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= Welcome to MWiki =
 
= Welcome to MWiki =
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== Theorem of the month ==
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== Theorems of the month ==
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=== Definition ===
  
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=== Universal multistep 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}(\mathbb{D})</math> centred on 0 on the domain <math>\mathbb{D} \subseteq {}^{\omega}\mathbb{C}</math> where <math>m, n \in {}^{\omega}\mathbb{N}^{*}</math> and <math>\eta_n^m := \underline{1}^{2^{\lceil m/n \rceil}}</math>. Then let <math>\delta_n^*f = \tilde{2}(f - f_n^*)</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 <math>n \in {}^{\nu}\mathbb{N}_{\le p}, k, m, p \in {}^{\nu}\mathbb{N}^{*}, {\downarrow}_{{}^\curvearrowright B} x \in\, ]0, 1[, x \in [a, b] \subseteq {}^{\omega}\mathbb{R}, y : [a, b] \rightarrow {}^{\omega}\mathbb{R}^q, f : [a, b]\times{}^{\omega}\mathbb{R}^{q \times n} \rightarrow {}^{\omega}\mathbb{R}^q, g_k({}^\curvearrowright x) := g_{\acute{k}}(x)</math>, and <math>g_0(a) = f(({}^\curvearrowleft)a, y_0, ... , y_{\acute{n}})</math>, the Taylor series of the initial value problem <math>y^\prime(x) = f(x, y(({}^\curvearrowright)^0 x), ... , y(({}^\curvearrowright)^{\acute{n}} x))</math> of order <math>n</math> implies <div style="text-align:center;"><math>y({}^\curvearrowright x) = y(x) + {\downarrow}_{{}^\curvearrowright}x{\pm}_{k=1}^{p}{\left (g_{p-k}(({}^\curvearrowright) x){+}_{m=k}^{p}{\widetilde{m!}\tbinom{\acute{m}}{\acute{k}}}\right )} + \mathcal{O}(({\downarrow}_{{}^\curvearrowright} x)^{\grave{p}}).\square</math></div>
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=== Representation theorem for integrals ===
  
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=== Goldbach’s theorem ===
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The Taylor series (see below) <math>f(z) \in \mathcal{O}(\mathbb{D})</math> centred on 0 on <math>\mathbb{D} \subseteq {}^{\omega}\mathbb{C}</math> gives for <math>\grave{m}, n \in {}^{\omega}\mathbb{N}^*</math><div style="text-align:center;"><math>{\uparrow}_0^z...{\uparrow}_0^{\zeta_2}{f(\zeta_1){\downarrow}\zeta_1\;...\;{\downarrow}\zeta_n} = \widetilde{n!} f(z\mu_n) z^n.\square</math></div>
  
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Every even whole number greater than 2 is the sum of two primes.
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=== Representation theorem for derivatives ===
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For <math>{}^{\widetilde{\nu}}\dot{\mathbb{C}} \subset \mathbb{D} \subseteq {}^{\omega}\mathbb{C},</math> the Taylor series<div style="text-align:center;"><math>f(z):=f(0) + {\LARGE{\textbf{+}}}_{m=1}^{\omega }{\widetilde{m!}\,{{f}^{(m)}}(0){z^m}},</math></div><math>\varepsilon := \tilde{2}^j\tilde{r}, j \in {}^{\omega}\mathbb{Z}, n = \epsilon^{\sigma} \in {}^{\omega}\mathbb{N}^{*}, u :=\epsilon^{\tilde{n} \hat{\underline{\pi}}}</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)=2^{jn}\acute{n}!{\LARGE{\textbf{+}}}_{k=1}^{n}{\delta_n^* f(\tilde{2}^j u^k)}.</math></div>
  
 
==== Proof: ====
 
==== Proof: ====
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Induction over all prime gaps until the maximally possible one each time.<math>\square</math>
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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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=== Foundation theorem ===
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== Reference ==
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<references />
  
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Only the postulation of the axiom of foundation that every nonempty subset <math>X \subseteq Y</math> contains an element <math>x_0</math> such that <math>X</math> und <math>x_0</math> are disjoint guarantees cycle freedom.
 
