The root of power law religion - Integral Post-Metaphysical Spirituality2022-10-07T09:29:34Zhttp://integralpostmetaphysics.ning.com/forum/topics/the-root-of-power-law-religion?feed=yes&xn_auth=noFrom this article:"A common g…tag:integralpostmetaphysics.ning.com,2019-08-31:5301756:Comment:771162019-08-31T14:02:06.083ZEdward theurj Bergehttp://integralpostmetaphysics.ning.com/profile/theurj
<p><span dir="ltr"><span class="_3l3x _1n4g"><span>From <a href="https://arxiv.org/pdf/1802.10582.pdf" rel="noopener" target="_blank">this</a> article:<br></br><br></br>"A common graph mining task is community detection, which seeks an unsupervised decomposition of a network into groups based on statistical regularities in network connectivity. Although many such algorithms exist, community detection’s No Free Lunch</span> <span>theorem implies that no algorithm can be optimal across all inputs. [...]…</span></span></span></p>
<p><span dir="ltr"><span class="_3l3x _1n4g"><span>From <a href="https://arxiv.org/pdf/1802.10582.pdf" target="_blank" rel="noopener">this</a> article:<br/><br/>"A common graph mining task is community detection, which seeks an unsupervised decomposition of a network into groups based on statistical regularities in network connectivity. Although many such algorithms exist, community detection’s No Free Lunch</span> <span>theorem implies that no algorithm can be optimal across all inputs. [...] We find that (i) algorithms vary widely in the number and composition of communities they find, given the same input; (ii) algorithms can be clustered into distinct high-level groups based on similarities of their outputs on real-world networks; (iii) algorithmic differences induce wide variation in accuracy on link-based learning tasks; and, (iv) no algorithm is always the best at such tasks across all inputs."</span></span></span></p> Power laws are a burr in my b…tag:integralpostmetaphysics.ning.com,2019-08-31:5301756:Comment:772132019-08-31T14:01:12.696ZEdward theurj Bergehttp://integralpostmetaphysics.ning.com/profile/theurj
<p><span dir="ltr"><span class="_3l3x _1n4g"><span>Power laws are a burr in my butt today. From <a href="https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.122.158303" rel="noopener" target="_blank">this</a> article:<br></br><br></br>"Many self-similar systems are scale invariant only in discrete steps. A blood vessel tends to branch into two smaller vessels, a fluid vortex into two or three smaller vortices, and the Sierpinski triangle is self-simila</span><span>r only by powers of two. These…</span></span></span></p>
<p><span dir="ltr"><span class="_3l3x _1n4g"><span>Power laws are a burr in my butt today. From <a href="https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.122.158303" target="_blank" rel="noopener">this</a> article:<br/><br/>"Many self-similar systems are scale invariant only in discrete steps. A blood vessel tends to branch into two smaller vessels, a fluid vortex into two or three smaller vortices, and the Sierpinski triangle is self-simila</span><span>r only by powers of two. These systems preserve relative proportions upon rescaling from one step to the next, but not upon arbitrary rescaling. This property is termed discrete-scale invariance or discrete renormalizability. It is a weaker condition than the continuous scale invariance underlying the Pareto distribution. Whereas strict scale invariance implies a power law and vice versa, discrete-scale invariance allows log-periodic modulations in the frequencies of observations that deviate from a pure power law such as Eq.(1). Such modulations are indeed observed in bronchial tube diameter, vortex ens-trophy, and financial asset prices."</span></span></span></p> Excerpts from The Number Sens…tag:integralpostmetaphysics.ning.com,2019-08-31:5301756:Comment:771152019-08-31T14:00:02.755ZEdward theurj Bergehttp://integralpostmetaphysics.ning.com/profile/theurj
<p><span dir="ltr"><span class="_3l3x _1n4g"><span>Excerpts from <a href="http://backspaces.net/temp/Spring2010Seminar/The%20Number%20Sense.pdf" rel="noopener" target="_blank"><em>The Number Sense</em></a> by Dehaene, pp. 242-45 are below. I'd say whether or not one believes in Platonic math, both it and formal math are abstract with a priori axioms divorced from concrete reality. The intuitionist or constructivist math he notes below, while</span> <span>accepting our innate categories of…</span></span></span></p>
