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.

References

1.  Devito, Carl L.  “A Non-Linear Model For Time.”  Astrophysics and Space Science

244 (1996):  357-369.

2.  Devito, Carl L.  “Time Scapes.”  Chaos, Solitons & Fractals  Vol. 9  No. 7  (1998):

1105-1114.