Table of Contents Author Guidelines Submit a Manuscript
Discrete Dynamics in Nature and Society
Volume 2008 (2008), Article ID 728420, 22 pages
http://dx.doi.org/10.1155/2008/728420
Research Article

A Wave-Spectrum Analysis of Urban Population Density: Entropy, Fractal, and Spatial Localization

Department of Urban and Economic Geography, College of Urban and Environmental Sciences, Peking University, Beijing 100871, China

Received 9 December 2007; Revised 15 May 2008; Accepted 16 August 2008

Academic Editor: Michael Batty

Copyright © 2008 Yanguang Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. C. Clark, “Urban population densities,” Journal of Royal Statistical Society, vol. 114, pp. 490–496, 1951. View at Google Scholar
  2. R. J. Smeed, “Road development in urban area,” Journal of the Institution of Highway Engineers, vol. 10, pp. 5–30, 1963. View at Google Scholar
  3. M. Batty and P. A. Longley, Fractal Cities: A Geometry of Form and Function, Academic Press, London, UK, 1994. View at Zentralblatt MATH
  4. D. S. Dendrinos and M. S. El Naschie, Eds., “Nonlinear dynamics in urban and transportation anaylysis,” Chaos, Soliton & Fractals, vol. 4, no. 4, pp. 497–617, 1994. View at Google Scholar
  5. P. Frankhauser, La Fractalite des Structures Urbaines, Economica, Paris, France, 1994.
  6. A.-L. Barabasi and E. Bonabeau, “Scale-free networks,” Scientific American, vol. 288, no. 5, pp. 50–59, 2003. View at Google Scholar
  7. Y. Chen and Y. Zhou, “Multi-fractal measures of city-size distributions based on the three-parameter Zipf model,” Chaos, Solitons & Fractals, vol. 22, no. 4, pp. 793–805, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. Y. Chen and Y. Zhou, “The rank-size rule and fractal hierarchies of cities: mathematical models and empirical analyses,” Environment and Planning B, vol. 30, no. 6, pp. 799–818, 2003. View at Publisher · View at Google Scholar
  9. D. S. Dendrinos, The Dynamics of Cities: Ecological Determinism, Dualism and Chaos, Routledge, London, UK, 1992.
  10. P. W. Anderson, “Is complexity physics? Is it science? What is it?” Physics Today, vol. 44, no. 7, pp. 9–11, 1991. View at Publisher · View at Google Scholar
  11. Y. Chen, Fractal Urban Systems: Scaling, Symmetry, and Spatial Complexity, Scientific Press, Beijing, China, 2008.
  12. Y. Chen and J. Liu, “Derivations of fractal models of city hierarchies using entropy-maximization principle,” Progress in Natural Science, vol. 12, no. 3, pp. 208–211, 2002. View at Google Scholar
  13. F. H. Wang and Y. Zhou, “Modeling urban population densities in Beijing 1982–1990: suburbanisation and its causes,” Urban Studies, vol. 36, no. 2, pp. 271–287, 1999. View at Publisher · View at Google Scholar
  14. J. Feng, “Modeling the spatial distribution of urban population density and its evolution in Hangzhou,” Geographical Research, vol. 21, no. 5, pp. 635–646, 2002 (Chinese). View at Google Scholar
  15. R. B. Banks, Growth and Diffusion Phenomena: Mathematical Frameworks and Application, vol. 14 of Texts in Applied Mathematics, Springer, Berlin, Germany, 1994. View at Zentralblatt MATH · View at MathSciNet
  16. R. Bussiere and F. Snickers, “Derivation of the negative exponential model by an entropy maximizing method,” Environment and Planning A, vol. 2, no. 3, pp. 295–301, 1970. View at Publisher · View at Google Scholar
  17. A. G. Wilson, Entropy in Urban and Regional Modelling, Pion Press, London, UK, 1970.
  18. A. G. Wilson, Complex Spatial Systems: The Modelling Foundations of Urban and Regional Analysis, Pearson Education, Singapore, 2000.
  19. B. H. Kaye, A Random Walk Through Fractal Dimensions, VCH Publishers, New York, USA, 1989.
  20. T. J. Zhu, Integral Transform in Engineering Mathematics, Higher Education Press, Beijing, China, 1991.
  21. S. D. Liu and S. K. Liu, An Introduction to Fractals and Fractal Dimension, Weather Press, Beijing, China, 1993.
  22. S. D. Liu and S. K. Liu, Solitary Wave and Turbulence, Shanghai Scientific and Technological Education, Shanghai, China, 1994.
