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The hydrostatic convention $b=f_0{\psi }_z$ may refer to full buoyancy or to its anomaly relative to $N^{2}z$. Subtracting the uniform reference profile changes $\psi$ by $-N^2{z}^2/(2f_0)$ and changes $q=f_0+{\nabla }_h^2\psi +(f_0^2/N^2){\psi }_{zz}$ by the uniform constant $-f_0$. Horizontal velocity and potential-vorticity gradients are unchanged, so the advection and stability dynamics are identical. Stating the convention avoids mistaking this reference constant for a dynamical gradient.

For a no-slip planar liquid film with an insoluble surfactant, lubrication theory gives $q=-h^3{p}_x/(3\mu )+h^2{\gamma }_x/(2\mu )$ and surface velocity $u_s=-h^2{p}_x/(2\mu )+h{\gamma }_x/\mu$. Mass conservation and conservation of insoluble surfactant on a moving interface give $h_t+q_x=0$ and $C_t+(C{u}_s{)}_x=0$. With $\gamma ={\gamma }_0-AC$ and small slopes, $p_x=\rho g{h}_x-(\gamma h_{xx}{)}_x$. The different coefficients in $q$ and $u_s$ are essential: Marangoni stress and hydrostatic return flow transport fluid and surfactant differently.

For an exterior thin film, circumferential and axial curvature together give $h_t+(h^3[h_{zzz}+h_z+G]{)}_z/3=0$ in appropriate dimensionless variables. The $h_z$ term destabilizes long waves through the Rayleigh–Plateau instability, whereas $h_{zzz}$ damps short waves. The linear growth rate is $(k^2-k^4)/3-iGk$, with fastest growth at $|k|=1/\sqrt{2}$. The gravity parameter is $G=\rho g{a}^2/(\gamma ϵ)$ when film thickness is scaled as $ϵah$ and axial distance by cylinder radius $a$.

For a full-rank Euclidean lattice, ${\Theta }_{\Lambda }(\tau )={\sum }_{x\in \Lambda }{e}^{\pi i|x{|}^2\tau }$ obeys ${\Theta }_{\Lambda }(\tau )=(-i\tau {)}^{-n/2}m(\Lambda {)}^{-1}{\Theta }_{{\Lambda }^{\vee }}(-1/\tau )$. Apply the Poisson summation formula for a Euclidean lattice and the complex Gaussian Fourier transform. The identity holds without assuming an integral or self-dual lattice.

For a full Euclidean lattice $\Lambda$ and a positive definite real symmetric matrix $A$, the Gaussian theta sum is ${\theta }_{\Lambda }(A)={\sum }_{\lambda \in \Lambda }{e}^{-\pi {\lambda }^{T}A\lambda }$. Its absolute convergence follows from Gaussian decay and the polynomial growth of the number of lattice points in a ball. The Poisson summation formula for a Euclidean lattice gives its anisotropic theta functional equation.

For a nonresonant charged-particle thermonuclear reaction rate, the thermal high-energy tail favors low $E$ while the Coulomb barrier penetration factor favors high $E$. Their product peaks in an intermediate energy window. Resonant reactions such as the Triple-alpha process require an additional resonance treatment.

In the language of ordered rings, the Theory of real closed fields has quantifier elimination. In its pure ring reduct the order is definable by nonzero squares, so model completeness survives, although quantifier elimination does not. Keeping these two language conventions separate is essential when constructing a model companion.

Over a finite field of odd cardinality $q$, a diagonal nondegenerate ternary quadratic form $A{x}^2+B{y}^2+C{z}^2$ has a zero with $z=1$: the sets $\{A{x}^2\}$ and $\{-C-B{y}^2\}$ both have $(q+1)/2$ elements and must intersect. This gives a nonzero nonsingular zero. Over a local integer ring with these coefficients units, the Hensel lemma lifts it to an isotropic vector in the fraction field.

A weakly terminal set is a set of objects $(W_i)$ of a category such that every object admits at least one morphism into some $W_i$. It is the dual of a weakly initial set.

