,bps,t,patid) Patient vital value taken v(p,t,patid) v(temp,t,patid) Taking pulse getpat(patient)=patid Prescr=get_Prescr(get_Treats(PatRecord)) pulse_wanted(prescr)=true PatRecord PatRecord Taking temperature getpat(patient)=patid Prescr=get_Prescr(get_Treats(PatRecord)) temp_wanted(prescr)=true V Patient Adding vital value to TC PatRecord patient getpat(v)=patid at ward TC=get_TC(PatRecord) v(cvp,t,patid) PatRecord Patient Patient Taking central venous pressure ch_patrecord (PatRecord,ch_TC(TC,V)) getpat(patient)=patid Prescr=get_Prescr(get_Treats(PatRecord)) cvp_wanted(prescr)=true quentargument. sorts:name,patient,patid,patrecord,... Fig.1:TheAlgebraicHigh-LevelNetVitalValuesMeasurement(VVM) WemerelystatethesortsandoperationsofVVM-Specusedexplicitlyinthesubse- PatRecord v(map,patid) Taking medium arterial pressure getpat(patient)=patid ward PatRecord Prescr=get_Prescr(get_Treats(PatRecord)) documents map_wanted(prescr)=true PatRecord v(io,t,patid) Measuring import/export getpat(patient)=patid tokensareelementsofavvm-spec-algebra.wehereconsiderthea-quotienttermalgebra(see[em85]),thealgebrageneratedaccordingtothespecicationovercarriersets opns:patient:name,sex,adress,patid!patient Inthefollowing,wegiveoneexemplarymarkingofthenetVVMexplicitly.Generally, getpat:patient!patid getpat:patrecord!patid getpatient:patid!patient Prescr=get_Prescr(get_Treats(PatRecord)) imp/exp_wanted(prescr)=true 8 Patient

9 warddocumentsrespectively. representedbythetokens(patient(smith;:::)anddontheplacespatientatwardand ThismarkingmeansthatthereisapatientSmithandhispatientrecordattheward (d;warddocuments)whered2apatrecordwithgetpat(d)=getpat(patient(smith;:::)). wecansupposethefollowingmarking(mvvm):(patient(smith;:::);patientatward) fornames,doctors,resourcesetc.assumingacarriersetaname=fsmith;miller;:::g LetNbeanAHLnet,undAbedenedasindenition2.3.ThesetCTofconsistent expressthenotionofafollowermarking,werstformalizewhichassignmentsmakethe Denition2.5(ConsistentTransitionAssignment) variablesofatransitionconsistentlyassigned: LetusnowformalizetheringbehaviourofAHLnets.Inordertobeableto 3 transitionassignmentsis awayfromthepredomainoraddedtothepostdomainwhenatransitionisring: CT=f(t;asg)jt2T;asg:Var(t)!A, Denition2.6(A-inducedFunctions) Here,Var(t)isthesetofallvarablesoccurringinpre(t),post(t)orcond(t).4 Nowwecandenethefunctionsthatgiveusthedataelementswhicharetaken suchthatthedataelementsinaundertheassignmentasg LetN,AandCTbedenedasabove.ThentheA-inducedfunctionspreA;postA: satisfytheequationsincond(t)g transitionresultsin: Denition2.7(EnabledTransition,FollowerMarking) CT!(AP)aredenedforall(t;asg)2CTbypreA(t;asg)=ASG(pre(t))and post(t;asg)=asg(post(t))withasga=(top(var(t))p)!(ap)denedas consistenttransitionassignmentaccordingtodenition2.5.then,thetransitiontis LetM2(AP)beamarkingaccordingtodenition2.3and(t;asg)2CTa ASGA(term;p)=(asg(term);p)forallp2Pandterm2TOP(Var(t)). Wearereadynowtodenethefollowermarkingthatis,themarkingtheringofa 4 suchthatthetransitiontakingbloodpressureisenabledandcomputethefollower LetMbethemarkingofnetVVMasgiveninexample2.4.Wewillgiveanassignment markingm0ofmthatisweletthetransitionre. Example2.8(FiringBehaviouroftheAHLNetVVM) by:m[t;asg>m0.thesetofallfollowermarkingsofmisdenotedby[m>.4 markingm0thenisconstructedbym0=m enabledunderthemarkingmfortheassignmentasgifasg(pre(t))m.thefollower LetVar(Takingbloodpressure)=fPatient;PatRecord;BPd;BPs;T;PatId;Prescrg. ASG(pre(t))ASG(post(t)),denoted Anassignmentasg:Var(Takingbloodpressure)!Aisgivenasfollows: asg(patrecord)=d;9 asg(patient)=patient(smith;:::;idsmith);

10 asg(bpd)=diastolicvalue; clarifytheirmeaning,assumingthecarriersetsofthecorrespondingsortscontainthese Remark:Fortheconcretemeasuredvaluesweheregiveconstantsassubstitutesto asg(prescr)=prescr(bloodpressure;:::) asg(patid)=idsmith; asg(bps)=systolicvalue; constantsaselements. asg(t)=timeofvvm; denition2.5becausetheequationsincond(takingbloodpressure)aresatised: bloodpressureiscontainedinthetermprescr(bloodpressure;:::),thereforetheboolean theprescriptionsheetoutofthelistoftreatmentsoutoftherecordofpatient'sdata Accordingtothealgebraicspecication,theequation getpat(patrecord)=getpat(patient)=getpat(patient(smith;:::;idsmith))=idsmith whichbelongstothesamepatientasthevariablepatidrefersto.here,theelement holds.thespecicationoftheselectoroperationsgetprescrandgettreatsyields (Takingbloodpressure;asg)isaconsistenttransitionassignmentasdenedin operationblpressurewantd(prescr)evaluatestotrue. asg,thatiswehaveasg(pre(takingbloodpressure))m: WeshownowthatthetransitionTakingbloodpressureisenabledunderMfor Thefollowermarkingiscomputedby =ASG((Patient;patientatward)(PatRecord;warddocuments) =(patient(smith;:::;idsmith);patientatward)(d;warddocuments) =(asg(patient);patientatward)(asg(patrecord);warddocuments) M Weseethattwotokensarereturnedtotheplacestheyaretakenfrom.Theonlynew tokeninthefollowermarkingisthedataelementrepresentingthemeasuredvalueson M0=M (v(diastolicvalue;systolicvalue;timeofvvm;idsmith);vitalvaluetaken) (patient(smith;:::;idsmith);patientatward)(d;warddocuments) [(patient(smith;:::;idsmith);patientatward)(d;warddocuments)] ASG(pre(Takingbloodpressure))ASG(post(Takingbloodpressure)) theplacevitalvaluetaken. sitionofnetsandforthecompatibilityofverticalandhorizontalstructuringtechniques. tiesofthiscategory,especiallyitscocompleteness,isanimportantbasisforthecompo- TogetherwithsuitableAHLnetmorphisms,AHLnetsformacategory.Theproper- 3 10

11 Denition2.9(MorphismsbetweenAHL-nets) A(transitionpreserving)AHLnetmorphismf:N1!N2betweentwoAHL-nets Ni=(SPECi;Pi;Ti;prei;posti;condi;Ai);i=1;2isgivenbyf=(fSPEC;fP;fT;fA), where {fspec:(sig1;e1)!(sig2;e2)isaspecicationmorphismwithf]spec(e1) suchthatthefollowingdiagramcommutescomponentwise(forpre-andpost-function): {fp:p1!p2andft:t1!t2arefunctionsonthesetsofplaces,resp.transitions. {(fspec;fa):a1!a2isageneralizedhomomorphisminthecategorygalgof generalizedalgebras,andfa:a1!vfspec(a2)isanisomorphismincat(spec1), E2,wheref]SPECistheextensionoffSPECtotermsandequations. thecategoryofspec1-algebras(fordetailssee[per95]). Pfin(EQNS(SIG1)) Pfin(f]SPEC) = ocond1 T1 ft post1 pre1/ / =(TOP(X)P1) Remark:AHLnetsandAHLnetmorphismsaredeningthecategoryAHLofalgebraic ThesetsofvariablesaredenedbyindexingaxedsetXi:=(Xfixs)s2Sifori=1;2. Pfin(EQNS(SIG2))T2 ocond2 post2(top(x)p2) pre2/ /(f]specfp) morphisms: high-levelnets(foraproofsee[per95]). MarkingsandSymbolicMarkings(termswithvariables)aremappedviathefollowing Denition2.10(MappingofMarkings) 4 Petrinets[PER95].Theideaistopresentrulesdenotingthereplacementofonesubnet Wewillnowreviewrule-basedmodicationasaverticalstructuringtechniquefor fs:(top1(x1)p1)!(top2(x2)p2):=(f#specfp) fm:(a1p1)!(a2p2):=((fspec;fa)fp) considertohavearulerwithaleft-handsidenetlthatisreplacedbyaright-hand byanotherwithoutchangingtheremainingpartofthewholenet.thishastheadvantageofalocaldescriptionofchangesinducingglobalchangeswithoutsideeects.we 4 sidenetr.thisrulecanbeappliedtosomenetn,yieldingthenewnetm.this DeletedarethosepartsofthenetLthatarenotintheimageofthemorphismK!L. byr=(l squares(1)and(2)indef.2.11inthecategoryahlofahlnetsandahlnetmorphisms.thenetcisthecontextnet(thatisnafterthedeletionofitemsbytherule Addingworkssymmetrically,allthosepartsofRareaddedthatarenotintheimage applicationofarule,calledtransformation,isdenotedbynr ofthemorphismk!r.thetransformationnr andbeforeadditionofthenewitemsfromr). K!R)whereK!LandK!RareinjectiveAHLnetmorphisms. =)Misdenedusingtwopushout =)M.Theruleisgiven 11

