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LATTICES, LAX Lemma Let linear

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$\mathit{\nu }\mathrm{\in }{\mathit{\pi }}_{\mathit{X}}^{\mathrm{*}}{\mathrm{\Omega }}^{\mathrm{1}\mathrm{,}\mathrm{0}}\mathrm{\left(}\mathit{X}\mathrm{\right)}$. ${{\displaystyle \sum }}_{\mathit{Q}\mathrm{\in }\stackrel{\mathrm{ˆ}}{\mathit{Y}}}\mathrm{R}\mathrm{e}\mathrm{s}\mathit{Q}$ A (0) $\mathit{\omega }\mathit{D}\mathrm{\left(}\mathit{t}\mathrm{\right)}$ $\mathrm{=}{\mathit{\pi }}_{\mathit{X}}^{\mathrm{-}\mathrm{1}}\mathrm{\left(}{\mathit{D}}_{\mathrm{1}}\mathrm{}\mathrm{\right(}\mathit{t}\mathrm{\left)}\mathrm{\right)}$. Ave: ${\mathrm{\Omega }}^{\mathrm{1}\mathrm{,}\mathrm{0}}\mathrm{\left(}\mathit{Y}\mathrm{\right)}\mathrm{\to }{\mathrm{\Omega }}^{\mathrm{1}\mathrm{,}\mathrm{0}}\mathrm{\left(}\mathit{Y}\mathrm{\right)}$

A A $\mathit{X}\mathit{X}\mathit{Y}\stackrel{\mathrm{ˆ}}{\mathit{Y}}\mathit{\omega }$, S. $\mathit{W}\mathrm{;}{\mathit{f}}_{\mathrm{1}}$. ${\mathit{D}}_{\mathrm{1}}\mathrm{\left(}\mathit{t}\mathrm{\right)}{\mathit{f}}_{\mathrm{1}}$ : $\mathit{X}\mathrm{\to }\mathrm{C}\mathit{\lambda }\mathrm{=}\mathrm{\Lambda }\mathrm{\circ }{\mathit{\pi }}_{\mathit{X}}\mathit{\omega }\mathrm{=}{\mathit{\pi }}_{\mathit{X}}^{\mathrm{*}}\mathit{\eta }\mathit{X}\mathrm{=}\mathit{Y}\mathrm{/}\mathit{S}$. ${\mathrm{\Omega }}^{\mathrm{1}\mathrm{,}\mathrm{0}}\mathrm{\left(}\mathit{Y}\mathrm{\right)}$

A ${\mathit{f}}_{\mathrm{1}}\mathrm{=}\mathit{z}{\mathit{f}}_{\mathrm{1}}$. $\mathit{f}\mathrm{=}\mathit{f}\mathrm{1}\mathrm{\circ }{\mathit{\pi }}_{\mathit{X}}$. $\mathit{g}\mathrm{\left(}\stackrel{\mathrm{ˆ}}{\mathit{Y}}\mathrm{\right)}\mathrm{=}\stackrel{\mathrm{ˆ}}{\mathit{Y}}\mathit{\omega }\mathrm{\in }\mathrm{k}\mathrm{e}\mathrm{r}$ Ave $\mathit{\omega }\mathrm{\in }\mathrm{k}\mathrm{e}\mathrm{r}$ Ave ${\mathit{\pi }}_{\mathit{X}}^{\mathrm{-}\mathrm{1}}\mathrm{\left(}{\mathit{X}}_{\mathrm{\infty }}\mathrm{\right)}$.

