Latest revision as of 02:03, 1 May 2024

Welcome to MWiki

Theorems of the month

Green's theorem

For some [math]\displaystyle{ h }[/math]-domain [math]\displaystyle{ \mathbb{D} \subseteq {}^{(\omega)}\mathbb{R}^{2} }[/math], infinitesimal [math]\displaystyle{ h = |{\downarrow}x|= |{\downarrow}y| = |\overset{\rightharpoonup}{\gamma}(s) - \gamma(s)| = \mathcal{O}({\tilde{\omega}}^{m}) }[/math], sufficiently large [math]\displaystyle{ m \in \mathbb{N}^{*}, (x, y) \in \mathbb{D}, \mathbb{D}^{-} := \{(x, y) \in \mathbb{D} : (x + h, y + h) \in \mathbb{D}\} }[/math], and a simply closed path [math]\displaystyle{ \gamma: [a, b[\rightarrow {\downarrow} \mathbb{D} }[/math] followed anticlockwise, choosing [math]\displaystyle{ \overset{\rightharpoonup}{\gamma}(s) = \gamma(\overset{\rightharpoonup}{s}) }[/math] for [math]\displaystyle{ s \in [a, b[, A \subseteq {[a, b]}^{2} }[/math], the following equation holds for sufficiently [math]\displaystyle{ \alpha }[/math]-continuous functions [math]\displaystyle{ u, v: \mathbb{D} \rightarrow \mathbb{R} }[/math] with not necessarily continuous [math]\displaystyle{ {\downarrow} u/{\downarrow} x, {\downarrow} u/{\downarrow} y, {\downarrow} v/{\downarrow} x }[/math] and [math]\displaystyle{ {\downarrow} v/{\downarrow} y }[/math]

[math]\displaystyle{ {\uparrow}_{\gamma }{(u\,{\downarrow}x+v\,{\downarrow}y)}={\uparrow}_{(x,y)\in {\mathbb{D}^{-}}}{\left( \tfrac{{\downarrow} v}{{\downarrow} x}-\tfrac{{\downarrow} u}{{\downarrow} y} \right){\downarrow}(x,y)}. }[/math]

Proof:

Only [math]\displaystyle{ \mathbb{D} := \{(x, y) : r \le x \le s, f(x) \le y \le g(x)\}, r, s \in {}^{(\omega)}\mathbb{R}, f, g : {\downarrow} \mathbb{D} \rightarrow {}^{(\omega)}\mathbb{R} }[/math] is proved, since the proof is analogous for each case rotated by [math]\displaystyle{ \check{\pi} }[/math]. Every [math]\displaystyle{ h }[/math]-domian is union of such sets. Simply showing

[math]\displaystyle{ {\uparrow}_{\gamma }{u\,{\downarrow}x}=-{\uparrow}_{(x,y)\in {\mathbb{D}^{-}}}{\tfrac{{\downarrow} u}{{\downarrow} y}{\downarrow}(x,y)}. }[/math]

is sufficient because the other relation is given analogously. Neglecting the regions of [math]\displaystyle{ \gamma }[/math] with [math]\displaystyle{ {\downarrow}x = 0 }[/math] and [math]\displaystyle{ s := h(u(r, g(r)) - u(t, g(t))) }[/math] shows

[math]\displaystyle{ -{\uparrow}_{\gamma }{u\,{\downarrow}x}-s={\uparrow}_{t}^{r}{u(x,g(x)){\downarrow}x}-{\uparrow}_{t}^{r}{u(x,f(x)){\downarrow}x}={\uparrow}_{t}^{r}{{\uparrow}_{f(x)}^{g(x)}{\tfrac{{\downarrow} u}{{\downarrow} y}}{\downarrow}y{\downarrow}x}={\uparrow}_{(x,y)\in {\mathbb{D}^{-}}}{\tfrac{{\downarrow} u}{{\downarrow} y}{\downarrow}(x,y)}.\square }[/math]

Singmaster's theorem

There are maximally 8 distinct binomial coefficients of the same value > 1.

