For unnormalized model posterior densities $h_j$, model-selection probabilities $s_{jk}$ and parameter-proposal densities $q_{jk}$, accept a proposal $(j,\theta )\to (k,{\theta }^{\prime })$ with the minimum of one and $h_k({\theta }^{\prime })s_{kj}({\theta }^{\prime })q_{kj}(\theta \mid {\theta }^{\prime })/[h_j(\theta )s_{jk}(\theta )q_{jk}({\theta }^{\prime }\mid \theta )]$. A deterministic matching-map formulation instead includes its explicit Jacobian determinant; it must not be counted again when already absorbed into a direct proposal density.

A pseudo-prior is a proper density assigned to a parameter that is inactive under a particular model. Integrating it out leaves that model's likelihood and marginal evidence unchanged. Its choice affects movement between model states, but not the intended marginal posterior over models.

A replicate is an instance of a treatment on a distinct experimental unit. independent replicates permit estimation of response variation between experimental units; subsamples within one experimental unit do not supply additional independent treatment replication. In the pot example, another pot receiving A supplies another A replicate, while another leaf from an existing A pot supplies another observational unit.

For $\beta =-b{g}^3$, ${\gamma }_{\varphi }=c{g}^2$, and $b>0$, one has ${\bar{g}}^2=g^2/(1+b{g}^2\log z)$. Integrating the field-renormalization anomalous dimension gives the displayed logarithmic factor. At large $z$, a nonzero free matching factor yields $d\sim C(\log z{)}^{c/b}$ in the printed RG sign convention.

Relative permeability divided by the phase's dynamic viscosity. In the phase Darcy law, $q_i=-K{\lambda }_i\nabla p_i$ without body forces. Ratios of mobilities determine fractional flow when the phases have a common pressure gradient.

The regular syntactic category has regular formulas-in-context as objects and provably functional regular relations as arrows. It has finite limits and stable image factorizations, with existential quantification supplying images. The regular coverage on this category presents the classifying topos of the theory.

If a regular theory proves an outer coherent sequent $\varphi ⊢{\bigvee }_{i=1}^n{\psi }_i$ with all formulas regular, it proves $\varphi ⊢{\psi }_i$ for some $i$. In the classifying topos, the corresponding subobjects cover the representable associated with $\varphi$. Irreducibility of that representable and full faithfulness of the syntactic Yoneda embedding produce the desired derivation.

For a subfunctor $F\prime \subseteq F$ of a regular-coverage sheaf, its closure consists of sections whose restriction belongs to $F\prime$ along one covering regular epimorphism. Pullbacks prove that local membership is a subfunctor; composition of witnessing covers proves the sheaf condition. It is the smallest subsheaf containing $F\prime$.

If a family of subsheaves covers $yA$, local membership of $1_A$ gives a regular epimorphism $\alpha :B\to A$ belonging to one subsheaf. The sheaf condition descends that section to $1_A$ in the same subsheaf. All maps into $A$ are restrictions of $1_A$, so that subsheaf is $yA$. Thus representable sheaves for the regular coverage are irreducible even for arbitrary unions.

The displayed metric on antipodal classes of the unit sphere induces its quotient topology. Triangle inequality follows by choosing the signs that realize each minimum and applying the Euclidean triangle inequality. Distinct classes have disjoint saturated unions of small balls, proving the quotient is Hausdorff; compactness follows from the continuous quotient map.

For $k>0$, horizontal Fourier transform gives $({\partial }_z^2-k^2)\tilde{\Phi }=4\pi G\tilde{\Sigma }\delta (z)$. Decay and continuity give an exponential on each side; its derivative jump fixes the coefficient. The uniform mode needs a separate linear vertical solution and a potential reference.

The matrix coefficients of a rational $G{L}_m$ action belong to its coordinate ring $\mathbb{C}[g_{ab},\det (g{)}^{-1}]$. Multiplying the action by a sufficiently large determinant twist clears all determinant denominators and produces a polynomial representation. Irreducible rational actions are indexed by decreasing integer highest weights, with negative coordinates allowed.

For finite-dimensional rational representations of complex $G{L}_m$ or $S{L}_m$, average a Hermitian inner product over the compact group $U(m)$ or $SU(m)$ using normalized Haar measure. Orthogonal complements become invariant under the compact group and its complexified Lie algebra. The latter generates the complex group, giving invariant complements and hence complete reducibility. This conclusion does not hold for arbitrary affine algebraic groups, such as the additive group.

After failures, a transmitter increases its random retry window by a multiplicative factor, so its access behavior depends on its collision history. This reduces immediate repeated contention, but its stability and delay properties depend on the exact rule and traffic model.

For finite positive measures $\mu ,\nu$, put $\rho =\mu +\nu$. The linear functional $g\mapsto \int gd\nu$ is bounded on real $L^2(\rho )$ by the Cauchy-Schwarz inequality. The Riesz representation theorem gives a measurable $h$ with $\nu (A)={\int }_{A}hd\rho$. Testing indicators of its negative and greater-than-one level sets shows $0\le h\le 1$ almost everywhere. Subtraction gives $\mu (A)={\int }_A(1-h)d\rho$. This constructs the densities without assuming the Radon-Nikodym theorem as a prior result.

For a series ${\sum }_n{a}_n{x}^{m_n}$ with distinct increasing integer exponents, define the ordinary coefficients to be $a_n$ at $m_n$ and zero elsewhere. The Cauchy-Hadamard theorem then givesThe exponent in the root is the actual degree $m_n$, not the index $n$. For $a_n=(n!{)}^2$ and $m_n=n^2$, $0\le 2\log (n!)/n^2\le 2\log n/n\to 0$, so $R=1$. Boundary behavior is a separate question.

Thermal electrons can transfer energy to lower-energy radiation photons through Compton scattering. In the nonrelativistic Thomson regime, the net cooling power per electron is $4{\sigma }_{T}c{U}_{\gamma }{k}_B(T_e-T_{\gamma })/(m_e{c}^2)$ for a blackbody photon bath. A sufficiently hot electron population therefore loses energy; a colder population is heated instead. The cosmic microwave background radiation energy density grows as $(1+z{)}^4$, making this channel more important for high-redshift diffuse ionized gas.

Dust grains emit thermal radiation after absorbing energy from radiation or collisions. Collisions can transfer thermal energy from dense gas to grains, whose radiation then cools the gas. This channel requires dust and does not operate in strictly pristine hydrogen-helium material without prior enrichment.

Hot electrons can transfer energy to cosmic microwave background photons through Compton scattering. For a fixed electron fraction $x_e$ and electron temperature well above the radiation temperature, the optically thin gas cooling time scales approximately as $(1+z{)}^{-4}/x_e$. Its volume loss scales with electron density, not with $n_H^2$, because the radiation density is independent of local gas density. The process tends toward the radiation temperature; it heats rather than cools electrons colder than that radiation.

For radial adiabatic stellar oscillations, the quotient is $\mathcal{R}[\xi ]=\int [\gamma p{r}^4|\xi \prime {|}^2+r^3((4-3\gamma )p)\prime |\xi {|}^2]dr/\int \rho r^4|\xi {|}^{2}dr$. Regular-centre and free-surface endpoint conditions remove the boundary term. The infimum is the fundamental squared frequency; any negative trial value proves radial instability. A variable stellar adiabatic exponent must remain inside the derivative.