| Description |
1 online resource (xvii, 313 pages : illustrations (somo color)) |
| Series |
Springer series in surface sciences ; 53 |
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Springer series in surface sciences ; 53
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| Contents |
880-01 Hexagonal Distributions in Economic Geography and Krugmans CorePeriphery Model -- Group-Theoretic Bifurcation Theory -- Agglomeration in Racetrack Economy -- Introduction to Economic Agglomeration on a Hexagonal Lattice -- Hexagonal Distributions on Hexagonal Lattice -- Irreducible Representations of the Group for Hexagonal Lattice -- Matrix Representation for Economy on Hexagonal Lattice -- Hexagons of Christaller and Losch: Using Equivariant Branching Lemma -- Hexagons of Christaller and Losch: Solving Bifurcation Equations |
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880-01/(S Machine generated contents note: 1. Hexagonal Distributions in Economic Geography and Krugman's Core-Periphery Model -- 1.1. Introduction -- 1.2. Christaller's Hexagonal Market Areas and Lösch's Hexagons -- 1.2.1. Christaller's Hexagonal Distributions -- 1.2.2. Lösch's Hexagons -- 1.3. Agglomeration in Southern Germany: Realistic Spatial Platform -- 1.4. Hexagons on Hexagonal Lattice: Idealized Spatial Platform -- 1.4.1. Hexagonal Lattice and Possible Hexagonal Distributions -- 1.4.2. Agglomeration Analysis for a Hexagonal Lattice -- 1.5. Appendix: Krugman's Core-Periphery Model -- 1.5.1. Basic Assumptions -- 1.5.2. Market Equilibrium -- 1.5.3. Spatial Equilibrium -- 1.5.4. Stability -- 1.6. Appendix: Preexisting Study of Self-organization of Cities -- 1.7. Summary -- References -- 2. Group-Theoretic Bifurcation Theory -- 2.1. Introduction -- 2.2. Bifurcation of the Two-Place Economy -- 2.2.1. Economic Understanding -- 2.2.2. Mathematical Treatment -- 2.3. Group Representation -- 2.3.1. Group -- 2.3.2. Basic Concepts of Group Representation -- 2.3.3. Irreducible Representation -- 2.3.4. Block-Diagonalization of Commuting Matrices -- 2.4. Group-Theoretic Bifurcation Analysis -- 2.4.1. Equilibrium Equation and Critical Point -- 2.4.2. Group Equivariance of Equilibrium Equation -- 2.4.3. Liapunov-Schmidt Reduction -- 2.4.4. Symmetry of Solutions -- 2.4.5. Equivariant Branching Lemma -- 2.5. Summary -- References -- 3. Agglomeration in Racetrack Economy -- 3.1. Introduction -- 3.2. Racetrack Economy and Symmetry -- 3.2.1. Equilibrium Equation -- 3.2.2. Equivariance of Equilibrium Equation -- 3.2.3. Distribution-Preserving Equilibria -- 3.3. Irreducible Representations -- 3.3.1. Irreducible Representations of the Dihedral Group -- 3.3.2. Irreducible Representations for Racetrack Economy -- 3.4. Bifurcation from Flat Earth Equilibria -- 3.4.1. Use of Block-Diagonalization -- 3.4.2. Use of Equivariant Branching Lemma -- 3.4.3. Use of the Bifurcation Equation -- 3.5. Other Bifurcations -- 3.5.1. Bifurcation from Distribution-Preserving Equilibria -- 3.5.2. Spatial Period Doubling Bifurcation Cascade -- 3.5.3. Transcitical Bifurcation -- 3.5.4. Hierarchical Bifurcations -- 3.6. Numerical Analysis of Racetrack Economy -- 3.6.1. Computational Bifurcation Analysis Procedure -- 3.6.2. Spatial Period Doubling Bifurcation Cascade -- 3.6.3. Spatial Period Multiplying Bifurcations -- 3.7. Summary -- References -- 4. Introduction to Economic Agglomeration on Hexagonal Lattice -- 4.1. Introduction -- 4.2. Core-Periphery Models -- 4.2.1. Equilibrium Equation -- 4.2.2. Iceberg Transport Technology -- 4.3. Framework of Group-Theoretic Bifurcation Analysis -- 4.3.1. Hexagonal Lattice and Symmetry -- 4.3.2. Equivariance of Equilibrium Equation -- 4.3.3. Bifurcation Analysis Procedure -- 4.4. Theoretical Predictions by