A collision or transient molecular pair can possess a dipole even when the isolated molecules do not. The resulting opacity can be important in a dense molecular hydrogen atmosphere. Pair absorption per unit length scales approximately as the product of the two partner number densities, so its mass absorption coefficient is generally density-dependent. It supplies continuum absorption between ordinary spectral lines.

Let a nonnegative martingale start at one, let $T$ be its first zero, and let $T_R$ be its first hit of $[R,\infty )$. Suppose $\mathbb{E}[(M_{n+1}-M_n{)}^2\mid {\mathcal{F}}_n]\ge {\sigma }^2{1}_{\{T>n\}}$ and $M_{T_R}\le \tau R$ on finite hits, for thresholds $R\ge 1$. Then $M_{T\wedge T_R\wedge n}^2-{\sigma }^2(T\wedge T_R\wedge n)$ is a submartingale. The bounded stopped values have mean one, so their second moment is at most $\tau R$. Consequently $\mathbb{E}(T\wedge T_R)\le (\tau R-1)/{\sigma }^2$. For $\tau \ge 1$ this implies the weaker bound ${\tau }^{2}R/{\sigma }^2$. The threshold restriction prevents an impossible overshoot requirement below the starting value.

An absolute wavenumber is a complex wavenumber at the selected zero-group velocity saddle of a dispersion relation. For the linear complex Ginzburg-Landau equation, $k_0=-U/[2(c_d-i)]$; the positive real part of the diffusion coefficient makes the Gaussian saddle accessible from the real Fourier contour.

The absolute growth rate is the exponential growth rate of a localized impulse at a fixed position in the chosen observation frame. Under the convention $e^{i(kx-\omega t)}$, it is the imaginary part of the physically selected absolute frequency. Positive values give absolute hydrodynamic instability; negative values can still coexist with convective hydrodynamic instability.

A time-independent ray Hamiltonian conserves the stationary-observer absolute frequency. A horizontally uniform background also conserves horizontal wavenumber. The intrinsic frequency $\hat{\omega }=\omega -kU(z)$ nevertheless changes when the ray moves through shear. These follow from the Hamiltonian ray-tracing equations and are essential to identifying a critical level of an internal gravity wave.

For odd $n\ge 3$, the dihedral group $D_{2n}$ of order $2n$ has rotation generator $r$ and reflection generator $s$, with $srs=r^{-1}$. The group commutator $[r,s]=r^{-2}$ generates the rotation subgroup, while the quotient by that subgroup is $C_2$. Hence the commutator subgroup is exactly $⟨r⟩$ and the abelianization is $C_2$. The only abelian quotients are $C_2$ and the trivial group.

For an Abelian p-form gauge field with local field strength $F=dA$, the exterior derivative identity $d^2=0$ gives $dF=0$. In a first-order duality formulation, this identity can be enforced with a Lagrange multiplier, whose elimination restores the local gauge potential.

A massless Abelian $p$-form in $D$-dimensional flat spacetime transforms on shell as an antisymmetric rank-$p$ tensor of the rotational little group $SO(D-2)$. Its physical state count is $\left(\genfrac{}{}{0ex}{}{D-2}{p}\right)$. A compatible real self-duality constraint halves this count. A four-dimensional three-form has no local propagating polarization.

Use metric $(+--)$, $D_{\mu }={\partial }_{\mu }-i{a}_{\mu }$ and Lagrangian $-f_{\mu \nu }{f}^{\mu \nu }/4+\overline{D_{\mu }\varphi }{D}^{\mu }\varphi /2-(1-|\varphi {|}^2{)}^2/8$. The Euler-Lagrange field equations are $D_{\mu }{D}^{\mu }\varphi +(|\varphi {|}^2-1)\varphi /2=0$ and ${\partial }_{\mu }{f}^{\mu \nu }=\mathrm{Im}(\bar{\varphi }{D}^{\nu }\varphi )$. Varying $\bar{\varphi }$ gives a kinetic contribution $-D_{\mu }{D}^{\mu }\varphi /2$ and potential contribution $(1-|\varphi {|}^2)\varphi /4$; variation of $a_{\nu }$ gives ${\partial }_{\mu }{f}^{\mu \nu }-\mathrm{Im}(\bar{\varphi }{D}^{\nu }\varphi )$. For static fields in temporal gauge $a_0=0$, the positive Bogomolny vortex equations imply the scalar equation because $(D_1-i{D}_2)(D_1+i{D}_2)=D_1^2+D_2^2+B$. They imply the spatial gauge equations because $\mathrm{Im}(\bar{\varphi }{D}_1\varphi )=-{\partial }_2|\varphi {|}^2/2$ and $\mathrm{Im}(\bar{\varphi }{D}_2\varphi )={\partial }_1|\varphi {|}^2/2$, while $B=(1-|\varphi {|}^2)/2$. The Gauss equation then vanishes identically.

