Periodic Functions On Non-Linear Temporal Models
By Alexia Puig
Over the course of this semester, I have been reading and working with two of Dr. Devito’s papers, “A Non-Linear Model For Time” and “Time Scapes.” These papers define time as a partially ordered set of instants. Since these instants cannot be added, there is a duration function defined in “A Non-Linear Model For Time” that assigns a non-negative real number for each pair of instants. This function is called duration or dur, for short. For example, if dur(x,y) = p, where p is the “length” between x and y (two instants), then p is also equal to dur(y,x). Therefore dur(x,y) = dur(y,x).
I have been looking at functions defined on time tracks and studying their mathematical properties. I am trying to extend some of the ideas of analysis to such functions such as periodicity and integrability. I was successful in extending periodicity using the translation function. This is defined as follows: let x and y be instants and x<y, then tp(x) = y, where p is the units of time past between x and y. A real number, p, is a period of the function f if f[tp(x)] = f(x) for all x. The set of all such numbers is denoted by P(f). We shall say that f is a periodic function if P(f) contains a non-zero number. I am currently still working on trying to interpret the integrals of these functions and have started by studying the Riemann-Stieltjes Integral, its properties and applications. The Riemann-Stieltjes Integral is defined as follows for two functions: Let g(x) and f(x) be real functions of x defined on the interval [a,b], where a £x £b. Then f(x) can be integrated with respect to g(x) as so: òabf(x)dg(x). The Mean Value Theorem can be applied to the Riemann-Stieltjes Integral: òabf(x)dg(x) = f(x)[g(b) –g(a)] where a £x £b. This will be useful for proving the integrability. The advantage here is that f(x) and g(x) are numbers while x is an instant.
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