中心地外文翻译.docx

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