<p><span dir="ltr"><span class="_3l3x _1n4g"><span>Excerpts from <a href="http://backspaces.net/temp/Spring2010Seminar/The%20Number%20Sense.pdf" target="_blank" rel="noopener"><em>The Number Sense</em></a> by Dehaene, pp. 242-45 are below. I'd say whether or not one believes in Platonic math, both it and formal math are abstract with a priori axioms divorced from concrete reality. The intuitionist or constructivist math he notes below, while</span> <span>accepting our innate categories of thought, are not the same as the image schema and basic categories of cognitive linguistics (and in fact are not referenced). But it bases this similar idea on the relation of math to our embodiment.<br/><br/>"Twentieth-century mathematicians have been profoundly divided over this fundamental issue concerning the nature of mathematical objects. For some, traditionally labeled 'Platonists,' mathematical reality exists in an abstract plane, and its objects are as real as those of everyday life. [...] For an epistemologist, a neurobiologist, or a neuropsychologist, the Platonist position seems hard to defend—as unacceptable, in fact, as Cartesian dualism is as a scientific theory of the brain."<br/><br/>"A second category of mathematicians, the 'formalists,' view the issue of the existence of mathematical objects as meaningless and void. For them, mathematics is only a game in which one manipulates symbols according to precise formal rules. Mathematical objects such as numbers have no relation to reality: They are defined merely as a set of symbols that satisfy certain axioms. [...] Though the formalist position may account for the recent evolution of pure mathematics, it does not provide an adequate explanation of its origins."<br/><br/>"A third category of mathematicians is thus that of the 'intuitionists' or 'constructivists,' who believe that mathematical objects are nothing but constructions of the human mind. In their view, mathematics does not exist in the outside world, but only in the brain of the mathematician who invents it. [...] Among the available theories on the nature of mathematics, intuitionism seems to me to provide the best account of the relations between arithmetic and the human brain. The discoveries of the last few years in the psychology of arithmetic have brought new arguments to support the intuitionist view. [...] These empirical results tend to confirm Poincare's postulate that number belongs to the 'natural objects of thought,' the innate categories according to which we apprehend the world. [...] Intuition about numbers is thus anchored deep in our brain. Number appears as one of the fundamental dimensions according to which our nervous system parses the external world."</span></span></span></p> Nonlinearity in Living System…tag:integralpostmetaphysics.ning.com,2019-08-31:5301756:Comment:771142019-08-31T13:58:26.762ZEdward theurj Bergehttp://integralpostmetaphysics.ning.com/profile/theurj
<p><span dir="ltr"><span class="_3l3x _1n4g"><span><em>Nonlinearity in Living Systems</em> is the title of a recently published <a href="https://www.frontiersin.org/research-topics/6983/nonlinearity-in-living-systems-theoretical-and-practical-perspectives-on-metrics-of-physiological-si?utm_source=fweb&utm_medium=nblog&utm_campaign=ba-sci-ebook-20191000" rel="noopener" target="_blank">e-book</a> by Frontiers in Applied Mathematics and Statistics. Here's an excerpt from the introductory…</span></span></span></p>
<p><span dir="ltr"><span class="_3l3x _1n4g"><span><em>Nonlinearity in Living Systems</em> is the title of a recently published <a href="https://www.frontiersin.org/research-topics/6983/nonlinearity-in-living-systems-theoretical-and-practical-perspectives-on-metrics-of-physiological-si?utm_source=fweb&utm_medium=nblog&utm_campaign=ba-sci-ebook-20191000" target="_blank" rel="noopener">e-book</a> by Frontiers in Applied Mathematics and Statistics. Here's an excerpt from the introductory editorial.<br/><br/>"The biological basis of physiological signals is incredibly complex. While many researches certa</span><span>inly appreciate molecular, cellular and systems approaches to unravel overall biological complexity, in the recent decades the interest for mathematical and computational characterization of structural and functional basis underlying biological phenomena gain wide popularity among scientists.