A Regional Core, Adjacent, Periphery Model for National Economic Geography Analysis

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transformed into a multi-region CAP model. The mathematical derivation of the national regional
geographic CAP model is presented in the following section.

4.2 The Mathematics of the CAP Model

Let U represent any country with a set of urban population density elements updi where i =
1... I. This set of population density elements is represented by:

U = {upd∖i = 1,..., I}                                                                 (1)

where i is the urban population density of a given urban area, and I is the total of all urban areas in a
country. It is possible to create three proper subsets of U, with the symbols C, A, and P, such that
C ⊂ U , A ⊂ U , and P ⊂ U , given the condition that C ≠ A ≠ P ≠ U. By using the extension theorem
of set theory, specific values of the elements from U can be assigned to the three respective subsets: C,
A, and P .Let the function φ(updi) be the criterion for the subset C, such that φ(updi) ∈ C. Subset C is
then characterised by the following condition:

φ(upd_i) ∈ C θ upd_i ∈ U ∩ φ(upd_i )           ∀i                           (2)

Thus each element updi in U that satisfies the criterion φ(updi) is assigned to the subset C. For subset
A, γ(updi) ∈ A, and is characterised by the following equation:

γ(upd_i) ∈ A θ upd_i ∈ U ∩ φ(upd_i) ∩ γ(upd_i)Θ_ca       ∀i                   (3)

Equation (3) states that every element updi, in U that satisfies the criteria γ(updi) and not the
criteria φ(updi) will be assigned to the subset A. Finally, the criterion for subset P is the same as for
subset A since a region that is two regions removed from the core can theoretically have the same
γ(updi) as an adjacent region. However, it is differentiated from an adjacent region by its geographic

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