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• Quantum q greater than 1
  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
• Quantum q less than 1
  ...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
• Delta derivative of product (1)
  ...ative of constant multiple|next=Delta derivative of product (2)}}: Theorem 1.20 (iii)
  579 bytes (80 words) - 05:45, 10 June 2016
• Mozyrska-Torres logarithm at 1
  ...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
• Mozyraska-Torres logarithm is negative on (0,1)
  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
• Mozyrska-Torres logarithm is positive on (1,infinity)
  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
• Mozyraska-Torres logarithm is positive on (1,infinity)
  #REDIRECT [[Mozyrska-Torres logarithm is positive on (1,infinity)]]
  67 bytes (7 words) - 15:21, 21 October 2017

Page text matches

• Main Page
  ...], 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/Quantum
  5 KB (665 words) - 01:55, 6 February 2023
• Exponential functions
  $$y'=y; y(s)=1.$$ $$y^{\Delta}=y;y(s)=1.$$
  839 bytes (127 words) - 20:55, 20 October 2014
• Nabla exponential
  $$\hat{\xi}_h(z) = -\dfrac{1}{h} \log(1-zh).$$ $$y^{\nabla} = py; y(s)=1.$$
  3 KB (538 words) - 01:11, 19 December 2016
• Delta exponential
  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
• Time scale
  # 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
  [[Delta derivative of product (1)]]<br /> ...on time scales|next=Delta differentiable implies continuous}}: Definition 1.10
  2 KB (249 words) - 15:19, 21 January 2023
• Real numbers
  |[[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
• Harmonic numbers
  ...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
• Convergence of time scales
  ...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
• Forward regressive
  $$1+\mu(t)p(t)\neq 0.$$
  253 bytes (42 words) - 12:58, 16 January 2023
• Paper:Martin Bohner/The logarithm on time scales
  :[[Delta exponential dynamic equation|$(1)$]]
  246 bytes (27 words) - 17:01, 11 February 2017
• Multiples of integers
  |$h_k(t,s) = \dfrac{1}{k!} \displaystyle\prod_{\ell=0}^{k-1}(t-\ell h-s)$ \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
• Integers
  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
• Closure of unit fractions
  ...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
• Quantum q greater than 1
  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
• Hilger imaginary part
  $$\mathrm{Im}_h(z)=\dfrac{\mathrm{Arg}(zh+1)}{h},$$
  701 bytes (105 words) - 15:40, 21 January 2023
• Hilger real part
  $$\mathrm{Re}_h(z)=\dfrac{|zh+1|-1}{h}.$$
  584 bytes (85 words) - 15:41, 21 January 2023
• Isolated points
  ...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
• Square integers
  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
• Gamma function
  ...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

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