J. Duhok Univ.,Engineering & Pure Science Vol.14, No.1 (2011)

J. Duhok Univ., Vol.14, No.1 (Pure and Eng. Sciences), 2011

CONTENTS -GoverningTheTerritoryInAPhaseOfGlobalization:TheIssueOfThe Territorial Planning And The Local Development Majed Ali Kareem........1 -On Cp-Open Sets And Two Classes Of FunctionsAlias B. Khalaf andShilan A. Mohammad ........9 -GeneticDiversityAssessmentAndVarietyIdentificationOfPeach(Prunus persica) From Kurdistan Region-Iraq Using Aflp Markers Shaymaa H. Ali ......17 - Existence And Uniqueness Solution For Nonlinear Volterra Integral Equation Raad. N. ButrisandAva Sh. Rafeeq.....25 -ProtectiveEffectsOfMelatonin,VitaminE,VitaminCAndTheir Combinations On Chronic Lead Induced Hypertensive Rats Ismail Mustafa Maulood.....30 -EngineeringClassificationAndIndexPropertiesOfTheRocksAtDerbandi Gomasbpan Suggested Dam Site Mohamed TahirA. Brifcani.......39 -SpectophotometricDeterminationOfParacetamoleViaOxidativeCoupling With Phenylephrine Ydrochloride In Pharmaceutical Preparations Firas Muhsen Al-Esawati and Raeed Megeed Qadir.....52 -CertainSpeciesOfMallophaga(BirdLice)OccuringOnDomesticPigeons (Columba Livia Domestica Gmelin, 1789) In Erbil City-Iraq Rezan Kamal Ahmed......58 -IncidenceOfBloodStreamInfectionInNeonateCareUnitInSulaimani Pediatric Teaching Hospital Sahand K. Arif and Golzar F. Abdulrahman .......63 -BioaccumulationOfSomeHeavyMetalsInTheTissuesOfTwoFishSpecies (Barbus luteus And Cyprinion macrostomum) In Greater Zab River- Iraq Nashmeel Said Khdhir, Lana S. Al-Alem and Shamall M.A. Abdullah .........71 -EccentricityOfTheHorizontalAxialRestraintForceForStraightAnd Cambered Beams Kanaan Sliwo Youkhanna Athuraia and. Riyadh Shafiq Al-Rawi .........78 -ThreeDimensionalRepresentationOfARemoteStructureUsingReflectorless Total Station Instrument Raad Awad KattanandSami Mamlook Gilyana..........87 - On Generalizations Of Regular Rings Abdullah M. Abdul-Jabbar..........100 J. Duhok Univ., Vol.14, No.1 (Pure and Eng. Sciences), 2011

-TheExistenceAndUniquenessSolutionForNonlinearSystemOfFractional Integro-Differential Equations Hussein J. Zekry...........106 -SpectrophotometricDeterminationOfPhenylephrineHydrochlorideIn Pharmaceutical Preparations Firas Muhsen Al-Esawati..........112 - Use Of Water Quality Index And Dissolved Oxygen Saturation As Indicators Of Water Pollution Of Erbil Wastewater Channel And Greater Zab River. Yahya A. Shekha and Jamal K. Al-Abaychi ...........119 - Flexural Analysis Of Fibrous Concrete GroundSquare SlabAzad A. Mohammed.............127 - Tests On Axially Restrained Ferrocement Slab Strips Azad Abdulkadr Mohammed and Yaman Sami Shareef .......138 - Some New Separation Axioms Zanyar A. Ameen And Ramadhan A. Muhammed..........156 - Some New Separation Axioms Zanyar A. Ameen

and Baravan A. Asaad............160 - Gamma Ray And Annealing Effects On The Energy Gap Of Galss AHMAD KHALAF MEHEEMEED and SULAIMAN HUSSEIN AL-SADOON ..............165 - Effect Of Long-Term Administration Of Melatonin, Vitamin E, Vitamin C And TheirCombinationsOnSomeLipidProfilesAndRenalFunctionTestsInRats Exposed To Lead Toxicity Almas M.R. Mahmud...............177 - Hyalomma aegyptium As A Dominant Tick On Certain Tortoises Of The Testudo graeca In Erbil Province-Kurdistan Region-IraqQaraman Mamakhidr Koyee.................186 - On Detectionof Feedback In The Time Series Sameera Abdulsalam Othman..................191 - The Singularity Of M-Connected Graph Payman A.Rashed.....................207 -AStudyOfNaupliarStagesOfMesocyclopsedaxForbes,1891(Copepoda: Cyclopoida) Luay A. Ali and Kazhal H. H. Rahim.....................217 -EffectOfSalicylicAcidOnSomeBiomassAndBiochemicalChangesOf Drought- Stressed Wheat (Triticum aestivum L. var. Cham 6) Seedlings Fakhriya M. Karimand Mohammed Q. Khursheed.........................223 - Bacteriological Study And Antibacterial ActivityOf Honey Against Some Pathogenic Bacteria Isolated From Burn Infections Suhaila N. Darogha and Ahmed A.Q.A.S. Al-Naqshbandi.........232 J. Duhok Univ., Vol.14, No.1 (Pure and Eng. Sciences), 2011

- Ann-Based Static Slip Power Recovery Control Of Wrim DriveAliA. Rasool and HilmiF. Ameen................................................242 -LowerBoundOfT-BlockingSetsInPg(2, q )AndExistenceOfMinimal Blocking Sets Of Size 16 And 17 In Pg(2,9) Abdul Khalik L.Yassen and Chinar A.Ahamed ......................253 - A Multiple Classifier System For Supervised Classification Of Remotely Sensed Data Ahmed AK. Tahir.............................................................................260 - Experimental Determination Of Paschen Curve And First Townsend Coefficient Of Nitrogen Plasma Discharge Sabah Ibrahim Wais..........................................................................274 - Numerical Solution Of Gray-Scott Model By A.D.M. And F.D.M. Saad A. Manaa And Chully M. R. ...........................................................281 -AnatomicalComparisonBetweenCissusRepens,CayratiaJaponica(Vitaceae) And Leea Aequata (Leeaceae)Chnar Najmaddin, Khatija Hussin, And Haja Maideen.........................290 -EffectsOfAcetamipridAndGlyphosatePesticidesOnTestisAndSerum Testosterone Level In Male MiceMahmoud Ahmed Chawsheen...........................................................299 - Minimal Blocking Sets In Pg(2,7) And Lower Bounds Of The Sixth And Seventh Blocking Sets. ChinarA. Kareem............................................................................207 J. Duhok Univ., Vol. 14, No.1 (Pure and Eng. Sciences), Pp 1-8, 2011 1 GOVERNING THE TERRITORY IN A PHASE OF GLOBALIZATION: THE ISSUE OF THE TERRITORIAL PLANNINGAND THE LOCAL DEVELOPMENT MAJED ALI KAREEM Urban & Regional Planning, University of Venice-Italy (Received: January 22, 2009; Accepted for publication: November 28, 2010) ABSTRACT One of the consequences of the present of the financial, communicative and decisional nets globalization is to make ineffective the traditional instruments for thedirect control of the territory by the public authorities. The territories arenowadaysstructured inover-regionalandover-nationalnetsand fluxeswhichtendtoestablishdirectrelations with the single local systems (i.e., towns, districts, regions, tourist resorts, etc).In this complex situation, the territorial planning shall thus and in first place propose itself as governance, that is tosayasgoverningnegotialprocessforthecooperativeandconflictualinteractionsbetweensubjectswhichare capable, for various reasons, to act on the territory and transform it. Thescopeofthisresearchistoinvestigateontheroleoftheterritorialplanning inrelationwith tworelational aspects: globaland local.Thetargetis thatto locateamethodologicalapproachabletohave anopenviewtothe territorial phenomenon in a globalized context; to define the territorial plannings role, procedures and instruments. PROBLEM DEFINITION heterritorybecomesevermore fragmentedinparts,eachoneofwhich tendstobecomeajunctionofover-local networksand,therefore,tofollowdifferent developmentroutesaccordingtothelong distancerelationsystemtowhichitbelongs.At thesametimeeachoneofthesesubjects dependseverlessfromthoserelationsof physicalproximitywiththecontiguous territories,whichweretheterritorialplannings existenceandoperativejustification.The proximityrelationscontinue,anyway,tobe important also and above all in order to optimize thelongdistancelocal relations; but just for this reasontheyrisktobesubordinatetoan exogenousrationalitywhichtendtoimpose itselfastheterritorialorganizationsprinciple also at a local level. In order to plan rationally a territory it should benecessarytocheckthisnetsbodywhich however,foritstrans-nationalnature,todayis not directly controllable by any public authority. Ontheotherendneithertheselongnetsnor theorganizationswhichoperatesthemcan directlycontroltheterritorieswhichtheyuseas anchoragesfortheirjunctionsandasphysical pathsoftheirfluxes.Theyinteractwiththe territoriesandtrytoobtaincompetitive advantagesthroughaseriesofnegotiations withthoseprivateandpublicsubjectswho,for various reasons, operate or have competences on a local level. WHAT ROLE FOR THETERRITORIAL PLAN? The role of the Territorial Plan1 today and inthenextfuture,mustbeplacedinthelocal developmentcontext2.Ithastotakeinto account,aboveall,thegrowingroleofthe municipalities-beingthebaselevelofthe territorygovernanceandofthepluralitiesof theinstitutionsinvolvedbythegrowing environmentalproblemssexpansionand complexity, but also by the ever major difficulty togovern thelocaleffects(Perulli, P.,2000),of decisionstakenelsewhereandwhicharetaken basingonpuresectorialrationalities.Itsrole seemstobeusefullysubdivisibleinthreemain directions:knowledgeandevaluation,strategic orientationandnetting,inbeingconflicts adjustment. a)Afirstfunctionconcernsthecognitiveand evaluationsupport which the territorialplancan supplytoallthesubjects,capable,forvarious reasons,toaffecttheterritorialandurban conditions and dynamics (Mazza L., 1997). Thisfunctionisimportantlocally,notonlythe localauthonomiescouldntbeefficaciously excercisedunlessonthebaseofanadequate knowledgeoftherealityintowhichtheyare askedtoweighheavily(andoftensucha knowledge is precluded to the Municipalities for territorial dimensions and technical, professional andadministrativeresources).Ingeneral,there couldbenotaneffectivedialoguebetweenthe variousinterested subjects unless itwould beon thebaseofdataandobjectivetiescommon T J. Duhok Univ., Vol. 14, No.1 (Pure and Eng. Sciences), Pp 1-8, 2011 2 knowledge,of thevalues andthestakes at play, reasonsofconflictandeffectsassociablewith thedifferentalternativescorrespondingtothe differentinterestswhichlegitimatelyconfront each other. b)Asecondfunctionconcernstheactions strategicorientation,exercisedbydifferent subjects in different sectors, susceptible to affect theconditionsandthedynamicofadistricts territory.Suchafunction,traditionallyentrusted totheterritorialplanning,assumestoday particularimportanceaboveallinrelationto emergingexigencies. As alreadynoted, the local systemsexploitationrequires,thenettingof resources,opportunities,projectsandinitiatives which,themselvesalone,couldntallowthe insertionofendogenousandselfpropulsive developments(RullaniE.,1997).Itmeans,in other terms, tofavour, on a wide territorial area, synergic interactions, complementarity relations, andproximityandcohesiontiesable tocreate a cooperativespherepropitioustothelocal systemsdurabledevelopmentandtothe strengthening of their own competitive capacity.Thedevelopmentprocessesrootinginthe specificstructuralconditionsofaspecific territoryinallitsnatural,historicalandcultural