If an inner race of radius $R$ rotates at angular velocity $\omega$ and an outer race is stationary, a ball of radius $r$ rolling at both contacts has centre speed $R\omega /2$ and spin $-R\omega /(2r)$. Its centre follows a circle of radius $R+r$. The angular momentum about the common race axis includes both orbital angular momentum and spin.

For the convention of balanced homodyne detection, assume independent signal and coherent reference states and ideal detectors. Write $b=\beta +\delta b$, with $\delta b$ in vacuum. The current difference splits into $2|\beta |X_{\vartheta }$ and $i(a^{†}\delta b-a\delta b^{†})$. Cross terms have zero expectation, and the second term has variance $⟨a^{†}a⟩$. Consequently the exact variance is $4|\beta {|}^2\mathrm{Var}{X}_{\vartheta }+⟨a^{†}a⟩$. Division by $4|\beta {|}^2$ approaches the signal optical quadrature variance when the photon-number correction is negligible.

HYPERNODULATION ABERRANT ROOT FORMATION1, a Lotus japonicus receptor kinase with a shoot role in systemic restriction of nodulation.

TOO MUCH LOVE, a root-localized negative regulator of symbiotic susceptibility targeted by miR2111 in Lotus japonicus.

A stainless steel with an austenitic matrix. Some grades are used in nuclear components, where void swelling and helium embrittlement can limit performance.

A central peak and two sidebands in the resonance fluorescence spectrum of a sufficiently strongly driven two-level system. The atom-field dressed state splitting produces sidebands near ${\omega }_L\pm \sqrt{{\Omega }^2+{\Delta }^2}$, where $\Omega$ is the resonant Rabi frequency. On resonance, changing the drive intensity changes the sideband spacing. Damping and detection conditions determine the resolved line shapes; the elastic coherent contribution must be distinguished from inelastic fluorescence.^[ref]

An experimental test drives an isolated transition with a narrow-band laser, collects fluorescence away from the intense drive beam, and scans a spectral analyzer while varying drive power. Resolving the sidebands requires spectral resolution and coupling frequency sufficient relative to the radiative linewidth. The predicted spectrum concerns transitions of the coupled system rather than simply the time dependence of its population.

For the same initial state, compression ratio and constant-volume heat input, the ideal Atkinson cycle adds net work during the extended expansion and subsequent constant-pressure return. The extra work is the integral of $p_{\mathrm{expansion}}-p_1$ over the added volume, which is positive. Equivalently, concavity of $x^{1/\gamma }$ gives $\gamma (r_p^{1/\gamma }-1)<r_p-1$ when $r_p>1$ and $\gamma >1$, so its efficiency exceeds the corresponding Otto cycle efficiency. The comparison fixes these shared conditions rather than asserting universal superiority under arbitrary engine constraints.

Set the chemical potential equal to the degenerate three-dimensional spin-degenerate Fermi energy, proportional to $n^{2/3}$, and keep the effective coupling and upper cutoff fixed. The asymmetric-cutoff BCS transition temperature scales as $n^{1/3}$ below $n_{\theta }$ and saturates above it. At low density the predicted $k_B{T}_c/\mu$ instead grows as $n^{-1/3}$, eventually invalidating the degenerate and logarithmic approximations. This is a prescribed cutoff model, not a self-consistent treatment of the BCS-BEC crossover.

Many short frames freeze atmospheric image fluctuations. Selecting the sharpest frames, registering them and combining them can improve angular resolution. Selection discards photons and can create a read-noise penalty, so adequate reference-source photon counts are essential. It does not guarantee that every high-order wavefront error disappears.

Calibrated Fourier statistics of short-exposure speckle patterns recover information suppressed by a long seeing-blurred exposure. A power spectrum constrains the modulus, while additional phase-sensitive statistics such as the bispectrum are needed for image reconstruction. The exposure must adequately freeze turbulence and the source must supply enough photons; noise and calibrator mismatch limit the reconstruction.

For three collectors whose measured pair phases are ${\varphi }_{ij}={\varphi }_{ij}^{\mathrm{source}}+{ϵ}_i-{ϵ}_j$, the oriented sum around the triangle cancels the collector-based errors. closure phase retains source phase information, including asymmetry, but does not cancel arbitrary baseline-dependent errors or fill missing spatial-frequency coverage.

If physical ordinate is $Y=M^2$ and abscissa $X=H/M$, slope $S=dY/dX$ and horizontal intercept $X_0$ give $b={\mu }_0/S$ and $a={\mu }_0{X}_0$. Any displayed squared-magnetisation scale must be converted before using S. For example, an ordinate unit $(10^4\mathrm{A}{\mathrm{m}}^{-1}{)}^2$ equals $10^8{\mathrm{A}}^2{\mathrm{m}}^{-2}$, not $10^4$. With a free-energy density, the resulting units are $[a]=\mathrm{H}{\mathrm{m}}^{-1}$ and $[b]=\mathrm{J}\mathrm{m}{\mathrm{A}}^{-4}$. Curvature or differing isotherm slopes limits the accuracy of a constant-coefficient fit.

Big-integer multiplication multiplies arbitrary-precision integers represented by arrays of digits. For $A={\sum }_i{a}_i{B}^i$ and $C={\sum }_j{c}_j{B}^j$, first compute the coefficient sums $s_k={\sum }_{i+j=k}{a}_i{c}_j$, then propagate carries in base $B$. Thus the main step is a convolution followed by carry handling.

The school method needs $O(n^2)$ digit multiplications for two $n$-digit inputs. Karatsuba multiplication splits each input into two halves and uses three recursive products instead of four, giving $O(n^{{\log }_{2}3})$ arithmetic work under the usual digit model. For larger inputs, transform-based methods accelerate the convolution. Which method is fastest depends on input size and implementation overhead.

The synthesized beam is the angular response produced by the sampled and weighted complex visibilities. Its characteristic width is set by the longest well-weighted projected interferometric baselines, of order $\lambda /b_{\max }$, while sidelobes reflect incomplete Fourier coverage. It differs from the individual antenna primary beam, which limits the field of view.

The proper radial particle horizon is the distance travelled by a null signal since the initial boundary, measured on the current time slice. A spherical patch with that radius has a physical diameter twice the radius. Dividing one or the other by an angular diameter distance therefore differs by a factor of two. A question using horizon size should state which linear extent is intended.

Interference of time-reversed multiple-scattering paths can enhance return probability and reduce conductance in a coherent disordered conductor with the appropriate symmetries. Magnetic fields, spin-orbit coupling and dephasing alter or suppress the correction; some symmetry classes exhibit antilocalization. This small-correction regime must be distinguished from strong Anderson localization.

A pair of distinct positive integers in which each is the proper-divisor sum of the other. The integers $220$ and $284$ form an amicable pair.

The electron-nucleus Coulomb-distortion factor multiplying an otherwise plane-wave beta decay spectrum. It depends on daughter charge, lepton charge and lepton energy. Its energy dependence means that a pure fifth-power phase-space rule is not generally exact.

With a momentum-independent allowed nuclear matrix element, negligible nuclear recoil and negligible neutrino mass, the electron spectrum in natural units is $d\Gamma /dp=CF(Z,E)p^2(E_0-E{)}^2$, where $E=\sqrt{p^2+m_e^2}$. Transforming to an energy spectrum requires the Jacobian determinant $dp/dE=E/p$.