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 t_{p}(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[t_{p}(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: ò_{a}^{b}f(x)dg(x). The Mean Value Theorem can be applied to the
Riemann-Stieltjes Integral: ò_{a}^{b}f(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.

3. Olmstead,
John M. H. __Advanced Calculus__. New York: Appleton-Century-Crofts, Inc.

1961.

4. Widder,
David Vernon. __The Laplace Transform__. London: Humphrey Milford Oxford

University
Press. 1946.

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