[15] Lowson, M.V. (1999). “Personal public transport.” Proceedings of the Institution of Civil Engineers: Transportation 135, 139–151.
[16] Mahajan, S. (2010). Street-Fighting Mathematics: The Art of Educated Guessing and Opportunistic Problem Solving, MIT Press, Cambridge, MA.
[17] Seifert, H.S. (1962). “The stop-light dilemma.” American Journal of Physics 30, 216–218.
[18] Chapman, S. (1942). “Should one stop or turn in order to avoid an automobile collision?” American Journal of Physics 10, 22–27.
[19] Ashdon, W.D. (1966). The Theory of Road Traffic Flow. Methuen, London.
[20] Smeed, R.J. (1961). The Traffic Problem in Towns. Manchester Statistical Society, Manchester, England.
[21] Smeed, R.J. (1963). “The road space required for traffic in towns.” Town Planning Review 33, 270–292.
[22] Smeed, R.J. (1963). “Road development in urban areas.” Journal of the Institute of Highway Engineers 10, 5–26.
[23] Smeed, R.J. (1965). “A theoretical model for commuter traffic in towns.” Journal of the Institute of Mathematics and its Applications 1, 208–225.
[24] Smeed, R.J. (1967). “The road capacity of city centers.” Highway Research Record 169, 22–29.
[25] Smeed, R.J. (1968). “Traffic studies and urban congestion.” Journal of Transport Economics and Policy 2, 33–70.
[26] Smeed, R.J. (1970). “The capacity of urban road networks.” Proceedings of the Australian Road Research Board 5, 10–28.
[27] Smeed, R.J. (1977). “Traffic in a linear town.” Proceedings of the 7th International Symposium on Transportation and Traffic Theory.
[28] von Foerster, H., Mora, P.M., and Amiot, L.W. (1960). “Doomsday: Friday 13 November, A.D. 2026.” Science 132, 1291–1295.
[29] Johansen, A. and Sornette, D. (2001). “Finite-time singularity in the dynamics of the world population, economic and financial indices.” Physica A 294, 465–502.
[30] Bracken, A.J., and Tuckwell, H.C. (1992). “Simple mathematical models for urban growth.” Proceedings of the Royal Society of London A 438, 171–181.
[31] Zipf, G.K. (1949). Human Behavior and the Principle of Least Effort. Harvard University Press, Cambridge, MA.
[32] Bettencourt, L.M.A., Helbing, J.L.D., Kühnert, C. and West, G.B. (2007). “Growth, innovation, scaling, and the pace of life in cities.” Proceedings of the National Academy of Sciences of the USA 104, 7301–7306; also Bettencourt, L.M.A., Lobo, J. and West, G.B. (2008). “Why are large cities faster? Universal scaling and self-similarity in urban organization and dynamics.” European Physical Journal B 63, 285–293.
[33] Ide, K., and Sornette, D. (2002). “Oscillatory finite-time singularities in finance, population and rupture.” Physica A 307, 63–106.
[34] Bosanquet, C.H., and Pearson, J.L. (1936). “The spread of smoke and gases from chimneys.” Transactions of the Faraday Society 32, 1249–1263.
[35] http://www.ias.ac.in/initiat/sci_ed/resources/chemistry/LightScat.pdf.
[36] Lynch, D., and Livingston, W. (2010). Color and Light in Nature. 2nd ed., reprint. Thule Scientific, Topanga, CA; p.26.
[37] http://antwrp.gsfc.nasa.gov/apod/ap040913.html.
[38] Minnaert, M. (1954). The Nature of Light and Colour in the Open Air. Dover, New York. (German edition, 1993, Light and Color in the Outdoors, Springer-Verlag, Berlin.)
[39] See website by Christian Fenn, “Rainbows in diverging light.” http://www.meteoros.de/rainbow/rbdive_1.htm.
[40] http://quirkynyc.com/2010/05/ahoy-lighthouses-litter-new-york-city/.
[41] http://www2.jpl.nasa.gov/sl9/.
[42] Banks, R.B. (1998). Towing Icebergs, Falling Dominoes, and Other Adventures in Applied Mathematics. Princeton University Press, Princeton, NJ.
[43] http://www.citymayors.com/statistics/largest-cities-area-125.html; see also http://www.worldatlas.com/citypops.htm.
[44] Taxicab geometry: http://en.wikipedia.org/wiki/Taxicab_geometry.
