Dyadic Cantor set and its kinetic and stochastic counterpart

Abstract

Firstly, we propose and investigate a dyadic Cantor set (DCS) and its kinetic counterpart where a generator divides an interval into two equal parts and removes one with probability $(1-p)$. The generator is then applied at each step to all the existing intervals in the case of DCS and to only one interval, picked with probability according to interval size, in the case of kinetic DCS. Secondly, we propose a stochastic DCS in which, unlike the kinetic DCS, the generator divides an interval randomly instead of equally into two parts. Finally, the models are solved analytically; an exact expression for fractal dimension in each case is presented and the relationship between fractal dimension and the corresponding conserved quantity is pointed out. Besides, we show that the interval size distribution function in both variants of DCS exhibits dynamic scaling and we verify it numerically using the idea of data-collapse.Comment: 8 pages, 6 figures, To appear in Chaos, Solitons & Fractal

Evolution through the stochastic dyadic Cantor Set: the uniqueness of mankind in the Universe

“…Both, simplistic but complex, the model accounts the importance of infinite possible paths, temporality with infinite continuous or discrete variables, and infinite choices over the entire set of sufficient iterations to observe each planet and life, with their unique characteristics and particular probabilities. The (S.D.C.S) with temporal and spatial randomness, models the stochastic, self-similar and fractal char-acteristics consistent with those of the stochastic, self-similar and fractal dimensions of the universe [Iovane et al, 2004;Hassan et al, 2014] With a single iteration, the (S.D.C.S) with temporal and spatial randomness manifest the straightforward argument from probability theory, of a simple binary choice with probability p. However, with more than one iteration, a binomial model is not continuous and precise, since p must change on every iteration, because nonlinear chaotic behaviours render long-term prediction and repeatability impossible in general [Kellert,1993]. An absolutely discrete model such as binomial or a single binary choice will not model stochastic, self-similar, fractal, deterministic, probabilistic, chaotic, discrete and continuous behaviours throughout the universe for the probability of finding intelligent life.…”

“…Through the self-similar and fractal properties on each step of iteration (Hassan et al 2014), the algorithm presents necessarily, a dimensional analysis that would also rise to the conclusion that generalities, may rise the expectations to the fact that intelligent life would exist. However, Fig.…”

“…The method can be applied by incorporating any evolutionary history of terrestrial planetary formation in the Universe within a set of stochastic, self-similar and fractal (Hassan et al 2014) dimensions. The precise probability path ( P Life ) for one particular evolutionary history of terrestrial planetary life will be produced through the iterative process of successes.…”

“…The probability path (P Lif e ) starts with any terrestrial planetary formation, evolving until it reaches its particular and unique configuration. Through the self-similar and fractal properties on each step of iteration [Hassan et al, 2014], the algorithm presents necessarily, a dimensional analysis that would rise also to the conclusion that generalities, may rise the expectations to the fact that intelligent life would exist. However, figure 2, illustrates the example of the precise path required, for the expectation of intelligent life appearance.…”

Multi-multifractality, dynamic scaling and neighbourhood statistics in weighted planar stochastic lattice

“…Finding dynamic scaling in any system has always represented progress for researchers as it implies that the phenomena that it represents is self-similar. One of us found such self-similarity in many different processes like the kinetics of aggregation, stochastic Cantor set and in complex network theory [23][24][25].…”