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__NOTOC__
 
= Welcome to MWiki =
 
= Welcome to MWiki =
 
== Theorem of the month ==
 
== Theorem of the month ==
Theorem: The intex method solves every solvable LP in <math>\mathcal{O}({\vartheta}^{3})</math>.
 
  
Proof and algorithm: First, we normalise and scale <math>{b}^{T}y - {d}^{T}x \le 0, Ax \le b</math> and <math>{A}^{T}y \ge d</math>. Let the ''height'' <math>h</math> have the initial value <math>{h}_{0} := |\text{min } \{{b}_{1}, ..., {b}_{m}, {-d}_{1}, ..., {-d}_{n}\}|/r</math> for the reduction factor <math>r \in \; [&frac12;, 1[</math>. Let the
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=== Universal multistep theorem ===
  
LP min <math>\{h \in [0, {h}_{0}] : x \in {}^{\omega}\mathbb{R}_{\ge 0}^{n}, y \in {}^{\omega}\mathbb{R}_{\ge 0}^{m}, {b}^{T}y - {d}^{T}x \le h, Ax - b \le (h, ..., h)^{T} \in {}^{\omega}\mathbb{R}_{\ge 0}^{m}, d - {A}^{T}y \le (h, ..., h)^{T} \in {}^{\omega}\mathbb{R}_{\ge 0}^{n}\}</math> have for <math>\underline{v} := {v}^{T}</math> the feasible interior starting point <math>v := ({\underline{x}, \underline{y}, h)}^{T} \in {}^{\omega}\mathbb{R}_{\ge 0}^{m+n+1}</math>, e.g. <math>({\underline{0}, \underline{0}, {h}_{0})}^{T}</math>.
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For <math>n \in {}^{\nu}\mathbb{N}_{\le p}, k, m, p \in {}^{\nu}\mathbb{N}^{*}, d_{\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 B x) := g_{\acute{k}}(x)</math> and <math>g_0(a) = f((\curvearrowleft B)a, y_0, ... , y_{\acute{n}})</math>, the Taylor series of the initial value problem <math>y^\prime(x) = f(x, y((\curvearrowright B)^0 x), ... , y((\curvearrowright B)^{\acute{n}} x))</math> of order <math>n</math> implies <div style="text-align:center;"><math>y(\curvearrowright B x) = y(x) - d_{\curvearrowright B}x\sum\limits_{k=1}^{p}{i^{2k} g_{p-k}((\curvearrowright B) x)\sum\limits_{m=k}^{p}{\widehat{m!}\binom{\acute{m}}{\acute{k}}}} + \mathcal{O}((d_{\curvearrowright B} x)^{\grave{p}}).\square</math></div>
  
It identifies the mutually dual LPs <math>\{{d}^{T}x : d \in {}^{\omega}\mathbb{R}^{n}, x \in {P}_{\ge 0}\}</math> and min <math>\{{b}^{T}y : y \in {}^{\omega}\mathbb{R}_{\ge 0}^{m}, {A}^{T}y \ge d\}</math>.
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=== Goldbach’s theorem ===
  
Hereon, we successively interpolate all <math>{v}_{k}^{*} := (\text{max } {v}_{k} + \text{min } {v}_{k})/2</math> until all <math>|\Delta{v}_{k}|</math> are sufficiently small. After that, we extrapolate then <math>v</math> via <math>{v}^{*}</math> into the boundary of the polytope. The <math>r</math>-fold of the distance exceeding <math>{v}^{*}</math> determines the new starting point <math>v</math>.
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Every even whole number greater than 2 is the sum of two primes.
  
If min<math>{}_{k} {h}_{k} t = 0</math> follows from <math>t :=</math> min<math>{}_{k} \Delta{h}_{k}</math>, we end. Then we start over until min <math>h = 0</math> or min <math>h > 0</math> is certain. Since <math>h</math> at least halves itself for each iteration step in <math>\mathcal{O}({\omega\vartheta}^{2})</math>, the strong duality theorem yields the result.<math>\square</math>
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==== Proof: ====
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Induction over all prime gaps until the maximally possible one each time.<math>\square</math>
  
== Recommended readings ==
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=== Foundation theorem ===
[http://www.epubli.de/shop/buch/Relil-Boris-Haase-9783844208726/11049 Relil - Religion und Lebensweg]
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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>
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== Recommended reading ==
  
 
[https://en.calameo.com/books/003777977258f7b4aa332 Nonstandard Mathematics]
 
[https://en.calameo.com/books/003777977258f7b4aa332 Nonstandard Mathematics]
  
 
[[de:Hauptseite]]
 
[[de:Hauptseite]]

Revision as of 01:05, 1 June 2021

Welcome to MWiki

Theorem of the month

Universal multistep theorem

For [math]\displaystyle{ n \in {}^{\nu}\mathbb{N}_{\le p}, k, m, p \in {}^{\nu}\mathbb{N}^{*}, d_{\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 B x) := g_{\acute{k}}(x) }[/math] and [math]\displaystyle{ g_0(a) = f((\curvearrowleft B)a, y_0, ... , y_{\acute{n}}) }[/math], the Taylor series of the initial value problem [math]\displaystyle{ y^\prime(x) = f(x, y((\curvearrowright B)^0 x), ... , y((\curvearrowright B)^{\acute{n}} x)) }[/math] of order [math]\displaystyle{ n }[/math] implies

[math]\displaystyle{ y(\curvearrowright B x) = y(x) - d_{\curvearrowright B}x\sum\limits_{k=1}^{p}{i^{2k} g_{p-k}((\curvearrowright B) x)\sum\limits_{m=k}^{p}{\widehat{m!}\binom{\acute{m}}{\acute{k}}}} + \mathcal{O}((d_{\curvearrowright B} x)^{\grave{p}}).\square }[/math]

Goldbach’s theorem

Every even whole number greater than 2 is the sum of two primes.

Proof:

Induction over all prime gaps until the maximally possible one each time.[math]\displaystyle{ \square }[/math]

Foundation theorem

Only the postulation of the axiom of foundation that every nonempty subset [math]\displaystyle{ X \subseteq Y }[/math] contains an element [math]\displaystyle{ x_0 }[/math] such that [math]\displaystyle{ X }[/math] und [math]\displaystyle{ x_0 }[/math] are disjoint guarantees cycle freedom.

Proof:

Set [math]\displaystyle{ X := \{x_m : x_0 := \{\emptyset\}, x_{\omega} := \{x_1\} }[/math] and [math]\displaystyle{ x_{\acute{n}} := \{x_n\} }[/math] for [math]\displaystyle{ m \in {}^{\omega}\mathbb{N} }[/math] and [math]\displaystyle{ n \in {}^{\omega}\mathbb{N}_{\ge 2}\} }[/math] .[math]\displaystyle{ \square }[/math]

Recommended reading

Nonstandard Mathematics