An abelian group in which no nonzero element has finite order. Free abelian groups are torsion-free, but torsion-free groups need not be free, as the additive rationals show. This algebraic meaning differs from the torsion-free connection in differential geometry.

A class of abelian groups closed under isomorphisms, subgroups, quotients and extensions. For the homotopical Serre-class theorems one also uses tensor/Tor closure and homology closure for Eilenberg–MacLane spaces. A homomorphism is an isomorphism modulo the class when its kernel and cokernel lie in the class.

A unit-modulus amplification root of a linear multistep method puts its test-equation parameter on this boundary locus, except when numerator and denominator both vanish. If the locus avoids the open left half-plane, the leading coefficient never vanishes there and the roots are initially inside the disk at one point there, continuity keeps them inside throughout that connected region. Checking zero-stability, imaginary-axis roots and common polynomial factors is still necessary. A boundary plot alone is not a proof of A-stability.

For signature $(+---)$ and future normal $n$, write $\Pi =n^{\mu }{\partial }_{\mu }\varphi$ and use the positive induced metric ${\gamma }_{ij}$ for $s^2={\gamma }^{ij}{D}_i\varphi D_j\varphi$. The Klein-Gordon scalar stress-energy tensor has normal-frame density $\rho =({\Pi }^2+s^2)/2+V$, momentum $J_i=-\Pi D_i\varphi$ and spatial stress $S_{ij}=D_i\varphi D_j\varphi +{\gamma }_{ij}[({\Pi }^2-s^2)/2-V]$. The spatial index convention and normal orientation must be specified.

After removing overweight items and handling free items, compare the feasible density-sorted prefix with the best feasible singleton. The fractional knapsack problem optimum is at most the sum of their profits, so the better solution has approximation ratio $1/2$.

At a closed point of a variety over an algebraically closed field, the cotangent space is ${\mathfrak{m}}_P/{\mathfrak{m}}_P^2$. The universal property of Kähler differentials identifies its dual vector space with $k$-derivations of the local ring into the residue field, hence with the Zariski tangent space. The Kähler differential sheaf has the same fibre because a derivation kills constants and products of two elements of the maximal ideal.

A local fermionic gauge symmetry shifting the gamma-trace part of the worldsheet gravitino, schematically $\delta {\chi }_{\mu }={\gamma }_{\mu }\rho$. Together with local worldsheet supersymmetry it permits the gravitino to be set to zero locally in superconformal gauge. Global spin structures and moduli still remain.

The local gauge choice $h_{\mu \nu }=e^{2\omega }{\eta }_{\mu \nu }$ and ${\chi }_{\mu }=0$ for an RNS string. Diffeomorphisms and Weyl symmetry fix the metric, while local supersymmetry and super-Weyl symmetry fix the gravitino. Stress-tensor and supercurrent constraints survive the gauge choice and eliminate longitudinal degrees of freedom. Higher-genus surfaces retain metric and spin-structure moduli.

For a centered free real scalar field, its vacuum time-ordered product has expectation $⟨T{\varphi }_1{\varphi }_2{\varphi }_3{\varphi }_4⟩=G_{12}{G}_{34}+G_{13}{G}_{24}+G_{14}{G}_{23}$, where $G_{ij}$ is the Feynman propagator between insertions $i,j$. These are the three complete pairings in the Wick theorem. All remaining normal-ordered products have zero vacuum expectation. No fermionic interchange signs occur for a bosonic field.

Relative to a timelike unit normal $n$, the electric part is $E_{ab}=C_{acbd}{n}^c{n}^d$. It describes the tidal component of the Weyl tensor measured by that observer. Antisymmetry in each curvature pair makes it spatial in both slots; pair symmetry and vanishing traces make it symmetric and trace-free.

In four spacetime dimensions, relative to a timelike unit normal $n$, define $B_{ab}=\frac{1}{2}{ϵ}_{acde}{C}^{de}{}_{bf}{n}^c{n}^f$, with the chosen spacetime metric volume tensor. This is the dual tidal part of the Weyl tensor. Normal contractions vanish by curvature and volume-tensor antisymmetry. Its sign depends on the specified orientation and dualization convention.