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263
Advanced Spectral Methods for Climatic Time Series
, 2001
"... The analysis of uni or multivariate time series provides crucial information to describe, understand, and predict climatic variability. The discovery and implementation of a number of novel methods for extracting useful information from time series has recently revitalized this classical eld of ..."
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Cited by 220 (32 self)
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The analysis of uni or multivariate time series provides crucial information to describe, understand, and predict climatic variability. The discovery and implementation of a number of novel methods for extracting useful information from time series has recently revitalized this classical eld of study. Considerable progress has also been made in interpreting the information so obtained in terms of dynamical systems theory.
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 69 (7 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.
ALMOST EVERY REAL QUADRATIC MAP IS EITHER REGULAR OR STOCHASTIC
, 1997
"... We prove uniform hyperbolicity of the renormalization operator for all possible real combinatorial types. We derive from it that the set of infinitely renormalizable parameter values in the real quadratic family Pc: x ↦ → x² + c has zero measure. This yields the statement in the title (where “ regul ..."
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Cited by 50 (4 self)
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We prove uniform hyperbolicity of the renormalization operator for all possible real combinatorial types. We derive from it that the set of infinitely renormalizable parameter values in the real quadratic family Pc: x ↦ → x² + c has zero measure. This yields the statement in the title (where “ regular ” means to have an attracting cycle and “stochastic” means to have an absolutely continuous invariant measure). An application to the MLC problem is given.
MPFUN: A Portable High Performance Multiprecision Package
, 1990
"... The author has written a package of Fortran routines that perform a variety of arithmetic operations and transcendental functions on floating point numbers of arbitrarily high precision, including large integers. This package features (1) virtually universal portability, (2) high performance, especi ..."
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Cited by 50 (4 self)
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The author has written a package of Fortran routines that perform a variety of arithmetic operations and transcendental functions on floating point numbers of arbitrarily high precision, including large integers. This package features (1) virtually universal portability, (2) high performance, especially on vector supercomputers, (3) advanced algorithms, including FFTbased multiplication and quadratically convergent algorithms for π and transcendental functions, and (4) extensive selfchecking and debug facilities that permit the package to be used as a rigorous system integrity test. Converting application programs to run with these routines is facilitated by an automatic translator program. This paper describes the routines in the package and includes discussion of the algorithms employed, the implementation techniques, performance results and some applications. Notable among the performance results is that this package runs up to 40 times faster than another widely used package on a RISC workstation, and it runs up to 400 times faster than the other package on a Cray supercomputer.
Multiprecision translation and execution of Fortran programs
 ACM Transactions on Mathematical Software
, 1993
"... This paper describes two Fortran utilities for multiprecision computation. The first is a package of Fortran subroutines that perform a variety of arithmetic operations and transcendental functions on floating point numbers of arbitrarily high precision. This package is in some cases over 200 times ..."
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Cited by 31 (12 self)
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This paper describes two Fortran utilities for multiprecision computation. The first is a package of Fortran subroutines that perform a variety of arithmetic operations and transcendental functions on floating point numbers of arbitrarily high precision. This package is in some cases over 200 times faster than that of certain other packages that have been developed for this purpose. The second utility is a translator program, which facilitates the conversion of ordinary Fortran programs to use this package. By means of source directives (special comments) in the original Fortran program, the user declares the precision level and specifies which variables in each subprogram are to be treated as multiprecision. The translator program reads this source program and outputs a program with the appropriate multiprecision subroutine calls. This translator supports multiprecision integer, real and complex datatypes. The required array space for multiprecision data types is automatically allocated. In the evaluation of computational expressions, all of the usual conventions for operator precedence and mixed mode operations are upheld. Furthermore, most of the Fortran77 intrinsics, such as ABS, MOD, NINT, COS, EXP are supported and produce true multiprecision values.
Analog Computation with Dynamical Systems
 Physica D
, 1997
"... This paper presents a theory that enables to interpret natural processes as special purpose analog computers. Since physical systems are naturally described in continuous time, a definition of computational complexity for continuous time systems is required. In analogy with the classical discrete th ..."
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Cited by 25 (0 self)
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This paper presents a theory that enables to interpret natural processes as special purpose analog computers. Since physical systems are naturally described in continuous time, a definition of computational complexity for continuous time systems is required. In analogy with the classical discrete theory we develop fundamentals of computational complexity for dynamical systems, discrete or continuous in time, on the basis of an intrinsic time scale of the system. Dissipative dynamical systems are classified into the computational complexity classes P d , CoRP d , NP d
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 23 (7 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.
The Dynamics of a Harmonically Excited System Having Rigid Amplitude Constraints, Part 1: Subharmonic Motions and Local Bifurcations,"
 ASME JOURNAL OF APPLIED MECHANICS,
, 1985
"... The simple model described in ..."
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