On the dimension of the compact invariant sets of certain non-linear maps

Mañé, Ricardo

doi:10.1007/bfb0091916

Cited by 338 publications

(209 citation statements)

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“…If d is adequately large, between the manifold M , generated by the points X(t) and the attractor U of the dynamic system that generated the time series, there is a diffeomorphism 3 . Takens-Mañé embedding theorem [78,79] states that, in order to obtain a faithful reconstruction of the system dynamics, it must be:…”

Section: Applicationsmentioning

confidence: 99%

Data dimensionality estimation methods: a survey

Camastra¹

2003

Pattern Recognition

236

172

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In this paper, data dimensionality estimation methods are reviewed. The estimation of the dimensionality of a data set is a classical problem of pattern recognition. There are some good reviews [1] in literature but they do not include more recent developments based on fractal techniques and neural autoassociators. The aim of this paper is to provide an up-to-date survey of the dimensionality estimation methods of a data set, paying special attention to the fractal-based methods.

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Section: Applicationsmentioning

confidence: 99%

Data dimensionality estimation methods: a survey

Camastra¹

2003

Pattern Recognition

236

172

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“…Mañé [33] demonstrates the existence of a dense set of injective projections for a compact subset of a Banach space. Ben-Artzi, Eden, Foias, and Nicolaenko [5] study the Hölder continuity of the inverse and establish sharp bounds on the Hölder exponent in the case X ⊂ R n .…”

Section: Conjecture 84 ([43]mentioning

confidence: 99%

Untitled

2005

Bull. Amer. Math. Soc.

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Abstract. Many problems in mathematics and science require the use of infinite-dimensional spaces. Consequently, there is need for an analogue of the finite-dimensional notions of 'Lebesgue almost every' and 'Lebesgue measure zero' in the infinite-dimensional setting. The theory of prevalence addresses this need and provides a powerful framework for describing generic behavior in a probabilistic way. We survey the theory and applications of prevalence.

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“…Indeed, as long ago as 1981 Mañé [45] proved an abstract result along these lines, and we use a strengthened version of his result due to Hunt and Kaloshin [33] in our proof. However, to our knowledge this is the first result which provides an experimentally realisable method of parametrisation.…”

Section: S(t)mentioning

confidence: 99%

Parametrising the attractor of the two-dimensional Navier–Stokes equations with a finite number of nodal values

Friz

Robinson

2001

Physica D: Nonlinear Phenomena

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We consider the solutions lying on the global attractor of the two-dimensional Navier-Stokes equations with periodic boundary conditions and analytic forcing. We show that in this case the value of a solution at a finite number of nodes determines elements of the attractor uniquely, proving a conjecture due to Foias and Temam. Our results also hold for the complex Ginzburg-Landau equation, the Kuramoto-Sivashinsky equation, and reaction-diffusion equations with analytic nonlinearities.

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On the dimension of the compact invariant sets of certain non-linear maps

Cited by 338 publications

References 0 publications

Data dimensionality estimation methods: a survey

Data dimensionality estimation methods: a survey

Untitled

Parametrising the attractor of the two-dimensional Navier–Stokes equations with a finite number of nodal values

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