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= Welcome to MWiki =
 
= Welcome to MWiki =
 
== Theorems of the month ==
 
== Theorems of the month ==
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=== Definition ===
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=== Green's theorem ===
  
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Let <math>f_n^*(z) = f(\eta_nz)</math> <em>sisters</em> of the Taylor series <math>f(z) \in \mathcal{O}(D)</math> centred on 0 on the domain <math>D \subseteq {}^{\omega}\mathbb{C}</math> where <math>m, n \in {}^{\omega}\mathbb{N}^{*}</math> and <math>\eta_n^m := i^{2^{\lceil m/n \rceil}}</math>. Then let <math>\delta_n^*f = (f - f_n^*)/2</math> the <em>halved sister distances</em> of <math>f.</math> For <math>\mu_n^m := m!n!/(m + n)!</math>, <math>\mu</math> and <math>\eta</math> form an calculus, which can be resolved on the level of Taylor series and allows an easy and finite closed representation of integrals and derivatives.<math>\triangle</math>
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For some <math>h</math>-domain <math>\mathbb{D} \subseteq {}^{(\omega)}\mathbb{R}^{2}</math>, infinitesimal <math>h = |{\downarrow}x|= |{\downarrow}y| = |\overset{\rightharpoonup}{\gamma}(s) - \gamma(s)| = \mathcal{O}({\tilde{\omega}}^{m})</math>, sufficiently large <math>m \in \mathbb{N}^{*}, (x, y) \in \mathbb{D}, \mathbb{D}^{-} := \{(x, y) \in \mathbb{D} : (x + h, y + h) \in \mathbb{D}\}</math>, and a simply closed path <math>\gamma: [a, b[\rightarrow {\downarrow} \mathbb{D}</math> followed anticlockwise, choosing <math>\overset{\rightharpoonup}{\gamma}(s) = \gamma(\overset{\rightharpoonup}{s})</math> for <math>s \in [a, b[, A \subseteq {[a, b]}^{2}</math>, the following equation holds for sufficiently <math>\alpha</math>-continuous functions <math>u, v: \mathbb{D} \rightarrow \mathbb{R}</math> with not necessarily continuous <math>{\downarrow} u/{\downarrow} x, {\downarrow} u/{\downarrow} y, {\downarrow} v/{\downarrow} x</math> and <math>{\downarrow} v/{\downarrow} y</math><div style="text-align:center;"><math>{\uparrow}_{\gamma }{(u\,{\downarrow}x+v\,{\downarrow}y)}={\uparrow}_{(x,y)\in {\mathbb{D}^{-}}}{\left( \tfrac{{\downarrow} v}{{\downarrow} x}-\tfrac{{\downarrow} u}{{\downarrow} y} \right){\downarrow}(x,y)}.</math></div>
  
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=== Speedup theorem for integrals ===
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==== Proof: ====
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Only <math>\mathbb{D} := \{(x, y) : r \le x \le s, f(x) \le y \le g(x)\}, r, s \in {}^{(\omega)}\mathbb{R}, f, g : {\downarrow} \mathbb{D} \rightarrow {}^{(\omega)}\mathbb{R}</math> is proved, since the proof is analogous for each case rotated by <math>\check{\pi}</math>. Every <math>h</math>-domian is union of such sets. Simply showing <div style="text-align:center;"><math>{\uparrow}_{\gamma }{u\,{\downarrow}x}=-{\uparrow}_{(x,y)\in {\mathbb{D}^{-}}}{\tfrac{{\downarrow} u}{{\downarrow} y}{\downarrow}(x,y)}.</math></div> is sufficient because the other relation is given analogously. Neglecting the regions of <math>\gamma</math> with <math>{\downarrow}x = 0</math> and <math>s := h(u(r, g(r)) - u(t, g(t)))</math> shows <div style="text-align:center;"><math>-{\uparrow}_{\gamma }{u\,{\downarrow}x}-s={\uparrow}_{t}^{r}{u(x,g(x)){\downarrow}x}-{\uparrow}_{t}^{r}{u(x,f(x)){\downarrow}x}={\uparrow}_{t}^{r}{{\uparrow}_{f(x)}^{g(x)}{\tfrac{{\downarrow} u}{{\downarrow} y}}{\downarrow}y{\downarrow}x}={\uparrow}_{(x,y)\in {\mathbb{D}^{-}}}{\tfrac{{\downarrow} u}{{\downarrow} y}{\downarrow}(x,y)}.\square</math></div>
  
