Fractional Calculus for Skeptics II

Quantifying Roughness

Bruce J. West, YangQuan Chen

• Publisher: CRC Press
• Year: 2026
• eBook ISBN: 9781040759325
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

This second volume of the multi-volume Fractional Calculus for Skeptics explores the concept of "roughness" and its applications in complex systems. It demonstrates how fractional calculus offers valuable insights into modeling real-world phenomena that traditional calculus struggles to address.

Building on the first volume's foundational work advocating a fractal/fractional calculus paradigm shift, this volume offers an in-depth exploration of the concept of "roughness," drawing parallels to what Benoît B. Mandelbrot originally intended to call the "Science of Roughness." The book introduces the novel concept of "body roughness" and demonstrates its practical applications through two case studies of "roughness-informed rehabilitation therapy" with closed-loop control systems. One study involves non-invasive, 40-Hz gamma-frequency sensory stimulation for Alzheimer's disease, and the other involves biologically variable ventilation using roughness-informed, closed-loop control. The four chapters are complemented by valuable appendices and essential chapter-end references, as well as insightful "take-home messages."

This book will appeal to students, researchers, and professionals in STEM fields, including engineering, physics, biology, biomedicine, control engineering, rehabilitation, big data, and machine learning.

Table of contents

• Cover Page
• Half-Title Page
• Series Page
• Title Page
• Copyright Page
• Dedication Page
• Contents
• Preface
• Author Bios
• List of Abbreviations
• Chapter 1 ▪ The Science of Roughness
• Chapter 2 ▪ The Rough Body Concepts
• Chapter 3 ▪ The Mathematics of Roughness
• Chapter 4 ▪ Roughness-Informed Rehabilitation Therapy
• Chapter 5 ▪ Rough Remarks on Complexity and Roughness
• Appendix A ▪ Fractal, Fractional and Other Calculi
• Appendix B ▪ Some Roughness Quantification Techniques
• Appendix C ▪ Fractal-Order Generalized Principle of Self-Support (FOGPSS)
• Index