Difference between revisions of "Closure of unit fractions"
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(Created page with "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 $\overline{\mathrm{overline}}$ deno...") |
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| − | |+$\mathbb{T}= | + | |+$\mathbb{T}= \overline{\left\{\dfrac{1}{n} \colon n \in \mathbb{Z}^+\right\}} $ |
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|Generic element $t\in \mathbb{T}$: | |Generic element $t\in \mathbb{T}$: | ||
| − | | Either $t=0$ or for some $n \in \mathbb{Z}, t = \dfrac{1}{n}$ | + | | Either $t=0$ or for some $n \in \mathbb{Z}^+, t = \dfrac{1}{n}$ |
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|Jump operator: | |Jump operator: | ||
Revision as of 17:53, 20 May 2014
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.
| 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: |
