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zq;k*${v8*n@&^?X*n4e!8Nk53I|vLc>R{6rKs3x|!1XFvJ^&wAf)*;Eo&j@WsE$z} z&Oj|41xindSiv1+SfBwl3GA9axcvMdH@3kF^Ox%WZ9=FOJ{$?S8V{8Whzigp3AHqU z78?KtFTYWs!P;;hkUwZ6z-$F7YWSr?unqyY)F97+;u}^41E+)Le30_N>(F;W#|FLn z1Rf+n*Fw~*G|(Xm*qp+y--FBxzw{3B6)el5!S3T9&-rtk8H^zaC<5hvKC2-3jZt&? ztU`_?lr~9&h|3kW@_#HO$P@qo diff --git a/doc/doc.tex b/doc/doc.tex index 463bf67..47d4d55 100644 --- a/doc/doc.tex +++ b/doc/doc.tex @@ -777,7 +777,7 @@ Weiterhin gilt für die Konstante $\tilde C_{\zeta_Q,j,k}$ \begin{align*} \tilde C_{\zeta_Q,j,k} & \leq 8 e \frac{c_1}{(c_2 \zeta_Q)^s} \max\{ \diam(T_j) , \diam(T_k)\}^{2-s} \left( 1 + \frac{2}{c_2\zeta_Q} \right)\sqrt{\abs{T_j}\abs{T_k}}, \end{align*} - das heißt für $s\leq2$ steigt die Konstante $\tilde C_{\zeta_Q,\j,k}$ gleichmäßig geschwächt. + das heißt für $s\leq2$ bleibt die Konstante $\tilde C_{\zeta_Q,j,k}$ gleichmäßig beschränkt. \end{sat} @@ -846,7 +846,7 @@ Wir wollen nun zeigen, dass die approximierende Matrix bezüglich der Frobenius- \begin{sat}\label{thm:sem:quad:EAV} - Seien $T_1,T_2,\ldots,T_n$ Rechtecke. Sei $\kappa : \R^3\times \R^3 \to \R$ eine asymptotisch glatte Kernfunktion mit Konstanten $c_1,c_2$ und Singularitätsordung $s \geq 0$. Sei $\tilde C_{\zeta_Q,j,k}$ wie in \eqref{math:sem:zetaQ:c} für $\zeta_Q$-zulässsige Rechtecke $T_j,T_k$ und für unzulässige sei $\tilde C_{\zeta_Q,j,k} = 0$ . Sei $A\in \R^{n \times n}$ eine Matrix, deren Einträge gegeben sind durch + Seien $T_1,T_2,\ldots,T_n$ Rechtecke. Sei $\kappa : \R^3\times \R^3 \to \R$ eine asymptotisch glatte Kernfunktion mit Konstanten $c_1,c_2$ und Singularitätsordung $0\leq s\leq 2$. Sei $\tilde C_{\zeta_Q,j,k}$ wie in \eqref{math:sem:zetaQ:c} für $\zeta_Q$-zulässsige Rechtecke $T_j,T_k$ und für unzulässige sei $\tilde C_{\zeta_Q,j,k} = 0$ . Sei $A\in \R^{n \times n}$ eine Matrix, deren Einträge gegeben sind durch \begin{align*} A_{jk} &= \int_{T_j} \int_{T_k} \kappa(\bs x,\bs y) ds_{\bs y} ds_{\bs x} @@ -956,7 +956,7 @@ Weiterhin gilt für die Konstante $\tilde C_{\zeta_E,j,k}$ \begin{align} \tilde C_{\zeta_E,j,k}&\leq 8 e \frac{c_1 }{(c_2 \zeta_E)^s} \max\{ \diam(T_j) , \diam(T_k)\}^{2-s} \left( 1 + \frac{\sqrt{2}}{c_2\zeta_E} \right)\sqrt{\abs{T_j}\abs{T_k} }, \end{align} -das heißt für $s\leq2$ steigt die Konstante $\tilde C_{\zeta_E,j,k}$ gleichmäßig geschwächt. +das heißt für $s\leq1$ bleibt die Konstante $\tilde C_{\zeta_E,j,k}$ gleichmäßig beschränkt. \end{sat} @@ -1041,7 +1041,7 @@ weshalb die Behauptung folgt. \end{defi} \begin{sat} \label{thm:sem:quad:EAE} - Seien $T_1,T_2,\ldots,T_n$ Rechtecke. Sei $\kappa : \R^3\times \R^3 \to \R$ eine asymptotisch glatte Kernfunktion mit Konstanten $c_1,c_2$ und Singularitätsordung $s \geq 0$. Sei $\tilde C_{\zeta_E,j,k}$ wie in \eqref{math:sem:zetaE:c} für $\zeta_E$-zulässsige Rechtecke $T_j,T_k$ und für unzulässige sei $\tilde C_{\zeta_E,j,k} = 0$ . Sei $A\in \R^{n \times n}$ eine Matrix, deren Einträge gegeben sind durch + Seien $T_1,T_2,\ldots,T_n$ Rechtecke. Sei $\kappa : \R^3\times \R^3 \to \R$ eine asymptotisch glatte Kernfunktion mit Konstanten $c_1,c_2$ und Singularitätsordung $0\leq s \leq 1$. Sei $\tilde C_{\zeta_E,j,k}$ wie in \eqref{math:sem:zetaE:c} für $\zeta_E$-zulässsige Rechtecke $T_j,T_k$ und für unzulässige sei $\tilde C_{\zeta_E,j,k} = 0$ . Sei $A\in \R^{n \times n}$ eine Matrix, deren Einträge gegeben sind durch \begin{align*} A_{jk} &= \int_{T_j} \int_{T_k} \kappa(\bs x,\bs y) ds_{\bs y} ds_{\bs x} -- 2.47.3