Difference between revisions of "Nabla exponential"
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Revision as of 09:16, 12 April 2015
The $\nabla$-exponential functions are examples of exponential functions on a time scale. Define the function $\hat{\xi}_{h} \colon \mathbb{C}_h \rightarrow \mathbb{Z}_h$ by $$\hat{\xi}_h(z) = \dfrac{1}{h} \log(1-zh).$$ Define the $\nabla$ exponential function for $s,t \in \mathbb{T}$ by $$\hat{e}_p(t,s) = \exp \left( \displaystyle\int_s^t \hat{\xi}_{\nu(\tau)}(p(\tau)) \nabla \tau \right).$$
Contents
Properties of $\nabla$-exponential functions
The function $\hat{e}_p(\cdot,s)$ is the unique solution of the initial value problem $$y^{\nabla} = py; y(s)=1.$$ For all $p,q \in \mathcal{R}_{\nu}$ and $t,s \in \mathbb{T}$,
- $\hat{e}_p(t,r)\hat{e}_p(r,s)=e_p(t,s)$ (semigroup property)
- $\hat{e}_0(t,s)=1, \hat{e}_p(t,t)=1$
- $\hat{e}_p(\rho(t),s)=(1-\nu(t)p(t))\hat{e}_p(t,s)$
- $\dfrac{1}{\hat{e}_p(t,s)}=\hat{e}_{\ominus_{\nu} p}(s,t)$
- $\hat{e}_p(t,s)\hat{e}_q(t,s)=\hat{e}_{p \oplus_{\nu} q}(t,s)$
- $\dfrac{\hat{e}_p(t,s)}{\hat{e}_q(t,s)} = \hat{e}_{p \ominus_{\nu} q}(t,s)$
- $\left( \dfrac{1}{\hat{e}_p(\cdot,s)} \right)^{\nabla} = -\dfrac{p(t)}{\hat{e}_p^{\rho}(\cdot,s)}$
Theorem: (Sign of the Nabla Exponential Function) Let $p \in \mathcal{R}_{\nu}$ and $s \in \mathbb{T}$.
i.) If $p \in \mathcal{R}_{\nu}^+$, then $\hat{e}_{p}(t,s) > 0$ for all $t \in \mathbb{T}$.
ii.) If $1-\nu(t)p(t) < 0$ for some $t \in \mathbb{T}_{\kappa}$, then
$$\hat{e}(\rho(t),s)\hat{e}_{p}(t,s)<0.$$
iii.) If $1-\nu(t)p(t) < 0$ for all $t \in \mathbb{T}$, then $\hat{e}_p(t,s)$ changes sign at every point of $\mathbb{T}$.
iv.) The exponential function $\hat{e}_p(\cdot,s)$ is a real-valued function that is never equal to zero.
Proof: proof goes here █
Theorem
If $q$ is continuous and $\mu$-regressive then $$e_q(t,s)=\hat{e}_{\frac{q^{\rho}}{1+q^{\rho}\nu}}(t,s)=\hat{e}_{\ominus_{\nu}(-q^{\rho})}(t,s),$$ where $e_q$ denotes the $\Delta$-exponential and $\hat{e}_q$ denotes the $\nabla$-exponential.
Proof
References
Theorem
If $p$ is continuous and $\nu$-regressive then $$\hat{e}_p(t,s)=e_{\frac{p^{\sigma}}{1-p^{\sigma}\nu}}(t,s)=e_{\ominus(-p^{\sigma})}(t,s),$$ where $\hat{e}_p$ denotes the $\nabla$-exponential and $e_p$ denotes the $\Delta$-exponential.
Proof
References
Examples
Let $p$ be a $\nu$-regressive function.
| $\mathbb{T}=$ | $\hat{e}_{p}(t,s)=$ |
| $\mathbb{R}$ | |
| $\mathbb{Z}$ | $\hat{e}_p(t,s)=\left\{ \begin{array}{ll} \displaystyle\prod_{k=s}^{t-1} \dfrac{1}{1-p(k)} &; t \gt s \\ 1 &; t=s \\ \prod_{k=t}^{s-1} (1-p(k)) &; t \lt s \end{array} \right.$ |
| $h\mathbb{Z}$ | $\hat{e}_p(t,s)=\left\{ \begin{array}{ll} \displaystyle\prod_{k=\frac{s}{h}}^{\frac{t}{h}-1} \dfrac{1}{1-hp(hk)} &; t \gt s \\ 1 &; t=s \\ \prod_{k=\frac{t}{h}}^{\frac{s}{h}-1} (1-hp(hk)) &; t \lt s \end{array} \right.$ |
| $\mathbb{Z}^2$ | |
| $\overline{q^{\mathbb{Z}}}, q > 1$ | |
| $\overline{q^{\mathbb{Z}}}, q < 1$ | |
| $\mathbb{H}$ |
Let $\alpha$ be a regressive constant.
| $\mathbb{T}=$ | $\hat{e}_{\alpha}(t,s)=$ |
| $\mathbb{R}$ | $\hat{e}_{\alpha}(t,s)=e^{\alpha(t-s)}$ |
| $\mathbb{Z}$ | $\hat{e}_{\alpha}(t,s)=\left( \dfrac{1}{1-\alpha} \right)^{t-s}$ |
| $h\mathbb{Z}$ | $\hat{e}_{\alpha}(t,s;h)=\left( \dfrac{1}{ 1-\alpha h} \right)^{\frac{t-s}{h}}$ |
| $\mathbb{Z}^2$ | |
| $\overline{q^{\mathbb{Z}}}, q > 1$ | $\hat{e}_{\alpha}(t,s;q)=\displaystyle\prod_{\xi \in [s,t)} \dfrac{1}{1-(q-1)\alpha \xi}; t>s$ |
| $\overline{q^{\mathbb{Z}}}, q < 1$ | |
| $\mathbb{H}$ |
References
Nabla dynamic equations on time scales
