Bruce J. West Libri

This text explores how both deterministic and random fractal phenomena evolve over time through the lens of fractional calculus. It aims to pinpoint the characteristics of complex physical phenomena that necessitate fractional derivatives or integrals to accurately describe temporal changes. The focus is on physical phenomena whose evolution is best captured by fractional calculus, particularly systems exhibiting long-range spatial interactions or long-time memory. Traditional analytic function theory often falls short in modeling such complexities; thus, the text illustrates how less familiar functions, like fractals, can effectively serve as models. Notably, fractal functions, such as the Weierstrass function—which lacks a traditional derivative—can be represented through fractional derivatives and framed as solutions to fractional differential equations. The text also discusses how conventional differential equation-solving techniques, including Fourier and Laplace transforms and Green's functions, can be adapted to accommodate fractional derivatives. Furthermore, it outlines a comprehensive strategy for understanding wave propagation in random media, the nonlinear responses of complex materials, and the transport fluctuations in heterogeneous materials, while explaining the limitations of historical techniques as phenomena grow increasingly intricate.

In life, we often face unavoidable complexities in terms of our ability to understand or influence outcomes. Some questions which arise due to these complexities are: Why can't the future be made certain? Why do the some people or events always end up at the center of controversy? Why do only a select few get ahead of their peers? Each question pertains to three central elements of complexities and these elements are: uncertainty, inequality and unfairness. Simplifying Complexity explains the scientific study of complex cognitive networks, as well as the methods scientists use to parse difficult problems into manageable pieces. Readers are introduced to scientific methodology and thought processes, followed by a discourse on perspectives on the three elements of complexity through concepts such as normal and non-normal statistics, scaling and complexity management. Simplifying Complexity combines basic cognitive science and scientific philosophy for both advanced students (in the fields of sociology, cognitive science, complex networks and change management) and for general readers looking for a more scientific guide to understanding and managing the nature of change in a complex world.

Complexity increases with increasing system size in everything from organisms to organizations. The nonlinear dependence of a system’s functionality on its size, by means of an allometry relation, is argued to be a consequence of their joint dependency on complexity (information). In turn, complexity is proven to be the source of allometry and to provide a new kind of force entailed by a system‘s information gradient. Based on first principles, the scaling behavior of the probability density function is determined by the exact solution to a set of fractional differential equations. The resulting lowest order moments in system size and functionality gives rise to the empirical allometry relations. Taking examples from various topics in nature, the book is of interest to researchers in applied mathematics, as well as, investigators in the natural, social, physical and life sciences. Contents Complexity Empirical allometry Statistics, scaling and simulation Allometry theories Strange kinetics Fractional probability calculus