Difference between revisions of "Forward circle minus"
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| − | 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 $\ominus_{\mu}$ | + | 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 |
| − | $$\ | + | $$\left( \ominus_{\mu} p \right)(t) = \dfrac{-p(t)}{1+p(t)\mu(t)}.$$ |
| + | Since the set of forward regressive functions [[forward regressive functions form a group|form a group]] $\left(\mathcal{R}(\mathbb{T},\mathbb{C}),\oplus_{\mu} \right)$ under [[circle plus]] with inverse operation $\ominus_{\mu}$, we define | ||
| + | $$p \ominus_{\mu} q = p \oplus_{\mu} (\ominus_{\mu} q).$$ | ||
=Properties= | =Properties= | ||
| + | {{:Forward regressive functions form a group}} | ||
{{:Circle minus inverse of circle plus}} | {{:Circle minus inverse of circle plus}} | ||
Revision as of 23:35, 31 May 2016
Let $\mathbb{T}$ be a time scale and let $p,q \in \mathcal{R}(\mathbb{T},\mathbb{C})$ be (forward) regressive functions . We define the (forward) circle minus operation $\ominus_{\mu} \colon \mathbb{T} \rightarrow \mathbb{T}$ by $$\left( \ominus_{\mu} p \right)(t) = \dfrac{-p(t)}{1+p(t)\mu(t)}.$$ Since the set of forward regressive functions form a group $\left(\mathcal{R}(\mathbb{T},\mathbb{C}),\oplus_{\mu} \right)$ under circle plus with inverse operation $\ominus_{\mu}$, we define $$p \ominus_{\mu} q = p \oplus_{\mu} (\ominus_{\mu} q).$$
Contents
Properties
Forward regressive functions form a group
Theorem
The circle minus $\ominus_h$ is the inverse operation of the circle plus operation $\oplus_h$. Moreover, $$z \ominus_h w = z \oplus_h (\ominus_h w).$$
