MediaWiki API result

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                        "user": "Tom",
                        "timestamp": "2023-02-06T01:55:48Z",
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                        "*": "This wiki is a resource for <strong>time scale calculus</strong>. 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 [[Real_numbers | $\\mathbb{T}=\\mathbb{R}$]] the resulting theory becomes [http://en.wikipedia.org/wiki/Differential_calculus differential calculus], when [[Multiples_of_integers | $\\mathbb{T}=\\mathbb{Z}$]] the resulting theory becomes [http://en.wikipedia.org/wiki/Difference_calculus difference calculus], and when [[Quantum q greater than 1 | $\\mathbb{T}=\\{1,q,q^2,\\ldots\\}, q>1$]], the resulting theory becomes the [https://en.wikipedia.org/wiki/Quantum_calculus $q$-calculus]. Time scales also include any closed subset of $\\mathbb{R}$, so more exotic sets such as the [http://en.wikipedia.org/wiki/Cantor_set Cantor set] are also subsumed in the theory.\n\nA 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 | $\\Delta$-derivative]] to classical differentiation or to taking a forward difference.\n\nSee the Python library [https://github.com/tomcuchta/timescalecalculus timescalecalculus] on GitHub and its [[timescalecalculus python library documentation|documentation]].\n\n<b><u>Registration</u></b>\nDue to a resurgence of automated spam bots, account registration and anonymous editing is currently disabled. Please contact Tom Cuchta (tomcuchta@gmail.com) to gain access to edit the wiki.\n\n=Time scales calculus=\n<center>{{:Time scales footer}}</center>\n<center>{{:Delta special functions footer}}</center>\n<center>{{:Hilger complex plane footer}}</center>\n{{:Delta inequalities footer}}\n\n[[Bilateral Laplace transform]]<br />\n[[Unilateral Laplace transform]]<br />\n[[Cauchy function]]<br />\n[[Chain rule]]<br />\n[[Unilateral convolution]]<br />\n[[Dense point]]<br />\n[[Disconjugate]]<br />\n[[Dynamic equation]]<br />\n[[Forward circle minus]]<br />\n[[Backward circle minus]]<br />\n[[Forward circle plus]]<br />\n[[Backward circle plus]]<br />\n[[Convergence of time scales]]<br />\n[[Dilation of time scales]]<br />\n[[Duality of delta and nabla | Duality of $\\Delta$ and $\\nabla$]]<br />\n[[Fractional calculus]]<br />\n[[Frequency roots]]<br />\n[[Generalized square]]<br />\n[[Generalized zero]]<br />\n[[Induction on time scales]]<br />\n[[L'Hospital's Rule]]<br />\n[[First mean value theorem]]<br />\n[[Pre-differentiable]]<br />\n[[Marks-Gravagne-Davis Fourier transform]]<br />\n[[Cuchta-Georgiev Fourier transform]]<br />\n[[rd-continuous]]<br />\n[[Forward regressive function]]<br />\n[[Regulated function]]<br />\n[[Riccati equation]]<br />\n[[Scattered point]]<br />\n[[Self-adjoint]]<br />\n[[Shifting problem]]<br />\n[[Variation of parameters]]<br />\n[[Wronskian]]<br />\n\n==$\\Delta$-calculus==\n[[delta_derivative | $\\Delta$-derivative]]<br />\n[[Delta heat equation | $\\Delta$ heat equation]]<br />\n[[delta_integral | $\\Delta$-integral]]<br />\n[[Delta Taylor's formula|$\\Delta$-Taylor's formula]]<br />\n[[Delta wave equation | $\\Delta$ wave equation]]<br />\n[[Directional Delta Derivative | Directional $\\Delta$ derivative]]<br />\n[[Partial Delta Derivative | Partial $\\Delta$ derivative]]<br />\n[[Partial Delta Dynamic Equations | Partial $\\Delta$ dynamic equations]]<br />\n\n==$\\nabla$-calculus==\n[[nabla_derivative | $\\nabla$-derivative]]<br />\n[[nabla integral | $\\nabla$-integral]]<br />\n\n==$\\Diamond_{\\alpha}$-calculus==\n[[diamond alpha derivative | $\\Diamond_{\\alpha}$-derivative]]<br />\n[[diamond alpha holder inequality | $\\Diamond_{\\alpha}$-H\u00f6lder inequality ]]<br />\n[[diamond alpha Jensen's inequality | $\\Diamond_{\\alpha}$-Jensen's inequality]]<br />\n[[diamond alpha Minkowski's inequality | $\\Diamond_{\\alpha}$-Minkowski's inequality]]<br />\n[[diamond integral | $\\Diamond$-integral]]<br />\n\n==Probability Theory==\n*[[Cumulant generating function]]\n*[[Cumulative distribution function]]\n*[[Probability density function]]\n*[[Joint time scales probability density function]]\n*[[Moment generating function]]\n*[[Expected value]]\n*[[Variance]]\n\n{{:Examples of time scales}}\n\n=Special functions on time scales=\n[[Delta cpq|$\\mathrm{c}_{pq}$]]<br />\n[[Delta chpq|$\\mathrm{ch}_{pq}$]]<br />\n[[Delta spq|$\\mathrm{s}_{pq}$]]<br />\n[[Delta shpq|$\\mathrm{sh}_{pq}$]]<br />\n[[Gamma function]]<br />\n[[Euler-Cauchy logarithm]]<br />\n[[Bohner logarithm]]<br />\n[[Jackson logarithm]]<br />\n[[Mozyrska-Torres logarithm]]<br />\n[[gaussian_bell | Gaussian bell]]<br />\n[[Uniform distribution]]<br />\n[[Exponential distribution]]<br />\n[[Gamma distribution]]<br />\n\n\n==$\\nabla$-calculus==\n[[Nabla cosine | $\\nabla \\widehat{\\cos}_p$]]<br />\n[[Nabla cosh | $\\nabla \\widehat{\\cosh}_p$]]<br />\n[[Nabla exponential | $\\nabla \\widehat{\\exp}$]]<br />\n[[Nabla hk|$\\nabla \\hat{h}_k$]]<br />\n[[Nabla gk|$\\nabla \\hat{g}_k$]]<br />\n[[Nabla sine | $\\nabla \\widehat{\\sin}_p$]]<br />\n[[Nabla sinh | $\\nabla \\widehat{\\sinh}_p$]]<br />"
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