Difference between revisions of "Forward regressive"

Revision as of 05:13, 18 May 2014

Let $\mathbb{T}$ be a time scale. Let $p \colon \mathbb{T} \rightarrow \mathbb{R}$. We say that $p$ is regressive if for all $t \in \mathbb{T}^{\kappa}$ $$1+\mu(t)p(t)\neq 0.$$ 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 with subtraction $(p \ominus q)(t) = (p \oplus (\ominus q))(t) = \dfrac{p(t)-q(t)}{1+\mu(t)q(t)}$.