  23. P. Bak, How Nature Works. The Science of Self-Organized Criticality, Springer, New York, NY, USA, 1996. View at Zentralblatt MATH · View at MathSciNet
  24. B. B. Mandelbrot, Multifractals and 1/f Noise: Wild Self-Affinity in Physics (1963–1976), Springer, New York, NY, USA, 1999. View at Zentralblatt MATH · View at MathSciNet
  25. J. Feder, Fractals, Physics of Solids and Liquids, Plenum Press, New York, NY, USA, 1988. View at Zentralblatt MATH · View at MathSciNet
  26. D. Saupe, “Random fractals in image synthesis,” in Fractals and Chaos, A. J. Crilly, R. A. Earnshaw, and H. Jones, Eds., pp. 89–118, Springer, New York, NY, USA, 1991. View at Google Scholar · View at MathSciNet
  27. R. F. Voss, “Fractals in nature: from characterization to simulation,” in The Science of Fractal Images, H.-O. Peitgen and Saupe D., Eds., pp. 21–70, Springer, New York, NY, USA, 1988. View at Google Scholar
  28. P. Bloomfield, Fourier Analysis of Time Series. An Introduction, Wiley Series in Probability and Statistics: Applied Probability and Statistics, John Wiley & Sons, New York, NY, USA, 2nd edition, 2000. View at Zentralblatt MATH · View at MathSciNet
  29. A. Einstein, “Quanten-mechanik und wirklichkeit [Quantum mechanics and reality],” Dialectica, vol. 2, no. 3-4, pp. 320–324, 1948. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  30. P. A. Longley, “Computer simulation and modeling of urban structure and development,” in Applied Geography: Principles and Practice, M. Pacione, Ed., Routledge, London, UK, 1999. View at Google Scholar
  31. M. Batty and Y. Xie, “Self-organized criticality and urban development,” Discrete Dynamics in Nature and Society, vol. 3, no. 2-3, pp. 109–124, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  32. J. Portugali, Self-Organization and the City, Springer, Berlin, Germany, 1999.
  33. Y. Chen and Y. Zhou, “Scaling laws and indications of self-organized criticality in urban systems,” Chaos, Solitons & Fractals, vol. 35, no. 1, pp. 85–98, 2008. View at Publisher · View at Google Scholar
  34. P. Frankhauser and R. Sadler, “Fractal analysis of agglomerations,” in Natural Structures: Principles, Strategies, and Models in Architecture and Nature, M. Hilliges, Ed., pp. 57–65, University of Stuttgart, Stuttgart, Germany, 1991. View at Google Scholar
  35. R. White and G. Engelen, “Urban systems dynamics and cellular automata: fractal structures between order and chaos,” Chaos, Solitons & Fractals, vol. 4, no. 4, pp. 563–583, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  36. L. Benguigui, D. Czamanski, M. Marinov, and Y. Portugali, “When and where is a city fractal?” Environment and Planning B, vol. 27, no. 4, pp. 507–519, 2000. View at Publisher · View at Google Scholar
  37. B. Ya. Ryabko, “Noise-free coding of combinatorial sources, Hausdorff dimension and Kolmogorov complexity,” Problemy Peredachi Informatsii, vol. 22, no. 3, pp. 16–26, 1986. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  38. M. Batty, “Spatial entropy,” Geographical Analysis, vol. 6, pp. 1–31, 1974. View at Google Scholar
  39. M. S. El Naschie, “Foreword: a very brief history of localization,” Chaos, Solitons & Fractals, vol. 11, no. 10, pp. 1479–1480, 2000. View at Publisher · View at Google Scholar
  40. M. Batty and K. S. Kim, “Form follows function: reformulating urban population density functions,” Urban Studies, vol. 29, no. 7, pp. 1043–1069, 1992. View at Publisher · View at Google Scholar
  41. F. X. Diebold, Elements of Forecasting, Thomson, Mason, Ohio, USA, 3rd edition, 2004.
  42. R. White and G. Engelen, “Cellular dynamics and GIS: modelling spatial complexity,” Geographical Systems, vol. 1, no. 3, pp. 237–253, 1994. View at Google Scholar
  43. R. White, G. Engelen, and I. Uljee, “The use of constrained cellular automata for high-resolution modelling of urban land-use dynamics,” Environment and Planning B, vol. 24, no. 3, pp. 323–343, 1997. View at Publisher · View at Google Scholar