12 Aruler=(L Denition2.11(RuleandTransformation) handsidesoftherule),anahlnetk(calledinterface)andtwoinjectiveahlnet morphismsl NL? (1) KandK!R. KC? K!R)consistsoftwoAHLnetsLandR(calledleftandright (2)-MR? viaruler=(l A(direct)transformationofanetNtoM rulesandcompatibilitywithhorizontalstructuringcanbefoundin[per95]. basedmodication.furtherresultsconcerningparallelandconcurrentapplicationof Thisdenitionisthetechnicalbasisfortheverticalstructuringtechniqueofrule- showninthediagraminthecategoryahl.4 isdenedusingtwopushoutsquares(1)and(2) K!R)atthematchL!N Example2.12(BloodHypertensionProblem) hypertension.inthiscasethedoctorshallbeinformed. Wenowwanttodescribetherenementstepaddinganexceptionincaseofblood thenetvvmdepictedingure1byanexceptionforbloodhypertension.foreach bloodpressurevaluetakenanadditionaltestforhypertensionisperformed.incaseof hypertensionthedoctorisnotied. bloodpressure.thecorrespondingalgebraicspecicationvvm?spechyperisgained isdeleted.additionally,therighthandsidenetrcontainstheplacesvaluesfor hypertensiontestanddoctor,andthetransitionsnotifyingdoctorandtaking Thetransformationruler:L TheinclusionmorphismK!LmeansthatthetransitionTakingbloodpressure K!Ringure2describestherenementof netcandtheadditionoftheplacesvaluesforhypertensiontest,doctor,andthe thespecicationvvm?spec. transitionsnotifyingdoctorandtakingbloodpressureyieldsthenetbex(short forbloodhypertensionexception). byaddingthenewoperationsandequationsusedforthetestofbloodhypertensionto ingure2:thedeletionofthetransitiontakingbloodpressureyieldsthecontext formalizesafetypropertiesasinvariants(temporallogicformulasusingthealwaysoperator"")overthemarkings.inthenetvvm,weconsiderthesafetyproperty Wenowintroducetheproblemofpreservationofsafetypropertiesbyrules.We TheapplicationofrulertothenetVVMyieldsthefollowingtransformationshown AddingvitalvaluetoTCthepatientrecordisonlyread,denotedbydoublearrows for(a;p)2apisanatomicformula(seedenition3.6). forsomed2apatrecordwithgetpat(d)=getpat(patient(smith;:::)).notethat(a;p) withtheinscriptionofavariableofsortpatrecord.thetransitionaddingvital operationchangestheinitialpatientidentity.thusafterringofanytransitionthe valuetotcchangestherecord,butbystructuralinductionwecanprovethatno Weinformallyarguethatthissafetypropertyholds.Foreachtransitionexcept (patient(smith;::::);patientatward)()(d;warddocuments) s.t.getpat(ai)=getpat(di)forai2apatientanddi2apatrecord. safetypropertystillholds. Moregenerally,weassumeamarkingofthenetVVM MVVM:=Pni=1(ai;patientatward)(di;warddocuments) 12

13 L K R Taking blood pressure Taking blood pressure VVM-Spec getpat(patient)=patid patient getpat(patient)=patid Prescr=get_Prescr(get_Treats(PatAkte)) Prescr=get_Prescr(get_Treats(PatAkte)) bl_pressure_wanted(prescr)=true ward bl_pressure_wanted(prescr)=true v(bpd,bps,t,patid) Patient v(bpd,bps,t,patid) Patient vital patient vital vital patient value at value value at taken PatRecord ward taken taken PatRecord ward V V V v(bpd,bps,t,patid) Adding vital value to TC Adding vital value to TC Adding vital value to TC values for getpat(v)=patid getpat(v)=patid getpat(v)=patid hypertension test TC=get_TC(PatRecord) TC=get_TC(PatRecord) TC=get_TC(PatRecord) v(bpd,bps,t,patid) PatRecord PatRecord ch_patrecord ch_patrecord Notifying the doctor ch_patrecord (PatRecord,ch_TC(TC,V)) (PatRecord,ch_TC(TC,V)) (PatRecord,ch_TC(TC,V)) getpat(v)=patid hypert(bpd,bps)=true Doctor ward documents ward ward documents documents doctor PatRecordVVM-Spec -?? VVM-Spechyper C? - Taking blood pressure Taking blood pressure getpat(patient)=patid Prescr=get_Prescr(get_Treats(PatRecord)) values for v(bpd,bps,t,patid) getpat(patient)=patid bl_pressure_wantd(prescr)=true hypertension Prescr=get_Prescr(get_Treats(PatRecord)) test v(bpd,bps,t,patid) bl_pressure_wanted(prescr)=true Patient v(bpd,bps,t,patid) v(bpd,bps,t,patid) PatRecord PatRecord Notifying the doctor Taking pulse Taking pulse getpat(v)=patid getpat(patient)=patid getpat(patient)=patid hypert(bpd,bps)=true Prescr=get_Prescr(get_Treats(PatRecord)) Prescr=get_Prescr(get_Treats(PatRecord)) pulse_wanted(prescr)=true pulse_wanted(prescr)=true Doctor v(p,t,patid) Patient v(p,t,patid) Patient PatRecord PatRecord doctor Taking temperature Taking temperature vital vital v(temp,t,patid) v(temp,t,patid) value getpat(patient)=patid value getpat(patient)=patid taken Prescr=get_Prescr(get_Treats(PatRecord)) taken Prescr=get_Prescr(get_Treats(PatRecord)) temp_wanted(prescr)=true temp_wanted(prescr)=true V Patient VVM-Spec V Patient Patient Adding vital value to TC Adding vital value to TC PatRecord patient PatRecord patient getpat(v)=patid at getpat(v)=patid VVM at ward ward TC=get_TC(PatRecord) v(cvp,t,patid) TC=get_TC(PatRecord) Figure2:VitalValueMeasurementwithHypertensionException VVM-Spechyper v(cvp,t,patid) PatRecord Patient PatRecord Patient Patient Patient ch_patrecord Taking central venous pressure ch_patrecord Taking central venous pressure (PatRecord,ch_TC(TC,V)) (PatRecord,ch_TC(TC,V)) getpat(patient)=patid getpat(patient)=patid Prescr=get_Prescr(get_Treats(PatRecord)) Prescr=get_Prescr(get_Treats(PatRecord)) cvp_wanted(prescr)=true cvp_wanted(prescr)=true PatRecord v(map,patid) PatRecord v(map,patid) Taking medium arterial pressure Taking medium arterial pressure 13 BEX PatRecord getpat(patient)=patid getpat(patient)=patid ward Prescr=get_Prescr(get_Treats(PatRecord)) ward PatRecord Prescr=get_Prescr(get_Treats(PatRecord)) documents PatRecord map_wanted(prescr)=true documents map_wanted(prescr)=true v(io,t,patid) PatRecord v(io,t,patid) Measuring import/export Measuring inport/export getpat(patient)=patid getpat(patient)=patid Prescr=get_Prescr(get_Treats(PatRecord)) Prescr=get_Prescr(get_Treats(PatRecord)) imp/exp_wanted(prescr)=true imp/exp_wanted(prescr)=true

