Difference between revisions of "Isolated points"

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*[[Square_integers | $\mathbb{Z}^2$]]
 
*[[Square_integers | $\mathbb{Z}^2$]]
 
*[[Harmonic_numbers | $\mathbb{H}$]]
 
*[[Harmonic_numbers | $\mathbb{H}$]]
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Revision as of 01:22, 22 May 2015

Let $\mathbb{T}$ be a time scale. We say that $\mathbb{T}$ is a time scale of isolated points if there exists $\epsilon > 0$ such that for all $t \in \mathbb{T}$, $\mu(t) \geq \epsilon$. Let $\mathbb{T}=\{\ldots,t_{-1},t_0,t_1,\ldots\}$ be a time scale of isolated points with $t_k > t_n$ iff $k>n$. Define the bijection $\pi \colon \mathbb{T} \rightarrow \mathbb{Z}$, $\pi(t_k)=k$.

$\mathbb{T}=\{\ldots,t_{-1},t_0,t_1,\ldots\}$, isolated points
Generic element $t\in \mathbb{T}$: For some $n \in \mathbb{Z}, t=t_n$
Jump operator: $\sigma(t)=\sigma(t_n)=t_{n+1}$
Graininess operator: $\mu(t)=\mu(t_n)=t_{n+1}-t_n$
$\Delta$-derivative: $f^{\Delta}(t)=f^{\Delta}(t_n) = \dfrac{f(t_{n+1})-f(t_n)}{t_{n+1}-t_n}$
$\Delta$-integral: $$\displaystyle\int_{t_s}^{t_n} f(\tau) \Delta \tau = \left\{ \begin{array}{ll} \displaystyle\sum_{k=s}^{n-1} \mu(t_k)f(t_k) &; n > s \\ 0 &; n=s \\ -\displaystyle\sum_{k=n}^{s-1} \mu(t_k) f(t_k) &; n < s \end{array} \right. $$
Exponential function: If $t_n > t_s$, $$\begin{array}{ll} e_p(t_n,t_s) &= \exp \left( \displaystyle\int_{t_s}^{t_n} \dfrac{1}{\mu(\tau)} \log(1 + p(\tau)) \Delta \tau \right) \\ &= \exp \left( \displaystyle\sum_{k=s}^{n-1} \log(1+\mu(t_k)p(t_k)) \right) \\ &= \displaystyle\prod_{k=s}^{n-1} \left( 1+\mu(t_k)p(t_k) \right) \\ \end{array}$$


Examples of time scales of isolated points

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