Random effects are crossed when an observation can belong to levels of two grouping factors without one grouping factor being nested within the other. Independent location intercepts and incubator slopes, for example, contribute separate terms to the observation covariance matrix.

Time is divided into slots, and a slot succeeds exactly when one station transmits. With backlog $N$ and independent attempt probability $p$, the success probability is $Np(1-p{)}^{N-1}$. Fluid approximation of slotted ALOHA replaces the large-system binomial probabilities by Poisson probabilities.

A thin radiative envelope with constant enclosed mass, luminosity and mean molecular weight, ideal gas pressure and opacity $\kappa ={\kappa }_0{\rho }^n{T}^{-s}$ has $a=n+1$, $b=n+s+4$, and $C=16\pi a_{\mathrm{rad}}cGM/[3{\kappa }_0(\mu m_u/k_B{)}^{n}L]$. Combining radiative diffusion in a star with hydrostatic equilibrium gives ${\nabla }_{\mathrm{rad}}=P^a/(C{T}^b)$, so $P^a-P_0^a=(aC/b)(T^b-T_0^b)$ for nonzero exponents. Zero exponents give logarithmic limits. The model must be stopped or modified if the Schwarzschild criterion predicts convection.

A constant radiative-diffusion coefficient and constant gravity give $T^4=T_0^4+4AP/g$. Thus ${\nabla }_{\mathrm{rad}}=AP/(g{T}^4)<1/4$ for $T_0>0$. A diatomic ideal gas has adiabatic temperature gradient $2/7$, so this idealized profile never becomes convective. A physical radiative-convective boundary requires additional opacity, flux, or thermodynamic variation.

Assume a thin disk has height-independent opacity ${\kappa }_{\mathrm{op}}$ and radiation-to-gas pressure ratio $\beta$, with vertical gravity $-{\Omega }_z^{2}z$. Hydrostatic equilibrium and radiative diffusion implyBalancing its derivative against viscous heating $\rho \nu q^2{\Omega }^2$ fixes the dynamic viscosityIt is independent of height under these assumptions. For a spherically symmetric potential, ${\Omega }_z=\Omega$. Integrating $\rho \nu$ through the full surface-to-surface thickness $2H$ gives $\bar{\nu }\Sigma =2H\rho \nu$, which determines the thickness from the radial accretion rate solution.

The zero-kick circular-parent threshold is $\beta =1/2$: below it grains are initially bound, at it their conservative release orbit is parabolic, and above it they are initially unbound. Poynting–Robertson drag can modify a marginal orbit, so the threshold is a statement about the instantaneous release orbit, not an exact capture criterion with dissipation included.

For spanning slab normals, apply the maximum of finitely many Rademacher linear forms bound to $u_i=(V{\Lambda }^{1/2}{)}^T{a}_i$, whose squared norms are $a_i^{T}X{a}_i\le 1$. Some sign vector has scaling denominator squared at most $2\log (2m)$. Rademacher rounding for a semidefinite relaxation then produces a feasible vector of squared norm at least the displayed fraction of the SDP optimum. This is an existence guarantee from the probabilistic method.

For independent Rademacher random variables and $\Vert u_i{\Vert }_2\le 1$, this bound holds for $\alpha >0$. The moment-generating function satisfies ${\prod }_j\cosh (t{u}_{ij})\le e^{t^2/2}$, with strict inequality for a nonzero vector and $t>0$. Applying the exponential Markov bound to both tails and then the union bound proves a strict failure bound; a zero vector has failure probability zero. At $\alpha =\sqrt{2\log (2m)}$, the success probability is strictly positive, ensuring a sign vector with all linear forms below that threshold.

The logit mean model $p_i=(1+e^{-x_i^T\beta }{)}^{-1}$ with working variance function $\varphi p_i(1-p_i)$ has the same coefficient estimates as ordinary binomial logistic regression, but estimates dispersion from residuals. For an actual individual binary random variable, $Y^2=Y$ forces variance $p(1-p)$, so a nonunit dispersion is a working specification rather than a new independent binary distribution.

