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25995 results in Abstract analysis

Page 966 of 1300

  • On a theorem of M. Rees for the matings of polynomials

  • Edited by Tan Lei, Université de Cergy-Pontoise
  • Book:
    The Mandelbrot Set, Theme and Variations
    Published online:
    05 August 2011
    Print publication:
    13 April 2000, pp 289-306

  • Summary

    Abstract

    We show the following theorem of M. Rees: if the formal mating of two postcritically finite polynomials is (weakly) equivalent to a rational map, then the topological mating is conjugate to the rational map.

    The dynamics of complex polynomials has been extensively studied and many results are obtained. For example, a polynomial of Thurston's type admits a description of the dynamics in terms of external angles, etc (see [DH1]). However, much less is known for rational maps in general (see [R] for degree two case). The mating is a way to construct a rational map or a branched covering of S2 from a pair of polynomials with connected Julia sets. When this is possible, we can understand the dynamics of the rational map in terms of that of the polynomials. The notion of the mating was also introduced by Douady and Hubbard [D].

    There are two notions of the mating —the formal mating and the topological mating. In fact there is another, the degenerate mating, which is a slightly modified version of the formal mating. To define the formal mating, we take two copies of ℂ each with the circle at infinity and make a topological sphere by gluing the copies together along the circles at infinity. Then define a mapping on the sphere using one polynomial on one copy and the other polynomial on the other copy.

  • Preface

  • Edited by Tan Lei, Université de Cergy-Pontoise
  • Book:
    The Mandelbrot Set, Theme and Variations
    Published online:
    05 August 2011
    Print publication:
    13 April 2000, pp xiii-xx

  • Summary

    Holomorphic dynamics is a subject with an ancient history: Fatou, Julia, Schroeder, Koenigs, Böttcher, Lattès, which then went into hibernation for about 60 years, and came back to explosive life in the 1980's.

    This rebirth is in part due to the introduction of a new theoretical tool: Sullivan's use of quasi-conformal mappings allowed him to prove Fatou's nowandering domains conjecture, thus solving the main problem Fatou had left open.

    But it is also due to a genuinely new phenomenon: the use of computers as an experimental mathematical tool. Until the advent of the computer, the notion that there might be an “experimental component” to mathematics was completely alien. Several early computer experiments showed great promise: the Fermi-Pasta-Ulam experiment, the number-theoretic computations of Birch and Swinnerton-Dyer, and Lorenz's experiment in theoretical meteorology stand out. But the unwieldiness of mainframes prevented their widespread use.

    The microcomputer and improved computer graphics changed that: now a mathematical field was behaving like a field of physics, with brisk interactions between experiment and theory.

    I mention computer graphics because faster and cheaper computers alone would not have had the same impact; without pictures, the information pouring out of mathematical computations would have remained hidden in a flood of numbers, difficult if not impossible to interpret. For people who doubt this, I have a story to relate. Lars Ahlfors, then in his seventies, told me in 1984 that in his youth, his adviser Lindelöf had made him read the memoirs of Fatou and Julia, the prize essays from the Académie des Sciences in Paris.

  • Propagation of fronts in a nonlinear fourth order equation

  • P. LORETI, R. MARCH
  • Journal:
    European Journal of Applied Mathematics / Volume 11 / Issue 2 / April 2000
    Published online by Cambridge University Press:
    01 April 2000, pp. 203-213

  • We consider a geometric motion associated with the minimization of a curvature dependent functional, which is related to the Willmore functional. Such a functional arises in connection with the image segmentation problem in computer vision theory. We show by using formal asymptotics that the geometric motion can be approximated by the evolution of the zero level set of the solution of a nonlinear fourth-order equation related to the Cahn–Hilliard and Allen–Cahn equations.

Page 966 of 1300