Time-scale calculus

In mathematics, time-scale calculus is a unification of the theory of difference equations with that of differential equations, unifying integral and differential calculus with the calculus of finite differences, offering a formalism for studying hybrid discrete–continuous dynamical systems. It has applications in any field that requires simultaneous modelling of discrete and continuous data. It gives a new definition of a derivative such that if one differentiates a function which acts on the real numbers then the definition is equivalent to standard differentiation, but if one uses a function acting on the integers then it is equivalent to the forward difference operator.

History

Time-scale calculus was introduced in 1988 by the German mathematician Stefan Hilger. However, similar ideas have been used before and go back at least to the introduction of the Riemann–Stieltjes integral which unifies sums and integrals.

Dynamic equations

Many results concerning differential equations carry over quite easily to corresponding results for difference equations, while other results seem to be completely different from their continuous counterparts. The study of dynamic equations on time scales reveals such discrepancies, and helps avoid proving results twice — once for differential equations and once again 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 (also known as a time-set), which may be an arbitrary closed subset of the reals. In this way, results apply not only to the set of real numbers or set of integers but to more general time scales such as a Cantor set.

Time scale

Time scale may refer to:

In astronomy and physics:

In cosmology and particle physics:

Rhythm

Rhythm (from Greek ῥυθμός, rhythmos, "any regular recurring motion, symmetry" (Liddell and Scott 1996)) generally means a "movement marked by the regulated succession of strong and weak elements, or of opposite or different conditions" (Anon. 1971, 2537). This general meaning of regular recurrence or pattern in time can apply to a wide variety of cyclical natural phenomena having a periodicity or frequency of anything from microseconds to millions of years.

In the performance arts rhythm is the timing of events on a human scale; of musical sounds and silences, of the steps of a dance, or the meter of spoken language and poetry. Rhythm may also refer to visual presentation, as "timed movement through space" (Jirousek 1995, ) and a common language of pattern unites rhythm with geometry. In recent years, rhythm and meter have become an important area of research among music scholars. Recent work in these areas includes books by Maury Yeston (Yeston 1976), Fred Lerdahl and Ray Jackendoff, Jonathan Kramer, Christopher Hasty (Hasty 1997), Godfried Toussaint (Toussaint 2005), William Rothstein, and Joel Lester (Lester 1986).