Abstracts Details

Recent studies of DNA sequence of letters A, C, G and T exhibit the inverse power law form frequency spectrum. Inverse power-law form of the power spectra of fractal spacetime fluctuations is generic to the dynamical systems in nature and is identified as self-organized criticality. In this study it is shown that the power spectra of the frequency distributions of bases A, C, G, T in the Human chromosome 1 DNA exhibit self-organized criticality. DNA is a quasicrystal possessing maximum packing efficiency in a hierarchy of spirals or loops. Self-organized criticality implies that non-coding *introns* may not be redundant, but serve to organize the effective functioning of the coding *exons* in the DNA molecule as a complete unit.

Dow Jones Index time series exhibit irregular or fractal fluctuations on all time scales from days, months to years. The apparently irregular (nonlinear) fluctuations are selfsimilar as exhibited in inverse power law form for power spectra of temporal fluctuations. Inverse power law form for power spectra of fractal fluctuations in space or time is generic to all dynamical systems in nature and is identified as self-organized criticality. Selfsimilarity implies long-range space-time correlations or non-local connections. It is important to quantify the total pattern of fractal fluctuations for predictability studies, e.g., weather and climate prediction, stock market trends, etc. The author has developed a general systems theory for universal quantification of the observed inverse power law spectra in dynamical systems. The model predictions are as follows.

- The power spectra of fractal fluctuations follow the universal and unique inverse power law form of the statistical normal distribution.
- The nonlocal connections or long-range correlations in space or time exhibited by the fractal fluctuations are signatures of quantum-like chaos in dynamical systems.
- The apparently irregular geometry of the fractal fluctuations forms the component parts of a unified whole precise geometrical pattern of the logarithmic spiral with quasiperiodic Penrose tiling pattern for the internal structure. Conventional power spectral analyses will resolve the logarithmic spiral pattern as an eddy continuum with progressive increase in eddy phase angle.
- Continuous periodogram power spectral analyses of normalised daily, monthly and annual Dow Jones Index for the past 100-years show that the power spectra follow the universal inverse power law form of the statistical normal distribution in agreement with model prediction.

The fractal fluctuations of the non-stationary Dow Jones Index time series therefore exhibit signature of quantum-like chaos on all time scales from days to years.

Continuous periodogram power spectral analyses of *fractal* fluctuations of frequency distributions of bases A, C, G, T in Drosophila DNA show that the power spectra follow the universal inverse power-law form of the statistical normal distribution. Inverse power-law form for power spectra of space-time fluctuations is generic to dynamical systems in nature and is identified as *self-organized criticality*. The author has developed a general systems theory, which provides universal quantification for observed *self-organized criticality* in terms of the statistical normal distribution. The long-range correlations intrinsic to *self-organized criticality* in macro-scale dynamical systems are a signature of quantumlike chaos. The *fractal* fluctuations self-organize to form an overall logarithmic spiral trajectory with the quasiperiodic *Penrose* tiling pattern for the internal structure. Power spectral analysis resolves such a spiral trajectory as an eddy continuum with embedded dominant wavebands. The dominant peak periodicities are functions of the *golden mean*. The observed *fractal* frequency distributions of the Drosophila DNA base sequences exhibit quasicrystalline structure with long-range spatial correlations or *self-organized criticality*. Modification of the DNA base sequence structure at any location may have significant noticeable effects on the function of the DNA molecule as a whole. The presence of non-coding *introns* may not be redundant, but serve to organize the effective functioning of the coding *exons* in the DNA molecule as a complete unit.

Fluid flows such as gases or liquids exhibit space-time fluctuations on all scales extending down to molecular scales. Such broadband continuum fluctuations characterise all dynamical systems in nature and are identified as selfsimilar fractals in the newly emerging multidisciplinary science of *nonlinear dynamics and chaos*. A cell dynamical system model has been developed by the author to quantify the fractal space-time fluctuations of atmospheric flows. The earth's atmosphere consists of a mixture of gases and obeys the gas laws as formulated in the *kinetic theory of gases* developed on probabilistic assumptions in 1859 by the physicist *James Clerk Maxwell*. An alternative theory using the concept of *fractals and chao*s is applied in this paper to derive the fundamental equation of the kinetic theory of ideal gases and the *Maxwell*'s distribution of molecular speeds.