14 sameargumentasabove. s.t.getpat(a)=getpat(d)fora2apatientandd2apatrecord. ifandonlyifthecorrespondingpatientrecordisattheward."andholdsduetothe thenetbex.thistransfershouldbeinducedbytheruler=(l Themoregeneralformulationofoursafetyproperty'VVMis NowthemainproblemisthetransferofthesafetypropertyfromthenetVVMto Thissafetypropertymeans"Atanytimewehave:thereissomepatientattheward [(a;patientatward)()(d;warddocuments)] property.wearelookingforproofrulesofthefollowingform: wehavetondapropertyoftherulesuchthatthetransformationpreservesthesafety somepropertyforr,vvmsatises'vvm BEXsatises'VVM K!R).Therefore, VVM L? fvvm KC? -BEX R? Themainideaofourapproachistouseaclass levelnets.inthispaperinsection3weshowthat onehandpreservesafetyproperties:asaresult ofmorphisms,calledplacepreserving,thatonthe theotherhand,placepreservingmorphismsarestableundertransformationswhichisshowninsection theideacanbetransferredtohigh-levelnets.on beenshownin[peu97]thatsafetypropertiesare preservedbyplacepreservingmorphismsforlow Petrinetztechnologie"(seepage1),ithasrecently ofacooperationwithinthedfg-forschergruppe thatfvvm:vvm!bexpreservessafetyproperties(theorem4.15).thuswehave thedesiredpropertysothatthefollowingproofruleholds: ThefactthatfVVM:L!Rpreservessafetyproperties(theorem3.17)alwaysimplies (rvvm;fvvm:l!r)preservessafetyproperties,vvmsatises'vvm formations. 4).Thus,wecantransfersafetypropertieviatrans- BEXsatises'VVM 3 14

15 transitionscouldaddordeletetokenson"old"(mapped)placesinanunpredictable themorphismandnoold(mapped)arcsaredeletedfromtheircontext.otherwisenew way.wethereforecallmorphismswiththesefeaturesplacepreserving.weshowin 3PreservingInvariantswithAHLNetMorphisms Inthissectionwedenemorphismspreservingsafetypropertiesofalgebraichigh-level nets.tobeabletopreservesafetyproperties(expressedviainvariantformulasonmarkings),wemusttakecarethatnonewarcsareaddedtothecontextofmappedplacesby whichwayplacepreservingmorphismspreserveinvariants.weformalizethenotions formulasarepreservedbyourmorphisms. notionsandnotationconventions. Denition3.1(Persistency) LetfSPEC:SPEC1!SPEC2beaspecicationmorphism.WecallfSPECpersistent inthesensethattop1(x1)=vfspec(top2(x2)). BeforedeningplacepreservingAHLnetmorphismsletusintroducesometechnical ofstaticandinvariantformulas,theirevaluationandtheirtranslationviamorphisms, denearestrictionofmarkingswrt.anahlnetmorphismandshowthatinvariant Cat(SPEC2).Ournotionofpersistencyisequivalentasthefreefunctorisuniquewrt. In[EM85],8.13thenotionpersistencyisdenedforthefreefunctorF:Cat(SPEC1)! Remark: VfSPECandwehaveespecially VfS(X2)=(X2fS(s1))s12S1=XfixfS(s1)=XfixS1=X1 4 Denition3.2(PreandPostDomainsofPlaces) ofpwith Letp2Pbeaplace.Wecall(term;p)thepredomainand(term;p)thepostdomain (term;p)=ftj(term;p)pre(t)g (term;p)=ftj(term;p)post(t)gand LetNi=(Pi;Ti;SPECi;prei;posti;condi;Ai);i2f1;2gbetwoAHLnetsaccordingto Denition3.3(PlacePreservingAHLNetMorphism) morphismasthefollowingdiagramshows: denition2.2.thenf=(fp;ft;fspec;fa):n1!n2isaplacepreservingahlnet NowwecandeneournotionofplacepreservingAHLnetmorphisms: 4 ithefollowingholds: Pfin(EQNS(SIG2))T2 Pfin(EQNS(SIG1)) Pfin(f]SPEC) (1) ocond1 ocond2 T1 ft post1 post2(top(x)p2) pre1/ pre2/ //(TOP(X)P1) 15 (f]specfp)

16 1.Preservationofringconditions:Diagram(1)commutes. 2.PlacePreservingCondition:Themorphismisplacepreservinginthesensethat 3.fT;fPandfSPECareinjectiveandfSPECispersistent(seedenition3.1). 4.EmbeddingCondition:N2isanembeddingofN1inthesensethattherecan therearenonewarcsaddedtomappedplaces: correspondingdomainsoftheoriginaltransition: bemoreplacesinthepreorpostdomainofamappedtransitionthaninthe (fs(term1;p1))=ft((term1;p1)) (fs(term1;p1))=ft((term1;p1)) 5.fA:A1?!VfSPEC(A2)isanisomorphisminAlg(SPEC1). fs(post1(t))post2(ft(t))forallt2t1 fs(pre1(t))pre2(ft(t))and diagramontherighthandsideindef.2.9yieldsapreservationoftransitionsinthe tionpreserving)ahlnetmorphismsasdenedindef.2.9.thecommutativityofthe sensethatnonewarcsareaddedtomappedtransitionsandnoold(mapped)arcsare deletedfromtheirpreandpostdomains.placepreservingmorphismsareingeneral nottransitionpreservingbecausecondition4indef.3.3expressesthatthepreandpost Notethedierencebetweenplacepreservingmorphisms(def.3.3)andthe(transi- 4 setofatransitioninn2cancontainmoreplacesthantheoriginaltransitioninn1.a icationmorphismfvvmspecisaninclusionasonlyoneoperationandoneequation timemerelyyieldsadisjointembeddingofn1inton2. Example3.4(PlacePreservingMorphisminHypertensionTest) TheinclusionsfVVMPandfVVMTaregivenimplicitlyusingnameidentity.Thespec- morphismf:n1!n2thatisplacepreservingandtransitionpreservingatthesame WesketchthatthemorphismfVVM:L!Rdeterminedbygure2isplacepreserving. opns:hypert:bloodpressuresystolicbloodpressurediastolic!bool eqns:hypert(bps,bpd)=(maxbpsbps)_(maxbpdbpd) persistent: VVM-Spechyper=VVM-Spec+ concerningthehypertensiontestareaddedinvvm-spechypersuchthatfvvmspecis DiastolichavebeendenedrenamingthesortNatofnaturalnumbers.Toeverysort stantsinvvm-speclikemaxbps:bloodpressuresystolicandmaxbpd:blood- PressureDiastolicaremeanttodenotecriticalvalues(here:maximalbloodpressure) denotingvitalvaluesthereareplausibilityborders(naturalnumberconstants)denedin VVM-Specthatallowtotestwheterthemeasuredvaluesarerealisticornot.Othercon- InthespecicationVVM-Spec,thesortsBloodPressureoSystolicandBloodPressure- 16

17 thecriticalvalueasgivenintheconstantsmaxbpsandmaxbpdrespectively. VVM-Spechypertotestwhetherthebloodpressurevalueactuallymeasuredliesbeyond valuesofthesesorts.thisoperationalsoisusedinthenewpartofthespecication thatindicateacriticalstateofthepatientwhosevitalvaluesaretaken.onthevital operationinvvm-spechyper,hypert,alwaysyieldstermsequivalenttotrueorfalse. ItisobviousthatfVVMSPECispersistentbecauseapplyingtheforgetfulfunctorVfSPEC (TOP2(X2)),weobtainatermalgebrathatisisomorphictoTOP1(X1)astheonlynew valuesortsdenedinvvm-spec,apreorderrelationisdenedthatallowstocompare shown,fortheotherplacesitisanalogous: (fvvms(v(bpd;bps;t;patid);vitalvaluetaken)) newarcsareadjacenttomappedplaces.fortheplacevitalvaluetakenthisisformally =ftakingbloodpressureg havethesameringconditionsastheiroriginalsinnetl.condition2issatisedasno preserving: Condition1issatisedbecausethetransitionsinnetRthatlieintheimageoffVVM Theconditionsofdenition3.3holdsuchthatthemorphismfVVM:L!Risplace analogously(fvvms(v;vitalvaluetaken))=fvvmt((v;vitalvaluetaken)) =fvvmt((v(bpd;bps;t;patid);vitalvaluetaken)) transitiontakingbloodpressureinrhasmoreplacesinitspostsetthantheoriginal Moreover,fVVM:L!Risanembedding(condition4)asnoarcsaredeleted. =fvvmt(ftakingbloodpressureg) phisms: Corollary3.5(PreservationofMarkingRelation) TakingbloodpressureinL. ThemorphismfVVMisnottransitionpreservinginthesenseofdef.2.9becausethe LetM;M02(A1P1)betwomarkingsoftheAHLnetN1.Then,MM0() fm(m)fm(m0)holdsduetothefactthatahlnetmorphismsaremonotonic. Thenextcorollaryshowshowtherelationofmarkingsispreservedbyourmor- 3 Denition3.6(StaticFormulas) backwardoperatorsareallowed. morphismsinaformalway. thatweusearestrictednotionas'ismerelyastaticformulawhereasin[mp92] morphismstobeabletoexpresssafetypropertiesandprovetheirpreservationvia Theinvariantformula'expressessafetypropertiesinthesenseof[MP92].Note WewillnowdeneformulasoverAHLnetmarkingsandtheirtranslationsvia StaticformulasdescribeastateofanAHLnet.Theyareconstructedsyntactically ofatomicformulas(a;p)denotingthemarkingofoneplacepwiththedataelement a2aandtheusuallogicconnectors^and:.thesetofstaticformulasoveranahl netisthesmallestsetofstringsavailablebyniteapplicationofthefollowingrules: For'1,'2staticformulas::'1;'1^'2arestaticformulas (a;p)2(ap):(a;p)isstaticformula 17

