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36
FeigenbaumCoulletTresser universality and Milnor's Hairiness Conjecture
, 1999
"... We prove the FeigenbaumCoulletTresser conjecture on the hyperbolicity of the renormalization transformation of bounded type. This gives the first computerfree proof of the original Feigenbaum observation of the universal parameter scaling laws. We use the Hyperbolicity Theorem to prove Milnor’s c ..."
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Cited by 51 (5 self)
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We prove the FeigenbaumCoulletTresser conjecture on the hyperbolicity of the renormalization transformation of bounded type. This gives the first computerfree proof of the original Feigenbaum observation of the universal parameter scaling laws. We use the Hyperbolicity Theorem to prove Milnor’s conjectures on selfsimilarity and “hairiness ” of the Mandelbrot set near the corresponding parameter values. We also conclude that the set of real infinitely renormalizable quadratics of type bounded by some N> 1 has Hausdorff dimension strictly between 0 and 1. In the course of getting these results we supply the space of quadraticlike germs with a complex analytic structure and demonstrate that the hybrid classes form a complex codimensionone foliation of the connectedness locus.
Double bubbles minimize
 Ann. of Math
"... The classical isoperimetric inequality in R 3 states that the surface of smallest area enclosing a given volume is a sphere. We show that the least area surface enclosing two equal volumes is a double bubble, a surface made of two pieces of round spheres separated by a flat disk, meeting along a sin ..."
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Cited by 21 (1 self)
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The classical isoperimetric inequality in R 3 states that the surface of smallest area enclosing a given volume is a sphere. We show that the least area surface enclosing two equal volumes is a double bubble, a surface made of two pieces of round spheres separated by a flat disk, meeting along a single circle at an angle of 120 ◦. 1.
The periodic points of renormalization
 Ann. Math
, 1998
"... Abstract. It will be shown that the renormalization operator, acting on the space of smooth unimodal maps with critical exponent α> 1, has periodic points of any combinatorial type. 1. ..."
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Cited by 15 (2 self)
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Abstract. It will be shown that the renormalization operator, acting on the space of smooth unimodal maps with critical exponent α> 1, has periodic points of any combinatorial type. 1.
ComputerAssisted Proofs in Analysis and Programming in Logic: A Case Study
 SIAM Review
, 1996
"... . 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 versi ..."
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Cited by 14 (1 self)
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. 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 proofs in analysis. Our proof is written in Prolog #Programming in logic#, a programming language which we found to be well suited for this purpose. In this paper we also giveanintroduction to Prolog, so that even a reader without prior exposure to programming should be able to verify the correctness of the proof. 1 Supported in Part by the National Science Foundation under GrantNo.DMS#9103590, and bytheTexas Advanced Research Program under Grant No. ARP#035. 2 Supported in Part by the Swiss National Science Foundation. Table of Contents 1. Introduction 2 2. Prolog 5 2.1. The Syntax of...
Expanding direction of the period doubling operator
 Commun. Math. Phys
, 1992
"... We prove that the period doubling operator has an expanding direction at the fixed point. We use the induced operator, a “PerronFrobenius type operator”, to study the linearization of the period doubling operator at its fixed point. We then use a sequence of linear operators with finite ranks to st ..."
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Cited by 10 (1 self)
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We prove that the period doubling operator has an expanding direction at the fixed point. We use the induced operator, a “PerronFrobenius type operator”, to study the linearization of the period doubling operator at its fixed point. We then use a sequence of linear operators with finite ranks to study this induced operator. The proof 1 is constructive. One can calculate the expanding direction and the rate of expansion of the period doubling operator at the fixed point. Contents §1 Introduction. §2 The Period Doubling Operator and the Induced Operator. §2.1 From the period doubling operator to the induced operator. §2.2 The induced operator Lϕ.
Proving Conjectures by Use of Interval Arithmetic
 Facius Axel: Perspective on Enclosure Methods
, 2001
"... Machine interval arithmetic has become an important tool in computer assisted proofs in analysis. Usually, an interval arithmetic computation is just one of many ingredients in such a proof. The purpose of this contribution is to highlight and to summarize the role of interval arithmetic in some out ..."
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Cited by 7 (0 self)
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Machine interval arithmetic has become an important tool in computer assisted proofs in analysis. Usually, an interval arithmetic computation is just one of many ingredients in such a proof. The purpose of this contribution is to highlight and to summarize the role of interval arithmetic in some outstanding results obtained in computer assisted analysis. 'Outstanding' is defined through the observation that the importance of a mathematical result is at least to some extent indicated by the fact that it has been formulated as a 'conjecture' prior to its proof.
Taylor Forms  Use and Limits
 Reliable Computing
, 2002
"... This review is a response to recent discussions on the reliable computing mailing list, and to continuing uncertainties about the properties and merits of Taylor forms, multivariate higher degree generalizations of centered forms. They were invented around 1980 by Lanford, documented in detail in 19 ..."
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Cited by 6 (0 self)
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This review is a response to recent discussions on the reliable computing mailing list, and to continuing uncertainties about the properties and merits of Taylor forms, multivariate higher degree generalizations of centered forms. They were invented around 1980 by Lanford, documented in detail in 1984 by Eckmann, Koch and Wittwer, and independently studied and popularized since 1996 by Berz, Makino and Hoefkens. A highlight is their application to the verified integration of asteroid dynamics in the solar system in 2001, although the details given are not sufficient to check the validity of their claims.
Interval Arithmetic In Quantum Mechanics
, 1996
"... x); y(0) = 1; y(1) = 0: (0:1) Define F(\Omega\Gamma = Z ` y(x) x \Gamma\Omega 2 x 2 ' 1 = 2 + dx; \Omega 2 (0 ;\Omega c ); where a+ = max(a; 0), and a number\Omega c will be defined at the beginning of Section 2. The function F(\Omega\Gamma depends smoothly on\Omega [41], and the main ..."
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Cited by 4 (0 self)
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x); y(0) = 1; y(1) = 0: (0:1) Define F(\Omega\Gamma = Z ` y(x) x \Gamma\Omega 2 x 2 ' 1 = 2 + dx; \Omega 2 (0 ;\Omega c ); where a+ = max(a; 0), and a number\Omega c will be defined at the beginning of Section 2. The function F(\Omega\Gamma depends smoothly on\Omega [41], and the main result in [21] is as follows: 2 Chapter 1 Theorem 0.1. F 00(\Omega\Gamma c ! 0 for all\Omega 2 (0 ;\Omega c ). This is a quantitative form of the nonperiodicity of almost all zeroenergy o