中心地外文翻译.docx
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中心地外文翻译.docx
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中心地外文翻译
Rank-SizeConstructionoftheCentralPlaceTheorybyFractalMethodandItsApplicationtotheYangtzeRiverDeltainChina
Abstract
TheCentralPlaceTheoryistheclassicaltheoryintheurbangeographyfieldaswellastheRank-SizeRule.TheZipf’slawistheimportantpracticalformulatodescribetheRank-SizeRule,buttherearestillsomeproblemswhentheZipf’sformulaisusedinurbansystem’ssizedistribution,suchasexperientialdiscussionsmorethantheoreticalstudiesandthetheoreticalvaluesfardifferentfromthepracticalvaluesandsoon.InordertoofferabettertheoreticalexplanationfortheRank-Sizeruleofregionalurbansystem,aRank-SizedeepconstructionofcityCentralPlaceTheoryusingfractalmethodisputforward.Giventhefractaldimensionofthesizesdistributionastooneentireurbansystem,thereasonableRank-Sizedistributioncouldbecalculatedbythemodel,andthemodeliscalledCentralRank-SizeRule.WecanpredictthefutureRank-Sizedistributionofaregion’surbansystemandestimatethevalueof“K”whichisdefinedinChristallerCenterPlaceModelbyourequation.Atlast,thecaseofthedeltaareaoftheYangtzeRiverinChinaisbeentakentoexplaintheapplicationofthemodel.
Keywords:
urbansystem;fractaltheory;CentralRank-Sizerule;CentralPlaceTheory
Ⅰ.INTRODUCTION
“Zipf’slawforcitiesisoneofthemostconspicuousempiricalfactsineconomics,orinthesocialsciencesgenerally”.Butasearlyas1980s,G.R.CarrollpointedoutthatthemostofthediscussionsaboutRank-SizeRulewereempiricalandlackedtheintercommunitythroughreviewedprevious70yearsstudyprogressofurbansizes’hierarchy.RosenandResnickinvestigatedthevalueoftheParetoexponentforasampleof44countriesandindicatedthatpopulationsinmostcountriesaremoreevenlydistributedthanwouldbepredictedbytherank-sizerule.MoredetailedstudiesoftheZipf’sLaw(e.g.Guerin-Pace’sstudyoftheurbansystemofFrancebetween1831and1990forcitieswithmorethan2000inhabitants)showthatestimates
TheCentralPlacetheoryisanothernotabletheoryaswellasRank-Sizeruleintheurbangeographyfield.Theformerislogicallydeducedfromsomeparticularhypothesizesbutlackprecisequantitytraitofurbansize,whilethelatterisinducedonbasesofmanyempiricalstatisticalcasesbutlackadeliberatespacestructure.Theremustbesomerelationshipsbetweenthetwoprominenttheoriessinceurbansystemisoneandthesameobjecttheystudyatnotwithstandingtheirmethodaredifferent.FractalmethodprovidesusagoodmeasuretosolveaboveproblembecauseChaostheoryisabridgebetweendeductionandinduction.S.L.Arlinghans(1985)hadfoundthatthemodelofCentralPlacetheoryimpliedfractalstructurecharacteristic.Andthehierarchyofurbansizehasbeenalsodemonstratedtopossessfractalstructurebythecreatoroffractaltheory.At1990sorevenearlier,ithasbeenshownthatafractaldistributionaccountsfortheParetoorlog-linearrank-sizedistribution.Inthispaper,wetrytobuildupanewmodelofurbansizehierarchyonthebaseofCentralPlaceTheoryinspiredbyfractalmethod,andwenameitCentralRank-SizeRule,andtakesthedeltaareaoftheYangtzeRiverinChinaforanexampletoexplainthedetailsofitsapplication.
II.THEORETICALFOUNDATION
A.CentralPlaceTheory
TheCentralPlacetheoryestablisheditselfasoneofthemostinfluentialtheoriesoftheoreticalgeographyandtheoreticalspatialeconomicanalysis.Withthehypothesizesofidealplaneandreasonableagent,Christallerpointoutthattheactivityplacesofhumanhaveobviouslycentricityanddifferenthierarchyduetothehinterlandareasize.Andallhinterlandareasofthecentralplacesatthesamehierarchicallevelformahexagonalcoveringoftheplane.Thesizeofthehinterlandareasincreasesfromthesmallest(onthelowertierofCentralPlacehierarchy)tothelargest(onthehighesttierofhierarchy)byaconstantnestingfactork,withwhich3,4,7playthemostimportantroleintheChristallerCentralPlacetheory:
theyexpressoneoftheChristallerthreeprinciples,namely,marketing(k=3),transportation(k=4)andadministrative(k=7)principles.Theschianhexagonallandscapeisthesuperpositionofallpossiblecoveringsofaplanebyhexagonswhosecentersarecoincidewiththeverticesofthetriangularlatticeandthesizesofmarketareas(nestingfactors)areintegers:
k=1,3,4,7,9,landscapewhichwaspurelydeducedusingthemethodsofmathematicsandeconomicsfirmedtheprestigeofCentralPlaceTheory.
