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The aim of this work is the analysis of multifractals in the context of Mohamed El Naschie’s ϵ(∞) Cantorian space–time applied to cosmology. As starting point we consider the results of the first author of the present paper describing scaling rules in nature, R(N)=(h/mnc)Nϕ. Then, we use multifractal analysis to show that the result, already develo...
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Agro-meteorological quantities are often in the form of time series, and knowledge about their temporal scaling properties is fundamental for transferring locally measured fluctuations to larger scales and vice versa. However, the scaling analysis of these quantities is complicated due to the presence of localized trends and nonstationarities. The...
Objective.The physiological activity of the heart is controlled and modulated mostly by the parasympathetic and sympathetic nervous systems. Heart rate variability (HRV) analysis is therefore used to observe fluctuations that reflect changes in the activity in these two branches. Knowing that acceleration and deceleration patterns in heart rate flu...
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Multifractal analysis, which mainly consists in estimating scaling exponents, has become a popular tool for empirical data analysis. Although widely used in different applications, the statistical performance and the reliability of the estimation procedures are still poorly known. Notably, little is known about confidence intervals, though they are...
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... Already with the eventual link of fractals with physical/natural phenomena, and also to the special types studied in the present work, there are special structures widely applied in the description of natural objects such as Moran types, self-similar, and generally iterated function systems which may be observed in clusters such as the passage from cosmological to nuclear objects where a form of hierarchy is observed, see [23][24][25][26]. ...
The concept of dimension is an important task in geometry. It permits a description of the growth process of objects. It may be seen as an invariant measure
characterizing the object. Fractal dimensions are a kind of invariants permitting essentially to describe the irregularity hidden in irregular objects, by providing a suitable growth law. Among fractal geometrical objects, Moran’s types play an important role in explaining many situations, in pure mathematics as the general context of Cantor’s, and in applied physics as a suitable context for studying scaling laws. In the present paper, some non-regular homogeneous Moran measures are investigated, by establishing some new sufficient conditions permitting an explicit computation of the relative multifractal dimensions of the level sets for which the classical formulation does not hold. Besides, the mutual singularity of the relative multifractal measures for the homogeneous Moran case with different multifractal dimensions is investigated. This is very important, as in quasi-all existing situations, the validity of the multifractal formalism passes through the equality of the multifractal Hausdorff dimension with the packing one.
... In this paper, we show how the new computational tools can be used to calculate the exact infinitesimal values of area of Sierpinski's carpet and volume of Menger's sponge. Calculation of numerical characteristics of fractal objects of this kind is particularly important in the context of E-Infinity theory (see [1,2,4,3,7,8,9,10,11,23]). ...
... Thus, this fact being obvious for finite values of n holds for infinite values of n, too. Finally, the importance of the possibility to have this kind of quantitative characteristics for E-Infinity theory (see [1,2,4,3,7,8,9,10,11,23]) has been emphasized in the paper. ...
Very often traditional approaches studying dynamics of self-similarity
processes are not able to give their quantitative characteristics at infinity
and, as a consequence, use limits to overcome this difficulty. For example, it
is well know that the limit area of Sierpinski's carpet and volume of Menger's
sponge are equal to zero. It is shown in this paper that recently introduced
infinite and infinitesimal numbers allow us to use exact expressions instead of
limits and to calculate exact infinitesimal values of areas and volumes at
various points at infinity even if the chosen moment of the observation is
infinitely faraway on the time axis from the starting point. It is interesting
that traditional results that can be obtained without the usage of infinite and
infinitesimal numbers can be produced just as finite approximations of the new
ones.
... In some cases, the distribution exhibits multi-scaling, with distinct and independent exponents governing the several moments of the distribution, a feature usually named as multifractality [3,4]. Multifractal distributions have been identified in a very broad range of scenarios and length scales including turbulence and chaos [5,6], high-energy physics [7], metal–insulator transition [8,9], quasi-crystals [10,11] and traffic flows [12], among others. Simple mathematical models which effectively reproduce the main features of the distribution of collective stochastic quantities without reference to the microscopic dynamics are of unquestionable value in clarifying the general mechanism behind the emergence of scale-free distributions. ...
We investigate non-Gaussian statistical properties of stationary stochastic signals generated by an analog circuit that simulates a random multiplicative process with weak additive noise. The random noises are originated by thermal shot noise and avalanche processes, while the multiplicative process is generated by a fully analog circuit. The resulting signal describes stochastic time series of current interest in several areas such as turbulence, finance, biology and environment, which exhibit power-law distributions. Specifically, we study the correlation properties of the signal by employing a detrended fluctuation analysis and explore its multifractal nature. The singularity spectrum is obtained and analyzed as a function of the control circuit parameter that tunes the asymptotic power-law form of the probability distribution function.
