Difference between revisions of "Isolated points"

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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 $\pi \colon \mathbb{T} \rightarrow \mathbb{Z}$, $\pi(t_k)=k$.
 
The set $h\mathbb{Z}=\{\ldots,-2h,-h,0,h,2h,\ldots\}$ of multiples of the integers is a [[time scale]].
 
  
 
{| class="wikitable"
 
{| class="wikitable"
|+$\mathbb{T}=\{\ldots,t_{-1},t_0,t_1,\ldots\}, isolated points$
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|+$\mathbb{T}=\{\ldots,t_{-1},t_0,t_1,\ldots\}$, 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 ==
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*[[Integers | $\mathbb{Z}$]]
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*[[Multiples_of_integers | $h\mathbb{Z}$]]
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*[[Square_integers | $\mathbb{Z}^2$]]
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*[[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 $\pi \colon \mathbb{T} \rightarrow \mathbb{Z}$, $\pi(t_k)=k$.

$\mathbb{T}=\{\ldots,t_{-1},t_0,t_1,\ldots\}$, 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:
Exponential function:


Examples of time scales of isolated points