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The Lindeberg replacement method, inspired by the Lindeberg condition, compares functions of two independent random vectors by replacing their coordinates one at a time. Matching moments cancel the corresponding terms in a Taylor expansion.

For prescribed values $F(x_i)$ at ordered points $x_0<\cdots <x_n$, the connect-the-dots interval map is the unique piecewise linear function obtained by linear interpolation between consecutive data points.

For closed intervals $J$ and $K$, write $J⟶K$ under $F$ when $F(J)\supseteq K$. The intermediate value theorem implies that there is a closed subinterval $L\subseteq J$ with $F(L)=K$.

A point $x$ is periodic for an interval map $F$ when $F^n(x)=x$ for some positive integer $n$. Its least such $n$ is its period, and the finite setis its periodic orbit.

For an interval map, this is the usual periodic point condition. Its least positive return time is the period.

For a smooth closed embedding $i:X\hookrightarrow Y$, the normal bundle fits into $0\to TX\to i^{\ast }TY\to N_{X/Y}\to 0$.

For a pure $k$-dimensional projective scheme $X\subseteq {\mathbb{P}}^m$, its degree is $k!$ times the leading coefficient of its Hilbert polynomial, equivalently ${\int }_X{H}^k$.

This extends degree of an algebraic variety to retain scheme multiplicities.

Chern classes are characteristic classes of vector bundles satisfying functoriality and the Whitney product formula $c(E)=c(E^{\prime })c(E^{\prime \prime })$ for a short exact sequence.

A $k$-cycle on a scheme $X$ is a finite integer linear combination of integral closed subschemes of dimension $k$. Their free abelian group is denoted $Z_k(X)$.

During inward migration, dust can be captured into an exterior mean-motion resonance with a planet. A phase-shifted conjunction then lets planetary torque replace angular momentum lost to Poynting–Robertson drag.

Poynting–Robertson drag is the tangential component of stellar radiation force caused by aberration in an orbiting dust grain's frame. It removes orbital energy and angular momentum and drives the grain inward.

The radiation-pressure coefficient is the ratio of outward stellar radiation pressure to inward stellar gravity on a dust grain. The grain therefore feels effective gravitational parameter $(1-\beta )G{M}_{\ast }$.

Successive symmetry-breaking bifurcations can pass stability from one boundary branch to an interior branch and then to a second boundary branch.

The force on a pipe junction follows by balancing inlet and outlet momentum fluxes and pressure forces.

An extension $R\subseteq A$ is integral when every element of $A$ is integral over $R$.

A Bézout domain is an integral domain in which every finitely generated ideal is principal. Equivalently, every pair $a,b$ has a greatest common divisor $d$ satisfying a Bézout identity $d=ra+sb$. A Noetherian Bézout domain is a principal ideal domain because all its ideals are finitely generated.

A principal ideal domain is an integral domain whose every ideal has one generator.

A greatest-common-divisor domain is an integral domain in which every two nonzero elements have a greatest common divisor, defined up to multiplication by a unit.

A nonzero nonunit $p$ in an integral domain is irreducible when every factorization $p=ab$ has at least one unit factor.

A nonzero nonunit $p$ in an integral domain is prime when $p\mid ab$ implies $p\mid a$ or $p\mid b$.

The field discriminant is the discriminant of any integral basis and controls the index of every suborder.