Search results
Create the page "$1" on this wiki! See also the search results found.
Page title matches
Page text matches
- ...entiable on $\mathbb{T}^{\kappa^n}$. Let $\alpha \in \mathbb{T}^{\kappa^{n-1}}, t\in\mathbb{T}$ then ...ha) f^{\Delta^k}(\alpha) + \displaystyle\int_{\alpha}^{\rho^{n-1}(t)} h_{n-1}(t,\sigma(\tau)) f^{\Delta^n}(\tau) \Delta \tau,$$516 bytes (83 words) - 17:05, 15 January 2023
- | $\cosh_1(t,0)=\dfrac{1}{2}\left( (1-h)^{\frac{t}{h}} + (1+h)^{\frac{t}{h}}\right) = \displaystyle\sum_{k=0}^{\infty} h_{2k}(t,0) $ |[[Quantum_q_greater_than_1 | $\overline{q^{\mathbb{Z}}}, q > 1$]]1 KB (169 words) - 14:13, 28 January 2023
- ...isplaystyle\int_0^{\infty} \int_0^{\infty} f_{X,Y}(x,y) \Delta y \Delta x=1$. ...umsystem.edu/xmlui/bitstream/handle/10355/29595/Matthews_2011.pdf?sequence=1 Probability theory on time scales and applications to finance and inequalit500 bytes (83 words) - 04:37, 6 March 2015
- $$\xi_h^{-1}(z)=\dfrac{e^{zh}-1}{h}.$$552 bytes (86 words) - 00:57, 30 May 2017
- ...$f_i,g_k \colon \mathbb{R} \rightarrow \mathbb{R}$ for $i=0,1,2$ and $k=0,1$. The Abel dynamic equation of the second kind is447 bytes (76 words) - 19:28, 5 April 2015
- \sin^{\Delta}_p(t,t_0) &= \dfrac{1}{2i} \dfrac{\Delta}{\Delta t} \left( e_{ip}(t,t_0) - e_{-ip}(t,t_0) \right) &= \dfrac{1}{2} (e_{ip}(t,t_0)+e_{-ip}(t,t_0)) \\601 bytes (104 words) - 21:28, 9 June 2016
- \cos_p^{\Delta}(t,t_0) &= \dfrac{1}{2} \dfrac{\Delta}{\Delta t} \Big(e_{ip}(t,t_0) + e_{-ip}(t,t_0) \Big) \\620 bytes (103 words) - 01:51, 6 February 2023
- Let $\mathbb{T}$ be a [[time scale]] and let $0\leq \alpha \leq 1$. The $\Diamond_{\alpha}$-derivative of a function $f \colon \mathbb{T} \ri $$\left| \alpha[f^{\sigma}(t)-f(s)]\eta_{ts} + (1-\alpha)[f^{\rho}(t)-f(s)]\mu_{ts}-f^{\Diamond_{\alpha}}\mu_{ts}\eta_{ts} \r2 KB (274 words) - 08:32, 12 April 2015
- where $i=\sqrt{-1}$.877 bytes (127 words) - 14:13, 28 January 2023
- ...{\displaystyle\prod_{k=t_0}^{t-1}1+ip(k) - \displaystyle\prod_{k=t_0}^{t-1}1-ip(k)}{2i}$ |[[Quantum_q_greater_than_1 | $\overline{q^{\mathbb{Z}}}, q > 1$]]597 bytes (86 words) - 18:39, 21 March 2015
- h_0(t,s;\mathbb{T})=1 \\ h_{k+1}(t,s;\mathbb{T})= \displaystyle\int_s^t h_{k}(\tau,s;\mathbb{T}) \Delta \ta817 bytes (135 words) - 14:13, 28 January 2023
- | $\dfrac{1}{k!} \displaystyle\prod_{\ell=0}^{k-1}(t-\ell h-s)$ |[[Quantum_q_greater_than_1 | $\overline{q^{\mathbb{Z}}}, q > 1$]]691 bytes (102 words) - 01:31, 24 September 2016
- |[[Quantum_q_greater_than_1 | $\overline{q^{\mathbb{Z}}}, q > 1$]] |[[Quantum_q_less_than_1 | $\overline{q^{\mathbb{Z}}}, q < 1$]]445 bytes (61 words) - 18:36, 21 March 2015
- where $q$ obeys $\dfrac{1}{p}+\dfrac{1}{q}=1$ with $p>1$. Then, ...\left( \displaystyle\int_a^b h(x)g^q(x) \Diamond_{\alpha} x \right)^{\frac{1}{q}}.$$732 bytes (122 words) - 12:36, 28 March 2015
