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Revision as of 16:46, 20 October 2014
This wiki is a resource for people who do research in time scale calculus. Time scale calculus is a unification and extension of differential and difference calculus in which one does calculus upon a set called a time scale $\mathbb{T}$. When $\mathbb{T}=\mathbb{R}$ the resulting theory is differential calculus but when $\mathbb{T}=\mathbb{Z}$ the resulting theory is difference calculus. Time scales also include any closed subset of $\mathbb{R}$, so more exotic sets such as the Cantor set are also considered in the theory.
Contents
How to get access to edit this wiki
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Calculus on time scales
- Abel's theorem
- List of time scales
- $\Delta$-derivative
- $\nabla$-derivative
- $\Diamond$-derivative
- $\Delta$-integration
- $\nabla$-integral
- $\Diamond$-integral
- Dynamic Equations
- Complex calculus on time scales
- Convergence of time scales
- Function spaces
- Laplace transform
- Fourier transform
- Regressive function
- Variation of parameters
- Wronskian
Examples of time scales
- The real line: $\mathbb{R}$
- The integers: $\mathbb{Z} = \{\ldots, -1,0,1,\ldots\}$
- Multiples of integers: $h\mathbb{Z} = \{ht \colon t \in \mathbb{Z}\}$
- Quantum numbers ($q>1$): $\overline{q^{\mathbb{Z}}}$
- Quantum numbers ($q<1$): $\overline{q^{\mathbb{Z}}}$
- Square integers: $\mathbb{Z}^2 = \{t^2 \colon t \in \mathbb{Z} \}$
- Harmonic numbers: $\mathbb{H}=\left\{\displaystyle\sum_{k=1}^n \dfrac{1}{k} \colon n \in \mathbb{Z}^+ \right\}$
- The closure of the unit fractions: $\overline{\left\{\dfrac{1}{n} \colon n \in \mathbb{Z}^+\right\}}$
- Isolated points: $\mathbb{T}=\{\ldots, t_{-1}, t_{0}, t_1, \ldots\}$
Inequalities
- Bernoulli inequality
- Bihari inequality
- Cauchy-Schwarz inequality
- Gronwall inequality
- Hölder inequality
- Jensen inequality
- Lyapunov inequality
- Minkowski inequality
- Opial inequality
- Wirtinger inequality
