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  
 
$$\mathbb{T}_1 \oplus \mathbb{T}_2=\{t_1+t_2 \colon t_1 \in \mathbb{T}_1, t_2 \in \mathbb{T}_2\}.$$
 
$$\mathbb{T}_1 \oplus \mathbb{T}_2=\{t_1+t_2 \colon t_1 \in \mathbb{T}_1, t_2 \in \mathbb{T}_2\}.$$
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=Examples=
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*$\{0,1\} \oplus \{4,5,10\} = \{4,5,6,10,11\}$
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*$\{0,2\} \oplus [0,1] = [0,1] \cup [2,3]$
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*$2\mathbb{Z} \oplus \mathbb{Z}=\mathbb{Z}$
  
 
=References=
 
=References=
 
[http://web.ecs.baylor.edu/faculty/gravagnei/archived/Fourier.pdf]
 
[http://web.ecs.baylor.edu/faculty/gravagnei/archived/Fourier.pdf]
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[[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

  • $\{0,1\} \oplus \{4,5,10\} = \{4,5,6,10,11\}$
  • $\{0,2\} \oplus [0,1] = [0,1] \cup [2,3]$
  • $2\mathbb{Z} \oplus \mathbb{Z}=\mathbb{Z}$

References

[1]