Difference between revisions of "Main Page"

From MWiki
Jump to: navigation, search
(Counter-directional theorem)
(Greatest-prime Criterion and Transcendence of Euler's Constant)
Line 1: Line 1:
 
__NOTOC__
 
__NOTOC__
 
= Welcome to MWiki =
 
= Welcome to MWiki =
−
== Theorem of the month ==
+
== Theorems of the month ==
−
=== Counter-directional theorem ===
+
=== Greatest-prime Criterion ===
  
−
If the path <math>\gamma: [a, b[ \, \cap \, C \rightarrow V</math> with <math>C \subseteq \mathbb{R}</math> passes the edges of every <math>n</math>-cube of side length d0 in the <math>n</math>-volume <math>V \subseteq {}^{(\omega)}\mathbb{R}^{n}</math> with <math>n \in \mathbb{N}_{\ge 2}</math> exactly once, where the opposite edges in all two-dimensional faces of every <math>n</math>-cube are traversed in reverse direction, but uniformly, then, for <math>D \subseteq \mathbb{R}^{2}, B \subseteq {V}^{2}, f = ({f}_{1}, ..., {f}_{n}): V \rightarrow {}^{(\omega)}\mathbb{R}^{n}, \gamma(t) = x, \gamma(\curvearrowright D t) = \curvearrowright B x</math> and <math>{V}_{\curvearrowright } := \{\curvearrowright B x \in V: x \in V, \curvearrowright B x \ne \curvearrowleft B x\}</math>, it holds that
+
If a real number may be represented as an irreducible fraction <math>\widehat{ap}b \pm \hat{s}t</math>, where <math>a, b, s</math>, and <math>t</math> are natural numbers, <math>abst \ne 0</math>, <math>a + s &gt; 2</math>, and the (second-)greatest prime number <math>p \in {}^{\omega }\mathbb{P}, p \nmid b</math> and <math>p \nmid s</math>, then <math>r</math> is <math>\omega</math>-transcendental.
  
 +
==== Proof: ====
 +
The denominator <math>\widehat{ap s} (bs \pm apt)</math> is <math>\ge 2p \ge 2\omega - \mathcal{O}({_e}\omega\sqrt{\omega}) &gt; \omega</math> by the prime number theorem.<math>\square</math>
 +
 +
=== Transcendence of Euler's Constant ===
 +
 +
For <math>x \in {}^{\omega }{\mathbb{R}}</math>, let be <math>s(x) := \sum\limits_{n=1}^{\omega}{\hat{n}{{x}^{n}}}</math> and <math>\gamma := s(1) - {_e}\omega = \int\limits_{1}^{\omega}{\left( \widehat{\left\lfloor x \right\rfloor} - \hat{x} \right)dx}</math> Euler's constant, where rearranging shows <math>\gamma \in \; ]0, 1[</math>.
  
−
<div style="text-align:center;"><math>\int\limits_{t \in [a,b[ \, \cap \, C}{f(\gamma (t)){{{{\gamma }'}}_{\curvearrowright }}(t)dDt}=\int\limits_{\begin{smallmatrix} (x,\curvearrowright B\,x) \\ \in V\times {{V}_{\curvearrowright}} \end{smallmatrix}}{f(x)dBx}=\int\limits_{\begin{smallmatrix} t \in [a,b[ \, \cap \, C, \\ \gamma | {\partial{}^{\acute{n}}} V \end{smallmatrix}}{f(\gamma (t)){{{{\gamma }'}}_{\curvearrowright }}(t)dDt}.</math></div>
+
If <math>{_e}\omega = s(\hat{2})\;{_2}\omega</math> is accepted, <math>\gamma \in {}^{\omega }\mathbb{T}_{\mathbb{R}}</math> is true with a precision of <math>\mathcal{O}({2}^{-\omega}\hat{\omega}\;{_e}\omega)</math>.
  
