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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

List of time scales

• Abel's theorem
• Calculus of variations
• Dynamic Equations
• Complex calculus on time scales
• Convergence of time scales
• Fractional calculus
• Function spaces
• Laplace transform
• L'Hospital's Rule
• Mean value theorem
• Fourier transform
• Regressive function
• Taylor's formula
• Variation of parameters
• Wronskian

$\Delta$-calculus

• $\Delta$-derivative
• $\Delta$-integral

$\nabla$-calculus

• $\nabla$-derivative
• $\nabla$-integral

$\Diamond_{\alpha}$-calculus

• $\Diamond_{\alpha}$-derivative
• $\Diamond_{\alpha}$-Jensen'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

1. The real line: $\mathbb{R}$
2. The integers: $\mathbb{Z} = \{\ldots, -1,0,1,\ldots\}$
3. Multiples of integers: $h\mathbb{Z} = \{ht \colon t \in \mathbb{Z}\}$
4. Quantum numbers ($q>1$): $\overline{q^{\mathbb{Z}}}$
5. Quantum numbers ($q<1$): $\overline{q^{\mathbb{Z}}}$
6. Square integers: $\mathbb{Z}^2 = \{t^2 \colon t \in \mathbb{Z} \}$
7. Harmonic numbers: $\mathbb{H}=\left\{\displaystyle\sum_{k=1}^n \dfrac{1}{k} \colon n \in \mathbb{Z}^+ \right\}$
8. The closure of the unit fractions: $\overline{\left\{\dfrac{1}{n} \colon n \in \mathbb{Z}^+\right\}}$
9. 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
• Markov inequality
• Minkowski inequality
• Opial inequality
• Tschebycheff inequality
• Wirtinger inequality

Special functions on time scales

• Polynomials
• Gamma function
• Hyperbolic functions
• Logarithms
• Trigonometric functions
• Gaussian bell

$\Delta$-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

• $\Diamond_{\alpha}$-$\cos_p$
• $\Diamond_{\alpha}$-$\cosh_p$
• $\Diamond_{\alpha}$-$e_p$
• $\Diamond$-$\sin_p$
• $\Diamond$-$\sinh_p$

Probability Distributions on time scales

• Uniform distribution
• Exponential distribution
• Gamma distribution