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 $t \in \mathbb{T}^{\kappa}$  
 
$$1+\mu(t)p(t)\neq 0.$$
 
$$1+\mu(t)p(t)\neq 0.$$
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 $t \in \mathbb{T}^{\kappa}$,
 
$$(p \oplus q)(t) = p(t)+q(t)+\mu(t)p(t)q(t).$$
 
We define the inverse operation of $\oplus$ by the formula
 
$$(\ominus p)(t) = -\dfrac{p(t)}{1+\mu(t)p(t)}.$$
 
The ordered pair $(\mathcal{R},\oplus)$ is an [[Abelian_group | Abelian group]] with subtraction
 
$$(p \ominus q)(t) = (p \oplus (\ominus q))(t) = \dfrac{p(t)-q(t)}{1+\mu(t)q(t)}.$$
 
  
==Related definitions==
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=See also=
*Let $\nu(We say that a function $p \colon \mathbb{T} \rightarrow \mathbb{R}$ is $\nu$-regressive if
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$$1-\nu(t)p(t) \neq 0$$
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=References=
for all $t \in \mathbb{T}_{\kappa}$. We use notation $\mathcal{R}_{\nu}$ for the set of $\nu$-regressive functions.
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*The set of positively regressive functions is
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[[Category:Definition]]
$$\mathcal{R}^+(\mathbb{T},X)=\{p \in \mathcal{R} \colon \forall t \in \mathbb{T}, 1+\mu(t)p(t)>0 \}.$$
 
*The set of negatively regressive functions is
 
$$\mathcal{R}^-(\mathbb{T},X)=\{p \in \mathcal{R} \colon \forall t \in \mathbb{T}, 1+\mu(t)p(t)<0 \}.$$
 
*Consider the [[dynamic_equations | dynamic equation]] $y^{\Delta \Delta}(t)+p(t)y^{\Delta}(t)+q(t)y(t)=f(t)$. We say this equation is regressive if $,p,q,f \in C_{rd}$ and for all $t \in \mathbb{T}$
 
$$1-\mu(t)p(t)+\mu^2(t)q(t)\neq 0.$$
 
*If we define the "circle dot" multiplication for $p \colon \mathbb{T} \rightarrow \mathbb{R}$ and $f \colon \mathbb{T} \rightarrow \mathbb{R}$ by
 
$$(f\odot p)(t) = \left\{ \begin{array}{ll} \dfrac{(1+\mu(t)p(t))^{f(t)}-1}{\mu(t)} &; \mu(t) > 0 \\
 
f(t) p(t) &; \mu(t)=0
 
\end{array} \right.$$
 
then if we restrict $f$ to real constant functions then the triple $(\mathcal{R}^+,\oplus,\odot)$ is a real [[vector_space | vector space]].
 

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