 
==== Proof: ====
 
==== Proof: ====
−
If two arbitrary squares are considered with common edge of length d0 included in one plane, then only the edges of <math>V\times{V}_{\curvearrowright}</math> are not passed in both directions for the same function value. They all, and thus the path to be passed, are exactly contained in <math>{\partial}^{\acute{n}}V.\square</math>
+
The (exact) integration of the geometric series yields <math>-{_e}(-\acute{x}) = s(x) + \mathcal{O}(\hat{\omega}{x}^{\grave{\omega}}/\acute{x}) + t(x)dx</math> for <math>x \in [-1, 1 - \hat{\nu}]</math> and <math>t(x) \in {}^{\omega }{\mathbb{R}}</math> such that <math>|t(x)| &lt; {\omega}</math>.
 +
 
 +
After applying Fermat's little theorem to the numerator of <math>\hat{p}(1 - 2^{-p}\,{_2}\omega)</math> for <math>p = \max\, {}^{\omega}\mathbb{P}</math>, the greatest-prime criterion yields the claim.<math>\square</math>
  
 
== Recommended reading ==
 
== Recommended reading ==

Revision as of 16:19, 30 September 2021

Welcome to MWiki

Theorems of the month

Greatest-prime Criterion

If a real number may be represented as an irreducible fraction [math]\displaystyle{ \widehat{ap}b \pm \hat{s}t }[/math], where [math]\displaystyle{ a, b, s }[/math], and [math]\displaystyle{ t }[/math] are natural numbers, [math]\displaystyle{ abst \ne 0 }[/math], [math]\displaystyle{ a + s > 2 }[/math], and the (second-)greatest prime number [math]\displaystyle{ p \in {}^{\omega }\mathbb{P}, p \nmid b }[/math] and [math]\displaystyle{ p \nmid s }[/math], then [math]\displaystyle{ r }[/math] is [math]\displaystyle{ \omega }[/math]-transcendental.

Proof:

The denominator [math]\displaystyle{ \widehat{ap s} (bs \pm apt) }[/math] is [math]\displaystyle{ \ge 2p \ge 2\omega - \mathcal{O}({_e}\omega\sqrt{\omega}) > \omega }[/math] by the prime number theorem.[math]\displaystyle{ \square }[/math]

Transcendence of Euler's Constant

For [math]\displaystyle{ x \in {}^{\omega }{\mathbb{R}} }[/math], let be [math]\displaystyle{ s(x) := \sum\limits_{n=1}^{\omega}{\hat{n}{{x}^{n}}} }[/math] and [math]\displaystyle{ \gamma := s(1) - {_e}\omega = \int\limits_{1}^{\omega}{\left( \widehat{\left\lfloor x \right\rfloor} - \hat{x} \right)dx} }[/math] Euler's constant, where rearranging shows [math]\displaystyle{ \gamma \in \; ]0, 1[ }[/math].

If [math]\displaystyle{ {_e}\omega = s(\hat{2})\;{_2}\omega }[/math] is accepted, [math]\displaystyle{ \gamma \in {}^{\omega }\mathbb{T}_{\mathbb{R}} }[/math] is true with a precision of [math]\displaystyle{ \mathcal{O}({2}^{-\omega}\hat{\omega}\;{_e}\omega) }[/math].

Proof:

The (exact) integration of the geometric series yields [math]\displaystyle{ -{_e}(-\acute{x}) = s(x) + \mathcal{O}(\hat{\omega}{x}^{\grave{\omega}}/\acute{x}) + t(x)dx }[/math] for [math]\displaystyle{ x \in [-1, 1 - \hat{\nu}] }[/math] and [math]\displaystyle{ t(x) \in {}^{\omega }{\mathbb{R}} }[/math] such that [math]\displaystyle{ |t(x)| < {\omega} }[/math].

After applying Fermat's little theorem to the numerator of [math]\displaystyle{ \hat{p}(1 - 2^{-p}\,{_2}\omega) }[/math] for [math]\displaystyle{ p = \max\, {}^{\omega}\mathbb{P} }[/math], the greatest-prime criterion yields the claim.[math]\displaystyle{ \square }[/math]

Recommended reading

Nonstandard Mathematics

Retrieved from "https://en.mwiki.de/index.php?title=Main_Page&oldid=186"