$\mathrm{\rfloor }\mathrm{i}{\mathrm{m}}_{\mathit{t}\mathrm{\to }\mathrm{0}}\mathrm{(}\mathrm{1}\mathrm{/}\mathit{t}\mathrm{)}{{\displaystyle \int }}_{\mathit{D}\mathrm{\left(}\mathrm{0}\mathrm{\right)}}^{\mathit{D}\mathrm{\left(}\mathit{t}\mathrm{\right)}}\mathit{\omega }\mathrm{=}{{\displaystyle \sum }}_{\mathit{Q}\mathrm{\in }\stackrel{\mathrm{ˆ}}{\mathit{Y}}}{\mathrm{Res}}_{\mathit{Q}}\mathrm{\left(}\mathit{\lambda }\mathrm{\right(}\mathrm{0}\mathrm{\left)}\mathrm{\right(}{\mathit{g}}^{\mathrm{-}\mathrm{1}}{\mathrm{)}}^{\mathrm{*}}\mathit{\omega }\mathrm{)}\mathrm{=}{{\displaystyle \sum }}_{\mathit{Q}\mathrm{\in }\stackrel{\mathrm{ˆ}}{\mathit{Y}}}{\mathrm{Res}}_{\mathit{g}\mathrm{\left(}\mathit{Q}\mathrm{\right)}}\mathrm{(}\mathit{\lambda }\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{\left(}{\mathit{g}}^{\mathrm{-}\mathrm{1}}{\mathrm{)}}^{\mathrm{*}}\mathit{\omega }\mathrm{\right)}$

${\mathrm{\Omega }}^{\mathrm{1}\mathrm{,}\mathrm{0}}\mathrm{\left(}\mathit{Y}\mathrm{\right)}\mathrm{=}\mathrm{k}\mathrm{e}\mathrm{r}$ Ave $\mathrm{\oplus }{\mathit{\pi }}_{\mathit{X}}^{\mathrm{*}}\mathrm{\left(}{\mathrm{\Omega }}^{\mathrm{1}\mathrm{,}\mathrm{0}}\mathrm{\right(}\mathit{X}\mathrm{\left)}\mathrm{\right)}$

$\mathrm{⊢}{\lim }_{\mathrm{\to }\mathrm{0}}\frac{\mathrm{1}}{\mathit{t}}{{\displaystyle \int }}_{\mathit{D}\mathrm{\left(}\mathrm{0}\mathrm{\right)}}^{\mathit{D}\mathrm{\left(}\mathit{t}\mathrm{\right)}}\mathit{\omega }\mathrm{=}{\lim }_{\mathit{t}\mathrm{\to }\mathrm{0}}\frac{\mathrm{1}}{\mathit{t}}\mathrm{ċ}\frac{\mathrm{1}}{\mathrm{\left|}\mathit{S}\mathrm{\right|}}{{\displaystyle \sum }}_{\mathit{g}\mathrm{\in }\mathit{S}}{{\displaystyle \int }}_{\mathit{g}\mathit{D}\mathrm{\left(}\mathrm{0}\mathrm{\right)}}^{\mathit{g}\mathit{D}\mathrm{\left(}\mathit{t}\mathrm{\right)}}\mathit{\omega }\mathrm{=}{\lim }_{\mathit{t}\mathrm{\to }\mathrm{0}}\frac{\mathrm{1}}{\mathit{t}}\mathrm{ċ}\frac{\mathrm{1}}{\mathrm{\left|}\mathit{S}\mathrm{\right|}}{{\displaystyle \int }}_{\mathit{D}\mathrm{\left(}\mathrm{0}\mathrm{\right)}}^{\mathit{D}\mathrm{\left(}\mathit{t}\mathrm{\right)}}{{\displaystyle \sum }}_{\mathit{g}\mathrm{\in }\mathit{S}}{\mathit{g}}^{\mathrm{*}}\mathit{\omega }\mathrm{=}{\lim }_{\mathit{t}\mathrm{\to }\mathrm{0}}\frac{\mathrm{1}}{\mathit{t}}{{\displaystyle \int }}_{\mathit{D}\mathrm{\left(}\mathrm{0}\mathrm{\right)}}^{\mathit{D}\mathrm{\left(}\mathit{t}\mathrm{\right)}}\mathrm{A}\mathrm{v}\mathrm{e}\mathit{\omega }\mathrm{=}\mathrm{0}$

