14
more than one adjacent and one periphery region is defined as a CAP cluster, which is
mathematically expressed as: 7
AP
CAPj = Cj ∩∑ Aj ∩∑ Pj (Θcp > Θca ≥ Θap ) = 0 (10)
j =1 j =1
where, CAPj represents a core, and a cluster of j adjacent, and periphery regions. These region types
are symbolised by: Cj core, Aj adjacent, Pj periphery. Distance from the core is represented by the
symbol θ. The expression in brackets states that the distance from the core to the periphery θCP is
greater than the distance from the core to the adjacent θCA, and the distance from the adjacent to the
periphery θAP is greater than, or equal to the distance from the core to the adjacent. The symbol
0 indicates that the regions are non-overlapping.
The CAP cluster is a multi-region CAP model. The number of first and second ring regions
around the core agglomerate determines the number of regions in the cluster. For example, if a core
agglomerate is contiguous to one adjacent and one periphery region such that j = 1 for both Aj and Pj,
this results in a basic three-region CAP cluster, as illustrated in Diagram 1. On the other hand, if a core
region is surrounded by three adjacent regions and two periphery regions, then Aj = 3, and Pj = 2, this
would provide us with a six-region model, with economic interaction occurring between the regions
due to their geographic proximity.
A multi-region country, Ui, can consist of a number of CAPj clusters, each with a varying
number of regions. An individual country then becomes the sum of its CAPj clusters, expressed as
follows:
CAP C A P
Ui =∑CAPj =∑Cj ∩∑Aj ∩∑Pj(θCP >θCA ≥θAP)=0 (10a)
j =1 j =1 j =1 j =1
7Equation (10) is developed from equation (9). Each CAP cluster is a union of administrative regions around a core region
that form a non-overlapping collective.
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