Under the energy-closed boundary assumptions of Backus' necessary condition for dynamo action, the magnetic energy equation gives ${\dot{E}}_B\le 2s(t)E_B$ after discarding nonnegative resistive dissipation. Integrating gives an upper bound by the time average of the largest spatial eigenvalue of the rate-of-strain tensor. Thus the field-amplitude exponent is at most its space-time supremum. The exponent of squared energy is twice the field-amplitude exponent.

The Bäcklund expansion produces local differential-polynomial currents. In the coordinates $\xi =m(x+t)/2$, $\eta =m(x-t)/2$, their charges are $\int (P_j+Q_j)dx$ with vanishing boundary flux. Derivative improvements contribute no new charge. The first nontrivial higher current can be written $P=u_{\xi \xi }^2-u_{\xi }^4/4$, $Q=\cos u{u}_{\xi }^2$. Continuing, and exchanging light-cone directions, produces the infinite higher-spin hierarchy characterizing classical integrability.

Mirroring the handedness of a helix reverses its axial translation-rotation coupling coefficients while preserving its pure translation and rotation resistances. With $\mathbf{t}=\cos \theta {\mathbf{e}}_z-\sin \theta {\mathbf{e}}_{\phi }$ for a left-handed helix and the force-on-body convention, $B=C=\ell a({\xi }_{\perp }-{\xi }_{\parallel })\sin \theta \cos \theta$.

Local resistive-force theory for a helix gives $AD-B^2={\ell }^2{a}^2{\xi }_{\parallel }{\xi }_{\perp }$. Positive local drag therefore guarantees a positive-definite axial resistance matrix and positive viscous power loss. The identity also keeps the denominator in counterrotating two-helix balances positive.

If $f$ is odd, its average $⟨f(A\cos \psi ){\sin }^2\psi ⟩$ vanishes under a half-period shift. Hence the averaged amplitude for position-dependent damping is constant at first order. Odd-power damping creates only even harmonics when multiplied by the leading oscillator velocity; these do not resonate with the fundamental harmonic. Higher-order drift is still possible.

Filtering the observations by $1-\varphi B$ gives covariance $A={\sigma }_z^2+(1+{\varphi }^2){\sigma }_w^2$ at zero, $C=-\varphi {\sigma }_w^2$ at lag one and zero elsewhere. Set $\nu =(A+\sqrt{A^2-4C^2})/2$ and $\vartheta =C/\nu$. Then $\nu (1+{\vartheta }^2)=A$, $\nu \vartheta =C$ and $|\vartheta |<1$. Applying $(1+\vartheta B{)}^{-1}$ defines actual white-noise innovations with variance $\nu$, proving an at-most-(1,1) ARMA representation without assuming Gaussianity.

The partial autocorrelation at lag $k$ is the correlation of $X_t$ and $X_{t-k}$ after their linear projections onto the intervening observations are removed. A causal autoregressive model of order $p$ has zero partial correlations after lag $p$. This complements the finite cutoff of the autocorrelation function for a moving-average model.

With radial pressure zero at $\psi =0$, an augmented density supplies $\rho ⟨v_r^2⟩={\int }_0^{\psi }\rho (r,q)dq$ and $\rho ⟨v_{\theta }^2⟩=(2r{)}^{-1}{\int }_0^{\psi }{\partial }_r[r^2\rho (r,q)]dq$. The derivative holds $q$ fixed. Their chain-rule combination satisfies the Spherical Jeans equation, but the physical density alone does not choose an augmented extension.

An upper atmospheric region above the tropopause in which shortwave heating can produce an atmospheric thermal inversion. Ozone on Earth, and methane and aerosols on giant planets, are examples of absorbers that help heat it. Its positive altitude temperature gradient makes it stable against homogeneous-gas thermal convection.

Atomic line cooling removes gas kinetic energy by collisional excitation followed by photon escape. It is strongest where atoms or ions have bound electrons and thermally accessible transitions. If photons are trapped, or if the relevant atoms are fully ionized, the simple optically thin line-cooling rate no longer gives the same loss.