Proof:

The existence is clear due to [math]\displaystyle{ \tbinom{3003}{1} = \tbinom{78}{2} = \tbinom{15}{5} = \tbinom{14}{6} }[/math] and the structure of Pascal's triangle. With [math]\displaystyle{ p \in {}^{\omega }{\mathbb{P}}, a,b ,c, d \in {}^{\omega }{\mathbb{N^*}}, \hat{a} \le r := p - b, \hat{a} \lt \hat{c} \le n := p - d, b \lt d }[/math] and [math]\displaystyle{ s \notin \mathbb{P} }[/math] for every [math]\displaystyle{ s \in [\max(r - \acute{a},\grave{n}), r] }[/math], Stirling's formula [math]\displaystyle{ {n!}^2\sim\pi(\hat{n}+\tilde{3}){(\tilde{\epsilon}n)}^{\hat{n}} }[/math] and the prime number theorem imply [math]\displaystyle{ \omega\tbinom{r}{a} \le {}_\epsilon\omega\tbinom{n}{c} }[/math] for [math]\displaystyle{ p \rightarrow \omega.\square }[/math]

@@ Line 2: / Line 2: @@
 = Welcome to MWiki =
 == Theorems of the month ==
-=== First fundamental theorem of exact differential and integral calculus for <abbr title="line integral">LI</abbr>s ===
+=== Green's theorem ===
-The function <math>F(z)={\uparrow}_{\gamma }{f(\zeta ){\downarrow}\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 G = [a, b[ \, \cap \, C</math>, and choosing <math>{}^\curvearrowright \gamma(x) = \gamma({}^\curvearrowright x)</math> is exactly differentiable, and for all <math>x \in G</math> and <math>z = \gamma(x)</math>
+For some <math>h</math>-domain <math>\mathbb{D} \subseteq {}^{(\omega)}\mathbb{R}^{2}</math>, infinitesimal <math>h = |{\downarrow}x|= |{\downarrow}y| = |\overset{\rightharpoonup}{\gamma}(s) - \gamma(s)| = \mathcal{O}({\tilde{\omega}}^{m})</math>, sufficiently large <math>m \in \mathbb{N}^{*}, (x, y) \in \mathbb{D}, \mathbb{D}^{-} := \{(x, y) \in \mathbb{D} : (x + h, y + h) \in \mathbb{D}\}</math>, and a simply closed path <math>\gamma: [a, b[\rightarrow {\downarrow} \mathbb{D}</math> followed anticlockwise, choosing <math>\overset{\rightharpoonup}{\gamma}(s) = \gamma(\overset{\rightharpoonup}{s})</math> for <math>s \in [a, b[, A \subseteq {[a, b]}^{2}</math>, the following equation holds for sufficiently <math>\alpha</math>-continuous functions <math>u, v: \mathbb{D} \rightarrow \mathbb{R}</math> with not necessarily continuous <math>{\downarrow} u/{\downarrow} x, {\downarrow} u/{\downarrow} y, {\downarrow} v/{\downarrow} x</math> and <math>{\downarrow} v/{\downarrow} y</math><div style="text-align:center;"><math>{\uparrow}_{\gamma }{(u\,{\downarrow}x+v\,{\downarrow}y)}={\uparrow}_{(x,y)\in {\mathbb{D}^{-}}}{\left( \tfrac{{\downarrow} v}{{\downarrow} x}-\tfrac{{\downarrow} u}{{\downarrow} y} \right){\downarrow}(x,y)}.</math></div>
-<div style="text-align:center;"><math>F^{\prime}(z) = f(z).</math></div>
+==== Proof: ====
+Only <math>\mathbb{D} := \{(x, y) : r \le x \le s, f(x) \le y \le g(x)\}, r, s \in {}^{(\omega)}\mathbb{R}, f, g : {\downarrow} \mathbb{D} \rightarrow {}^{(\omega)}\mathbb{R}</math> is proved, since the proof is analogous for each case rotated by <math>\check{\pi}</math>. Every <math>h</math>-domian is union of such sets. Simply showing <div style="text-align:center;"><math>{\uparrow}_{\gamma }{u\,{\downarrow}x}=-{\uparrow}_{(x,y)\in {\mathbb{D}^{-}}}{\tfrac{{\downarrow} u}{{\downarrow} y}{\downarrow}(x,y)}.</math></div> is sufficient because the other relation is given analogously. Neglecting the regions of <math>\gamma</math> with <math>{\downarrow}x = 0</math> and <math>s := h(u(r, g(r)) - u(t, g(t)))</math> shows <div style="text-align:center;"><math>-{\uparrow}_{\gamma }{u\,{\downarrow}x}-s={\uparrow}_{t}^{r}{u(x,g(x)){\downarrow}x}-{\uparrow}_{t}^{r}{u(x,f(x)){\downarrow}x}={\uparrow}_{t}^{r}{{\uparrow}_{f(x)}^{g(x)}{\tfrac{{\downarrow} u}{{\downarrow} y}}{\downarrow}y{\downarrow}x}={\uparrow}_{(x,y)\in {\mathbb{D}^{-}}}{\tfrac{{\downarrow} u}{{\downarrow} y}{\downarrow}(x,y)}.\square</math></div>
