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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.

A nonstationary spacetime lacks the stationary time-translation symmetry in the domain being considered. Without a preferred suitable timelike Killing vector field, the classical geometry alone does not determine a unique positive-frequency solution subspace for quantization.

For a smooth real phase with an isolated nondegenerate interior critical point $q$, smooth amplitude $a$, and no competing boundary contribution, the leading contribution is $2\pi {\lambda }^{-1}a(q)|\det H{|}^{-1/2}{e}^{i\lambda \varphi (q)+i\pi \mathrm{sgn}(H)/4}$, where $H$ is the Hessian matrix. Orthogonal diagonalization gives two one-dimensional Fresnel integrals. A degenerate critical point or a stationary boundary needs its own local analysis.

In an integral $\int a(q)e^{i\lambda \beta (q)}dq$, the real-valued function $\beta$ is the phase. Its stationary points satisfy $\nabla \beta =0$; its Hessian matrix determines the local quadratic oscillation and the signature factor in stationary phase method. Degenerate stationary points and boundaries require separate analysis.

If the first two phase derivatives vanish at an endpoint but the third does not, an oscillatory contribution is concentrated on a width proportional to $N^{-1/3}$. For a smooth nonzero endpoint oscillatory integral amplitude, rescaling reduces the leading contribution to a cubic oscillatory integral. Its constant follows from the Gamma function Fourier integral. This is a degenerate form of the stationary phase method, and a quadratic formula is insufficient.

For $u,v\ge 0$ with $u+v>0$, multiplication by the sum of the square roots givesThis identity exposes small differences hidden by cancellation. For example, $\sqrt{n^2+n}-n=1/(\sqrt{1+1/n}+1)\to 1/2$. It also converts normalized square-root increments into terms comparable with the harmonic series.

For a spherically symmetric gradient field regular at the origin and satisfying Poisson equation, the divergence theorem on a ball determines its radial magnitude from the enclosed source integral. It depends only on the source at smaller radii. Regularity excludes the source-free but origin-singular addition $Cx/|x{|}^3$; without that condition, solving the local equation on a punctured domain leaves this extra term undetermined.

Here $\mu =\cos \theta$ and $a_{\ell ,m}=\sqrt{({\ell }^2-m^2)/[(2\ell +1)(2\ell -1)]}$. This follows by combining the associated Legendre function recurrence $(2\ell +1)\mu P_{\ell }^m=(\ell -m+1)P_{\ell +1}^m+(\ell +m)P_{\ell -1}^m$ with the normalized spherical harmonic constants. It gives the adjacent-multipole couplings in a photon Boltzmann hierarchy.

A spin-weight-$s$ field on an oriented sphere acquires a phase under rotation of its local tangent-frame basis. Spin-weighted spherical harmonics form the angular basis for such fields. The polarization combinations $Q\pm iU$ carry opposite spin weights of magnitude two; ordinary scalar functions have spin weight zero.