[...] We witnessed wide range applications of nonlinear quantitative analysis that produced measures such as fractal dimension, power law scaling, Hurst exponent, Lyapunov exponent, approximate entropy, sample entropy, Lempel–Ziv complexity as well as other metric. [...] Also there is another more theoretical challenge of contemporary nonlinear signal measurements, especially including fractal-based methods. The question of choosing the right method and its possible adjustment in order for the results of the analysis to be as accurate as possible is the persistent problem.[...] We seek to bring together the recent practical and theoretical advances in the development and application of nonlinear methods or narrower fractal-based methods for characterizing the complex physiological systems at multiple levels of organization. [...] A comprehensive understanding of advantages and disadvantages of each method, especially between its mathematical assumptions and real-world applicability, can help to find out what is at stake regarding the above aims and to direct us toward more fruitful application of nonlinear measures and statistics in physiology and biology in general."<br/><br/>Excerpts from this article in the ebook, "Measures and metrics of biological signals."<br/><br/>"With the growing complexity of the applied mathematical concepts, we are approaching some serious issues of foundations of Mathematics. Before that, let us mention that the symbol ∞ does not represent infinity uniquely since Cantor's discoveries in 1873, when he showed that arithmetical and geometric infinity, i.e., natural numbers and real line are different infinite quantities. As a consequence, infinity has been scaled in terms of pairwise different cardinal numbers. However, the size of this scale is enormous; it cannot be coded by any set. This was the creation of Set theory, and the beginning of the studies of foundations of Mathematics, which is probably never ending."<br/><br/>"We learned that Mathematical theories, packed around their axioms can be at the same level of logical certainty, while obviously impossible mixed together since with colliding axioms.[...] Let us just say that AC (Axiom of Choice) is very much needed in the foundations of Mathematics, but there are alternatives. [...] Some of the functions close to the above-examined fractals are complex enough to open the fundamental issue. [...] On the other hand, we can stay on the flat Earth and deal only with short approximation of the phenomena, avoiding entering the zone of the complex Mathematics and its fundamental issues. Yet, as proved by Goedel, we cannot escape the hot issues even remaining only in Arithmetic, nor in any theory containing its copy (like Geometry)." <br/></span></span></span></p> From this piece, which feeds…tag:integralpostmetaphysics.ning.com,2019-08-31:5301756:Comment:772122019-08-31T13:56:58.103ZEdward theurj Bergehttp://integralpostmetaphysics.ning.com/profile/theurj
<p><span dir="ltr"><span class="_3l3x _1n4g"><span>From <a href="https://www.nature.com/articles/d41586-019-00083-3?fbclid=IwAR0VGlfvffxI_jlK0yQ_yFQfuC8G9pf2mFtSriQtSTfGIqUeRdLUipii_bY" rel="noopener" target="_blank">this</a> piece, which feeds my thesis:<br></br><br></br>"[...] a paradox known as the continuum hypothesis. Gödel showed that the statement cannot be proved either true or false using standard mathematical language. [...It] efficiently boils down to a question in the theory of sets…</span></span></span></p>
<p><span dir="ltr"><span class="_3l3x _1n4g"><span>From <a href="https://www.nature.com/articles/d41586-019-00083-3?fbclid=IwAR0VGlfvffxI_jlK0yQ_yFQfuC8G9pf2mFtSriQtSTfGIqUeRdLUipii_bY" target="_blank" rel="noopener">this</a> piece, which feeds my thesis:<br/><br/>"[...] a paradox known as the continuum hypothesis. Gödel showed that the statement cannot be proved either true or false using standard mathematical language. [...It] efficiently boils down to a question in the theory of sets [...Cantor] was not ab</span><span>le to prove this continuum hypothesis, and nor were many mathematicians and logicians who followed him."<br/><br/>"Gödel [...] showed that the continuum hypothesis cannot be proved either true or false starting from the standard axioms — the statements taken to be true — of the theory of sets, which are commonly taken as the foundation for all of mathematics. Gödel and Cohen’s work on the continuum hypothesis implies that there can exist parallel mathematical universes that are both compatible with standard mathematics — one in which the continuum hypothesis is added to the standard axioms and therefore declared to be true, and another in which it is declared false.</span></span></span></p> This chart/image helps with s…tag:integralpostmetaphysics.ning.com,2019-08-31:5301756:Comment:772112019-08-31T13:55:01.108ZEdward theurj Bergehttp://integralpostmetaphysics.ning.com/profile/theurj
