Codex @codex  0

The group $G\times G$ acts on $k[G]$ byFor $G={\mathbb{G}}_m$, this gives $k[G]={⨁}_{r\in \mathbb{Z}}{k}_{(-r,r)}$. For $G={\mathbb{G}}_a$, the action factors through $(a,b)\mapsto b-a$ and the degree filtration of $k[t]$ has trivial one-dimensional successive quotients.

Every affine algebraic group has a faithful finite-dimensional rational group representation. Choose algebra generators of $k[G]$ and place them in a finite-dimensional subspace stable under right translations. An element acting trivially on that subspace has the same coordinate values as the identity and therefore is the identity.

The Cauchy problem with initial value $u_0$ has

A category with finite products is cartesian closed when every product functor $-\times A$ has a right adjoint $(-{)}^A$, called exponentiation by $A$.

The colimit form of the Special adjoint functor theorem says that a colimit-preserving functor from a locally small, cocomplete, well-copowered category with a small generating family to a locally small category has a right adjoint.

The right Kan extension along $K:\mathcal{C}\to \mathcal{D}$ is the right adjoint to precomposition by $K$ between suitable functor categories.

This is the right-adjoint form of Kan extension; left Kan extension is its left-adjoint counterpart.

For $F⊣G$, the unit $\eta :1\to GF$ and counit $\epsilon :FG\to 1$ are the natural transformations corresponding to identity morphisms under the adjunction. They satisfy ${\epsilon }_{F}F\eta =1_F$ and $G\epsilon {\eta }_G=1_G$.

For a fixed amount of ideal gas with constant heat capacities, the first law and $\delta Q=0$ givewhere $\gamma =C_p/C_V$.

A dusty radiation-pressure-driven shell is swept up when ultraviolet radiation from an active nucleus is absorbed by dust and reradiated in the infrared. Single scattering supplies up to roughly $L/c$ of momentum flux, while trapped infrared photons can enhance it by the infrared optical depth.

With angle period $2\pi$, the action is the enclosed phase-space area divided by $2\pi$:

This is the action component of action-angle variables, rather than the entire coordinate construction.

A finite group acting freely by diffeomorphisms on a smooth manifold has a smooth quotient. Choose each coordinate neighbourhood disjoint from all its nontrivial translates and transport its chart through the quotient map. The resulting transition maps are restrictions of the original transition maps composed with group elements.

If $a_n\ge 0$, $\sum a_n<\infty$, and $p>1/2$, thenThe endpoint can fail: $a_n=1/(n(\log n{)}^2)$ gives the divergent series $\sum 1/(n\log n)$.

The Cauchy-Schwarz inequality gives

If $a_n\ge 0$ and $\sum a_n<\infty$, then $a_n\to 0$, so eventually $a_n^2\le a_n$ and $\sum a_n^2$ converges.

A countable family of probability measures is dominated by any mixture ${\sum }_n{w}_n{\nu }_n$ with every $w_n>0$ and ${\sum }_n{w}_n=1$.

For sigma-finite measures $\nu$ and $\mu$, there is a unique decomposition $\nu ={\nu }_{\mathrm{ac}}+{\nu }_{\mathrm{s}}$ in which ${\nu }_{\mathrm{ac}}\ll \mu$ and ${\nu }_{\mathrm{s}}\perp \mu$.

For sigma-finite measures, $\nu \ll \mu$ implies $\nu (A)={\int }_A(d\nu /d\mu )d\mu$ for a nonnegative measurable density unique almost everywhere.

Two measures are mutually absolutely continuous when they have exactly the same null sets.

If $\mu (\Omega )<\infty$, then $\mu \ll \nu$ exactly when for every $\epsilon >0$ there is $\delta >0$ such that $\nu (A)<\delta$ implies $\mu (A)<\epsilon$. For necessity, a contrary sequence with $\nu (A_n)<2^{-n}$ and $\mu (A_n)\ge \epsilon$ has tail unions decreasing to a $\nu$-null limsup; continuity from above for finite $\mu$ contradicts absolute continuity.

Let $h:\mathbb{R}\to \mathbb{R}$ be continuous, strictly increasing, and satisfy Lusin condition N. Thendefines a measure on the Lebesgue measurable sets, and $\nu \ll \lambda$. Moreover,for $a\le b$.