A Rigorous ODE Solver and Smale’s 14th Problem

Tucker, Warwick

doi:10.1007/s002080010018

Cited by 399 publications

(307 citation statements)

References 21 publications

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“…(11) can be analyzed with the Poincare section and the first return map. 38 Parameter c has been used to classify certain behavior of the Lorenz Eq. (11).…”

Section: Numerical Resultsmentioning

confidence: 99%

Molecular nonlinear dynamics and protein thermal uncertainty quantification

Xia

Guo

2014

Chaos

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This work introduces molecular nonlinear dynamics (MND) as a new approach for describing protein folding and aggregation. By using a mode system, we show that the MND of disordered proteins is chaotic while that of folded proteins exhibits intrinsically low dimensional manifolds (ILDMs). The stability of ILDMs is found to strongly correlate with protein energies. We propose a novel method for protein thermal uncertainty quantification based on persistently invariant ILDMs. Extensive comparison with experimental data and the state-of-the-art methods in the field validate the proposed new method for protein B-factor prediction. V C 2014 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4861202]Protein folding produces characteristic and functional three-dimensional structures from unfolded polypeptides or disordered coils. [1][2][3][4][5] The emergence of extraordinary complexity in the protein folding process poses astonishing challenges to theoretical modeling and computer simulations. 6,7 The present work introduces molecular nonlinear dynamics (MND), or molecular chaotic dynamics, as a theoretical framework for describing and analyzing protein folding. We represent the dynamics of macromolecular particles (i.e., atoms or coarse-grained superatoms) by a set of intrinsically chaotic oscillators. A geometry to topology mapping is employed to create driving and response relations among chaotic oscillators. We unveil the existence of intrinsically low dimensional manifolds (ILDMs) in the chaotic dynamics of folded proteins. Additionally, we reveal that the transition from disordered to ordered conformations in protein folding increases the transverse stability of the ILDM. Stated differently, protein folding reduces the chaoticity of the nonlinear dynamical system, and a folded protein has the best ability to tame chaos. Furthermore, we bring to light the connection between the ILDM stability and the thermodynamic stability, which enables us to quantify the disorderliness and relative energies of folded, misfolded, and unfolded protein states. Finally, we exploit chaos for protein uncertainty quantification and develop a robust chaotic algorithm for the prediction of Debye-Waller factors, or temperature factors, of protein structures.

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“…(11) can be analyzed with the Poincare section and the first return map. 38 Parameter c has been used to classify certain behavior of the Lorenz Eq. (11).…”

Section: Numerical Resultsmentioning

confidence: 99%

Molecular nonlinear dynamics and protein thermal uncertainty quantification

Xia

Guo

2014

Chaos

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“…Moreover, this attractor is sensitive to initial conditions and can not be destroyed by small perturbations of the original flow, that is to say it is robust. Finally, Tucker [38,39] proved the existence and robustness of the Lorenz attractor and, as a consequence of the method of his proof, showed that these models do describe the behaviour of (3.1). For more information on the history of the subject and the construction of the geometric models, we refer the reader to Araujo, Pacifico and Viana [4,40] and references therein.…”

Section: Applicationsmentioning

confidence: 98%

Decay of correlations and laws of rare events for transitive random maps

Araújo

Aytaç

2017

Nonlinearity

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Abstract. We show that a uniformly continuous random perturbation of a transitive map defines an aperiodic Harris chain which also satisfies Doeblin's condition. As a result, we get exponential decay of correlations for suitable random perturbations of such systems. We also prove that, for transitive maps, the limiting distribution for Extreme Value Laws (EVLs) and Hitting/Return Time Statistics (HTS/RTS) is standard exponential. Moreover, we show that the Rare Event Point Process (REPP) converges in distribution to a standard Poisson process.

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“…17-digit precision) to prove that the Lorenz equations support a "strange" attractor. Berkeley mathematician Steven Smale had previously included this as one of the most challenging problems for the 21st century [32].…”

Section: Computer-assisted Solution Of Smale's 14th Problemmentioning

confidence: 99%

High-Precision Arithmetic in Mathematical Physics

Bailey

Borwein

2015

Mathematics

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For many scientific calculations, particularly those involving empirical data, IEEE 32-bit floating-point arithmetic produces results of sufficient accuracy, while for other applications IEEE 64-bit floating-point is more appropriate. But for some very demanding applications, even higher levels of precision are often required. This article discusses the challenge of high-precision computation, in the context of mathematical physics, and highlights what facilities are required to support future computation, in light of emerging developments in computer architecture.

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A Rigorous ODE Solver and Smale’s 14th Problem

Cited by 399 publications

References 21 publications

Molecular nonlinear dynamics and protein thermal uncertainty quantification

Molecular nonlinear dynamics and protein thermal uncertainty quantification

Decay of correlations and laws of rare events for transitive random maps

High-Precision Arithmetic in Mathematical Physics

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