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Статті в журналах з теми "Self-similar and multifractal streams"

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Автор: Grafiati

Опубліковано: 30 травня 2022

Оновлено: 31 травня 2022

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1

Huillet, T., and B. Jeannet. "Multifractal formalism for self-similar bridges." Journal of Physics A: Mathematical and General 31, no. 11 (March 20, 1998): 2567–90. http://dx.doi.org/10.1088/0305-4470/31/11/008.

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2

Barral, Julien, and Stéphane Seuret. "Renewal of singularity sets of random self-similar measures." Advances in Applied Probability 39, no. 01 (March 2007): 162–88. http://dx.doi.org/10.1017/s0001867800001658.

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Анотація:

This paper investigates new properties concerning the multifractal structure of a class of random self-similar measures. These measures include the well-known Mandelbrot multiplicative cascades, sometimes called independent random cascades. We evaluate the scale at which the multifractal structure of these measures becomes discernible. The value of this scale is obtained through what we call the growth speed in Hölder singularity sets of a Borel measure. This growth speed yields new information on the multifractal behavior of the rescaled copies involved in the structure of statistically self-similar measures. Our results are useful in understanding the multifractal nature of various heterogeneous jump processes.

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3

Barral, Julien, and Stéphane Seuret. "Renewal of singularity sets of random self-similar measures." Advances in Applied Probability 39, no. 1 (March 2007): 162–88. http://dx.doi.org/10.1239/aap/1175266474.

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Анотація:

This paper investigates new properties concerning the multifractal structure of a class of random self-similar measures. These measures include the well-known Mandelbrot multiplicative cascades, sometimes called independent random cascades. We evaluate the scale at which the multifractal structure of these measures becomes discernible. The value of this scale is obtained through what we call the growth speed in Hölder singularity sets of a Borel measure. This growth speed yields new information on the multifractal behavior of the rescaled copies involved in the structure of statistically self-similar measures. Our results are useful in understanding the multifractal nature of various heterogeneous jump processes.

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4

Taqqu, Murad S., Vadim Teverovsky, and Walter Willinger. "Is Network Traffic Self-Similar or Multifractal?" Fractals 05, no. 01 (March 1997): 63–73. http://dx.doi.org/10.1142/s0218348x97000073.

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This paper addresses the question of whether self-similar processes are sufficient to model packet network traffic, or whether a broader class of multifractal processes is needed. By using the absolute moments of aggregate traffic measurements, we conclude that measured local-area network (LAN) and wide-area network (WAN) traffic traces, with the sample means subtracted, are well modeled by random processes that are either exactly or asymptotically self-similar.

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5

Dyskin, A. V. "Multifractal properties of self-similar stress distributions." Philosophical Magazine 86, no. 21-22 (July 21, 2006): 3117–36. http://dx.doi.org/10.1080/14786430500421490.

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6

Mandelbrot, Benoit B., Carl J. G. Evertsz, and Yoshinori Hayakawa. "Exactly self-similar left-sided multifractal measures." Physical Review A 42, no. 8 (October 1, 1990): 4528–36. http://dx.doi.org/10.1103/physreva.42.4528.

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7

Yu, Jing Hu, and Di He Hu. "Multifractal Decomposition of Statistically Self-Similar Sets." Acta Mathematica Sinica, English Series 17, no. 3 (July 2001): 507–16. http://dx.doi.org/10.1007/s101140100121.

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8

Yu, Jing Hu, and Di He Hu. "Multifractal Decomposition of Statistically Self-Similar Sets." Acta Mathematica Sinica, English Series 17, no. 3 (July 2001): 507–16. http://dx.doi.org/10.1007/pl00011627.

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9

AOUIDI, JAMIL, and ANOUAR BEN MABROUK. "MULTIFRACTAL ANALYSIS OF SOME WEIGHTED QUASI-SELF-SIMILAR FUNCTIONS." International Journal of Wavelets, Multiresolution and Information Processing 09, no. 06 (November 2011): 965–87. http://dx.doi.org/10.1142/s0219691311004407.

