Abstract
At the beginning of Part II, several fundamental issues are presented and major findings of Part II are previewed. The equilibrium equation of variants of core–periphery models and the framework for theoretical treatment of the economic model on the hexagonal lattice are given. As a summary of the theoretical results in Chaps. 5–9, a list of bifurcating hexagonal distributions and the associated sizes of the hexagonal lattice are provided. The existence of hexagonal distributions is demonstrated by numerical bifurcation analysis for a specific core–periphery model. These distributions are the ones envisaged by central place theory and also envisaged to emerge by Krugman, 1996 for a core–periphery model in two dimensions. The missing link between central place theory and new economic geography has thus been discovered.
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Notes
- 1.
Hexagonal distributions of Christaller and Lösch (Christaller, 1933 [4] and Lösch, 1940 [10]) are introduced in Sect. 1.2 and Krugman’s new economic geographical model is given in Sect. 1.5.
- 2.
Neary, 2001, p. 551 [11] stated “Perhaps it will prove possible to extend the Dixit–Stiglitz approach to a two-dimensional plain.” Stelder, 2005 [14] conducted a simulation of agglomeration for cities in Europe using a grid of points. Barker, 2012 [2] extended the racetrack geometry to two dimensions and conducted a simulation and compared the results with real cities.
- 3.
The term of hexagonal lattice is commonly used in many fields of mathematical sciences, although it is also called regular-triangular lattice.
- 4.
- 5.
- 6.
These relations can be derived based on geometrical consideration; for an algebraic understanding of these relations, see (7.11)–(7.14) in Sect. 7.3. Note that ⟨r, s⟩ is isomorphic to the dihedral group D6 and ⟨p 1, p 2⟩ is isomorphic to the direct product \(\mathbb{Z}_{n} \times \mathbb{Z}_{n}\) of two cyclic groups \(\mathbb{Z}_{n}\) (denoted as C n in Sect. 2.3). The group G is in fact the semidirect product of D6 by \(\mathbb{Z}_{n} \times \mathbb{Z}_{n}\), i.e., \(G =\mathrm{ D}_{6} \ltimes (\mathbb{Z}_{n} \times \mathbb{Z}_{n})\); see Sect. 2.3.1 for this concept and notation. The structure of this group is discussed in Sect. 6.2, and its irreducible representations in Sect. 6.3.
- 7.
The concrete form of T(g) is given in Sects. 7.2 and 7.3 .
- 8.
Although a large number of bifurcation points exist on this line, the theoretical development in Chap. 8 was useful in pinpointing a bifurcation point that produces a specified hexagonal pattern.
- 9.
For the two-level hierarchy, for example, each second-level center is shared by two neighboring market areas. In effect, \(6/2 = 3\) second-level centers exist in the market area, thereby leading to N 1: N 2 = 1: 3.
- 10.
In the secondary bifurcated solution FG, six of the 12 third-level centers had slightly larger population than the other six, but they are considered to be identical herein.
- 11.
In this case with n = 7, the expression \(\langle r,p_{1}^{3}p_{2},p_{1}^{-1}p_{2}^{2}\rangle\) can be reduced to a simpler form \(\langle r,p_{1}^{3}p_{2}\rangle\).
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Ikeda, K., Murota, K. (2014). Introduction to Economic Agglomeration on Hexagonal Lattice. In: Bifurcation Theory for Hexagonal Agglomeration in Economic Geography. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54258-2_4
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