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- Let $q>1$. The set $\overline{q^{\mathbb{Z}}}=\{0, \ldots, q^{-2}, q^{-1}, 1, q, q^2, \ldots \}$ of quantum numbers is a [[time scale]]. |[[Derivation of forward jump for T=Quantum q greater than 1|derivation]]5 KB (814 words) - 14:49, 15 January 2023
- ...q<1$. The set $\overline{q^{\mathbb{Z}}}=\{0, \ldots, q^{2}, q^{1}, 1, q^{-1}, q^{-2}, \ldots \}$ of quantum numbers is a [[time scale]]. |[[Derivation of forward jump for T=Quantum q greater than 1|derivation]]4 KB (735 words) - 00:45, 9 September 2015
- ...ative of constant multiple|next=Delta derivative of product (2)}}: Theorem 1.20 (iii)579 bytes (80 words) - 05:45, 10 June 2016
- ...nd at least one other point $t$ such that $0< t < 1$. Then $L_{\mathbb{T}}(1)=0$, where $L_{\mathbb{T}}$ denotes the [[Mozyrska-Torres logarithm]].480 bytes (63 words) - 15:28, 21 October 2017
- Let $\mathbb{T}$ be a [[time scale]]. If $t \in (0,1) \cap \mathbb{T}$, then $L_{\mathbb{T}}(t) < 0$, where $L_{\mathbb{T}}$ den ...res logarithm is increasing|next=Mozyrska-Torres logarithm is positive on (1,infinity)}}462 bytes (59 words) - 15:13, 21 January 2023
- Let $\mathbb{T}$ be a time scale. If $t \in (1,\infty) \cap \mathbb{T}$, then $L_{\mathbb{T}}(t) > 0$, where $L_{\mathbb{R ...r2 = Delfim F. M. Torres|prev=Mozyraska-Torres logarithm is negative on (0,1)|next=Mozyrska-Torres logarithm composed with forward jump}}465 bytes (61 words) - 15:13, 21 January 2023
- #REDIRECT [[Mozyrska-Torres logarithm is positive on (1,infinity)]]67 bytes (7 words) - 15:21, 21 October 2017
Page text matches
- ...], and when [[Quantum q greater than 1 | $\mathbb{T}=\{1,q,q^2,\ldots\}, q>1$]], the resulting theory becomes the [https://en.wikipedia.org/wiki/Quantum5 KB (665 words) - 01:55, 6 February 2023
- $$y'=y; y(s)=1.$$ $$y^{\Delta}=y;y(s)=1.$$839 bytes (127 words) - 20:55, 20 October 2014
- $$\hat{\xi}_h(z) = -\dfrac{1}{h} \log(1-zh).$$ $$y^{\nabla} = py; y(s)=1.$$3 KB (538 words) - 01:11, 19 December 2016
- File:Integerexponential,a=2,s=-1plot.png|Graph of $e_2(t,-1;\mathbb{Z})$. File:Integerexponential,a=2,s=1plot.png|Graph of $e_2(t,1;\mathbb{Z})$.4 KB (689 words) - 14:12, 28 January 2023
- # The integers: [[Integers | $\mathbb{Z} = \{\ldots, -1,0,1,\ldots\}$]] # Quantum numbers ($q>1$): [[Quantum_q_greater_than_1 | $\overline{q^{\mathbb{Z}}}$]]4 KB (545 words) - 14:47, 15 January 2023
- [[Delta derivative of product (1)]]<br /> ...on time scales|next=Delta differentiable implies continuous}}: Definition 1.102 KB (249 words) - 15:19, 21 January 2023
- |[[Derivation of delta sin sub 1 for T=R|derivation]] |[[Derivation of delta cos sub 1 for T=R|derivation]]5 KB (842 words) - 15:55, 15 January 2023