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In this paper, a multifractal analysis of some non-self-similar functions based on the superposition of finite number of weighted quasi-self-similar ones ∑iωiFi is developed. In general, such superpositions do not yield neither a self-similar nor a quasi-self-similar outcome. Furthermore, there are two main problems that appear. Firstly, a phenomenon of regularity compensation may exist. Secondly, the computation of the spectrum of singularities and therefore the validity of the multifractal formalism based on the possibility of constructing Gibbs measures fail. In this paper, we propose to study such problems by conducting a multifractal analysis of such combinations and to check the validity of the multifractal formalism in the case where there is no compensation of regularity. Furthermore, we compute the box dimension of the associated graphs and provide some examples. The paper in its full subject re-considers the results of Ref. 3 in the quasi-self-similar case.

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10

Bruggeman, Cameron, Kathryn E. Hare, and Cheuk Yu Mak. "Multifractal spectrum of self-similar measures with overlap." Nonlinearity 27, no. 2 (January 16, 2014): 227–56. http://dx.doi.org/10.1088/0951-7715/27/2/227.

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11

Riedi, R. "An Improved Multifractal Formalism and Self-Similar Measures." Journal of Mathematical Analysis and Applications 189, no. 2 (January 1995): 462–90. http://dx.doi.org/10.1006/jmaa.1995.1030.

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12

O'Neil, Toby C. "The Multifractal Spectrum of Quasi Self-Similar Measures." Journal of Mathematical Analysis and Applications 211, no. 1 (July 1997): 233–57. http://dx.doi.org/10.1006/jmaa.1997.5458.

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13

Lau, Ka-Sing, and Sze-Man Ngai. "Second-order self-similar identities and multifractal decompositions." Indiana University Mathematics Journal 49, no. 3 (2000): 0. http://dx.doi.org/10.1512/iumj.2000.49.1789.

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14

Olsen, L., and N. Snigireva. "Multifractal spectra of in-homogenous self-similar measures." Indiana University Mathematics Journal 57, no. 4 (2008): 1789–844. http://dx.doi.org/10.1512/iumj.2008.57.3622.

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15

Falconer, K. J. "The multifractal spectrum of statistically self-similar measures." Journal of Theoretical Probability 7, no. 3 (July 1994): 681–702. http://dx.doi.org/10.1007/bf02213576.

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16

Shankar, U., C. P. Pearson, V. I. Nikora, and R. P. Ibbitt. "Heterogeneity in catchment properties: a case study of Grey and Buller catchments, New Zealand." Hydrology and Earth System Sciences 6, no. 2 (April 30, 2002): 167–84. http://dx.doi.org/10.5194/hess-6-167-2002.

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Анотація:

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

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17

BEN MABROUK, ANOUAR. "STUDY OF SOME NONLINEAR SELF-SIMILAR DISTRIBUTIONS." International Journal of Wavelets, Multiresolution and Information Processing 05, no. 06 (November 2007): 907–16. http://dx.doi.org/10.1142/s0219691307002105.

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We study some properties of some self-similar distributions constructed on a nonlinear way. We use wavelets to characterize such properties and to check the validity of the multifractal formalism in some cases.

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18

OLSEN, L. "SYMBOLIC AND GEOMETRIC LOCAL DIMENSIONS OF SELF-AFFINE MULTIFRACTAL SIERPINSKI SPONGES IN ℝd". Stochastics and Dynamics 07, № 01 (березень 2007): 37–51. http://dx.doi.org/10.1142/s0219493707001925.

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In this paper we study the multifractal structure of a certain class of self-affine measures known as self-affine multifractal Sierpinski sponges. Multifractal analysis studies the local scaling behaviour of measures. In particular, multifractal analysis studies the so-called local dimension and the multifractal spectrum of measures. The multifractal structure of self-similar measures satisfying the Open Set Condition is by now well understood. However, the multifractal structure of self-affine multifractal Sierpinski sponges is significantly less well understood. The local dimensions and the multifractal spectrum of self-affine multifractal Sierpinski sponges are only known provided a very restrictive separation condition, known as the Very Strong Separation Condition (VSSC), is satisfied. In this paper we investigate the multifractal structure of general self-affine multifractal Sierpinski sponges without assuming any additional conditions (and, in particular, without assuming the VSSC).