aspects,requirestheconstructionofimages, visionsandlongtermstrategiesabletoforesee and, if possible, to anticipate the cumulative and indirecteffectsof thein being dynamics and the programmable actions to control same.Theseexigenciesemphasizetheusefulness ofthereferringscenarios,continuouslyand flexibly adapted to the changes of theeconomic, territorialandenvironmental conditions, and the opportunitytodirecttheinterestofthevarious actors and economical and institutional available resourcestowardssomeintegratedprojectsof strategic prominence. c)Athirdfunction,moreproperlyanddirectly ruler,concernstheprotectionoftheover-local interests which are of specific competence of the district. Such competence, to be precise, through thelegislativereforms,bothnationaland regional(CastellsM.,1997),concernscertainly sometypicalcontentswhichcannotbe adequatelytreatedonlywithinlocalscale(i.e., the municipal area), like those which concern the wholeterritoryorganization,theintermediate scaleinfrastructuralsystems,thesoilprotection andthehydraulic,hydrogeologicaland hydroforestalarrangement,aswellasthe institution of parks and natural reservoirs. Beside, suchcompetencemustbebetterdefined for at least two aspects. Fromoneside it isnecessary toconsiderthe role that a wide areas (a whole region) planning is requested to develop and to ensure the respect andtheexploitationoftheterritorialstructural characters.Certainlyalsothoserelevanttothe landscapeandhistorical-culturalcharacteristics and the territory ecological infrastructure.Fromtheotheritoccursthatnotnecessarily theterritorialplanningrulingactionexpresses itselfwithimmediatelybindingandprevailing regulationson thepossible differentrulesof the localorothersectorialplans.Thelegislative systemsoftheterritorialplansshowthatthe rulingefficacycan beentrustedmorefrequently toguidelinesortonegotiatedregulationswhich responsibilizethelocalpower,especiallywhen thesubjecttobeprotectedisconsistingin propertiesorresourceswhoseprecise determinationrequiresprobingorspecifications whichcanbemoreprofitablycarriedoutona localarea.Itconcernstworelatedaspects:asa matteroffacttheconspicuouswideningofthe contentsandoftheterritorialplansapplication fieldcouldntbeproposableandjuridically sustainableshouldsuchcontentstranslateinto tight and immediately binding rules. Onthismatteritcanbeaffirmedthatthe territorialplansspecificitycomesfromits capacitytotreatcertainterritorialsubjects (MagnaghiA.2005)whichassumeathematic, problematic,projectual,managerialandruling specificityduetothefactofbeing conceptualizedtodistrictlevel(forinstance: ecologicalnets,greenparks,housingnets,shortrangetouristcircuits,localwork markets etc). PRELIMINARY CONDITIONS Theproceduresfortheterritorialplan drawing represent the mean to reach the creation ofnew,sharedrulesfortheterritory managementanditschanges,ofnewstatutes fortheuseoftheavailableenvironmental resourceswhichcanbemadeownbythe citizenswholivethereandbytheorganizations whichrepresentthem.Thewideareaterritorial planning (i.e., be it a region or a district) remains oneofthe fewpointsofview fromwhichthe problemtomanage thechangein a sociallyand environmentallysustainablewaycanbeseized and treated at certain conditions:J. Duhok Univ., Vol. 14, No.1 (Pure and Eng. Sciences), Pp 1-8, 2011 3 1.tore-anouncethetechnocraticattitudefor definingtheoptimuminfavourofthe constructionofvariousopenalternativeson whichtoconfrontwithalltheinterested subjects. 2.thatthemethodofcomparisonbetweenthe alternativesandtheirconsequencesatasystem levelbeactuallydonealsobythesectorial policies promoted by the municipal or districtual authorities,utilizingforsuchpurposesthemost commoncognitiveresourcesandthespecific instruments of the planning itself. 3.Thatnewrepresentationandcommunication formsbeexperimentedfortheterritorial consequencesofwhatagreedbetweensocial, economical and institutional actors. 4. that there must be full consciousness by all the participantsthattheplanningisacontinuous, fatiguingandcomplexprocess;theplanisa contractwhichtheactors,presentonacertain territory,undertakeforitstransformation;once concluded it is indispensable to start to think and to build the future forms of new contracts which will rule what not foreseen or not foreseeable up tothatmoment,thatwillimprovethe representationoftheinterestswhichare consideredexcluded;whichwillincludenew knowledge. Theplanningprocessiscertainlymore importantthantheplan,buttheexistenceofa planchangestheprocess,becauseitrequiresa firm and clear political projectuality, its potential contribute to the setting up for a better agenda of theregionalpolicies(i.e.,wellasoftheir comprehensivecoherentdirection)isvery important. Theformthattheterritorialplanassumesin takingintoaccounttheexigenciesandthe perspectives up to now recalled is the following: Astrategicreferencescenarioforthewhole territorydenotesthelocalidentities3andit indicates the desirable development lines. To the indication of a series of strategic projects inactuationoftheplan,withtheconcourseof theotherbodies,isassignedthetaskof deepeningthepossiblesolutionsandthe feasibilityconditionsforwhatitconcerns emergency problems. Arulingsystemwhichintroducesinnovative agreementsprocedures,limitstherules,recalls thedirectives,foreseesthemunicipalities involvementinthemanagementofprotected areasaswellasinothermattersofover-local interest. Thegeneraltargetthattheplanassumesis thereachingofanenvironmentalandsocial sustainability4 forthewholeterritory,thatisto sayformsofdevelopmentwhichareableto safeguardandincreasethenaturalandsocial resourcesandtheareasspecificidentitiesand by the thrust of a cooperative5 approach. From control to self-control: the choice is that topromote,inlieuofthecontrolandofthetie, newagreementsinstrumentssuchasformsof self-controlbetweenlocalsubjectsinthe decisionmoment (Magnani, A. 2000). It iswhat today goes under the name of governance: the governmentofthecooperativeandconflictual interactionsbetweentheactorswhoactinthe territoryandwhotransformit,insteadofthe directgovernmentoftheterritoryssmallsingle pieces. Theeffortisthattodefinethereciprocal autonomies(amongstthevariousauthorities), adaptedintoanormativesystemwhichstrongly limitstherules;asasupporttoprojectuality formsongeneralmatters, alsowith the scopeto buildprojectscapabletoacquireexternal resources. THE PLAN APPROACH TO THE TERRITORIAL PROBLEMS Withinthenegotialprocedurethestrategic andrulingfunctionsexplicateterritorialeffects (transformation,explotation,conservation, protectionetc),notdirectlybutthrough collectivelocalsubjects.Inorderthatthis happensitisnecessarythatsuchsubjectsmight act as collective actors on territorial basis, i.e. as territoriallocalsystems 6.Withthisexpression areintendedpublicandprivatesubject'slocal aggregations(ornets)abletoorganizethem andtoorganizetheirownterritorytointeract withexternalsubjectsandthusrealizing common shared projects. Thenegotial process with the local collective subjectsmustthereforebeseenbothasan operationalandlatentidentitieshearingphase andasoperationalidentitiesconstruction moments(PorterM.,1987),aroundover-local scale territorial projects. In such a way the plans promotionandplanningroleisexplicated.It goesintoeffectthroughknot(setupofthe localsystems)andnetpolicies(connectionof morelocalsystemsaroundover-localprojects). Underthispointofviewthelocalsystems(and therefore alsothelocal identitieswith the above J. Duhok Univ., Vol. 14, No.1 (Pure and Eng. Sciences), Pp 1-8, 2011 4 mentionedmeaning)impliesinter-municipal relations as well as with local sectorial bodies. Inorder to talkof local territorial system it isnecessarythatcertainpublicandprivate subjectsselfrepresentthemselvesinaprojectual perspective,aslocalsubjectsnetswith interfacefunctionsbetweenthelocalmilieu resourcesandover-localsubjectsnets.Itis necessarythereforethatthehorizontalties, whichensuretheterritorialsystemsinternal cohesion,derive,atleastinpart,fromthe relationsthatthesubjectswhocomposeithave with the local, specific milieu, meaning a natural andhistoric-culturalconditionsstableand localizedwhole,seenaspossibledevelopment projectsandlocalterritorialre-qualification possible intakes. Underthisperspectivetheplanplacesitself towardsthelocalsystems,asclaimedby Castells(1977),itsdirectionsandrulesmust translateintolocalprojectsandactions(of knot,ofnet)bytheactorswhoconstituteit in order to obtain the desired territorial effects. Theteritorysusefulknowledgefortheplan isofpolitical-operationaltyperatherthan technical-operational.Thepolitical-operational territorialknowledgeconcernstheidentification of thelocal systems andof the actors (and more ingeneralthesubjects)whocomposesthem; theirrelationswiththeterritory;thespecific rationalitieswhichrulesuchrelationsandthus thelocalorganizativeprinciplesofsaid territories. Startingfromtheseconsiderations,such knowledgecanstartfromobjectiveanalysises andaboveallfromareasonedinventoryofthe projectspromotedatsub-districtlevelbythe variouspublicand private subjects,butifforms itself,aboveall,through thenegotialinteraction withthelocalsystemsanditbuildsitselfinthe territorialplansfullfilment.Howeverthis doesntpreventtodefineamapofthese differents,possibleaggregationswhichisalsoa mapofthelocalidentitiesandthebasisforthe identitiesandthestatutesoftheplaces definition.Thelocalidentity,inaplansperspective, cannotbaseitselfonlyonapassivesenseof belonging,foundedontheterritorys aesthetical-symbolic characters, but a resource to be exploited.This resource derives from the local subjects capacitytoconnectbetweenthem,toself-organizethemselvesinordertoevidencetheir territorysresourcesintheinteractionwith externalsubjects(Dematteis,1997).Withouta commonprojectofsuchatypethereisnotan active,operativelocalidentity.Inthissenseit canbesaidthattheterritorialPlannotonly utilizesthecollective,localidentitiesasa passive resource, but also operates to build them, tolet them pass from a latent and potential state to a real and operational one. PLAN ORGANIZATION THROUGH ACTIONS AND PROJECTS Thisplanformemphasizesitsprocedural aspectsinadoubledirection, activating through theprojects,thelocalinstitutions,andin proposingtheindicationofaseriesofreal projectsofpriorityimportance.Inthiswaythe planconcludesitselfwiththestartofa complexactionsprocedurewhich,foreach project,activatesasystemoflocalandover-local pertinent actors. ThePlan,therefore,startsactuallywithits formalconclusion,supplying,besidethe strategicscenarioofreferenceandthe accomplishmentrules,theindicationofaseries ofspecificprojectsprominentforits achievement, comprehsensiveof the local actors andtheextra-localinstitutionalandfinancial contributionswhichareneededfortheir