The positive separation parameter in $-{\nabla }^2\varphi =B_g^2\varphi$ for a specified fundamental spatial mode and boundary conditions. The corresponding material parameter must match it for a source-free steady solution.

Differentiating the screened-channel velocity gives $x_{\max }=d-{\kappa }^{-1}\mathrm{arsinh}[(\cosh \kappa d-1)/(\kappa d)]$. For $\kappa d\gg 1$, $x_{\max }\simeq \log (\kappa d)/\kappa$. The outer flow declines almost linearly to the passive no-slip wall rather than forming a symmetric plug.

Zero wall charge constrains the jump in normal electric displacement, while fixing a wall potential is a separate electrical condition. Taking the external normal field to vanish gives a zero normal derivative at an ideal uncharged wall. A grounded wall may carry induced charge even though its imposed potential is zero.

Growth of planetesimal random orbital motions through gravitational encounters. The word viscous does not denote molecular fluid viscosity here. Stirring competes with gas drag, collisions and dynamical friction in setting the encounter speeds that control gravitational focusing.

The region of a disk from which a planetary embryo can efficiently accrete solids. Its width depends on gravitational encounters and the surrounding orbital population. Approximating that width by a multiple of the Hill radius gives a useful local isolation mass.

For a short applied perturbation $V(t)$ in a Quantum Hamiltonian, define $K=\int V(t)dt$. Since $V$ has energy units, $K$ has action units. In the ideal impulsive limit, during which the unperturbed evolution is negligible, the quantum state changes by $\exp (-iK/\hslash )$ if the perturbation commutes with itself at different times. More general pulses require time ordering.

Writing the impulse as $V(t)=K\delta (t-t_0)$ makes the dimensions explicit: the Dirac delta function in time has inverse-time units. For an off-diagonal two-state impulse with coupling $K_{21}=k$, the transition probability is ${\sin }^2(|k|/\hslash )$. The ratio $|k|/\hslash$, rather than $|k|$ alone, is the dimensionless pulse strength. Integrating across the full pulse gives its full area; an artificial factor of one-half must not be introduced by treating the pulse as an endpoint without specifying a regularization.

Adjustment of ocean and sediment carbonate chemistry through dissolution or burial of carbonate minerals. Following a carbon-dioxide perturbation, additional carbonate dissolution can increase alkalinity and permit further ocean uptake. The adjustment is much slower than surface gas exchange, operating over millennial timescales.

For a fractional feature amplitude $d$, equal independent in- and out-of-eclipse samples with $N_{\gamma }$ stellar photons each have fractional difference noise approximately $\sqrt{2/N_{\gamma }}$. The ideal significance is $d\sqrt{N_{\gamma }/2}$. Correlated systematics, stellar variability and unequal exposure budgets change this result.

The distribution of received radiative flux or emitted luminosity across frequency or photon energy. Plotting $\nu F_{\nu }$ against $\log \nu$ makes the area equal to the integrated radiative flux, since $F_{\nu }d\nu =\nu F_{\nu }d\log \nu$. A convention such as $F_{\nu }$, $\nu F_{\nu }$ or $F_{\lambda }$ must be specified before comparing spectral slopes.

An absent otherwise-fast gamma transition can support a spin assignment that forbids it. It is not an absolute selection rule: a nuclear reduced matrix element may be small even when angular momentum and parity allow the mode. Meaningful limits require detection thresholds, branching normalization and competing-channel information. Assuming ordinary unhindered strengths can yield a useful conventional assignment, but that assumption must be distinguished from strict observed-line constraints.

The motion of particles occupying the momentum sea of a Fermi gas, especially nucleons in an atomic nucleus. It persists at zero temperature and broadens scattering kinematics relative to a stationary-nucleon approximation.

For a product of bosonic number states, direct and exchange contractions correctly count distinct modes. They overcount repeated-mode terms because $⟨a^{†2}{a}^2⟩=N(N-1)$, while Wick theorem would give $2N^2$. Subtracting the excess yields the displayed exact two-point correction. At higher order the occupation decrements carry falling factorials. This distinction is essential when comparing a fixed-number condensate with independent thermal modes.

For a number-conserving Gaussian pairing state, six permutations give this expression. Each transposition has sign $ϵ$, positive for bosons and negative for fermions; the two three-cycles have positive sign for either statistics. Equivalently it is the permanent or determinant of the three-by-three one-body density matrix. For bosonic number states, repeated-mode factorial corrections must be included instead of assuming that Wick theorem.

A single-component chaotic Gaussian bosonic field has normalized coincident normally ordered correlation $n!$: every pairing contributes the same product of densities. A same-component fermionic field instead has zero coincident correlations for $n\ge 2$ by exclusion. These are state-dependent statements: a coherent bosonic field has factor one, and a single-mode number state has $N^{\underline{n}}/N^n$. Ratios to distant unnormalized correlations additionally require the compared densities to agree and distant correlations to factorize.

For two identical ideal coils with self-inductance $L$ and signed mutual inductance $M$, the magnetic energy isThe symmetric matrix has eigenvectors $(1,1)$ and $(1,-1)$ with eigenvalues $L+M$ and $L-M$. Strict positivity for every nonzero current vector therefore requires $L>|M|$; nonnegative energy alone permits $L=|M|$. A singular limiting matrix has a zero-energy current combination, so formulas dividing by its eigenvalues require separate treatment. Reversing one coil's winding convention changes the sign of $M$ and exchanges the two combinations.

An upper mass $3m$ and lower mass $m$ joined by equal light strings of length $\ell$ have small-angle mass and stiffness matrices $m{\ell }^2\left(\begin{array}{cc}4& 1\\ 1& 1\end{array}\right)$ and $mg\ell \left(\begin{array}{cc}4& 0\\ 0& 1\end{array}\right)$. The squared frequencies are $2g/(3\ell )$ and $2g/\ell$, with angular eigenvectors $(1,2)$ and $(1,-2)$ respectively. Equal-angle release decomposes with coefficients $3\alpha /4$ and $\alpha /4$. The angles are measured from vertical, not between strings.

For $f(w)=-aw+b{w}^n$, with $a,b>0$ and integer $n>1$, a positive localized static pulse has $C^{n-1}=a(n+1)/(2b)$ and $B=(n-1)\sqrt{a}/2$. Writing the same profile as $[A{\cosh }^2(Bx){]}^{1/(1-n)}$ instead gives $A=2b/[a(n+1)]$. These are different amplitude conventions for the same solution. A Lorentz boost moves the pulse at speed $\tanh \beta$, preserves its peak height and contracts its width by $1/\cosh \beta$. Existence alone does not imply dynamical stability.

The only diagonalizable nilpotent matrix is the zero matrix. Indeed, if $A^p=0$ and $Av=\lambda v$ with $v\ne 0$, then ${\lambda }^p=0$, so $\lambda =0$. A basis of eigenvectors would make every diagonal entry zero and therefore $A=0$. Nonzero nilpotent matrices require nontrivial Jordan blocks, even if their characteristic polynomial has only one distinct root.

The binomial theorem for the commuting matrices $I,N$ proves the formula for nonnegative integers: every term containing $N^2$ vanishes. Since $(I+N)(I-N)=I$, the matrix inverse is $I-N$; applying the same argument to its powers gives the negative integers as well. The same identity shows that $I+N$ is invertible even when the nilpotent matrix $N$ is nonzero. In dimension two, the example $N=\left(\begin{array}{cc}0& p\\ 0& 0\end{array}\right)$ describes repeated shear mappings, whose displacement parameters add under composition.

The nilpotency index is the least positive power at which a nilpotent matrix becomes zero.