<p><span dir="ltr"><span class="_3l3x"><span><a href="https://www.cecan.ac.uk/sites/default/files/2018-06/The%20Visual%20Communication%20of%20Complexity%20-%20May2018%20-%20EcoLabs.pdf?fbclid=IwAR1fHVdNwl6LPxoGoDJLof_wcpO2EP2GQInHzixHJ0klqI1sMM859oQQ020" rel="noopener" target="_blank">This</a> chart/image helps with some of the ideas above. E.g., #16 noting multiple scales and levels interacting, reminding me of Mascolo's quote above.…</span></span></span></p>
<p><span dir="ltr"><span class="_3l3x"><span><a href="https://www.cecan.ac.uk/sites/default/files/2018-06/The%20Visual%20Communication%20of%20Complexity%20-%20May2018%20-%20EcoLabs.pdf?fbclid=IwAR1fHVdNwl6LPxoGoDJLof_wcpO2EP2GQInHzixHJ0klqI1sMM859oQQ020" target="_blank" rel="noopener">This</a> chart/image helps with some of the ideas above. E.g., #16 noting multiple scales and levels interacting, reminding me of Mascolo's quote above.</span></span></span> <span dir="ltr"><span class="_3l3x"><span>Although #15 claims that "complex systems are often nested hierarchies." Which of course depends on what we mean by nested hierarchies. If by that it means fractals with scale-invariance, then that per above is rare, not often. Also note that the chart is similar to Edwards' chart of the categories of lenses.</span></span></span></p>
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<p><span dir="ltr"><span class="_3l3x"><span><img class="spotlight" alt="No photo description available." style="width: 356px; height: 400px;" src="https://scontent-atl3-1.xx.fbcdn.net/v/t1.0-9/67788807_2280753902022661_2439640109591560192_n.jpg?_nc_cat=104&_nc_oc=AQmdn7bRe61CeehETUyaATKaW6NNvai-_Hxauk_tMLfbzl30aPB58s6CbsDsPxUU6AxkuXNl9zBeXNahopiNXGzR&_nc_ht=scontent-atl3-1.xx&oh=e3dfd818d38ce3168f0ffb27c6e4707f&oe=5E069907"/></span></span></span></p>
<p></p> This morning I was re-reading…tag:integralpostmetaphysics.ning.com,2019-08-31:5301756:Comment:773092019-08-31T13:52:39.589ZEdward theurj Bergehttp://integralpostmetaphysics.ning.com/profile/theurj
<p><span dir="ltr"><span class="_3l3x _1n4g"><span>This morning I was re-reading a <a href="https://wiki.p2pfoundation.net/Collective_Enlightenment_Through_Postmetaphysical_Eyes?fbclid=IwAR04NjIPjTF3s7iRmPEoEoK-p45H4UIOdE-NI0YUAlLzQLTaiJWA0Sqk_wU" rel="noopener" target="_blank">paper</a> I co-authored with Michel Bauwens, which reminded me of the above discussion. From footnote 2 of that article:<br></br><br></br>Bryant (2011b) discusses how Bhaskar sees the difference between the transcendent and…</span></span></span></p>
<p><span dir="ltr"><span class="_3l3x _1n4g"><span>This morning I was re-reading a <a href="https://wiki.p2pfoundation.net/Collective_Enlightenment_Through_Postmetaphysical_Eyes?fbclid=IwAR04NjIPjTF3s7iRmPEoEoK-p45H4UIOdE-NI0YUAlLzQLTaiJWA0Sqk_wU" target="_blank" rel="noopener">paper</a> I co-authored with Michel Bauwens, which reminded me of the above discussion. From footnote 2 of that article:<br/><br/>Bryant (2011b) discusses how Bhaskar sees the difference between the transcendent and transcendental. The former assu</span><span>mes a metaphysical foundation for knowledge as described above. Transcendental deduction bypasses such a framing by speculating on what virtual preconditions must be supposed for knowledge to be possible. The virtual by this definition is multiple and immanent without any need of a transcendent, metaphysical underpinning. Bryant (2008) explores this in depth in another book about Deleuze.<br/><br/>Nobuhara (1998) asserts that for Hartshorne relative (r) terms are the basis of absolute (a) terms, noting: "As the concrete includes and exceeds the abstract." The ever-changing relative domain includes within itself the abstract absolute. He defines the absolute as supremely relative, or surrelative.