Multifractal spectra and multifractal rigidity for horseshoes

1997 ◽  
Vol 3 (1) ◽  
pp. 33-49 ◽  
Author(s):  
L. Barreira ◽  
Ya. Pesin ◽  
J. Sehmeling
Keyword(s):  
Multifractal Spectra ◽  
Multifractal Rigidity

Study of the structural degradation of steel 15Kh5M upon continuous duty

2020 ◽  
Vol 86 (1) ◽  
pp. 38-43
Author(s):  
Vladimir A. Kim ◽  
Valeriya V. Lysenko ◽  
Anna A. Afanaseva ◽  
Khasan I. Turkmenov
Keyword(s):  
Grain Boundaries ◽  
Structural Changes ◽  
Structural Transformations ◽  
Structural Degradation ◽  
Structural Arrangement ◽  
Dispersed Particles ◽  
Complex Process ◽  
Multifractal Spectra ◽  
Steel 15Kh5m ◽  
Complex Indicators

Structural degradation of the material upon long-term thermal and force impacts is a complex process which includes migration of the grain boundaries, diffusion of the active elements of the external and technological environment, hydrogen embrittlement, aging, grain boundary corrosion and other mechanisms. Application of the fractal and multifractal formalism to the description of microstructures opens up wide opportunities for quantitative assessment of the structural arrangement of the material, clarifies and reveals new aspects of the known mechanisms of structural transformations. Multifractal parameterization allows us to study the processes of structural degradation from the images of microstructures and identify structural changes that are hardly distinguishable visually. Any quantitative structural indicator can be used to calculate the multifractal spectra of the microstructure, but the most preferable is that provides the maximum range of variation in the numerical values of the multifractal components. The results of studying structural degradation of steel 15Kh5M upon continuous duty are presented. It is shown that structural degradation of steel during operation under high temperatures and stresses is accompanied by enlargement of the microstructural objects, broadening of the grain boundaries and allocation of the dispersed particles which are represented as point objects in the images. The processes of structural degradation lead to an increase in the range of changes in the components of the multifractal spectra. High values of complex indicators of structural arrangement indicate to an increase in heterogeneity and randomness at the micro-scale level, but at the same time, to manifestation of the ordered combinations of individual submicrostructures. Those structural transformations adapt the material to external impacts and provide the highest reliability and fracture resistance of the material.

scholarly journals Regularity of multifractal spectra of conformal iterated function systems

2011 ◽  
Vol 363 (01) ◽  
pp. 313-313 ◽  
Author(s):  
Johannes Jaerisch ◽  
Marc Kesseböhmer
Keyword(s):  
Iterated Function Systems ◽  
Iterated Function ◽  
Multifractal Spectra

scholarly journals Dynamical multifractal zeta-functions and fine multifractal spectra of graph-directed self-conformal constructions

2015 ◽  
Vol 36 (6) ◽  
pp. 1922-1971 ◽  
Author(s):  
V. MIJOVIĆ ◽  
L. OLSEN
Keyword(s):  
General Class ◽  
Zeta Functions ◽  
Continuous Functions ◽  
Precise Information ◽  
Multifractal Spectra ◽  
Conformal Measures

We introduce multifractal pressure and dynamical multifractal zeta-functions providing precise information of a very general class of multifractal spectra, including, for example, the fine multifractal spectra of graph-directed self-conformal measures and the fine multifractal spectra of ergodic Birkhoff averages of continuous functions on graph-directed self-conformal sets.

Stability of the multifractal spectra by transformations of discrete series

2007 ◽  
Author(s):  
A. Corvalán ◽  
E. Serrano
Keyword(s):  
Discrete Series ◽  
Multifractal Spectra

scholarly journals Partition zeta functions, multifractal spectra, and tapestries of complex dimensions

2015 ◽  
pp. 267-322 ◽  
Author(s):  
Kate E. Ellis ◽  
Michel L. Lapidus ◽  
Michael C. Mackenzie ◽  
John A. Rock
Keyword(s):  
Zeta Functions ◽  
Complex Dimensions ◽  
Multifractal Spectra

scholarly journals Heterogeneity in catchment properties: a case study of Grey and Buller catchments, New Zealand

2002 ◽  
Vol 6 (2) ◽  
pp. 167-184 ◽  
Author(s):  
U. Shankar ◽  
C. P. Pearson ◽  
V. I. Nikora ◽  
R. P. Ibbitt
Keyword(s):  
New Zealand ◽  
Fractal Dimensions ◽  
Affine Scaling ◽  
Stream Networks ◽  
Ecological Processes ◽  
Spectral Parameters ◽  
Scaling Behaviour ◽  
Multifractal Spectra ◽  
Box Counting Method ◽  
Landscape Patches

Abstract. The scaling behaviour of landscape properties, including both morphological and landscape patchiness, is examined using monofractal and multifractal analysis. The study is confined to two neighbouring meso-scale catchments on the west coast of the South Island of New Zealand. The catchments offer a diverse but largely undisturbed landscape with population and development impacts being extremely low. Bulk landscape properties of the catchments (and their sub-basins) are examined and show that scaling of stream networks follow Hack’s empirical rule, with exponents ∼0.6. It is also found that the longitudinal and transverse scaling exponents of stream networks equate to νl ≈0.6 and νw≈ 0.4, indicative of self-affine scaling. Catchment shapes also show self-affine behaviour. Further, scaling of landscape patches show multifractal behaviour and the analysis of these variables yields the characteristic parabolic curves known as multifractal spectra. A novel analytical approach is adopted by using catchments as hydrological cells at various sizes, ranging from first to sixth order, as the unit of measure. This approach is presented as an alternative to the box-counting method as it may be much more representative of hydro-ecological processes at catchment scales. Multifractal spectra are generated for each landscape property and spectral parameters such as the range in α (Holder exponent) values and maximum dimension at α0, (also known as the capacity dimension Dcap), are obtained. Other fractal dimensions (information Dinf and correlation Dcor) are also calculated and compared. The dimensions are connected by the inequality Dcap≥Dinf≥Dcor. Such a relationship strongly suggests that the landscape patches are heterogeneous in nature and that their scaling behaviour can be described as multifractal. The quantitative parameters obtained from the spectra may provide the basis for improved parameterisation of ecological and hydrological models. Keywords: fractal, multifractal, scaling, landscape, patchiness

scholarly journals Multifractal spectra for random self-similar measures via branching processes

2011 ◽  
Vol 43 (01) ◽  
pp. 1-39
Author(s):  
J. D. Biggins ◽  
B. M. Hambly ◽  
O. D. Jones
Keyword(s):  
Random Number ◽  
Branching Processes ◽  
Multifractal Spectrum ◽  
Compact Set ◽  
Similar Measure ◽  
Random Mass ◽  
Random Fractal ◽  
Multifractal Spectra ◽  
Self Similar ◽  
Random Division

Start with a compact setK⊂Rd. This has a random number of daughter sets, each of which is a (rotated and scaled) copy ofKand all of which are insideK. The random mechanism for producing daughter sets is used independently on each of the daughter sets to produce the second generation of sets, and so on, repeatedly. The random fractal setFis the limit, asngoes to ∞, of the union of thenth generation sets. In addition,Khas a (suitable, random) mass which is divided randomly between the daughter sets, and this random division of mass is also repeated independently, indefinitely. This division of mass will correspond to a random self-similar measure onF. The multifractal spectrum of this measure is studied here. Our main contributions are dealing with the geometry of realisations inRdand drawing systematically on known results for general branching processes. In this way we generalise considerably the results of Arbeiter and Patzschke (1996) and Patzschke (1997).

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