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- ...{Z}^+$, and $[k]_{\mathbb{T}}$ is constant on $\mathbb{T}^+$ for all $k\in[1,n]\bigcap \mathbb{Z}^+$, then ...bb{T}}\left( [n]_{\mathbb{T}};s \right) = \dfrac{[n-1]_{\mathbb{T}}!}{s^{n-1}},$$494 bytes (72 words) - 18:07, 15 January 2023
- 1&; n=0 \\ \displaystyle\prod_{j=1}^n [j]_{\mathbb{T}} &; n=1,2,\ldots435 bytes (60 words) - 12:53, 16 January 2023
- [n-1]_{\mathbb{T}} \boxplus_{\mu} 1 &; n=1,2,\ldots464 bytes (63 words) - 12:53, 16 January 2023
- $$\Gamma_{\mathbb{T}}\left(x \boxplus_{\mu} 1;s\right) = \dfrac{x}{s} \Gamma_{\mathbb{T}}(x;s),$$387 bytes (58 words) - 18:08, 15 January 2023
- $$\Gamma_{\mathbb{T}}(1;s)=1,$$242 bytes (32 words) - 17:57, 15 January 2023
- :1. Introduction ::2.1. Relevant time scales1 KB (119 words) - 16:01, 15 January 2023
- If $\mathbb{T}$ is a [[time scale]], $a,b \in \mathbb{T}$ with $a<b$, $p>1$, and $f,g \colon [a,b]\cap \mathbb{T}\rightarrow \mathbb{R}$ are continuou ...+ \left( \displaystyle\int_a^b |g(x)|^p \Diamond_{\alpha} x \right)^{\frac{1}{p}},$$566 bytes (94 words) - 15:17, 21 January 2023
- :1. Unifying Continuous and Discrete Analysis ::Theorem 3.1: [[Delta Hölder inequality]] <br />892 bytes (95 words) - 22:44, 10 February 2017
- |[[Quantum_q_greater_than_1 | $\overline{q^{\mathbb{Z}}}, q > 1$]] |[[Quantum_q_less_than_1 | $\overline{q^{\mathbb{Z}}}, q < 1$]]2 KB (316 words) - 15:21, 21 January 2023
- <td><center>[[Quantum q greater than 1|$\huge\overline{q^{\mathbb{Z}}}$]]<br /> [[Quantum q greater than 1|Quantum, $q>1$]]</center></td>1 KB (211 words) - 22:29, 23 February 2016
- $$p(t)=\ominus(t \odot 1).$$ |$2^{\frac{-t(t-1)}{2}}$1 KB (193 words) - 15:03, 21 January 2023
- File:Arccos.png|Graph of $\mathrm{arccos}$ on $[-1,1]$. |[[Quantum_q_greater_than_1 | $\overline{q^{\mathbb{Z}}}, q > 1$]]6 KB (884 words) - 07:54, 1 June 2016
- |[[Quantum_q_greater_than_1 | $\overline{q^{\mathbb{Z}}}, q > 1$]] |[[Quantum_q_less_than_1 | $\overline{q^{\mathbb{Z}}}, q < 1$]]582 bytes (98 words) - 07:57, 23 March 2015
- $$e_{\alpha} \geq 1 + \alpha(t-s),$$539 bytes (70 words) - 15:45, 21 January 2023
- $$y(t) \leq G^{-1} \left[ G(\beta) + \displaystyle\int_a^t p(\tau) \Delta \tau \right]$$799 bytes (124 words) - 00:36, 15 September 2016
- ...{\frac{1}{p}} \left(\displaystyle\int_a^b |g(t)|^q \Delta t \right)^{\frac{1}{q}}$$ where $p>1$ and $q = \dfrac{p}{p-1}$.637 bytes (97 words) - 00:36, 15 September 2016
- Let $a,b \in \mathbb{T}$ and $p>1$. For [[continuity | rd-continuous]] $f,g \colon [a,b] \cap \mathbb{T} \rig ...\frac{1}{p}}+ \left( \displaystyle\int_a^b |g(t)|^p \Delta t\right)^{\frac{1}{p}}.$$644 bytes (96 words) - 00:38, 15 September 2016