realization. Inthiswaytheplanobtainsaprocedural continuitywhichallowstoverifythestrategic scenarioduringthecourseoftheprojects fullfilmentandifthecasetocorrectit.Each projecthasanidealreferencearea(variable geometry)andaninternalandexternalactors system.Thestrategicvalueprojectisgenerallya projectwithmultidisciplinaryandmultisectorial integratedcharacter(HaleyP.1997),inwhich great relevance is given to the virtuous synergies betweensectorialactions;thisimpliesthe overcomingofthesectorplanningandagreat willtocooperatebytheassessorships(at regionallevel,butalsoatdistrictualand municipalone) tocreatead-hocinter-assessorial coordinationsforeach projectsformulationand management. Inordertomakeeasiertheverification procedure for the single, new actions, but also to helptheconstructionofadynamicstragetic scenario and sustainable in its widest meaning, it is fundamental the use and the improvement of a polyvalentevaluationmodel,akindof J. Duhok Univ., Vol. 14, No.1 (Pure and Eng. Sciences), Pp 1-8, 2011 5 metaprojectamongsttheproposedstrategic projects.Theconstructionandtheuseofapolyvalent evaluationmodelgotowardsthedirectionto conceivethestrategicplanningprocessasan intelligentguideofsectorialandpreciseactions beingalreadyoperational,bethempoliciesor works:evidencingandexploitingthepositive energies existing in the territory. In any case it is thus necessary a cognitive apparatus (a record of theprojectsandoftheactionsbeingcarriedout intheprojectsarea,beitinstitutionalornot) whichcouldallowtheevaluationand,caseby case,exploitation,correction,integration (Piroddi E., 1999). Thetargetisthatofselectingprojectsand policieswhichmightcontributetotheincrease oftheterritorialandenvironmentalquality.If thedevelopment sustainability depends from the equilibriumandthesynergiesbetween economical, territorial,environmental and social transformations,toincreasetheterritorial patrimony,itisjustontheinter-sectorial relationsthat thesingleactions coherencemust besearched.Theconstructionofapolyvalent evaluationmodelofpolicies,plansandprojects referred toeach strategic project area constitutes aprominentelementoftheproposedplanning methodology.Inthispicturetheevaluation which means to attribute to a project or to a plan qualityorcriticitycharacteritsicsbecomes intrisecallytiedtothedecisionandprojectual action,makingexplicitandverifiablethe projectualchoicestowardsthelocalterritorial impact optimization criteria. THE PROSPECT OF THE PLAN IN LOCAL DEVELOPMENT It is typically a planning activity of integrated type,inthesensethatitpointstoexploitthe effects that derivefrom putting in a net different sector'spoliciesandinterventionsdemand crucial (Mazza L., 1977). It is a creative process, inwhicheachinvolvedsubject,bearerofa specificdefinitionoftheproblems,ofthe priorityandthedevelopmentnecessity, contributes toelaborate the basic orientation and themissionsofthecommunity.Inthissenseit intends to activate and thisconstitutes perhaps itsmostimportantresultanactorsself-reflectionprocess(ForesterJ.1989,Porter M.1987) about the future of a territory.Theplanhastherefore,asaim,the constructionofadocumentwhichcan individuatetheproblems,theopportunities,a territorys development targets7 and scenarios.Certainlytheplantakestheterritoryasits applicationfield(MazzaL.,1997),butitlooks towardsthetown asthe possible policiesspace andthereforefromtimetotimeitsreference changes.Itcanbeaspecificdimensionbecause itisrecognizedbythelocalactorsasworthof particularattention(requalificationofthe historical town) orin a wider sphere, referred to thedifferentdevelopmentgeographies(therole of the town in a territorial context).The plan reference territory thereforeisnot a databutasequence(construction),itdepends fromtheplacestowardwhichtheactors attentionisdrawnandfromthelevelatwhich the questions that they put can be treated.Inorder torespond tothechallengesthat the futuredelineatesitisnecessarytotakeinto consideration some principles. 1.The first principle refers to the assumption of a pragmaticapproach,whichdoesntwaitthe completionof a comprehesive projecttobeable tooperate,but which startsto workin thesense oftheanticipationofthatgeneralproject. Betweencomprehensiveprojectanddetails choices it is necessary to establish a co-evolution connection,inthesensethatthesecond contributetodefinethefirstbutthatfromthis they are also conditioned. Letstakethecaseofthehistoricaltown.It deals with a towns area that seems to require the activation of a regeneration complex policy (that istosaymadeofdifferentinterventionsand integratedbetweenthem)whichcantwaitthe conclusion of the general variants iter tobestarted,butthat,instead,cansupplyto theGeneralTownPlaninterestingtestelements and,moreingeneral,usefulindicationstothe urbanpoliciesonhowtoplanmultidimensional interventions.Thestrategicplanintendsto considerthepragmaticapproachasitsown orientationprinciple,indicatingthoseactions whichcanbeimmediatelystartedorthatitis necessary to discuss for their relevance. 2. The second principle is that of subsidiarity, as themodalitytodefinetherelationsbetweenthe institutionalsubjectsand,moreingeneral, betweenthepublicpoliciesactors.The subsidiarityconcernstheappointmentof competences towards those subjectswho are the nearesttothetreatmentoftheproblems(be thesepublicorprivate)andthereforean assumption of responsibility by them. J. Duhok Univ., Vol. 14, No.1 (Pure and Eng. Sciences), Pp 1-8, 2011 6 Itdealswithaprinciplealreadyacquired, bothintheadministrationstheoryandinthe realpractices.Itimpliesthesubstitutionofthe hierarchicprincipletendingtocooperation.It consistsinanapproachwhichhastopermeate theactivityofthewholepublicadministration, butinwhichitispossibletoindicatethe testingspriorityfields,thosepoliciesorthose interventionsonwhichtostartworkinginthis sense. Againthegoverningofthewidearea relationsseemsoneofthemorepertinent,as wellasthepoliciesforthedevelopments promotion and the same cultural policies. In fact itdealswithworksituationswhichputatplay actorspluralities, ofdifferentnatureandplaced atdifferentdecisionallevels,inwhichitis crucial,thecapacitytogoverncomplex decisionalorganizations,bothverticaland horizontal. 3.Thethirdprincipletiedtothepreceeding oneisthepublicrolerequalificationthat todaymustbeableabovealltodevelopa coordinatingfunction,ofprojectuality promotion,oflocal resources activation. In fact the subsidiarityhorizon doesnt imply the public subjectwithdrawal,butadeepchangeofits actionscharacters.Whatthepublicrole requalification principle suggests is to certainly walkastepbehindtowardsthedirectintake relationwiththesocietyanditsproblems,but alsobeingpotentiallyabletoovercomethe reductiveadministrativelogicsandtoinvestthe own resources efficicaciously. 4.Thefourthprincipledoesntreferonhowto dothingsbutrathertowhattodo.Itisthe principle of the research of the urbanity meant astownscharactertopreserveandto consolidate.Asurbanitywemeanthe compresenceofdifferentusesandfunctionsin thetown(startingfromthehistorictown),the accessibility to all the services which are offered bytheterritory,thestrenghteningoftheties betweenthepartsasguaranteeoftheterritorys general good operation. CONCLUSION Theproposedapproachtotheplanning process,ofwhichtheplansdesignconstitutes and intermediate and temporary phase, intends to contributetothedebateaddressedtothe planningroleandtoitsworkinginstrumentsin the localdevelopment. In this debate, also under theperspectiveofthefederalismsreformina localcontext,awholeofwidelyshared principles is compared:1.Theprincipleofdevelopmentsustainability mustbeintendedinitswidestmeaningof political,socialandculturalsustainability,of economicalself-sustainabilityandterritorial exploitation. 2.Theprincipleofsubsidiarityandof responsibilizationwhichimpliesnotonlyan actualstrengtheningofthelocalpowersanda directinvolvementofthelocalactorsinthe choiceswhichmayconcernthem,butalsoa more coherent and transparent distribution of the government and management responsibilities, on all levels and on all sectors. 3. The principle of solidarity, of cohesion and of inclusionwhichobligestoacontinuous comparison between specificorlocalissuesand generalinterests,betweencompetitivereasons and cooperative exigencies. Suchprinciplespush,jointly,towardsadeep renewal of the methods, of the approaches and of thesameplanningconceptions,puttinginto growingevidencetheexigencyofthedialogue andofthecooperationbetweenthevarious insititutionalsubjectsandthesocialactors involved in the territorial changes.Inthisresearchworker,thebaseofthe analyticapproachisconstitutedbythe individualizationofthefollowinginnovation points: a.newformsoftheplan:thatistosaythe elaborationofnewtownplanninginstruments (newplans,contentsandtargetsconstruction modalities,new structuresof the rulessystem); definitionandutilizationoftheconceptsof equalization,compensation,sustainable development,dimensioning,compensative acquisition,fromwhosecorrectdevelopment dependsthepossiblesolutionsoftown plannings historical knots. b.programmaticinstrumentsforthesocial-ecomomicdevelopment:thepredispositionof plansandstrategicdocumentsforthelocal development; c.complexprograms:thetestingofthe proceduresoftheurbanprojectandofthe integratedprogramsasovercomingofthe traditionalformsofthetownplanning instruments actuation; the relation with thenew formof the townplanninginstrumentsandwith the programmatic means for the socio-ecomomic development. Thenecessityofacooperativeapproachto theterritorysmanagementandplanningis J. Duhok Univ., Vol. 14, No.1 (Pure and Eng. Sciences), Pp 1-8, 2011 7 particularlyemphasized,inourcase,bythe actualsituationofaterritoryinvolvedin considerableglobalizationprocesses.Inthis perspectivetheplanningprocesscannotexhaust itselfintheattempttocoordinatethe municipalitiesinstitutionalaction,sincean authenticcooperationmustbaseitselfonthe self-government of the local realities and thus on the social actors permanent agreeement.Thisimpliestheattempttoletgrowthe territorialsujectivityinviewoftheexploitation ofthelocalsystemsandoftheterritorial identity. Forthisaimitisnecessarythatthe affirmation of the collective subjectivities, which hassomehowalreadyexpresseditselfwiththe adhesionof a plurailityof actors, institutionalor not,mightturnintoasharedprojectual engagementwhichcanexpressitsself-organizingcapacitywithactionsproposal which havetorespondtothelocalexpectationsand interests. Theindicationtoarticulatetheplanprocess strenghteningthelocaldecisionsystemsgoes towardsthedirectioninwhichthelocal initiativeshavetheaimtostrenghtenthe autonomouscapacityofaspecificareatolook foritsowndevelopmentsystem.Inspecific, the plan indications identified as follows: a-Itsuppliesspecificknowlegdebecauseit thematizesandvisualizesfactsandproblemsat systemicaggregationlevelwhichoftenslips awayfromtheattentionoftheaforesaid interlocutors; b-Itsuggeststotheirproblemspossible alternativesolutions,solutionswhichjustderive from the capacity to see and to think the territory onadifferentscaleandinanycasemore complex,topointoutthe problemsintermsnot purelyquantitative,toseizethepossible synergies with other subjects action, etc etc. Insuchawayitisbuiltanenvironment favourabletothedevelopmentstartingfrom eachterritoryspeculiaritiesandwealths,inthe globalcompetitionageareevidencedthe advantagesofterritorialguidelinewhich, engagingthetypicalresourcesaimsatthe qualityanddifferenceoftheofferofproducts and services. Notes 1.Theterritorialplan,meantasscientific definitionmoment,neutral,ofanidealterritory organizationwithinaclearandfirmcontextfor thedistributionoftheadministrative,financial andpoliticalresourcesamongstthevarious governmentslevels,formulatesideal hypothesesfromthepointofviewofthewhole rationalityoftheterritorysuse.Thereforeno to the homologation, but research and promotion ofintegrationchancesbasedontheexploitation of a territorys differences. 