For an integer $n\ge 2$, its smallest prime factor $\mathrm{spf}(n)$ is the least prime divisor of $n$. If $n$ is prime, it equals $n$; if $n$ is composite, it is at most $\sqrt{n}$, because otherwise every product of two prime factors would exceed $n$.

For example, $\mathrm{spf}(91)=7$. A sieve can tabulate smallest prime factors for all integers up to a bound. Repeatedly replacing $n$ by $n/\mathrm{spf}(n)$ then produces its prime factorization. The value at one is a sentinel convention in implementations, since one has no prime factor.

A spatially varying pressure exerts net force per unit volume $-\nabla p$ on a small fluid element. Its contribution to acceleration is $-\nabla p/\rho$. Pressure increases in the direction opposite to this force.

A high-pressure apparatus in which several anvils compress a relatively large specimen. Compared with a diamond anvil cell, it typically offers greater sample volume for synthesis and bulk measurements, with less optical access and a lower ultimate pressure range.

A general transcription factor complex containing TATA-binding protein and associated subunits, involved in recognition and assembly at many RNA polymerase II core promoters.

A general transcription factor contributing to assembly and positioning of RNA polymerase II in the preinitiation complex.

A general transcription factor complex containing activities involved in promoter opening and regulation of RNA polymerase II; some components also function in DNA repair.

A new solid phase formed within an existing solid matrix. Its morphology reflects anisotropic interfacial energy, coherency strain and growth kinetics.

Resistance to dislocation motion caused by the energetic cost of disrupting chemical order, for example creating an antiphase boundary in a gamma prime phase precipitate.

A dislocation unable to shear an obstacle can bow around it and leave a loop. Balancing the Peach-Koehler force against the tension of a curved dislocation gives the critical scale $\Delta \tau \simeq 2T_{\ell }/(b\ell )$, where $\ell$ is the free obstacle span on the slip plane. Centre pitch and free span are not interchangeable when particle size is appreciable.

For one spherical particle per representative volume $L^3$, a volume fraction $f$ implies $d=L(6f/\pi {)}^{1/3}$. A mean nearest-neighbor distance in a random population does not determine this number density exactly, and an atomic thermodynamic phase fraction requires a molar-volume conversion before it is used as $f$.

An initial rapid accumulation of product followed by slower turnover. It can reveal a fast first conversion followed by a slower recycling step; converting burst amplitude to active-enzyme concentration requires the appropriate mechanistic assumptions.

Time-independent crossed electric and magnetic fields can carry a nonzero energy flux $\mathbf{S}=\mathbf{E}\times \mathbf{H}$. A steady coaxial circuit is an example: radial electric and azimuthal magnetic fields carry energy axially from source to load. Static energy density does not imply zero flux.

The Poynting flux through an oriented surface is the normal component of the electromagnetic energy-flow vector. Integrating it over area gives outward electromagnetic power. A scale estimate of the form energy density times propagation speed must still specify the field and geometric regime.

An Achilles number is a powerful number greater than one that is not a perfect power. In its prime factorizationevery prime exponent satisfies $e_i\ge 2$, and $\gcd (e_1,\dots ,e_k)=1$. The first condition is precisely the definition of a powerful number. For the second, $n=m^r$ with integers $m,r>1$ exactly when $r$ divides every prime exponent; hence a greatest common divisor greater than one is precisely the obstruction.

For example, $72=2^3{3}^2$ is an Achilles number because $\gcd (3,2)=1$. By contrast, $144=2^4{3}^2=12^2$ is powerful but is a perfect square, so it is not an Achilles number.

Every positive integer has a unique factorization $n=as$ with $a$ powerful, $s$ squarefree, and $\gcd (a,s)=1$: assign each prime power of exponent at least two to $a$, and each prime of exponent one to $s$. In general $s$ differs from the squarefree part, which instead retains every prime of odd exponent.

The number of symmetry-equivalent directed reciprocal-lattice normals belonging to a powder-reflection family. For a cubic crystal, $\{111\}$ has multiplicity eight, corresponding to four unoriented plane orientations; hexagonal $002$ has multiplicity two. Multiplicity contributes to integrated powder intensity alongside the structure factor, angular factors and sample-dependent corrections.

Named after John Lennard-Jones, this is a pair potential energy combining steep short-range repulsion with nonretarded London dispersion force attraction. Its minimum is $U=-ϵ$ at $r=2^{1/6}\sigma$, found by differentiation. The power twelve is a convenient model for repulsion, rather than an exact consequence of the Pauli exclusion principle. The power six in the attractive energy follows from second-order coupling of atomic electric dipoles, before retardation matters.

A barrier to gene flow acting after fertilization, through reduced survival or fertility of hybrids. Test both sexes and later generations: survival of first-generation offspring does not establish their fertility.

Reduced survival or fertility appearing in later-generation or backcross descendants of hybrids, even when first-generation offspring are fertile. genetic recombination can expose harmful combinations previously masked in the first generation.