- ...i}{s_i \neq \sigma_i(t_i)}} \dfrac{f(t_1,\ldots,t_{i-1},\sigma_i(t_i),t_{i+1},\ldots,t_n)-f(t_1,\ldots,t_n)}{\sigma_i(t_i)-s_i}$$ ...mic equations on time scales|2006|Billy Jackson||prev=|next=}}: Definition 11 KB (185 words) - 14:12, 28 January 2023
- $$(f \circ g)^{\Delta}(t) = \left\{ \displaystyle\int_0^1 f'(g(t)+h\mu(t)g^{\Delta}(t)) dh \right\} g^{\Delta}(t)$$1 KB (154 words) - 18:37, 6 April 2015
- $$e_q(t,s)=\hat{e}_{\frac{q^{\rho}}{1+q^{\rho}\nu}}(t,s)=\hat{e}_{\ominus_{\nu}(-q^{\rho})}(t,s),$$392 bytes (58 words) - 22:22, 9 June 2016
- $$\hat{e}_p(t,s)=e_{\frac{p^{\sigma}}{1-p^{\sigma}\nu}}(t,s)=e_{\ominus(-p^{\sigma})}(t,s),$$366 bytes (54 words) - 22:22, 9 June 2016
- \hat{h}_{n+1}(t,s)=\displaystyle\int_s^t \hat{h}_n(\tau,s) \nabla \tau. ...ne{q^{\mathbb{Z}}}, q>1$, then $\hat{h}_k(t,s)=\displaystyle\prod_{r=0}^{k-1} \dfrac{q^rt-s}{\sum_{j=0}^r q^j}$.529 bytes (101 words) - 07:02, 14 April 2015
- $$\mathrm{E}_{\mathbb{T}}(X)=\dfrac{1}{\lambda}.$$230 bytes (28 words) - 14:07, 28 January 2023
- $$\mathrm{Var}_{\mathbb{T}}(X)=\dfrac{1}{\lambda^2},$$280 bytes (33 words) - 14:06, 28 January 2023
- # The integers: [[Integers | $\mathbb{Z} = \{\ldots, -1,0,1,\ldots\}$]] # Quantum numbers ($q>1$): [[Quantum_q_greater_than_1 | $\overline{q^{\mathbb{Z}}}$]]878 bytes (119 words) - 07:22, 29 April 2015
- ...displaystyle\int_s^t \displaystyle\lim_{h \rightarrow 0} \dfrac{1}{h} \log(1 + hp(\tau)) d\tau \right) \\ ...\left( \displaystyle\int_s^t \displaystyle\lim_{h \rightarrow 0} \dfrac{1}{1+hp(\tau)} p(\tau) d\tau \right) \\389 bytes (63 words) - 19:13, 29 April 2015
- ...s of [[Delta exponential|$e_p$]], it is clear that $\sin_p(t,s) = \dfrac{1-1}{2i} = 0$. Furthermore if $t>s$, then ...frac{\displaystyle\prod_{k=s}^{t-1}1+ip(k) - \displaystyle\prod_{k=s}^{t-1}1-ip(k)}{2i}.$$466 bytes (92 words) - 00:46, 22 May 2015
- $$L(t,s)=\displaystyle\int_{s}^t \dfrac{1}{\tau + 2\mu(\tau)} \Delta \tau.$$653 bytes (87 words) - 15:15, 21 January 2023
- ...e numbers including $1$ and at least one other point $t$ such that $0< t < 1$. For $t \in \mathbb{T} \cap (0,\infty)$, define $$L_{\mathbb{T}}(t) = \displaystyle\int_1^t \dfrac{1}{\tau} \Delta \tau.$$1 KB (128 words) - 18:56, 11 December 2017
- :1. The scalar case ::[[Jackson logarithm|Definition 1.1, (1.1)]]325 bytes (36 words) - 17:30, 11 February 2017
- $$-\dfrac{1}{h} < \mathrm{Re}_h(z) < \infty,$$298 bytes (43 words) - 12:59, 19 August 2017
- e_p(t,s) &= \exp \left( \displaystyle\int_s^t \dfrac{1}{\mu(\tau)} \log(1+\mu(\tau)p(\tau)) \Delta \tau \right) \\ &= \exp \left( \displaystyle\sum_{k=\pi(s)}^{\pi(t)-1} \log(1+\mu(t_k)p(t_k)) \right) \\417 bytes (78 words) - 23:35, 9 June 2015
- \hat{e}_p(t,s) &= \exp \left(-\displaystyle\int_s^t \dfrac{1}{\nu(\tau)} \log(1-\nu(\tau)p(\tau)) \Delta \tau \right) \\ &= \exp \left(-\displaystyle\sum_{k=\pi(s)+1}^{\pi(t)} \log(1-\nu(t_k)p(t_k)) \right) \\440 bytes (85 words) - 23:39, 9 June 2015