18 ofastaticformulaunderthemarkingmisdenedasfollows: LetM2(AP)beamarkingandlet'1and'2bestaticformulas.Theevaluation LetNbeanAHLnetand'astaticformulaoverN.Then'isaninvariantformula. Denition3.7(InvariantFormula) Mj='1^'2()(Mj='1)^(Mj='2) Mj=:'1():(Mj='1) Mj='1()'1Mfor'1=(a;p) LetM2(AB)beamarkingofN.Theinvariantformula'holdsinNunderM i'holdsinallstatesreachablefromm: Mj='()8M02[M>:M0j=' anahlnetn2viaplacepreservingahlnetmorphisms: Denition3.8(TranslationofFormulas) Letf=(fP;fT;fSPEC;fA):N1!N2beaplacepreservingAHLnetmorphism.Then, thetranslationtfofformulasisgivenasfollows,wherefmisdenedasindef.2.10: WenowcandeneatranslationofformulasoveranAHLnetN1toformulasover 4 Tf('1^'2)=Tf('1)^Tf('2) Tf(')=Tf(') Tf(:')=:Tf(') Tf(')=fM(')for'=(a;p)2(A1P1) Letf:N1!N2beaplacepreservingAHLnetmorphism,M12(A1P1)amarking markingm22(a2p2).letusdenethenotionofatranslatedmarkingm2viathe Denition3.9(RestrictionofMarking) notionofarestrictionofthemarkingm2withrespecttof: Next,weexplainhowatranslatedformulaTf(')isevaluatedunderatranslated ofn1andm22(a2p2)amarkingofn2 s.t. ThentherestrictionM2jfofthemarkingM2tothenetN1withrespecttofisgiven asfollows:m2jf:=m1 M2=fM(M1)Pmj=1j(aj;pj) withj(aj;pj)=2fm(a1p1) teristicsofrestrictionsofmarkings(denition3.9): Thefollowinglemma3.10andthecorollaries3.11and3.12describesomecharac- M2jfiswell-denedduetotheinjectivityoftheunderlyingmorphisms. 18 4

19 LetM22(A2P2)beamarkingoftheAHLnetN2withf:N1!N2beingaplace Lemma3.10(CharacterizationofRestriction) preservingahlnetmorphism.therestrictionm2jfofthemarkingm2tothenetn1 withrespecttofischaracterizedasfollows: Proof: Condition1issatisedduetothedenitionofM2jf.Weprovethesatisfactionofcondition2bycontradiction: 2.M2jfisthelargestofallpossiblemarkingsofN1satisfyingcondition1: 8M012(A1P1):fM(M01)M2=)M01M2jf fm(m2jf)m2 1.ThetranslationofmarkingM2jfviafispartofM2: Letf:N1!N2beaplacepreservingAHLnetmorphismandM2(A1P1)a Corollary3.11(IdempotencyofRestriction) thatm0m2jf. (a;p)2(a1p1)suchthatm0=(a;p)pni=1ki(ai;pi).thus,fm(a;p)m,that is((fspec;fa)(a);fp(p))mandso(a;p)m2jfwhichcontradictsourassumption LetM0Pni=1ki(ai;pi)withfM(M0)M.Thenthereisatleastonetoken markingofnetn1.thenwehave(fm(m))jf=m 4 duetolemma3.10. Corollary3.12(RestrictionandMonoidalOperators) Letf:N1!N2beaplacepreservingAHLnetmorphismandM12(A1P1)and M22(A2P2)bemarkingsofthenetsN1andN2.Then,duetolemma3.10,the followingholds:(m1m2)jf=m1jfm2jf (1) "newparts"oftheembeddingn2areconnectedto"oldparts"(objectsintheimage off)atmostviatransitionsofthe"oldpart"andnotviaplaces,asthenextlemma OurAHLnetmorphismsareplacepreserving(seedenition3.3).Thisimpliesthat M1M2=)M1jfM2jf M2)jf=M1jf M2jfforM1M2 (3) (2) shows: Lemma3.13(RestrictionandPlacePreservingMorphisms) Letf:N1!N2beaplacepreservingAHLnetmorphism.Thenalltransitionsin theimageoffinn2haveatleastthoseplacesintheirpre(post)domainthatare translationsofthepre(post)domainoftheoriginaltransitionsinn1: 19

20 ThetransitionsinN2thatarenotintheimageoffcannothaveplacesintheirpre (post)domainthataretranslatedplacesfromn1viafp: (1)forallt22T2withfT(t1)=t2: (2)forallt22T2nfT(T1)(i)pre2(t2)jf=and (ii)post2(t2)jf=post1(t1) (i)pre2(t2)jf=pre1(t1)and Proof: (1)Weshowthat(a)pre2(fT(t1))jfpre1(t1)and (b)pre2(ft(t1))jfpre1(t1) (ii)post2(t2)jf= Proofof(a):pre1(t1)=fS(pre1(t1))jf(corollary3.11) fs(term1;p1)pre2(ft(t1))jf=)ft(t1)2(fs(term1;p1))(denition3.2) Proofof(b)(bycontradiction): Letpre1(t1)(term1;p1)=pre2(fT(t1))jf.Then pre2(ft(t1))jf(denition3.3,part3) Thiscontradictsourassumptionthat(term1;p1)isnotinthepredomainoft1. Theproofof(1)(ii)isanalogous. =)t12(term1;p1)(asftisinjective) =)(term1;p1)pre1(t1)(denition3.2) =)ft(t1)2ft((term1;p1))(denition3.3,part1) (2)Let(term2;p2)pre2(t2)withfS(term1;p1)=(term2;p2).Then t22(term2;p2)=)t22(fs(term1;p1)) WeshowthatatranslatedvariableassignmentintheembeddingN2restrictedtoN1 Theproofof(2)(ii)isanalogous. Thiscontradictsourassumptionthatt22T2nfT(T1). Thenextlemmaconcernsthepreservationofvariableassignmentsbymorphisms. =)t22ft((term1;p1)) symbolicmarking: wrt.f:n1!n2isthesameastheoriginalvariableassignmentoftherestricted 20 4

Radius and Diameter of Circles (A)

Radius and Diameter of Circles (A) Radius and Diameter of Circles (A) C=40.841 ft C=57.805 mm C=15.708 ft C=33.929 cm Radius and Diameter of Circles (A) Answers C=40.841 ft C=57.805 mm 6.5 ft 9.2 mm 13.0 ft 18.4 mm C=15.708 ft C=33.929

More information

Name: OMRON M3 HEM-7200-E Upper Arm Blood Pressure Monitor

Name: OMRON M3 HEM-7200-E Upper Arm Blood Pressure Monitor -NEW PRODUCT INFORMATION SHEET- Specifications subject to change Name: OMRON M3 HEM-7200-E Upper Arm Blood Pressure Monitor Product USPs: Full size display with all parameters at once Comfort inflation:

