Suppose a three-dimensional Debye model is normalized to $3N_m$ vibrational modes in volume $V$. Its low-temperature heat-capacity coefficient is the one displayed, and its spherical wavevector cutoff obeys $k_D^3=6{\pi }^2{N}_m/V$. The Debye temperature is ${\Theta }_D=\hslash v_D{k}_D/k_B$. At fixed measured $\beta$ and volume, changing the counting convention by $N_m\mapsto c{N}_m$ raises both ${\Theta }_D$ and $k_D$ by $c^{1/3}$; the inferred average sound speed is unchanged.

For a mole of formula units containing $r$ atoms each, a conventional full atomic-mode count uses $N_m=r{N}_A$ and tends to lattice heat capacity $3rR$ at high temperature. A reduced convention using $N_m=N_A$ instead has three effective modes per formula unit. These are different continuum normalizations, not a claim that the real crystal has only three modes per formula unit. A real multi-atom primitive cell has three acoustic branches and additional optical branches; a Debye sphere preserving the full atomic count is an effective representation of that spectrum, rather than its literal first Brillouin zone.

The effective isotropic acoustic speed used to replace separate branches in the Debye model. With atomic number density $n$, counting three branches gives $k_D=(6{\pi }^{2}n{)}^{1/3}$ and ${\omega }_D=v_D{k}_D=k_B{\theta }_D/\hslash$. Matching the low-frequency density of states for one longitudinal and two transverse branches gives the displayed inverse-cube average. A Debye temperature alone does not determine both individual sound speeds.

The 1964 report established quantum interference in superconducting tunneling. It is an experimental counterpart to the phase-sensitive current predicted for coupled superconductors.^[ref]

A diffraction reflection enabled by a larger chemical or structural repeat than that of a reference parent structure. In ordered $L{1}_2$ $A_{3}B$, sites $(0,0,0)$ and the three face centres give $F=f_B+f_A[(-1{)}^{h+k}+(-1{)}^{h+l}+(-1{)}^{k+l}]$. Equal-parity indices give $f_B+3f_A$; mixed parity gives $f_B-f_A$. The latter vanish when the two scattering factors are equal.

The scattering amplitude assigned to one atom or ion as a function of momentum transfer, normalized relative to a stated scattering convention. X-ray and electron factors are different; multiplying by basis phase factors produces the total unit-cell amplitude.

With $e^{i\omega t}$ phasors, critical damping gives $\Theta (\omega )=F_0/({\omega }_0+i\omega {)}^2$, amplitude $F_0/({\omega }_0^2+{\omega }^2)$ and phase $-2\arctan (\omega /{\omega }_0)$. The response strongly suppresses alternating currents well above the mechanical frequency.

A transition from effective nucleate boiling toward impaired surface wetting and substantially poorer heat removal in the relevant boiling regime. It is a concern when a heated fuel surface can become much hotter at an otherwise similar imposed heat load.

A disk-gas trajectory that turns from one side of a planet's orbital radius to the other in the corotating frame. Its angular-momentum exchange contributes to the corotation torque; repeated libration without restoring gradients can saturate that contribution.

An entropy contrast carried by gas executing a horseshoe turn creates a density asymmetry around an embedded planet. The resulting torque can oppose the usual negative wave torque. Its persistence depends on thermal transport and the replenishment of the background entropy gradient.

Repeated mixing or libration in an isolated corotation region can erase the gradient that drives its torque. Viscosity and thermal diffusion can help restore relevant gradients; excessive diffusion also changes the response. Saturation concerns the sustained torque, not cessation of all disk waves.

A network layer aggregating nearby activations, for example by a maximum or mean. Pooling can reduce spatial size but does not by itself guarantee arbitrary translation invariance.

A former southern supercontinent. Its changing connections constrain historical biogeography, without implying that every shared lineage originated by vicariance.

For constant supply, negligible seed mass and fixed emitting radius, $M=\dot{M}t$ and $L_{\mathrm{acc}}\propto t$. Thermal emission with $L=4\pi r_0^2{\sigma }_{\mathrm{SB}}{T}_{\mathrm{eff}}^4$ therefore gives the quarter-power law. It describes an effective surface temperature. A homologous ideal-gas interior at the same fixed radius instead has virial temperature proportional to mass and hence to time; its absolute temperature requires a structural model.

Electronic compensation of the time-dependent reference geometric delay multiplies each complex visibility by an opposite reference phase. A stationary point source at the phase centre then has constant corrected phase despite Earth rotation. antenna pointing alone does not perform this correction. Off-axis sources retain the relative fringe evolution that encodes their angular offsets.

With constant surface brightness over a centred angular rectangle of widths ${\Theta }_x,{\Theta }_y$, and a nearly uniform primary beam over that rectangle, the Fourier transform separates into two one-dimensional integrals. The result is $V(u,v)=S_{\nu }\mathrm{sinc}(\pi {\Theta }_{x}u)\mathrm{sinc}(\pi {\Theta }_{y}v)$, where $\mathrm{sinc}x=\sin x/x$. The zero-spacing value is the integrated spectral flux density, while the first axial null is at $|u|=1/{\Theta }_x$. Long baselines therefore do not measure the total flux of a resolved rectangle.

A centred uniform source of angular width $\Delta \alpha$ has normalized visibility $V/V(0)=\mathrm{sinc}(\pi b_p\Delta \alpha /\lambda )$, with $b_p$ the projected baseline along that angular coordinate and $\mathrm{sinc}x=\sin x/x$. The result is the integral of the point-source complex phase over source brightness. Its first null occurs at width $\lambda /|b_p|$; negative values are phase reversals rather than negative brightness.

If $D=[A,B]$ commutes with $A$, then $[A^n,B]=nD{A}^{n-1}$. A convergent power series therefore gives $[F(A),B]=D{F}^{\prime }(A)$ on a common domain where termwise operations are justified. For unbounded linear operators, the domain condition is essential.

For linear operators on a common invariant domain, $[A,BC]=[A,B]C+B[A,C]$. The identity follows by adding and subtracting $BAC$ in the commutator.

The single-site average in molecular field $z_{c}JS$ is $S=\coth x-1/x$, where $x=\beta z_{c}JS$. Thus $\tau x=\coth x-1/x$ with $\tau =k_{B}T/(z_{c}J)$. The small-$x$ slope $1/3$ gives $k_B{T}_c^{\mathrm{MF}}=z_{c}J/3$. On a square lattice $z_c=4$. This finite mean-field transition does not imply actual positive-temperature magnetization for the ideal two-dimensional short-range isotropic model, where the Mermin-Wagner theorem applies.

The Chudnovsky series is the rapidly converging identityThe absolute ratio of successive summands tends to $1728/640320^3$, approximately $6.58\times 10^{-15}$, giving about fourteen new decimal digits per term asymptotically. Keeping only $k=0$ gives $\pi \approx 640320^{3/2}/(12\cdot 13591409)$. The factorial ratio, not a direct geometric approximation of a circle, controls the numerical convergence.^[ref]