2. Itmust be rather thought inlocal systemsnet terms(MagnaniA.2000,RattiR.1997)which, coordinatingandnettingbythemselves,create synergies,thatistosaytheyincreasethewhole wealth,notonlyeconomicalbutalsosocialand cultural,atdisposalofanareathatissubjectto this insitutional competence of the Region.AccordingtoDematteis(1995,46)thelocal developmentisalwaysthecombinationof somethingwhichisfixedwith somethingwhich ismobile:thepotentialspecificresourcesofa territory with the overlocal nets. Thisgivesspace,anyway,ontheterritory,to variousdevelopmentrelations,thatistosayto differenttypesofcombinationsbetweenglobal nets,localnetsandterritorysresources.There arearchitectureswhichhaveamajor endogenouscomponent,thusamoreorless strong local identity is noticed (identity meant as selforganizing capacity, that is to say as specific principlesoflocalorganization).Therarethose which,instead,stronglydependfrom organizations, from overlocal nets. 3. The local identity is meant as a resource, from theeconomicpointofview,astheterritorys competitiveadvantage,fromthesocialand politicalpointofview(thatistosaythe autonomyofthelocal)andfromthecultural pointofview,asculturalvariety,ofaspecific territory (Poli D. 1998, Magnaghi A. 2005 . 4.Inaglobalizationandgrowingcompetition agetopromotethesustainabilitymeansbefore all to strenghten localidentities and peculiarities (Rullani E.1994), indispensable values to be able toplaceoneselfon themarketofferingproducts improbablysubjecttoworldcompetition.To buildlocalsystemsisindispensableto competewithcontinuityagainsttheeconomic trendss changes. 5. The necessity of a cooperative approach to the territorysmanagemnentandplanningis particularlyemphasizedbytheactualsituation of a territory involved in important globalization processes (Dematteis G. 1995) . 6.QuotedinDematteis(1997),thesearepublic and private local subjects self-governing forms. Theselocalself-governingformsarethosewho allowsthemobilizationodendogenous J. Duhok Univ., Vol. 14, No.1 (Pure and Eng. Sciences), Pp 1-8, 2011 8 resources,properoftheseterritories,which otherwisecouldntbeutilizedandthatfeeda circularaccumulationprocessofnewresources ontheterritorythroughpositivecompetitions, thatistosaycompetitionsthroughwhichnew resources and new externalities are created. 7. Theyrepresentsthelocal specificitymeant as aresource(Castells,M.1997)whichcanbe exploitedforitsgreatestpart,onlythroughthe negotiationofthelocalactorswhostartthat cumulative,circularprocesswhichisthelocal development and which allows, on the economic ground, to transform these potential resources in valueswhich can be exported, to attract external humanandcognitiveresources;capitalsand ivestments for these resources exploitation. REFERENCES - Becattini G., Rullani E. (1994), Sistema locale e mercato globale,inBecattiniG.,VaccS.,Prospettive deglistudidieconomiaepoliticaindustrialein Italia, Franco Angeli - Milano. -BramantiA.,MaggioniM.A.(1997),Ladinamicadei sistemiproduttiviterritoriali:teorie,tecniche, politiche, Franco Angeli - Milano. -BramantiA.,SennL.(1997),Cambiamentostrutturale, connessionilocaliegovernanceneisistemi produttiviterritoriali,Ladinamicadeisistemi produttiviterritoriali:teorie,tecniche,politiche, Franco Angeli - Milano.-Castells,M.,(1997),ThePowerofIdentit,Oxford, Blackwell, publishers Ltd. -CastellsM.,(1997),TheRiseoftheNetworkSociety, Oxford, Blackwell Publishers Ltd.-DematteisG.,(1997),"Lecittcomenodidireti:la transizioneurbanainunaprospettivaspaziale",in G.DematteiseP.Bonavero,Ilsistemaurbano italiano nello spazio unificato europeo, Bologna, Il Mulino, pp. 15-35. - Dematteis G., (1995), "Sistemi locali e reti globali: il problema del radicamento territoriale", Archivio di studi urbani e regionali, v.24, n.53, pp.39-52. - Forester J., (1989), Planning in the Face of Power , The Regents of the University of California. GrandinettiR.,RullaniE.(1996),Impresatransnazionale edeconomiaglobale,LaNuovaItaliaScientifica- Roma.- Haley P., (1997), Collaborative Planning, London Mac Millan . -HarveyD.,(1997),Theconditionof Postmodernity,Basil,1990MagnaniA.(2000),Il progetto locale, Torino, Bollati Boringieri. -MagnaghiA.(2005),Larappresentazioneidentitariadel territorio:atlanti,codici,figure,paradigmiperil progettolocale,Alinea,FirenzeMaskellP., MalbergA.(1997),Apprendimentolocalizzatoe competitivitindustriale,Ladinamicadeisistemi produttiviterritoriali:teorie,tecniche,politiche, Franco Angeli - Milano. -MazzaL.,(1997),Trasformazionedelpiano,Franco Angeli, Milano - Perulli P., (2000), La citt delle reti, Torino, Bollati Boringhieri. - Piroddi E., (1999), Le forme del piano urbanistico, Milano, Franco Angeli. -PoliD.,(1998),Ilterritoriofraidentite rappresentazione.Progettodelluogocome biografiaterritoriale,TesidiDottoratoin Progettazioneurbana,territorialeeambientale, Universit di Firenze, IX ciclo, - Porter M., (1980), Competitive Strategy, New York.- Porter M.( 1985), Competitive Advantage - New York.- Porter M.( 1987), Competizione globale, ISEDI - Torino.- Porter M.( 1987), The competitive Advantage of Nations - New York.-PriscoM.R.,SilvaniA.(1997),Lamisurazionedel capitaleumanonellepolitichetecnologiche: l'esperienzadelleareemenofavorite,inBramanti A. , Maggioni-RabellottiR.(1997),ExternalEconomiesand CooperationinIndustrialDistricts,MacMillan press ltd, Houndmills,-RattiR.(1997),Lospazioattivo:unarisposta paradigmaticaaldibattitolocale-globale,La dinamicadeisistemiproduttiviterritoriali:teorie, tecniche, politiche, Franco Angeli - Milano.- Rullani E.( 1994), Sistema locale e mercato globale: una risposta,inBecattiniG.,VaccS.,Prospettive deglistudidieconomiaepoliticaindustrialein Italia, Franco Angeli - Milano. -RullaniE.(1997),Pilocaleepiglobale:verso uneconomiapostfordistadelterritorio,La dinamicadeisistemiproduttiviterritoriali:teorie, tecniche, politiche, Franco Angeli - Milano. J. Duhok Univ., Vol.14, No.1 (Pure and Eng. Sciences), Pp 9-16, 2011 9 ON CP-OPEN SETS AND TWO CLASSES OF FUNCTIONS ALIAS B. KHALAF * andSHILAN A. MOHAMMAD ** * Dept. of Math., Faculty of Science, University of Duhok, Kurdistan Region-Iraq ** Dept. of Math., Faculty of Science, University of Zakho, Kurdistan Region-Iraq(Received: October 1, 2009; Accepted for publication: November 3, 2010) ABSTRACT In this paper we introduce the concept of cp-openset and study some of its properties. Further we introduce the cp-continuous and cp-open functions and investigate the basic propertied. Necessary and sufficient condition of 1P -closed graph and cp-continuous function were found. KEYWORDS: c-open set, cp-open set, cp-continuous and cp-open functions, 1P -closed graph. 1. INTRODUCTION AND PRELIMINARIES

hroughoutthepresentpaper,XandY denotetopologicalspacesinwhichno separationaxiomisassumed.LetAbeasubset of X,we denote theinterior and theclosure of a setAbyint(A)andcl(A) respectively. Asubset A is said to be preopen [10] (resp. semi open[7]) setifAintcl(A)(resp.Aclint(A)).The complement of a preopen set is called preclosed. Theintersectionofallpreclosedsetscontaining A is called the preclosure of A and is denoted by pcl(A).ThepreinteriorofAisdefinedasthe unionofallpreopensetscontainedinAandis denotedbypint(A).Thefamilyofallpreopen setsofXisdenotedbyPO(X)andthe setofall preopensetcontainingxXisdenotedby PO(X,x).Theunionofanyfamilyofpreopen setsispreopen.AsubsetNofXis preneighbourhood[12]ofapointxofXifthere existsapreopensetUcontainingxwithUN and it is denoted by pN (x). AsstatedbyMashhouretal.[10],Katetov madesomecommentsonthepaper[9]tofind conditionsunderwhichtheintersectionofany two preopen sets is preopen. Mashhourtogetherwithothersofferedan answertothisremarkintheformofa theorem[10, Theorem 2.3]. Definition1.1[16].Aspace(X, )willbesaid to have the property P if the closure is preserved underfiniteintersectionorequivalently,ifthe closureof intersectionof any two subsetsequals the intersection of their closures. Lemma1.2[16].Fromtheabovedefinitionit readily follows that if a space X has the property P,thentheintersectionofanytwopreopensets ispreopen. Asa consequenceofthisPO(X) isa topology for X and it is finer than . Lemma1.3[4].LetAandX0besubsetsofa space X. If APO(X) and X0 is semi open in X, then AX0 PO(X0), Lemma 1.4[2]. If AYX and Y is a preopen setinX,thenAPO(X)ifandonlyif APO(Y). Definition 1.5. A function f : XYis called: 1-preirresolute[15] iff1 (V)PO(X)for each preopen set V of Y. 2-M-preopen[16] if theimageofeverypreopen set in X is a preopen set in Y, 3-M-preclosd[13]iftheimageofevery preclosed set in X is a preclosed set in Y. Definition1.6[1].Letf :XYbeany function,thegraphofthefunction f isdenoted by G( f ) andissaidto be 1P -closedif foreach (x,y)G( f ),thereexistUPO(X,x)and VPO(Y,y) with (U V) G( f )= . Weprovedifthegraphoff is 1P -closed and X has the property P, then the inverse image ofastronglycompactsubsetinYispreclosed set in X. Definition1.7[4].AspaceXissaidtobe submaximal if every dense subset of X is open.Lemma 1.8[4].A space Xis submaximalif and only if every preopen set is open. Definition 1.9. A space X is: 1-pre-1T [11]ifforeverydistinctpointsx,yof X,thereexistsUPO(X,x)notcontainingy and V PO(X, y) not containingx, 2- 2T [17](resp.pre-2T [11])ifforeverydistinct pointsx,yofX,thereexisttwodisjoint open(resp.preopen)setseachcontainingoneof them, T J. Duhok Univ., Vol.14, No.1 (Pure and Eng. Sciences), Pp 9-16, 2011 10 3-prenormal[13]ifeachpairofdisjoint preclosedsetsofXthereexistdisjointpreopen sets each containing one of them, 4- p-regular[15] if for each closed set F and each pointxeF,thereexistdisjointpreopensetsU and V such that xeU and F cV, 5-stronglycompact[5]ifeverypreopencover has a finite subcover, 6-Locallyp-compact[8]ifforeveryelementof X,thereexistsanopensetcontainingthem which is strongly compact of X. Lemma1.10[5].Everypreclosedsubsetofa strongly compact space is strongly compact. The following definition appeared in [3] and [6]. Definition 1.11. A subset Aof a space X is said to be a c-open set ifAis an open set and X\Ais compact. 