Difference between revisions of "Closure of unit fractions"

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Revision as of 01:23, 22 May 2015

The set $\overline{\left\{\dfrac{1}{n} \colon n \in \mathbb{Z}^+\right\}}=\left\{ 0,1,\dfrac{1}{2},\dfrac{1}{3},\ldots \right\}$, where the $\overline{\mathrm{overline}}$ denotes topological closure of this set in the usual topology on $\mathbb{R}$ is a time scale.

Closureunitfractionstimescale.png

$\mathbb{T}= \overline{\left\{\dfrac{1}{n} \colon n \in \mathbb{Z}^+\right\}} $
Generic element $t\in \mathbb{T}$: Either $t=0$ or for some $n \in \mathbb{Z}^+, t = \dfrac{1}{n}$
Jump operator: $\sigma(t) = \left\{ \begin{array}{ll} \dfrac{1-t}{t} &; t>1 \\ 0 &; t=0 \end{array} \right.$
Graininess operator: $\mu(t) = \left\{ \begin{array}{ll} \dfrac{1-t-t^2}{t} &; t>0 \\ 0 &; t=0 \end{array} \right.$
$\Delta$-derivative: $f^{\Delta}(t) = \left\{ \begin{array}{ll} \dfrac{t}{1-t} \left[ f \left(\dfrac{1-t}{t} \right)-f(t) \right] &; t>0 \\ \displaystyle\lim_{h \rightarrow 0} \dfrac{f(h)-f(0)}{h} &; t=0 \end{array} \right.$
$\Delta$-integral:
Exponential function:

Examples of time scales
$\Huge\mathbb{R}$
Real numbers
$\Huge\mathbb{Z}$
Integers
$\Huge{h\mathbb{Z}}$
Multiples of integers
$\Huge\mathbb{Z}^2$
Square integers
$\Huge\mathbb{H}$
Harmonic numbers
$\Huge\mathbb{T}_{\mathrm{iso}}$
Isolated points
$\Huge\sqrt[n]{\mathbb{N}_0}$
nth root numbers
$\Huge\mathbb{P}_{a,b}$
Evenly spaced intervals
$\huge\overline{q^{\mathbb{Z}}}$
Quantum, $q>1$
$\huge\overline{q^{\mathbb{Z}}}$
Quantum, $q<1$
$\overline{\left\{\dfrac{1}{n} \colon n \in \mathbb{Z}^+\right\}}$
Closure of unit fractions
$\Huge\mathcal{C}$
Cantor set
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