This text describes how fractal phenomena, both deterministic and random, change over time, using the fractional calculus. The intent is to identify those characteristics of complex physical phenomena that require fractional derivatives or fractional integrals to describe how the process changes over time. The discussion emphasizes the properties of physical phenomena whose evolution is best described using the fractional calculus, such as systems with long-range spatial interactions or long-time memory. In many cases, classic analytic function theory cannot serve for modeling complex phenomena; "Physics of Fractal Operators" shows how classes of less familiar functions, such as fractals, can serve as useful models in such cases. Because fractal functions, such as the Weierstrass function (long known not to have a derivative), do in fact have fractional derivatives, they can be cast as solutions to fractional differential equations. The traditional techniques for solving differential equations, including Fourier and Laplace transforms as well as Green's functions, can be generalized to fractional derivatives. Physics of Fractal Operators addresses a general strategy for understanding wave propagation through random media, the nonlinear response of complex materials, and the fluctuations of various forms of transport in heterogeneous materials. This strategy builds on traditional approaches and explains why the historical techniques fail as phenomena become more and more complicated.

In Chapter One we review the foundations of statistieal physies and frac tal functions. Our purpose is to demonstrate the limitations of Hamilton's equations of motion for providing a dynamical basis for the statistics of complex phenomena. The fractal functions are intended as possible models of certain complex phenomena; physical.systems that have long-time mem ory and/or long-range spatial interactions. Since fractal functions are non differentiable, those phenomena described by such functions do not have dif ferential equations of motion, but may have fractional-differential equations of motion. We argue that the traditional justification of statistieal mechan ics relies on aseparation between microscopic and macroscopie time scales. When this separation exists traditional statistieal physics results. When the microscopic time scales diverge and overlap with the macroscopie time scales, classieal statistieal mechanics is not applicable to the phenomenon described. In fact, it is shown that rather than the stochastic differential equations of Langevin describing such things as Brownian motion, we ob tain fractional differential equations driven by stochastic processes."

Former Arizona Ranger, Sam Allen is trying to convince himself he can be content retiring to a rocking chair but destiny has other ideas. A Son's Blood is the story of Sam's pursuit of the Mexican killers of his son. An educated man, the product of southern aristocracy, Sam was a commander in the Confederate Army, and had returned from the war to find his parents dead and the plantation taken by carpet baggers. Now a widower, he is surprised to learn that the killers of his son were vaqueros working for his wife's Mexican cousins. The fugitive Mexicans, led by the patrician Alejandro Herrera, are confronted by Apaches led by the fierce warriors, Geronimo and Juh. At the same time, Sam is wounded by an Apache warparty and rescued by Papago warriors. While Alejandro and his men are stripped of everything they own and forced to walk naked across the unrelenting desert, Sam is taken to a Papago village where he is nursed back to health by Shashani, a former lover. The rekindling of their relationship doesn't deter the old Ranger from returning to the chase as soon as he's able. Captured by the Apache Kid in an unforgettable battle in Cazador Canyon near the Mexico Border, Sam is taken to the El Mundo Rancho, where he is confronted by Alejandro and his powerful father, the Baron of Sonora. The ensuing duel is a not-to-be-missed battle between a strong cocky youth and a wiser more experienced man. What makes A Son's Blood stand alone among adventure stories isn't only that Sam Allen is older than the typical western hero. It's the author's intimate relationship to the desert, a place his family settled a century-and-a-half ago. Names, descriptions, characters, even judicious use of Papago, Apache and Spanish languages gives one a personal feeling of old Arizona. In the words of the Tombstone Epitaph newspaper, "The characters and events of the Arizona border country come alive as you ride the vengeance trail with a veteran Arizona Ranger . . . you're behind-the-scenes at the O.K. Corral shootout, in the desert with Geronimo and the last of the wild free-roaming Apaches-it's a book that puts you into the heart of Old Arizona in a personal way."