Isolated points
From timescalewiki
Let $\mathbb{T}$ be a time scale. We say that $\mathbb{T}$ is a time scale of isolated points if there exists $\epsilon > 0$ such that for all $t \in \mathbb{T}$, $\mu(t) \geq \epsilon$. Let $\mathbb{T}=\{\ldots,t_{-1},t_0,t_1,\ldots\}$ be a time scale of isolated points with $t_k > t_n$ iff $k>n$. Define the bijection $\pi \colon \mathbb{T} \rightarrow \mathbb{Z}$, $\pi(t_k)=k$.
| Generic element $t \in \mathbb{T}$: | for some $n \in \mathbb{Z}$, $t=t_n$ | |
| Forward jump: | $\sigma(t_n)=t_{n+1}$ | derivation |
| Forward graininess: | $\mu(t_n)=t_{n+1}-t_n$ | derivation |
| Backward jump: | $\rho(t_n)=t_{n-1}$ | derivation |
| Backward graininess: | $\nu(t_n)=t_{n}-t_{n-1}$ | derivation |
| $\Delta$-derivative | $f^{\Delta}(t_n)=\dfrac{f(t_{n+1})-f(t_n)}{t_{n+1}-t_n}$ | derivation |
| $\nabla$-derivative | $f^{\nabla}(t_n)=\dfrac{f(t_n)-f(t_{n-1})}{t_{n}-t_{n-1}}$ | derivation |
| $\Delta$-integral | $\displaystyle\int_{s}^{t} f(\tau) \Delta \tau= \left\{ \begin{array}{ll} -\displaystyle\sum_{k=\pi(t)}^{\pi(s)-1} \mu(t_k)f(t_k) &; t<s, \\ 0 &; t=s, \\ \displaystyle\sum_{k=\pi(s)}^{\pi(t)-1} \mu(t_k)f(t_k) &; t>s \end{array}\right.$ | derivation |
| $\nabla$-integral | $\displaystyle\int_s^t f(\tau) \nabla \tau=\left\{ \begin{array}{ll} -\displaystyle\sum_{k=\pi(t)+1}^{\pi(s)} \nu(t_k)f(t_k) &; t<s, \\ 0&; t=s, \\ \displaystyle\sum_{k=\pi(s)+1}^{\pi(t)} \nu(t_k)f(t_k) &; t>s \end{array} \right.$ | derivation |
| $h_k(t,s)$ | $h_k(t,s)=$ | derivation |
| $\hat{h}_k(t,s)$ | $\hat{h}_k(t,s)=$ | derivation |
| $g_k(t,s)$ | $g_k(t,s)=$ | derivation |
| $\hat{g}_k(t,s)$ | $\hat{g}_k(t,s)=$ | derivation |
| $e_p(t,s)$ | $e_p(t,s)=$ | derivation |
| $\hat{e}_p(t,s)$ | $\hat{e}_p(t,s)=$ | derivation |
| Gaussian bell | $\mathbf{E}(t)=$ | derivation |
| $\mathrm{sin}_p(t,s)=$ | $\sin_p(t,s)=$ | derivation |
| $\mathrm{\sin}_1(t,s)$ | $\sin_1(t,s)=$ | derivation |
| $\widehat{\sin}_p(t,s)$ | $\widehat{\sin}_p(t,s)=$ | derivation |
| $\mathrm{\cos}_p(t,s)$ | $\cos_p(t,s)=$ | derivation |
| $\mathrm{\cos}_1(t,s)$ | $\cos_1(t,s)=$ | derivation |
| $\widehat{\cos}_p(t,s)$ | $\widehat{\cos}_p(t,s)=$ | derivation |
| $\sinh_p(t,s)$ | $\sinh_p(t,s)=$ | derivation |
| $\widehat{\sinh}_p(t,s)$ | $\widehat{\sinh}_p(t,s)=$ | derivation |
| $\cosh_p(t,s)$ | $\cosh_p(t,s)=$ | derivation |
| $\widehat{\cosh}_p(t,s)$ | $\widehat{\cosh}_p(t,s)=$ | derivation |
| Gamma function | $\Gamma_{\mathbb{T}_{\mathrm{iso}}}(x,s)=$ | derivation |
| Euler-Cauchy logarithm | $L(t,s)=$ | derivation |
| Bohner logarithm | $L_p(t,s)=$ | derivation |
| Jackson logarithm | $\log_{\mathbb{T}_{\mathrm{iso}}} g(t)=$ | derivation |
| Mozyrska-Torres logarithm | $L_{\mathbb{T}_{\mathrm{iso}}}(t)=$ | derivation |
| Laplace transform | $\mathscr{L}_{\mathbb{T}_{\mathrm{iso}}}\{f\}(z;s)=$ | derivation |
| Hilger circle | derivation |