- \sigma(t) &= \sigma \left( \displaystyle\sum_{k=1}^{n} \dfrac{1}{k} \right) \\ &= \displaystyle\sum_{k=1}^{n+1} \dfrac{1}{k} \\184 bytes (25 words) - 06:28, 16 June 2015
- \mu(t) &= \mu \left( \displaystyle\sum_{k=1}^{n} \dfrac{1}{k} \right) \\ &= t + \dfrac{1}{n+1} - t \\154 bytes (23 words) - 06:32, 16 June 2015
- ...quation $y^{\Delta}(t)=p(t)y(t);y(s)=1$ to get the equation $y(\sigma(t))=(1+hp(t))y(t)$. From this it is clear that ...+hp(s))y(s)=(1+hp(s))=\displaystyle\prod_{k=\frac{s}{h}}^{\frac{s+h}{h}-1} 1+hp(hk),$$1 KB (282 words) - 04:38, 27 July 2015
- ...n $y^{\nabla}(t)=p(t)y(t);y(s)=1$ to get the equation $y(t)=\dfrac{y(t-h)}{1-hp(t-h)}$. From this it is clear that ...{1-hp(s+h)}=\displaystyle\prod_{k=\frac{s+h}{h}}^{\frac{s+h}{h}} \dfrac{1}{1-hp(hk)},$$1 KB (283 words) - 04:47, 27 July 2015
- $$\mathbb{C}_h = \left\{ z \in \mathbb{C} \colon z \neq -\dfrac{1}{h} \right\},$$650 bytes (94 words) - 12:45, 6 June 2023
- $$\mathbb{R}_h = \left\{ z \in \mathbb{R} \colon z > -\dfrac{1}{h} \right\},$$668 bytes (93 words) - 15:40, 21 January 2023
- The Hilger alternating axis is defined for $h>1$ by $$\mathbb{A}_h = \left\{z \in \mathbb{R} \colon z < -\dfrac{1}{h} \right\},$$645 bytes (90 words) - 15:38, 21 January 2023
- ...\left\{ z \in \mathbb{C}_h \colon \left| z + \dfrac{1}{h} \right| = \dfrac{1}{h} \right\},$$707 bytes (96 words) - 15:40, 21 January 2023
- *$\{0,1\} \oplus \{4,5,10\} = \{4,5,6,10,11\}$ *$\{0,2\} \oplus [0,1] = [0,1] \cup [2,3]$500 bytes (64 words) - 15:27, 21 January 2023
- |[[Quantum_q_greater_than_1 | $\overline{q^{\mathbb{Z}}}, q > 1$]] |[[Quantum_q_less_than_1 | $\overline{q^{\mathbb{Z}}}, q < 1$]]2 KB (273 words) - 14:11, 28 January 2023
- $$\mathring{\iota} \omega = \dfrac{e^{2\pi i \omega}-1}{h},$$ where $i=\sqrt{-1}$.1 KB (195 words) - 15:40, 21 January 2023
- $$\left( \ominus_{\mu} p \right)(t) = \dfrac{-p(t)}{1+p(t)\mu(t)}.$$523 bytes (77 words) - 15:26, 21 January 2023
- |[[Quantum_q_greater_than_1 | $\overline{q^{\mathbb{Z}}}, q > 1$]] |[[Quantum_q_less_than_1 | $\overline{q^{\mathbb{Z}}}, q < 1$]]1 KB (215 words) - 14:51, 21 January 2023
- :1. Introduction ::(1.1) [[Laplace transform]]487 bytes (55 words) - 14:48, 21 January 2023
- The proof is the same as [[Derivation of delta exponential T=hZ]] with $h=1$.77 bytes (14 words) - 01:13, 19 February 2016
- Let $n \in \{1,2,\ldots\}$. The set $\sqrt[n]{\mathbb{N}_0}=\{0,1,\sqrt[n]{2}, \sqrt[n]{3},\ldots\}$ is a [[time scale]]. |$\sigma(t)=\sqrt[n]{t^n+1}$5 KB (807 words) - 00:56, 11 December 2016
- $$e_0(t,s;\mathbb{T})=1,$$257 bytes (37 words) - 21:31, 9 June 2016
- $$e_p(t,t;\mathbb{T})=1,$$323 bytes (47 words) - 22:19, 9 June 2016
- ...b{T}$, and let $k$ be a nonnegative integer. Then for all $0 \leq n \leq k-1$,370 bytes (60 words) - 01:58, 10 June 2016
- $$h_k(t,s;\mathbb{T})=(-1)^kg_k(s,t;\mathbb{T}),$$342 bytes (59 words) - 15:40, 22 September 2016
