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Galactic synchrotron emission is nonthermal radio radiation from relativistic electrons spiralling in the Galactic magnetic field. Its broad latitude distribution reflects cosmic-ray transport and the magnetic halo.

If $g\in L^1(a,b)$, then $F(x)=F(a)+{\int }_a^{x}g(s)ds$ is absolutely continuous and satisfies $F^{\prime }=g$ almost everywhere. Conversely, every absolutely continuous function has this form.

If two unital operations on one set share their identity and satisfy $(a\ast b)\cdot (c\ast d)=(a\cdot c)\ast (b\cdot d)$, then the operations coincide and are commutative.

A functor $F:\mathcal{C}\to \mathcal{D}$ is full and faithful when every mapis a bijection.

The functor category has functors $\mathcal{C}\to \mathcal{D}$ as objects and natural transformations as morphisms.

If $K/k$ is a finitely generated field extension, write $K=\mathrm{Frac}(A)$ for a finitely generated integral $k$-algebra $A$. The projective closure of the affine variety $\mathrm{Spec}A$ is an irreducible projective variety with function field $K$.

For an irreducible variety $X$ over a field $k$,Consequently birational irreducible varieties have the same dimension.

If $R$ is a nonzero commutative ring and $R^m\cong R^n$, reduction modulo a maximal ideal produces isomorphic vector spaces of dimensions $m$ and $n$. Hence $m=n$.

For a nonzero commutative scalar ring, every two bases of a finite-rank free module have the same cardinality. This cardinality is the rank.

A module $F$ is free on a set $S$ exactly when every function from $S$ to any module $N$ extends uniquely to an $R$-module homomorphism $F\to N$.

A basis $S$ of an $R$-module $M$ is a subset such that every element of $M$ has a unique expression as a finite $R$-linear combination of elements of $S$.

For $A$ terms with phase ratio $q$, the amplitude factor is $(1-q^A)/(1-q)$, with limiting value $A$ when $q=1$.

If the fourth-order Peano kernel of the four-point endpoint formula is nonnegative, its sharp sup-norm error constant is its integral. Evaluating the error on $(x+1{)}^4/24$ gives

If ${\Delta }^{k+1}f=0$, then $f$ is an integer linear combination of $1,\left(\genfrac{}{}{0ex}{}{n}{1}\right),\dots ,\left(\genfrac{}{}{0ex}{}{n}{k}\right)$ whenever $f$ is integer-valued. Successive differences recover the coefficients.

A discrete antiderivative of $f$ is a function $F$ satisfying $\Delta F=f$. Pascal's identity gives

When the source function varies approximately linearly with optical depth, the emergent intensity obeys $I_{\nu }(0,\mu )\simeq S_{\nu }({\tau }_{\nu }=\mu )$.

The relation is an approximation within radiative transfer, not the entire transfer theory.

For current $J=aP-\kappa \nabla P/2$, a scalar potential exists when the one-form ${\kappa }^{-1}a$ is exact. Then $D=\kappa /2$, $a=-D\nabla V$, and $J=-PD\nabla (V+\log P)$.

For unit edge resistances, the energy of a flow iswhere every unoriented edge is counted once.

The strength of a source-to-sink flow is the net amount leaving the source, equivalently the net amount entering the sink.

The divergence of a flow at a vertex is its total outward flow. Flow conservation means that this divergence vanishes at every interior vertex.