Group-Theoretic Analysis -- 4.4.1. Christaller's Hexagonal Distributions -- 4.4.2. Lösch's Hexagonal Distributions -- 4.5. Numerical Examples of Hexagonal Economic Agglomeration -- 4.5.1. Christaller's k = 3 System and a Lösch's Hexagon -- 4.5.2. Christaller's k = 4 System and Lösch's Hexagons -- 4.5.3. Christaller's k = 7 System -- 4.6. Summary -- References -- 5. Hexagonal Distributions on Hexagonal Lattice -- 5.1. Introduction -- 5.2. Infinite Hexagonal Lattice -- 5.2.1. Basis Vectors -- 5.2.2. Hexagons of Christaller and Lösch -- 5.3. Description of Hexagonal Distributions -- 5.3.1. Parameterization of Hexagonal Distributions -- 5.3.2. Types of Hexagonal Distributions -- 5.4. Finite Hexagonal Lattice -- 5.5. Groups Expressing the Symmetry -- 5.5.1. Symmetry of the Finite Hexagonal Lattice -- 5.5.2. Subgroups for Hexagonal Distributions -- 5.5.3. Conjugate Patterns -- 5.6. Summary -- References -- 6. Irreducible Representations of the Group for Hexagonal Lattice -- 6.1. Introduction -- 6.2. Structure of the Group for the Hexagonal Lattice -- 6.3. List of Irreducible Representations -- 6.3.1. Number of Irreducible Representations -- 6.3.2. One-Dimensional Irreducible Representations -- 6.3.3. Two-Dimensional Irreducible Representations -- 6.3.4. Three-Dimensional Irreducible Representations -- 6.3.5. Four-Dimensional Irreducible Representations -- 6.3.6. Six-Dimensional Irreducible Representations -- 6.3.7. Twelve-Dimensional Irreducible Representations -- 6.4. Derivation of Irreducible Representations -- 6.4.1. Method of Little Groups -- 6.4.2. Orbit Decomposition and Little Groups -- 6.4.3. Induced Irreducible Representations -- 6.5. Summary -- References -- 7. Matrix Representation for Economy on Hexagonal Lattice -- 7.1. Introduction -- 7.2. Simple Example of Representation Matrix -- 7.3. Representation Matrix -- 7.4. Irreducible Decomposition -- 7.4.1. Simple Examples -- 7.4.2. Analysis for the Finite Hexagonal Lattice -- 7.5. Transformation Matrix for Irreducible Decomposition -- 7.5.1. Formulas for Transformation Matrix -- 7.5.2. Proof of Proposition 7.1 -- 7.6. Geometrical Implication and Computational Use of Transformation Matrix -- 7.6.1. Hexagonal Patterns Associated with Transformation Matrix -- 7.6.2. Use in Computational Bifurcation Analysis -- 7.7. Summary -- References -- 8. Hexagons of Christaller and Lösch: Using Equivariant Branching Lemma -- 8.1. Introduction -- 8.2. Theoretically Predicted Bifurcating Hexagonal Distributions -- 8.2.1. Symmetry of Bifurcating Hexagonal Distributions -- 8.2.2. Hexagons Engendered by Direct Bifurcations -- 8.3. Procedure of Theoretical Analysis: Recapitulation -- 8.3.1. Bifurcation and Symmetry of Solutions -- 8.3.2. Use of Equivariant Branching Lemma -- 8.4. Bifurcation Point of Multiplicity 2 -- 8.5. Bifurcation Point of Multiplicity 3 -- 8.6. Bifurcation Point of Multiplicity 6 -- 8.6.1. Representation in Complex Variables -- 8.6.2. Isotropy Subgroups -- 8.6.3. Hexagons of Type V -- 8.6.4. Hexagons of Type M -- 8.6.5. Hexagons of Type T -- 8.6.6. Possible Hexagons for Several Lattice Sizes -- 8.7. Bifurcation Point of Multiplicity 12 -- 8.7.1. Representation in Complex Variables -- 8.7.2. Outline of Analysis -- 8.7.3. Isotropy Subgroups -- 8.7.4. Existence of Bifurcating Solutions -- 8.7.5. Hexagons of Type V -- 8.7.6. Hexagons of Type M -- 8.7.7. Hexagons of Type T -- 8.7.8. Possible Hexagons for Several Lattice Sizes -- 8.7.9. Appendix: Construction of the Function φ -- 8.7.10. Appendix: Proofs of Propositions 8.12, 8.14, and 8.15 -- 8.8. Summary -- References -- 9. Hexagons of Christaller and Lösch: Solving Bifurcation Equations -- 9.1. Introduction -- 9.2. Bifurcation and Symmetry of Solutions -- 9.3. Bifurcation