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(Cauchy's integral theorem, fundamental theorem of algebra and Newton’s method)
(Prime number and Gelfond-Schneider theorem)
 
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= Welcome to MWiki =
 
= Welcome to MWiki =
 
== Theorems of the month ==
 
== Theorems of the month ==
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=== Cauchy's integral theorem ===
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=== Prime number theorem ===
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Given the <abbr title="neighbourhood relation">NR</abbr>s <math>B \subseteq {D}^{2}</math> and <math>A \subseteq [a, b]</math> for some <math>h</math>-domain <math>D \subseteq {}^{\omega}\mathbb{C}</math>, infinitesimal <math>h</math>, <math>f \in \mathcal{O}(D)</math> and a <abbr title="closed path">CP</abbr> <math>\gamma: [a, b[\rightarrow \partial D</math>, choosing <math>{}^\curvearrowright \gamma(t) = \gamma({}^\curvearrowright t)</math> for <math>t \in [a, b[</math> gives
 
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<div style="text-align:center;"><math>{\uparrow}_{\gamma }{f(z){\downarrow}z}=0.</math></div>
 
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'''Proof:''' By the Cauchy-Riemann differential equations and Green's theorem, with <math>x := \text{Re} \, z, y := \text{Im} \, z, u := \text{Re} \, f, v := \text{Im} \, f</math> and <math>{D}^{-} := \{z \in D : z + h + \underline{h} \in D\}</math>, it holds that
 
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<div style="text-align:center;"><math>{\uparrow}_{\gamma }{f(z){\downarrow}z}={\uparrow}_{\gamma }{\left( u+\underline{v} \right)\left( {\downarrow}x+{\downarrow}\underline{y} \right)}={\uparrow}_{z\in {{D}^{-}}}{\left( \left( \tfrac{{\downarrow} \underline{u}}{{\downarrow} x}-\tfrac{{\downarrow} \underline{v}}{{\downarrow} y} \right)-\left( \tfrac{{\downarrow} v}{{\downarrow} x}+\tfrac{{\downarrow} u}{{\downarrow} y} \right) \right){\downarrow}(x,y)}=0.\square</math></div>
 
  
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=== Fundamental theorem of algebra ===
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For <math>\pi(x) := |\{p \in {\mathbb{P}_{\le x}} : x \in {}^{\omega}{\mathbb{R}}\}|</math>, it holds that <math>\pi(\omega) = \widetilde{{_e}\omega}\omega + \mathcal{O}({_\epsilon}\omega\;{\omega}^{\tilde{2}})</math>.
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Every non-constant polynomial <math>p \in {}^{(\omega)}\mathbb{C}</math> has at least one complex root.
 
  
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'''Indirect proof:''' By performing an affine substitution of variables, reduce to the case <math>\widetilde{p(0)} \ne \mathcal{O}(\iota)</math>. Suppose that <math>p(z) \ne 0</math> for all <math>z \in {}^{(\omega)}\mathbb{C}</math>.
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==== Proof: ====
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From intervals of fix length <math>y \in {}^{\omega}{\mathbb{R}_{>0}}, \check{y}</math> set-2-tuples of prime numbers are formed such that the first interval has the unchanged representative prime number density and the second interval is empty, then the interval with the second most prime number density is followed by the second least one etc. The Stirling formula suggests the prime gap <math>n = {\epsilon}^{\sigma} = \mathcal{O}({_\epsilon}(n!))</math>.
  
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Since <math>f(z) := \widetilde{p(z)}</math> is holomorphic, it holds that <math>f(\tilde{\iota}) = \mathcal{O}(\iota)</math>. By the mean value inequality <math>|f(0)| \le {|f|}_{\gamma}</math> for <math>\gamma = \partial\mathbb{B}_{r}(0)</math> and arbitrary <math>r \in {}^{(\omega)}\mathbb{R}_{&gt;0}</math>, and hence <math>f(0) = \mathcal{O}(\iota)</math>, which is a contradiction (and hence exactly <math>z(m) = m</math> holds).<math>\square</math>
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For induction basis <math>n = 2</math> resp. 3, the hypothesis states the first interval to contain <math>x_n/{_\epsilon}x_n</math> primes for <math>n \in {}^{\omega}{\mathbb{N}_{\ge2}}</math> and <math>x_4 \in [2, 4[</math>. Stepping from <math>x_n</math> to <math>x_n^2</math> finds <math>\pi(x_n^2) = \pi(x_n) \check{x}_n</math> primes only from <math>\pi(x_n) = x_n/{_\epsilon}x_n</math>. The average prime gap is <math>{_\epsilon}x_n</math>, the maximal one <math>{_\epsilon}x_n^2</math> and the maximal <math>x_n^2</math> to <math>x_n</math> behaves like <math>\omega</math> to <math>{\omega}^{\tilde{2}}.\square</math>
  
