Meine LaTeX-Spielwiese: p p r i m :⇔ ( p ∈ N ∧ p > 1 ) : ∄ a , b ∈ Z : a , b ≠ 1 ∧ a , b ≠ p : a b = p {\displaystyle p\,prim:\Leftrightarrow (p\in \mathbb {N} \land p>1):\nexists a,b\in \mathbb {Z} :a,b\neq 1\land a,b\neq p:ab=p} ∑ k = 0 n ( n k ) p k ( 1 − p ) n − k = 1 {\displaystyle \sum _{k=0}^{n}{\binom {n}{k}}p^{k}(1-p)^{n-k}=1} ∑ k = 0 n k ( n k ) p k ( 1 − p ) n − k = n p {\displaystyle \sum _{k=0}^{n}k{\binom {n}{k}}p^{k}(1-p)^{n-k}=np} ( ( a i , j ) ) ∈ K m × n , a i , j = { 1 wenn i = j b e l . wenn j < m 0 sonst {\displaystyle ((a_{i,j}))\in K^{m\times n},a_{i,j}={\begin{cases}1&{\text{wenn }}i=j\\bel.&{\text{wenn }}j<m\\0&{\text{sonst}}\end{cases}}}
Sei Φ : Ω → R k , Ω ⊂ R n offen , Φ ∈ C ′ ( Ω ) {\displaystyle {\textrm {Sei}}\ \Phi :\Omega \to \mathbb {R} ^{k},\Omega \subset \mathbb {R} ^{n}\ {\textrm {offen}},\Phi \in {\mathcal {C}}^{\prime }(\Omega )} [ x ∈ R ∧ rg ( D Φ ( x ) ) = dim ( D Φ ( x ) R n ) = n − dim ( Ker ( D Φ ( x ) ) ) = m {\displaystyle \left[x\in {\text{R}}\land {\text{rg}}(D\Phi (x))=\dim(D\Phi (x)\mathbb {R} ^{n})=n-\dim({\text{Ker}}(D\Phi (x)))=m\right.} ⇒ ∃ ρ > 0 ∀ y ∈ B ρ ( x ) : rg ( D Φ ( y ) ) ≤ m ] {\displaystyle \left.\Rightarrow \exists \rho >0\forall y\in B_{\rho }(x):{\text{rg}}(D\Phi (y))\leq m\right]} Falls ∃ ρ > 0 ∀ y ∈ B ρ ( x ) , sodass rg ( D Φ ( y ) ) = rg ( D Φ ( x ) ) {\displaystyle {\text{Falls}}\ \exists \rho >0\forall y\in B_{\rho }(x),{\text{sodass rg}}(D\Phi (y))={\text{rg}}(D\Phi (x))} ⇒ lokal sind sowohl Φ − 1 ( Φ ( x ) ) , als auch Φ ( B ρ ( x ) ) lokal glatte F l a ¨ c h e n {\displaystyle \Rightarrow {\text{lokal sind sowohl}}\ \Phi ^{-1}(\Phi (x)),{\text{als auch}}\ \Phi (B_{\rho }(x))\ {\text{lokal glatte}}\ \mathrm {Fl{\ddot {a}}chen} } Falls rg ( e D Φ ( x ) ) = rg e = rg ( D Φ ( x ) ) , e e = e , e ∈ R k × k {\displaystyle {\text{Falls rg}}(e\,D\Phi (x))={\text{rg}}\,e={\text{rg}}(D\Phi (x)),\ ee=e,\ e\in \mathbb {R} ^{k\times k}} ⇒ ∃ ε > 0 : e | Φ ( B ε ( x ) ) lokal invertierbar mit C ′ -Inversem und e ( Φ ( B ε ( x ) ) ) offen in e ( R k ) {\displaystyle \Rightarrow \exists \varepsilon >0:\left.e\right|_{\Phi (B_{\varepsilon }(x))}\ {\text{lokal invertierbar mit}}\ {\mathcal {C}}^{\prime }{\text{-Inversem und}}\ e(\Phi (B_{\varepsilon }(x)))\ {\text{offen in}}\ e(\mathbb {R} ^{k})} Falls rg ( D Φ ( x ) e ~ ) = rg e ~ = rg ( D Φ ( x ) ) , e ~ e ~ = e ~ , e ~ ∈ R n × n {\displaystyle {\text{Falls rg}}(D\Phi (x){\tilde {e}})={\text{rg}}\,{\tilde {e}}={\text{rg}}(D\Phi (x)),\ {\tilde {e}}{\tilde {e}}={\tilde {e}},\ {\tilde {e}}\in \mathbb {R} ^{n\times n}} ⇒ ∃ ε > 0 : Id − e ~ | B ε ( x ) ∩ Φ − 1 ( Φ ( x ) ) lok. inv. mit C ′ -Inv. und ( Id − e ~ ) ( B ε ( x ) ∩ Φ − 1 ( Φ ( x ) ) ) offen in ( Id − e ~ ) ( R n ) {\displaystyle \Rightarrow \exists \varepsilon >0:\left.{\text{Id}}-{\tilde {e}}\right|_{B_{\varepsilon }(x)\cap \Phi ^{-1}(\Phi (x))}\ {\text{lok. inv. mit}}\ {\mathcal {C}}^{\prime }{\text{-Inv. und}}\ ({\text{Id}}-{\tilde {e}})(B_{\varepsilon }(x)\cap \Phi ^{-1}(\Phi (x)))\ {\text{offen in}}\ ({\text{Id}}-{\tilde {e}})(\mathbb {R} ^{n})}
![{\displaystyle \left.\Rightarrow \exists \rho >0\forall y\in B_{\rho }(x):{\text{rg}}(D\Phi (y))\leq m\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a601cdee437500e624cd049dffc6106b0721adb)
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