Lanford has shown in 1982 that Feigenbaum’s functional equation has an analytic solution. We show that this solution is a polynomial time computable function. This implies in particular that the so-called first Feigenbaum constant is a polynomial time computable real number.

We prove the Feigenbaum-Coullet-Tresser conjecture on the hyperbolicity of the renormalization transformation of bounded type. This gives the first computer-free proof of the original Feigenbaum observation of the universal parameter scaling laws. We use the Hyperbolicity Theorem to prove Milnor’s conjectures on self-similarity and “hairiness” of the Mandelbrot set near the corresponding parame...

In this paper we present a computer{assisted proof of the existence of a solution for the Feigenbaum equation '(x) = 1 '('( x)): There exist by now various such proofs in the literature. Although the one presented here is new, the main purpose of this paper is not to provide yet another version, but to give an easy{to{read and self contained introduction to the technique of computer{assisted pr...

We give examples of infinitely renormalizable quadratic polynomials Fc : z 7→ z + c with stationary combinatorics whose Julia sets have Hausdorff dimension arbitrary close to 1. The combinatorics of the renormalization involved is close to the Chebyshev one. The argument is based upon a new tool, a “Recursive Quadratic Estimate” for the Poincaré series of an infinitely renormalizable map. Stony...

particles down to a distance scale on the order of 18 10 m. Despite its widespread acceptance, SM operates with a large number of arbitrary parameters whose physical origin is presently unknown. Our work suggests that the generation structure of at least some SM parameters stems from the chaotic regime of renormalization group flow. Invoking the universal route to chaos in systems of nonlinear ...

Over the past six years, there have been some interesting trends in expert system activity worldwide (Lee, Liebowitz, and Chae 1996; Liebowitz 1996, 1994a, 1994b, 1991; Liebowitz et al. 1996; CantuOrtiz 1991; Lee et al. 1991; Suen 1991; Zarri 1991). An excellent way to gain a global perspective on expert system technology, applications, and management is to examine the world congresses on exper...

The study of dynamics in the plane has recently seen a surge in interest due to three recent breakthroughs: the Sullivan-McMullen-Lyubich proof of the Feigenbaum Universality, the introduction by O. Schramm of SLE processes, and the work of S. Smirnov on percolation. The fields of Holomorphic Dynamics, SLE, and Conformal Field Theory (CFT) are now seen to be closely linked, the glue being provi...

Lindsay, R.K., B.G. Buchanan, E.A. Feigenbaum and J. Lederberg, DENDRAL: a case study of the first expert system for scientific hypothesis formation, Artificial Intelligence 61 (1993) 209-261. The DENDRAL Project was one of the first large-scale programs to embody the strategy of using detailed, task-specific knowledge about a problem domain as a source of heuristics, and to seek generality thr...

Recently, Takahashi has introduced the James and von Neumann-Jordan type constants. In this paper, we present some sufficient conditions for uniform normal structure and therefore the fixed point property of a Banach space in terms of the James and von Neumann-Jordan type constants and the Ptolemy constant. Our main results of the paper significantly generalize and improve many known results in...