For an open-shell species, distinguish a singly occupied spatial orbital from a vacant spin orbital. Adding an electron to an oxygen molecule can fill the opposite-spin vacancy in a pi-star singly occupied molecular orbital even though the next completely empty spatial orbital is sigma-star.

A molecular orbital whose occupation changes bonding between the selected centres relatively little. In a real complex, weak bonding or antibonding contributions may replace exact nonbonding character.

A molecular orbital near the boundary between occupied and unoccupied levels. The highest occupied molecular orbital and LUMO are central examples. Their energies, occupancies, spatial form and symmetry help predict which new chemical bonds can form when fragments approach.

The simplest whole-number chemical atom ratio of a substance.

A model assigning the same evolutionary rate to the relevant branches. Expected branch length is rate multiplied by duration; scaling times and inverse-scaling the rate preserves the sequence likelihood.

A model allowing evolutionary rates to vary among branches. The distribution and dependence of rates affect inferred times and must be distinguished from absolute-time calibration.

Use of dated fossil evidence to constrain evolutionary times. Geological age, phylogenetic placement, stem versus crown assignment and uncertainty all affect the constraint.

A two-dimensional electron gas supplied by remote donors in aluminium gallium arsenide and confined on the gallium arsenide side of a heterojunction. An undoped spacer separates ionized donors from mobile carriers, reducing impurity scattering. A conduction-band band offset and self-consistent band bending create the confinement well.

For invertible residue classes $a_1,\dots ,a_n$, all their modular inverses can be recovered with one inversion and a linear number of multiplications. Store prefix products $P_i=a_1\cdots a_i$, invert $P_n$ once, and sweep backward: if the current inverse is $P_i^{-1}$, then $a_i^{-1}=P_i^{-1}{P}_{i-1}$ and $P_{i-1}^{-1}=P_i^{-1}{a}_i$. The same construction works in any commutative group.

The computation of an integer $x$ satisfying $ax\equiv 1(\mathrm{mod}m)$. An inverse exists exactly when $\gcd (a,m)=1$. The extended Euclidean algorithm produces such an $x$ from a Bezout identity $ax+my=1$.

If $ax\equiv 1(\mathrm{mod}m)$, the update $x^{\prime }=x(2-ax)$ satisfies $a{x}^{\prime }\equiv 1(\mathrm{mod}{m}^2)$ because $1-a{x}^{\prime }=(1-ax{)}^2$. For odd $a$, starting with $x=1$ modulo $2$ and repeatedly doubling the number of correct bits computes its inverse modulo a power of two.

Computing the remainder $ab\mathrm{mod}m$ while working with bounded integer representatives. Repeated doubling and addition can avoid forming an overflowing product: process the binary digits of $b$, double the current contribution modulo $m$, and add it to the result whenever the current binary digit is 1. The additions themselves must also use an overflow-safe implementation.

A modular square root of a residue $a$ modulo $m$ is a residue $x$ with $x^2\equiv a(\mathrm{mod}m)$. Such a root need not exist or be unique. Modulo seven, $2$ has the two roots $3,4$, while $3$ has none.

For an odd prime $p$, each nonzero quadratic residue has exactly two roots, $x$ and $-x$, because $X^2-a$ has at most two roots in a field. A composite modulus can have more: $1$ has four roots modulo eight. For coprime factors of $m$, the Chinese remainder theorem combines local roots into all roots modulo $m$.

For an integer $a$ and a positive integer modulus $m$, the modulo operation returns the unique least nonnegative residue $r$ satisfying $a=qm+r$ for an integer $q$ and $0\le r<m$. Equivalently, $r=a-m\lfloor a/m\rfloor$. For example, $-3\mathrm{mod}5=2$. Some programming languages instead use a remainder with the sign of the dividend, so their negative-input convention must be checked before identifying an operator with this mathematical definition.

Modular division by $b$ modulo $m$ means multiplication by a modular inverse $b^{-1}$. This is defined uniquely for every numerator exactly when $\gcd (b,m)=1$. For example, $3^{-1}\equiv 5(\mathrm{mod}7)$, so $2/3\equiv 2\cdot 5\equiv 3(\mathrm{mod}7)$.

If $b$ is not invertible, solve the linear congruence $bx\equiv a(\mathrm{mod}m)$ directly. It has solutions exactly when $d=\gcd (b,m)$ divides $a$, and then has $d$ solutions modulo $m$. Thus $2x\equiv 2(\mathrm{mod}6)$ gives $x\equiv 1,4(\mathrm{mod}6)$; division by two cannot select a unique residue. Cancellation instead reduces the modulus to $m/d$.

Dynamic remodeling of a mitochondrial biological population through division and merger. Fusion permits content mixing, while fission supports distribution and can segregate portions for mitophagy.

A scalar sum of missing transverse momentum magnitude and selected high-energy transverse momenta, used as a hard-event-scale diagnostic in some supersymmetry searches. Which jets and leptons enter must be defined. It is not an invariant mass or the solid-state effective mass, and no universal equality with a produced particle mass holds.