The primordial atomic cooling curve describes hydrogen-helium gas without heavy elements. Near the hydrogen excitation threshold it rises sharply, then shows hydrogen and helium excitation/ionization features. Bound-state losses weaken when both elements become highly ionized. At sufficiently high temperatures thermal bremsstrahlung gives a slowly rising tail. Composition, density convention and ionization assumptions fix the numerical normalization.

For a nonrelativistic ideal gas with $n_{\mathrm{tot}}$ thermal particles per volume, divide its thermal energy $3n_{\mathrm{tot}}{k}_{B}T/2$ by the optically thin volume radiation loss. In the $n_H^2$ cooling convention this becomes $3\chi k_{B}T/(2n_H\Lambda )$, where $\chi =n_{\mathrm{tot}}/n_H$. Density scaling and ionization-dependent particle count must be kept consistent.

A set-theoretic tree of height ${\omega }_2$, levels of size at most ${\aleph }_1$, and no cofinal branch. The Continuum hypothesis supplies one through the minimal-walk tree.

An Aronszajn tree that is a union of countably many tree antichains. Equivalently, it admits a map into a countable set that is injective on each chain.

An Aronszajn tree with no uncountable tree antichain. A normal splitting Suslin tree yields a Suslin line by a lexicographic ordering followed by Dedekind completion.

Given coherent coinfinite injections into omega, use all restrictions $e_{\alpha }\upharpoonright \beta$, where $\beta \le \alpha <{\omega }_1$, ordered by proper extension. Every level is countable because its nodes differ finitely from a fixed function on a countable domain. An uncountable chain in a partial order would have unbounded domain heights, and its union would inject ${\omega }_1$ into $\omega$, which is impossible.

The harmonic mean of a nonempty finite list of positive real numbers is the multiplicative inverse of the arithmetic mean of their inverses:For two inputs, $H=2x_1{x}_2/(x_1+x_2)$. Replacing each input by $H$ preserves the sum of inverses, making it useful when a fixed amount divided by a rate determines a cost or duration. This positive-input definition avoids division by zero; negative inputs need not retain the usual averaging or ordering properties. ^[ref].

Travel two legs of equal length $d$ at positive speeds $v_1,v_2$, with no stops. Their travel times are $d/v_1$ and $d/v_2$, so the average speed isFor two $60\mathrm{km}$ legs at $30$ and $60km/h$, the times are two hours and one hour. The average speed is $120/3=40km/h$. The same speeds maintained for one hour each instead cover $90\mathrm{km}$ in two hours, giving the arithmetic mean $45km/h$. A slower equal-distance leg occupies more time, so the data being averaged must include what is held equal. This is total distance traveled divided by elapsed time; a return journey can have zero average velocity while keeping a positive average speed. ^[ref].

The arithmetic-geometric mean inequality applied to the inverses gives $H\le G$; together with $G\le A$ it yields the comparison in geometric mean. For two inputs the arithmetic–harmonic gap has the exact formFor $30$ and $60$ the gap is five. With one positive input held fixed and another approaching zero, the harmonic mean approaches zero too: a very slow fixed-distance leg dominates the travel time. Unequal leg lengths instead require a weighted harmonic mean.

The inverse germs of $w^3-3w$ over $\mathbb{C}$ form the connected surface $\mathbb{C}\setminus \{-1,1\}$. Its projection is surjective and locally biholomorphic, but the fibres have three points generically and one point at each critical value $\pm 2$, so it is not an evenly covered topological covering.

For a weak force depending only on position, the amplitude average is a periodic total derivative and vanishes. The leading harmonic amplitude is constant, while the phase may drift. The method of multiple scales agrees with the exact conservation of energy for bounded orbits in the effective potential. A cubic position force on the right side of the oscillator equation lowers the frequency at first order when its coefficient is positive.

When the weak force depends only on velocity, the phase average is a periodic total derivative and vanishes. The leading phase is constant, while the amplitude evolves according to the remaining average. The sign of that drift determines damping or growth and must be checked from the force, rather than inferred from its velocity dependence alone.