Difference between revisions of "Time scale"
From timescalewiki
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== Examples of time scales == | == Examples of time scales == | ||
| − | # The real line $\mathbb{R}$ | + | # The real line: $\mathbb{R}$ |
| − | # The integers $\mathbb{Z} = \{\ldots, -1,0,1,\ldots\}$ | + | # The integers: $\mathbb{Z} = \{\ldots, -1,0,1,\ldots\}$ |
| − | # Multiples of integers $h\mathbb{Z} = \{ht \colon t \in \mathbb{Z}\}$ | + | # Multiples of integers: $h\mathbb{Z} = \{ht \colon t \in \mathbb{Z}\}$ |
| − | # Harmonic numbers $\mathbb{H}=\left\{\displaystyle\sum_{k=1}^n \dfrac{1}{k} \colon n \in \mathbb{Z}^+ \right\}$ | + | # Harmonic numbers: $\mathbb{H}=\left\{\displaystyle\sum_{k=1}^n \dfrac{1}{k} \colon n \in \mathbb{Z}^+ \right\}$ |
| + | # The closure of the unit fractions: $\overline{\left\{\dfrac{1}{n} \colon n \in \mathbb{Z}^+\right\}}$ | ||
Revision as of 02:46, 18 May 2014
A time scale is a set $\mathbb{T} \subset \mathbb{R}$ which is closed under the standard topology of $\mathbb{R}$. Given a time scale we define the jump operator $\sigma \colon \mathbb{T} \rightarrow \mathbb{T}$ by the formula $$\sigma(t) := \inf \left\{ x \in \mathbb{T} \colon x > t \right\}.$$ The graininess operator is the function $\mu \colon \mathbb{T} \rightarrow \mathbb{R}^+ \cup \{0\}$ is defined by the formula $$\mu(t) := \sigma(t)-t.$$
Examples of time scales
- The real line: $\mathbb{R}$
- The integers: $\mathbb{Z} = \{\ldots, -1,0,1,\ldots\}$
- Multiples of integers: $h\mathbb{Z} = \{ht \colon t \in \mathbb{Z}\}$
- Harmonic numbers: $\mathbb{H}=\left\{\displaystyle\sum_{k=1}^n \dfrac{1}{k} \colon n \in \mathbb{Z}^+ \right\}$
- The closure of the unit fractions: $\overline{\left\{\dfrac{1}{n} \colon n \in \mathbb{Z}^+\right\}}$
