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An equilibrium is Lyapunov stable if every neighbourhood contains a smaller neighbourhood whose forward trajectories remain in the original neighbourhood for all time.

Expansion reduces bolometric flux by one factor of $1+z$ from photon energy and one from arrival rate, in addition to inverse-area dilution.

The nonrelativistic limit takes characteristic speeds much smaller than $c$. Lorentz transformations then reduce to Galilean transformations at leading order.

The Lorentz group consists of real matrices preserving the Minkowski metric: ${\Lambda }^T\eta \Lambda =\eta$.

To first order around flat spacetime,and

In four dimensions, the trace-reversed perturbation isTrace reversal simplifies the linearized Einstein equations.

The linearized Riemann tensor defines an operator on two-forms byUnlike the metric perturbation itself, this operator is invariant under linearized gauge transformations.

An affine recurrence has the form $x_{n+1}=A{x}_n+b$ and is solved by a matrix geometric sum.

For a sequence $(u_n)$, the ordinary generating function isMultiplying a linear recurrence by $x^n$ and summing converts shifts of the sequence into polynomial multiples of $G$.

The simplex method moves between adjacent basic feasible solutions along improving edges of a feasible polytope until no improving pivot remains.

Every linear maximization program has a dual minimization program whose feasible objective values bound the primal values.

Linear-fractional programming optimizes a ratio $c^{T}x/d^{T}x$ over a polyhedron on which the denominator is positive.

For $Ax=b$, $x\ge 0$, a feasible vector $x$ is basic when the columns $A_i$ indexed by its positive coordinates are linearly independent. Equivalently, enlarge those columns to a basis, set every nonbasic coordinate to zero, and solve the resulting square system.

The residual variance after projection onto one factor is $\mathrm{Var}(Y)-\mathrm{Cov}(X,Y{)}^2/\mathrm{Var}(X)$, a covariance Schur complement.

A scalar cosmological perturbation transforms as a scalar under spatial rotations and can be represented using scalar metric potentials, density and pressure perturbations, and the longitudinal part of the fluid velocity.

A mode of comoving wavenumber $k$ and physical wavelength $2\pi a/k$ crosses a conformal horizon of physical size $ac\tau$ when

During matter domination in a flat FLRW metric, the cosmological continuity equation gives $\rho \propto a^{-3}$, and the Friedmann equation gives $a\propto {\tau }^2$ and $8\pi G\rho a^2=12/{\tau }^2$. The CDM density equation in a matter-radiation universe, with radiation neglected, is therefore ${\delta }^{\prime \prime }+2{\delta }^{\prime }/\tau -6\delta /{\tau }^2=0$. Its Euler-Cauchy equation powers satisfy $(p-2)(p+3)=0$, proving the displayed general solution. The growing density contrast is proportional to the scale factor, while the decaying mode is proportional to $a^{-3/2}$. This all-scale form uses the CDM-comoving synchronous gauge; other density slices require their corresponding gauge transformation.

During radiation domination in a flat expanding universe, $a(t)\propto t^{1/2}$ and $H=1/(2t)$.

If a smooth component dominates the expansion but does not cluster, pressureless matter still satisfiesAs ${\bar{\rho }}_m$ becomes negligible, Hubble friction leaves a constant mode and a decaying mode, so structure growth freezes.