[45] http://www.sci.csuhayward.edu/statistics/Resources/Essays/BinPois.htm and https://mywebspace.wisc.edu/adjacob1/soc_361/handouts/poisson%20handout.pdf
[46] See http://www.atoptics.co.uk/.
[47] Tatum, J.B. (1997). “The capture cross-section of earth for errant asteroids.” Journal of the Royal Astronomical Society of Canada 91, 276–278.
OTHER RELEVANT LINKS
Chapter 15: Adapted from the question posed by Nathan Keyfitz, “How many people have lived on the Earth?” See also http://www.prb.org/Articles/2002/HowManyPeopleHaveEverLived
onEarth.aspx.
Chapter 18: See also http://brenocon.com/blog/2009/05/zipfs-law-and-world-city-populations/ (an excellent link) and http://opinionator.blogs.nytimes.com/2009/05/19/math-and-the-city/ (an informative article by Steven Strogatz).
Chapter 25: http://www.solarviews.com/eng/comet/appendc.htm (by Calvin Hamilton).
GENERAL REFERENCES
Adam, J.A. (1988). “Complementary levels of description in applied mathematics—III: equilibrium models of cities.” Mathematical and Computer Modelling 10, 321–339.
Adam, J.A. (2006). Mathematics in Nature: Modeling Patterns in the Natural World. Paperback ed. Princeton University Press, Princeton, NJ.
Adam, J.A. (2011). A Mathematical Nature Walk. Paperback ed. Princeton University Press, Princeton, NJ.
Adams, G.U. (1987). “Smeed’s law: some further thoughts.” Traffic Engineering and Control 28, 70–73.
Alvarez, L.W., Alvarez, W., Asaro, F., and Michel, H.V. (1980). “Extraterrestrial Cause for the Cretaceous-Tertiary Extinction.” Science 208, 1095–1108.
Amson, J.C. (1972). “Equilibrium models of cities: 1. An axiomatic theory.” Environment and Planning 4, 429–444.
Amson, J.C. (1973). “Equilibrium models of cities: 2. Single-species cities.” Environment and Planning 5, 295–338.
Angel, S., and Hyman, G.M. (1970). “Urban velocity fields.” Environment and Planning 2, 211–224.
Angel, S., and Hyman, G.M. (1976). Urban Fields. Pion, London.
Apsimon, H.G. (1958). “Note 2754. A repeated integral.” Mathematical Gazette 42, 52.
Atkinson, B.W. (1968). “A Preliminary examination of the possible effects of London’s urban area on the distribution of thunder rainfall 1951–1960.” Transactions of the Institute of British Geographers 44, 97–108.
Batty, M. (1971). “Modelling cities as dynamic systems.” Nature 231, 425–428.
Batty, M., Carvalho, R., Hudson-Smith, A., Milton, R., Smith, D., and Steadman, P. (2008). “Scaling and allometry in the building geometries of Greater London.” European Physical Journal B 63, 303–314.
Billah, K.Y., and Scanlan, R.H. (1991). “Resonance, Tacoma Narrows bridge failure, and undergraduate physics textbooks.” American Journal of Physics 59, 118–124.
Boas, R. (2002). Guest essay, “Kinematics of jogging.” pp. 351–352 in M. J. Strauss, G. L. Bradley and K. J. Smith, Calculus. 3rd ed. Prentice-Hall, Englewood Cliffs, NJ.
Brakman, S., Garretsen, H., and van Marrewijk, C. (2009). The New Introduction to Geographical Economics. Cambridge University Press, Cambridge. 2nd ed.
Cohen, J.E. (1995). “Population growth and Earth’s human carrying capacity.” Science 269, 341–346.
Clayden, A.W. (1891). “On Brocken spectres in a London fog.” Quarterly Journal of the Royal Meteorological Society 17, 209–216.
Crawford, F.S. (1988). “Rainbow dust.” American Journal of Physics 56, 1006–1009.
Drew, D.R. (1968). Traffic Flow Theory and Control. McGraw-Hill, New York.
Einhorn, S.J. (1967). “Polar vs. rectangular road networks,” Operations Research 35, 546–548.
Ehrlich, R. (1993). The Cosmological Milkshake. Rutgers University Press, New Brunswick, NJ.
Fairthorne, D.B. (1964). “The distances between random points in two concentric circles.” Biometrika 51, 275–277.
Fairthorne, D.B. (1965). “The distance between pairs of points in towns of simple geometrical shape.” Proceedings of the
Second International Symposium on Theory of Traffic flow (J. Almond, ed.). OECD, Paris, 391–406.
