Difference between revisions of "Forward regressive"

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Let $\mathbb{T}$ be a [[time_scale | time scale]]. Let $p \colon \mathbb{T} \rightarrow \mathbb{R}$. We say that $p$ is ''regressive'' if for all $t \in \mathbb{T}^{\kappa}$  
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Let $\mathbb{T}$ be a [[time_scale | time scale]]. Let $p \colon \mathbb{T} \rightarrow \mathbb{C}$. We say that $p$ is forward regressive if for all $t \in \mathbb{T}^{\kappa}$  
 
$$1+\mu(t)p(t)\neq 0.$$
 
$$1+\mu(t)p(t)\neq 0.$$
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We let $\mathcal{R}(X,Y)$ denote the set of regressive functions $p \colon X \rightarrow Y$. Let $p,q \in \mathcal{R}$ and define the "circle plus" operation $\oplus \colon \mathcal{R} \times \mathcal{R} \rightarrow \mathcal{R}$ by the formula, for $t \in \mathbb{T}^{\kappa}$,
 
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$$(p \oplus q)(t) = p(t)+q(t)+\mu(t)p(t)q(t).$$
 
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We define the inverse operation of $\oplus$ by the formula
 
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$(\ominus p)(t) = -\dfrac{p(t)}{1+\mu(t)p(t)}$.
 
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The ordered pair $(\mathcal{R},\oplus)$ is an [[Abelian_group | Abelian group]] with subtraction $(p \ominus q)(t) = (p \oplus (\ominus q))(t) = \dfrac{p(t)-q(t)}{1+\mu(t)q(t)}$.
 
  
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==Related definitions==
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=See also=
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The set of positively regressive functions is
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$$\mathcal{R}^+(\mathbb{T},X)=\{p \in \mathcal{R} \colon \forall t \in \mathbb{T}, 1+\mu(t)p(t)>0 \}.$$
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=References=
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The set of negatively regressive functions is
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$$\mathcal{R}^-(\mathbb{T},X)=\{p \in \mathcal{R} \colon \forall t \in \mathbb{T}, 1+\mu(t)p(t)<0 \}.$$
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[[Category:Definition]]

Latest revision as of 12:58, 16 January 2023

Let $\mathbb{T}$ be a time scale. Let $p \colon \mathbb{T} \rightarrow \mathbb{C}$. We say that $p$ is forward regressive if for all $t \in \mathbb{T}^{\kappa}$ $$1+\mu(t)p(t)\neq 0.$$

See also

References

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