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The Taylor series (see below) <math>f(z) \in \mathcal{O}(D)</math> centred on 0 on <math>D \subseteq {}^{\omega}\mathbb{C}</math> gives for <math>\grave{m}, n \in {}^{\omega}\mathbb{N}^*</math><div style="text-align:center;"><math>\int\limits_0^z...\int\limits_0^{\zeta_2}{f(\zeta_1)\text{d}\zeta_1\;...\;\text{d}\zeta_n} = \widehat{n!} f(z\mu_n) z^n.\square</math></div>
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=== Singmaster's theorem ===
  
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=== Speedup theorem for derivatives ===
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There are maximally 8 distinct binomial coefficients of the same value > 1.
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For <math>\mathbb{B}_{\hat{\nu}}(0) \subset  D \subseteq {}^{\omega}\mathbb{C},</math> the Taylor series<div style="text-align:center;"><math>f(z):=f(0) + \sum\limits_{m=1}^{\omega }{\widehat{m!}\,{{f}^{(m)}}(0){z^m}},</math></div><math>b_{\varepsilon n} := \hat{\varepsilon}\,\acute{n}! = 2^j, j, n \in {}^{\omega}\mathbb{N}^{*}, \varepsilon \in ]0, r^n[, {{d}_{\varepsilon k n}}:={{\varepsilon}^{{\hat{n}}}}{e}^{\hat{n}k\tau i}</math> and <math>f</math>'s radius of convergence <math>r \in {}^{\nu}{\mathbb{R}}_{&gt;0}</math> imply<div style="text-align:center;"><math>{{f}^{(n)}}(0)=b_{\varepsilon n}\sum\limits_{k=1}^{n}{\delta_n^* f({{d}_{\varepsilon k n}})}.</math></div>
 
  
 
==== Proof: ====
 
==== Proof: ====
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Taylor's theorem<ref name="Remmert">[[w:Reinhold Remmert|<span class="wikipedia">Remmert, Reinhold</span>]]: ''Funktionentheorie 1'' : 3., verb. Aufl.; 1992; Springer; Berlin; ISBN 9783540552338, S. 165 f.</ref> and the properties of the roots of unity.<math>\square</math>
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The existence is clear due to <math>\tbinom{3003}{1} = \tbinom{78}{2} = \tbinom{15}{5} = \tbinom{14}{6}</math> and the structure of Pascal's triangle. With <math>p \in {}^{\omega }{\mathbb{P}}, a,b ,c, d \in {}^{\omega }{\mathbb{N^*}}, \hat{a} \le r := p - b, \hat{a} < \hat{c} \le n := p - d, b < d</math> and <math>s \notin \mathbb{P}</math> for every <math>s \in [\max(r - \acute{a},\grave{n}), r]</math>, Stirling's formula <math>{n!}^2\sim\pi(\hat{n}+\tilde{3}){(\tilde{\epsilon}n)}^{\hat{n}}</math> and the prime number theorem imply <math>\omega\tbinom{r}{a} \le {}_\epsilon\omega\tbinom{n}{c}</math> for <math>p \rightarrow \omega.\square</math>
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== Reference ==
 
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<references />
 
  
 
