For a small perturbation of a stationary horizon, the linearized Null Raychaudhuri equation gives $\delta {\theta }^{\prime }=-8\pi \delta T_{kk}$. With vanishing boundary term $[\lambda \delta \theta ]$, integration by parts gives $\delta A=8\pi \int \lambda \delta T_{kk}d\lambda d{A}_0$. Combining this with affine horizon-generator scaling converts area response to horizon Killing-energy flux.

In Newtonian gauge in cosmology, the Free-streaming photon Boltzmann equation is $\dot{\Theta }+\mathbf{e}\cdot \nabla \Theta =\dot{\varphi }-\mathbf{e}\cdot \nabla \psi$. Angular averaging and the photon angular temperature moments give the displayed continuity equation. Its dipole yields the photon Euler equation, with the sign of the anisotropic stress term determined by the stress convention.

With $B_i^{(\pm )}=i{B}^{(\pm )}{m}_i^{(\pm )}/\sqrt{2}$ and the same convention for baryon velocity, choose ${\Theta }^{(\pm )}={\sum }_{\ell \ge 1}(-i{)}^{\ell }\sqrt{2\pi /(2\ell +1)}{\Theta }_{\ell }^{(\pm )}{Y}_{\ell ,\pm 1}$. The spherical-harmonic streaming recurrence givesThis is the temperature-only Thomson scattering hierarchy without polarization feedback, with optical depth to the observer satisfying $\dot{\tau }<0$. A photon multipole normalization change changes the denominators and the dipole-source prefactor. The lower coupling vanishes at $\ell =1$, so no vector monopole occurs.

A direction-dependent fractional photon temperature change. For a thermal spectrum, $f(ϵ)=\bar{f}(ϵ)-\Theta ϵ{\bar{f}}^{\prime }(ϵ)$ is its linear expansion at fixed comoving energy. For a nonthermal spectrum the same ansatz is a brightness dilation, rather than necessarily a thermodynamic temperature. In unweighted Legendre polynomial multipoles, $\Theta ={\sum }_{\ell }(-i{)}^{\ell }{\Theta }_{\ell }{P}_{\ell }$, the photon density, velocity and anisotropic stress are ${\delta }_{\gamma }=4{\Theta }_0$, $v_{\gamma }=-{\Theta }_1$ and ${\Pi }_{\gamma }=-3{\Theta }_2/5$.

If a differentiable dispersion relation $\omega (\mathbf{k})$ is homogeneous of degree zero at nonzero $\mathbf{k}$, the Euler homogeneous function theorem gives $\mathbf{k}\cdot {\nabla }_{\mathbf{k}}\omega =0$. Therefore phase velocity $\omega \mathbf{k}/k^2$ is orthogonal to group velocity ${\nabla }_{\mathbf{k}}\omega$. The result applies exactly to limiting inertial waves; finite compressibility can alter the dispersion.

The lever rule finds phase fractions from a horizontal tie line on a phase diagram. For solid and liquid concentrations $C_s,C_l$ and bulk concentration $\overline{C}$, conservation gives $\overline{C}=(1-\varphi )C_s+\varphi C_l$. Thus $\varphi =(\overline{C}-C_s)/(C_l-C_s)$ is the liquid fraction, provided $C_s\le \overline{C}\le C_l$. For pure solid, $\varphi =\overline{C}/C_l$.

Computing conductance from a smoothed gradient gives $u_t=\mathrm{div}(c(|\nabla (G_{\sigma }\ast u)|)\nabla u)$. The image edge detector is nonlocal, while the local highest-order diffusion has positive coefficient when conductance is positive. A smooth kernel, lower conductance bound and appropriate regularity/boundary hypotheses support a well-posed regularized filter.

For unit-variance innovations, orthogonality to the past givesConsequently ${\varphi }_{\nu }={\gamma }_{\nu }(1)/V_{\nu -1}$ and ${\sigma }_{\nu }^2=V_{\nu }-{\gamma }_{\nu }(1{)}^2/V_{\nu -1}$. Replacing autocovariance by matched sample estimates yields seasonal regression estimators. The unit innovation variance is necessary to identify the noise scale separately.

For bond percolation on the square lattice, write $h_n={\mathbb{P}}_p(0\leftrightarrow (n,0))$ and $g_k={\mathbb{P}}_p(0\leftrightarrow \partial [-k,k{]}^2)$. A maximal-probability boundary graph vertex can be placed at $(k,j)$ by symmetry. Reflection about $x=k$ identifies its connection probability from $0$ with that from $(2k,0)$. The Harris-FKG inequality yields $(g_k/(8k){)}^2\le h_{2k}\le g_{2k}$ for $k\ge 1$.