More information

WIRELESS SENSOR NETWORK INTEGRATING WITH CLOUD COMPUTING FOR PATIENT MONITORING

WIRELESS SENSOR NETWORK INTEGRATING WITH CLOUD COMPUTING FOR PATIENT MONITORING WIRELESS SENSOR NETWORK INTEGRATING WITH CLOUD COMPUTING FOR PATIENT MONITORING S. Janani Devi 1, G. S. Sreetha Devi 2, G. M. Tamil Selvan 3 SPG Scholar Bannari Amman Institute of Technology, Alathukombai,

More information

MY TYPE 2 DIABETES NUMBERS

MY TYPE 2 DIABETES NUMBERS BLOOD SUGAR MANAGEMENT GUIDE MY TYPE 2 DIABETES NUMBERS Understanding and Tracking the ABCs of Type 2 Diabetes 1 BLOOD MY TYPE SUGAR 2 DIABETES MANAGEMENT ABC NUMBERS GUIDE When you have type 2 diabetes,

More information

Checking Vitals in the Optometric Office. Optometric Pretesting. Vital Signs 9/23/2014. Common Tests at the Eye Clinic

Checking Vitals in the Optometric Office. Optometric Pretesting. Vital Signs 9/23/2014. Common Tests at the Eye Clinic Checking Vitals in the Optometric Office Presented by: Cassidy Obermark, O.D. Missouri Optometric Association Conference 2014 Optometric Pretesting Common Tests at the Eye Clinic Patient history Acuity

More information

Knee Arthroscopy: Information for Patients and Carers

Knee Arthroscopy: Information for Patients and Carers Directorate of Surgical Care Department of Orthopaedics Knee Arthroscopy: Information for Patients and Carers What is an Arthroscopy? Arthroscopy is the examination of the inside of a joint, using a special

More information

Patient Details. Surgery Details. Surgeon's Notes to Surgery Theaters Sister. Accomodation Details. Pre Payment Details

Patient Details. Surgery Details. Surgeon's Notes to Surgery Theaters Sister. Accomodation Details. Pre Payment Details Patient Details Given Name Ανδρέας Middle Name Surname Ρούσος Gender M Date of Birth 12/12/1996 Age 19 Telephone 99824411 Next of Kin Name Next of Kin Telephone Language (if not Greek or English) Surgery

More information

Ďě Ž ť č ď ť ď ú ď ť ě Ě ň Ě ě ú ň ž ú ú Ú ú ú Ě ň é é ž ú ž Ť Ť Ť ú ň Ď ú ň ď Ě ú É ž ř ú ě ň ý Ě ň ý ň ň Ť ř ď ř ň ú Ť ě ř ě ý Š Ú Ú ň ň ú Ó Ú ň Ň Ů ž ú ň Č ř ř ú É ě ň ú Ž ý ú ú Ú Ú ť ž ž ď ý ž ď ž

More information

Finite dimensional C -algebras

Finite dimensional C -algebras Finite dimensional C -algebras S. Sundar September 14, 2012 Throughout H, K stand for finite dimensional Hilbert spaces. 1 Spectral theorem for self-adjoint opertors Let A B(H) and let {ξ 1, ξ 2,, ξ n

More information

Summary of EWS Policy for NHSP Staff

Summary of EWS Policy for NHSP Staff Summary of EWS Policy for NHSP Staff For full version see CMFT Intranet Contact Sister Donna Egan outreach coordinator bleep 8742 Tel: 0161 276 8742 Introduction The close monitoring of patients physiological

More information

Lecture 4: Properties of and Rules for

Lecture 4: Properties of and Rules for Lecture 4: Properties of and Rules for Asymptotic Big-Oh, Big-Omega, and Big-Theta Notation Georgy Gimel farb COMPSCI 220 Algorithms and Data Structures 1 / 13 1 Time complexity 2 Big-Oh rules Scaling

More information

Procedure: Hazardous Materials Medical Support and Rehabilitation Functions

Procedure: Hazardous Materials Medical Support and Rehabilitation Functions Procedure: HAZARDOUS MATERIALS MEDICAL SUPPORT Purpose: This standard operating procedure requires that a medical support function be designated to the Hazardous Materials Group during all operations within

More information

ANESTHESIA - Medicare

ANESTHESIA - Medicare ANESTHESIA - Medicare Policy Number: UM14P0008A2 Effective Date: August 19, 2014 Last Reviewed: January 1, 2016 PAYMENT POLICY HISTORY Version DATE ACTION / DESCRIPTION Version 2 January 1, 2016 Under

More information

CONSEQUENCES OF THE SYLOW THEOREMS

CONSEQUENCES OF THE SYLOW THEOREMS CONSEQUENCES OF THE SYLOW THEOREMS KEITH CONRAD For a group theorist, Sylow s Theorem is such a basic tool, and so fundamental, that it is used almost without thinking, like breathing. Geoff Robinson 1.

More information

Early Warning Scores (EWS) Clinical Sessions 2011 By Bhavin Doshi

Early Warning Scores (EWS) Clinical Sessions 2011 By Bhavin Doshi Early Warning Scores (EWS) Clinical Sessions 2011 By Bhavin Doshi What is EWS? After qualifying, junior doctors are expected to distinguish between the moderately sick patients who can be managed in the

More information

Cardiac Catheterisation. Cardiology

Cardiac Catheterisation. Cardiology Cardiac Catheterisation Cardiology Name: Cardiac catheterisation Version: 1 Page 1 of 7 Contents Page Number(s) 1. Introduction 3 2. Management pre operative 3 3. Management post operative 5 4. Discharge

More information

Instruction Manual Blood Pressure Monitor ZSBP-101

Instruction Manual Blood Pressure Monitor ZSBP-101 Contents Instruction Manual Blood Pressure Monitor ZSBP-101 1. Introduction..1 2. Medical Disclaimer.... 1 3. Intended use... 1 4. Blood Pressure Chart....2 5. Parts identification.. 4 6. Battery Installation

More information

Women s Health Laparoscopy Information for patients

Women s Health Laparoscopy Information for patients Women s Health Laparoscopy Information for patients This leaflet is for women who have been advised to have a laparoscopy. It outlines the common reasons doctors recommend this operation, what will happen

More information

Oregon Health Study. Report of Findings. When we measured your weight, it was .

Oregon Health Study. Report of Findings. When we measured your weight, it was <X, Look- up>. Oregon Health Study Report of Findings Dear , Recently, as part of the Oregon Health Study, you had a health screening. This letter is to inform you about the results of that screening. While these

More information

ZA-12. Temperature - Liquidus + 45 o C (81 o C) Vacuum = 90mm

ZA-12. Temperature - Liquidus + 45 o C (81 o C) Vacuum = 90mm Ragonne Fluidity, Inches Zn-Al Impact 38 34 30 26 22 18 14 No. 3 Zn-Al ZA-8 Liquidius ZA-12 Temperature - Liquidus + 45 o C (81 o C) Vacuum = 90mm Zn-Al (0.01-0.02 percent mg) ZA-27 10 0 2 4 6 8 10 12

More information

Principles and Techniques of Blood Pressure Measurement

Principles and Techniques of Blood Pressure Measurement Principles and Techniques of Blood Pressure Measurement Introduction Blood pressure assessment is an integral part of clinical practice. Routinely, a patient s blood pressure is obtained at every physical

More information

Telehealth and the Homebound Heart Failure Patient

Telehealth and the Homebound Heart Failure Patient Telehealth and the Homebound Heart Failure Patient By Karen Malin Garfield, RN, BSN 104 HEART 2010 The Official Guide to a Strong Heart and Healthy Lifestyle PTS Article Heart2010_Suncrest.indd 1 Health

More information

Solutions for Practice problems on proofs

Solutions for Practice problems on proofs Solutions for Practice problems on proofs Definition: (even) An integer n Z is even if and only if n = 2m for some number m Z. Definition: (odd) An integer n Z is odd if and only if n = 2m + 1 for some

More information

TAKE-AWAY GAMES. ALLEN J. SCHWENK California Institute of Technology, Pasadena, California INTRODUCTION

TAKE-AWAY GAMES. ALLEN J. SCHWENK California Institute of Technology, Pasadena, California INTRODUCTION TAKE-AWAY GAMES ALLEN J. SCHWENK California Institute of Technology, Pasadena, California L INTRODUCTION Several games of Tf take-away?f have become popular. The purpose of this paper is to determine the

More information

Lecture 11. Shuanglin Shao. October 2nd and 7th, 2013

Lecture 11. Shuanglin Shao. October 2nd and 7th, 2013 Lecture 11 Shuanglin Shao October 2nd and 7th, 2013 Matrix determinants: addition. Determinants: multiplication. Adjoint of a matrix. Cramer s rule to solve a linear system. Recall that from the previous