Atmospheric flows exhibit long-range spatiotemporal correlations manifested as the fractal geometry to the global cloud cover pattern concomitant with inverse power-law form for power spectra
of temporal fluctuations on all space-time scales ranging from turbulence (centimetres-seconds) to climate (kilometres-years). Long-range spatiotemporal correlations
are ubiquitous to dynamical systems in nature and are identified as signatures of self-organized criticality. Standard models in meteorological theory cannot explain satisfactorily the observed self-organized criticality in atmospheric flows. Mathematical models for simulation and prediction of atmospheric flows are nonlinear and do not possess analytical solutions. Finite precision
computer realizations of nonlinear models give unrealistic solutions because of deterministic chaos, a direct consequence of round-off error
growth in iterative numerical computations. Recent studies show that round-off error doubles on an average for each iteration of iterative computations. Round-off error propagates to the mainstream computation and gives unrealistic solutions in numerical weather prediction (NWP) and climate models, which incorporate thousands of iterative computations in long-term numerical integration schemes. A general systems theory
model for atmospheric flows developed by the author predicts the observed self-organized criticality as intrinsic to quantumlike chaos
in flow dynamics. The model provides universal quantification for self-organized criticality in terms of the golden ratio τ (≈1.618). Model predictions are in agreement with a majority of observed spectra of time series of several standard meteorological and climatological data sets representative of disparate climatic regimes. Universal spectrum for natural climate variability
rules out linear trends. Man-made greenhouse gas related atmospheric warming
would result in intensification of natural climate variability, seen immediately in high-frequency fluctuations such as QBO
and ENSO and even shorter timescales. Model concepts and results of analyses are discussed with reference to possible prediction of climate change. Model concepts, if correct, rule out unambiguously, linear trends in climate. Climate change
will only be manifested as increase or decrease in the natural variability. However, more stringent tests of model concepts and predictions are required before applications to such an important issue as climate change. The cell dynamical system model for atmospheric flows is a general systems theory
applicable in general to all dynamical systems in other fields of science, such as, number theory, biology, physics and botany.
A straightforward simple proof is given that dark energy is the natural conse-quence of a quantum disentanglement physical process. Thus while the ordinary energy density of the cosmos is equal to half that of Hardy’s quantum probability of Entanglement i.e. where , the density of cosmic dark energy is consequently one minus divided by two i.e. . This result is in full agreement with all the numerous previous theoretical predictions as well as being in remarkable agreement with the overwhelming majority of cosmic accurate measurements and observations.
The topological speed of light which may be used to compute the density of ordinary energy and dark energy of the cosmos is replaced by dimensionless quantity taken from Special Relativity. The said quantity may be interpreted as akin to time dilation ergo a notion topologically equivalent to the speed of the passing of time or the difference of elapsed time between two events in Einstein’s Relativity Theory. This results via Newton’s kinetic energy into the well-known observationally confirmed and accurately measured 4.5 and 95.5 percent of ordinary and dark Cosmic Energy density respectively.
The four-dimensional character of Einstein’s spacetime is generally accepted in mainstream physics as beyond reasonable doubt correct. However the real problem is when we require scale invariance and that this spacetime be four-dimensional on all scales. It is true that on our classical scale, the 4D decouples into 3D plus one time dimension and that on very large scale only the curvature of spacetime becomes noticeable. However the critical problem is that such spacetime must remain 4D no matter how small the scale we are probing is. This is something of crucial importance for quantum physics. The present work addresses this basic, natural and logical requirement and shows how many contradictory results and shortcomings of relativity and quantum gravity could be eliminated when we “complete” Einstein’s spacetime in such a geometrical gauge invariant way. Concurrently the work serves also as a review of the vast Literature on E-Infinity theory used here.
Transfer matrix method gives information about underlying dynamics of a multifractal. In the present studies, transfer matrix method is applied to multifractal properties of a cherenkov image from which probabilities of electromagnetic components are obtained.
In this paper, we introduce vector-valued non-separable higher-dimensional wavelet packets with an arbitrary integer dilation factor. An approach for constructing vector-valued higher-dimensional wavelet packet bases is proposed. Their characteristics are investigated by means of harmonic analysis method, matrix theory and operator theory, and three orthogonality formulas concerning the wavelet packets are presented. Orthogonal decomposition relation formulas of the space L2(Rn)p are derived by designing a series of subspaces of the vector-valued wavelet packets. Moreover, several orthonormal wavelet packet bases of L2(Rn)p are constructed from the wavelet packets. Relation to some physical theories such as E-infinity theory and multifractal theory is also discussed.
