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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 $\mathbb{T}$ of real numbers called a time scale. When $\mathbb{T}=\mathbb{R}$ the resulting theory becomes differential calculus but when $\mathbb{T}=\mathbb{Z}$ the resulting theory becomes difference calculus. Time scales also include any closed subset of $\mathbb{R}$, so more exotic sets such as the Cantor set are also subsumed in the theory.
A result proven in time scale calculus implies the result for all choices of $\mathbb{T}$ so a result in time scale calculus immediately implies the result in differential calculus, the same result in difference calculus, the same result in $q$-calculus, the same result in calculus on the Cantor set, and countless others. For an example of this phenomenon, see the familiar properties of the $\Delta$-derivative to classical differentiation or to taking a forward difference.
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Calculus on time scales
- Abel's theorem
- Calculus of variations
- Chain rule
- Dynamic Equations
- Complex calculus on time scales
- Convergence of time scales
- [[Duality of $\Delta$ and $\nabla$ calculi]]
- Fractional calculus
- Function spaces
- Induction on time scales
- Laplace transform
- L'Hospital's Rule
- Mean value theorem
- Fourier transform
- Regressive function
- Taylor's formula
- Variation of parameters
- Wronskian
$\Delta$-calculus
- Completely delta differentiable
- $\Delta$-Bernoulli inequality
- $\Delta$-Bihari inequality
- $\Delta$-Cauchy-Schwarz inequality
- $\Delta$-derivative
- $\Delta$-Gronwall inequality
- $\Delta$ heat equation
- $\Delta$-Hölder inequality
- $\Delta$-integral
- $\Delta$-Jensen inequality
- $\Delta$-Lyapunov inequality
- $\Delta$-Markov inequality
- $\Delta$-Minkowski inequality
- $\Delta$-Opial inequality
- $\Delta$-Tschebycheff inequality
- $\Delta$-Wirtinger inequality
- $\Delta$ wave equation
- Directional $\Delta$ derivative
- Partial $\Delta$ derivative
- Partial $\Delta$ dynamic equations
$\nabla$-calculus
$\Diamond_{\alpha}$-calculus
- $\Diamond_{\alpha}$-derivative
- $\Diamond_{\alpha}$-Hölder inequality
- $\Diamond_{\alpha}$-Jensen's inequality
- $\Diamond_{\alpha}$-Minkowski's inequality
- $\Diamond$-integral
Probability Theory
- Cumulant generating function
- Cumulative distribution function
- Moments
- Probability density function
- Joint time scales probability density function
- Moment generating function
- Expected value
- Variance
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: $\mathbb{T}=\{\ldots, t_{-1}, t_{0}, t_1, \ldots\}$
Special functions on time scales
$\Delta$-calculus
- $\Delta$-$\cos_p$
- $\Delta$-$\cosh_p$
- $\Delta$-$e_p$
- $\Delta$-$h_k$
- $\Delta$-$g_k$
- $\Delta$-$\sin_p$
- $\Delta$-$\sinh_p$
$\nabla$-calculus
- $\nabla$-$\widehat{\cos}_p$
- $\nabla$-$\widehat{\cosh}_p$
- $\nabla$-$\hat{e}_p$
- $\nabla$-$\hat{h}_k$
- $\nabla$-$\hat{g}_k$
- $\nabla$-$\widehat{\sin}_p$
- $\nabla$-$\widehat{\sinh}_p$
$\Diamond_{\alpha}$-calculus
- $\Diamond_{\alpha}$-$\cos_p$
- $\Diamond_{\alpha}$-$\cosh_p$
- $\Diamond_{\alpha}$-$e_p$
- $\Diamond$-$\sin_p$
- $\Diamond$-$\sinh_p$