2. SOME PROPERTIES OFCP-OPENSETS Definition 2.1. A subset A of a space X is said to be a cp-opensetifAisapreopensetandX\Ais strongly compact. The complementof a cp-open setissaidtobeacp-closedset.Thefamilyof allcp-open(resp.cp-closed)setsisdenotedby CPO(X)(resp. CPC(X)) . Theclassofc-openandcp-opensetsisnot comparableasshowninthefollowingtwo examples: Example2.2.LetX={a,b,c}witht ={ | ,X, {a}}, then {a, b} is a cp-open set, but it isnot a c-open set. Example2.3.(- ,0) (1,)isac-openset, but it is not a cp-open set. Theorem2.4.IfaspaceXissubmaximal,then thefamilyofc-openandcp-opensetsare equivalent. Proof. It is obvious from Lemma 1.8. Lemma 2.5. The union of any family of cp-open sets is a cp-open set. Proof.Let{oU : A e o }beanyfamilyofcp-openset,sinceoU isapreopensetand X\ oU = X\oU foreachA e o , byLemma 1.10, it is strongly compact. Hence {oU: A e o } is cp-open set. The intersection of two cp-open sets need not be cp-open as seen by the following example: Example2.6.LetX={a,b,c}withtopology t ={ | , X, {a, b}}, then {a, c} and {b, c} are two cp-open sets in X , but {a, c} {b, c}={c} is not cp-open set. Fromtheaboveexamplewenoticethatthe family of all cp-open sets need not be a topology on X. Theorem2.7.IfaspaceXhasthepropertyP, then cp-open sets is a topology on X. Proof.SinceXhasthepropertyP,thenby Lemma 1.2,theintersectionof anytwopreopen setsispreopenanduniontwostronglycompact isstronglycompact.Therefore,theintersection ofanytwocp-opensetsiscp-open.Itfollows that cp-open sets is a topology on X. ItisclearthatifaspaceXisfinite,thenthe familyofcp-opensetsandpreopensetsare coincident. Definition2.8.AsubsetNofaspaceXiscp- neighborhoodofapointx,ifNcontainsacp-open setwhichis containing x andit isdenoted bycpN (x).Theorem2.9.Let(X, t )beanytopological space, and A is any subset of X. A is cp-open set if andonlyifforevery xin Xthereexistsacp-open set xGsuch thatxe xGcA. Prood.LetasubsetAofXbeacp-openset containing x, then xeAcA. Conversely.LetAbeanysubsetofXand assumethatthereexistsacp-openset xGcontainingxsuchthatxexGcA.Hence A= {xG :xeA},sobyLemma2.5,Aiscp-open set. Corollary 2.10. Let A be a subset of a space X. Aiscp-opensetifandonlyifitiscp-neighborhood of each of its points.Definition 2.11. A point xeX is said to be a cp-interior point of A if there exists a cp-open set U containingxsuchthatUcA.Thesetofallcp-interior points of Ais said to be cp-interior of A and denoted by cp-int(A). Herewegivesomepropertiesofcp-interior operator on a set Proposition2.12.ForanysubsetAandBofa topologicalspaceX.Thefollowingstatements are true: 1.The cp-interior of A is the union of all cp-open sets which are contained in A, 2.cp-int(A) is cp-open set contained in A, 3.cp-int(A)isthelargestcp-opensetcontained in A,4.Aiscp-open setif and onlyifcp-int(A)=A,it follows that cp-int(cp-int(A))=cp-int(A), 5.cp-int(A) cA, 6.If AcB, then cp-int(A) ccp-int(B), 7.cp-int(A) cp-int(B) ccp-int(A B), 8.cp-int(A B)c cp-int(A) cp-int(B). J. Duhok Univ., Vol.14, No.1 (Pure and Eng. Sciences), Pp 9-16, 2011 11 Weonlyprovepart(3),theproofoftheother parts is obvious. Proof.(3).LetGbeanycp-opensetinX containingx,byCorollary2.10,Aisacp-neighborhoodofx,thisshowsthatxisacp-interior of A. Hence xecp-int(A), so xeGccp-int(A) cA.Wegetcp-int(A)containsallcp-opensubsetofAanditistherefore,thelargest cp-open subset of A. Ingeneralcp-int(A) cp-int(B) = cp-int(A B),andcp-int(A B) = cp-int(A) cp-int(B) as shown by the following two examples: Example 2.13. Let X={a, b, c, d} with topology t ={ | , X, {c}, {a, d}, {a, c, d}} Take A={a, d} andB={b,c}.cp-int(A B)=Xcp-int(A)={a, d}andcp-int(B)={c},socp-int(A) cp-int(B) = cp-int(A B) Example2.14.LetX={a,b,c,d}andt ={ | , X,{a, b}, {a, b, c}}. Take A={b, c} and B={a, c, d},thenA B={c},cp-int(A B)=| butcp-int(A)={b,c}andcp-int(B)={a,c,d}.Socp-int(A) cp-int(B)={b}andhencecp-int(A B) = cp-int(A) cp-int(B) . Itfollowsthat,cp-int(A)=cp-int(B)doesnot implythatA=B.ThisisshownbyExample4.9 in [12]. Definition2.15.LetAbeasubsetofa topologicalspace(X, t ).ApointxeXissaidto beacp-limitpointofAifeverycp-openset containing x contains a point of A different from x. The set of all cp-limit points of A is called cp-derived set and denoted by cp-d(A). Lemma 2.16. LetA be a subset of a topological space(X,t ).Aiscp-closedifandonlyifA contains all of its cp-limit points. Proof.AssumethatAiscp-closedsetandletp isacp-limitpointofAwhichbelongstoX\A. ThenX\Aisacp-opensetcontainingthecp-limit point of A. Therefore, we get X\A contains an element of A which is contradiction. Conversely.AssumethatAcontainsallofits cp-limitpoints.ForeachxeX\Athereexistsa cp-opensetUcontainingxsuchthatU A= | , thatis,xeUcX\A,byTheorem2.9,X\Aisa cp-open set. Hence A is acp-closed set. Somepropertiesofcp-derivedsetare mentioned in the following results: Proposition 2.17.For any subsetsAandBofa topological space X, thenwehave the following properties. 1. If AcB, then cp-d(A) c cp-d(B), 2. xecp-d(A) implies xecp-d(X\A), 3.cp-d(A) cp-d(B) c cp-d(A B), 4.cp-d(A B) ccp-d(A) cp-d(B). Proof. (3) and (4) follow from (1). Theequalitydoesnotholdin(3)and(4)as shown by Examples 4.12 and 4.13 in [12]. Definition 2.18. The intersection of all cp-closed setscontainingAiscalledthecp-closureofA and it is denoted by cp-cl(A). Herewegivesomepropertiesofcp-closure of the set. Proposition2.19.ForanysubsetEandFofa topologicalspaceX.Thefollowingstatements are true. 1. Ec cp-cl(E), 2. cp-cl(E) is cp-closed set in X containing E, 3. cp-cl(E) isthe smallest cp-closed setcontaining E, 4. E is cp-closed if and only if cp-cl(E)=E, then cp- cl(cp-cl(E)) =cp-cl(E), 5.IfEcF, thencp-cl(E) ccp-cl(F), 6.cp-cl(A) cp-cl(B) ccp-cl(A B), 7.cp-cl(A B) c cp-cl(A)cp-cl(B). Generally,equalityin(6)and(7)doesnot holdsasshownbyExamples4.22and4.23in [12]. Icp-cl(A)=cp-cl(B)doesnotimplyA=Bthisis shown in Example 4.18 in [12]. Theorem2.20.LetXbeaspaceandAbeany subset of X, then A cp-d(A) is cp-closed. Proof.LetxeA cp-d(A).Thisimpliesthat xeAandxecp-d(A).Sincexecp-d(A)there existsa cp-openset xG of xwhich containsno pointofAotherthanxbutxeA.So xGcontainsnopointofA,whichimplies xGcX\A.Again xG is a cp-opensetofeach ofits points. But xGdoes not contain any point ofA,nopoint of xG can be acp-limitpointof A. Thennopointof xG canbelong to cp-d(A), so xGcX\cp-d(A)implies xexGcX\A X\cp-d(A)=X\(A cp-d(A)), byTheorem2.9,X\(A cp-d(A))isacp-open set, so A cp-d(A) is a cp-closed set. Corollary2.21.ForanysubsetAofa topological space X. we havecp-cl(A) =A cp-d(A). Proof.SincebyTheorem2.20,A cp-d(A)is cp-closedcontainingAandcp-cl(A)isthe smallest cp-closed containing A implies thatcp-cl(A) cA cp-d(A).Hencecp-cl(A)=A cp-d(A). J. Duhok Univ., Vol.14, No.1 (Pure and Eng. Sciences), Pp 9-16, 2011 12 Corollary2.22.LetEbeasetinaspaceX.A point xeX is in the cp-closure of E if and only if E U= | , for every cp-open set U containing x. Proof.Letxecp-cl(A),sobyCorollary2.21, eitherxeAorxecp-d(A)andinbothcase E U= | ,foreverycp-opensetUcontainingx. Conversely.SupposethatUisanycp-openset containingxsuchthatE U= | thisimplies that xeE, then xecp-cl(E). Theorem2.23.ForanysubsetAofa topologicalspaceX.Thefollowingstatements are true: 1. cp-cl(A) = X\cp-int(X\A), 2. X\cp-cl(A) = cp-int(X\A), 3. cp-int(A) = X\cp-cl(X\A), 4. X\cp-int(A) = cp-cl(X\A). Weonlyprovepart(i)andtheotherpartsare similar. Proof.ForanyxeX,xecp-cl(A)andby Corollary 2.22A U= | foreach cp-open setUcontainsxxeUcX\Axecp-int(X\A) xeX\cp-int(X\A). Definition2.24.Thesetofallpointsneitherin the cp-interior of A nor in the cp-interior of X\A iscalledcp-boundaryofAanddenotedbycp-b(A). Somepropertiesofcp-boundaryare mentioned in the following results: Theorem2.25.ForanysubsetAofa topologicalspaceX.cp-b(A)=cp-cl(A)\cp-int(A). Proof.sincecp-b(A)=X\(cp-int(A) cp-int(X\A))=X\cp-int(A) X\cp-int(X\A)=cp-cl(A) X\cp-int(A)=cp- cl(A) \cp-int(A) . Corollary2.26.ForanysubsetAofa topological space X, the following are true: 1. If A is cp-closed, then cp-b(A)= A\cp-int(A), 2. If A is cp-open, then cp-b(A)= cp-cl(A)\A, 3. If A is both cp-open and cp-closed, then cp- b(A)=| , 4. A is cp-open if and only if cp-b(A) A= | . That is cp-b(A) cp-int(A) = | , 5. A is cp-closed if and only if cp-b(A) cA, 6. If A is cp-closed and cp-int(A)= | , then cp- b(A)=A, 7. cp-cl(A)=cp-int(A) cp-b(A). We only prove parts(4), (5) and (7), the proof of other parts are obvious. Proof.(4).LetAbeacp-openset.Thencp-b(A)=cp-cl(A)\AcX\Aimpliesthatcp-b(A) A=| . Conversely.Ifcp-b(A) A= | .SoA cp-cl(A) X\cp-int(A) = |implies AX\cp-int(A) =| .ThusAcX\X\cp-int(A)=cp-int(A)buton other hand cp-int(A) cA. It follows that A is cp-open set. (5).LetAbeacp-closed.Thencp-b(A)=A\cp-int(A) cA. Conversely.Ifcp-b(A) cA,thencp-b(A) X\A= | impliesthatcp-cl(A) X\cp-int(A) X\A= | andhencecp-cl(A) cp-int(X\A) X\A= | ,thencp-cl(A) X\A=| . Therefore,cp-cl(A) cAbutonotherhandAc cp-cl(A). It follows that A is cp-cdosed. (7).cp-int(A) cp-b(A)=cp-int(A) cp-cl(A)\cp-int(A)= cp-cl(A). Proposition2.27.ForanysubsetAofa topological space X, the following are true.1. cp-b(A) is cp-closed, 2. cp-b(A)=cp-b(X\A), 3. cp-b(cp-b(A)) ccp-b(A), 4. cp-b(cp-int(A)) ccp-b(A), 5. cp-b(cp-cl(A)) ccp-b(A). Proof.(1).cp-cl(cp-b(A))=cp-cl(cp-cl(A) cp-cl(X\A)) ccp-cl(cp-cl(A)) cp-cl(cp-cl(X\A))=cp-b(A). Therefore A is cp-closed. (2).cp-b(A)=X\(cp-int(A) cp-int(X\A))=X\(cp-int(X\A) cp-int(A))=cp-b(X\A). (3).cp-b(cp-b(A))=cp-cl(cp-b(A)) cp-cl(X\cp-b(A)) ccp-cl(cp-b(A))=cp-b(A). (4).cp-b(cp-int(A))=cp-cl(cp-int(A))\cp-int(cp-int(A))=cp-cl(cp-int(A))\cp-int(A) ccp-cl(A) \cp-int(A) = cp-b(A). (5).cp-b(cp-cl(A))=cp-cl(cp-cl(A))\cp-int(cp-cl(A))=cp-cl(A))\cp-int(cp-cl(A)) ccp-cl(A)\cp-int(A)=cp-b(A). 