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=== Newton’s method ===
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=== Gelfond-Schneider theorem ===
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Demanding above <math>f(\curvearrowright z)=f(z)+f^\prime(z){\downarrow}z=0</math> implies <math>z_{\grave{n}} := z_n-{f^\prime(z_n)}^{-1}f(z_n)</math> if <math>{f^\prime(z_n)}^{-1}</math> is invertible resulting in quadratic convergence close to a zero.<math>\square</math>
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It holds <math>a^b \notin {}_{\omega}^{\omega} \mathbb{A}_\mathbb{C}</math> where <math>a, c \in {}^{\omega} \mathbb{A}_\mathbb{C} \setminus \mathbb{B}</math> and infinitesimal <math>\varepsilon, b \in {}^{\omega}\mathbb{A}_\mathbb{C} \setminus {}_{\omega}^{\omega}\mathbb{R}</math>.
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==== Proof: ====
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The minimal polynomials <math>p</math> (and <math>q</math>) of <math>c^r</math> resp. <math>c^{r\pm\varepsilon} = a^b</math> for maximal <math>r \in {}_{\omega}^{\omega}\mathbb{R}_{>0}</math> and <math>f = p\;(q)</math> lead to the contradiction <math>{}^1f(c^{r(\pm\varepsilon)}) \ne 0 = (f(c^r) - f(c^{r\pm\varepsilon})) / (c^r - c^{r\pm\varepsilon}) = {}^1f(c^{r(\pm\varepsilon)}).\square</math>
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== MWiki has moved ==
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The new URL is: [https://en.hwiki.de/maths.html HWiki]
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== Recommended reading ==
  
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== Recommended readings ==
 
 
[https://en.calameo.com/books/003777977258f7b4aa332 Nonstandard Mathematics]
 
[https://en.calameo.com/books/003777977258f7b4aa332 Nonstandard Mathematics]
  
 
[[de:Hauptseite]]
 
[[de:Hauptseite]]

Latest revision as of 18:01, 31 July 2024

Welcome to MWiki

Theorems of the month

Prime number theorem

For [math]\displaystyle{ \pi(x) := |\{p \in {\mathbb{P}_{\le x}} : x \in {}^{\omega}{\mathbb{R}}\}| }[/math], it holds that [math]\displaystyle{ \pi(\omega) = \widetilde{{_e}\omega}\omega + \mathcal{O}({_\epsilon}\omega\;{\omega}^{\tilde{2}}) }[/math].

Proof:

From intervals of fix length [math]\displaystyle{ y \in {}^{\omega}{\mathbb{R}_{\gt 0}}, \check{y} }[/math] set-2-tuples of prime numbers are formed such that the first interval has the unchanged representative prime number density and the second interval is empty, then the interval with the second most prime number density is followed by the second least one etc. The Stirling formula suggests the prime gap [math]\displaystyle{ n = {\epsilon}^{\sigma} = \mathcal{O}({_\epsilon}(n!)) }[/math].

For induction basis [math]\displaystyle{ n = 2 }[/math] resp. 3, the hypothesis states the first interval to contain [math]\displaystyle{ x_n/{_\epsilon}x_n }[/math] primes for [math]\displaystyle{ n \in {}^{\omega}{\mathbb{N}_{\ge2}} }[/math] and [math]\displaystyle{ x_4 \in [2, 4[ }[/math]. Stepping from [math]\displaystyle{ x_n }[/math] to [math]\displaystyle{ x_n^2 }[/math] finds [math]\displaystyle{ \pi(x_n^2) = \pi(x_n) \check{x}_n }[/math] primes only from [math]\displaystyle{ \pi(x_n) = x_n/{_\epsilon}x_n }[/math]. The average prime gap is [math]\displaystyle{ {_\epsilon}x_n }[/math], the maximal one [math]\displaystyle{ {_\epsilon}x_n^2 }[/math] and the maximal [math]\displaystyle{ x_n^2 }[/math] to [math]\displaystyle{ x_n }[/math] behaves like [math]\displaystyle{ \omega }[/math] to [math]\displaystyle{ {\omega}^{\tilde{2}}.\square }[/math]

Gelfond-Schneider theorem

It holds [math]\displaystyle{ a^b \notin {}_{\omega}^{\omega} \mathbb{A}_\mathbb{C} }[/math] where [math]\displaystyle{ a, c \in {}^{\omega} \mathbb{A}_\mathbb{C} \setminus \mathbb{B} }[/math] and infinitesimal [math]\displaystyle{ \varepsilon, b \in {}^{\omega}\mathbb{A}_\mathbb{C} \setminus {}_{\omega}^{\omega}\mathbb{R} }[/math].

Proof:

The minimal polynomials [math]\displaystyle{ p }[/math] (and [math]\displaystyle{ q }[/math]) of [math]\displaystyle{ c^r }[/math] resp. [math]\displaystyle{ c^{r\pm\varepsilon} = a^b }[/math] for maximal [math]\displaystyle{ r \in {}_{\omega}^{\omega}\mathbb{R}_{\gt 0} }[/math] and [math]\displaystyle{ f = p\;(q) }[/math] lead to the contradiction [math]\displaystyle{ {}^1f(c^{r(\pm\varepsilon)}) \ne 0 = (f(c^r) - f(c^{r\pm\varepsilon})) / (c^r - c^{r\pm\varepsilon}) = {}^1f(c^{r(\pm\varepsilon)}).\square }[/math]

MWiki has moved

The new URL is: HWiki

Recommended reading

Nonstandard Mathematics

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