████

██████████ ████ (██)

The ██████████ ████ (or ██████ ████ or ███████) of a ████ of ███████, is the ███ of ███ of the ███████ ███████ by █████ █████. █████████, the ████ of a ██████ ${\displaystyle {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}},{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}},\dots ,{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}$, ███████ ███████ by ${\displaystyle \overline{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}$, is the ███ of the ███████ ██████ ███████ by the ██████ of █████ in the ██████.

${\displaystyle \overline{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}=\frac{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}\sum _{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}=\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}^{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}=\frac{{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}+{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}+\cdots +{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}$

For ███████, the ██████████ ████ of ████ ██████: █, ██, ██, ██, ██ is:

${\displaystyle \frac{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}+\mathrm{<span\; word-length="2"\; class="word\; redacted">██</span>}+\mathrm{<span\; word-length="2"\; class="word\; redacted">██</span>}+\mathrm{<span\; word-length="2"\; class="word\; redacted">██</span>}+\mathrm{<span\; word-length="2"\; class="word\; redacted">██</span>}}{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}=\frac{\mathrm{<span\; word-length="3"\; class="word\; redacted">███</span>}}{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}=\mathrm{<span\; word-length="2"\; class="word\; redacted">██</span>.}}$

█████████ ████ (██)

The █████████ ████ is an ███████ that is ██████ for ████ of ████████ ███████, that are ███████████ █████████ to █████ ███████ (as is the ████ with █████ of ██████) and not █████ ███ (as is the ████ with the ██████████ ████):

${\displaystyle \overline{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}={\left(\prod _{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}=\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}^{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}\right)}^{\frac{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}={\left({\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}\cdots {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}\right)}^{\frac{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}}$

For ███████, the █████████ ████ of ████ ██████: █, ██, ██, ██, ██ is:

${\displaystyle (\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\times \mathrm{<span\; word-length="2"\; class="word\; redacted">██</span>}\times \mathrm{<span\; word-length="2"\; class="word\; redacted">██</span>}\times \mathrm{<span\; word-length="2"\; class="word\; redacted">██</span>}\times \mathrm{<span\; word-length="2"\; class="word\; redacted">██</span>}{)}^{\frac{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}=\sqrt[\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}]{\mathrm{<span\; word-length="2"\; class="word\; redacted">██</span>}\mathrm{<span\; word-length="3"\; class="word\; redacted">███</span>}\mathrm{<span\; word-length="3"\; class="word\; redacted">███</span>}}=\mathrm{<span\; word-length="2"\; class="word\; redacted">██</span>.}}$

████████ ████ (██)

The ████████ ████ is an ███████ which is ██████ for ████ of ███████ which are ███████ in ████████ to some ████, as in the ████ of █████ (█.█., ████████ per ████ of ████):

${\displaystyle \overline{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}=\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}{\left(\sum _{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}=\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}^{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}\frac{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}\right)}^{-\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}$

For ███████, the ████████ ████ of the ████ ██████: █, ██, ██, ██, ██ is

${\displaystyle \frac{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{\frac{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}+\frac{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{\mathrm{<span\; word-length="2"\; class="word\; redacted">██</span>}}+\frac{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{\mathrm{<span\; word-length="2"\; class="word\; redacted">██</span>}}+\frac{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{\mathrm{<span\; word-length="2"\; class="word\; redacted">██</span>}}+\frac{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{\mathrm{<span\; word-length="2"\; class="word\; redacted">██</span>}}}=\frac{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{\frac{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}=\mathrm{<span\; word-length="2"\; class="word\; redacted">██</span>.}}$

If ██ have ████ █████ that can █████ a ████ of a ███████ ████ in ████████████ █, ██, ██, ██, and ██ ███████, ████ the ████████ ████ of ${\displaystyle \mathrm{<span\; word-length="2"\; class="word\; redacted">██</span>}}$ █████ ██ that █████ ████ █████████ █████ ███████ ████████ will ████ at the ████ ████ as ████ █████ that can ████ █████ the ████ in ${\displaystyle \mathrm{<span\; word-length="2"\; class="word\; redacted">██</span>}}$ ███████.

████████████ between ██, ██, and ██

██, ██, and ██ of ███████████ ████ ███████ ███████ █████ ████████████:

${\displaystyle \mathrm{A}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\ge \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\ge \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$

████████ █████ if ███ the ████████ of the █████ ██████ are █████.

████ of a ███████████ ████████████

The ████ of a ███████████ ████████████ is the ████-███ ██████████ ███████ █████ of a ██████ ████████ ██████ that ████████████. If the ██████ ████████ is ███████ by ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$, ████ the ████ is ████ █████ as the ████████ █████ of ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ (███████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})}$). For a ████████ ███████████ ████████████, the ████ is █████ by ${\displaystyle \sum \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})}$, █████ the ███ is █████ over ███ ████████ ██████ of the ██████ ████████ and ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})}$ is the ███████████ ████ ████████. For a ██████████ ████████████, the ████ is ${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$, █████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})}$ is the ███████████ ███████ ████████. In ███ █████, █████████ █████ in which the ████████████ is ███████ ████████ ███ ██████████, the ████ is the ████████ ████████ of the ██████ ████████ with ███████ to ███ ███████████ ███████. The ████ ████ not █████ or be ██████; for some ███████████ █████████████ the ████ is ████████ (+∞ or −∞), █████ for ██████ the ████ is █████████.

█████ ████

The ███████████ ████, ████ █████ as the █████ ████ or ██████ ████, █████████ ███████ █████ █████. It is ███████ for ████████ ███████ ${\displaystyle {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}},\dots ,{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}$ by

${\displaystyle {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}({\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}},\dots ,{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}})={\left(\frac{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}\sum _{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}=\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}^{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}^{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}\right)}^{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}/\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}.}$

This, as a ████████ of ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$, is ████ ███████ on ${\displaystyle \mathbb{<span\; word-length="1"\; class="word\; redacted">█</span>}\setminus \{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\}}$, but can be ████████ ████████████ to ${\displaystyle \mathbb{<span\; word-length="1"\; class="word\; redacted">█</span>}\cup \{-\mathrm{\infty },+\mathrm{\infty }\}}$. By ████████ █████████ ██████ for ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$, █████ ████ █████ █████ are █████████.

█████-██████████ ████

A ███████ ████████ to the █████ ████ is the ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$-████, ████ █████ as the █████-██████████ ████. For an █████████ ████████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}:\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\to \mathbb{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ on an ████████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\subset \mathbb{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ and ████ ███████ ${\displaystyle {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}},\dots ,{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}\in \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ ██ ██████ █████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$-████ as

${\displaystyle {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}({\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}},\dots ,{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}})={\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}^{-\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}\left(\frac{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}\sum _{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}=\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}^{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\left({\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}\right)\right).}$

By ████████ █████████ █████████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$, █████ ████ █████ █████ are █████████.