For commutative $R$-algebras, their module tensor product becomes a commutative algebra by $(a\otimes b)(a^{\prime }\otimes b^{\prime })=a{a}^{\prime }\otimes b{b}^{\prime }$. Its universal property is that pairs of compatible $R$-algebra maps from $A$ and $B$ into a commutative algebra $C$ correspond to maps $A{\otimes }_{R}B\to C$. This coproduct of algebras gives, contravariantly, the fibre product of schemes on affine charts.

For a homomorphism of commutative rings $R\to A$ and two $A$-modules, the tensor product of modules over $A$ is the quotient of the tensor product of modules over $R$ imposing all relations $am\otimes n=m\otimes an$. The quotient map is $A$-linear when $A$ acts through the first factor on the source. This expresses the extra balancing imposed by change of rings.

A symmetric tensor is unchanged by every permutation of its slots. A symmetric second-rank tensor is the rank-two case; symmetry for higher rank requires invariance under every such slot exchange.

Under an orthogonal coordinate transformation $R$, a Cartesian pseudotensor has the usual tensor transformation law multiplied by $\det R$. Thus proper rotation matrices act as on ordinary tensors, while an orientation-reversing change contributes an extra minus sign. Interpreted as a rank-three pseudotensor, the Levi-Civita symbol has the same numerical components in every orthonormal frame, because its ordinary three-index transformation already contributes a factor $\det R$.

A type-$(1,1)$ tensor identifies with an endomorphism of a finite-dimensional vector space. Composing two such maps produces another type-$(1,1)$ tensor, because $(X,\eta )\mapsto \eta (B(A(X)))$ is a multilinear map. In output-first components the result is $C^a{}_b=B^a{}_c{A}^c{}_b$; the contracted index implements composition, and a basis change conjugates the corresponding matrices.

A Cartesian second-rank tensor has component matrix $T$ in each orthonormal basis, with transformation $T\prime =RT{R}^T$ under an orthogonal component change $R$. Both indices transform; this is why assigning arbitrary matrices independently in different bases does not define a tensor. The determinant, trace, and Frobenius inner product with another such tensor are unchanged by these basis transformations.

A Cartesian second-rank tensor invariant under every proper rotation preserving an unoriented axis $n$ has this form. Rotations about the axis give a planar block $aI+bJ$ and eliminate mixed axial-planar entries. A half-turn about a transverse axis reverses $J$, forcing $b=0$. If only rotations about the oriented axis are imposed, this planar antisymmetric term may survive.

For $n>1$, a tempered distribution $T$ on ${\mathbb{R}}^n$ is radial when $T\circ R=T$ for every $R\in SO(n)$, where $⟨T\circ R,\phi ⟩=⟨T,\phi \circ R^t⟩$. This agrees with the usual radial function definition for regular smooth distributions, because the special orthogonal group acts transitively on spheres.

For $T\in {\mathcal{S}}^{\prime }$ and $\psi \in \mathcal{S}$, define $(T\ast \psi )(x)=⟨T_y,\psi (x-y)⟩$. Translations are smooth in the Schwartz space, so every derivative is ${\partial }^{\alpha }(T\ast \psi )(x)=⟨T,{\partial }^{\alpha }\psi (x-\cdot )⟩$. The finite-seminorm continuity estimate for $T$ implies $|{\partial }^{\alpha }(T\ast \psi )(x)|\le C_{\alpha }(1+|x|{)}^m$ for one $m$. Thus the result is a smooth function defining a tempered distribution, but it need not be a Schwartz function: $1\ast \psi =1$ when $\int \psi =1$. If both factors are radial, testing the rotation action on the translated $\psi$ shows that the convolution is radial.

An extra relativistic neutrino that decouples before ordinary neutrinos misses two stages of plasma heating. Until ordinary decoupling, entropy conservation gives a temperature ratio $(43/(4g_d){)}^{1/3}$. Electron-positron annihilation then heats photons but neither neutrino population, multiplying the ratio by $(4/11{)}^{1/3}$. Here $g_d$ counts the coupled bath at the extra species' decoupling, with no subsequent exotic entropy injection.

For a continuously differentiable vector-valued function whose domain contains the segment from $x$ to $x+h$, apply the fundamental theorem of calculus componentwise to $t\mapsto F(x+th)$. The resulting integral matrix is valid even when there is no common intermediate point giving a vector mean-value formula. For a score function, it integrates the Hessian matrix along the segment to a candidate estimate.