Tracing out one mode leaves geometric occupation probabilities with $q=|\lambda {|}^2$. In an oscillator basis of frequency $\omega$, they are Boltzmann weights with $k_{B}T=\hslash \omega /\ln (1/q)$. The joint state can be pure while each reduced state is mixed.

For two exactly coplanar circular orbits around the same star, transit impact parameters scale as semi-major axis, hence as $P^{2/3}$. Any overlap by the outer planet requires $b_2<1+k_2$. In the point-planet limit the period ratio is bounded by $b_1^{-3/2}$. Tangency has zero duration, so this is a geometric supremum; practical detection also requires sufficient signal-to-noise ratio and survey coverage.

Assume Kepler's third law, a negligible planet mass, and $M_{\star }/M_{\odot }=R_{\star }/R_{\odot }$. Define $A_{\odot }=[(GM{)}_{\odot }{P}^2/(4{\pi }^2){]}^{1/3}/R_{\odot }$. The total duration of a circular planetary transit determines $a/R_{\star }$, giving the displayed stellar radius. Unknown orbital eccentricity, transit dilution or a different stellar mass-radius calibration introduces additional uncertainty.

For a circular orbit, define $k=R_p/R_{\star }$, $A=a/R_{\star }$ and $s=\sin (\pi T_{14}/P)$. Inverting contact geometry gives $b^2=[(1+k{)}^2-A^2{s}^2]/(1-s^2)$. The inferred value is conditional on the stellar scale, the total-duration convention and the orbital model; it must satisfy $0\le b<1+k$. A non-grazing complete crossing additionally has $b<1-k$.

The total duration $T_{14}$ runs from first to fourth contact; the fully superimposed duration $T_{23}$ runs from second to third contact. Circular-orbit formulas contain $1+k$ for the former and $1-k$ for the latter. Confusing the two changes inferred impact parameter, even if the period and transit depth are unchanged.

In occupied bulk regions where density-gradient energy is negligible, the stationary relation $g\rho =\mu -V$ gives speed of sound $c=\sqrt{g\rho /m}$. Empty regions do not have an imaginary physical speed of sound; the bulk condensate formula is simply inapplicable there.

A thermal-stability argument that holds dust properties fixed can fail near a sublimation front. A rise in temperature can destroy grains, changing the absorption and emission efficiencies, the dust mass and the optical depth through which radiation reaches neighboring material. The heating and cooling laws then change along with the temperature.

Write the net heating schematically as $Q(T,a,{\rho }_d)$, where $a$ is grain size and ${\rho }_d$ is dust density. The response along an evolving state includes $Q_{a}da/dT$ and $Q_{{\rho }_d}d{\rho }_d/dT$, not only the fixed-grain derivative $Q_T$. These additional terms can reinforce or oppose the perturbation. The sign depends on geometry, radiation spectrum and dust–gas coupling, so the fixed-property result in thermal stability of externally heated blackbody dust does not by itself establish the stability of a sublimation front.

A therapsid belonging to the group containing mammaliaforms and close extinct relatives. Enlargement of the dentary, differentiation of teeth and changes in the palate record a mosaic of feeding transformations.

For diffusion starting at the centre of $(-L,L)$ with absorbing endpoints, only even-parity spatial eigenfunctions contribute. Integrating their cosine expansion gives the displayed series. The leading late-time term has prefactor $4/\pi$ and decay rate ${\pi }^{2}D/(4L^2)$; its first relative correction is $-\frac{1}{3}{e}^{-2{\pi }^{2}Dt/L^2}$.

For an initially intact link with no rebinding, survival probability obeys $dp/dt=-k(F)p$. A force-tilted activation barrier with force $F=ft$, constant $f>0$ and $p(0)=1$ integrates to the displayed law. The rupture-force probability density is $k(F)p(F)/f$, not $p(F)$ itself. The no-rebinding approximation requires separated partners to leave the capture region before appreciable reassociation; it is not guaranteed by the rate equation alone.

In a minimal SU(5) scalar multiplet, the weak doublet and a colored triplet share a unified representation. The doublet must remain near the electroweak scale while the triplet is much heavier, notably to suppress proton decay. Arranging this splitting without an unprotected cancellation is a model-building problem distinct from merely embedding the Standard Model gauge group.

Pair creation and annihilation in two independent oscillator modes obey $[K_{-},K_{+}]=2K_3$ and $[K_3,K_{\pm }]=\pm K_{\pm }$. The Hermitian components $K_1=(K_{+}+K_{-})/2$ and $K_2=(K_{+}-K_{-})/(2i)$ satisfy $[K_1,K_2]=-i{K}_3$. Multiplying all three Hermitian components by $i$ gives the real Lie bracket signs of SU(1,1). Both occupations change equally, so their difference is conserved. This realizes the algebra used in two-mode squeezed states.

The three-dimensional static screening function is finite at $q=2k_F$, but its derivative diverges logarithmically. Expanding the explicit static Lindhard dielectric function gives the title's nonanalytic term. The singularity comes from particle–hole excitations connecting opposite parts of the Fermi surface. It produces a Friedel density tail from a derivative singularity; analogous electronic response can cause a Kohn anomaly in a coupled lattice vibration.

A spin valve with two ferromagnetic layers of different magnetic coercivities, allowing separate reversal without requiring one layer to be rigidly exchange-pinned. A continuous weakly coupling spacer permits a controllable antiparallel state.

Fluctuations that limit the detectable field of a magnetoresistive read sensor. Thermal magnetization noise, domain jumps, electronic thermal noise and resistance fluctuations compete with the field-dependent signal; magnetic bias and adequate anisotropy stability are important.

An energy gap is a positive separation between a ground-state sector and its lowest excitations, with the sector, momentum and thermodynamic limit specified. A local positive curvature of an auxiliary field is not alone a proof of a dynamical gap: the kinetic terms and conjugate variables determine the excitation poles.

For subcritical incidence, a reflected SH-wave loses energy into the faster substrate. Postcritical incidence has an evanescent substrate tail and no outgoing normal energy flux in the ideal lossless model. Repeated reflections can then interfere to form Love waves. Individual reflected rays are contributions to the field; normal-mode frequencies require the reflection-phase condition and are not supplied by a geometrical travel-time curve alone.

The orientation factor converting the magnitude of uniaxial stress to resolved shear stress. It lies between zero and $1/2$ because the slip-plane normal and slip direction are perpendicular.

In an ideal single-slip tensile test neglecting elastic stretch, the specimen end-to-end vector expressed in crystal coordinates changes as $\mathbf{r}={\mathbf{r}}_0+\gamma (\mathbf{n}\cdot {\mathbf{r}}_0)\mathbf{s}$, with unit slip direction $\mathbf{s}$ and slip-plane normal $\mathbf{n}$. It moves towards the slip direction along a great circle. The perpendicular component is unchanged, so $L\sin \lambda =L_0\sin {\lambda }_0$. Secondary slip changes this kinematics; equal Schmid factors determine onset only if the systems have equal current critical stresses.

The RSK insertion shape is the integer partition giving the row lengths of the insertion tableau $P$ in the Robinson-Schensted-Knuth correspondence. The recording tableau has the same shape. For ordinary row insertion, a new entry replaces the first strictly larger entry in a row, bumping it to the next row; if none exists it is appended.

Inserting the word $312$ gives rows $(1,2)$ and $(3)$, so the shape is $(2,1)$. Under this convention, the first row length is the length of a longest weakly increasing subsequence, while the first column length is the length of a longest strictly decreasing subsequence. The shape summarizes ordered structure even when the original word is much longer.