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19

Millán, G., and G. Lefranc. "Development of Multifractal Models for Self-Similar Traffic Flows." IFAC Proceedings Volumes 46, no. 24 (September 2013): 114–17. http://dx.doi.org/10.3182/20130911-3-br-3021.00095.

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20

Jaffard, S. "Multifractal Formalism for Functions Part II: Self-Similar Functions." SIAM Journal on Mathematical Analysis 28, no. 4 (July 1997): 971–98. http://dx.doi.org/10.1137/s0036141095283005.

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21

Radivilova, Tamara, Lyudmyla Kirichenko, Maksym Tawalbeh, Petro Zinchenko, and Vitalii Bulakh. "THE LOAD BALANCING OF SELF-SIMILAR TRAFFIC IN NETWORK INTRUSION DETECTION SYSTEMS." Cybersecurity: Education, Science, Technique 3, no. 7 (2020): 17–30. http://dx.doi.org/10.28925/2663-4023.2020.7.1730.

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The problem of load balancing in intrusion detection systems is considered in this paper. The analysis of existing problems of load balancing and modern methods of their solution are carried out. Types of intrusion detection systems and their description are given. A description of the intrusion detection system, its location, and the functioning of its elements in the computer system are provided. Comparative analysis of load balancing methods based on packet inspection and service time calculation is performed. An analysis of the causes of load imbalance in the intrusion detection system elements and the effects of load imbalance is also presented. A model of a network intrusion detection system based on packet signature analysis is presented. This paper describes the multifractal properties of traffic. Based on the analysis of intrusion detection systems, multifractal traffic properties and load balancing problem, the method of balancing is proposed, which is based on the funcsioning of the intrusion detection system elements and analysis of multifractal properties of incoming traffic. The proposed method takes into account the time of deep packet inspection required to compare a packet with signatures, which is calculated based on the calculation of the information flow multifractality degree. Load balancing rules are generated by the estimated average time of deep packet inspection and traffic multifractal parameters. This paper presents the simulation results of the proposed load balancing method compared to the standard method. It is shown that the load balancing method proposed in this paper provides for a uniform load distribution at the intrusion detection system elements. This allows for high speed and accuracy of intrusion detection with high-quality multifractal load balancing.

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22

Biggins, J. D., B. M. Hambly, and O. D. Jones. "Multifractal spectra for random self-similar measures via branching processes." Advances in Applied Probability 43, no. 01 (March 2011): 1–39. http://dx.doi.org/10.1017/s0001867800004663.

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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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23

LOU, ManLi, and YaHao WU. "The gauge of multifractal components for some self-similar sets." SCIENTIA SINICA Mathematica 44, no. 3 (February 1, 2014): 221–34. http://dx.doi.org/10.1360/012014-17.

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24

Testud, B. "Phase transitions for the multifractal analysis of self-similar measures." Nonlinearity 19, no. 5 (April 13, 2006): 1201–17. http://dx.doi.org/10.1088/0951-7715/19/5/009.

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25

Feng, De-Jun, and Ka-Sing Lau. "Multifractal formalism for self-similar measures with weak separation condition." Journal de Mathématiques Pures et Appliquées 92, no. 4 (October 2009): 407–28. http://dx.doi.org/10.1016/j.matpur.2009.05.009.

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26

Fourcade, B., and A. M. S. Tremblay. "Anomalies in the multifractal analysis of self-similar resistor networks." Physical Review A 36, no. 5 (September 1, 1987): 2352–58. http://dx.doi.org/10.1103/physreva.36.2352.

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27

Biggins, J. D., B. M. Hambly, and O. D. Jones. "Multifractal spectra for random self-similar measures via branching processes." Advances in Applied Probability 43, no. 1 (March 2011): 1–39. http://dx.doi.org/10.1239/aap/1300198510.