- ...style\sum_{k=1}^n \dfrac{1}{k} \colon n \in \mathbb{Z}^+ \right\} = \left\{1,\frac{3}{2},\frac{11}{6},\frac{25}{12},\frac{137}{60},\frac{49}{20},\frac{ ...hen for some positive integer $n$, $t=H_n=\displaystyle\sum_{k=0}^n \dfrac{1}{k}$.5 KB (717 words) - 00:38, 9 September 2015
- ...ndard metric $d(x,y)=|x-y|$ but an equivalent bounded metric $d(x,y)=\min\{1,|x-y|\}$. It [http://books.google.com/books?id=UrsHbOjiR8QC&pg=PA161&lpg=PA &= \max \left\{ 0, 1 \right\} \\4 KB (659 words) - 03:18, 26 April 2015
- $$1+\mu(t)p(t)\neq 0.$$253 bytes (42 words) - 12:58, 16 January 2023
- :[[Delta exponential dynamic equation|$(1)$]]246 bytes (27 words) - 17:01, 11 February 2017
- |$h_k(t,s) = \dfrac{1}{k!} \displaystyle\prod_{\ell=0}^{k-1}(t-\ell h-s)$ \displaystyle\prod_{k=\frac{t}{h}}^{\frac{s}{h}-1} \dfrac{1}{1+hp(hk)} &; t < s \\5 KB (819 words) - 15:55, 15 January 2023
- The set $\mathbb{Z}=\{\ldots,-2,-1,0,1,2,\ldots\}$ of integers is a [[time scale]]. |$\sigma(t)=t+1$5 KB (867 words) - 01:14, 19 February 2016
- ...c{1}{n} \colon n \in \mathbb{Z}^+\right\}}=\left\{ 0,1,\dfrac{1}{2},\dfrac{1}{3},\ldots \right\}$, where the $\overline{\mathrm{overline}}$ denotes topo |+$\mathbb{T}=\overline{\left\{\dfrac{1}{n} \colon n \in \mathbb{Z}^+\right\}}$5 KB (785 words) - 22:32, 23 February 2016
- Let $q>1$. The set $\overline{q^{\mathbb{Z}}}=\{0, \ldots, q^{-2}, q^{-1}, 1, q, q^2, \ldots \}$ of quantum numbers is a [[time scale]]. |[[Derivation of forward jump for T=Quantum q greater than 1|derivation]]5 KB (814 words) - 14:49, 15 January 2023
- $$\mathrm{Im}_h(z)=\dfrac{\mathrm{Arg}(zh+1)}{h},$$701 bytes (105 words) - 15:40, 21 January 2023
- $$\mathrm{Re}_h(z)=\dfrac{|zh+1|-1}{h}.$$584 bytes (85 words) - 15:41, 21 January 2023
- ...l $t \in \mathbb{T}$, $\mu(t) \geq \epsilon$. Let $\mathbb{T}=\{\ldots,t_{-1},t_0,t_1,\ldots\}$ be a [[time_scale | time scale]] of isolated points with |$\sigma(t_n)=t_{n+1}$5 KB (870 words) - 23:20, 9 June 2015
- The set $\mathbb{Z}^2 = \{0,1,4,9,16,\ldots\}$ of square integers is a [[time scale]]. |$\sigma(t)=t+2\sqrt{t}+1$4 KB (616 words) - 01:27, 22 May 2015
- ...displaystyle\int_0^{\infty} p_{f \boxminus_{\mu}1}(\eta,s) e_{\ominus_{\mu}1}^{\sigma}(\eta,0) \Delta \eta,$$ |$\displaystyle\int_0^{\infty} \left( \dfrac{\tau}{s} \right)^{x-1}e^{-\tau} \mathrm{d}\tau$2 KB (299 words) - 12:53, 16 January 2023
- ...{Z}^+$, and $[k]_{\mathbb{T}}$ is constant on $\mathbb{T}^+$ for all $k\in[1,n]\bigcap \mathbb{Z}^+$, then ...bb{T}}\left( [n]_{\mathbb{T}};s \right) = \dfrac{[n-1]_{\mathbb{T}}!}{s^{n-1}},$$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