The spacing intervals of adjacent Riemann zeta zeros (nontrivial) exhibit fractal (irregular) fluctuations generic to dynamical systems in nature such as fluid flows, heart beat patterns, stock market price index, etc., and are associated with unpredictability or chaos. The power spectra of such fractal space-time fluctuations exhibit inverse power-law form and signify long-range correlations, identified as *self-organized criticality*. A cell dynamical system model developed by the author for turbulent fluid flows provides a unique quantification for the observed power spectra in terms of the statistical normal distribution, such that the variance represents the statistical probability densities. Such a result that the additive amplitudes of eddies when squared, represent the statistical probabilities is an observed feature of the subatomic dynamics of quantum systems such as an electron or photon. *Self-organized criticality* is therefore a signature of quantum-like chaos in dynamical systems. The model concepts are applicable to all real world (observed) and computed (mathematical model) dynamical systems. Continuous periodogram analyses of the fractal fluctuations of Riemann zeta zero spacing intervals show that the power spectra follow the unique and universal inverse power-law form of the statistical normal distribution. The Riemann zeta zeros therefore exhibit quantum-like chaos, the spacing intervals of the zeros representing the energy (variance) level spacings of quantum-like chaos inherent to dynamical systems in nature. The cell dynamical system model is a general systems theory applicable to dynamical systems of all size scales.

Recent studies by mathematicians and physicists have identified a close association between the distribution of prime numbers and quantum mechanical laws governing the subatomic dynamics of quantum systems such as the electron or the photon. It is now recognised that Cantorian fractal space-time fluctuations characterise dynamical systems of all space-time scales ranging from the microscopic subatomic dynamics to macro-scale turbulent fluid flows such as atmospheric flows. The spacing intervals of adjacent prime numbers also exhibit fractal (irregular) fluctuations generic to dynamical systems in nature. The apparently irregular (chaotic) fractal fluctuations of dynamical systems, however, exhibit self-similar geometrical pattern and are associated with inverse power-law form for the power spectrum. Self-similar fluctuations imply long-range space-time correlations identified as self-organized criticality. A cell dynamical system model for atmospheric flows developed by the author gives the following important results: (a) Self-organized criticality is a signature of quantum-like chaos (b) The observed self-organized criticality is quantified in terms of the universal inverse power-law form of the statistical normal distribution (c) The spectrum of fractal fluctuations is a broadband continuum with embedded dominant eddies. The cell dynamical system model is a general systems theory applicable to all dynamical systems (real world and computed) and the model concepts are applied to derive the following results for the observed association between prime number distribution and quantum-like chaos. (i) Number theoretical concepts are intrinsically related to the quantitative description of dynamical systems. (ii) Continuous periodogram analyses of different sets of adjacent prime number spacing intervals show that the power spectra follow the model predicted universal inverse power-law form of the statistical normal distribution. The prime number distribution therefore exhibits self-organized criticality, which is a signature of quantum-like chaos. (iii) The continuum real number field contains unique structures, namely, prime numbers, which are analogous to the dominant eddies in the eddy continuum in turbulent fluid flows.

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*Can. J. Phys.*, V68, pp. 831-841 (1990).

The complex spaciotemporal patterns of atmospheric flows that result from the cooperative existence of fluctuations ranging in size from millimetres to thousands of kilometres are found to exhibit long-range spacial and temporal correlations. These correlations are manifested as the self-similar fractal geometry of the global cloud cover pattern and the inverse power-law form for the atmospheric eddy energy spectrum. Such long-range spaciotemporal correlations are ubiquitous in extended natural dynamical systems and are signatures of deterministic chaos or self-organized criticality. In this paper, a cell dynamical system model for atmospheric flows is developed by consideration of microscopic domain eddy dynamical processes. This nondeterministic model enables formulation of a simple closed set of governing equations for the prediction and description of observed atmospheric flow structure characteristics as follows. The strange-attractor design of the field of deterministic chaos in atmospheric flows consists of a nested continuum of logarithmic spiral circulations that trace out the quasi-periodic Penrose tiling pattern, identified as the quasi-crystalline structure in condensed matter physics. The atmospheric eddy energy structure follows laws similar to quantum mechanical laws. The apparent wave-particle duality that characterize quantum mechanical laws is attributed to the bimodal phenomenological form of energy display in the bidirectional energy flow that is intrinsic to eddy circulations, e.g., formation of clouds in updrafts and dissipation of clouds in downdrafts that result in the observed discrete cellular geometry of cloud structure.