-==== Proof ====
+=== Singmaster's theorem ===
-<math>{\downarrow}F(z)</math> <math>={\uparrow}_{t\in [d,x] \cap C}{f(\gamma (t)){{\gamma }^{\prime}}(t){\downarrow}t}-{\uparrow}_{t\in [d,x[ \, \cap \, C}{f(\gamma (t)){{\gamma }^{\prime}}(t){\downarrow}t}</math> <math>={\uparrow}_{x}{f(\gamma (t))\tfrac{\gamma ({}^\curvearrowright t)-\gamma (t)}{{}^\curvearrowright t-t}{\downarrow}t}</math> <math>=f(\gamma (x)){{\gamma}^{\prime}}(x){\downarrow}x=</math> <math>\,f(\gamma (x))({}^\curvearrowright\gamma (x)-\gamma (x))</math> <math>=f(z){\downarrow}z.\square</math>
-=== Second fundamental theorem of exact differential and integral calculus for <abbr title="line integral">LI</abbr>s ===
+There are maximally 8 distinct binomial coefficients of the same value > 1.
-Conditions above imply with <math>\gamma: G \rightarrow {}^{(\omega)}\mathbb{K}</math> that
+==== Proof: ====
+The existence is clear due to <math>\tbinom{3003}{1} = \tbinom{78}{2} = \tbinom{15}{5} = \tbinom{14}{6}</math> and the structure of Pascal's triangle. With <math>p \in {}^{\omega }{\mathbb{P}}, a,b ,c, d \in {}^{\omega }{\mathbb{N^*}}, \hat{a} \le r := p - b, \hat{a} < \hat{c} \le n := p - d, b < d</math> and <math>s \notin \mathbb{P}</math> for every <math>s \in [\max(r - \acute{a},\grave{n}), r]</math>, Stirling's formula <math>{n!}^2\sim\pi(\hat{n}+\tilde{3}){(\tilde{\epsilon}n)}^{\hat{n}}</math> and the prime number theorem imply <math>\omega\tbinom{r}{a} \le {}_\epsilon\omega\tbinom{n}{c}</math> for <math>p \rightarrow \omega.\square</math>
-<div style="text-align:center;"><math>F(\gamma (b))-F(\gamma (a))={\uparrow}_{\gamma }{{F^{\prime}}(\zeta ){\downarrow}\zeta }.</math></div>
+== Recommended reading ==
-==== Proof ====
-<math>F(\gamma (b))-F(\gamma (a))</math> <math>={+}_{t\in G}{F({}^\curvearrowright\,\gamma (t))}-F(\gamma (t))</math> <math>={+}_{t\in G}{{{F}^{\prime}}(\gamma (t))({}^\curvearrowright\,\gamma (t)-\gamma (t))}</math> <math>={\uparrow}_{t\in G}{{F^{\prime}}(\gamma (t)){{\gamma }^{\prime}}(t){\downarrow}t}</math> <math>={\uparrow}_{\gamma }{{F_{{}^\curvearrowright }^{\prime}}(\zeta ){\downarrow}\zeta }.\square</math>
-=== Approximation theorem ===
-The derivatives <math>f^{(s)}(x) \in {}^{\omega}\mathbb{R}</math> for <math>x \in {}^{\omega}\mathbb{R}</math> allow computing the interpolating function <math>g(x) := {+}_{r=0}^{\acute{m}}{\chi_{]x_r, x_{\grave{r}}[}(x)((x_{\grave{r}}-x)p_r(x)+(x-x_r)p_{\grave{r}}(x))/(x_{\grave{r}}-x_r)}+{+}_{r=0}^m{\chi_{\{x_r\}}(x)p_r(x)}</math> for <math>m, n \in {}^{\nu}\mathbb{N}</math> and <math>p_r(x) := {+}_{s=0}^n{f^{(s)}(x_r){(x-x_r)}^s/s!}</math> in <math>\mathcal{O}(\sigma mn)</math> where <math>f^{(s)}(x_r) = g^{(s)}(x_r)</math> holds for every <math>x_r \in {}^{\omega}\mathbb{R}</math>. Replace in the complex case <math>{}^{\omega}\mathbb{R}</math> by <math>{}^{\omega}\mathbb{C}</math> and put <math>x = \gamma(t) \in {}^{\omega}\mathbb{C}</math> for the path <math>\gamma(t)</math> where <math>t \in {}^{\omega}\mathbb{R}.\square</math>
-== Recommended readings ==
 [https://en.calameo.com/books/003777977258f7b4aa332 Nonstandard Mathematics]
 [[de:Hauptseite]]

Difference between revisions of "Main Page"

Latest revision as of 02:03, 1 May 2024

Welcome to MWiki

Theorems of the month

Green's theorem

Proof:

Singmaster's theorem

Proof:

Recommended reading

Navigation menu

Personal tools

Namespaces

Variants

Views

More

Search

Navigation

Tools

In other languages