Difference between revisions of "Shifting problem"

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(Created page with "Let $\mathbb{T}$ be a time scale, $t_0 \in \mathbb{T}$, and $f \colon [t_0,\infty) \cap \mathbb{T} \rightarrow \mathbb{C}$. The shifting problem is the following partial dyn...")
 
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\end{array} \right.$$
 
\end{array} \right.$$
 
The solution $\hat{f}$ of the shifting problem is called the shift of $f$ (also called the delay of $f$).
 
The solution $\hat{f}$ of the shifting problem is called the shift of $f$ (also called the delay of $f$).
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{| class="wikitable"
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|+Time Scale Shift
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|-
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|$\mathbb{T}$ | Shift
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|
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|-
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|[[Real_numbers | $\mathbb{R}$]]
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|$\hat{f}(t,s)=f(t-s)$
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|-
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|[[Integers | $\mathbb{Z}$]]
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|
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|-
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|[[Multiples_of_integers | $h\mathbb{Z}$]]
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|
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|-
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| [[Square_integers | $\mathbb{Z}^2$]]
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|
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|-
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|[[Quantum_q_greater_than_1 | $\overline{q^{\mathbb{Z}}}, q > 1$]]
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|
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|-
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|[[Quantum_q_less_than_1 | $\overline{q^{\mathbb{Z}}}, q < 1$]]
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|
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|-
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|[[Harmonic_numbers | $\mathbb{H}$]]
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|
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|}

Revision as of 14:35, 8 February 2016

Let $\mathbb{T}$ be a time scale, $t_0 \in \mathbb{T}$, and $f \colon [t_0,\infty) \cap \mathbb{T} \rightarrow \mathbb{C}$. The shifting problem is the following partial dynamic equation for $t,s \in \mathbb{T}$: $$\left\{ \begin{array}{ll} \dfrac{\partial \hat{f}}{\Delta t}(t,\sigma(s))=-\dfrac{\partial \hat{f}}{\Delta s}(t,s)& ; t \geq s \geq t_0, \\ \hat{f}(t,t_0)=f(t)&; t \geq t_0. \end{array} \right.$$ The solution $\hat{f}$ of the shifting problem is called the shift of $f$ (also called the delay of $f$).

Time Scale Shift
Shift
$\mathbb{R}$ $\hat{f}(t,s)=f(t-s)$
$\mathbb{Z}$
$h\mathbb{Z}$
$\mathbb{Z}^2$
$\overline{q^{\mathbb{Z}}}, q > 1$
$\overline{q^{\mathbb{Z}}}, q < 1$
$\mathbb{H}$