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Анотація:

Start with a compact set K ⊂ Rd. This has a random number of daughter sets, each of which is a (rotated and scaled) copy of K and all of which are inside K. 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 set F is the limit, as n goes to ∞, of the union of the nth generation sets. In addition, K has 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 on F. The multifractal spectrum of this measure is studied here. Our main contributions are dealing with the geometry of realisations in Rd and 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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28

Olsen, L. "A lower bound for the symbolic multifractal spectrum of a self-similar multifractal with arbitrary overlaps." Mathematische Nachrichten 282, no. 10 (October 2009): 1461–77. http://dx.doi.org/10.1002/mana.200610179.

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29

AOUIDI, JAMIL, and ANOUAR BEN MABROUK. "A WAVELET MULTIFRACTAL FORMALISM FOR SIMULTANEOUS SINGULARITIES OF FUNCTIONS." International Journal of Wavelets, Multiresolution and Information Processing 12, no. 01 (December 2013): 1450009. http://dx.doi.org/10.1142/s021969131450009x.

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In this paper, a wavelet multifractal analysis is developed which permits to characterize simultaneous singularities for a vector of functions. An associated multifractal formalism is introduced and checked for the case of functions involving self similar aspects.

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30

MABROUK, ANOUAR BEN. "WAVELET ANALYSIS OF NONLINEAR SELF-SIMILAR DISTRIBUTIONS WITH OSCILLATING SINGULARITY." International Journal of Wavelets, Multiresolution and Information Processing 06, no. 03 (May 2008): 447–57. http://dx.doi.org/10.1142/s0219691308002410.

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Анотація:

In the present paper, the properties of some nonlinear self-similar distributions involving both Hölder and oscillating singularities are studied. Wavelets are applied to prove that such distributions do not satisfy the classical multifractal formalism whereas they satisfy a grand canonical one. The paper in its subject is an extension of our work in Ref.7.

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31

BAEK, IN-Soo. "SIMPLE APPROACH TO MULTIFRACTAL SPECTRUM OF A SELF-SIMILAR CANTOR SET." Communications of the Korean Mathematical Society 20, no. 4 (October 1, 2005): 695–702. http://dx.doi.org/10.4134/ckms.2005.20.4.695.

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32

Baek, In-Soo. "MULTIFRACTAL SPECTRUM IN A SELF-SIMILAR ATTRACTOR IN THE UNIT INTERVAL." Communications of the Korean Mathematical Society 23, no. 4 (October 31, 2008): 549–54. http://dx.doi.org/10.4134/ckms.2008.23.4.549.

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33

Maulik, Krishanu, and Sidney Resnick. "The Self‐Similar and Multifractal Nature of a Network Traffic Model." Stochastic Models 19, no. 4 (November 2003): 549–77. http://dx.doi.org/10.1081/stm-120025404.

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34

FALCONER, KENNETH, and TONY SAMUEL. "Dixmier traces and coarse multifractal analysis." Ergodic Theory and Dynamical Systems 31, no. 2 (February 2, 2010): 369–81. http://dx.doi.org/10.1017/s0143385709001102.

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AbstractWe show how multifractal properties of a measure supported by a fractal F⊆[0,1] may be expressed in terms of complementary intervals of F and thus in terms of spectral triples and the Dixmier trace of certain operators. For self-similar measures this leads to a non-commutative integral over F equivalent to integration with respect to an auxiliary multifractal measure.

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35

OLSEN, L. "Empirical multifractal moment measures and moment scaling functions of self-similar multifractals." Mathematical Proceedings of the Cambridge Philosophical Society 133, no. 3 (November 2002): 459–85. http://dx.doi.org/10.1017/s0305004102006199.