<br/><br/>Another way of approaching the asymmetrical relationship between the relative and the absolute is through basic categories and image schema as elucidated by Lakoff (1999). Recall that these prototypes are in the middle of classical categorical hierarchies, between the most general and the most particular. Basic categories are the most concrete way we have of relating to and operating within the environment. Thus both the more particular and more general categories are more abstract. And yet our usual way of thinking is that the more particular the category the more concrete or relative the object it represents is and vice versa.<br/><br/>Which is indeed related to the absolute being asymmetrically dependent on the relative, if by relative we mean those concrete image schema which are the basis of more abstract derivations. It's easy to confuse them because our 'common sense' associates the more concrete objects of the world with the most particular objects on our constructed hierarchies; the same for the most abstract and ephemeral of thoughts, which do not seem physical or material. And yet these hierarchies are not constructed that way, instead being from the middle up and down via image schema and basic categories.<br/><br/>Such things are unconscious and not readily apparent. So of course we can 'reason' from both the bottom-up and top-down in such hierarchies if we associate the relative with the most particular and the absolute with the most general or abstract. But we do so from the most concrete of image schema, the actual relative, while the top and bottom of the usual, classical hierarchy are the most abstract.</span></span></span></p> Also now an Integral World ar…tag:integralpostmetaphysics.ning.com,2019-08-31:5301756:Comment:773082019-08-31T13:51:26.892ZEdward theurj Bergehttp://integralpostmetaphysics.ning.com/profile/theurj
<p>Also now an Integral World article <a href="http://integralworld.net/berge11.html?fbclid=IwAR2ZZAjDKy3nUJFdgqQbrjVjZgeJql6YAdr8wOa7cnh81IeAi5eNRL46_HA" target="_blank" rel="noopener">here</a>.</p>
<p>Also now an Integral World article <a href="http://integralworld.net/berge11.html?fbclid=IwAR2ZZAjDKy3nUJFdgqQbrjVjZgeJql6YAdr8wOa7cnh81IeAi5eNRL46_HA" target="_blank" rel="noopener">here</a>.</p> I also reminded of this Masco…tag:integralpostmetaphysics.ning.com,2019-08-31:5301756:Comment:771132019-08-31T13:50:07.794ZEdward theurj Bergehttp://integralpostmetaphysics.ning.com/profile/theurj
<p><span dir="ltr"><span class="_3l3x _1n4g"><span>I also reminded of <a href="https://www.researchgate.net/publication/263752474_The_Concept_of_Domain_in_Developmental_Analyses_of_Hierarchical_Complexity?fbclid=IwAR19nfCdl37ed-ewN2qMNtN3gCyYeNgOjiOjTs6t_LovEOCf3qr5x2xzIlE" rel="noopener" target="_blank">this</a> Mascolo quote:<br></br><br></br>"Although it is possible to identify particular tasks and activities that operate within particular domains of thinking, feeling, or acting in everyday life,…</span></span></span></p>
<p><span dir="ltr"><span class="_3l3x _1n4g"><span>I also reminded of <a href="https://www.researchgate.net/publication/263752474_The_Concept_of_Domain_in_Developmental_Analyses_of_Hierarchical_Complexity?fbclid=IwAR19nfCdl37ed-ewN2qMNtN3gCyYeNgOjiOjTs6t_LovEOCf3qr5x2xzIlE" target="_blank" rel="noopener">this</a> Mascolo quote:<br/><br/>"Although it is possible to identify particular tasks and activities that operate within particular domains of thinking, feeling, or acting in everyday life, most tasks involve an integration of multiple task domains. […] Higher-order skills emerge from the constructive differentiation and i</span><span>nter-coordination of skill elements from diverse task domains. […] Viewed in this way, it becomes clear that development takes place in a multidirectional web of pathways (Fischer and Bidell, 2006) rather than a unidirectional ladder. […] Developing skills do not move in a ﬁxed order of steps in a single direction, but they develop in multiple directions along multiple strands that weave in and out of each other in ontogenesis, the developmental history of the person (or other organism)" (336-37).</span></span></span></p> I responded:I did mention mul…tag:integralpostmetaphysics.ning.com,2019-08-31:5301756:Comment:771122019-08-31T13:48:57.532ZEdward theurj Bergehttp://integralpostmetaphysics.ning.com/profile/theurj
<p><span dir="ltr"><span class="_3l3x _1n4g"><span>I responded:<br></br><br></br>I did mention multi-fractals to the Yahoo Adult Development group. E.g., from this [1] article:</span><span><br></br><br></br>"Multifractals are more highly advanced mathematical structures: fractals of fractals. They arise from fractals 'interwoven' with each other in an appropriate manner and in appropriate proportions. Multifractals are not simply the sum of fractals and cannot be divided to return back to their original…</span></span></span></p>
