This book concerns the development of a theory of complex phenomena; using such concepts as fractals; chaos; and fractional derivatives; but; most important; the idea of an allometric control process is developed. In summary the theory attempts to explain why the distribution in the intensity of wars is the same as the relative frequency of the number of words used in languages and the number of species evolved over time from one or a few remote ancestors. The theory also describes the similarity in the variability of the number of births to teens in Texas to the number of sexual partners in homosexual liaisons. The data in both of the aforementioned categories are shown to have long-term memory; and it is this memory that also gives rise to inverse power laws in such physiological phenomena as the interbeat interval distribution of the human heart; the interstride interval distribution in the human gait; and memory in DNA sequences.
A tale of uncertain knowledge; systems perspective of complexity; fractal random processes; modelling random time series; on chance and necessity; averages and expected values; variances and correlations; power laws, dimensional analysis and scaling; Bernoulli and memory; simple random walks and diffusion; scaling, geometry and memory; inverse power-law distributions; fractal random walks; Levy distributions and scaling; inverse power-law correlations; ensembles, signals and noise; spectra and noise; generalized Weierstrass function; how certain can you be?; scaling in aggregated data sets; linear regression; auto regressive integrated moving average (ARIMA); fractional differences and inverse power laws; haphazard human heart; willy-nilly walking; gamboling genes; scaling sexual schemes; the nonlinear perspective; dynamics and geometry; mapping and chaos; memory and intermittency; intermittency and Levy statistics; strange dynamics; ART and teen births; Brown versus Weierstrass; fractional operations; fractional calculus model of Creep; fractional relaxation; fractional Langevin equation; fractional diffusion equation; the end of our tale.