X and the City: Modeling Aspects of Urban Life (89 page)

[
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.

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).

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.