Differentiate the Grahame equation for a planar symmetric monovalent electrolyte with respect to surface potential at fixed reservoir concentration and temperature. The result is the displayed capacitance per unit area. It approaches $\epsilon {\kappa }_D$ in the Debye–Hückel approximation. A compact surface layer can contribute a separate series capacitance; point-ion packing assumptions also fail at sufficiently high near-surface concentration.^[ref]

The reduced chi-squared divides a weighted squared-residual statistic by its degrees of freedom. For an identifiable Gaussian linear model with $N$ independent observations, known correct errors and $p$ fitted coefficients, $\nu =N-p$ and its expectation is one. An observed value close to one is compatible with the model but does not prove it. Unknown systematic errors, dependent residuals or fitted noise scales change this interpretation.

A phigital number is a number represented using golden ratio base, with digits weighted by powers of $\phi =(1+\sqrt{5})/2$. The word describes the representation rather than a separate kind of arithmetic object.

For example, $10.01_{\phi }=\phi +{\phi }^{-2}=2$. Since $100_{\phi }=011_{\phi }$, distinct strings can represent the same value unless a normalization convention is imposed. Finite phigital expansions belong to $\mathbb{Z}[\phi ]$, because ${\phi }^{-1}=\phi -1$, so even negative powers can be reduced to integer combinations of $1,\phi$.

The bony-vertebrate lineage including ray-finned vertebrates and Sarcopterygii; descendants remain members even after extensive modification of their ancestral skeleton.

An Early Silurian jawed vertebrate known from mineralized tooth whorls, illustrating the information carried by isolated hard parts.

The cartilaginous-fish clade, including sharks, ray fish and chimaeras. Extant lack of extensive bone does not demonstrate that their ancestors never possessed it.

A member of a traditionally recognized assemblage of Palaeozoic spiny jawed fishes. Many analyses resolve these taxa as a succession on the Chondrichthyes stem rather than a separate complete clade.

An armoured jawed fish in a traditional Palaeozoic assemblage. Whether the assemblage is monophyletic or a succession of branches affects ancestral state reconstruction.

An Early Devonian stem jawed vertebrate combining an extensively ossified dermal roof with informative braincase anatomy.

A member of an extinct armoured jawless lineage commonly placed on the gnathostome stem outside osteostracans and jawed vertebrates.

A member of an extinct armoured jawless lineage with differentiated pectoral fins and usually recovered close to jawed vertebrates.

Take $\Psi =⟨\hat{\psi }⟩$ normalized so that $|\Psi {|}^2$ is condensate density. The coherent-field expectation of a dilute contact-interacting Bose Hamiltonian gives the grand-potential functional $\int [{\hslash }^2|\nabla \Psi {|}^2/(2m_B)+(V_{\mathrm{ext}}-\mu )|\Psi {|}^2+g_B|\Psi {|}^4/2]d^{3}r$. Its leading microscopic coefficients are therefore those displayed, with positive s-wave scattering length $a_s$ for repulsion. Varying with respect to ${\Psi }^{\ast }$ gives the stationary Gross–Pitaevskii equation. Thermal and quantum fluctuations can renormalize the coefficients; these mean-field identities are not exact universal critical coefficients.

For the stipulated spherical solid–liquid capillarity model, $T_m(r)=T_m(\infty )[1-2\gamma /(r\Delta H_V)]$, where $\gamma$ is the interfacial free energy and $\Delta H_V$ the latent heat density. It follows by matching $g_l-g_s\simeq \Delta H_V[T_m(\infty )-T]/T_m(\infty )$ to $2\gamma /r$. Support contacts, free surfaces, anisotropy, premelting and atomistic structure can modify this relation. A continuum extrapolation to negative melting temperatures is not physical.

Dot the steady linear-drag momentum equation with the velocity. The Coriolis force does no work, so pressure-gradient work balances frictional dissipation. In a strictly geostrophic flow the wind is along isobars and pressure work vanishes. Cross-isobar motion supplies the positive pressure work needed to maintain a dissipative equilibrium.

A weighted geometric series multiplies successive powers by a specified weight sequence. Polynomial weights can be handled by differentiating the geometric series inside its radius of convergence. For $|r|<1$,Apply the operator $rd/dr$ once or twice to $(1-r{)}^{-1}$. At $r=1/2$, the first identity gives $\sum n/2^n=2$. The convergence restriction is essential; substitution at $r=1$ is invalid. Finite weighted sums can instead be obtained by differentiating the finite geometric-sum identity, which has no infinite-series interchange.

If complete bounce distances obey $D_{n+1}=r{D}_n$ with $0\le r<1$, their sum is $D_1/(1-r)$. The first observed distance alone does not identify $r$, and a fixed energy-retention fraction need not imply the same distance-retention fraction when turning-point spring compression varies.

Applying a single-mode squeeze operator to a thermal state gives the displayed principal optical-quadrature variances. Their uncertainty product exceeds the pure-state minimum when $n_{\mathrm{th}}>0$. Sufficient squeezing is needed to put the quiet optical quadrature below the vacuum reference.

Here number-conserving means invariant under a global particle-number quantum phase rotation: the field mean and anomalous averages $⟨\psi \psi ⟩$ vanish. It does not mean a sharp total particle number. All normal correlations are obtained by Wick pairings of the one-body density matrix. Independent thermal bosonic modes satisfy this rule; generic nonvacuum bosonic number states do not. A Slater determinant is the analogous fermionic pairing state.

The Bareiss algorithm is fraction-free Gaussian elimination for exact determinants. For an integer matrix, put $A^{(0)}=A$ and $p_0=1$. At step $k$, choose a nonzero pivot element $p_k=a_{kk}^{(k-1)}$ and update the trailing entries bySet the eliminated entries below the pivot to zero. The determinant identities behind the method make division by $p_{k-1}$ exact, so intermediate entries remain integers. Without row exchanges, the last pivot $a_{nn}^{(n-1)}$ is $\det A$; row exchanges change its sign in the usual way. A zero pivot needs an exchange before continuing.

For $A=\left(\begin{array}{ccc}2& 1& 0\\ 1& 2& 1\\ 0& 1& 2\end{array}\right)$, the first trailing block becomes $\left(\begin{array}{cc}3& 2\\ 2& 4\end{array}\right)$. The last update is $(3\cdot 4-2\cdot 2)/2=4$, giving the determinant without rational intermediate entries. Exact division is different from having no division at all.^[ref]

Let the relevant rest-frame photon-photon pair production energy product be $K={ϵ}_1^{\prime }{ϵ}_2^{\prime }$ for a specified collision-angle prescription. For fixed observed high-energy photon energy, ${ϵ}_1^{\prime }={ϵ}_{1,\mathrm{obs}}/D$, so the target has ${ϵ}_2^{\prime }=DK/{ϵ}_{1,\mathrm{obs}}$. Observing the same internally produced target population supplies a second Doppler factor, giving the displayed result. An external stationary radiation field requires its own angular transformation instead.