The percolation susceptibility is $\chi (p)=1+{\sum }_{m\ge 1}{\sum }_{y\in \partial {\Lambda }_m}{\tau }_p(0,y)$. Finiteness gives a boundary sum $a<1$. The weighted BK boundary-splitting estimate then gives $g_k\le a^{\lfloor k/m\rfloor }$. The strict bound $g_1=1-(1-p{)}^{2d}<1$ absorbs the finitely many smaller radii into a positive exponential rate with unit prefactor. At $p=0$, all positive-radius connection probabilities are zero.

For independent bond percolation on a transitive locally finite lattice, let $c_n$ count rooted self-avoiding walks and let $\mu =\lim c_n^{1/n}$ be the connective constant. An infinite open percolation cluster supplies an open walk of every length. The union bound gives $\theta (p)\le c_n{p}^n$, which tends to zero when $p\mu <1$. Thus $p_c\ge 1/\mu$.

Independent bond percolation on the cubic lattice has exponential connection decay at every $p<p_c$ and positive percolation probability at every $p>p_c$. The finite-set criterion for percolation sharpness gives decay below its auxiliary threshold. The Margulis–Russo formula and exploration from a finite box boundary give $g_n^{\prime }(p)\ge (1-g_n(p)){\inf }_{S\ni 0}{\phi }_p(S)/(p(1-p))$, with the infimum over interior finite sets. Above the auxiliary threshold the infimum is at least one; integrating gives positive percolation probability and identifies the two thresholds. This theorem does not identify the value of $p_c$ for a particular lattice.

An infinite percolation cluster is an open connected component of a graph with infinitely many graph vertices. On a countable translation-invariant lattice, a zero root percolation probability implies that no infinite percolation cluster exists almost surely, by the countable union over all graph vertices. A positive root probability implies almost-sure existence by translation ergodicity of Bernoulli percolation; uniqueness is an additional theorem.

For observed counts $O_{ij}$ and positive expected counts $E_{ij}$ in a contingency table, the statistic is $X^2={\sum }_{ij}(O_{ij}-E_{ij}{)}^2/E_{ij}$. Under common category probabilities or statistical independence, estimate $E_{ij}$ by the product of the row and column totals divided by the grand total. With large expected counts, this gives the test statistic for the Pearson chi-squared test of homogeneity or the Pearson chi-squared test of independence, with asymptotic chi-squared distribution on $(r-1)(c-1)$ degrees of freedom.

An internally finite subset of $\{0,\dots ,c-1\}$ can be represented in Peano arithmetic by a product of distinct prime numbers: multiply the $e$th prime number exactly when $e$ belongs to the subset. Divisibility by that prime recovers membership. Arithmetic induction proves existence of this code even for a definable subset with a nonstandard cutoff in a nonstandard model of Peano arithmetic.

Declustering groups dependent threshold exceedances into extreme-event clusters and retains a representative, commonly the cluster maximum. It aims to make the events used in a peaks-over-threshold method approximately independent. The event rate must be estimated consistently with the retained clusters when computing return levels.

Pull back along loop evaluation $S^1\times \Omega B\to B$ and take the slant product with the circle's fundamental class. This degree-lowering operation kills positive-degree cup products. It commutes with stable cohomology operations such as Steenrod squares and identifies a transgressing base class with its fiber class in the path-loop fibration.

A dilute, vertically mixed particle-laden gravity current with no fluid entrainment, bed erosion or carrier-fluid leakage obeys volume conservation $q=uh=\text{constant}$. With downward settling velocity magnitude $v_s$, an absorbing bed gives $qd{g}^{\prime }/dx=-v_s{g}^{\prime }$. The variable-buoyancy shallow water equations conserve the kinematic momentum flux including hydrostatic pressure,These equations describe a steady interior supplied continuously from upstream, not the moving nose of a finite-volume release. The depth follows one of the depth branches of a steady depositing gravity current.

For slowly depositing particles, $B_x=-v_{s}B/(uh)$ and local adjustment gives $u\sim {\beta }_{\ast }{B}^{1/3}$ with the constant-speed entraining gravity current on a slope coefficient. At leading order $h\sim h_0+E(x-x_0)$ and $B^{1/3}=B_o^{1/3}-[v_s/(3{\beta }_{\ast }E)]\ln [1+E(x-x_0)/h_0]$. Its validity requires settling slow compared with the current and, for small $E$, compared with entrainment.

The integrated budget is ${\partial }_t(g^{\prime }h)+B_x=S_B-v_s{g}^{\prime }$. A steady spatial increase needs a source $S_B$ exceeding deposition, for example sediment resuspension or lateral particle input. In unsteady flow, local depletion can produce $B_x>0$ even without particle addition.