An isomorphism of categories is a functor with a strictly inverse functor. It is equivalently bijective on objects and on each hom-set. An equivalence of categories only requires inverse composites up to invertible natural transformations. For example, the category of partial functions and the category of pointed sets are equivalent but their actual object collections prevent an isomorphism: the former has one zero object, the empty set, while the latter has distinct singleton pointed set objects that are all zero objects.

A functor $F:\mathcal{C}\to \mathcal{D}$ is essentially surjective when every object of $\mathcal{D}$ is isomorphic to $FA$ for some $A\in \mathcal{C}$. Together with full and faithful, this characterizes an equivalence of categories under the appropriate axiom of choice convention. For large categories, choosing a quasi-inverse on all objects requires a universe or a class-choice convention.

A sequence $(x_n)$ in $\mathbb{R}/\mathbb{Z}$ is an equidistributed sequence if and only if $N^{-1}{\sum }_{n=0}^{N-1}{e}^{2\pi im{x}_n}\to 0$ for every nonzero integer $m$. Approximation of continuous functions by trigonometric polynomials proves the criterion.

The quartic interaction $g_1{\varphi }_1^4+g_2{\varphi }_2^4+\lambda {\varphi }_1^2{\varphi }_2^2$ includes the Gaussian fixed point, single and double Ising model Wilson-Fisher fixed points, and an O(N) model fixed point with $N=2$. A rotation of two decoupled identical Ising models produces a second coordinate representation of their renormalization-group fixed point.

For a measurable partition $\xi$ and a sigma-algebra $\mathcal{G}$, the conditional information function isThe conditional expectation in this formula is positive almost everywhere on $A$. Without conditioning, the information is $I_{\mu }(\xi )(x)=-\log \mu (\xi (x))$.

For a sinusoidal Eliassen–Palm flux whose vertical divergence is constant in $-D<z<D$ and zero elsewhere, the residual mean circulation is determined by a forced one-dimensional modified Helmholtz equation. With $v^{\ast }=C^{\prime }(z)\sin (\pi y/L)$ and $w^{\ast }=-(\pi /L)C(z)\cos (\pi y/L)$, a forcing proportional to $-G_0\sin (\pi y/L)/(2D)$ givesThe response is uniquely selected by decay of the residual circulation at both vertical infinities. An Eulerian mean vertical flow can persist in the undamped wave region even where this residual response vanishes.

An energetic charge carrier creates an additional electron–hole pair. In a multiplication register this is stochastic, so the output charge fluctuates even when the input electron number is fixed.

Random multiplication increases the noise relative to deterministic gain. If the output from one input electron has mean $g$ and variance $g^2$, a Poisson distribution of input electrons with mean $\mu$ produces output variance $2g^2\mu$, by the law of total variance. The high-gain analog-mode excess-noise factor is $\sqrt{2}$.

In the displayed iterated order, with the $n$ sum evaluated first, $G_2(z)=({\pi }^2/3)E_2(z)$. The cosecant partial-fraction identity ${\sum }_n(w+n{)}^{-2}={\pi }^2{\csc }^2(\pi w)$ gives the positive-$m$ contribution $-4{\pi }^2{\sum }_{r\ge 1}r{q}^{mr}$; negative $m$ gives the same contribution. Including the $m=0$ term yields ${\pi }^2/3-8{\pi }^2{\sum }_{n\ge 1}{\sigma }_1(n)q^n$. The full lattice series does not have absolute convergence, so this order must not be replaced by arbitrary rearrangement.

A generalized eigenfunction solves the eigenvalue equation for a linear operator in an enlarged space, often a space of distributions, although it may fail to belong to the underlying Hilbert space. A plane wave $e^{ikx}$ on the whole real line is a generalized momentum eigenstate, with momentum $\hslash k$, and is not square-integrable. This usage differs from a generalized eigenvector in a Jordan chain, where a higher power of $L-\lambda$ vanishes rather than $L-\lambda$ itself.

For a stationary Markov chain and a scalar observable with summable autocorrelation function $\rho (k)$, the asymptotic variance of its sample mean is $\mathrm{Var}(h){\tau }_{\mathrm{int}}/n$. Thus the effective sample size of a Markov chain is approximately $n/{\tau }_{\mathrm{int}}$. Different observables can have very different autocorrelation times.

