List the $k$ distinct prime divisors of $r$ in increasing order. The $j$th is at least $j+1$, and $k\le {\log }_{2}r$. The Euler totient function product formula gives $\phi (r)/r\ge {\prod }_{j=1}^{k}j/(j+1)=1/(k+1)$. This elementary bound is sufficient to show inverse-polynomial success probability for coprime phase labels in quantum order finding.

An isoperimetric constraint fixes the value of an integral while optimizing another integral over functions. A constant Lagrange multiplier incorporates the constraint into the variational integrand, and the resulting Euler-Lagrange equation must be supplemented by the appropriate natural boundary conditions for a free endpoint. The name includes integral constraints more general than prescribed geometric length.

Motion on a sphere with speed proportional to $\sin \theta$ has travel-time line element proportional to $\sqrt{d{\theta }^2/{\sin }^2\theta +d{\varphi }^2}$. With $z=\mathrm{arsinh}(\cot \theta )$, this becomes the flat cylinder metric $\sqrt{d{z}^2+d{\varphi }^2}$. Shortest lifted paths are straight lines. Nonvertical lines give $\cot \theta =\sinh (A\varphi +B)$; vertical lines are meridians of constant $\varphi$. The poles lie at infinite travel-time distance.

A Euclidean ray from the origin with direction $v\ne 0$ is the geometric half-line $\{tv:t\ge 0\}$. Points in the strictly positive quadrant lie on rays with positive slope.

Placement of geometric objects without interior overlap. A packing fraction measures how much area or volume they occupy. Circle packing and sphere packing are examples.

The union of the two infinite lateral conical sheets meeting at a common vertex is a double circular cone. The displayed example has axis $z$ and semi-angle $\pi /4$. It is smooth away from the vertex, where its defining gradient is nonzero. A right circular cone truncated on one side is a related bounded solid; this surface includes both axial directions and has no base disks.

The hour angle of a direction is the westward angle from the upper celestial meridian to its hour circle. In a consistent equinox convention, $h=\mathrm{LST}-\alpha$ modulo one turn, where $\alpha$ is right ascension and LST is local sidereal time.

Infection transferred between individuals in the same generation, often modeled by a law of mass action term proportional to susceptible and infected populations. This contrasts with vertical transmission from parent to offspring.

Infection passed from a parent to its offspring. In a population model, the corresponding infected births remain in the infected compartment rather than entering the susceptible compartment.

A mean-field coupling of the Poisson equation to an ideal Boltzmann distribution of mobile ions. In Gaussian units with uniform dielectric constant $ϵ$, ${\nabla }^2\varphi =-(4\pi /ϵ)[{\rho }_f+{\sum }_i{q}_i{n}_{i,0}{e}^{-\beta q_i\varphi }]$. The potential reference and chemical potentials determine the normalization factors. It neglects discrete-ion correlations and assumes local thermal equilibrium. A counterion-only model is not the same equation as a symmetric salt reservoir.

For a chosen reference charge $q$, the separation at which its Coulomb interaction energy equals $k_{B}T$. The charge-$q$ convention is ${\ell }_B=q^2/(4\pi {ϵ}_{\mathrm{abs}}{k}_{B}T)$ in SI, or ${\ell }_B=\beta q^2/ϵ$ in Gaussian units. The commonly tabulated value uses the elementary charge; changing counterion valence requires tracking that reference convention. It sets the strength of counterion condensation.

The Debye–Hückel approximation linearizes mobile-ion Boltzmann densities about a neutral bulk electrolyte. It is valid for ionic potential energies small compared with $k_{B}T$. The inverse Debye–Hückel screening length satisfies ${\kappa }_D^2={\sum }_i{n}_i{z}_i^2{e}^2/(\epsilon k_{B}T)$. The quadratic screened charging free energy includes both electric field energy and the linearized ionic-response term $\epsilon {\kappa }_D^2{\varphi }^2/2$.

Across a charged dielectric interface the tangential electric field is continuous and the normal displacement jump is the fixed surface charge density. Without a dipole sheet the electrostatic potential is continuous. For a charge sheet with equal permittivity on both sides, its normal potential derivative has jump $-{\sigma }_{\mathrm{fixed}}/\epsilon$. Setting the field to zero on one side is an additional physical boundary assumption, as in a conductor or a negligible-exterior-admittance idealization.

Bremsstrahlung is electromagnetic radiation emitted when a charged particle accelerates during interaction with another charged particle or field. The microscopic process $e^{-}+Z\to e^{-}+Z+\gamma$ creates a photon, while its inverse absorbs one. Thus it contributes to a photon-number collision source in the photon number current, unlike ordinary Compton scattering. Thermal bremsstrahlung averages this emission over a thermal distribution of charged particles.

The electromagnetic scattering solution for a sphere describes the transition between Rayleigh scattering and geometrical optics. Grain-size-dependent efficiencies can produce structure in the radiation-pressure coefficient.

The nonrelativistic radiated power of an accelerated point charge is $P=q^2{\mu }_0{a}^2/(6\pi c)$ in SI units. Integrating its dipole radiation pattern over solid angle gives this result; relativistic motion requires the Lienard generalization.

Elastic energy is the recoverable energy stored by deforming an elastic material. In the small-deflection model of an elastic filament, the bending contribution is a positive quadratic functional of the curvatures, such as $\frac{1}{2}\int (A{y}^{\prime \prime 2}+B{z}^{\prime \prime 2})dx$. Work by applied loads must also be included when minimizing the total potential energy.

An elastic filament is a slender elastic body modeled by a centerline and bending rigidities. Its elastic energy penalizes curvature. In a small-deflection representation $(x,y(x),z(x))$, the bending energy is quadratic in $y^{\prime \prime }$ and $z^{\prime \prime }$. Compression can cause Euler buckling of an elastic filament.

For an Eisenhart-Duval lift, reduction along the null isometry fixes a nonzero conserved $p_u=t\prime$ and projects geodesics to $(t,\mathbf{x})$. One never divides by $g_{uu}=0$. When $\mathbf{E}=-\nabla U$ and $\nabla \times \mathbf{A}=\mathbf{B}$, the reduced equation is $\ddot{\mathbf{x}}=\mathbf{E}+2\mathbf{B}\times \dot{\mathbf{x}}$. Every trajectory admits a null lift by choosing $\dot{u}=U+2\mathbf{A}\cdot \dot{\mathbf{x}}-\frac{1}{2}|\dot{\mathbf{x}}{|}^2$.

A time-independent gauge transformation $\mathbf{A}\mapsto \mathbf{A}+\nabla \chi$ in an Eisenhart-Duval lift is the coordinate change $u\mapsto u+2\chi$. It preserves the metric one-form $du-2\mathbf{A}\cdot d\mathbf{x}-Udt$ and adds only $-2d\chi /dt$ to the reduced Lagrangian.