Two identical circuits with self-inductance $L$, capacitance $C$ and mutual inductance $\mathcal{M}$ have same-sign and opposite-sign charge modes with ${\omega }_{\pm }^2=1/[C(L\pm \mathcal{M})]$. The inductance matrix acts as the mass matrix; positivity requires $|\mathcal{M}|<L$.

For a conservative elastic body, the stiffness operator is a self-adjoint operator under compatible boundary conditions. The dynamical operator ${\rho }^{-1}\hat{H}$ is self-adjoint in the mass inner product $⟨\mathbf{u},\mathbf{v}{⟩}_{\rho }=\int \rho {\mathbf{u}}^{\ast }\cdot \mathbf{v}dV$. Choosing mass-normalized normal modes converts the field equation into independent forced oscillators. If the normal modes have mass norm $N_{\alpha }$ instead, each projected force must be divided by $N_{\alpha }$.

Three identical repelling charges in a one-dimensional harmonic trap have a translation mode $(1,1,1)$, a breathing mode $(1,0,-1)$, and a symmetric middle-against-outer mode $(1,-2,1)$. Their frequencies are $\omega$, $\sqrt{3}\omega$, and $\sqrt{29/5}\omega$. The translation frequency is unchanged by internal Coulomb forces. Linearizing each pair potential produces a positive spring coefficient $2q^2/(4\pi {ϵ}_0{d}^3)$.

An isotropic planar harmonic oscillator of positive charge in a perpendicular magnetic field has two circular normal modes of opposite handedness. With ${\omega }_c=qB/m>0$ and ${\omega }_0^2=k/m$, the signed angular frequencies have the displayed form for trajectories $x+iy\propto e^{i\omega t}$. A static electric field shifts the equilibrium without changing these frequencies. The clockwise mode has the larger frequency magnitude.

For free-end masses $m,\alpha m,m$ joined by two identical springs, the displayed squared frequencies have displacement vectors $(1,1,1)$, $(1,0,-1)$ and $(1,-2/\alpha ,1)$. Distinct eigenvectors are orthogonal in the mass inner product. The zero mode carries translation, while the middle mass does not move in the antisymmetric internal mode. Fixed end springs would give a different spectrum.

The fundamental mode is the lowest-frequency oscillatory mode for the specified geometry, constraints and, in a propagation problem, horizontal wavenumber. Its frequency need not be a universal constant of the material. In a waveguide it forms the lowest dispersion branch; rigid zero-frequency motions, when present, must be distinguished from oscillations.

An overtone is an oscillatory mode above the fundamental mode in the same mode family. Waveguide branches are compared at fixed horizontal wavenumber. Their frequencies need not be integer multiples of the fundamental frequency. In a scalar layered guide, additional vertical nodes distinguish successive overtones.

A normality assumption specifies a normal distribution for the relevant observations, errors, or latent effects. These are different assumptions: normal latent random effects do not imply that the observed response is normal. Estimated regression residuals or fitted random effects are not independent copies of the underlying errors or effects.

Commuting input field factors permit the response kernel to be symmetrized under interchange of input index-frequency pairs. For example ${\chi }_{ijk}^{(2)}({\omega }_s;{\omega }_1,{\omega }_2)={\chi }_{ikj}^{(2)}({\omega }_s;{\omega }_2,{\omega }_1)$. This does not authorize exchanging indices while keeping unequal frequency arguments in their old slots, nor exchanging the output index without further assumptions.

Far from relevant resonances, when frequency dispersion of the nonlinear response can be neglected, a susceptibility can acquire full permutation symmetry among spatial indices. For second order this reduces a general symmetric-input tensor's 18 components to 10 before material point-group reductions. Mere transparency does not establish this approximation. The name refers to D. A. Kleinman.^[ref]

An isotropic fourth-rank polar tensor has the displayed form. If its three input slots are fully interchangeable at the same relevant frequency arguments, then $a=b=c$, giving ${\chi }_{xxxx}=3{\chi }_{xxyy}$. For mixed-sign or unequal frequencies, intrinsic permutation exchanges the arguments too; it need not make all coefficients at one fixed ordering equal.

A thermogenic tissue of many placental mammals. It should not be assumed to be the ancestral heat-producing mechanism in every synapsid.

The bitwise exclusive or of a list of nonnegative integers. For heap sizes in ordinary Nim, a zero nim-sum characterizes a losing position under the normal-play convention.

A Nim heap of size $n$ contains $n$ counters and permits removing any positive number, leaving a heap of any size $0\le k<n$. Under the normal-play convention, its Grundy number is $n$: the reachable values are exactly $0,\dots ,n-1$, whose mex is $n$.

For several heaps, a turn changes exactly one heap. Their combined value is the Nim-sum of the heap sizes, so heaps of sizes three and three form a losing position even though either heap alone is winning. The ability to remove arbitrarily many counters is part of the rule; a bounded-removal subtraction game has different values.

The ordered $L{1}_2$ phase based on $\mathrm{N}{\mathrm{i}}_3\mathrm{Al}$ in many nickel-based superalloys. A matrix translation $a⟨110⟩/2$ disrupts its order and can create an antiphase boundary.

A turbine blade cast as one crystal to remove high-temperature grain boundary sliding and grain-boundary cavitation pathways. This does not eliminate dislocations or all sources of creep.

A sinusoidal surface-density component ${\Sigma }_k\cos (kx)$ on $y=0$, with $k>0$ and translational invariance in the remaining direction, produces the displayed gravitational potential. Laplace's equation away from the sheet separates into a cosine in $x$ and a decaying exponential in $|y|$. Continuity across the sheet excludes a potential jump, while integrating Poisson equation for Newtonian gravity across it gives $[{\Phi }_y{]}_{0^{-}}^{0^{+}}=4\pi G{\Sigma }_k\cos (kx)$. For a potential $C\cos (kx)e^{-k|y|}$ that derivative jump is $-2kC\cos (kx)$, giving $C=-2\pi G{\Sigma }_k/k$.

More generally a specified jump $J\cos (kx)$ gives potential $-J\cos (kx)e^{-k|y|}/(2k)$. Each spatial Fourier mode decays on the length scale $1/k$, so fine density structure has a shorter-range field. The sinusoid changes sign and is naturally interpreted as a density perturbation component. General sheet distributions are built by superposing their Fourier modes; decay alone does not restrict a harmonic potential to a single mode.

The mass-shedding limit is the spin at which equatorial surface matter can no longer remain attached in equilibrium. Newtonian gravity gives a Keplerian frequency scale at the surface. Rotation-induced deformation, general relativity and the equation of state change the numerical bound. A magnetic-equilibrium formula is not valid indefinitely as the exterior magnetosphere shrinks inside the star.

A neutron emitted following a radioactive transformation of a precursor, rather than promptly in the original nuclear fission. The waiting time reflects the precursor lifetime. It is distinct from the later detection of a prompt neutron after scattering or moderation.

A neutron released on the short timescale associated with the fission event, rather than after a precursor radioactive transformation. Prompt describes emission timing, rather than the neutron speed after subsequent scattering.