${{\displaystyle \sum }}_{\mathrm{!}\mathrm{2}\mathrm{\in }\stackrel{\mathrm{ˆ}}{\mathit{Y}}}{\mathrm{Res}}_{\mathit{Q}}\mathit{\lambda }\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathit{\omega }\mathrm{=}{{\displaystyle \sum }}_{\mathit{Q}\mathrm{\in }\stackrel{\mathrm{ˆ}}{\mathit{Y}}}{\mathrm{Res}}_{\mathit{Q}}\mathrm{\left(}\mathit{\lambda }\mathrm{\right(}\mathrm{0}\mathrm{)}\mathrm{\circ }\mathit{g}\mathrm{\circ }\mathit{\omega }\mathrm{)}$ (for $\mathit{g}\mathrm{\in }\mathit{S}$

${{\displaystyle \sum }}_{\mathrm{2}\mathrm{\in }\stackrel{\mathrm{ˆ}}{\mathit{Y}}}$ tes A (0) $\mathit{\omega }\mathrm{=}{{\displaystyle \sum }}_{\mathit{Q}\mathrm{\in }\stackrel{\mathrm{ˆ}}{\mathit{Y}}}\frac{\mathrm{1}}{\mathrm{\left|}\mathit{S}\mathrm{\right|}}{{\displaystyle \sum }}_{\mathit{g}\mathrm{\in }\mathit{S}}{\mathrm{Res}}_{\mathit{Q}}$ (A (0) ( ${\mathit{g}}^{\mathrm{-}\mathrm{1}}{\mathrm{)}}^{\mathrm{*}}\mathit{\omega }$) $\mathrm{=}{{\displaystyle \sum }}_{\mathit{Q}\mathrm{\in }\stackrel{\mathrm{ˆ}}{\mathit{Y}}}{\mathrm{Res}}_{\mathit{Q}}$( $\mathit{\lambda }\mathrm{\left(}\mathrm{0}\mathrm{\right)}$ (Ave $\mathit{\omega }\mathrm{)}$) $\mathrm{=}\mathrm{0}$.

${\lim }_{{\mathrm{\}}}_{\mathrm{\to }\mathrm{0}}}\frac{\mathrm{1}}{\mathit{t}}{{\displaystyle \int }}_{\mathit{D}\mathrm{\left(}\mathrm{0}\mathrm{\right)}}^{\mathit{D}\mathrm{\left(}\mathit{t}\mathrm{\right)}}\mathit{\omega }\mathrm{=}\mathrm{\left|}\mathit{S}\mathrm{\right|}{\lim }_{\mathit{t}\mathrm{\to }\mathrm{0}}\frac{\mathrm{1}}{\mathit{t}}{{\displaystyle \int }}_{{\mathit{D}}_{\mathrm{1}}\mathrm{\left(}\mathrm{0}\mathrm{\right)}}^{{\mathit{D}}_{\mathrm{1}}\mathrm{\left(}\mathit{t}\mathrm{\right)}}\mathit{\eta }\mathrm{=}\mathrm{\left|}\mathit{S}\mathrm{\right|}{{\displaystyle \sum }}_{\mathit{P}\mathrm{\in }\overline{\mathit{X}}}{\mathrm{Res}}_{\mathit{P}}\mathrm{\Lambda }\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathit{\eta }\mathrm{=}{{\displaystyle \sum }}_{\mathit{Q}\mathrm{\in }\stackrel{\mathrm{ˆ}}{\mathit{Y}}}{\mathrm{Res}}_{\mathit{Q}}\mathit{\lambda }\mathrm{\left(}\mathrm{0}\mathrm{\right)}{\mathit{\pi }}_{\mathit{X}}^{\mathrm{*}}\mathit{\eta }$,

note

obtain of of of of of of of On on orbit order other over

17. 18) 297 4. flow (by (since )), $\mathrm{\square }$

Proof. proposition. $\mathrm{P}\mathrm{R}\mathrm{Y}\mathrm{M}\mathrm{-}\mathrm{T}\mathrm{J}\mathrm{U}\mathrm{R}\mathrm{I}\mathrm{N}$

quotient

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