3. CP-CONTINUOUS AND CP-OPEN FUNCTIONS Inthissection,weintroducetheconceptof cp-continuousandcp-openfunctionsbyusing cp-opensetsalsowegivesomepropertiesand characterizations of such functions. Definition3.1.Thefunctionf :XYiscp-continuousatapoint xeXiffor each xeXand each cp-open set V ofY containingf (x), there existsapreopensetUofXcontainingxsuch thatf (U) cV.Iff iscp-continuousatevery point of X, then it is called cp-continuous. J. Duhok Univ., Vol.14, No.1 (Pure and Eng. Sciences), Pp 9-16, 2011 13 Theorem3.2.Letf :XYbe any function. If f isapreirresolutefunction,thenitiscp-continuous function. Proof. For each xeX andeach cp-open set V of Ycontaining f (x).ThenVispreopenset containingf (x).Sincef ispreirresolute function,thereexistsapreopensetUofX containingxsuchthatf (U) cV.Hencef is cp-continuous. Theconverse of Theorem 3.2 isnot true as it is shown in the following example: Example3.3.Letf fromRwithusual topologyontoRwithco-finitetopologybethe identityfunction.Thenf iscp-continuousbut itisnotpreirresolutebecause(0,1]ispreopen setinRwithco-finitetopologyandinverse image (0, 1] is (0, 1] which isnot preopen set in R with usual topology. Theorem3.4.If theco-domainof cp-continuous functionf :XY is strongly compact, then it is preirresolute. Proof. For each xeX andeach preopen set Vof Y containing f (x), so Y\V is preclosed and Y\V isasubsetofstronglycompactY,thenby Lemma 1.10, Y\V is strongly compact. Sincefiscp-continuousthereexistsapreopensetUof Xcontainingxsuchthatf (U) cVitfollows that fis preirresolute. Theorem3.5.Forafunction f :XY,the following statements are equivalent:1.fis cp-continuous, 2.f1 (V) is preopen set in X, for each cp-open set V in Y, 3.f1 (E)ispreclosedsetinX,foreach cp-closed set E in Y, 4.f (pcl(A)) ccp-cl(f (A)),foreach subset A of X,5.pcl(f1 (B)) cf1 (cp-cl(B)),for each subset B of Y, 6.f1 (cp-int(B)) cpint(f1 (B)),for each subset B of Y, 7.cp-int(f (A)) cf (pint(A)),foreach subset A of X. Proof. (1) (2). Let V be any cp-open set inY. Wehave to show thatf1 (V)is preopen setin X. Let xef1 (V), thenf (x)eV. By (i), there existsapreopensetUofXcontainingxsuch thatf (U) cVwhichimpliesthat xeUcf1 (V)thenf1 (V)= {U: xef1 (V) } is preopen set in X. (2) (3).LetEbeanycp-closedsetinY,then Y\Eisacp-opensetofY.By(ii), f1 (Y\E)=X\ f1 (E)ispreopensetinXand hencef1 (E) is preclosed set in X. (3) (4).LetAbeanysubsetofX,then f (A) ccp-cl( f (A))andAcf1 (cp-cl( f (A))).Sincecp-cl( f (A))iscp-closedset inY.By(iii),wehavef1 (cp-cl( f (A))is preclosedsetinX.Sopcl(A) cf1 (cp-cl( f (A)))andhencef (pcl(A)) ccp-cl( f (A)). (4) (5).LetBbeanysubsetofY,then f1 (B)isasubsetofX.By(iv),wehave f (pcl( f1 (B))) ccp-cl( f ( f1 (B)))=cp-cl(B). It follows that pcl( f1 (B)) cf1 (cp-cl(B)). (5) (6).LetBbeany subsetofY, thenapply (v)toY\B,thenpcl( f1 (Y\B)) cf1 (cp-cl(Y\B)) pcl(X\ f1 (B)) cf1 (Y\cp-int(B)) X\pint((B)) cX\ f1 (cp-int(B)) f1 (cp-intB) cpintf1 (B)). (6) (7).LetAbe any subsetof X,thenf (A) isasubsetofY.By(vi),f1 (cp-int( f (A)) c pint( f1 ( f (A)))=pint(A).Itfollowsthatcp-int( f (A))cf ( pint(A)). (7) (1). Let xeX and lef V be any cp-open set ofYcontainingf (x),thenxef1 (V) and f1 (V)isasubsetofX.By(vii),cp-int ( f ( f1 (V))) cf (pint( f1 (V))).Socp-int(V) cf (pint( f1 (V))),sinceVisacp-openset.ThenVcf (pintf1 (V)))implies thatf1 (V) cpint( f1 (V))andhence f1 (V)ispreopensetinXwhichcontainsx and clearlyf ( f1 (V)) cV. Definition 3.6. A function f : X Y is said to beacp-openfunctionifforeachxeXandfor eachpreopensetUcontainingx,thereexistsa cp-opensetVcontainingf (x)suchthat Vcf (U). J. Duhok Univ., Vol.14, No.1 (Pure and Eng. Sciences), Pp 9-16, 2011 14 Theorem3.7.Forafunction f :XY,the following statements are equivalent: 1.fis a cp-open function, 2.TheimageofeverypreopensetinXis cp-open in Y, 3.f (pint(A)) ccp-int(f (A))foreach subset A of X, 4.pint(f1 (B)) cf1 (cp-int(B))for each subset B of Y, 5.f1 (cp-cl(B))cpcl(f1 (B)) for each subset B of Y, 6.cp-cl(f (A)) cf (pcl(A))foreach subset A of X. Proof.(1) (2). LetUbeanypreopenset in X, weshowthatf (U)iscp-opensetinY.Let f (x)ef (U)impliesxeU.sincef iscp-openfunction,thereexistscp-opensetVinYsuchthatf (x)eVcf (U).Itfollowsthat f (U)= {V:f (x)ef (U) }, by Lemma 2.5, f (U) is cp-open set in Y. (2) (3).LetAbeanysubsetofX.Since pint(A) cAandpint(A)isapreopen setin X. by(ii),f (pint(A))isacp-opensetinY.So f (pint(A))ccp-intf (A). (3) (4).LetBbeanysubsetofY,then f1 (B)isasubsetofX.By(iii), f (pint(f1 (B))) ccp-int f (f1 (B))=cp-int(B). It follows that pint(f1 (B))cf1 (cp-int(B)). (4) (5).LetBbeanysubsetofY,soY\B isa subsetofY.By(iv),pint(f1 (Y\B)) cf1 (cp-int(Y\B))which impliesthatpint(X\ f1 (B)) cf1 (Y\cp-cl(B)),thenX\pcl(f1 (B)) cX\ f1 (cp-cl(B)).Thisshowsthatf1 (cp-cl(B)) cpcl(f1 (B)). (5) (6).LetAbeany subsetofX,thenf (A) isasubsetofY.By(v),weobtainf1 (cp-cl(f (A))) cpcl(f1 ( f (A)))=pcl(A).Hence cp-cl(f (A)) cf (pcl(A)). (6) (1).Foreach xeX.LetU beany preopen set in X containing x, then X\U is a subset of X. By(vi),cp-cl(f (X\U)) cf (pcl(X\U))= f (X\U)andhencef (X\U)=Y\ f (U)iscp-closed.Therefore,f (U)iscp-openset containingf (x) andf (U) cf (U). Hence fis a cp-open function.Theorem3.8.Iff :XYisacp-open function, thenit isM-preopen, and theconverse is also true if the space Y is strongly compact. Proof.LetU be any preopen set in X. Sincef iscp-open,byTheorem3.7(ii),f (U)iscp-opensetinYwhichisalsopreopensetinY. Hencef is M-preopen function. Conversely. Let Y be a strongly compact and let UbeanypreopensetinX.Sincef isM-preopen,thenf (U)ispreopensetinYandby Lemma1.10,Y\ f (U)isstronglycompact.So by Theorem 3.7(ii),f is cp-open. Ingeneraltheconverseofabove theoremisnottrueasitisshowninthe following example: Example3.9.Letf fromRwithusual topologyontoRwithco-finitetopologybethe identity function. Thenfis M-preopen, but it is not cp-open because(0, 1)ispreopen setin R with usual topology and image of(0, 1) is (0, 1) whichisnotcp-opensetinRwithco-finite topology. Theorem3.10.Letf beaM-preopen(resp.M-preclosed)functionfromXontoYandlet g :YZbeanyfunctionsuchthat g o f :X Ziscp-continuous.Theng iscp-continuous. Proof.LetVbecp-open(resp.preclosed strongly compact) subsetof Z. Since g o fis cp-continuous,then( g o f )1 (V)ispreopen( resp.preclosed)subsetofX.Sincef isaM-preopen(resp.M-preclosed)impliesthat f (( g o f )1 (V))= f ( f1 ( g1 (V)))= g1 (V) is a preopen(resp. preclosed) set in Y. Hence gis cp-continuous. Theorem3.11.Letf :XYbeafunction. Then the following statements are true: 1.Iff iscp-continuousandAisasemi open subset of X, then so isf \A: A Y, 2.If {Uo:o inA} isapreopenset ofX and if foreacho , of = f \Uo is cp-continuous, then so isf . Proof (1). Let V be a cp-open set of Y. Sincefiscp-continuous,thenf1 (V)isapreopenset andso( f \A)1 = f1 (V) A, byLemma 1.3, J. Duhok Univ., Vol.14, No.1 (Pure and Eng. Sciences), Pp 9-16, 2011 15 isapreopensubsetofA.Hencef \Aiscp-continuous. (2).LetVbeacp-opensetofY.Then f1 (V)={ 1 of (V):o inA}andsince eachof iscp-continuous,itfollowsthat each1 of (V)isapreopensetinUoandby Lemma1.4, 1 of (V)isapreopensetinX.So f1 (V) is a preopen set on X. Theorem3.12.Iff :XYispreirresolute functionandg :YZiscp-continuous,then g o f : X Z is cp-continuous. Proof. Let V be any cp-open set in Z. Sincegis cp-continuous,sog1 (V)ispreopensetinY. Sincef ispreirresolute,thenweobtain f1 ( g1 (V))=( g o f )1 (V)isapreopen set in X. Henceg o fis cp-continuous. Theorem3.13.Letf :XYbeM-preopen andg :YZbeacp-openfunction,then g o fis cp-open function. Proof.LetUbe any preopenset in X. SincefisM-preopen,thenf (U)ispreopensetinY. sinceg iscp-openfunction,sog ( f (U))= ( g o f )(U)iscp-opensetinZ.ByTheorem 3.7(ii),g o fis a cp-open function. Theorem 3.14. Letf : X Y andg : Y Z be any two function, then. 1.Ifg o f iscp-openfunction andf is preirresolute, theng is a cp-open function, 2.Ifg o f ispreirresoluteandf isM-preopensurjective,theng iscp-continuousfunction, 3.Ifg o f iscp-openfunctionandg is cp-continuousinjective, thenf isaM-preopen function, 4.Ifg o f iscp-continuousfunction and g iscp-openinjective,thenf is preirresolute function. Proof. (1). Let V be any preopen set in Y. Since f ispreirresolute,thenf1 (V)ispreopenset inX.Sinceg o f iscp-openfunction,so ( g o f )(f1 (V))=g ( f (f1 (V)))=g (V) is cp-open setin Z. By Theorem 3.7(ii), gis a cp-open function. (2). Let V be any cp-open set in Z, thus it is also preopen set.Sinceg o f is preirresolute, then ( g o f )1 (V)ispreopensetinX.Sincef is M-preopensurjective,impliesthat f (( g o f )1 (V))=f ( f1 ( g1 (V))) = g1 (V)ispreopensetinY.ByTheorem 3.5(ii), gis cp-continuousfunction, (3). Let U be any preopen set in X,g o fis cp-openfunction,thenbyTheorem3.7(ii),( g o f )(U)isacp-opensetinZ.Sinceg is cp-continuous injective, sog1 (( g o f )(U)))= g1 ( g ( f (U)))=f (U)isapreopensetinY. Hencefis M-preopen function. (4).LetVbeanypreopensetinY.Sinceg is cp-open function, sog (V) is a cp-open set in Z. Sinceg o f iscp-continuousandg is injectivefunction,then( g o f )1 ( g (V))= f1 ( g1 ( g (V)))=f1 (V)isa preopenset in X. Hencefis a preirresolute function.Theorem3.15.Let f :XYbeacp-continuous,M-preclosedfunctionfroma prenormalspaceXontoaspaceY.IfeitherX or Y is pre-T1, then Y is pre-T2. Proof(i).Yispre-T1.Lety1,y2 eYand y1 = y2.So{y1},{y2}arepreclosedstrongly compactsubsetofY,byTheorem3.5(iii),we havef1 (y1)andf1 (y2)arepreclosed subsetofaprenormalspaceX,thenthereexist twodisjointpreopensetsU1andU2inX containingthem.Sinceafunction f isM-preclosed,thesetV1=Y\ f (X\U1)and V2=Y\ f (X\U2) are preopen set in Y . Also are disjointandcontainingy1andy2respectively, so Y is pre-T2. (ii).Xispre-T1, Letf (x) be a pointofY.{x} is preclosedin X. Sincefis M-preclosed, then { f (x)} is a preclosed set ofY. Hence Yis pre- T1 and the proof is complete by part(i). Theorem 3.16. Letf : X Y be any function with 1P -closed graph, X hasthe property P and Yisstronglycompact,thenf iscp-continuous. Proof. For each xeX and each cp-open set Vof Ycontaining f (x),thenVispreopenset containing f (x).HenceY\Vispreclosedand Y\V is a subset of strongly compact Y, so Y\V is stronglycompact.Since f hasa 1P -closed J. Duhok Univ., Vol.14, No.1 (Pure and Eng. Sciences), Pp 9-16, 2011 16 graph,then 1 f (Y\V)ispreclosedandby Theorem3.5(iii),wegetf isacp-continuous function. Theorem3.17.Ifafunction f :XYiscp-continuousandYislocallyp-compact,p-regular, 2Tspace, thenfhas1P -closed graph. Proof. Let (x, y)eG( f ), then f (x) = y. Since Y isa 2T space,thenthereexistsanopensetV1 containingyandf (x)ecl(V1).SinceYis locally p-compact, p-regular space, there exists a preopensetVinYsuchthat yeVcpcl(V) cV1 and V1 is strongly compact, byLemma1.10,pcl(V)isstronglycompact. Therefore,Y\pcl(V)iscp-opensetinY containingf (x).Sincef iscp-continuous, thereexistsapreopensetUcontainingxsuch thatf (U) cY\pcl(V)whichimplies f (U) pcl(V)= | andhenceweobtainf (U) V= | .Itfollowsthatf has 1P -closed graph. REFERENCES N. Bandyopadhyay and P. Bhattacharyya, Function with preclosed graph, Bull. Malays. Math. Sci. Soc., 28(1-2)(2005), 87-93. N. Bandyopadhyay and P. Bhattacharyya, On function with strongly preclosed graph, Soochow. J. of Math.,32(1)(2006), 77-95. H.S. Behnam. Some results about c-continuous functions and c-dimension functions, Mosul University M.SC. Thesis 1984. J. Dontchev, Surveyof preopen sets, The Proceedings oftheYatsushiroTopologicalConference, (1998) 1-18. S. Jafari and T.Noiri, More on strongly compactSpaces,ConferenceofTopologyanditsApplication (Topo 2000) Miami University, USA, A.B. Khalaf. Closed, compact sets and some dimension function, Mosul University M.SC. Thesis 1982. N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly,70(1963), 36-41. A.S. Mashhoure, M.E. Abd El-Monsof, I.A. Hasanein, and T. Noiri. Strongly compact spaces, Delta J. Sci.,8(1)(1984), 30-46. A.S. Mashhoure, M.E. Abd El-Monsof, and S.N. El- Deeb. On precontinuous and weak precontinuousmappings, Proc. Math. and Phys. Soc. Egypt, 53(1982), 47-53. A.S. Mashhoure, M.E. Abd El-Monsof, and S.N. El- Deeb. On pretopological spaces, Bull. Math. Soc.Sci. Math. R.S. Roumanie (N.S.). 