Difference between revisions of "Forward circle minus"

From timescalewiki
Jump to: navigation, search
 
(4 intermediate revisions by the same user not shown)
Line 1: Line 1:
Let $\mathbb{T}$ be a [[time scale]] and let $p,q \in \mathcal{R}(\mathbb{T},\mathbb{C})$ be [[forward regressive function| (forward) regressive functions ]]. We define the (forward) circle minus operation $\ominus_{\mu} \colon \mathbb{T} \rightarrow \mathbb{T}$ by  
+
Let $\mathbb{T}$ be a [[time scale]] and let $p,q \in \mathcal{R}(\mathbb{T},\mathbb{C})$ be [[forward regressive function| (forward) regressive functions ]]. We define the (forward) circle minus operation by  
 
$$\left( \ominus_{\mu} p \right)(t) = \dfrac{-p(t)}{1+p(t)\mu(t)}.$$
 
$$\left( \ominus_{\mu} p \right)(t) = \dfrac{-p(t)}{1+p(t)\mu(t)}.$$
Since the set of forward regressive functions [[forward regressive functions form a group|form a group]] $\left(\mathcal{R}(\mathbb{T},\mathbb{C}),\oplus_{\mu} \right)$ under [[circle plus]] with inverse operation  $\ominus_{\mu}$, we define
+
Often in the literature, the subscript is suppressed.
$$p \ominus_{\mu} q = p \oplus_{\mu} (\ominus_{\mu} q).$$
 
  
 
=Properties=
 
=Properties=
Line 10: Line 9:
 
=See Also=
 
=See Also=
 
[[Delta exponential]]<br />
 
[[Delta exponential]]<br />
 +
 +
=References=
  
 
[[Category:Definition]]
 
[[Category:Definition]]

Latest revision as of 15:26, 21 January 2023

Let $\mathbb{T}$ be a time scale and let $p,q \in \mathcal{R}(\mathbb{T},\mathbb{C})$ be (forward) regressive functions . We define the (forward) circle minus operation by $$\left( \ominus_{\mu} p \right)(t) = \dfrac{-p(t)}{1+p(t)\mu(t)}.$$ Often in the literature, the subscript is suppressed.

Properties

Forward regressive functions form a group
Circle minus inverse of circle plus

See Also

Delta exponential

References