For a Boussinesq approximation, constant buoyancy frequency $N$, and buoyancy heating rate $R$, the quasi-geostrophic potential-vorticity equation becomes $D_{g}q/Dt=(f_0/N^2)R_z$, where $q={\nabla }_h^2\psi +(f_0^2/N^2){\psi }_{zz}+\beta y$. Heating changes the vertical gradient of buoyancy and therefore produces a potential-vorticity tendency proportional to its height derivative.

Every quasi-coherent sheaf on an affine scheme has zero higher sheaf cohomology. Together with the equivalence between modules and quasi-coherent sheaves on an affine scheme, this makes affine varieties the basic acyclic pieces for computing cohomology. An affine open cover of a separated scheme has affine finite intersections, so the acyclic cover theorem identifies its Čech cohomology with sheaf cohomology.

The unassisted classical capacity is the supremum of classical bit rates achievable with asymptotically vanishing message error. It allows entanglement among different channel inputs, but no preshared sender-receiver entanglement. The Holevo-Schumacher-Westmoreland theorem gives its regularized entropy expression.

The Holevo capacity of a quantum channel is the maximum one-use output Holevo quantity,It is the unassisted classical communication limit for product state encodings with collective output measurements. Arbitrary entangled inputs require the regularized expression in the Holevo-Schumacher-Westmoreland theorem.

Along the dyadic partitions of an interval of $[0,1]$, the sums ${\sum }_i|{\Delta }_{i}M{|}^p$ of a continuous local martingale tend to zero in probability for $p>2$. If $0<p<2$ and these sums are eventually bounded almost surely, then $M$ is constant on $[0,1]$: compare them with the sums defining quadratic variation, using the maximum increment, which tends to zero by uniform continuity.

For homomorphisms $i_j:H\to G_j$, the pushout of groups is $(G_1\ast G_2)/⟨⟨i_1(h)i_2(h{)}^{-1}:h\in H⟩⟩$. It has the universal property for homomorphisms out of $G_1,G_2$ agreeing on $H$. When the maps $i_j$ are injective, it is the amalgamated free product, with the factors and common subgroup embedded.

For a rank-$r$ complex vector bundle and $u=c_1({\lambda }^{\ast })$, the cohomology of its projective bundle is free over the cohomology of the base on $1,u,\dots ,u^{r-1}$. Consequently pullback is injective, which yields the splitting principle for complex vector bundles by iteration.

For probit regression with $s_i=2Y_i-1$ and prior $N(0,{\sigma }^{2}I)$, the posterior score control variate isFor $h(\beta )=\Phi (x_{\ast }^T\beta )$, its covariance with the score is $-x_{\ast }\mathbb{E}[\varphi (x_{\ast }^T\beta )]$. It is nonzero whenever $x_{\ast }\ne 0$, ensuring strict improvement by the optimal control variate.

For a probability density function $f$ on $\mathbb{R}$ with Lipschitz bound $L>0$, the triangular lower envelope $f(x+u)\ge (f(x)-L|u|{)}_{+}$ gives $f(x{)}^2\le L$. If $f$ is strictly positive everywhere, the inequality is strict. Equality forces the entire probability density function to coincide with the triangular envelope, which vanishes outside a finite interval.

A probability density is log-concave when its logarithm is a concave function on its support. Tangent lines to a differentiable concave log density lie above its graph, giving exponential envelopes for rejection sampling.

For a strictly positive probability density function with Lipschitz bound $L$, $p_x(r)={\int }_{x-r}^{x+r}f(y)dy$ satisfies $|p_x(r)-2rf(x)|\le L{r}^2$. If $0<s<1$ and $s\le f(x{)}^2/L$, monotonicity gives $p_x^{-1}(s)\le s/f(x)$ and hence $|2f(x)p_x^{-1}(s)-s|\le L{s}^2/f(x{)}^2$. The Lipschitz density height bound prevents extending this local argument to every $s\in (0,1)$ by assuming $f(x)\ge \sqrt{L}$ for a strictly positive density.