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On becoming familiar with difference equations and their close re
lation to differential equations, I was in hopes that the theory of
difference equations could be brought completely abreast with that
for ordinary differential equations. [HUGH L. TURRITTIN, My
Mathematical Expectations, Springer Lecture Notes 312 (page 10),
1973] A major task of mathematics today is to harmonize the
continuous and the discrete, to include them in one comprehensive
mathematics, and to eliminate obscurity from both. [E. T. BELL, Men
of Mathematics, Simon and Schuster, New York (page 13/14), 1937]
The theory of time scales, which has recently received a lot of
attention, was introduced by Stefan Hilger in his PhD thesis [159]
in 1988 (supervised by Bernd Aulbach) in order to unify continuous
and discrete analysis. This book is an intro duction to the study
of dynamic equations on time scales. Many results concerning
differential equations carryover quite easily to corresponding
results for difference equations, while other results seem to be
completely different in nature from their continuous counterparts.
The study of dynamic equations on time scales reveals such
discrepancies, and helps avoid proving results twice, once for
differential equa tions and once for difference equations. The
general idea is to prove a result for a dynamic equation where the
domain of the unknown function is a so-called time scale, which is
an arbitrary nonempty closed subset of the reals.
The study of dynamic equations on a measure chain (time scale) goes
back to its founder S. Hilger (1988), and is a new area of still
fairly theoretical exploration in mathematics. Motivating the
subject is the notion that dynamic equations on measure chains can
build bridges between continuous and discrete mathematics. Further,
the study of measure chain theory has led to several important
applications, e.g., in the study of insect population models,
neural networks, heat transfer, and epidemic models. Key features
of the book: * Introduction to measure chain theory; discussion of
its usefulness in allowing for the simultaneous development of
differential equations and difference equations without having to
repeat analogous proofs * Many classical formulas or procedures for
differential and difference equations cast in a new light * New
analogues of many of the "special functions" studied * Examination
of the properties of the "exponential function" on time scales,
which can be defined and investigated using a particularly simple
linear equation * Additional topics covered: self-adjoint
equations, linear systems, higher order equations, dynamic
inequalities, and symplectic dynamic systems * Clear, motivated
exposition, beginning with preliminaries and progressing to more
sophisticated text * Ample examples and exercises throughout the
book * Solutions to selected problems Requiring only a first
semester of calculus and linear algebra, Dynamic Equations on Time
Scales may be considered as an interesting approach to differential
equations via exposure to continuous and discrete analysis. This
approach provides an early encounter with many applications in such
areas as biology, physics, and engineering. Parts of the book may
be used in a special topics seminar at the senior undergraduate or
beginning graduate levels. Finally, the work may serve as a
reference to stimulate the development of new kinds of equations
with potentially new applications.
These poems remind us that we are all in the thick of things, the
rich and complicated givens. Moving fluently from subjects as
diverse as the surface of Europa to a tiny spider in a tear of
wallpaper, from Pythagoras at Tyre to the wings of a dragonfly,
they are in love with the world and the deep seriousness of living.
Often lavish themselves, they reflect that fact that the author is
a visual artist, as well as a poet of insightful and sustained
imagination.
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