For density operators, Klein's inequality gives $D(\rho \Vert \sigma )\ge 0$, with equality exactly when $\rho =\sigma$. If the support of a positive operator $\rho$ is not contained in that of $\sigma$, the quantum relative entropy is $+\infty$.

For a rigid body rotating around its third principal axis, the anisotropic part of its second mass moment tensor oscillates at angular frequency $2\Omega$. The quadrupole formula gives the displayed luminosity. Equal transverse principal moments remove this rotating quadrupole and its leading gravitational wave emission.

The iterative soft-thresholding algorithm applies the proximal gradient method to $\Vert x{\Vert }_1+\frac{1}{2}\Vert Ax-y{\Vert }_2^2$:Here $S_{\tau }$ is coordinatewise soft thresholding. The objective is convex and coercive, so iterates converge to a minimizer in finite dimensions; uniqueness requires additional hypotheses.

For a small category $\mathcal{C}$, an object $P$ of $[\mathcal{C},\mathbf{Set}]$ is an irreducible projective when $\mathrm{Nat}(P,-)$ preserves coproduct in a category objects and epimorphisms. Preservation of coproducts means every map from $P$ into a coproduct factors uniquely through one summand. Such objects are precisely retracts of representable functors; when the indexing category $\mathcal{C}$ is a Cauchy-complete category, they are representable. To see the retract, split the canonical epimorphism from a coproduct of representables using projectivity, and use coproduct preservation to factor the splitting through one summand.

For a nonzero module, the projective dimension ${\mathrm{pd}}_{R}M$ is the shortest length of a projective resolution of the module $M$, or infinity if no finite projective resolution exists. Equivalently, it is the supremum of degrees with nonzero ${\mathrm{Ext}}_R^n(M,N)$ as $N$ ranges over all modules. This equivalence follows by dimension shifting and the characterization of projective modules by vanishing ${\mathrm{Ext}}^1$.

For a generic filter $H$ in product forcing, let $G_0,G_1$ be its coordinate projections. They are filters in an ordered set. Given $p\in G_0$ and $q\in G_1$, choose separate pairs witnessing their membership and then a common strengthening in $H$. Upward closure puts $(p,q)$ in $H$, proving the displayed equality. Every ground-model dense factor set lifts to a dense product set, proving genericity of each projection over the ground model.

If $H=G_0\times G_1$ is generic for product forcing over $M$, then $G_1$ is generic over $M[G_0]$. For a name $\dot{D}$ for a dense subset of the second factor, take $p_0\in G_0$ forcing density. Ground-model pairs with first coordinate incompatible with $p_0$, or with first coordinate below $p_0$ forcing the second into $\dot{D}$, form a dense product set. The existential clause of syntactic forcing and the membership clause for a canonical forcing name give the density witnesses. The product filter meets this set, cannot use the incompatible case, and hence its second projection meets ${\dot{D}}^{G_0}$.

For observations or orbit points $x_0,\dots ,x_{N-1}$, the empirical measure is ${\nu }_N=N^{-1}{\sum }_{j=0}^{N-1}{\delta }_{x_j}$. Integrating a test function gives its sample average. For orbit points $x_j=T^{j}x$, the difference between $T_{\ast }{\nu }_N$ and ${\nu }_N$ telescopes to $({\delta }_{T^{N}x}-{\delta }_x)/N$, so on a compact metric space with continuous $T$, every weak limit is invariant.

A uniform prior on a finite-volume parameter region assigns constant probability density there and zero outside. On $[0,1]$ it is $\mathrm{Beta}(1,1)$, giving a beta distribution posterior after binomial distribution data. Uniformity depends on the coordinate: a nonlinear reparametrization introduces a Jacobian and generally does not preserve a uniform density.

A primordial bispectrum is squeezed when one external Fourier mode has much smaller wavenumber than the other two. Momentum conservation then makes the two short wavevectors almost opposite. This configuration measures the response of short-scale correlations to a long perturbation.