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Let Si: ℝd → ℝd for i = 1, …, n be contracting similarities, and let (p1, …, pn) be a probability vector. Let K and μ be the self-similar set and the self-similar measure associated with (Si,pi)i. For q ∈ ℝ and r > 0, define the qth covering moment and the qth packing moment of μ by[formula here]where the infimum is taken over all r-spanning subsets E of K, and the supremum is taken over all r-separated subsets F of K. If the Open Set Condition (OSC) is satisfied then it is well known that[formula here]where β(q) is defined by [sum ]ipqirβi(q) = 1 (here ri denotes the Lipschitz constant of Si). Assuming the OSC, we determine the exact rate of convergence in (*): there exist multiplicatively periodic functions πq, Πq: (0,∞) → ℝ such that[formula here]where ε(r) → 0 as r[searr ]0. As an application of (**) we show that the empirical multi-fractal moment measures converges weakly:[formula here]where, for each positive r, Er is a (suitable) minimal r-spanning subset of K and Fr is a (suitable) maximal r-separated subset of K, and [Hscr ]q,β(q)μ and [Pscr ]q,β(q)μ are the multifractal Hausdorff measure and the multifractal packing measure, respectively.

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36

Xiao, Jiaqing, and Wu Min. "Empirical multifractal moment measures of self-similar measure for q < 0*." Monatshefte für Mathematik 156, no. 2 (July 9, 2008): 175–85. http://dx.doi.org/10.1007/s00605-008-0563-z.

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37

Shui-cao, Zheng, and Hu Di-he. "A note on the multifractal decomposition of directed graph self similar sets." Wuhan University Journal of Natural Sciences 9, no. 3 (May 2004): 269–72. http://dx.doi.org/10.1007/bf02907876.

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38

Dettmann, C. P., and N. E. Frankel. "potential theory and analytic properties of self-similar fractal and multifractal distributions." Journal of Statistical Physics 72, no. 1-2 (July 1993): 241–75. http://dx.doi.org/10.1007/bf01048049.

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39

MOCTEZUMA, R. E., and JORGE GONZÁLEZ-GUTIÉRREZ. "MULTIFRACTAL STRUCTURE IN SAND DRAWINGS." Fractals 28, no. 01 (January 30, 2020): 2050004. http://dx.doi.org/10.1142/s0218348x20500048.

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Анотація:

The construction of an abstract expressionist artwork is driven by chaotic mechanisms that sculpt multifractal characteristics. Jackson Pollock’s paintings, for example, arise due to the random process of depositing drops and jets of paint on a canvas. However, most of the paintings and drawings try to recreate with fidelity common forms, natural landscapes, and the human figure. Accordingly, in the context of the formation of statistically self-similar objects, a question persists: will it be possible to find some vestige of multifractal structure in drawings or paintings whose elaboration process tries to avoid chaos? In this work, we scrutinize into several artistic drawings in sand to answer this intriguing question. These pieces of art are elaborated using craters, furrows, and sand piles; and some of them are inscribed on the Representative List of the Intangible Cultural Heritage of Humanity. We prove that the sand drawings analyzed here are multifractal objects. This finding suggests that a piece of visual art, which may initially appear ordered, contains many components distributed at different degrees of self-similarity that substantially increase the structural complexity.

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40

Shmerkin, Pablo. "A Modified Multifractal Formalism for a Class of Self-similar Measures with Overlap." Asian Journal of Mathematics 9, no. 3 (2005): 323–48. http://dx.doi.org/10.4310/ajm.2005.v9.n3.a3.

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41

Ben Slimane, M. "Multifractal Formalism for Self-Similar Functions Under the Action of Nonlinear Dynamical Systems." Constructive Approximation 15, no. 2 (April 1999): 209–40. http://dx.doi.org/10.1007/s003659900105.

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42

Aung, Han, Nir Mandelker, Daisuke Nagai, Avishai Dekel, and Yuval Birnboim. "Kelvin–Helmholtz instability in self-gravitating streams." Monthly Notices of the Royal Astronomical Society 490, no. 1 (July 17, 2019): 181–201. http://dx.doi.org/10.1093/mnras/stz1964.

