Difference between revisions of "Dilation of time scales"
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Let $\mathbb{T}_1$ and $\mathbb{T}_2$ be [[time scale|time scales]]. Their dilation, $\mathbb{T}_1 \oplus \mathbb{T}_2$, is a time scale defined by | Let $\mathbb{T}_1$ and $\mathbb{T}_2$ be [[time scale|time scales]]. Their dilation, $\mathbb{T}_1 \oplus \mathbb{T}_2$, is a time scale defined by | ||
$$\mathbb{T}_1 \oplus \mathbb{T}_2=\{t_1+t_2 \colon t_1 \in \mathbb{T}_1, t_2 \in \mathbb{T}_2\}.$$ | $$\mathbb{T}_1 \oplus \mathbb{T}_2=\{t_1+t_2 \colon t_1 \in \mathbb{T}_1, t_2 \in \mathbb{T}_2\}.$$ | ||
| + | |||
| + | =Examples= | ||
| + | *$\{0,1\} \oplus \{4,5,10\} = \{4,5,6,10,11\}$ | ||
| + | *$\{0,2\} \oplus [0,1] = [0,1] \cup [2,3]$ | ||
| + | *$2\mathbb{Z} \oplus \mathbb{Z}=\mathbb{Z}$ | ||
=References= | =References= | ||
[http://web.ecs.baylor.edu/faculty/gravagnei/archived/Fourier.pdf] | [http://web.ecs.baylor.edu/faculty/gravagnei/archived/Fourier.pdf] | ||
| + | |||
| + | [[Category:Definition]] | ||
Latest revision as of 15:27, 21 January 2023
Let $\mathbb{T}_1$ and $\mathbb{T}_2$ be time scales. Their dilation, $\mathbb{T}_1 \oplus \mathbb{T}_2$, is a time scale defined by $$\mathbb{T}_1 \oplus \mathbb{T}_2=\{t_1+t_2 \colon t_1 \in \mathbb{T}_1, t_2 \in \mathbb{T}_2\}.$$
Examples
- $\{0,1\} \oplus \{4,5,10\} = \{4,5,6,10,11\}$
- $\{0,2\} \oplus [0,1] = [0,1] \cup [2,3]$
- $2\mathbb{Z} \oplus \mathbb{Z}=\mathbb{Z}$
