For $a\propto t^{1/2}$ and $\rho =3/(32\pi G{t}^2)$, the specified equation $\ddot{\Delta }+2H\dot{\Delta }=(32\pi G\rho /3)\Delta$ reduces to $\ddot{\Delta }+t^{-1}\dot{\Delta }=t^{-2}\Delta$. Its two modes are $t$ and $t^{-1}$. This is a property of that equation; pressure-gradient and gauge choices must be specified before extending it to general radiation perturbations.

A linear combination of atomic orbitals chosen to transform under an irreducible representation of the molecular point group. Relative orbital signs must be transformed as well as the positions of their centres. Such combinations organize a Hamiltonian operator into symmetry blocks; combinations of the same symmetry generally still mix and need not themselves be molecular orbitals.

Gauss-Jordan elimination uses elementary row operations to reduce a matrix to reduced row echelon form. After choosing and normalizing each pivot, eliminate entries both below and above it. In a consistent linear system, pivot variables are then expressed directly in terms of free variables.

For example, the augmented matrix with rows $(1,1\mid 3)$ and $(1,-1\mid 1)$ reduces to $(1,0\mid 2)$ and $(0,1\mid 1)$, giving $(x,y)=(2,1)$. Applied to $[A\mid I]$ for an invertible square matrix, it produces $[I\mid A^{-1}]$. For numerical solution of one system, Gaussian elimination with back substitution usually needs less work, and pivoting is still necessary for numerical reliability.

An organized light-harvesting and reaction-center assembly supporting photosynthetic energy conversion.

An antenna transfers excitation to a reaction centre, where charge separation initiates electron transfer. This distinguishes capture of a photon from conservation of its energy in a redox reaction. In oxygenic photosynthesis, two photosystems act in a connected electron-transfer chain. ^[ref]

Transfer of energy from an optical electromagnetic wave into matter. Electronic absorption depends on transition matrix elements, occupied and empty states, and the joint density of states. Bound excitons can create resonances below an independent-particle continuum edge.

The principal part consists of all negative-power terms of a Laurent series about its center. On a punctured neighborhood of an isolated singularity it is zero for a removable singularity, a nonzero finite sum for a pole, and an infinite series for an essential singularity. A pole's order is the largest $N$ with $a_{-N}\ne 0$. At infinity use the local variable $w=1/z$: negative powers of $w$ correspond to positive powers of $z$. A Laurent series on an annulus with strictly positive inner radius may have a principal part without the function being defined close enough to the center to classify an isolated singularity there.

Let a finite directed acyclic graph have weighted edges, sources $a_1,\dots ,a_r$ and sinks $b_1,\dots ,b_r$. Let $M_{ij}$ be the sum of products of edge weights over directed paths from $a_i$ to $b_j$. The Lindström-Gessel-Viennot lemma says that $\det M$ is the signed weighted sum of vertex-disjoint path families, with sign given by the permutation pairing sources to sinks. Expand the determinant: families with an intersection cancel by exchanging the tails of two paths at the first shared vertex, a sign-reversing involution. In planar arrangements where only the identity pairing can be disjoint, the determinant directly counts disjoint lattice paths.

For $N$ indistinguishable adsorbates on $M$ equivalent sites, at most one per site and no lateral interactions, $Z_N=\left(\genfrac{}{}{0ex}{}{M}{N}\right)e^{-N{\epsilon }_s/(k_{B}T)}$. The binomial coefficient already accounts for indistinguishability; an additional division by $N!$ would count it twice.

In laser ablation, concentrated optical energy removes or disrupts material locally. Cutting a loaded cellular structure and observing recoil can probe its mechanics, but the recoil also depends on dissipation and surrounding constraints.

A laser based on an excited molecular species. ArF systems commonly provide 193 nm radiation for projection lithography.