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Erschienen in:
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2018 | OriginalPaper | Buchkapitel

Pandora Box of Multifractals: Barely Open?

verfasst von : Daniel Schertzer, Ioulia Tchiguirinskaia

Erschienen in: Advances in Nonlinear Geosciences

Verlag: Springer International Publishing

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Abstract

Three decades ago, multifractals were a major breakthrough in nonlinear geophysics by providing a general framework to understand, analyze, and simulate fields that are extremely inhomogeneous over a wide range of space-time scales. They have remained on the forefront of nonlinear methodologies, but they are still far from being used or even developed to their full extent. Indeed, they have been too often limited to scalar-valued fields, whereas the relevant geophysical fields are vector fields. This chapter therefore gives new insights on current developments to overcome this limitation. This is done in an inductive manner. For instance, it takes hold on simple considerations on “spherical” and “hyperbolic” rotations to introduce step by step the Clifford algebra of Lévy stable generators of multifractal vectors that have both universal statistical and robust algebraic properties.

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Vorheriges Kapitel Studying the Complexity of Rainfall Within California Via a Fractal Geometric Method
Nächstes Kapitel Complex Networks and Hydrologic Applications
Fußnoten
1
One may note that they were originally called “coquaternions” (Cookie, 1849) and more recently “split quaternions” (Rosenfeld 1988) or “pseudo-quateriions” (Yaglom, 1968), whreas the name quasi-quaternions was used by Okubo (1978) for a more involved non associative algebra.
 
2
The choice of sub-index 2 to distinguish the quaternions from the pseudo-quaternions is due to the fact that their matrix representation corresponds to 2 × 2 block matrices of the latter (see Eq.9).
 
3
Euler is known for introducing the complex exponential notation (α = 1) and Moivre for the identity corresponding to integer α’s. Equation (10) summarizes both contributions.
 
4
Similar to that defined in special and general relativity, i.e. the path of a light flash emanating from a single event.
 
5
Due to common usage, we are compelled to use curly brackets both for anti-commutators and linear spans, but this should not introduce any confusion.
 
6
The moment order q and the index q of a Clifford algebra have nothing else in common, except to be the same alphabetical letter due to respective usages.
 
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Metadaten
Titel
Pandora Box of Multifractals: Barely Open?
verfasst von
Daniel Schertzer
Ioulia Tchiguirinskaia
Copyright-Jahr
2018
Buch
Advances in Nonlinear Geosciences

Print ISBN: 978-3-319-58894-0

Electronic ISBN: 978-3-319-58895-7

Copyright-Jahr: 2018

DOI
https://doi.org/10.1007/978-3-319-58895-7_25