== Recommended reading ==
 
== Recommended reading ==

Latest revision as of 02:03, 1 May 2024

Welcome to MWiki

Theorems of the month

Green's theorem

For some [math]\displaystyle{ h }[/math]-domain [math]\displaystyle{ \mathbb{D} \subseteq {}^{(\omega)}\mathbb{R}^{2} }[/math], infinitesimal [math]\displaystyle{ h = |{\downarrow}x|= |{\downarrow}y| = |\overset{\rightharpoonup}{\gamma}(s) - \gamma(s)| = \mathcal{O}({\tilde{\omega}}^{m}) }[/math], sufficiently large [math]\displaystyle{ m \in \mathbb{N}^{*}, (x, y) \in \mathbb{D}, \mathbb{D}^{-} := \{(x, y) \in \mathbb{D} : (x + h, y + h) \in \mathbb{D}\} }[/math], and a simply closed path [math]\displaystyle{ \gamma: [a, b[\rightarrow {\downarrow} \mathbb{D} }[/math] followed anticlockwise, choosing [math]\displaystyle{ \overset{\rightharpoonup}{\gamma}(s) = \gamma(\overset{\rightharpoonup}{s}) }[/math] for [math]\displaystyle{ s \in [a, b[, A \subseteq {[a, b]}^{2} }[/math], the following equation holds for sufficiently [math]\displaystyle{ \alpha }[/math]-continuous functions [math]\displaystyle{ u, v: \mathbb{D} \rightarrow \mathbb{R} }[/math] with not necessarily continuous [math]\displaystyle{ {\downarrow} u/{\downarrow} x, {\downarrow} u/{\downarrow} y, {\downarrow} v/{\downarrow} x }[/math] and [math]\displaystyle{ {\downarrow} v/{\downarrow} y }[/math]

[math]\displaystyle{ {\uparrow}_{\gamma }{(u\,{\downarrow}x+v\,{\downarrow}y)}={\uparrow}_{(x,y)\in {\mathbb{D}^{-}}}{\left( \tfrac{{\downarrow} v}{{\downarrow} x}-\tfrac{{\downarrow} u}{{\downarrow} y} \right){\downarrow}(x,y)}. }[/math]

Proof:

Only [math]\displaystyle{ \mathbb{D} := \{(x, y) : r \le x \le s, f(x) \le y \le g(x)\}, r, s \in {}^{(\omega)}\mathbb{R}, f, g : {\downarrow} \mathbb{D} \rightarrow {}^{(\omega)}\mathbb{R} }[/math] is proved, since the proof is analogous for each case rotated by [math]\displaystyle{ \check{\pi} }[/math]. Every [math]\displaystyle{ h }[/math]-domian is union of such sets. Simply showing

[math]\displaystyle{ {\uparrow}_{\gamma }{u\,{\downarrow}x}=-{\uparrow}_{(x,y)\in {\mathbb{D}^{-}}}{\tfrac{{\downarrow} u}{{\downarrow} y}{\downarrow}(x,y)}. }[/math]

is sufficient because the other relation is given analogously. Neglecting the regions of [math]\displaystyle{ \gamma }[/math] with [math]\displaystyle{ {\downarrow}x = 0 }[/math] and [math]\displaystyle{ s := h(u(r, g(r)) - u(t, g(t))) }[/math] shows

[math]\displaystyle{ -{\uparrow}_{\gamma }{u\,{\downarrow}x}-s={\uparrow}_{t}^{r}{u(x,g(x)){\downarrow}x}-{\uparrow}_{t}^{r}{u(x,f(x)){\downarrow}x}={\uparrow}_{t}^{r}{{\uparrow}_{f(x)}^{g(x)}{\tfrac{{\downarrow} u}{{\downarrow} y}}{\downarrow}y{\downarrow}x}={\uparrow}_{(x,y)\in {\mathbb{D}^{-}}}{\tfrac{{\downarrow} u}{{\downarrow} y}{\downarrow}(x,y)}.\square }[/math]

Singmaster's theorem

There are maximally 8 distinct binomial coefficients of the same value > 1.

Proof:

The existence is clear due to [math]\displaystyle{ \tbinom{3003}{1} = \tbinom{78}{2} = \tbinom{15}{5} = \tbinom{14}{6} }[/math] and the structure of Pascal's triangle. With [math]\displaystyle{ p \in {}^{\omega }{\mathbb{P}}, a,b ,c, d \in {}^{\omega }{\mathbb{N^*}}, \hat{a} \le r := p - b, \hat{a} \lt \hat{c} \le n := p - d, b \lt d }[/math] and [math]\displaystyle{ s \notin \mathbb{P} }[/math] for every [math]\displaystyle{ s \in [\max(r - \acute{a},\grave{n}), r] }[/math], Stirling's formula [math]\displaystyle{ {n!}^2\sim\pi(\hat{n}+\tilde{3}){(\tilde{\epsilon}n)}^{\hat{n}} }[/math] and the prime number theorem imply [math]\displaystyle{ \omega\tbinom{r}{a} \le {}_\epsilon\omega\tbinom{n}{c} }[/math] for [math]\displaystyle{ p \rightarrow \omega.\square }[/math]

Recommended reading

Nonstandard Mathematics

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