The orbit-averaged Poynting–Robertson drag equations preserve $C=a(1-e^2)e^{-4/5}$ for $0<e<1$. This relates shrinking semi-major axis to decreasing orbital eccentricity; a circular orbit is treated by its separate limiting solution.

For a spherical grain with constant radiation-pressure coefficient $\beta$, describe its orbit using the reduced central attraction $G{M}_{\star }(1-\beta )$ and treat Poynting–Robertson drag to first order in $v/c$. In that approximation the orbit-averaged apsidal precession isThe dissipative force has a radial component proportional to $-2\dot{r}/r^2$ and a transverse component proportional to $-v_{\theta }/r^2$, as obtained from the first-order radiation-force expression.^[ref]

The cancellation follows by orbit averaging: radial velocity is odd in true anomaly, whereas radius, transverse speed and the time-weighting factor are even. The terms contributing to the change of apsidal angle integrate to zero over a full orbit. Semi-major axis and orbital eccentricity can still decrease. Instantaneous osculating angles can oscillate, and nonspherical forces or additional planetary perturbations need not share this cancellation; a circular orbit has no defined apsidal direction.

For the Poynting–Robertson drag invariant $C$ and $A=\beta G{M}_{\star }/c$, the orbit-averaged time to the point-star limit isThis tends to $a_0^2/(4A)$ for a circular orbit and to $4Q_0^{1/2}{q}_0^{3/2}/(5A)$ for a highly eccentric orbit. Stellar radius, sublimation, collisions, and failure of orbital averaging can terminate the evolution earlier.

A Cartier divisor is ample when its associated line bundle is ample. The Nakai–Moishezon criterion and Kleiman's criterion characterize this condition numerically.

A line bundle is nef when its degree on every integral complete curve is nonnegative. Equivalently a Cartier divisor $D$ is nef when $D\cdot C\ge 0$ for every such curve. On a projective scheme, adding a positive ample class to a nef class makes it ample; taking limits in the intersection product shows that a nef divisor has nonnegative top self-intersection number.

Two Cartier divisors are numerically equivalent when they have equal intersection numbers with every integral complete curve. A numerically trivial divisor has zero intersection with every such curve. Unlike linear equivalence of Cartier divisors, numerical equivalence does not require the same line bundle.

A Cartier divisor $D$ on an integral $n$-dimensional projective variety is big when $h^0(X,mD)\ge c{m}^n$ for some $c>0$ and infinitely many positive integers $m$. Equivalently, its Iitaka dimension is $n$. Kodaira's lemma characterizes bigness by an ample-plus-effective decomposition, and the birational linear system criterion for bigness shows why this growth captures the full variety.

A Cartier divisor is semiample if some positive multiple is a basepoint-free divisor, equivalently its divisor line bundle has a globally generated positive power. Such a power defines a Kodaira map $f$ with $\mathcal{O}(mD)=f^{\ast }\mathcal{O}(1)$. Semiampleness implies nefness but need not imply ampleness: a fibre divisor of a morphism to a curve is a basic example.

For an integer $0\le k\le n$, the fantope is the convex hull of rank-$k$ orthogonal projection matrices in ${\mathbb{R}}^n$. Equivalently, it consists of real symmetric matrices with eigenvalues in $[0,1]$ and matrix trace $k$. To prove the equivalence, apply the spectral theorem for real symmetric matrices and express the eigenvalue vector as a convex combination of the zero-one extreme points of the capped simplex. The sum of the largest eigenvalues is its support function.

For $\varphi h_t=D_m(h^m{h}_x{)}_x$ on $0<x<L$, with $h(0,t)=0$ and zero right-hand volume flux, a separation of variables giveswhere the positive profile satisfiesIts boundary volume flux per unit width isThese profiles are exact separated solutions and describe the leading long-time discharge for a broad class of positive initial data. The virtual origin $t_0$ depends on the initial profile; it does not make the separated solution an exact representation of every initial condition.