More information

P R E F E I T U R A M U N I C I P A L D E J A R D I M

P R E F E I T U R A M U N I C I P A L D E J A R D I M C O N T R A T O N 7 8 / 2 0 1 4 C o n t r a t o d e P r e s t a ç ã o d e S e r v i ç o s A d v o c a t í c i o s q u e e n t r e s i c e l e b r a m o M u n i c í p i o d e J A R D I M - M S e A IR E

More information

Robotic Partial Nephrectomy. Department of Urology Information for patients

Robotic Partial Nephrectomy. Department of Urology Information for patients Robotic Partial Nephrectomy Department of Urology Information for patients i Introduction You have recently been told that you need to have a tumour removed from your kidney. The standard operation to

More information

SOME PROPERTIES OF FIBER PRODUCT PRESERVING BUNDLE FUNCTORS

SOME PROPERTIES OF FIBER PRODUCT PRESERVING BUNDLE FUNCTORS SOME PROPERTIES OF FIBER PRODUCT PRESERVING BUNDLE FUNCTORS Ivan Kolář Abstract. Let F be a fiber product preserving bundle functor on the category FM m of the proper base order r. We deduce that the r-th

More information

Stenosis Surveillance 2009

Stenosis Surveillance 2009 5 Diamond Patient Safety Program Stenosis Surveillance 2009 *This presentation was collaboratively developed by the Mid-Atlantic Renal Coalition (MARC) and the ESRD Network of New England for the 5-Diamond

More information

Chapter 7. Homotopy. 7.1 Basic concepts of homotopy. Example: z dz. z dz = but

Chapter 7. Homotopy. 7.1 Basic concepts of homotopy. Example: z dz. z dz = but Chapter 7 Homotopy 7. Basic concepts of homotopy Example: but γ z dz = γ z dz γ 2 z dz γ 3 z dz. Why? The domain of /z is C 0}. We can deform γ continuously into γ 2 without leaving C 0}. Intuitively,

More information

University of Huddersfield Repository

University of Huddersfield Repository University of Huddersfield Repository Atkin, Leanne and Shirlow, K. Understanding and applying compression therapy Original Citation Atkin, Leanne and Shirlow, K. (2014) Understanding and applying compression

More information

Health Condition Alarm System

Health Condition Alarm System Health Condition Alarm System Maiga Chang 1, Ebenezer Aggrey 1, Mehadi Sayed 2, and Kinshuk 1 1 School of Computing and Information Systems, Athabasca University, Canada maiga@ms2.hinet.net, aggreyeb@shaw.ca,

More information

Large induced subgraphs with all degrees odd

Large induced subgraphs with all degrees odd Large induced subgraphs with all degrees odd A.D. Scott Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, England Abstract: We prove that every connected graph of order

More information

What is meant by Vital Signs? Py_lWGY80zqMv8e&index=1

What is meant by Vital Signs?  Py_lWGY80zqMv8e&index=1 Vital Signs What is meant by Vital Signs? http://www.youtube.com/watch?v=hq4jceedpa0&list=plhihquuxqydjpisgp- Py_lWGY80zqMv8e&index=1 Vital Signs Measurements of the body s most basic functions 4 main

More information

Q-PERFECT GROUPS AND UNIVERSAL Q-CENTRAL EXTENSIONS

Q-PERFECT GROUPS AND UNIVERSAL Q-CENTRAL EXTENSIONS Publicacions Matemátiques, Vol 34 (1990), 291-297. Q-PERFECT GROUPS AND UNIVERSAL Q-CENTRAL EXTENSIONS RONALD BROWN Abstract Using results of Ellis-Rodríguez Fernández, an explicit description by generators

More information

Blood Pressure How to Measure Ahmed Khashaba, MD

Blood Pressure How to Measure Ahmed Khashaba, MD Blood Pressure How to Measure Ahmed Khashaba, MD Cardiology Department Ain Shams University Why..! Control of Hypertension begins with accurate BP Measurement Blood pressure measurement is often considered

More information

Study of Wireless Sensor Networks and their application for Personal Health Monitoring. Abstract

Study of Wireless Sensor Networks and their application for Personal Health Monitoring. Abstract Study of Wireless Sensor Networks and their application for Personal Health Monitoring. Author 1 Mr. Parag Jawarkar, Author 2 Mrs. Shweta Lambat Abstract Our paper studied Wireless Sensor Network Application

More information

Disability claim Attending physician s statement of disability

Disability claim Attending physician s statement of disability To avoid any delays in the assessment of this claim, the Claimant s statement and the Employer s statement must be submitted. Any cost for information to support your claim will be the policy owner s responsibility.

More information

The Role of The Consultant, The Doctor and The Nurse. Mr Gary Kitching Consultant in Emergency Medicine Foundation Training Programme Director

The Role of The Consultant, The Doctor and The Nurse. Mr Gary Kitching Consultant in Emergency Medicine Foundation Training Programme Director The Role of The Consultant, The Doctor and The Nurse. Mr Gary Kitching Consultant in Emergency Medicine Foundation Training Programme Director Objective To provide an overview of your role as a junior

More information

National Early Warning Score and associated Education Programme CASE STUDY 3

National Early Warning Score and associated Education Programme CASE STUDY 3 National Early Warning Score and associated Education Programme CASE STUDY 3 Case Study 3 Case 3 (Atrial Fibrillation) The important things to get across in this case are: Sick patients must always go

More information

Project 4.2.1: Heart Rate

Project 4.2.1: Heart Rate Project 4.2.1: Heart Rate Introduction Even before you were born, one of the first things your doctor did when you went for an office visit was listen to your heart. Your heart rate, the number of times

More information

Expanding brackets and factorising

Expanding brackets and factorising Chapter 7 Expanding brackets and factorising This chapter will show you how to expand and simplify expressions with brackets solve equations and inequalities involving brackets factorise by removing a

More information

Two classes of ternary codes and their weight distributions

Two classes of ternary codes and their weight distributions Two classes of ternary codes and their weight distributions Cunsheng Ding, Torleiv Kløve, and Francesco Sica Abstract In this paper we describe two classes of ternary codes, determine their minimum weight

More information

Blood Pressure Medication. Barbara Pfeifer Diabetes Programs Manager

Blood Pressure Medication. Barbara Pfeifer Diabetes Programs Manager United Indian Health Services Blood Pressure Medication Titration Program Barbara Pfeifer Diabetes Programs Manager Why a Medication titration program? Despite the many antihypertensive medications available,

More information

This leaflet is intended to provide answers to common questions that you may have regarding fibroid embolisation

This leaflet is intended to provide answers to common questions that you may have regarding fibroid embolisation Introduction This leaflet is intended to provide answers to common questions that you may have regarding fibroid embolisation It is almost certain that you are having the fibroid embolisation done as a

More information

Having a Trans-Arterial Embolisation

Having a Trans-Arterial Embolisation Having a Trans-Arterial Embolisation Delivering the best in care UHB is a no smoking Trust To see all of our current patient information leaflets please visit www.uhb.nhs.uk/patient-information-leaflets.htm

More information

4.4 Mathematical induction

4.4 Mathematical induction 4.4 Mathematical induction This section treats three kinds of induction: Ordinary or weak mathematical induction (which you are probably already familiar with), strong mathematical induction, and structural

More information

DEFINITIONS: 1. FACTORIAL: n! = n (n 1) (n 2) ! = 1. n! = n (n 1)! = n (n 1) (n 2)! (n + 1)! = (n + 1) n! = (n + 1) n (n 1)!

