Difference between revisions of "Isolated points"
From timescalewiki
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Let $\mathbb{T}$ be a [[time_scale | 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 | time scale]] of isolated points with $t_k > t_n$ iff $k>n$. Define the bijection $\pi \colon \mathbb{T} \rightarrow \mathbb{Z}$, $\pi(t_k)=k$. | Let $\mathbb{T}$ be a [[time_scale | 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 | time scale]] of isolated points with $t_k > t_n$ iff $k>n$. Define the bijection $\pi \colon \mathbb{T} \rightarrow \mathbb{Z}$, $\pi(t_k)=k$. | ||
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{| class="wikitable" | {| class="wikitable" | ||
| − | |+$\mathbb{T}=\{\ldots,t_{-1},t_0,t_1,\ldots\}, isolated points | + | |+$\mathbb{T}=\{\ldots,t_{-1},t_0,t_1,\ldots\}$, isolated points |
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|Generic element $t\in \mathbb{T}$: | |Generic element $t\in \mathbb{T}$: | ||
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| + | == Examples of time scales of isolated points == | ||
| + | *[[Integers | $\mathbb{Z}$]] | ||
| + | *[[Multiples_of_integers | $h\mathbb{Z}$]] | ||
| + | *[[Square_integers | $\mathbb{Z}^2$]] | ||
| + | *[[Harmonic_numbers | $\mathbb{H}$]] | ||
Revision as of 01:05, 26 May 2014
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 $\pi \colon \mathbb{T} \rightarrow \mathbb{Z}$, $\pi(t_k)=k$.
| 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: | |
| Exponential function: |