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ABSTRACT Self-gravitating gaseous filaments exist on many astrophysical scales, from sub-pc filaments in the interstellar medium to Mpc scale streams feeding galaxies from the cosmic web. These filaments are often subject to Kelvin–Helmholtz Instability (KHI) due to shearing against a confining background medium. We study the non-linear evolution of KHI in pressure-confined self-gravitating gas streams initially in hydrostatic equilibrium, using analytic models and hydrodynamic simulations, not including radiative cooling. We derive a critical line mass, or mass per unit length, as a function of the stream Mach number and density contrast with respect to the background, μcr(Mb, δc) ≤ 1, where μ = 1 is normalized to the maximal line mass for which initial hydrostatic equilibrium is possible. For μ < μcr, KHI dominates the stream evolution. A turbulent shear layer expands into the background and leads to stream deceleration at a similar rate to the non-gravitating case. However, with gravity, penetration of the shear layer into the stream is halted at roughly half the initial stream radius by stabilizing buoyancy forces, significantly delaying total stream disruption. Streams with μcr < μ ≤ 1 fragment and form round, long-lived clumps by gravitational instability (GI), with typical separations roughly eight times the stream radius, similar to the case without KHI. When KHI is still somewhat effective, these clumps are below the spherical Jeans mass and are partially confined by external pressure, but they approach the Jeans mass as μ → 1 and GI dominates. We discuss potential applications of our results to streams feeding galaxies at high redshift, filaments in the ISM, and streams resulting from tidal disruption of stars near the centres of massive galaxies.

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43

Douzi, Zied, and Bilel Selmi. "The mutual singularity of the relative multifractal measures." Nonautonomous Dynamical Systems 8, no. 1 (January 1, 2021): 18–26. http://dx.doi.org/10.1515/msds-2020-0123.

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Abstract M. Das proved that the relative multifractal measures are mutually singular for the self-similar measures satisfying the significantly weaker open set condition. The aim of this paper is to show that these measures are mutually singular in a more general framework. As examples, we apply our main results to quasi-Bernoulli measures.

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44

Giona, Massimiliano, Manuela Giustiniani, and Oreste Patierno. "Projected Measures: A Simple Way to Characterize Fractal Structures and Interfaces." Fractals 05, no. 02 (June 1997): 295–308. http://dx.doi.org/10.1142/s0218348x97000280.

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The properties of projected measures of fractal objects are investigated in detail. In general, projected measures display multifractal features which play a role in the evolution of dynamic phenomena on/through fractal structures. Closed-form results are obtained for the moment hierarchy of model fractal interfaces. The distinction between self-similar and self-affine interfaces is discussed by considering the properties of multifractal spectra, the orientational effects in the behavior of the moment hierarchies, and the scaling of the corresponding Fourier transforms. The implications of the properties of projected measures in the characterization of transfer phenomena across fractal interfaces are briefly analyzed.

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45

LAMMERING, BIRGER. "SLICES OF MULTIFRACTAL MEASURES AND APPLICATIONS TO RAINFALL DISTRIBUTIONS." Fractals 08, no. 04 (December 2000): 337–48. http://dx.doi.org/10.1142/s0218348x00000391.

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We discuss the relationship between the multifractal functions of a plane measure and those of slices or sections of the measure with a line. Motivated by recent mathematical ideas about the relationship between measures and their slices, we formulate the "slice hypothesis," and consider the theoretical limitations of this hypothesis. We compute the multifractal functions of several standard self-similar and self-affine measures and their slices to examine the validity of the slice hypothesis. We are particularly interested in using the slice hypothesis to estimate multifractal properties of spatial rainfall fields by analyzing rainfall data representing slices of rainfall fields. We consider how rainfall time series at a fixed site and slices of composite radar images can be used for this purpose, testing this on field data from a radar composite in the USA and on appropriate time series.

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46

VENEZIANO, DANIELE. "BASIC PROPERTIES AND CHARACTERIZATION OF STOCHASTICALLY SELF-SIMILAR PROCESSES IN Rd." Fractals 07, no. 01 (March 1999): 59–78. http://dx.doi.org/10.1142/s0218348x99000086.

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The classical notion of self-similarity (ss) for random X(t) as invariance under the group of positive affine transformations {X→ arX, t→rt; ar>0} is extended by allowing ar to be a random variable. The resulting property of "stochastic self-similarity" (sss) is applied to both ordinary and generalized random processes in Rd, d≥1. The class of sss processes seems to correspond to that of multifractal processes (the latter are variously defined in the literature). The spectral measures of ordinary and generalized sss processes are themselves stochastically self-similar. Two characterizations of ss processes by Lamperti are extended to the sss case and several basic properties of ordinary and generalized sss processes are derived.