Floor, C. (1980). “Rainbows and haloes in lighthouse beams.” Weather 35, 203–208.
Floor, C. (1982). “Optic phenomena and optical illusions near lighthouses.” Meteorologische Zeitschrift 32, 229–233.
Foster, J.H., and Pedersen, J.J. (1980). “On the reflective property of ellipses.” American Mathematical Monthly 87, 294–297.
Garwood, F., and Tanner, J.C. (1958). “Note 2800. On note 2754: a repeated integral.” Mathematical Gazette 52, 292–293.
Gerlough, D.L. (1955). Use of Poisson Distribution in Highway Traffic. ENO Foundation for Highway Traffic Control, Saugatuck, CT.
Giordano, F.R., Weir, M.D., and Fox, W.P. (2003). A First Course in Mathematical Modeling. 3rd ed. Thomson Brookes/Cole, Pacific Grove, CA.
Gislén, L., and Mattsson, J.O. (2003). “Observations and simulations of some divergent-light halos.” Applied Optics 42, 4269–4279.
Gislén, L., and Mattsson, J.O. (2007). “Tabletop divergent-light halos.” Physics Education 42, 579–584.
Großmann, M., Schmidt, E. and Haußmann, A. (2011). “Photographic evidence for the third-order rainbow.” Applied Optics 50, F134–F141.
Haberman, R. (1977). Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow, Prentice-Hall, Englewood Cliffs, NJ.
Haight, F.A. (1964). “Some probability distributions associated with commuter travel in a homogeneous circular city.” Operations Research 12, 964–975.
Harsch, J., and Walker, J.D. (1975). “Double rainbow and dark band in searchlight beam.” American Journal of Physics 43, 453.
Herman, R., and Gardels, K. (1963). “Vehicular traffic flow.” Scientific American 209, 35–43.
Hobbs, F.D., and Richardson, B.D. (1967). Traffic Engineering, vol. 2. Pergamon Press, Oxford.
Holroyd, E.M. (1969). “Polar and rectangular road networks for circular cities.” Transportation Science 3, 86–88.
Hunt, J.C.R. (1971). “The effect of single buildings and structures.” Philosophical Transactions of the Royal Society of London A 269, 457–467.
Ioannides, Y.N., and Overman, H.G. (2003). “Zipf’s law for cities: an empirical examination.” Regional Science and Urban Economics 33, 127–137.
Ishikawa, H. (1980). “A new model for the population density distribution in an isolated city.” Geographical Analysis 12, 223–235.
Jacobs, J. (1984). Cities and the Wealth of Nations. Vintage, New York.
Jones, T.R., and Potts, R.B. (1962). “The measurement of acceleration noise—a traffic parameter.” Operations Research 10, 745–763.
Kerensky, O.A.(1971). “Bridges and other large structures.” Philosophical Transactions of the Royal Society of London A 269, 343–351.
Keyfitz, N. (1976). Applied Mathematical Demography. Wiley, New York.
Kilminster, C.W. (1976) “Population in cities.” Mathematical Gazette 60, 11–24.
Krause, E.F. (1986). Taxicab Geometry: An Adventure in Non-Euclidean Geometry. Dover, New York.
Lew, J.S., Frauenthal, J.C., and Keyfitz, N. (1978). “On the average distances in a circular disc.” SIAM Review 20, 584–592.
Lowry, W.P. (1967). “The climate of cities.” Scientific American 217, 15–23.
Mair, W.A., and Maull, D.J. (1971). “Aerodynamic behaviour of bodies in the wakes of other Bodies.” Philosophical Transactions of the Royal Society of London A 269, 425–437.
Makse, H.A., Andrade, J.S., Jr., Batty, M., Havlin S. and Stanley, H.E. (1998). “Modeling urban growth patterns with correlated percolation.” Physical Review E 58, 7054–7062.
Makse, H.A., Havlin, S., and Stanley, H.E. (2002). “Modelling urban growth patterns.” Nature 377, 608–612.
Mallmann, A.J., Hock, J.L., and Greenler, R.G. (1998). “Comparison of sun pillars with light pillars from nearby light sources.” Applied Optics 37, 1441–1449.
Mattsson, J.O., Nordbeck, S., and Rystedt, B. (1971). “Dewbows and fogbows in divergent light,” No. 11 of Lund Studies in Geography, Series C, Lund University, Lund, Sweden.
Mattsson, J.O. (1973). “ ‘Sun-sun’ and light-pillars of street lamps.”Weather 28, 66–68.
