This volume delineates the use of fractal patterns and measures of fractal dimensions in describing and understanding general aspects of biology, particularly human physiology. After describing the ubiquitous nature of fractal phenomena, the authors give examples of the properties of fractals in space and time. Proceeding from mathematical definitions, they develop detailed practical methods for assessing the fractal characteristics of wave forms varying with time, tissue density variation, and surface irregularities. Most importantly, the authors show how fractal variation defines internal spatial or temporal correlations within the fractal system or object. Simple, recursively applied rules can give rise to complex biological structures by a variety of methods. This suggests that genetic rules govern the general structuring of an organism, while rules implied by interactions at the biochemical, cellular, and tissue levels govern ontogenic development and therefore play the major role in the growth of an organism. Chaos, or non-linear dynamics, is introduced as a stimulating way to examine biological behavior at the cellular and whole animal levels, even though proof of the chaotic nature of normal physiologic events is as yet meager. The later chapters give sets of examples of structural and behavioral fractal phenomena in nerve and muscle, in the cardiovascular and respiratory systems and in growth processes. Why molecular interactions and complex systems give rise to fractals is explored and related to the ideas of emergent properties of systems operating at high levels of complexity.

Fractal Physiology (Methods in Physiology)

Science & Math

Bassingthwaighte, James B.

Liebovitch, Larry S.

West, Bruce J.

Biological Sciences

American Physiological Society / Oxford

Fractional Calculus View of Complexity: Tomorrow’s Science

This book is not a text devoted to a pedagogical presentation of a specialized topic nor is it a monograph focused on the author's area of research. It accomplishes both these things while providing a rationale for why the reader ought to be interested in learning about fractional calculus. This book is for researchers who has heard about many of these scientifically exotic activities, but could not see how they fit into their own scientific interests, or how they could be made compatible with the way they understand science. It is also for beginners who have not yet decided where their scientific talents could be most productively applied. The book provides insight into the long-term direction of science and show how to develop the skills necessary to successfully do research in the twenty-first century.

CRC-Press

One of my favorite quotes is from a letter of Charles Darwin (1887): "I have long discovered that geologists never read each other's works, and that the only object in writing a book is proof of earnestness, and that you do not form your opinions without undergoing labour of some kind. " It is not clear if this private opinion of Darwin was one that he held to be absolutely true, or was one of those opinions that, as with most of us, coincides with our "bad days," but is replaced with a more optimistic view on our "good days. " I hold the sense of the statement to be true in general, but not with regard to scientists never reading each other's work. Even if that were true however, the present essay. would still have been written as a proof of earnestness. This essay outlines my personal view of how nonlinear mathematics may be of value in formulating models outside the physical sciences. This perspective has developed over a number of years during which time I have repeatedly been amazed at how an "accepted" model would fail to faithfully characterize the full range of avail able data because of its implicit or explicit dependence on linear concepts. This essay is intended to demonstrate how linear ideas have come to dominate and therefore limit a scientist's ability to understand any given class of phenomena.

An Essay on the Importance of Being Nonlinear (Lecture Notes in Biomathematics)

Mathematics

Springer, Verlag

You can never step in the same river twice, goes the old adage of philosophy. An observation on the transitory nature of fluids in motion, this saying also describes the endless variations researchers face when studying human movement. Understanding these biodynamics-why the wirewalker doesn't fall-requires a grasp of the constant fluctuations and fine tunings which maintain balance in the complex, fluid system of human locomotion. Taking a comprehensive approach to the phenomenon of locomotion, Biodynamics: Why the Wirewalker Doesn't Fall integrates physical laws and principles with concepts of fractals, chaos, and randomness. In so doing, it formulates a description of both the large-scale, smooth aspects of locomotion and the more minute, randomized mechanisms of this physiological process. Ideal for beginners in this subject, Biodynamics provides an elegant explanation without assuming the reader's understanding of complex physical principles or mathematical equations. Chapter topics include: * Dimensions, measurement, and scaling * Mechanics and dynamics * Biometrics * Conservation of momentum * Biomechanics * Bioelectricity * Bioenergetics * Fluid mechanics and dynamics * Data analysis * Biostatistics Packed with problem sets, examples, and original line drawings, Biodynamics is an invaluable text for advanced undergraduates, graduate students, and instructors in medicine, biology, physiology, biophysics, and bioengineering.

Biodynamics: Why the Wirewalker Doesn't Fall

Griffin, Lori A.

Wiley-Liss

Complex Webs synthesises modern mathematical developments with a broad range of complex network applications of interest to the engineer and system scientist, presenting the common principles, algorithms, and tools governing network behaviour, dynamics, and complexity. The authors investigate multiple mathematical approaches to inverse power laws and expose the myth of normal statistics to describe natural and man-made networks. Richly illustrated throughout with real-world examples including cell phone use, accessing the Internet, failure of power grids, measures of health and disease, distribution of wealth, and many other familiar phenomena from physiology, bioengineering, biophysics, and informational and social networks, this book makes thought-provoking reading. With explanations of phenomena, diagrams, end-of-chapter problems, and worked examples, it is ideal for advanced undergraduate and graduate students in engineering and the life, social, and physical sciences. It is also a perfect introduction for researchers who are interested in this exciting new way of viewing dynamic networks.

Complex Webs: Anticipating the Improbable

Grigolini, Paolo

Cambridge University Press

This book is the first of its kind on fractional calculus (FC), dedicated to advocating for FC in STEM education and research.Fractional calculus is increasingly used today, but there remains a core population of skeptics regarding the utility of this "new" calculus. This book is intended for those who are skeptical about the need for fractional calculus to describe dynamic complex networks and must be convinced of its use on a case-by-case basis. It is a one-stop resource to rapidly read and replace the appropriate skepticism with new knowledge. It offers compelling reasons from the perspectives of the physical, social, and life sciences as to why fractional calculus is needed when addressing the complexity of an underlying STEM phenomenon. The six chapters are accompanied by useful and essential appendices and chapter-end references. Each includes new (fractional-order) ways of thinking about statistics, complexity dynamics, and what constitutes a solution to a complexity science problem.The book will appeal to students and researchers in all STEM-related fields, such as engineering, physics, biology and biomedicine, climate change, big data, and machine learning. It is also suitable for general readers interested in these fields.

Fractional Calculus for Skeptics I: The Fractal Paradigm (Fractional Order Thinking in Exploring the Frontiers of STEM)

Chen, YangQuan

On the Fractal Language of Medicine (Fractional Order Thinking in Exploring the Frontiers of STEM)

Mutch, W. Alan C.

This text describes the statistcal behavior of complex systems and shows how the fractional calculus can be used to model the behavior. The discussion emphasizes physical phenomena whose evolution is best described using the fractional calculus, such as systems with long-range spatial interactions or long-time memory. The book gives general strategies for understanding wave propagation through random media, the nonlinear response of complex materials, and the fluctuations of heat transport in heterogeneous materials.

Physics of Fractal Operators (Institute for Nonlinear Science)

Bologna, Mauro

Reference

Springer

Fractional Calculus View of Complexity: Tomorrow's Science

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