Search results
Create the page "$1" on this wiki! See also the search results found.
Page title matches
- 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\}$$ <strong>Theorem (L'Hospital's Rule 1):</strong> Assume $f,g$ $\Delta$-differentiable on $\mathbb{T}$ and for som2 KB (360 words) - 08:11, 8 February 2015
- ...entiable on $\mathbb{T}^{\kappa^n}$. Let $\alpha \in \mathbb{T}^{\kappa^{n-1}}, t\in\mathbb{T}$ then ...ha) f^{\Delta^k}(\alpha) + \displaystyle\int_{\alpha}^{\rho^{n-1}(t)} h_{n-1}(t,\sigma(\tau)) f^{\Delta^n}(\tau) \Delta \tau,$$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}),$$342 bytes (59 words) - 15:40, 22 September 2016
- $$\mathrm{c}_{pq}^2(t,s;\mathbb{T})+\mathrm{s}_{pq}^2(t,s;\mathbb{T})=1,$$371 bytes (54 words) - 00:41, 15 September 2016
- [http://web.mst.edu/~bohner/sample.pdf Chapters 1-3 hosted by Martin Bohner]<br /> :Chapter 1. The Time Scales Calculus4 KB (374 words) - 15:27, 15 January 2023
- ...ev=Delta derivative of sum|next=Delta derivative of product (1)}}: Theorem 1.20 (ii)524 bytes (70 words) - 05:45, 10 June 2016
- ...a derivative of product (1)|next=Delta derivative of reciprocal}}: Theorem 1.20 (iii)572 bytes (79 words) - 05:44, 10 June 2016
- ...vative|delta differentiable]], and $f(t)f(\sigma(t)) \neq 0$. Then $\dfrac{1}{f}$ is delta differentiable and $$\left( \dfrac{1}{f} \right)^{\Delta}(t) = -\dfrac{f^{\Delta}(t)}{f(t)f(\sigma(t))},$$623 bytes (88 words) - 15:19, 21 January 2023
- ...Delta}(t)=\displaystyle\sum_{j=0}^{m-1} (\sigma(t)-\alpha)^j (t-\alpha)^{m-1-j},$$ ...nt|next=Delta derivative of reciprocal of classical polynomial}}: Theorem $1.24(i)$600 bytes (88 words) - 15:06, 25 September 2016
- ...}$, and define $g\colon \mathbb{T} \rightarrow \mathbb{R}$ by $g(t)=\dfrac{1}{(t-\alpha)^m}$. Then ...splaystyle\sum_{j=0}^{m-1} \dfrac{1}{(\sigma(t)-\alpha)^{m-j}(t-\alpha)^{j+1}},$$583 bytes (84 words) - 12:48, 23 September 2016
- $$\xi_h(z)=\dfrac{1}{h} \mathrm{Log}(1+zh),$$675 bytes (98 words) - 00:52, 30 May 2017
- [[Mozyrska-Torres logarithm at 1]]<br /> [[Mozyraska-Torres logarithm is negative on (0,1)]]<br />714 bytes (81 words) - 15:32, 21 October 2017
- ...ime scale including $1$ and at least one other point $t$ such that $0< t < 1$. The following formula holds: $$L_{\mathbb{T}}^{\Delta}(t) = \dfrac{1}{t},$$500 bytes (65 words) - 15:28, 21 October 2017
- ...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
- :1 Elements of the Time Scale Calculus ::1.1 Forward and Backward Jump Operators, Graininess Function5 KB (497 words) - 02:57, 20 December 2017
- Let $\mathbb{T}=\left\{0,\dfrac{1}{3},\dfrac{1}{2},\dfrac{7}{9},1,2,3,4,5,6,7 \right\}$.<br /> >>> ts=tsc.timescale([0,Fraction(1,3),Fraction(1,2),Fraction(7,9),1,2,3,4,5,6,7],'documentation example')4 KB (494 words) - 14:41, 4 December 2018
- ...garithm on time scales|2005|Martin Bohner|next=Euler-Cauchy logarithm}}: $(1)$588 bytes (76 words) - 17:02, 11 February 2017
- ...s logarithm on the reals|next=Mozyraska-Torres logarithm is negative on (0,1)}}438 bytes (59 words) - 15:13, 21 January 2023
- 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
- ...thor2 = Delfim F. M. Torres|prev=Mozyrska-Torres logarithm is positive on (1,infinity)|next=Euler-Cauchy logarithm}}541 bytes (71 words) - 15:13, 21 January 2023
- :1. Introduction ::Lemma 2.1710 bytes (78 words) - 20:19, 22 January 2023
- :1 Preliminaries ::1.1 Delta Calculus1 KB (129 words) - 02:37, 20 December 2017
- :1. Introduction650 bytes (71 words) - 14:57, 21 October 2017
- ...a Mozyrska|author2 = Delfim F. M. Torres|prev=Mozyrska-Torres logarithm at 1|next=Mozyrska-Torres logarithm is increasing}}534 bytes (68 words) - 15:14, 21 January 2023
- :Chapter 1 ::1.0 Introduction1 KB (134 words) - 02:43, 20 December 2017
- ...) + \displaystyle\sum_{k=1}^{n-1} \mu(t_k) \displaystyle\prod_{m=0}^{k-1} (1+z\mu(t_m)).$$366 bytes (56 words) - 15:28, 21 January 2023
- $$\mathrm{DFT}\{f\}(z_k)=\displaystyle\sum_{k=0}^{N-1} x(t_k) \overline{e_{z_n}(t_k)} \mu(t_k),$$522 bytes (90 words) - 04:00, 26 February 2018
- :1. Introduction and motivation ::[[Partial Delta Derivative|Definition 1]]876 bytes (94 words) - 15:00, 15 January 2023
- ...\{f\}(z;s) = \displaystyle\sum_{k=-\infty}^{\infty} \dfrac{f(k)}{(1+iz)^{k+1-s}}$ ...\}(z;s) = h\displaystyle\sum_{k=-\infty}^{\infty} \dfrac{f(hk)}{(1+ihz)^{k+1-\frac{s}{h}}}$2 KB (244 words) - 20:18, 22 January 2023
- :1. Introduction550 bytes (58 words) - 02:33, 16 January 2023
- ...delta derivative|delta differentiable]] and for all $\ell \in \{0,\ldots,k-1\}$, $\displaystyle\lim_{t \rightarrow \pm \infty} f^{\Delta^{\ell}}(t)e_{\o633 bytes (89 words) - 16:50, 15 January 2023
- :1. Introduction477 bytes (60 words) - 17:47, 15 January 2023
- $$1+\mu(t)p(t) > 0.$$287 bytes (44 words) - 12:57, 16 January 2023