28(76)(1984), 39-45. G. B. Navalagi, Further Properties of pre-0T , pre-1T and pre-2T Spaces, Topology Atlaspreprints #428. G. B. Navalagi, pre-neighbourhoods, The Proceedings of the Yatsushiro Topological Conference, (1998), 1-18. G. B. Navalagi, Definition Bank in general topology, Topology Atlaspreprints #422. T.Noiri, Onu -preirresolute functions, Acta Math. Hungar, 95(4) (2002), 287-298. T.Noiri, Almost p-Regular Spaces and some Functions, Acta Math. Hungar, 79(3)(1998), 207-2160 R. Paul and P. Bhattacharyya. On pre-Urysohn spaces, Bull. of the Malaysian Math. Soc.(Second series), 22(1999), 23-34. S. Willard, General Topology, Addison-Wesley publishing company , 1970. cp . cp . 1P . ( cp ) ( cp ) . 1P .J. Duhok Univ., Vol.14, No.1 (Pure and Eng. Scienes), Pp 17-24, 2011 17 GENETIC DIVERSITY ASSESSMENT AND VARIETY IDENTIFICATION OF PEACH (Prunus persica) FROM KURDISTANREGION-IRAQ USING AFLP MARKERS SHAYMAA H. ALIScientific Research Center, University of Duhok. Kurdistan Region-Iraq (Received: January 25, 2010; Accepted for publication: February 27, 2011) ABSTRACT The peach (Prunus persica) is an important member of the Rosaceae family, which contains many fruit, nut, and ornamentalspecies.Ithasabasicchromosomenumberof8.Amplifiedfragmentlengthpolymorphisms(AFLP) markerswereusedtodeterminethelevelofgeneticdiversity,geneticrelationships,andfingerprintingofpeach varieties cultivated in Kurdistan- Iraq. A total of 21 samples have been collected from different districts of Kurdistan includingDuhok,ErbilandSulaymani.ThesampleswereanalyzedbyusingAFLPmarkers.Twoprimer combinationsgeneratedatotalof124bandsandamongthem109(87.9%)werepolymorphic.UsingUPGMA clusteringanalysismethodbasedonthesimilaritycoefficient,varietieswereseparatedintothreemajorgenetic clusters.Thefirstgeneticclustermostlyincludes(Korneetasfer,Floredasin,Motaembkornetmobaker(ahmer), Sprink time, Nectar 4, Nectar 6, Ahmer myse, Badree, Mskee, Tenee, Esmailly, Migrant, Abo-zalma, Silverking). The second genetic cluster includes (Read heaven, Sharly shapor). While the third genetic cluster contains (Zard, Elberta, Zaefaran,Dixired,j.h.hale).Geneticdistanceamong21peachvarietieswererangedfrom0.0073to0.8572.The lowest genetic distance (0.0073) was found between varieties(Tenee) and (Esmailly) which were collected from Erbil, whereas the highest genetic distance (0.8572) was found between varieties num (Badree) and (Elberta) collected from Duhok and Sulaymani respectively. The results obtained in this study may assist peach cultivation and peach breeding programs in the region. KEYWORDS: - Peach (Prunus persica), Genetic Diversity, AFLP-Markers. INTRODUCTION hepeach(Prunuspersica)isoneofthe speciesofgenus,Prunus,nativeto Chinathatbearsanediblejuicyfruit.Itisa deciduous tree growing to 5-10m tall, and it is an importantmemberoftheRosaceaefamily, whichcontainsmanyfruit,nut,andornamental species having a basic chromosome number of 8 (Wang et al. 2002). The scientific name persica, alongwiththeword"peach"itselfandits cognatesinmanyEuropeanlanguages,derives from anearly European belief that peaches were nativetoPersia.Themodernbotanical consensusisthattheyoriginateinChina,and were introduced to Persia and theMediterranean regionalongtheSilkRoadbeforeChristian times (Huxley 1992). Polymerasechainreaction(PCR)-based methods for genetic diversity analyses have been developed,suchasrandomamplified polymorphicDNA(RAPD),amplifiedfragment lengthpolymorphism(AFLP),andintersimple sequencerepeat(ISSR/SSR).Eachtechniqueis not only differed in principal, but also in the type andamountofpolymorphismdetected.AFLP techniqueisbasedontheselectivePCR amplificationofrestrictionfragmentsfroma totaldigestofgenomicDNA.Thetechnique involvesthreebasicsteps:(1)restrictionofthe DNAandligationofoligonucleotideadapters, (2)selectiveamplificationofsetsofrestriction fragments,and(3)gelanalysisoftheamplified fragments (Vos et al. 1995). Recently,theuseofAmplifiedfragment length polymorphisms (AFLP) in genetic marker technologies has become the main tool due to its capabilitytodiscloseahighnumberof polymorphicmarkersbysinglereaction,high throughput,andcosteffective(Jonesetal. 1997). AFLP have beenwidely utilizedmarkers forconstructinggeneticlinkagemapsand geneticdiversityanalysis.Itisauseful techniqueforbreederstoaccelerateplant improvementforavarietyofcriteria,byusing moleculargeneticmapstoundertakemarker-assistedselectionandpositionalcloningfor specialcharacters.Molecularmarkersaremore reliableforgeneticstudiesthanmorphological characteristics because the environment does not affect them (Vos et al. 1995).AFLPmarkershavesuccessfullybeenused foranalyzinggeneticdiversityinsomeother plantspecies.Ithasbeenproventhemost efficient technique estimating diversity in barley (Russeletal.1997),providesdetailedestimates ofthegeneticvariationofpapaya(Kimetal. 2002), and have been used to analyze the genetic T J. Duhok Univ., Vol.14, No.1 (Pure and Eng. Scienes), Pp 17-24, 2011 18 diversityof various plants such as tea (Laiet al. 2001),eggplant(Maceetal.1999),peach (Manubensetal.1999),apple(Guolaoetal. 2001),rapeseed(Lombardetal.1999),wild radish (Man and Ohnishi 2002), Musa sp. (Wong et al. 2001; Ude et al. 2002), peanut (Herselman, 2003),soybean(Udeetal.2003),andmaize (Lbberstedt et al. 2000). A little work has been done on peach using AFLP molecular techniques forevaluatinggeneticdiversityinrelatedness with geographical origin. Theobjectivesof thisstudy are to useAFLP markersforvarietalidentificationandto estimategeneticrelationshipsamongthepeach varieties from Kurdistan region of Iraq. MATERIALS AND METHODS Sample Collection Leafsamplesofthelocalpeachvarieties werecollectedfromdifferentdistrictsin Kurdistan region and analyzed forAFLP. These sampleswereobtainedfromDuhok,Erbiland Sulaymani.Thevarietiesofpeachselectedfor thisstudywere:Korneetasfer,Floredasin, Motaem bkornet mobaker (Ahmer), Sprink time, Nectar4,Nectar6,Ahmermyse,Badree, Mskee,Tenee,Esmailly,Migrant,Abo-zalma, Silverking,Readheaven,Sharlyshapor,Zard, Elberta, Zaefaran, Dixired, and J. H. Hale. DNA Extraction Fromeachvariety,approximately3gof youngleaf tissuewascollected andgrounded to afinepowderusingliquidnitrogen.DNAwas extracted asreported byWeigandetal., (1993). Thismethodwasbasedontheuseof10mlof pre-heated(60 oc)2xCTABextractionbuffer (2xCTAB,5MNaCl,1MTris-HCl,0.5M EDTA),mixedwell,andincubatedat60cin shakingwaterbath.After30minofincubation, themixturewasextractedwith anequalvolume ofcholoroform/isoamylalcohol(24:1,v/v).The mixturewasthencentrifuged(at4000rpmfor 30min).The aqueous phasewastransferredinto anothertubeandprecipitatedwith0.66volume of isopropanol, and then TE- buffer was added to dissolve the nucleic acids.ThesamplesofDNAobtainedwereloaded ontoa0.8%agarosegel,andDNA concentrationwasestimatedbycomparingthe florescence with DNA standard. PCR Amplification of AFLP- primers TheAFLP analysiswas performed according toVosetal.(1995)methodwithminor modifications. Initially, genomic DNA (500ng of each sample) were digested with 5U each of two restrictionenzymes,Tru91(recognitionsite 5TTAA3)andPstI(recognitionsite 5CTGCAG3)in30lafinalvolumeof reactionmixcontaining,1xonephor-allbuffer (pharmacia Bioteh, Uppsala, Sweden) incubating threehoursat37oC.TheDNAfragmentswere thenligatedwithPstIandTru91adapter.This wasachievedbyadding50pmolofTru91-adapter,5pmolofPstI-adapter,inareaction containing1UofT4-DNAligase,1mMrATPin 1xonephore-allbufferandincubatingfor3 hoursat37oC.Afterligation,thereaction mixturewasdilutedto1:5usingsteriledistilled water.Pre-selectivePCRamplificationwas performedinareactionvolumeof20l containing50ngofeachoftheoligonucleotid primers(P00,M43)correspondingtotheTru91 and PstI adapters(P00 primer corresponding for Pst1adapterandM43primercorrespondingfor Mse1(Tru91)adapter),2loftemplate-DNA, 1UTaqDNApolymerase,1xPCRbufferand 5mM dNTPs, in a final volume 20l.The PCR reaction was performed in a thermal cycler using following temperature: 30 cyclesof 30secat94C,6at60C,1minat72C.After that, the pre-amplification product was diluted to 1:5and2lusedastemplateforselective amplification.Selectiveamplificationwas conductedusingTru91andPst1primers combinations.ThePre-amplificationand selective amplification primer combinationsthat usedinthisstudyare(P101+M181, P101+M184).Amplificationwasperformedin thermocyclerprogrammedfor36cycleswith thefollowingcycleprofile:a30secDNA denaturationstepat94C,30secannealingstep (seebelow)anda1minextensionstepat72C. The annealing temperature was varied in the first few cycleitwas65C; ineach subsequent cycle forthenext12cycleitwasreducedby0.7C (touchdownPCR),andfortheremaining23 cycles,itwas56C.Theselectiveamplification productswereloadedonto6%denaturating polyacrylamidgels,andDNAfragmentswere visualizedbysilverstainingkit(Promega, Madison,Wis)asdescribedbythesupplier, silver-stained gelswere scaned to capture digital images of the gels after air dryin. Data analysis Totalbandswerescoredvisuallyand polymorphicbandswererecordedforpresence (1)orabsence(0).Thepolymorphicadaptwas usedtoestimateJaccardcoefficientof dissimilarity(Rohlf,1993).Thesimilarity coefficientwasusedforconstructionof dendrogrambaseonUnweightedPair-Group Method Arithmetic (UPGMA). The dissimilarity coefficientestimationanddedrogram constructionwereperformedusingNTSYS-pc ver. 1.8 software (Rohlf, 1993). J. Duhok Univ., Vol.14, No.1 (Pure and Eng. Scienes), Pp 17-24, 2011 19 RESULTS & DISCUSSION Figure 1 shows a typical AFLP gel image for the21varietiesstudiedwiththeprimer combinations (P101/M181) and (P101/M184). In thisstudygeneticfingerprinting,phylogenetic diversityandgeneticdistanceofpeachvarieties