Every proper ideal in a Noetherian ring is a finite intersection of primary ideals. This existence theorem does not give uniqueness of the individual components: embedded primary components may vary.

The inner Shakura--Sunyaev thin disk is supported mainly by radiation pressure and has nearly constant electron-scattering opacity. Equating radiative diffusion with local dissipation and imposing vertical hydrostatic equilibrium gives the characteristic half-thicknessThe radiation-supported height approaches a plateau away from the inner boundary. The formal zero at that boundary is outside the reliable radiation-dominated approximation. At an Eddington accretion rate defined using efficiency $0.1$, this height can make the inner flow only marginally thin.

For an unstable Taylor–Goldstein equation mode in a finite channel, take $a=1$ in the power-transformed Taylor–Goldstein energy identity. Its imaginary part givesThus the real phase velocity lies strictly between the extremes of a nonconstant smooth shear profile. Equality would force the regular eigenfunction to vanish on an interval, hence everywhere by uniqueness for its ordinary differential equation.

A smooth inviscid stratified parallel flow with gradient Richardson number at least $1/4$ everywhere has no exponentially growing two-dimensional normal modes. Put $a=1/2$ in the power-transformed Taylor–Goldstein energy identity and take its imaginary part:The integral is positive for a nonzero mode when the numerator is nonnegative, forcing $c_i=0$. This is a modal stability theorem, not a prohibition on transient growth. Maslowe's review discusses the theorem and the role of critical layers.

The polynomial ring $\mathbb{Z}[X]$ is a unique factorization domain by Gauss lemma for polynomials, but its ideal $(2,X)$ is not principal. Any generator would divide both 2 and $X$, hence be a unit: a divisor of 2 has degree zero, and a constant dividing $X$ is $\pm 1$. Yet evaluation at zero followed by reduction modulo 2 annihilates the ideal and not 1. It embeds in the principal ideal domain $\mathbb{Q}[X]$, illustrating that the property need not pass to a subring.

If $\Psi (R,0)={\Psi }_1+{\Psi }_0(R/R_0{)}^{\delta }$ and the surface inclination to the vertical is a radius-independent $\alpha$, then $B_z=\delta {\Psi }_0{R}^{\delta -2}/R_0^{\delta }$ and $B_R=B_z\tan \alpha$. The solenoidal vector field condition givesThe constant offset in the poloidal magnetic flux function has no effect on the magnetic field.

If $f:M\to N$ is a map between closed connected oriented manifolds of the same dimension and its degree is nonzero in a coefficient field $F$, then $f^{\ast }:H^{\ast }(N;F)\to H^{\ast }(M;F)$ is injective. For a nonzero class $a$, Poincare duality supplies $b$ with $⟨a⌣b,[N]⟩\ne 0$, and naturality gives

For a closed oriented $d$-dimensional manifold over a field $F$, the cup product pairingis a perfect pairing between complementary degrees. The cap product with the fundamental class is an isomorphism by Poincare duality, and the universal coefficient theorem for cohomology over $F$ identifies the complementary cohomology with the full dual space of the resulting homology. Evaluation is therefore a perfect pairing. Taking the top-degree part of the same formula defines a nondegenerate bilinear form on the full graded cohomology. A boundary requires relative groups and Poincare-Lefschetz duality instead.

For fully ionized pure hydrogen of density $n(r)=n_0(r/r_0{)}^{-\beta }$ and constant Case B recombination coefficient $\alpha$, the total recombination rate isThe central integral converges only for $\beta <3/2$; an envelope extending to infinity has finite recombination capacity only for $\beta >3/2$. At $\beta =3/2$, each logarithmic radius interval contributes equally. A cored power-law density profile removes the central singularity.

For a Kepler orbit with pericentre distance $q$, apocentre distance $Q$, and semi-major axis $a=(q+Q)/2$, a uniform mean anomaly givesThe two radial passages per orbital period are included, and ${\int }_q^Q{p}_r(r)dr=1$.

Uniform mean anomaly implies a line probability inversely proportional to the orbital speed. For small orbital eccentricity and true anomaly $f$, it is proportional to $1-e\cos f$.

The scale at which collinear contributions are assigned to parton distribution functions rather than the hard scattering coefficient. The separate factors depend on this scale, while a consistently calculated physical differential scattering cross-section does not.

For a three-dimensional droplet with concentration jump $\Delta \varphi$, constant mobility $M$, and quasi-static exterior chemical potential,The first formula solves the exterior Laplace equation. The second follows from conservation: the total outward flux $4\pi MR({\mu }_s-{\mu }_{\infty })$ removes excess composition at the rate $-4\pi R^2\Delta \varphi \dot{R}$. For a symmetric mixture, $\Delta \varphi =2{\varphi }_B$ and the Gibbs--Thomson relation gives ${\mu }_s=\gamma /({\varphi }_{B}R)$.

