In a Grothendieck topos, an object is a quotient of a decidable object when it has an epic cover by one. These objects are closed under subobjects, small coproducts, quotients and finite limits. Their union inside any fixed object gives the largest subobject of this kind, defining a coreflective subcategory. The induced idempotent comonad is left exact, so the full subcategory is a topos.

A dbar equation in a complex spectral parameter, rather than the original spatial coordinate. In spectral reconstruction, its right-hand side is transformed data. A Cauchy-Pompeiu formula in the spectral plane reconstructs the auxiliary function; an asymptotic coefficient or the original differential equation then recovers the spatial field.

With permeability $K$, depth $d$, kinematic viscosity $\nu$ and the relevant effective thermal diffusivity ${\kappa }_T$, the Darcy Rayleigh number compares buoyancy-driven Darcy advection to thermal diffusion. It differs from the clear-fluid Rayleigh number by replacing $d^3$ with $Kd$. Effective heat capacity and porosity factors may be absorbed into the definition of ${\kappa }_T$ and the thermal-time scale, and must be kept consistent with the chosen Darcy law convention.

In a unit square with prescribed conductive boundary temperatures and impermeable boundaries, the stream function convention $\mathbf{u}=(-{\psi }_z,0,{\psi }_x)$ gives linear equations $\Delta \psi =R{\theta }_x$, ${\theta }_t={\psi }_x+\Delta \theta$, with $\psi =\theta =0$ on all sides. The velocity equation follows by taking the vertical-plane curl of Darcy law, and the temperature equation by linearizing the heat equation about $1-z$. Unlike a laterally unbounded Darcy layer, the conducting side boundaries produce a two-dimensional real marginal eigenspace at $R_c=8{\pi }^2$.

A rectangular initial displacement with zero initial velocity splits into two equal half-height copies moving in opposite directions. The D'Alembert formula reads $u(x,t)=[u_0(x+ct)+u_0(x-ct)]/2$. Overlap adds the heights; separated copies each retain the original width. For discontinuous fronts the formula defines a weak solution of the wave equation.

A cyclic subspace generated by $x$ under a linear operator $T$ is the span of its successive iterates $x,Tx,T^{2}x,\dots$. In finite dimension it is already generated by the first $n$ iterates. The first linear dependence expresses the next iterate in previous ones and proves invariance without using the Cayley-Hamilton theorem.

An element killed by the next differential of a cochain complex. Its class modulo coboundaries is a cohomology class.

For simply connected $X$ with finitely generated integral homology in every degree, the Hurewicz theorem modulo a Serre class gives finitely generated homotopy groups. Realize finite generating sets of successive relative homotopy groups by attaching finitely many cells at each stage. The resulting weak equivalence has finitely many cells in each dimension.

The four-current combines charge density and current density into the four-vector $J^{\mu }=(c\rho ,\mathbf{J})$. With $x^{\mu }=(ct,\mathbf{x})$, its four-divergence vanishes by the charge continuity equation: ${\partial }_{\mu }{J}^{\mu }={\partial }_t\rho +\nabla \cdot \mathbf{J}=0$. It is the source in the covariant Maxwell equations.

For open subsets $U,V$ of a topological space $Y$, the relative cup product givesCompatibility with the maps to absolute cohomology identifies its image with the ordinary cup product. More generally the construction requires an excisive pair of subsets. The open-set case can be proved using subdivision to compute on chains subordinate to the open cover.

The Kolmogorov distance between two probability distributions with cumulative distribution functions $F,G$ is ${\sup }_{z\in \mathbb{R}}|F(z)-G(z)|$.

Neglect metric fluctuations and derivatives of $\dot{\bar{\varphi }}/H$, so $\delta X_1=-2\bar{X}\dot{\zeta }/H$ and $\delta X_2=\bar{X}[{\dot{\zeta }}^2-a^{-2}(\nabla \zeta {)}^2]/H^2$. The coefficient of ${\dot{\zeta }}^3$ in the cosmic-time action is $C=-a^3[2{\bar{X}}^2{P}_{,XX}+(4/3){\bar{X}}^3{P}_{,XXX}]/H^3$. This unfactored form is valid even when $P_{,XX}=0$ and a factored ratio would be undefined.

If $P_n$ are the increasing selected integers in the Cramér model, then almost surely ${\mathrm{limsup}}_n(P_{n+1}-P_n)/(\log P_n{)}^2\le 1$. A zero block of length $(1+ϵ)(\log k{)}^2$ after $k$ has summable probability, so the Borel-Cantelli first lemma excludes these blocks eventually.

When $d_j$ events occur at the same recorded time, the Breslow approximation to Cox partial likelihood uses the factor $\exp ({\sum }_{i\in D_j}{x}_i^T\beta )/({\sum }_{i\in R_j}{e}^{x_i^T\beta }{)}^{d_j}$, where $R_j$ is the risk set and $D_j$ the event set. It approximates the ordering of tied events rather than treating their within-tie order as known.

An observed event counting process has conditional increment $\mathbb{E}(d{N}_i(t)\mid {\mathcal{H}}_{t-})=Y_i(t)h_i(t)dt$, where $Y_i$ is the at-risk process. The filtration records the past observed events, censoring, and covariates; the integral of $Y_i{h}_i$ is the compensator of a counting process.

A predictable intensity $\lambda (t)$ is the derivative of an absolutely continuous compensator of a counting process: $N(t)-{\int }_0^t\lambda (u)du$ is a local martingale. It represents the conditional event rate given the immediately preceding history. In survival analysis the individual intensity is an at-risk indicator multiplied by its hazard function.

The dimensionless line-charge coupling $\eta =\lambda {\ell }_B/q$, with negative cylinder line charge $-\lambda$ and counterions of charge $q>0$. Here the Bjerrum length uses the same reference charge $q$. The mean-field infinite-dilution threshold is $\eta =1$. If ${\ell }_B$ is instead tabulated for elementary charge $e$ and counterions have charge $ze$, the corresponding coupling is $z\lambda {\ell }_B/e$.

The positive mobile charge per unit axial length accumulated from cylinder radius $a$ to $r$ is $Q(r)=2\pi {\int }_a^{r}s\rho (s)ds$. For fixed negative line charge $-\lambda$, Gauss's law gives signed radial electric field $E_r=2[-\lambda +Q(r)]/(ϵr)$ in Gaussian units. Its absolute value is the field magnitude. An integral over a local infinite-dilution profile measures condensed charge, rather than every ion in a finite neutral cell.

In the mean-field infinite-dilution cylinder model, the fraction of neutralizing charge that remains localized is $f_c=\max (1-1/\eta ,0)$, where $\eta$ is the Manning parameter. For $\eta >1$, integrating the cylindrical counterion-only Poisson-Boltzmann profile gives $Q_c=\lambda -q/{\ell }_B$. Below threshold, the local limiting density vanishes even though every finite neutral cell contains the full counterion charge.

A finite Cartesian product of countable sets is countable. For a fixed finite exponent $m$, index tuples in ${\mathbb{N}}^m$ can be listed by increasing maximum coordinate, with each group finite. If $B$ is countably infinite and $m>0$, its power $B^m$ is also countably infinite.