DEFINITIONS: 1. FACTORIAL: n! = n (n 1) (n 2) ! = 1. n! = n (n 1)! = n (n 1) (n 2)! (n + 1)! = (n + 1) n! = (n + 1) n (n 1)! MATH 2000 NOTES ON INDUCTION DEFINITIONS: 1. FACTORIAL: n! = n (n 1) (n 2)... 3 2 1 0! = 1 n! = n (n 1)! = n (n 1) (n 2)! (n + 1)! = (n + 1) n! = (n + 1) n (n 1)! 2. SUMMATION NOTATION: f(i) = f(1) + f(2)

More information

GROUP ALGEBRAS. ANDREI YAFAEV

GROUP ALGEBRAS. ANDREI YAFAEV GROUP ALGEBRAS. ANDREI YAFAEV We will associate a certain algebra to a finite group and prove that it is semisimple. Then we will apply Wedderburn s theory to its study. Definition 0.1. Let G be a finite

More information

Driving evidence-based decision-making with predictive analytics

Driving evidence-based decision-making with predictive analytics Emma Grundy IBM Predictive Analytics Solution Architect for SPSS Software Driving evidence-based decision-making with predictive analytics Business Analytics software To understand God's thoughts we must

More information

Core Measures SEPSIS UPDATES

Core Measures SEPSIS UPDATES Patricia Walker, RN-BC, BSN Evidence Based Practice Manager Quality Management Services UCLA Health System, Ronald Reagan Medical Center Core Measures SEPSIS UPDATES Sepsis Core Measures Bundle Requirements

More information

ON INDUCED SUBGRAPHS WITH ALL DEGREES ODD. 1. Introduction

ON INDUCED SUBGRAPHS WITH ALL DEGREES ODD. 1. Introduction ON INDUCED SUBGRAPHS WITH ALL DEGREES ODD A.D. SCOTT Abstract. Gallai proved that the vertex set of any graph can be partitioned into two sets, each inducing a subgraph with all degrees even. We prove

More information

Medical Records Training Manual for EMR

Medical Records Training Manual for EMR Medical Records Training Manual for EMR ENTERPRISE MEDICAL RECORD (EMR) The MEDITECH Enterprise Medical Record (EMR) collects, stores, and displays clinical data such as lab results, transcribed reports,

More information

Proofs by induction, Alphabet, Strings [1] Proofs by Induction

Proofs by induction, Alphabet, Strings [1] Proofs by Induction Proofs by induction, Alphabet, Strings [1] Proofs by Induction Proposition: If f(0) = 0 and f(n + 1) = f(n) + n + 1 then, for all n N, we have f(n) = n(n + 1)/2 Let S(n) be f(n) = n(n + 1)/2 We prove S(0)

More information

PAH. Salman Bin AbdulAziz University College Of Pharmacy 22/01/35

PAH. Salman Bin AbdulAziz University College Of Pharmacy 22/01/35 Salman Bin AbdulAziz University College Of Pharmacy PAH Therapeutics II PHCL 430 Ahmed A AlAmer PharmD R.W. is a 38-year-old obese woman who presents with increasing symptoms of fatigue and shortness of

More information

MEASURING AND RECORDING BLOOD PRESSURE

MEASURING AND RECORDING BLOOD PRESSURE MEASURING AND RECORDING BLOOD PRESSURE INTRODUCTION The blood pressure, along with the body temperature, pulse, and respirations, is one of the vital signs. These measurements are used to quickly, easily,

More information

GUIDELINES FOR HOSPITALS WITH NEONATAL INTENSIVE CARE SERVICE : REGULATION 4 OF THE PRIVATE HOSPITALS AND MEDICAL CLINICS REGULATIONS [CAP 248, Rg 1] I Introduction 1. These Guidelines serve as a guide

More information

Levels of Critical Care for Adult Patients

Levels of Critical Care for Adult Patients LEVELS OF CARE 1 Levels of Critical Care for Adult Patients STANDARDS AND GUIDELINES LEVELS OF CARE 2 Intensive Care Society 2009 All rights reserved. No reproduction, copy or transmission of this publication

More information

What should I expect before the procedure?

What should I expect before the procedure? The British Association of Urological Surgeons 35-43 Lincoln s Inn Fields London WC2A 3PE Phone: Fax: Website: E- mail: +44 (0)20 7869 6950 +44 (0)20 7404 5048 www.baus.org.uk admin@baus.org.uk PROCEDURE-

More information

Blood Transfusion. Red Blood Cells White Blood Cells Platelets

Blood Transfusion. Red Blood Cells White Blood Cells Platelets Blood Transfusion Introduction Blood transfusions are very common. Each year, almost 5 million Americans need a blood transfusion. Blood transfusions are given to replace blood lost during surgery or serious

More information

Having a Fibroid Embolisation

Having a Fibroid Embolisation Having a Fibroid Embolisation Information for Patients In this leaflet: Introduction 2 What is fibroid embolisation? 2 Why do I need fibroid embolisation? 2 Who has made the decision?. 2 Who will be doing

More information

Careers in Nursing. Kris Hart RN-C, FNP MN

Careers in Nursing. Kris Hart RN-C, FNP MN Careers in Nursing Kris Hart RN-C, FNP MN What is a Profession? Profession is a calling that requires special knowledge, skill and preparation. An occupation that requires advanced knowledge and skills

More information

AT500 Magnetostrictive Level Transmitter. Compact magnetostrictive liquid level transmitter for direct insertion K-TEK Products

AT500 Magnetostrictive Level Transmitter. Compact magnetostrictive liquid level transmitter for direct insertion K-TEK Products Data sheet DS/AT500-EN Rev. M AT500 Magnetostrictive Level Transmitter Compact magnetostrictive liquid level transmitter for direct insertion K-TEK Products Features Mounts from Top of Tank High Resolution

More information

CODING AND COMPLIANCE NEW APPOINTMENT AND REAPPOINTMENT MODULE FOR ANESTHESIA FACULTY

CODING AND COMPLIANCE NEW APPOINTMENT AND REAPPOINTMENT MODULE FOR ANESTHESIA FACULTY CODING AND COMPLIANCE NEW APPOINTMENT AND REAPPOINTMENT MODULE FOR ANESTHESIA FACULTY ANESTHESIA BILLING: MUST BE DOCUMENTED AS: Personally performed: you perform the case without a resident or a CRNA

More information

Central Venous Lines (CVP)

Central Venous Lines (CVP) 2011 Central Venous Lines (CVP) Central Venous Line This pamphlet is about a central venous pressure (CVP) line and why it may be needed. We would like to encourage you to read this pamphlet. The nurses

More information

G = G 0 > G 1 > > G k = {e}

G = G 0 > G 1 > > G k = {e} Proposition 49. 1. A group G is nilpotent if and only if G appears as an element of its upper central series. 2. If G is nilpotent, then the upper central series and the lower central series have the same

More information

Section IV.21. The Field of Quotients of an Integral Domain

Section IV.21. The Field of Quotients of an Integral Domain IV.21 Field of Quotients 1 Section IV.21. The Field of Quotients of an Integral Domain Note. This section is a homage to the rational numbers! Just as we can start with the integers Z and then build the

More information

17. Inner product spaces Definition 17.1. Let V be a real vector space. An inner product on V is a function

17. Inner product spaces Definition 17.1. Let V be a real vector space. An inner product on V is a function 17. Inner product spaces Definition 17.1. Let V be a real vector space. An inner product on V is a function, : V V R, which is symmetric, that is u, v = v, u. bilinear, that is linear (in both factors):

More information

MEDICAL/CERTIFIED MEDICAL ASSISTANT

MEDICAL/CERTIFIED MEDICAL ASSISTANT MEDICAL/CERTIFIED MEDICAL ASSISTANT Occ. Work Prob. Effective Last Code No. Class Title Area Area Period Date Action 4547 Medical Assistant 12 442 6 mo. 07/15/12 Rev. 0000 Certified Medical Assistant 12

More information

Algebraic Geometry. Keerthi Madapusi

Algebraic Geometry. Keerthi Madapusi Algebraic Geometry Keerthi Madapusi Contents Chapter 1. Schemes 5 1. Spec of a Ring 5 2. Schemes 11 3. The Affine Communication Lemma 13 4. A Criterion for Affineness 15 5. Irreducibility and Connectedness

More information

AMPUTATION OF THE PENIS (PARTIAL OR COMPLETE) FOR CANCER INFORMATION FOR PATIENTS

AMPUTATION OF THE PENIS (PARTIAL OR COMPLETE) FOR CANCER INFORMATION FOR PATIENTS The British Association of Urological Surgeons 35-43 Lincoln s Inn Fields London WC2A 3PE Phone: Fax: Website: E-mail: +44 (0)20 7869 6950 +44 (0)20 7404 5048 www.baus.org.uk admin@baus.org.uk AMPUTATION

More information

OEM MAXNIBP Frequently Asked Questions

OEM MAXNIBP Frequently Asked Questions Frequently Asked Questions Why does the monitor sometimes inflate the BP cuff, then shortly thereafter reinflate the cuff? How will I know if the monitor is experiencing motion artifact during a measurement?

