From 4aa3ecd2c9d67172c2f90063aa965f48a3127be6 Mon Sep 17 00:00:00 2001 From: LipArcanjo Date: Thu, 25 Jul 2024 18:04:48 -0300 Subject: [PATCH] add inver matrix and pick --- pdf-generator/main.typ | 14 ++++++++++++-- 1 file changed, 12 insertions(+), 2 deletions(-) diff --git a/pdf-generator/main.typ b/pdf-generator/main.typ index 8864255..5a5dba2 100644 --- a/pdf-generator/main.typ +++ b/pdf-generator/main.typ @@ -40,13 +40,22 @@ while the `m5dsum` of `hash.sh` is `9cd12`. +== Matrix Inverse 2x2 + +$A = mat(a,b;c,d)$, $A^(-1) = 1/(a d-b c)mat(d,-b;-c,a)$ + +== Pick Theorem + +$A = i + b/2 -1$, where i is the number of lattice points inside the polygon +and b is the number on boundary. Work for simple polygon. + == Burnside Lemma Let $G$ be a finite group that acts on set $X$. Let $X^g := { x in X | g.x = x }$. The number of orbits $|X slash G|$ is $|X slash G| = 1/(|G|) sum_(g in G) |X^g| $ -== Lucas' Theorem +== Lucas Theorem For prime p, $binom(n, m) = product_(i=0)^n binom(n_i, m_i) space (mod p)$ @@ -78,7 +87,7 @@ $vec(0,0,delim: "[") = 1, vec(n,0,delim: "[") = vec(0,n,delim: "[") = 1$ == Planar Graph -If G has k connected components, then $n-m+f=k+1$. +If G has k connected components, then $n-m+f=k+1$. $m <= 3n - 6$. If G has no triangles, $m ≤ 2n − 4$. @@ -111,3 +120,4 @@ $1^3+2^3+.. = (n^2(n+1)^2)/4$ | $1^4+2^4+... = (n(n+1)(2n+1)(3n^2+3n+1))/30$ = Code // Script will be used to insert code here +