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. To define a partially ordered set, let I be an infinite set and let ≤ be a relation on the elements of I.[1] Then ≤ is a partial ordering of I, and the pair (I,≤) is a partially ordered set if:

a) for every i Î I we have i≤i

b) if i and j are in I and we have both i≤j and j≤i, then i = j

c) if i, j and k are in I and we have i≤j and j≤k, then i≤k

There is a duration function that assigns a non-negative real number for each pair of comparable instants.[2] This function is called duration or dur, for short and has the following properties:

1) dur(*x,y*) = p, where p is the “time” between
x and y (two instants)

2) dur(*x,y*) = dur(*y,x*), and if x=y, then
dur(*x,y*) = 0

3) if y is between x and z, meaning x≤y≤z or z≤y≤x,
then dur(*x,z*) = dur(*x,y*) + dur*(y,z*).

I have been looking at functions defined on time tracks and studying their mathematical properties. A time track, T, is a non-empty subset of the infinite set of instants and has the following properties[3]:

a) Any two instants on T can be compared.

b)
If x, y, and z are on T and y is between x and z, then dur(*x,z*) = dur(*x,y*)

+ dur*(y,z*).

c) Given y on T, and a positive, real number p, there are exactly 2 instants

x and z on T such that dur(*x,y*) = p and dur(*y,z*)
= p.

I am trying to extend the idea of periodicity and integrability to functions defined on time tracks. Since instants cannot be added, this requires some new ideas. I was successful in extending periodicity using the translation function. This is defined as follows:

Definition 1: There is a
translation function, t_{p}, for any fixed number p, which maps T to T
as follows:[4]

1)
t_{0}(x)
= x for all x Î T

2)
if
p>0, t_{p}(x)
= y where y is the unique instant on T such that x<y and dur(*x,y*) = p

3)
if
p<0, t_{p}(x) = z where z is the unique instant on T such that
z<x and dur(*z,x*) = |p|

Then, a real
number p is a period of the function f if *f[t _{p}(x)] = f(x)* for
all

Lemma 1: If p and q are contained in the set P(f), then so are -p, p+q and p–q

Proof : Suppose
that *f[t _{-p}(x)] = f(y)* where y is the instant in the past such
that dur(

Now to prove that p+q and p–q are also included in the set of periods for f, when p and q are both periods:

*f[t _{p}(x)]
= f(x)* and

*f[t _{p+q}(x)]
= f[t_{p}[t_{q} (x)]] = f[t_{q}(x)] = f(x)* for all

\p+q Î P(f)

*f[t _{p-q}(x)]
= f[t_{p}[t-_{q} (x)]] = f[t-_{q}(x)] = f(x)* for all

\p-q Î P(f)

Corollary 1: If p is contained in the set P(f) then so is np, where n is an integer. When p is the smallest, positive member of P(f), these two sets coincide.

To prove that np is also a period contained in the set:

Since this is clearly true for n=1, I will need to prove it for n>1.

Suppose that (n-1)p Î P(f) and consider np. Since both 1p and (n-1)p are in P(f),

then 1p + np = np is in P(f) by Lemma 1

Now to prove that the sets nq Î P(f) and q Î P(f) coincide:

Def.: *f[t _{p}(x)]
= f(x) *for all

Suppose that *f[t _{q}(x)]
= f(x)* for all

Then p<q, so divide q = np + r, where r is the remainder (0£r<p)

*f[t _{q}(x)]
= f[t_{np+r}(x)] = f[t_{np} [t_{r}(x)]]*

Since p is a period so is np as shown above.

Then *f[t _{np}
[t_{r}(x)]] = f[t_{r}(x)] *for all x, so r is a period.

But since 0£r<p, and p is the smallest positive period, r must be 0.

If q<0, then -q Î P(f) and since -q is positive, -q = np for some integer n. But then

q = (-n)p , so r can be proven to be 0 for q<0 also.

I have also been interpreting the integrals of these functions and 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, (a=x_{0}<x_{1}<x_{2}<…….<x_{n}=b).
The limit of: _{}, as max|x_{j} – x_{j+1}|
® 0, where x_{j-1}≤x_{j}*≤x_{j},
is denoted by: _{}, and is called the integral of f
with respect to g.[5]
There is a Mean Value Theorem for the Riemann-Stieltjes Integral: _{} = f(x)[g(b) –g(a)]
where a£x £b.[6]
This will be useful below. The advantage here is that f(x) and g(x) are
numbers while x is an instant.

Let g be a
function such that dur(*x,y*) = |g(x)-g(y)|. Such a function is defined
in Dr. Devito’s paper as: D(s(p_{1}), s(p_{2})) =
|p_{1}-p_{2}|.[7]
Instead of “s”, I shall use “g”

Lemma 2: Let f be a
continuous and periodic function on the real line, and let a>0 be any
element contained in P(f), then for any real x and y, _{} = _{}.

Proof:_{} *= _{} *–

Now _{}= _{}

First note that g[t_{α}(u)]
= g(u) because α is a constant and g(x) – g(y) = dur*(x,y)*, so g[t_{α}(x)]–
g[t_{α}(y)] = dur*(x,y)*

Then_{}*= _{}*

So_{} = _{}

Therefore_{}= _{}

Lemma 3: If a continuous function has arbitrarily small periods, it is a constant

Proof: Consider
the integral: _{}_{}

Suppose a and b
are instants where a=x_{0}<x_{1}<x_{2}<…….<x_{n}=b.

Then the integral
can be represented by the sum _{}

* *where x_{j-1}£ x_{j}*£ x_{j }

The limit of this
sum as max[g(x_{j}) – g(x_{j-1})] goes to 0 is defined to be _{}which is the
Riemann-Stieltjes Integral.

Then using the Mean Value Theorem: m[g(b)-g(a)] £_{}£M[g(b)-g(a)] where

m stands for minimum and M stands for maximum.

So _{}≤ _{}≤_{}

Now a = dur*(t _{α}(x),x)*
= g[t

Then *1/[ g(x+α _{n})
– g(x)]*

where x£ x£x+α_{n}

Since a_{n} Î P(f), then for
any x: _{} 1/*a*_{n}*ò*_{}* = f(x)*

Then for any x and
y: * 1/**a*_{n}*ò*_{}* = 1/**a** _{n }*so f(x) = f(y)

Therefore, f is a constant.

Corollary 2: Any __non-constant__,
continuous, periodic function has a smallest positive period

Proof: *P(f[t _{α}(x)])
= P(f(x)) *for all

To prove that a_{0} Î P_{+}(f),
I shall argue by contradiction. If a_{0} is not in this
set, then there must be some {a_{n} } Í P_{+}(f)
such that * _{}*a

\a_{0} Î P_{+}(f)

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 V. __Advanced Calculus__. Englewood Cliffs, N.J.: Prentice-Hall, Inc.

1947.

5. Widder, David Vernon. __The ____Laplace____ Transform__. London: Humphrey Milford Oxford

University Press. 1946.

** **

[1] Devito, Carl L. “Time Scapes.”

[2]
Devito, Carl L. “A Non-Linear Model For Time.”**
**

[3]
Devito, Carl L. “A Non-Linear Model For Time.”**
**

[4]
Devito, Carl L. “A Non-Linear Model For Time.”**
**

[5]
Widder, David Vernon. __The ____Laplace____
Transform__.

[6]
Olmstead, John M. H. __Advanced Calculus__.**
**

[7] Devito, Carl L. “Time Scapes.”