# Difference between revisions of "Timescalecalculus python library documentation"

Line 6: | Line 6: | ||

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

<pre> >>> from timescalecalculus import *</pre> | <pre> >>> from timescalecalculus import *</pre> | ||

− | + | ||

Right now, a [[time scale]] in this library can consist of only a finite list of numbers. Fraction types are available. | Right now, a [[time scale]] in this library can consist of only a finite list of numbers. Fraction types are available. | ||

Let $\mathbb{T}=\left\{0,\dfrac{1}{3},\dfrac{1}{2},\dfrac{7}{9},1,2,3,4,5,6,7 \right\}$.<br /> | Let $\mathbb{T}=\left\{0,\dfrac{1}{3},\dfrac{1}{2},\dfrac{7}{9},1,2,3,4,5,6,7 \right\}$.<br /> | ||

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

+ | |||

+ | ==Forward jump and graininess== | ||

The [[forward jump]] $\sigma$ can be computed:<br /> | The [[forward jump]] $\sigma$ can be computed:<br /> | ||

$\sigma(0)=\dfrac{1}{3}$ | $\sigma(0)=\dfrac{1}{3}$ | ||

Line 25: | Line 27: | ||

Fraction(1, 6)</pre> | Fraction(1, 6)</pre> | ||

+ | ==Backward jump and graininess== | ||

The [[backward jump]] $\rho$ can be used:<br /> | The [[backward jump]] $\rho$ can be used:<br /> | ||

$\rho(1)=\dfrac{7}{9}$ | $\rho(1)=\dfrac{7}{9}$ | ||

Line 51: | Line 54: | ||

<pre>>>> dexpf(lambda x: 1, 3, 1, ts) | <pre>>>> dexpf(lambda x: 1, 3, 1, ts) | ||

4</pre> | 4</pre> | ||

+ | |||

+ | =Special functions= | ||

+ | ==Delta exponential $e_p(t,s)$== | ||

+ | Let $\mathbb{T}=\{1,2,3,4,5\}$. Then the [[delta exponential]] $e_1(3,1)$ is given by | ||

+ | $$e_1(3,1) = \displaystyle\prod_{k=1}^2 1+\mu(k)(1)=2^{2}=4$$ | ||

+ | <pre>>>> ts=[1,2,3,4,5] | ||

+ | >>> dexpf(3,1,ts) | ||

+ | >>> dexpf(lambda x: 1,3,1,ts) | ||

+ | 4</pre> | ||

+ | The function $e_2(3,1)$ is given by | ||

+ | $$e_2(3,1) = \displaystyle\prod_{k=1}^2 1+\mu(k)(2)=3^2=9$$ | ||

+ | <pre>>>> dexpf(lambda x: 2,3,1,ts) | ||

+ | 9</pre> | ||

+ | The function $e_{\mathrm{id}}(3,1)$ (where $\mathrm{id}$ is the [https://en.wikipedia.org/wiki/Identity_function identity function] on $\mathbb{T}$) is given by | ||

+ | $$e_{\mathrm{id}}(3,1)=\displaystyle\prod_{k=1}^2 1+\mu(k)k=(1+(1)(1))(1+(1)(2))=6$$ | ||

+ | <pre>>>> dexpf(lambda x: x,3,1,ts) | ||

+ | 6</pre> | ||

+ | The function $e_{e_1(\cdot,1)}(3,1)$ is given by | ||

+ | $$e_{e_1(\cdot,1)}(4,1)=\displaystyle\prod_{k=1}^3 1+\mu(k)e_1(k,1)=(1+(1)e_1(1,1))(1+(1)e_1(2,1))(1+(1)e_1(3,1))=(2)(3)(5)=30$$ | ||

+ | <pre>>>> (1+dexpf(lambda x: 1,1,1,ts))*(1+dexpf(lambda x: 1,2,1,ts))*(1+dexpf(lambda x: 1,3,1,ts)) | ||

+ | 30</pre> |

## Revision as of 11:05, 24 December 2016

This is the documentation for the Python repository timescalecalculus.

## Contents

# The basics

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

>>> from timescalecalculus import *

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]

## Forward jump and graininess

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)

## Backward jump and graininess

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

# Special functions

## Delta exponential $e_p(t,s)$

Let $\mathbb{T}=\{1,2,3,4,5\}$. Then the delta exponential $e_1(3,1)$ is given by $$e_1(3,1) = \displaystyle\prod_{k=1}^2 1+\mu(k)(1)=2^{2}=4$$

>>> ts=[1,2,3,4,5] >>> dexpf(3,1,ts) >>> dexpf(lambda x: 1,3,1,ts) 4

The function $e_2(3,1)$ is given by $$e_2(3,1) = \displaystyle\prod_{k=1}^2 1+\mu(k)(2)=3^2=9$$

>>> dexpf(lambda x: 2,3,1,ts) 9

The function $e_{\mathrm{id}}(3,1)$ (where $\mathrm{id}$ is the identity function on $\mathbb{T}$) is given by $$e_{\mathrm{id}}(3,1)=\displaystyle\prod_{k=1}^2 1+\mu(k)k=(1+(1)(1))(1+(1)(2))=6$$

>>> dexpf(lambda x: x,3,1,ts) 6

The function $e_{e_1(\cdot,1)}(3,1)$ is given by $$e_{e_1(\cdot,1)}(4,1)=\displaystyle\prod_{k=1}^3 1+\mu(k)e_1(k,1)=(1+(1)e_1(1,1))(1+(1)e_1(2,1))(1+(1)e_1(3,1))=(2)(3)(5)=30$$

>>> (1+dexpf(lambda x: 1,1,1,ts))*(1+dexpf(lambda x: 1,2,1,ts))*(1+dexpf(lambda x: 1,3,1,ts)) 30