Infinitesimal contact transformations and their applications to abstract growth patterns
Author:
Igor Frolow
Journal:
Quart. Appl. Math. 37 (1980), 389-399
MSC:
Primary 92A05; Secondary 51N05, 53A05, 58C28, 73P05
DOI:
https://doi.org/10.1090/qam/564730
MathSciNet review:
564730
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: Several abstract mathematical models of two-dimensional objects or “organisms” are introduced. The objects are abstractions, created by the combination of pattern-theoretic generators. They are deformed by what are shown to be infinitesimal contact transformations. Analytical methods are utilized to determine some of the resulting forms and a computer is programmed to display graphically the developing patterns.
V. G. Boltianskii, Ogibaiushchaia, Gosudarstvennoe Izdatelstvo Fizico-Mathematicheskoy Literaturi, Moscow, 1961
- C. Carathéodory, Calculus of variations and partial differential equations of the first order. Part I: Partial differential equations of the first order, Holden-Day, Inc., San Francisco-London-Amsterdam, 1965. Translated by Robert B. Dean and Julius J. Brandstatter. MR 0192372
- Luther Pfahler Eisenhart, Contact transformations, Ann. of Math. (2) 30 (1928/29), no. 1-4, 211–249. MR 1502878, DOI https://doi.org/10.2307/1968276
- Igor Frolow, ABSTRACT GROWTH PATTERNS, ProQuest LLC, Ann Arbor, MI, 1978. Thesis (Ph.D.)–Brown University. MR 2628113
- U. Grenander, Pattern synthesis, Springer-Verlag, New York-Heidelberg, 1976. Lectures in pattern theory, Vol. 1; Applied Mathematical Sciences, Vol. 18. MR 0438838
J. S. Huxley, Problems of relative growth, Methuen, London, 1932
A. Lindenmayer, Mathematical models for cellular interactions in development, Part I: filaments with one-sided inputs, J. Theor. Biol. 18, 280–299 (1968)
A. Lindenmayer, Mathematical models for cellular interactions in development, Part II: simple and branching filaments with two-sided inputs, J. Theor. Biol. 18, 300–315 (1968)
- N. Rashevsky, Mathematical biophysics: Physico-mathematical foundations of biology., Dover Publications, Inc., New York, 1960. 3rd revised ed. 2 vols. MR 0118542
- Robert Rosen, Morphogenesis, Foundations of mathematical biology, Vol. II: Cellular systems (dedicated to the memory of Nicolas Rashevsky), Academic Press, New York, 1972, pp. 1–77. MR 0408871
M. S. Steinberg, Reconstruction of tissues by dissociated cells, in Mathematical models for cell rearrangement (G. D. Mostow, ed.), 1975, 82–99
- René Thom, Stabilité structurelle et morphogénèse, W. A. Benjamin, Inc., Reading, Mass., 1972 (French). Essai d’une théorie générale des modèles; Mathematical Physics Monograph Series. MR 0488155
- D’Arcy Wentworth Thompson, On Growth and Form, New edition, Cambridge University Press, Cambridge, England, 1942. MR 0006348
A. M. Turning, The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. London B237, 37–72 (1952)
L. Wolpert, Positional information and pattern formation, in Toward a theoretical biology, Vol. III (C. H. Waddington, ed.). 1970, 198–230
V. G. Boltianskii, Ogibaiushchaia, Gosudarstvennoe Izdatelstvo Fizico-Mathematicheskoy Literaturi, Moscow, 1961
C. Carathéodory, Calculus of variations and partial differential equations of the first order, Part I (translated by R. B. Dean and J. J. Brandstatter), Holden-Day, San Francisco, 1965
L. P. Eisenhart, Contact transformations, Annals Math. 2 series 30, 211–249 (1928)
I. Frolow, Abstract growth patterns, Ph.D. Thesis, Brown University, 1978
U. Grenander, Pattern synthesis—lectures in pattern theory, Vol. I, Springer-Verlag, New York, 1976
J. S. Huxley, Problems of relative growth, Methuen, London, 1932
A. Lindenmayer, Mathematical models for cellular interactions in development, Part I: filaments with one-sided inputs, J. Theor. Biol. 18, 280–299 (1968)
A. Lindenmayer, Mathematical models for cellular interactions in development, Part II: simple and branching filaments with two-sided inputs, J. Theor. Biol. 18, 300–315 (1968)
N. Rashevsky, Mathematical biophysics: physico-mathematical foundations of biology, Vol. I, 3rd ed., Dover, New York, 1960
R. Rosen, Morphogenesis, in Foundations of mathematical biology, Vol. II (R. Rosen, ed.), 1972, 1–77
M. S. Steinberg, Reconstruction of tissues by dissociated cells, in Mathematical models for cell rearrangement (G. D. Mostow, ed.), 1975, 82–99
R. Thom, Structural stability and morphogenesis (translated by D. H. Fowler), W. A. Benjamin, Reading, Massachusetts, 1975
D’Arcy Thompson, On growth and form, 2nd ed., Cambridge University Press, 1942
A. M. Turning, The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. London B237, 37–72 (1952)
L. Wolpert, Positional information and pattern formation, in Toward a theoretical biology, Vol. III (C. H. Waddington, ed.). 1970, 198–230
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
92A05,
51N05,
53A05,
58C28,
73P05
Retrieve articles in all journals
with MSC:
92A05,
51N05,
53A05,
58C28,
73P05
Additional Information
Article copyright:
© Copyright 1980
American Mathematical Society