- ...[a,b] \cap \mathbb{T}} \dfrac{M(t)}{M^{\sigma}(t)} \right) \right]^{\frac{1}{2}} \right\}^2.$$962 bytes (160 words) - 00:39, 15 September 2016
- ...}(\cdot,s)=0; \hat{y}(\sigma(s),s)=0, \hat{y}^{\Delta}(\sigma(s),s)=\dfrac{1}{p(\sigma(s))}.$$498 bytes (85 words) - 22:17, 27 June 2015
- $$\dfrac{1}{e_p(t,s;\mathbb{T})}=e_{\ominus p}(s,t;\mathbb{T}),$$408 bytes (64 words) - 22:20, 9 June 2016
- $$e_p(\sigma(t),s;\mathbb{T})=(1+\mu(t)p(t))e_p(t,s;\mathbb{T}),$$ ...ev=Delta derivative at right-dense|next=Delta derivative of sum}}: Theorem 1.16 (iv)645 bytes (97 words) - 06:08, 10 June 2016
- Supposing \( f \) is differentiable, given that, by Definition 1.10, $ \epsilon^* > 0 $ there is a neighborhood \( U \) \( (U = (t - \delta, ...gative image) and considering (without loss of generality) \( \epsilon^* < 1 \), it follows that4 KB (666 words) - 01:14, 15 March 2022
- e_p(t,s) &= \exp \left( \displaystyle\int_{s}^{t} \dfrac{1}{\mu(\tau)} \log(1 + p(\tau)) \Delta \tau \right) \\ &= \exp \left( \displaystyle\sum_{k=s}^{t-1} \log(1+p(k)) \right) \\512 bytes (92 words) - 19:33, 29 April 2015
- g_0(t,s)=1 \\ g_{k+1}(t,s)=\displaystyle\int_s^t g_k(\sigma(\tau),s) \Delta \tau.812 bytes (134 words) - 14:13, 28 January 2023
- \dfrac{1}{\sigma(b)-a} &; a \leq t \leq b \\515 bytes (74 words) - 01:22, 30 September 2018
- ...bb{R}$ with $\lambda > 0$ and define $\Lambda_0(t,t_0)=0, \Lambda_1(t,t_0)=1$ and $$\Lambda_{k+1}(t,t_0) = -\displaystyle\int_{t_0}^t (\ominus \lambda)(\tau) \Lambda_k(\sig626 bytes (90 words) - 14:11, 28 January 2023
- $$\displaystyle\int_0^{\infty} f(t) \Delta t = 1.$$ ...umsystem.edu/xmlui/bitstream/handle/10355/29595/Matthews_2011.pdf?sequence=1 Probability theory on time scales and applications to finance and inequalit578 bytes (80 words) - 21:44, 14 April 2015
- The cylinder strip $\mathbb{Z}_h$ is defined for $h>1$ by461 bytes (70 words) - 00:50, 30 May 2017
- ...=Ian A. Gravagne|author3=Robert J. Marks II|prev=findme|next=findme}}: $(3.1)$ ...plications|2021|Tom Cuchta|author2=Svetlin Georgiev|prev=|next=}}: Section 1927 bytes (130 words) - 15:12, 21 January 2023
- $$L_{\epsilon}(\infty) := \left\{t \in \mathbb{T} \colon t > \dfrac{1}{\epsilon} \right\}$$ <strong>Theorem (L'Hospital's Rule 1):</strong> Assume $f,g$ $\Delta$-differentiable on $\mathbb{T}$ and for som2 KB (360 words) - 08:11, 8 February 2015
- ...entiable on $\mathbb{T}^{\kappa^n}$. Let $\alpha \in \mathbb{T}^{\kappa^{n-1}}, t\in\mathbb{T}$ then ...ha) f^{\Delta^k}(\alpha) + \displaystyle\int_{\alpha}^{\rho^{n-1}(t)} h_{n-1}(t,\sigma(\tau)) f^{\Delta^n}(\tau) \Delta \tau,$$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