More information

Victims Compensation Claim Status of All Pending Claims and Claims Decided Within the Last Three Years

Victims Compensation Claim Status of All Pending Claims and Claims Decided Within the Last Three Years Claim#:021914-174 Initials: J.T. Last4SSN: 6996 DOB: 5/3/1970 Crime Date: 4/30/2013 Status: Claim is currently under review. Decision expected within 7 days Claim#:041715-334 Initials: M.S. Last4SSN: 2957

More information

The Challenge of Aero-medical Critical Care Transport. Focus on critical cases air-ambulance only

The Challenge of Aero-medical Critical Care Transport. Focus on critical cases air-ambulance only The Challenge of Aero-medical Critical Care Transport. Focus on critical cases air-ambulance only In-flight 8,000 Ft. Logistics of intrahospital transfers Is the transfer absolutely necessary? What are

More information

Heart Failure Clinical Pathway

Heart Failure Clinical Pathway Patient & Family Guide 2016 Heart Failure Clinical Pathway www.nshealth.ca Heart Failure Clinical Pathway Your hospital stay will follow a written care plan called a Clinical Pathway. The pathway is a

More information

Body cavities. Body Planes

Body cavities. Body Planes Body cavities Body Planes Directional terms http://homepage.smc.edu/wissmann_paul/anatomy1textbook/1anatomytextch1.html abdomen abdominal front of elbow antecubital arm brachial groin inguinal armpit axillary

More information

1.4.4 Oxyhemoglobin desaturation

1.4.4 Oxyhemoglobin desaturation Critical Care Therapy and Respiratory Care Section Category: Clinical Section: Clinical Monitoring Title: Monitoring of Patients Undergoing Conscious Sedation Policy #: 09 Revised: 05/00 1.0 DESCRIPTION

More information

The mid-segment of a triangle is a segment joining the of two sides of a triangle.

The mid-segment of a triangle is a segment joining the of two sides of a triangle. 5.1 and 5.4 Perpendicular and Angle Bisectors & Midsegment Theorem THEOREMS: 1) If a point lies on the perpendicular bisector of a segment, then the point is equidistant from the endpoints of the segment.

More information

Simple Graphs Degrees, Isomorphism, Paths

Simple Graphs Degrees, Isomorphism, Paths Mathematics for Computer Science MIT 6.042J/18.062J Simple Graphs Degrees, Isomorphism, Types of Graphs Simple Graph this week Multi-Graph Directed Graph next week Albert R Meyer, March 10, 2010 lec 6W.1

More information

Heart Failure EXERCISES. Ⅰ. True or false questions (mark for true question, mark for false question. If it is false, correct it.

Heart Failure EXERCISES. Ⅰ. True or false questions (mark for true question, mark for false question. If it is false, correct it. Heart Failure EXERCISES Ⅰ. True or false questions (mark for true question, mark for false question. If it is false, correct it. ) 1. Heart rate increase is a kind of economic compensation, which should

More information

Factors Affecting Blood Pressure and Heart Rate

Factors Affecting Blood Pressure and Heart Rate Factors Affecting Blood Pressure and Heart Rate Pamela Kay Runyan Chester Nimitz Academy; San Antonio, TX Research Host: Dr. Vernon S. Bishop and Dr. Jeremiah Herlihy The University of Texas Health Center

More information

Healthy Blood Pressure Healthy Heart Beat. Initiated by the World Hypertension League

Healthy Blood Pressure Healthy Heart Beat. Initiated by the World Hypertension League Healthy Blood Pressure Healthy Heart Beat Initiated by the World Hypertension League MAY 17, 2013 What is Hypertension? Hypertension is most commonly known as High Blood Pressure. It is a chronic medical

More information

Effective homotopy of the fiber of a Kan fibration

Effective homotopy of the fiber of a Kan fibration Effective homotopy of the fiber of a Kan fibration Ana Romero and Francis Sergeraert 1 Introduction Inspired by the fundamental ideas of the effective homology method (see [5] and [4]), which makes it

More information

Chair of Software Engineering. Software Verification. Assertion Inference. Carlo A. Furia

Chair of Software Engineering. Software Verification. Assertion Inference. Carlo A. Furia Chair of Software Engineering Software Verification Assertion Inference Carlo A. Furia Proving Programs Automatically The Program Verification problem: Given: a program P and a specification S = [Pre,

More information

Properties of BMO functions whose reciprocals are also BMO

Properties of BMO functions whose reciprocals are also BMO Properties of BMO functions whose reciprocals are also BMO R. L. Johnson and C. J. Neugebauer The main result says that a non-negative BMO-function w, whose reciprocal is also in BMO, belongs to p> A p,and

More information

The Coalition of Orange County Community Clinics Information Technology Activities

The Coalition of Orange County Community Clinics Information Technology Activities The Coalition of Orange County Community Clinics Information Technology Activities A case study on the pursuit of HIT in Community Clinic Healthcare. 1 Mike Matull 3/7/2005 Director of Information Technology,

More information

Great-West G R O U P. Short Term Disability Income Benefits Employee s Statement

Great-West G R O U P. Short Term Disability Income Benefits Employee s Statement Great-West G R O U P Short Term Disability Income Benefits Employee s Statement Employee s Statement Short Term Disability Income Benefits This guide contains the forms you need to apply for disability

More information

One wire. Many possibilities.

One wire. Many possibilities. One wire. Many possibilities. OptoWire One. The one wire you need for pre- and post-procedure FFR and device delivery. Medical Experience an entirely new level of productivity with OptoWire One Reliable

More information

Mini slide, Series MSN narrow version Ø 6-16 mm double-acting with magnetic piston cushioning: elastic with integrated ball rail guide

Mini slide, Series MSN narrow version Ø 6-16 mm double-acting with magnetic piston cushioning: elastic with integrated ball rail guide Piston rod cylinder uide cylinders ini slide, Series SN 1 Ambient temperature min./max. +0 C / +60 C edium Compressed air ax. particle size 5 µm Oil content of compressed air 0 mg/m³ - 1 mg/m³ Pressure

More information

IV.2 (b) Higher level programming concepts for URMs Anton Setzer

IV.2 (b) Higher level programming concepts for URMs Anton Setzer CS 275 Automata and Formal Language Theory Course Notes Additional Material Part IV: Limits of Computation Chapt. IV.2: The URM IV.2 (a) Definition of the URM IV.2 (b) Higher level programming concepts

More information

Arterial Blood Gases, Digital Pulse Oximetry, and Routine Blood Work. By John R. Goodman BS RRT

Arterial Blood Gases, Digital Pulse Oximetry, and Routine Blood Work. By John R. Goodman BS RRT Arterial Blood Gases, Digital Pulse Oximetry, and Routine Blood Work By John R. Goodman BS RRT Patients with chronic lung disease frequently are tested to determine their lung function. The lungs basically

More information

Mesenteric Angiography

Mesenteric Angiography Information for patients Mesenteric Angiography Sheffield Vascular Institute Northern General Hospital You have been given this leaflet because you need a procedure known as a Mesenteric Angiogram. This

More information

Challenge of FUJIFILM in Medical ICT

Challenge of FUJIFILM in Medical ICT Challenge of FUJIFILM in Medical ICT Sep.10, 2014 FORWARD-LOOKING STATEMENTS Forward-looking statements, such as those relating to earnings forecasts and other projections contained in this material, are

More information

Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan

Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 3 Binary Operations We are used to addition and multiplication of real numbers. These operations combine two real numbers

More information

Understanding Diseases and Treatments with Canadian Real-world Evidence

Understanding Diseases and Treatments with Canadian Real-world Evidence Understanding Diseases and Treatments with Canadian Real-world Evidence Real-World Evidence for Successful Market Access WHITEPAPER REAL-WORLD EVIDENCE Generating real-world evidence requires the right

More information

Information for patients. Angiography. Northern General Hospital

Information for patients. Angiography. Northern General Hospital Information for patients Angiography Northern General Hospital You have been given this information booklet because you need to have a test known as Angiography. This leaflet explains more about angiography

More information

SECTION 2: ADMISSION & RISK ASSESSMENT AUDIT INDICATORS

SECTION 2: ADMISSION & RISK ASSESSMENT AUDIT INDICATORS AUDIT INDICATORS Each record should be audited against each question. Where the record does not contain the required elements for audit, instructions are given below. Question 1 should be completed for

More information

A Family Caregiver s Planner for Care at Home

A Family Caregiver s Planner for Care at Home Family Caregiver Guide A Family Caregiver s Planner for Care at Home When Home Care Services Start It is important that you plan to be present at the first home care visit. You have important information

More information