fromKurdistanregionwasevaluatedbyusing AFLPmarkers.Thereweresomestudieshad beendonetoestimatediversityofpeach cultivars in Lebanon using microsattalite primer, (Chalak et al., 2006), AFLP markers also used to evaluate genetic diversity of ornamental peaches, (Donglinetal.,2005).Table4summarizethe valuesofgeneticdistanceof21peachvarieties from different sources and locations. Thegeneticdistancevaluesrangedfrom (0.0073to0.8572).Itwasclearthatthelowest geneticdistance(0.0073)wasfoundbetween varieties(Tenee)and(Esmailly)whichwere collected from Erbil, whereas the highest genetic distance(0.8572)wasfoundbetweenvarietiesBadree)and(Elberta)werecollectedfrom DuhokandSulaymanirespectivelymeansthat the similarity between them is very low. The dendrogram based on UPGMA produced threemajor clusters as shown in (Figure 2). The first genetic cluster mostly consisted of (Korneet Asfer,Floredasin,MotaemBkornetMobaker (Ahmer),SprinkTime,Nectar4,Nectar6, AhmerMyse,Badree,Mskee,Tenee,Esmailly, Migrant,Abo-zalma,Silverking).Thesecond geneticclusterconsistof(ReadHeaven,Sharly Shapor). Whilethe thirdgenetic cluster consists of (Zard, Elberta, Zaefaran, Dixired, J. H. Hale). ThetotalnumberofamplifiedDNAfragments maymakethesevarietiescomesinseparated groups.Studyingthemorphologyofthese varieties,itisnotedthattheyhavesome charactersthatareclosetoeachother,for example,theshapeandcoloroffruits.Sub-clustersseparated thevarietiesand form distinct genetic diversity among clusters. Thegeneticrelationshipamongthecultivars basedonmolecularmarkeranalysiswillbe usefulforvarietalidentificationandinfurther geneticimprovement.Itwillalsoprovide supportforselectionofparentsforcrossingin order to broaden the genetic base of the breeding programs(ThormanandOsborn,1992).Estimationofgeneticrelationshipswillhelpto preventgeneticerosionwithinvarietiesby selectingalargenumberofdifferentclonesof eachvariety(Rhl,etal.,2000).Resultsofthis studywillprovideguidanceforfuture germplasm collection, conservation and breeding of peach. Table (1): Primer name and their sequences used for AFLP analysis Table (2): Name and sampling region of the peach varieties used No.Name of VarietiesLocationNumber of Varieties 1.Korneet Asfer, Floredasin, Motaem Bkornet mobaker (ahmer), Sprink time, Nectar 4, Nectar 6, Ahmer myse, Badree Duhok8 2.Mskee, tenee, Esmailly, Migrant, Abo-zalma, SilverkingErbil6 3.Read heaven, Sharly shapor, Zard, Elberta, Zaefaran, Dixired, J. H. haleSulaymani7 Total21 Table (3): Total number of bands, number of polymorphic bands and theirpercentage as amplified by the two primer combinations. AFLP primer CombinationNumber of Amplified Bands Number of Polymorphic Bands Percentage of Polymorphic Bands P101/M181574985.9% P101/M184676089.5% Total12410987.9% No.Pre selective primer (5------3)Selective primer (5-----3) 1POOGACTGCGTACATGCAGP101GACTGCGTACATGCAGAACG 2M43GATGAGTCCTGAGTAAATAM181GATGAGTCCTGAGTAACCCC 3M184GATGAGTCCTGAGTAACCGA J. Duhok Univ., Vol.14, No.1 (Pure and Eng. Scienes), Pp 17-24, 2011 20 A B Fig. (1): AFLP Gel image of 21 peach varieties produced by primer combination (P101/M181) and (P101/M184). ((1.Korneet asfer 2. Floredasin 3. Motaem bkornet mobaker (ahmer) 4. Sprink time 5. Nectar 4 6. Nectar 6 7. Ahmer myse 8. Badree 9. Mskee 10. Tenee 11. Esmailly 12. Migrant 13. Abo-zalma 14. Silverking 15. Read heaven 16. Sharly shapor 17. Zard 18. Elberta 19. Zaefaran 20. Dixired 21. j. h. hale). J. Duhok Univ., Vol.14, No.1 (Pure and Eng. Scienes), Pp 17-24, 2011 21 Table (4): Genetic distance (Jaccard coefficient) between the peach varieties Fig. (2): The genetic relationship between peach varieties as estimated by AFLP markers analysis. J. Duhok Univ., Vol.14, No.1 (Pure and Eng. Scienes), Pp 17-24, 2011 22 REFERENCES -WangY.,GeorgiL.L.,ReighardG.L.,ScorzaR.,and AbbottA.G.(2002).GeneticMappingofthe evergrowingGeneinPeach[Prunuspersica(L.) Batsch].2002TheAmericanGeneticAssociation 93:352358. - Vos P, Hogers R, Bleeker M, Reijians M, Lee T, Hornes M,FrijtersA,PotJ,PelemanJ,KuiperM,and Zaneau M, 1995. AFLP: a new technique for DNA fingerprinting. 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Lebanese Science Journal, Vol. 7, No. 2 - Donglin,H.; Zuoshuang, Z.; Donglin, Z.; Qixiang, Z. And Jianhua, L. (2005). Genetic relationship of ornamental Peach determined using AFLP-Markers. HortScience 40(6):1782-1786. J. Duhok Univ., Vol.14, No.1 (Pure and Eng. Scienes), Pp 17-24, 2011 23 AFLP - . AFLP . AFLP . DNA ( 421 ) , ( 406 ) ( %89 ) ( 42 ) ( 4224 % ) . UPGMA . ( Korneet asfer, Floredasin, Motaem bkornet mobaker (ahmer), Sprink time, Nectar 4, Nectar 6, Ahmer myse, Badree, Mskee, Tenee, Esmailly, Migrant, Abo-zalma, Silverking ) ( Read heaven, Sharly shapor ) ( Zard, Elberta, Zaefaran, Dixired, j. h. hale .) 020040 025242 . ( 40 ) ( 44 ) , ( 5 ) ( 45 ) . . J. Duhok Univ., Vol.14, No.1 (Pure and Eng. Scienes), Pp 17-24, 2011 24 AFLP ( Prunuspersica ) 5 . AFLP , AFLP . 24 , , AFLP 421 406 ( 54 % ) 42 42 .% UPGMA ( , , ( ) , , 1 , 3 , , , , , , , , ) . ( ) . ( , , , ) . 020040 - 025242 ( ) , ( 025242 ) . . J. Duhok Univ., Vol.14, No.1 (Pure and Eng. Sciences), Pp 25-29, 2011 25 EXISTENCE AND UNIQUENESS SOLUTION FOR NONLINEAR VOLTERRA INTEGRAL EQUATION RAAD. N. BUTRIS* andAVA SH. RAFEEQ** *Dept. ofMathematics, Faculty of Education, University ofZakho ,Kurdistan Region-Iraq **Dept. ofMathematics, Faculty of Science, University ofDuhok ,Kurdistan Region-Iraq (Received: February 14, 2010; Accepted for publication: November 28, 2010) ABSTRACT Inthispaper,westudytheexistence anduniquenesssolutionfornonlinearVolterra integral equation,byusing both methods ( Picard Approximation ) and (Banach Fixed Point Theorem). Also these methods could be developed and extended throughout the study. KEYWORDS: Existence and UniquenessSolution; Volterra Integral Equation;Non-linear; Picard Approximation;Banach Fixed Point Theorem. INTRODUCTION ntegralequationsareencounteredin variousfieldsofscienceandnumerous applications(oscillationtheory,fluiddynamics, electrical engineering, etc.). Exact(closed-form)solutionsofintegral equationsplayanimportantroleintheproper understandingofqualitativefeaturesofmany phenomenaandprocessesinvariousareasof naturalscience.Lotsofequationsofphysics, chemistryandbiologycontainfunctionsor parameters which are obtained from experiments andhencearenotstrictlyfixed.Therefore,itis expedienttochoosethestructureofthese functionssothatitwouldbeeasiertoanalyze andsolvetheequation.Asapossibleselection criterion, one may adopt the requirement that the modelintegralequationadmitasolutionina closedform.Exactsolutionscanbeusedto verifytheconsistencyandestimateerrorsof variousnumerical,asymptotic,andapproximate methods. Recently, [2,3,6]. Pachpztte[5]studiedtheglobalexistenceof solutionsofsomevolterraintegralandintegro-differential equations ofthe form 0( ) ( ) ( , ) ( , ( )) ,txt h t kt s gsx s ds = +}and'0( ) ( , ( ), ( , ) ( , ( )) ),tx t f t xt kt s gsx s ds =}with initial condition(0)ox x = . Tidke [7]investigated the existence of global solutionstofirst-orderinitial-valueproblems, withnon-localconditionfornonlinearmixed Volterra-Fredholmintegrodifferentialequations in Banach spaces of the form. '0 0( ) ( , ( ), ( , , ( )) , ( , , ( )) )t bx t f t xt kt sx s ds h t sx s ds =} } with non-local condition (0) ( )ox gx x = + . Considerthefollowingnonlinearsystemof Volterra integral equations which has the form : ( , ) ( ) ( , ( , ), ( , ) ( , ( , )) ,t so o o oaxt x Ft f s x s x Gs g x x d t t t t= +} } ( )( )( , ( , )) ) ,bsoasg x x d ds t t t}(1) where nx D R e c Disaclosedandbounded domain subset of Euclidean space nR .Let the vectors functions 1 2( , , , ) ( ( , , , ), ( , , , ),..., ( , , , ))nf t x y z f t x y z f t x y z f t x y z =1 2( , ) ( ( , ), ( , ),..., ( , ))ngt x g t x g t x g t x =and 1 2( ) ( ( ), ( ),..., ( ))o on o oF t F t F t F t =are defined and continuous in the domain 1 2 1 2( , , , ) [ , ] ( , ) t x y z a b D D D D D D e c (2) where 1D and 2D areclosedandbounded domains subsets of Euclidean spacemR . Supposethatthefunctions( , , , ) f t x y z and ( , ) gt x satisfy the following inequalities : ( , , , ) , ( , ) f t x y z M gt x N s s (3) 1 1 1 2 2 2 1 2 1 2( , , , ) ( , , , ) f t x y z f t x y z Kx x Ly y s + 1 2Qz z + (4) 1 2 1 2( , ) ( , ) gt x gt x Hx x s (5) for all 1 2 1 2 1[ , ] , , , , , , , t a b x x x D y y y D e e e1 2 2, , z z z D e . whereMandNarepositiveconstantvectors andK,L,QandHarepositiveconstant matrices.LetG(t,s)isan(n n)positive matrixwhichisdefinedandcontinuousinthe I J. Duhok Univ., Vol.14, No.1 (Pure and Eng. Sciences), Pp 25-29, 2011 26 domain[ , ] [ , ] a b a b satisfyingthefollowing condition: ( )( , ) , , 0t sG t s e s >(6) wherea s t b s s s s s and also [ , ] [ , ]max ( ) ( ) , . max .t a b t a bh b t a te e= = , wherea(t)andb(t)arecontinuousfunctions defined on the domain (2). We defined the non-empty sets 1 12 2( )( )( )fffD D Mb aD D HMb aD D hHMb a= = `= )(7)

Furthermore,wesupposethatthegreatest eigenvalue maxof the matrix: ( ( ))( ) K HL Qh b aA = + + ,doesnotexceed unity ,i.e : max1. < PICARD APPROXIMATION METHOD Thestudyoftheexistenceanduniqueness solutionofVolterra integral equation (1)will be introduced by the theorems : Theorem 1. ( Existence Theorem ) Let( , , , ) f t x y z ,( , ) gt x and( )oF t bevector functionswhicharedefinedandcontinuouson thedomain(2),satisfytheinequalities(3),(4) and(5),alsoG(t,s)isdefinedandcontinuous in[ , ] [ , ] a b a b ,satisfiesthecondition (6),then the sequence of functions : 1( , ) ( ) ( , ( , ), ( , ) ( , ( , )) ,t so o o oam m mx t x Ft f s x s x Gs g x x d t t t t+= +} }

( )( )( , ( , )) ) ,bsm oasg x x d ds t t t}(9) with ( , ) ( ) , 1, 2,...o o o ox t x F t x m = = =convergent uniformly on the domain : ( , ) [ , ] ( , )f ft x a b D D e c (10)tothelimitfunction( , )ox t xwhichis satisfying the integral equation:( , ) ( ) ( , ( , ), ( , ) ( , ( , )) ,t so o o oaxt x Ft f s x s x Gs g x x d t t t t= +} } ( )( )( , ( , )) ) ,bsoasg x x d ds t t t}(11) with0( , ( ) ( ) )ox t x F t Mb a s (12) and 10 0( , ( , ( ) . ) ) ( )mmx t x x t x Mb a E s A A (13)Proof:Considerthesequenceoffunctions 1 0 2 0 0( , ( , ( , ), ), ... , ) , ...mx t x x t x x t xdefined by recurrencerelation(9),eachfunctionofthese sequence is continuous in t , x . From(9) whenm = 0and using (3), we have 1 0( , ( ) ( ) ( , , ) ( , ) ( , ) ,taso o o ox t x F t F t f s x Gs g x d t t t + =} } ( )( )( ) ( , ) )obsoast g x d ds F t t } ( )( )( , , ( , ) ( , ) , ( , ) )tas bso o oasf sx Gs g x d g x d ds t t t t ts} } } so that 1 0( , ( ) ) ( )ox t x F t Mb a s (14) By using (5) and (14), we get 1 0 0 0 1 0( , ( , ( , ) ) ( , ) ( , ))t