Among subsets of the $n$-dimensional hypercube graph with $a$ vertices, initial segments of the binary order on the discrete cube minimize the edge boundary. Its exact minimum is $na-2{\sum }_{j=0}^{a-1}{s}_2(j)$. This strengthens the entropy lower bound on the same boundary.

The vertical eddy diffusion coefficient parametrizes turbulent transport in an atmosphere by a diffusive flux proportional to an abundance gradient. Over distance $L$, the corresponding eddy mixing time is $L^2/K_{zz}$. Its units are area per time, so $1{\mathrm{m}}^2{\mathrm{s}}^{-1}=10^4\mathrm{c}{\mathrm{m}}^2{\mathrm{s}}^{-1}$.

An isolated compact dust grain of radius $r_d$ and material density ${\rho }_d$ has opacity per unit dust mass ${\kappa }_d=3Q_{\mathrm{pr}}/(4{\rho }_d{r}_d)$, where $Q_{\mathrm{pr}}$ is its radiation-pressure efficiency. Its Eddington luminosity is $4\pi GMc/{\kappa }_d$. Matching the electron-scattering opacity gives $r_{d,\mathrm{crit}}=3Q_{\mathrm{pr}}/(4{\rho }_d{\kappa }_{\mathrm{es}})$, approximately $1.9\mathrm{cm}$ for $Q_{\mathrm{pr}}=1$, ${\rho }_d=1\mathrm{g}\mathrm{c}{\mathrm{m}}^{-3}$, and ${\kappa }_{\mathrm{es}}=0.4\mathrm{c}{\mathrm{m}}^2{\mathrm{g}}^{-1}$. At that electron-scattering luminosity, smaller grains feel a net outward force and larger grains remain attracted. This is an isolated-grain force balance, not the opacity of an arbitrary gas-dust mixture.

If the relative acceleration is $\ddot{\mathbf{r}}=-GM\mathbf{r}/r^3+\mathbf{f}$, with constant masses, then the specific orbital energy $\epsilon$, specific angular momentum $\mathbf{h}$ and eccentricity vector obeyThe inverse-square-force terms cancel in each derivative. These identities express perturbing work and torque, and track the instantaneous change of the ellipse's shape and orientation.

For a semi-infinite stratified fluid with thermal wind $U=\Lambda z$ above a gently sloping boundary $z=\alpha y$, a zero-interior-potential vorticity perturbation decays as $e^{-\mu z}$, where $\mu =N|k|/|f_0|$. The boundary buoyancy perturbation obeys a single oscillatory equation with phase velocity $c=(\Lambda -\alpha N^2/f_0)/\mu$. The topographic and thermal-wind contributions cancel when the boundary follows a background isopycnal.

For a set family $\mathcal{A}\subseteq \mathcal{P}([n])$, its downward closure is $\downarrow \mathcal{A}=\{S:\text{some }A\in \mathcal{A}\text{ has }S\subseteq A\}$. It is the smallest down-set containing $\mathcal{A}$, just as the upward closure of a set family is the smallest up-set containing it. The two closures allow Harris' inequality to control sizes of a cross-Sperner family pair.

The antiparticle of the down quark has electric charge $+1/3$ and baryon number $-1/3$.

The ensemble average $\overline{\omega }$ uniquely minimizes ${\sum }_j{p}_{j}D({\omega }_j\Vert \rho )$ over density operators $\rho$. Donald's identity separates this objective into a constant plus $D(\overline{\omega }\Vert \rho )$, and nonnegativity of quantum relative entropy proves the claim. The minimum is the Holevo quantity $S(\overline{\omega })-{\sum }_j{p}_{j}S({\omega }_j)$, equivalently the quantum mutual information of the associated classical-quantum state.

A single-box up-move changes a partition of an integer by adding one to row $i$ and subtracting one from row $j>i$, provided the result remains a partition. It increases exactly the partial sums ending between rows $i$ and $j-1$. A partition $\mu$ dominates $\lambda$ precisely when it is obtainable from $\lambda$ by a sequence of these moves: fill the first deficient row from the row at the first return of the cumulative deficit to zero. This preserves the partition inequalities and decreases the distance to $\mu$.