Difference between revisions of "Main Page"
From timescalewiki
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==Special functions on time scales== | ==Special functions on time scales== | ||
*[[polynomials | Polynomials]] | *[[polynomials | Polynomials]] | ||
| + | *[[Gamma function]] | ||
| + | *[[hyperbolic_functions | Hyperbolic functions]] | ||
| + | *[[logarithms | Logarithms]] | ||
| + | *[[trig_functions | Trigonometric functions]] | ||
| + | *[[gaussian_bell | Gaussian bell]] | ||
| + | |||
| + | ===$\Delta$-calculus=== | ||
*[[Delta cosine | $\Delta$-$\cos_p$]] | *[[Delta cosine | $\Delta$-$\cos_p$]] | ||
*[[Delta cosh | $\Delta$-$\cosh_p$]] | *[[Delta cosh | $\Delta$-$\cosh_p$]] | ||
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*[[Delta sine | $\Delta$-$\sin_p$]] | *[[Delta sine | $\Delta$-$\sin_p$]] | ||
*[[Delta sinh | $\Delta$-$\sinh_p$]] | *[[Delta sinh | $\Delta$-$\sinh_p$]] | ||
| − | + | ||
| − | + | ===$\nabla$-calculus=== | |
| − | |||
*[[Nabla cosine | $\nabla$-$\cos_p$]] | *[[Nabla cosine | $\nabla$-$\cos_p$]] | ||
*[[Nabla cosh | $\nabla$-$\cosh_p$]] | *[[Nabla cosh | $\nabla$-$\cosh_p$]] | ||
| Line 75: | Line 81: | ||
*[[Nabla sine | $\nabla$-$\sin_p$]] | *[[Nabla sine | $\nabla$-$\sin_p$]] | ||
*[[Nabla sinh | $\nabla$-$\sinh_p$]] | *[[Nabla sinh | $\nabla$-$\sinh_p$]] | ||
| − | *[[ | + | |
| − | *[[ | + | ===$\Diamond_{\alpha}$-calculus=== |
| + | *[[Diamond alpha cosine | $\Diamond_{\alpha}$-$\cos_p$]] | ||
| + | *[[Diamond alpha cosh | $\Diamond_{\alpha}$-$\cosh_p$]] | ||
| + | *[[Diamond exponential | $\Diamond_{\alpha}$-$e_p$]] | ||
| + | *[[Diamond sine | $\Diamond$-$\sin_p$]] | ||
| + | *[[Diamond sinh | $\Diamond$-$\sinh_p$]] | ||
===Probability Distributions on time scales=== | ===Probability Distributions on time scales=== | ||
Revision as of 08:10, 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
$\Delta$-calculus
$\nabla$-calculus
$\Diamond_{\alpha}$-calculus
- $\Diamond_{\alpha}$-$\cos_p$
- $\Diamond_{\alpha}$-$\cosh_p$
- $\Diamond_{\alpha}$-$e_p$
- $\Diamond$-$\sin_p$
- $\Diamond$-$\sinh_p$
