With $P=W^{\prime }$ a nonconstant polynomial, positive area measure and Berezin integration oriented positively, a holomorphic polynomial insertion in $S=|P{|}^2-P^{\prime }{\theta }_1{\theta }_2-\overline{P^{\prime }}{\bar{\theta }}_1{\bar{\theta }}_2$ equals $\pi {\sum }_a{m}_{a}f(z_a)$, where $P(z_a)=0$ and $m_a$ is the multiplicity of a root. Rescaling $P$ by $t>0$ leaves the insertion fixed and localizes it as $t\to \infty$.

At a vacuum obeying F-flatness and D-flatness, charged scalar field expectation values can break an internal gauge symmetry while preserving supersymmetry. For each broken generator, a massless vector multiplet combines with a chiral superfield into a massive N=1 vector multiplet. The vector absorbs one real Goldstone boson; a real scalar remains, and the gaugino mixes with a higgsino. All four bosonic and four fermionic on-shell states have a common mass.

In four spacetime dimensions, power-counting renormalizability restricts an ungauged chiral action to a canonical quadratic Kähler potential, up to transformations, and a superpotential of degree at most three. Terms of larger degree belong to an effective field theory.

The gravitino is the spin-$3/2$ gauge field of local supersymmetry and the partner of the graviton. A massless four-dimensional gravitino has two physical helicities, $\pm 3/2$; a massive one has four, adding $\pm 1/2$. The super-Higgs mechanism supplies these extra states from the goldstino.

When local supersymmetry is spontaneously broken, the gravitino absorbs the goldstino and becomes massive. The two goldstino states provide its longitudinal spin-$1/2$ polarizations. With only global supersymmetry breaking, the goldstino remains a physical massless particle. An ordinary supersymmetric Higgs mechanism instead makes an internal spin-one gauge boson massive and can leave supersymmetry unbroken.

A dilatino is a spin-one-half fermionic partner in a supergravity theory containing a dilaton. Type IIA and type IIB theories each contain two Majorana-Weyl dilatini with eight physical polarizations apiece. Their chirality assignments differ between the two theories.

Massless type IIA supergravity is the nonchiral ten-dimensional maximal supergravity with metric, Kalb–Ramond field, dilaton, Ramond-Ramond one-form and three-form potentials, two opposite-chirality Majorana-Weyl gravitini, and two opposite-chirality Majorana-Weyl dilatini. Its massless spectrum has $128$ bosonic and $128$ fermionic polarizations. The two supercharges have opposite ten-dimensional chirality.

Type IIB supergravity is the chiral ten-dimensional maximal supergravity with metric, Kalb–Ramond field, dilaton, Ramond-Ramond zero-form, two-form and four-form potentials, and a self-dual five-form field strength. Two same-chirality Majorana-Weyl gravitini and two opposite-to-gravitino-chirality Majorana-Weyl dilatini complete its $128+128$ massless spectrum.

The ungauged maximal nine-dimensional supergravity zero-mode multiplet has one metric, three vectors, two two-forms, one three-form and three scalars, giving $128$ bosonic states. Two Majorana gravitini and four Majorana spin-one-half fields give $128$ fermionic states. Flat-circle reductions of massless type IIA and type IIB theories, and a flat-two-torus reduction of eleven-dimensional supergravity, give this same nonchiral spectrum.

For a condensate $\psi =\sqrt{n}{e}^{iS}$ with kinetic operator $-{\hslash }^2{\nabla }^2/(2m)$, its particle current is $n\mathbf{u}$ with the displayed velocity. This follows by subtracting the Gross–Pitaevskii equation from its complex conjugate. More generally, in units where the kinetic operator is $-D{\nabla }^2$, the current velocity is $2D\nabla S$. The complex argument gradient and current velocity coincide only when $D=1/2$.

A centered random vector is sub-Gaussian with variance proxy ${\tau }^2$ when the displayed inequality holds for every deterministic vector $a$. Equivalently every scalar inner product is a sub-Gaussian random variable with proxy ${\tau }^2\Vert a{\Vert }_2^2$. Independence of coordinates is not required. For vectors $a_j$ with Euclidean norms at most one, the union bound gives ${\max }_j|⟨a_j,g⟩|\le \tau \sqrt{2\log (2m/\delta )}$ with probability at least $1-\delta$. This assumption alone is weaker than concentration for all Lipschitz functions.

For a regular self-adjoint Sturm-Liouville problem with positive weight $w$ and self-adjoint boundary conditions, eigenfunctions with distinct eigenvalues are orthogonal in the weighted inner product $\int w\overline{u}v$. Integration by parts proves this by equating $\int \overline{u}Lv$ and $\int \overline{Lu}v$.

In a local orthonormal tetrad, anisotropic stress is the trace-free spatial part of the stress-energy tensor, conventionally denoted ${\pi }^{ij}$. Some linear cosmological perturbation theory conventions instead use ${\Pi }^{ij}=-{\pi }^{ij}$ and write $T^{ij}=P{\delta }^{ij}-{\Pi }^{ij}$. The sign must be specified before using a photon Euler equation. The photon quadrupole determines the linear photon anisotropic stress.

Momentum density is momentum per unit physical volume. In a local orthonormal tetrad of the measuring observer it is $q^{\hat{i}}=T^{\hat{0}\hat{i}}$. A perfect fluid in general relativity moving at a small velocity relative to that observer has $q^{\hat{i}}=(\rho +P)v^{\hat{i}}$ to first order. For isotropic background photons, $P=\rho /3$, so the factor is $4\rho /3$ rather than $\rho$. Integrating the kinetic stress-energy tensor gives $q^{\hat{i}}=\int f{p}^{\hat{i}}{d}^{3}p$.

A pressureless fluid with proper number density $n_0$, rest mass $m$ and four-velocity $u$ has tensor $m{n}_0{u}^{\mu }{u}^{\nu }$. In its local rest frame there is energy density but no spatial stress. Particle-number conservation and the Lorentz force make its divergence the electromagnetic force density.

For axisymmetric incompressible flow without swirl, $u_r=-{\psi }_z/r$ and $u_z={\psi }_r/r$. These identities enforce incompressibility, and $2\pi \psi$ measures axial volume flux from the symmetry axis when its additive constant is chosen there.

A stellar atmosphere is the outer region that determines the escaping spectrum and matches the interior stellar structure to its radiation boundary conditions. A photosphere is the layer near unit optical depth, not a zero-temperature boundary. The inward intensity at the outer surface vanishes when there is no external illumination.

A stellar population is an ensemble of stars described by its formation times, initial masses, composition and present stellar evolution states. Number fractions must be distinguished from mass-weighted fractions. Shared formation times can correlate the states of components of a binary star even when their initial masses have independence.

Outside a spherical star, its gravitational acceleration is $-G{M}_{\star }\mathbf{r}/r^3$. It provides the dominant central force for Kepler orbits in a star-dominated planetary system.

A statistic is a measurable function of observed data whose definition does not involve an unknown parameter. It may be scalar or vector valued. Sufficient statistics summarize all information about a parameter in the conditional-distribution sense.