timescalecalculus python library documentation

This is the documentation for the Python repository timescalecalculus.

The basics

After extracting the files, open a Python instance in its folder and type

 >>> from timescalecalculus import *

Time scale basics

Right now, a time scale in this library can consist of only a finite list of numbers. Fraction types are available. Let $\mathbb{T}=\left\{0,\dfrac{1}{3},\dfrac{1}{2},\dfrac{7}{9},1,2,3,4,5,6,7 \right\}$.

>>> ts=[0,Fraction(1,3),Fraction(1,2),Fraction(7,9),1,2,3,4,5,6,7]

The forward jump $\sigma$ can be computed:
$\sigma(0)=\dfrac{1}{3}$

>>> sigma(0,ts)
Fraction(1, 3)

$\sigma(4)=5$

>>> sigma(4,ts)
5

$\sigma(7)=7$

>>> sigma(7,ts)
7

The forward graininess $\mu$ can be computed:
$\mu \left( \dfrac{1}{3} \right)=\dfrac{1}{2}-\dfrac{1}{3}=\dfrac{1}{6}$

>>> mu(Fraction(1,3),ts)
Fraction(1, 6)

The backward jump $\rho$ can be used:
$\rho(1)=\dfrac{7}{9}$

>>> rho(1,ts)
Fraction(7, 9)

$\rho(3)=2$

>>> rho(3,ts)
2

$\rho(0)=0$

>>> rho(0,ts)
0

The backward graininess $\nu$ can be computed:
$\nu\left( \dfrac{7}{9} \right)=\dfrac{7}{9}-\dfrac{1}{2}=\dfrac{5}{18}$

>>> nu(Fraction(7,9),ts)
Fraction(5, 18)

Delta-derivative

The delta derivative works as expected. The delta derivative of a constant is zero:

>>> dderivative(lambda x: 1,5,ts)
0

and obeying the delta derivative of squaring function, we see

>>> dderivative(lambda x: x*x,5,ts)
11

The delta exponential is supported. For example if $\mathbb{T}=\{1,2,3,4,5,6,7\}$ then $e_1(3,1)=(1+\mu(1))(1+\mu(2))=(2)(2)=4$ which is correctly computed:

>>> dexpf(lambda x: 1, 3, 1, ts)
4