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(Three-cube theorem and Fickett's theorem)
(Fundamental theorems)
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= Welcome to MWiki =
 
= Welcome to MWiki =
 
== Theorems of the month ==
 
== Theorems of the month ==
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=== Three-Cube Theorem ===
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First fundamental theorem of exact differential and integral calculus for line integrals: The function <math>F(z)=\int\limits_{\gamma }{f(\zeta )dB\zeta }</math> where <math>\gamma: [d, x[ \, \cap \, C \rightarrow A \subseteq {}^{(\omega)}\mathbb{K}, C \subseteq \mathbb{R}, f: A \rightarrow {}^{(\omega)}\mathbb{K}, d \in [a, b[ \, \cap \, C</math>, and choosing <math>\curvearrowright B \gamma(x) = \gamma(\curvearrowright D x)</math> is exactly <math>B</math>-differentiable, and for all <math>x \in [a, b[ \, \cap \, C</math> and <math>z = \gamma(x)</math>
  
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By Fermat’s little theorem, <math>k \in {}^{\omega }{\mathbb{Z}}</math> is sum of three cubes if and only if
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<div style="text-align:center;"><math>F' \curvearrowright B(z) = f(z).</math></div>
  
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<div style="text-align:center;"><math>k=(n - a)^3 + n^3 + (n + b)^3 = 3n^3 - a^3 + b^3+ 3c \ne \pm 4\mod 9</math></div>
 
  
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and <math>a, b, c, d, m, n \in {}^{\omega }{\mathbb{Z}}</math> implies both <math>(a^2 + b^2)n - (a - b)n^2 = c =: dn</math> and <math>m^2 = n^2 - 4(b^2 - bn + d)</math> for <math>2a_{1,2} = n \pm m.\square</math>
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Proof: <math>dB(F(z))=\int\limits_{t\in [d,x] \, \cap \, C}{f(\gamma (t)){{{{\gamma }'}}_{\curvearrowright }}D(t)dDt}-\int\limits_{t\in [d,x[ \, \cap \, C}{f(\gamma (t)){{{{\gamma }'}}_{\curvearrowright }}D(t)dDt}=\int\limits_{x}{f(\gamma (t))\frac{\gamma (\curvearrowright Dt)-\gamma (t)}{\curvearrowright Dt-t}dDt}=f(\gamma (x)){{{\gamma }'}_{\curvearrowright }}D(x)dDx=\,f(\gamma (x))(\curvearrowright B\gamma (x)-\gamma (x))=f(z)dBz.\square</math>
  
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=== Fickett's Theorem ===
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Second fundamental theorem of exact differential and integral calculus for line integrals: According to the conditions from above, we have with <math>\gamma: [a, b[ \, \cap \, C \rightarrow {}^{(\omega)}\mathbb{K}</math> that
  
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For any relative positions of two overlapping congruent rectangular <math>n</math>-prisms <math>Q</math> and <math>R</math> with <math>n \in {}^{\omega }\mathbb{N}_{\ge 2}</math> and <math>m := 2n - 1</math>, it can be stated for the exact standard measure <math>\mu</math>, where <math>\mu</math> for <math>n = 2</math> needs to be replaced by the Euclidean path length <math>L</math>, that:
 
  
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<div style="text-align:center;"><math>\hat{m} &lt; r := \mu(\partial Q \cap R)/\mu(\partial R \cap Q) &lt; m.</math></div>
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<div style="text-align:center;"><math>F(\gamma (b))-F(\gamma (a))=\int\limits_{\gamma }{{{{{F}'}}_{\curvearrowright }}B(\zeta )dB\zeta }.</math></div>
  
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==== Proof: ====
 
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Since the underlying extremal problem has its maximum for rectangles with the side lengths <math>s</math> and <math>s + 2d0</math>, min <math>r = s/(3s - 2d0) \le r \le</math> max <math>r = (3s - 2d0)/s</math> holds. The proof for <math>n &gt; 2</math> is analogous.<math>\square</math>
 