Using the positive polar molecular field $h_i=\delta F/\delta p_i$ and velocity gradient ${\partial }_i{v}_j$, the reversible order-parameter stress consists of a distortion contribution and rotational and alignment contributions:For a local free-energy density, a representative is ${\Sigma }_{ij}^{(1)}=(f-p_k{h}_k){\delta }_{ij}-f_{{\partial }_i{p}_k}{\partial }_j{p}_k$. The Euler-Lagrange equation verifies its divergence. Integration by parts gives $\delta F=\int {\Sigma }_{ij}^p{\partial }_i{u}_j$ for an incompressible displacement without surface work; the corresponding mechanical power has the opposite sign. Pressure contributions may be reassigned in an incompressible flow.

Amputation removes the external propagator factors from a momentum-space correlation function. For a connected four-point function, its diagrammatic external lines each carry a full two-point function; multiplying by the corresponding inverse two-point kernels leaves an interaction kernel.

This operation helps separate propagation to and from the interaction from the interaction itself. It is part of the connection between Green functions and scattering amplitudes, although physical external states also require on-shell limits and the appropriate pole-residue normalization. An amputated connected function need not be one-particle irreducible: it may still contain internal lines whose removal disconnects it. External-line removal and internal irreducibility are distinct operations.

At a heavy-quark mass $m$, leading-order continuity gives ${\beta }_{-}\ln (m/{\Lambda }_{-})={\beta }_{+}\ln (m/{\Lambda }_{+})$. Therefore $\boxed{{\Lambda }_{+}=m({\Lambda }_{-}/m{)}^{{\beta }_{-}/{\beta }_{+}}}$. This is a change of the strong-coupling scale parametrization across a change of active flavour number, not a discontinuity of the matched QCD coupling. For four to five active flavours in $SU(3)$, ${\beta }_{-}=25/3$ and ${\beta }_{+}=23/3$, hence ${\Lambda }_5={\Lambda }_4(m_b/{\Lambda }_4{)}^{-2/23}$.

For an odd map between spheres $f:S^n\to S^m$ and its quotient $\bar{f}$, there is an isomorphism ${\gamma }_n\cong {\bar{f}}^{\ast }{\gamma }_m$ of real tautological line bundles. With $x$ a unit vector, send $tx$ to $tf(x)$; replacing $x$ by $-x$ and $t$ by $-t$ gives the same vector. This proves both linearity on fibers and well-definedness. Therefore ${\bar{f}}^{\ast }{w}_1({\gamma }_m)=w_1({\gamma }_n)$. The mod-two cohomology ring of real projective space then implies $n\le m$, since a vanishing $(m+1)$st power must pull back to a vanishing power.

Suppose a continuous map $g:S^n\times S^n\to S^n$ changes sign on negating either input. Its quotient $\bar{g}:{\mathbb{RP}}^n\times {\mathbb{RP}}^n\to {\mathbb{RP}}^n$ pulls back the real tautological line bundle to the tensor product of vector bundles $p_1^{\ast }{\gamma }_n\otimes p_2^{\ast }{\gamma }_n$. The fiber map sends $(sx)\otimes (ty)$ to $stg(x,y)$. The First Stiefel–Whitney class of a tensor product of real line bundles consequently gives ${\bar{g}}^{\ast }a=a_1+a_2$. The Künneth theorem and the mod-two cohomology ring of real projective space implyEvery interior binomial coefficient in row $n+1$ must be even. By binomial coefficients with even interior terms, $n=2^k-1$ for some $k\ge 0$. This is a necessary condition, not an assertion of existence in every such dimension.

With velocity gradient $A_{ij}={\partial }_i{v}_j$, set $\Omega =(A-A^T)/2$. A vector rigidly rotating with the material has material derivative $D_t\mathbf{p}=-\Omega \mathbf{p}$, so its corotational derivative vanishes. Its unit rotational coefficient is fixed by an objective time derivative; adding $-\xi D\mathbf{p}$ describes the material's flow alignment of a polar order parameter.

For a real vector $x_0$ with support $S$, uniqueness in basis pursuit is equivalent to $|⟨\mathrm{sgn}(x_{0,S}),h_S⟩|<\Vert h_{S^c}{\Vert }_1$ for every nonzero $h\in \ker A$. The supporting-line inequality for the absolute value proves sufficiency. Taking small positive and negative multiples of $h$ before any active sign changes proves necessity.