A hanging mass $m$ pulls the centre of a uniform disc of mass $M$ up an incline of angle $\theta$, using a light string and light frictionless pulley. If the disc rolls without slipping and the string is parallel to the slope, its effective translational mass is $M+I/a^2=3M/2$. With downward hanging acceleration $q$, $q=g(m-M\sin \theta )/(m+3M/2)$. Static friction has signed downslope component $Mq/2$ and supplies the rolling torque. The acceleration threshold is unchanged from a sliding block, although rotational inertia reduces the magnitude.

For a homogeneous canonical inflaton oscillating near $V=m^2{\varphi }^2/2$, use reduced-Planck units and require $m\gg H$. The slowly varying ansatz $\varphi =A(t)\cos mt$ gives $H_m^2\simeq m^2{A}^2/6$. The Nonincreasing Hubble rate for a canonical inflaton becomesCycle averaging gives ${\dot{H}}_m=-3H_m^2/2$, so $H_m=2/[3(t-t_{\ast })]$. Integrating the rapidly varying term while holding its coefficient approximately constant gives the displayed correction. It is smaller than the mean by $O(H_m/m)$. Although the correction oscillates, the total Hubble parameter remains nonincreasing. Equal average kinetic and potential energies give pressureless-matter-like expansion, with amplitude $A\propto a^{-3/2}$.

For a detached double-lined binary, radial velocities supply an orbital length scale times $\sin i$, while eclipse geometry supplies inclination and relative radii. Together they determine physical stellar radii. Comparing them with angular radii from a calibrated surface-brightness/colour relation gives distance. The orbital scale is geometric, but the angular-radius calibration, extinction and third light add astrophysical assumptions.^[ref]

Geology reconstructs the composition, structure and history of the solid Earth and other rocky bodies. Its observations include rock textures, fossils, layer relationships, deformation and numerical ages. A cross-cutting intrusion, for example, must be younger than the rock it cuts, even before either has a numerical date.

Physical models connect those relationships to processes such as plate tectonics, sedimentation, metamorphism and erosion. Distinguishing a directly observed structure from the process proposed to explain it is essential: similar present-day rocks can have different formation and alteration histories.

The outer compositional shell above the mantle. Crustal thickness and density variations contribute to surface topography and gravity; crust and mechanically defined lithosphere are different boundaries.

The rocky shell below the crust and above the core. It transmits S waves on seismic time scales while undergoing very slow viscous deformation over geological time scales.

A dung-beetle species in which quantitative-genetic and microbial-transmission studies motivate experimentally separating maternal provisioning pathways.

A dung-beetle species in which pedestal-associated microbiota have been experimentally linked to developmental benefits under some conditions.

A fly species with experimentally characterized sex-ratio drive and suppressor systems.

Drosophila persimilis is a fruit-fly species closely related to Drosophila pseudoobscura. Comparisons between them have been used to study reproductive isolation, including differences in mating and the consequences of hybridization.

For normalized localized states related by reflection and real overlap $s=⟨{\varphi }_L|{\varphi }_R⟩$, the displayed superpositions are normalized even and odd states. The even state is lower for the lowest tunnelling pair; the odd state has one central node. Small overlap explains nearly identical probability densities in the two distant wells, but the relative phase remains different. A localized one-well state is a superposition of the pair rather than an exact stationary parity eigenstate.

A sequence of individual localized electron detections can accumulate a two-path interference distribution when the apparatus maintains coherence. The 1989 experiment led by Akira Tonomura used a field-emission source, an electron biprism, electron lenses and a position-sensitive electron counter. It realized two coherent paths without literal machined double slits. Low occupation of the instrument excludes interactions between different electrons as the origin of the fringes.^[ref]

The ground-state probability of both electrons occupying one site is $P_{\mathrm{double}}=\frac{1}{2}[1-U/\sqrt{U^2+16t^2}]$. It equals the derivative of the relative ground energy with respect to $U$. It tends to zero for $|t|/U\to 0$ and to $1/2$ for $|t|/U\to \infty$.

For nonzero real vectors in a Euclidean space, their angle $\theta \in [0,\pi ]$ is determined by the displayed dot product formula. The Cauchy-Schwarz inequality puts the quotient in $[-1,1]$, so the inverse cosine is defined. Perpendicular vectors have angle $\pi /2$, parallel vectors pointing the same way have angle zero, and those pointing in opposite directions have angle $\pi$. A zero vector has no uniquely defined direction and hence no angle by this definition.

Let $A$ be a real positive-definite matrix and let the columns of $U$ be an orthonormal basis of a two-dimensional linear subspace in three-dimensional Euclidean space. Points in that central plane are $x=Uq$, with $|x{|}^2=|q{|}^2$. Restricting the ellipsoid $x^{T}Ax=1$ gives $q^T(U^{T}AU)q=1$. The compressed matrix $U^{T}AU$ is a positive-definite matrix, and its extreme eigenvalues give the distance formulas in the title by the Rayleigh quotient. The two extremal axes are the images under $U$ of the corresponding eigenvectors. This restriction reduces a problem with two Lagrange multipliers to a two-dimensional quadratic form.

A spanning surface that cuts a charging wire carries free current $I$. A surface passing through the capacitor gap instead carries displacement flux $\int {\partial }_t\mathbf{D}\cdot d\mathbf{A}=\dot{Q}=I$. Their common boundary has the same magnetic circulation only when both terms are included in Ampère's law.

For a uniform ohmic material under harmonic forcing, the magnitudes obey $|J_{\mathrm{cond}}|/|J_{\mathrm{disp}}|=\sigma /(\omega ϵ)$. A large ratio supports a local conduction-dominated approximation; it does not remove the displacement term from Maxwell's exact continuity relation.

If a thin capacitor interior is modeled with axially uniform axisymmetric fields, Maxwell's equations give $r^{-1}(r{E}_z^{\prime }{)}^{\prime }+(\omega /c{)}^2{E}_z=0$. The solution regular at the axis is proportional to $J_0(\omega r/c)$. Normalizing to a central sinusoidal voltage gives $E_z=-(V_0/L)J_0(\omega r/c)\sin \omega t$ in this ideal interior model. It can change sign across the radius when propagation matters. Feed, edge and radiation boundary conditions are needed for a full driven-device solution.

The dimensionless displacement strength for the equal-frequency single-vibration model in displaced-oscillator Franck-Condon progression. It equals the mean number of excited vibrational quanta in its zero-temperature overlap distribution.