Point of Multiplicity 2 -- 9.3.1. Irreducible Representation -- 9.3.2. Equivariance of Bifurcation Equation -- 9.3.3. Bifurcating Solutions -- 9.4. Bifurcation Point of Multiplicity 3 -- 9.4.1. Irreducible Representation -- 9.4.2. Equivariance of Bifurcation Equation -- 9.4.3. Bifurcating Solutions -- 9.5. Bifurcation Point of Multiplicity 6 -- 9.5.1. Representation in Complex Variables -- 9.5.2. Equivariance of Bifurcation Equation -- 9.5.3. Bifurcating Solutions -- 9.6. Bifurcation Point of Multiplicity 12 -- 9.6.1. Representation in Complex Variables -- 9.6.2. Equivariance of Bifurcation Equation -- 9.6.3. Bifurcating Solutions -- 9.7. Summary -- References |
| Summary |
This book contributes to an understanding of how bifurcation theory adapts to the analysis of economic geography. It is easily accessible not only to mathematicians and economists, but also to upper-level undergraduate and graduate students who are interested in nonlinear mathematics. The self-organization of hexagonal agglomeration patterns of industrial regions was first predicted by the central place theory in economic geography based on investigations of southern Germany. The emergence of hexagonal agglomeration in economic geography models was envisaged by Krugman. In this book, after a brief introduction of central place theory and new economic geography, the missing link between them is discovered by elucidating the mechanism of the evolution of bifurcating hexagonal patterns. Pattern formation by such bifurcation is a well-studied topic in nonlinear mathematics, and group-theoretic bifurcation analysis is a well-developed theoretical tool. A finite hexagonal lattice is used to express uniformly distributed places, and the symmetry of this lattice is expressed by a finite group. Several mathematical methodologies indispensable for tackling the present problem are gathered in a self-contained manner. The existence of hexagonal distributions is verified by group-theoretic bifurcation analysis, first by applying the so-called equivariant branching lemma and next by solving the bifurcation equation. This book offers a complete guide for the application of group-theoretic bifurcation analysis to economic agglomeration on the hexagonal lattice |
| Bibliography |
Includes bibliographical references and index |
| Notes |
English |
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Print version record |
| Subject |
Economic geography -- Mathematical models.
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Industrial clusters.
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Bifurcation theory.
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BUSINESS & ECONOMICS -- Economics -- General.
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BUSINESS & ECONOMICS -- Reference.
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Ingénierie.
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Bifurcation theory
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Economic geography -- Mathematical models
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Industrial clusters
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| Form |
Electronic book
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| Author |
Murota, Kazuo, 1955- author.
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| ISBN |
9784431542582 |
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4431542582 |
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4431542574 |
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9784431542575 |
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4431542590 |
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9784431542599 |
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