Latest revision as of 02:03, 1 May 2024

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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 1: / Line 1: @@
 __NOTOC__
 = Welcome to MWiki =
-== Theorem of the month ==
+== Theorems of the month ==
-The intex method solves every solvable LP in <math>\mathcal{O}({\vartheta}^{3})</math>.
+=== Green's theorem ===
-== Proof and algorithm ==
+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>
-First, normalise and scale <math>{b}^{T}y - {c}^{T}x \le 0, Ax \le b</math> as well as <math>{A}^{T}y \ge c</math>. Let the <em>height</em> <math>h</math> have the initial value <math>h_0 := s |\min \; \{b_1, ..., b_m, -d_1, ..., -d_n\}|</math> for the <em>elongation factor</em> <math>s \in \, ]1, 2]</math>.</br>
-The 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 - {c}^{T}x \le h, Ax - b \le (h, ..., h)^T \in {}^{\omega}\mathbb{R}_{\ge 0}^{m}, c - {A}^{T}y \le (h, ..., h)^T \in {}^{\omega}\mathbb{R}_{\ge 0}^{n}\}</math> has <math>k</math> constraints and the feasible starting point <math>(x_0, y_0, h_0/s)^{T} \in {}^{\omega}\mathbb{R}_{\ge 0}^{m+n+1}</math>, e.g. <math>(0, 0, h_0/s)^{T}</math>.</br>
-It identifies the mutually dual LPs max <math>\{{c}^{T}x : c \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 c\}</math>.
-Let the point <math>p := (x, y, h)^T</math> approximate the subpolytope’s centre of gravity <math>P^*</math> as <math>p_k^* := (\min p_k + \max p_k)/2</math> until <math>{|| \Delta p ||}_{1}</math> is sufficiently small. Here <math>x</math> takes precedence over <math>y</math>. Then extrapolate <math>p</math> via <math>{p}^{*}</math> into <math>\partial P^*</math> as <math>u</math>. Put <math>p := p^* + (u - p^*)/s</math> to shun <math>\partial P^*</math>. Hereon approximate <math>p</math> more deeply again as centre of gravity. After optionally solving all LPs min<math>{}_{k} {h}_{k}</math> by bisection methods for <math>{h}_{k} \in {}^{\omega}\mathbb{R}_{\ge 0}</math> in <math>\mathcal{O}({\vartheta}^{2})</math> each time, <math>v \in {}^{\omega}\mathbb{R}^{k}</math> may be determined such that <math>v_k := \Delta{p}_{k} \Delta{h}_{k}/r</math> and <math>r :=</math> min<math>{}_{k} \Delta{h}_{k}</math>. Simplified let <math>|\Delta{p}_{1}| = ... = |\Delta{p}_{m+n}|</math>.
+==== 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>
-Here min <math>{h}_{m+n+1}</math> may be solved for <math>p^* := p + tv</math> where <math>t \in {}^{\omega}\mathbb{R}_{\ge 0}</math> and <math>{v}_{m+n+1} = 0</math>. If min<math>{}_{k} {h}_{k} r = 0</math> follows, end, otherwise start over until min <math>h = 0</math> or min <math>h &gt; 0</math> is certain. If necessary, relax the constraints temporarily by the same small modulus.</br>
+=== Singmaster's theorem ===
-Since almost every iteration step in <math>\mathcal{O}({\omega\vartheta}^{2})</math> halves <math>h</math> at least, the strong duality theorem yields the result.<math>\square</math>
+There are maximally 8 distinct binomial coefficients of the same value > 1.
+==== 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>
+== Recommended reading ==
-== Recommended readings ==
 [https://en.calameo.com/books/003777977258f7b4aa332 Nonstandard Mathematics]
 [[de:Hauptseite]]

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Latest revision as of 02:03, 1 May 2024

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Green's theorem

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