  
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== Recommended reading ==
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Proof: <math>F(\gamma (b))-F(\gamma (a))=\sum\limits_{t\in [a,b[ \, \cap \, C}{F(\curvearrowright B\,\gamma (t))}-F(\gamma (t))=\sum\limits_{t\in [a,b[ \, \cap \, C}{{{{{F}'}}_{\curvearrowright }}B(\gamma (t))(\curvearrowright B\,\gamma (t)-\gamma (t))}=\int\limits_{t\in [a,b[ \, \cap \, C}{{{{{F}'}}_{\curvearrowright }}B(\gamma (t)){{{{\gamma }'}}_{\curvearrowright }}D(t)dDt}=\int\limits_{\gamma }{{{{{F}'}}_{\curvearrowright }}B(\zeta )dB\zeta }.\square</math>
  
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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]]

Revision as of 20:55, 30 November 2021

Welcome to MWiki

Theorems of the month

First fundamental theorem of exact differential and integral calculus for line integrals: The function [math]\displaystyle{ F(z)=\int\limits_{\gamma }{f(\zeta )dB\zeta } }[/math] where [math]\displaystyle{ \gamma: [d, x[ \, \cap \, C \rightarrow A \subseteq {}^{(\omega)}\mathbb{K}, C \subseteq \mathbb{R}, f: A \rightarrow {}^{(\omega)}\mathbb{K}, d \in [a, b[ \, \cap \, C }[/math], and choosing [math]\displaystyle{ \curvearrowright B \gamma(x) = \gamma(\curvearrowright D x) }[/math] is exactly [math]\displaystyle{ B }[/math]-differentiable, and for all [math]\displaystyle{ x \in [a, b[ \, \cap \, C }[/math] and [math]\displaystyle{ z = \gamma(x) }[/math]

[math]\displaystyle{ F' \curvearrowright B(z) = f(z). }[/math]


Proof: [math]\displaystyle{ dB(F(z))=\int\limits_{t\in [d,x] \, \cap \, C}{f(\gamma (t)){{{{\gamma }'}}_{\curvearrowright }}D(t)dDt}-\int\limits_{t\in [d,x[ \, \cap \, C}{f(\gamma (t)){{{{\gamma }'}}_{\curvearrowright }}D(t)dDt}=\int\limits_{x}{f(\gamma (t))\frac{\gamma (\curvearrowright Dt)-\gamma (t)}{\curvearrowright Dt-t}dDt}=f(\gamma (x)){{{\gamma }'}_{\curvearrowright }}D(x)dDx=\,f(\gamma (x))(\curvearrowright B\gamma (x)-\gamma (x))=f(z)dBz.\square }[/math]

Second fundamental theorem of exact differential and integral calculus for line integrals: According to the conditions from above, we have with [math]\displaystyle{ \gamma: [a, b[ \, \cap \, C \rightarrow {}^{(\omega)}\mathbb{K} }[/math] that


[math]\displaystyle{ F(\gamma (b))-F(\gamma (a))=\int\limits_{\gamma }{{{{{F}'}}_{\curvearrowright }}B(\zeta )dB\zeta }. }[/math]


Proof: [math]\displaystyle{ F(\gamma (b))-F(\gamma (a))=\sum\limits_{t\in [a,b[ \, \cap \, C}{F(\curvearrowright B\,\gamma (t))}-F(\gamma (t))=\sum\limits_{t\in [a,b[ \, \cap \, C}{{{{{F}'}}_{\curvearrowright }}B(\gamma (t))(\curvearrowright B\,\gamma (t)-\gamma (t))}=\int\limits_{t\in [a,b[ \, \cap \, C}{{{{{F}'}}_{\curvearrowright }}B(\gamma (t)){{{{\gamma }'}}_{\curvearrowright }}D(t)dDt}=\int\limits_{\gamma }{{{{{F}'}}_{\curvearrowright }}B(\zeta )dB\zeta }.\square }[/math]

Recommended readings

Nonstandard Mathematics

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