B.FractalTheory
Fractaltheoryderivedfrom1970sisfairlyanewfieldinvolvingtheknowledgeofnaturescience,socialscienceandphilosophy.Itinvestigatesandrevealsthedeep-seatedrulesunderthenaturalandsocialphenomenaandattemptstomodelthecomplexprocessbysearchingforthesimpleprocessunderneathonthebaseoffractalgeometry.Theirregularshapesinnatureorabstractsocialeconomystructures,suchascoastline,rivers,urbansystem,thatcannotberepresentedbyclassicalEuclideangeometryaretheobjectsoffractalresearch.Almostallfractalsareatleastpartiallyself-similar.Thismeansthatapartofthefractalisidenticaltotheentirefractalitselfexceptsmaller.Theessentialoffractalsisself-similarquantifiedbyfractaldimension.
Thedistributionofcity-sizepossessesthetraitofself-similarandanurbanhierarchycanbecharacterizedbyafractaldimensionalityorarecursiverelationmathematically.Astoaregion,givenanurbansizertocalculatethenumberN(r)ofthecitiessurpassingthesizer,thenwecandefinethefollowingform:
N(r)=k-r
wherekisacoefficient,Ddenotesthefractaldimensionoftheurbansizedistribution.
III.THEFRAMEWORKOFCENTRALRANK-SIZERULE
A.Hypotheses
ExceptingthetwosupposesofidealplaneandreasonableagentreferringtoCentralPlaceTheory,sinceourmodelherewillbebuiltonitsbase,therearetwopointsneedtoclear:
first,thereshouldbeasingleregionwithoneprimarycity.Iftherearetwoormoreprimarycitieswithalmostequalsize,thenwetakethemasthesecondlevelcitiesandsupposetheprimarywasexisting;second,inourmodel,thesamerankcitiessurroundingtheupcentralcityhaveanequalsize,andthesizemeanstheaveragesizeofthecitiesatthesamelevel.Soitisdeferentwiththemeanofrank-sizeinZipf’slaw.
B.InferenceofCentralRank-SizeRule
Ithasbeenprovedthatthecentralplacemodelpossessesfractalcharacterorself-similar.So,fractalmathematicscouldbeusedtoexplainourmodelwhosespatialstructureisalmostsamewithcentralplacemodel.GivenPrrepresentstheaveragesizeoftheurbanrankr,andN(r)denotesthenumberofcitieswhosesizessurpassPr,wecanget:
N(r)=
(1)
WherekisthenestingfactorofChristaller’smodel.Thenaccordingtoourhypothesesweknow:
N
(1)=1
(2)
Basingonthefractaltheoryexpressionbyequation
(1),givenagaugeofurbansizePrtocalculatethenumberN(r)ofcitieswhosesizesaregreaterthanPr,therelationshipcanbegivenas:
N(r)=
(3)
isconstant,Disfractaldimension.Givenr=1intheformula(3),yieldafunction
N(l)=
(4)
Combineabovefunctions
(1)-(4),apolynomialparametricmodelcanbewrittenasfollows:
Pr=
(5)
Theformula(5)istheexpressionoftheCentralRank-SizeRuleforurbansystem.AndthefractaldimensionDcouldbederivedfromformula(3)byconvertingittologarithmformthenusingthepracticaldatum.
C.ApplicationoftheModel
Usingtheformula(5),giventheprimarycitysizeofanurbansystem,wecanworkouttheotherurbanranks’sizesunderdifferentvaluesofChristaller’sKafterreckoningoutthefractaldimensionDoftheurbansystem.LiuJisheng(1998)hasprovedthemeaningofthefractaldimensionofurbannetwork:
D=1,meansthesizeratiooftheprimarycityandthelattermostcityopportunelyequalsthenumberofcitiesinaregion,whichiscalledrestricteddistributionofcity-sizebyCarroll;D<1,meansaprimarydistributionwithpolarizedcity-size;D>1,meanssizedistributionismoreequilibratingandmostcitiesareinmidst;D→0,thereisonlysinglecityinaregion;D→∞,everycityintheregionhasequalsize.Astoaregion,givenD=1,theurbanrank-sizedistributionunderdifferentvaluesofKisfollowingTab.1.TheDofeveryregionalurbansystemisdifferentbecausediscrepantdevelopmentstage.Soinpractice,thesizesofurbanrankscouldbecalculatedoutaccordingtoitsdevelopmentstagebytakingtheDofitself.AndthedistributionprincipleinChristaller’smodeloftheregionalurbansystemcanbeestimatedbycomparingtheidealandpracticaldistributiontoo.
TABLEI.THECENTERDISTRIBUTEDMODELOFURBANRANK-SIZEWHEND=1
P1(primacysize)
Rank
1
2
3
4
……
r
K
3
/3
/7
/15
……
/(1+2+…+
)
4
/4
/19
/40
……
/(1+3+…+
)
7
/7
/43
/259
……
/(1+6+…+
)
ByemployingCentralPlaceTheoryandRank-SizeRuleinspiringbyFractalTheory,abovemodelgivesanexplanationthatthetwoclassicaltheoriesabouturbansystemareabsolutelyinterrelatedandenhancethetheorizationofrank-sizerulebyformulatingitbutnotjustinducingfrompractices.Ourmodelhassomeadvantages:
First,byintroducingthefractaldimensionD,therationalrank-sizedistributionofaregionalurbansystemaccordingtoitsurbanizationlevelcouldbecalculatedout.Soithasmorepracticability;
Second,theintroductionofthefractaldimensionofthescalesothatinter-regionalurbansystemisbothlaterallycomparablelevel,andcan
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