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| − | *[[diamond derivative | $\ | + | *[[diamond alpha derivative | $\Diamond_{\alpha}$-derivative]] |
*[[delta_integral | $\Delta$-integral]] | *[[delta_integral | $\Delta$-integral]] | ||
*[[nabla integral | $\nabla$-integral]] | *[[nabla integral | $\nabla$-integral]] | ||
Revision as of 08:01, 10 March 2015
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
- $\Diamond_{\alpha}$-derivative
- $\Delta$-integral
- $\nabla$-integral
- $\Diamond$-integral
- Dynamic Equations
- Complex calculus on time scales
- Convergence of time scales
- Function spaces
- Laplace transform
- L'Hospital's Rule
- Mean value theorem
- Fourier transform
- Regressive function
- Taylor's formula
- Variation of parameters
- Wronskian
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: $\mathbb{T}=\{\ldots, t_{-1}, t_{0}, t_1, \ldots\}$
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
- $\Delta$-$\cos_p$
- $\Delta$-$\cosh_p$
- $\Delta$-$e_p$
- $\Delta$-$\sin_p$
- $\Delta$-$\sinh_p$
- Gamma function
- Hyperbolic functions
- Logarithms
- $\nabla$-$\cos_p$
- $\nabla$-$\cosh_p$
- $\nabla$-$\hat{e}_p$
- $\nabla$-$\sin_p$
- $\nabla$-$\sinh_p$
- Trigonometric functions
- Gaussian bell
