Rationality and periodicity

[AntonioLao]

A function depending on the time parameter, [math]t[/math], is most often that of a periodic function where the condition for periodicity is the existence of a period, [math]T[/math], such that

[math]f(t)=f(t + T)[/math]

is always true. But the existence of the period [math]T[/math] implies the rationality of inverse of [math]T[/math], which is called the frequency of occurrence of the period and in such a way that the ratio of any two different frequencies must always be a rational number. Hence the necessary condition for periodicity is rationality.

This condition also implies that frequencies can only take on positive integers excluding the whole number 0 for all strong cases, while for weak cases of the condition, the frequencies can be identically zero. It is these weak cases that can be applied to the universe as a whole.

[Guille]

[math]f(t)=f(t + T)[/math]

doesn't this mean that T=0? If it does, why use it?

[AntonioLao]

doesn't this mean that T=0? If it does, why use it?

one example of periodic functions taken from Fourier analysis is the following
[math]f(t)=cos\frac{1}{3}(t+T)+cos\frac{1}{4}(t+T)=cos\f rac{t}{3}+cos\frac{t}{4}[/math]
by trigonometric manipulations. [math]T=24\pi[/math]

[Guille]

What do you mean by trigonometric malipulations? and what are the ones done here (if it's not too dificult)?

[AntonioLao]

the following sites give examples of trigonometric formulas

http://mathworld.wolfram.com/Trigono...nFormulas.html

http://mathworld.wolfram.com/Double-AngleFormulas.html

http://mathworld.wolfram.com/Half-AngleFormulas.html

the example i gave in post#3 is based on the trig fact that

[math]cos(\theta + 2\pi m)= cos\theta[/math]

[Guille]

[math]cos(\theta + 2\pi m)= cos\theta[/math]

I understand it, but what does m represent?

[AntonioLao]

I understand it, but what does m represent?

for the strong case of the periodic condition, m must be a positive integer, but for the weak case, m can also be zero.

[mkirkpatrick]

Without the "weak cases" the stronger ones could not survive?The strong need the weak
more than vice-vera I think!


regards michael.

[AntonioLao]

Without the "weak cases" the stronger ones could not survive?The strong need the weak more than vice-vera I think!

the weakest force is still called gravity.

[mkirkpatrick]

the weakest force is still called gravity.

Is that not another name for the "strange attractor?


Regards michael.