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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."
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