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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 flow, also called its flow value, is net outflow from $s$, equivalently net inflow to $t$:The notation $|f|$ denotes this signed value, rather than an absolute value.^[ref]

The equality follows by summing the divergence of a flow over all vertices: the internal vertices contribute zero by flow conservation, while every edge cancels between its two endpoints. Hence $\mathrm{div}f(s)=-\mathrm{div}f(t)$. For the running flow, $|f|=3+2=2+3=5$.

Net outflow matters even when the source has incoming edges. As a separate example, give $s\to a$, $a\to s$ and $s\to t$ flow network edge capacities $2,2,3$ and flow amounts $2,2,3$. At $a$, flow conservation reads $2=2$. The source sends $5$ units along outgoing edges but receives $2$ back, so the flow value is $5-2=3$, exactly the amount arriving at $t$. The two-edge network circulation through $a$ raises total edge usage without increasing net delivery.

The divergence of a flow at a vertex $v$ is net outward flow. For original nonnegative edge amounts and their signed net record, respectively,Positive divergence of a flow means net supply, negative divergence of a flow means net demand, and zero is flow conservation. This is the same sign convention as flow balance.

In the running example, the divergences of a flow at $(s,a,b,t)$ are $(5,0,0,-5)$. Adding the four divergences of a flow gives zero: every edge contributes positively at one endpoint and negatively at the other. In a finite flow network, this cancellation explains why net supply at $s$ equals net demand at $t$.

The fractional part of a real number isIt is the amount left after removing the lower enclosing integer, even for negative inputs. Thus $\{-2.7\}=-2.7-(-3)=0.3$. It is not the signed string of digits obtained by truncation toward zero.

Translation by an integer leaves the fractional part unchanged: $\{x+m\}=\{x\}$. It is therefore a periodic function with period one, increasing from zero toward one on each unit interval, then returning to zero at the next integer. In Euclidean division $N=kq+r$ with $0\le r<k$, $\{N/k\}=r/k$; for $102/20$, this is $2/20=0.1$.

The notation is useful for following repeated motion modulo an integer. In normality and equidistribution under integer multiplication, the values $\{K^{n}x\}$ shift the digits of a positional notation; the fractional part discards the leading integer portion without losing the remaining digits. Its braces must be distinguished by context from the braces denoting a set.

The continuity equation givesDuring expansion with $\rho +3P<0$, the product $\rho a^2$ grows and $|{\Omega }^{-1}-1|\propto (\rho a^2{)}^{-1}$ decreases.

A nonzero commutative ring $R$ has the property that $M{\otimes }_{R}N=0$ forces $M=0$ or $N=0$ exactly when $R$ is local with maximal ideal $\mathfrak{m}$ and $M=\mathfrak{m}M$ forces $M=0$ for every $R$-module. Reduction modulo $\mathfrak{m}$ turns a tensor product into a tensor product of vector spaces.

If an $R$-algebra $A$ is flat as an $R$-module, then for ideals $I,J\subseteq R$,Tensor the exact sequence $0\to I\cap J\to I\oplus J\to I+J\to 0$ with $A$.

Over a principal ideal domain, a module is flat exactly when it is torsion-free. Hence the tensor product of two torsion-free modules over a principal ideal domain is torsion-free: both tensor functors are exact, so their composite is exact.

An $R$-module $M$ is flat if and only if $M_{\mathfrak{p}}$ is flat over $R_{\mathfrak{p}}$ for every prime ideal $\mathfrak{p}$, equivalently for every maximal ideal.

A flat module $M$ is faithfully flat when $N{\otimes }_{R}M=0$ implies $N=0$. Every field extension is faithfully flat over its base field.

For $N(\mu ,v)$, the one-observation score is $((x-\mu )/v,-1/(2v)+(x-\mu {)}^2/(2v^2))$ and the information matrix is $\mathrm{diag}(v^{-1},(2v^2{)}^{-1})$.