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47

Hulton, N. R. J., and M. J. Mineter. "Modelling self-organization in ice streams." Annals of Glaciology 30 (2000): 127–36. http://dx.doi.org/10.3189/172756400781820561.

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AbstractThe EISMINT II experiments revealed the tendency for idealized model ice sheets to produce spatially variable flow under certain uniform thermal, mass-balance and topographic boundary conditions. Warm, fast-flowing streams with enhanced creep were separated by zones of colder, slower flow. Similar but different spatial patterns of differentiated flow were produced by all authors. We present further experiments that explore the formation and function of such ice streams at higher modelled resolutions. These are explored by the use of flat, but stochastically rough (10 m amplitude) beds, idealized, parallel-sided model ice sheets and models of finer (12.5 and 5 km) resolutions. Ice streams self-organize irregularly, but with consistent typical spacings which vary with thermal and miss-balance boundary conditions. More radial features are produced at finer scales indicating a dependency on the grid resolution used although this is not linear; at finer resolutions streams occupy increasingly more gridcells. This variation in scale may be related to the finer resolution of the warm/cold streaming/non-streaming boundary. The numerical solution of the thermodynamic ice equation is also highly sensitive to the orthogonality of the model grid. A major deficiency is that the numerical solution appears to fail where the flow is parallel to the grid axes, suggesting that artificial diffusion in the numerical scheme helps to smooth streams lying across the axes directions. The inclusion of sliding produces fewer, more concentrated, flow features, but these also display a level of scale-dependent organization. The spatial arrangement of such streams adjusts in response to the global mass flux of the ice sheet between "warm" and "cold" flow end-member. The results point to a mechanism in which ice sheets respond to climate by altering the large-scale arrangement of their flow patterns.

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48

Mondragón, R. J., J. M. Pitts, and D. K. Arrowsmith. "Chaotic intermittency-sawtooth map model of aggregate self-similar traffic streams." Electronics Letters 36, no. 2 (2000): 184. http://dx.doi.org/10.1049/el:20000184.

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YAO, CAN-ZHONG. "SELF-SIMILARITY PROPERTIES OF THE INDUSTRIAL COMPETITION NETWORKS." Fractals 22, no. 01n02 (March 2014): 1450002. http://dx.doi.org/10.1142/s0218348x14500029.

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It has been shown that many complex networks share self-similar properties. To reveal the ubiquitous characteristics that many real networks may have in common, we first improve the box covering algorithm and use it to measure several real industrial competition networks. As a result, we find that some real industrial competition networks illustrate the fractal properties. As the improved box covering algorithm could not reflect the structure characteristic of the industrial competitive network efficiently, and the scale of the network namely the average length path is not enough for quantifying the fractal properties, we use the clustering coefficient as the scale measure index and present a systematic analysis of some real industrial competition networks from the perspective of multifractal features. We find that many real industrial competition networks can also be characterized by the multifractal features. Finally, with a motif-hierarchical model, we simulate how the fractal structure of the industrial networks may be formed, and to some extent verify that the disassortative process is an important self-organizing mechanism in industry system.

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50

Xiao, JiaQing, and YouMing He. "Multifractal Structure of the Divergence Points of Some Homogeneous Moran Measures." Advances in Mathematical Physics 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/161756.

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The pointxfor which the limitlimr→0⁡(log⁡μBx,r/log⁡r)does not exist is called divergence point. Recently, multifractal structure of the divergence points of self-similar measures has been investigated by many authors. This paper is devoted to the study of some Moran measures with the support on the homogeneous Moran fractals associated with the sequences of which the frequency of the letter exists; the Moran measures associated with this kind of structure are neither Gibbs nor self-similar and than complex. Such measures possess singular features because of the existence of so-called divergence points. By the box-counting principle, we analyze multifractal structure of the divergence points of some homogeneous Moran measures and show that the Hausdorff dimension of the set of divergence points is the same as the dimension of the whole Moran set.

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