<p><span dir="ltr"><span class="_3l3x _1n4g"><span>I responded:<br/><br/>I did mention multi-fractals to the Yahoo Adult Development group. E.g., from this [1] article:</span><span><br/><br/>"Multifractals are more highly advanced mathematical structures: fractals of fractals. They arise from fractals 'interwoven' with each other in an appropriate manner and in appropriate proportions. Multifractals are not simply the sum of fractals and cannot be divided to return back to their original components, because the way they weave is fractal in nature. The result is that in order to see a structure similar to the original, different portions of a multifractal need to expand at different rates. A multifractal is therefore non-linear in nature."<br/><br/>And from this paper[2]:<br/><br/>"Self-organized criticality (SOC) purports to build multi-scaled structures out of local interactions. Evidence of scaling in various domains of biology may be more generally understood to reflect multiplicative interactions weaving together many disparate scales. The self-similarity of power-law scaling entails homogeneity: fluctuations distribute themselves similarly across many spatial and temporal scales. However, this apparent homogeneity can be misleading, especially as it spans more scales. Reducing biological processes to one power-law relationship neglects rich cascade dynamics. We review recent research into multifractality in executive-function cognitive tasks and propose that scaling reflects not criticality but instead interactions across multiple scales and among fluctuations of multiple sizes."<br/><br/>Some of the references I've provided show that cognitive functioning, and natural phenomenon generally, operate via multifractal cascades, not linear, repeated, monofractal similarities. The latter are an imposition created by formal mathematics under the guise of ideal Platonic forms and/or ideal Aristotelian rules and categories. As much is admitted in this MHC paper (pp. 113-15) [3]. Even advanced maths [4] operate via such cascades and not formal necessary and sufficient conditions that fit into tidy, reiterated sets. It seems to me that any model of complexity should be based on how dynamic systems actually operate rather than trying to fit them into a formal ideal.<br/><br/>The crux of the issue [5]:<br/><br/>"Hierarchical organization is a cornerstone of complexity and multifractality constitutes its central quantifying concept. For model uniform cascades the corresponding singularity spectrum are symmetrical while those extracted from empirical data are often asymmetric."<br/><br/>[1] <a target="_blank" href="https://www.sciencedaily.com/releases/2016/01/160121110913.htm?fbclid=IwAR01K3YoHxkBcfbfdr6sencuwEWG7UXXo7lssDLwqVgle8ylVmX4jGWadsE" rel="nofollow noopener">https://www.sciencedaily.com/rel.../2016/01/160121110913.htm</a><br/>[2] <a target="_blank" href="https://www.frontiersin.org/articles/10.3389/fphys.2012.00102/full?fbclid=IwAR0Nq4DW936K3eitzI69DvabvHuOvXAjReH6qPBTBUG5AoQTpOEHBF5j28A" rel="nofollow noopener">https://www.frontiersin.org/.../10.../fphys.2012.00102/full</a><br/>[3] <a target="_blank" href="https://l.facebook.com/l.php?u=https%3A%2F%2Fwww.dareassociation.org%2Fdocuments%2FGWOF_A_330277%2520Introduction.pdf%3Ffbclid%3DIwAR2b9mj9lKlco6uVc_kvRWwxw8xRmbGBX5wrVGRhIBE58ioanhiXhtyZKuY&h=AT3_ICyaBlwWDCok1b73byTZ6icgkbyeD0JuxJxPGruJDPJ68cNAfQZ9cKgjk-VnM2b2ELa-S-KKEaiSQXF3XOmvTn5Eaij0k0B8uYrHBI-fz8WHjm7Dx6c_C_vPaym4YQSwzpBLrovt5jDnnaP5" rel="nofollow noopener">https://www.dareassociation.org/.../GWOF_A_330277...</a><br/>[4] <a target="_blank" href="https://en.wikipedia.org/wiki/Multifractal_system?fbclid=IwAR01K3YoHxkBcfbfdr6sencuwEWG7UXXo7lssDLwqVgle8ylVmX4jGWadsE" rel="nofollow noopener">https://en.wikipedia.org/wiki/Multifractal_system</a><br/>[5] <a target="_blank" href="https://l.facebook.com/l.php?u=https%3A%2F%2Farxiv.org%2Fpdf%2F1503.02405.pdf%3Ffbclid%3DIwAR3LshH4Pul8RtJlLK9KviOXBvrD_Shr5lXdBcOqOXit5on3M2jDu8aVeJw&h=AT0yiPr0kofLEiOeLrCMAZW8Mdbdk1jotopiQbMkZhYtqetRMuMCpa-d3tT3pVg8O1ArrDKSY7TUNfiqjjBo4uid1bRLdiBdriUYWz9VXiooEPWlSYwBD_Xj_5vMHA4n6fXKup4-w0oMCRL1vv02" rel="nofollow noopener">https://arxiv.org/pdf/1503.02405.pdf</a></span></span></span></p>
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