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==== Proof: ====
 
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Set <math>X := \{x_m : x_0 := \{\emptyset\}, x_{\omega} := \{x_1\}</math> and <math>x_{\acute{n}} := \{x_n\}</math> for <math>m \in {}^{\omega}\mathbb{N}</math> and <math>n \in {}^{\omega}\mathbb{N}_{\ge 2}\}</math> .<math>\square</math>
 
 
== Recommended reading ==
 
== Recommended reading ==
  

Revision as of 23:05, 31 March 2024

Welcome to MWiki

Theorems of the month

Definition

Let [math]\displaystyle{ f_n^*(z) = f(\eta_nz) }[/math] sisters of the Taylor series [math]\displaystyle{ f(z) \in \mathcal{O}(\mathbb{D}) }[/math] centred on 0 on the domain [math]\displaystyle{ \mathbb{D} \subseteq {}^{\omega}\mathbb{C} }[/math] where [math]\displaystyle{ m, n \in {}^{\omega}\mathbb{N}^{*} }[/math] and [math]\displaystyle{ \eta_n^m := \underline{1}^{2^{\lceil m/n \rceil}} }[/math]. Then let [math]\displaystyle{ \delta_n^*f = \tilde{2}(f - f_n^*) }[/math] the halved sister distances of [math]\displaystyle{ f. }[/math] For [math]\displaystyle{ \mu_n^m := m!n!/(m + n)! }[/math], [math]\displaystyle{ \mu }[/math] and [math]\displaystyle{ \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]\displaystyle{ \triangle }[/math]

Representation theorem for integrals

The Taylor series (see below) [math]\displaystyle{ f(z) \in \mathcal{O}(\mathbb{D}) }[/math] centred on 0 on [math]\displaystyle{ \mathbb{D} \subseteq {}^{\omega}\mathbb{C} }[/math] gives for [math]\displaystyle{ \grave{m}, n \in {}^{\omega}\mathbb{N}^* }[/math]

[math]\displaystyle{ {\uparrow}_0^z...{\uparrow}_0^{\zeta_2}{f(\zeta_1){\downarrow}\zeta_1\;...\;{\downarrow}\zeta_n} = \widetilde{n!} f(z\mu_n) z^n.\square }[/math]

Representation theorem for derivatives

For [math]\displaystyle{ {}^{\widetilde{\nu}}\dot{\mathbb{C}} \subset \mathbb{D} \subseteq {}^{\omega}\mathbb{C}, }[/math] the Taylor series

[math]\displaystyle{ f(z):=f(0) + {\LARGE{\textbf{+}}}_{m=1}^{\omega }{\widetilde{m!}\,{{f}^{(m)}}(0){z^m}}, }[/math]

[math]\displaystyle{ \varepsilon := \tilde{2}^j\tilde{r}, j \in {}^{\omega}\mathbb{Z}, n = \epsilon^{\sigma} \in {}^{\omega}\mathbb{N}^{*}, u :=\epsilon^{\tilde{n} \hat{\underline{\pi}}} }[/math] and [math]\displaystyle{ f }[/math]'s radius of convergence [math]\displaystyle{ r \in {}^{\nu}{\mathbb{R}}_{>0} }[/math] imply

[math]\displaystyle{ {{f}^{(n)}}(0)=2^{jn}\acute{n}!{\LARGE{\textbf{+}}}_{k=1}^{n}{\delta_n^* f(\tilde{2}^j u^k)}. }[/math]

Proof:

Taylor's theorem[1] and the properties of the roots of unity.[math]\displaystyle{ \square }[/math]

Reference

  1. ↑ Remmert, Reinhold: Funktionentheorie 1 : 3., verb. Aufl.; 1992; Springer; Berlin; ISBN 9783540552338, S. 165 f.

Recommended reading

Nonstandard Mathematics

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