Mattsson, J.O. (1974). “Experiments on horizontal haloes in divergent light,” Weather 29, 148–150.
Mattsson, J.O. (1978). “Experiments on the horizontal circle in divergent light.” Meteorologische Zeitschrift 28, 123–125.
Mattsson, J.O. (1998). “Concerning haloes, rainbows and dewbows in divergent light.” Weather 53, 176–181.
Mattsson, J.O., Bärring, L., and Almqvist, E. (2000). “Experimenting with Minnaert’s cigar.” Applied Optics 39, 3604–3611.
Mattsson, J.O., and Bärring, L. (2001). “Heiligenschein and related phenomena in divergent Light.” Applied Optics 40, 4799–4806.
Medda, F., Nijkamp, P., and Rietveld, P. (2003). “Urban land use for transport systems and city Shapes.” Geographical Analysis 35, 46–57.
Memory, J.D. (1973). “Kinematics problem for joggers.” American Journal of Physics 41, 1205–1206.
Mills, E.S. (1970). “Urban density functions.” Urban Studies 7, 15–20.
Montroll, E.W., and Badger, W.W. (1974). Introduction to Quantitative Aspects of Social Phenomena. Gordon and Breach, New York.
Monteith, J.L. (1954). “Refraction and the spider.” Weather 9, 140–141.
Myrup, L.O. (1969). “A numerical model of the urban heat island.” Journal of Applied Meteorology 8, 908–918.
Newling, B.E. (1969). “The spatial variation of urban population densities.” Geographical Review 59, 242–252.
Olfe, D.B., and Lee, R.L. (1971). “Linearized calculations of urban heat island convection effects.” Journal of Atmospheric Sciences 28, 1374–1388.
Oke, T.R. (1973). “City size and the urban heat island.” Atmospheric Environment 7, 769–779.
Pearce, C.E.M. (1974). “Locating concentric ring roads in a city.” Transportation Science 8, 142–168.
Preston-Whyte, R.A. (1970). “A spatial model of an urban heat island.” Journal of Applied Meteorology 9, 571–573.
Puu, T. (1978). “Towards a theory of optimal roads.” Regional Science and Urban Economics 8, 203–226.
Rosenau, H. (1983). The Ideal City: Its Architectural Evolution in Europe. Routledge and Kegan Paul, London.
Shankland, R.S. (1968). “Rooms for speech and music.” Physics Teacher 6, 443–449.
Shoemaker, E.M. (1983). “Asteroid and comet bombardment of the earth.” Annual Review of Earth and Planetary Sciences 11, 461–494.
Sitte, C. (1889, 1965). City Planning, According to Artistic Principles. Random House, New York.
Tan, T. (1966). “Road networks in an expanding circular city.” Operations Research 14, 607–613.
Theusner, M. (2011). “Photographic observation of a natural fourth-order rainbow.” Applied Optics 50, F129–F133.
Vaughan, R.J., et al. (1972.) “Traffic characteristics as a function of the distance to the town center.” Traffic Engineering Control 14, 224–227.
Vergara, W.C. (1959). Mathematics in Everyday Things. Harper, New York.
Walker, J.D. (1975). The Flying Circus of Physics with Answers. Wiley, New York.
Walker, J.D. (1976). “Multiple rainbows from single drops of water and other liquids.” American Journal of Physics 44, 421–433.
Wardrop, J.G. (1952). “Road paper: Some theoretical aspects of road traffic research.” Proceedings of the Institution of Civil Engineers, Part II, 1, 325–362.
Wardrop, J.G. (1968). “Journey speed and flow in central urban areas.” Traffic Engineering Control 9, 528–532.
Watson, G.N. (1959). “Note 2871: A quadruple integral.” Mathematical Gazette 43, 280–283.
Webster, F.V. (1958). “Traffic signal settings.” Technical paper 39, Road Research Laboratory, D.S.I.R.
West, G.B. (1999). “The origin of universal scaling laws in biology.” Physica A 263, 104–113
.
Williamson, S.J. (1973). Fundamentals of Air Pollution. Addison-Wesley, Reading, MA.
Wolke, R.L. (2000). What Einstein Told His Barber. Bantam Doubleday Dell, New York.
Yeo, G.F. (1964). “Traffic delays on a two-lane road.” Biometrika 51, 11–15.
INDEX
acceleration “noise,” 85–87
X and the City: Modeling Aspects of Urban Life Page 25