A powder-diffraction camera with photographic film wrapped around a cylindrical specimen axis. Diffraction cones intersect the film in paired curved lines; angular position along the cylindrical film differs from the $R=L\tan 2\theta$ relation for a forward flat plate. Back reflections can also be recorded.

In variable projection, a prediction has the form $\mathbf{G}(\mathbf{q})\mathbf{m}$, nonlinear in $\mathbf{q}$ but linear in $\mathbf{m}$. For each trial $\mathbf{q}$, solve the inner weighted least squares problem for $\mathbf{m}$, then optimize the resulting residual over $\mathbf{q}$. This reduces the nonlinear dimension without pretending that the whole inverse problem is linear.

When frequency and phase are known, sinusoidal amplitude and offset fitting is a two-parameter linear regression. For equal known errors $\sigma$, profiling out the offset gives $K=\sum (s_i-\bar{s})(y_i-\bar{y})/\sum (s_i-\bar{s}{)}^2$ and $\mathrm{Var}(K)={\sigma }^2/\sum (s_i-\bar{s}{)}^2$. Identical phases cannot separate amplitude from offset. Unknown phase instead requires independent sine and cosine coefficients.

Independent observations with known errors add a positive-semidefinite Fisher information matrix $G$ to the existing information $F$. Writing $F+G=F^{1/2}(I+F^{-1/2}G{F}^{-1/2})F^{1/2}$ shows its inverse covariance cannot increase in any parameter direction. Improvement can be zero along an uninformative direction. This statement does not assume that data-dependent residual variance estimates or model systematic errors must decrease.

For a proposed gamma-transition graph, fix the ground-state energy and fit the remaining level energies with a linear incidence matrix. Loops give consistency constraints and correlated excitation-energy errors. Different transition graphs can have the same minimized residuals if they rearrange the same energy-sum identities; a small chi-squared is not then a unique level assignment.

For a regular implicit surface $f(x,y,z)=0$ with $f_z\ne 0$, the implicit function theorem gives a local graph $z=h(x,y)$. Its upward vector area element is $(-h_x,-h_y,1)dxdy$. Differentiating $f(x,y,h(x,y))=0$ yields $h_x=-f_x/f_z$, $h_y=-f_y/f_z$, proving the formula. Division by $f_z$ makes the final component positive independently of the sign of the original gradient. At a point where $f_z=0$, another projection or parametrization is needed.

The thermodynamic identity $(\partial U/\partial V{)}_T=T(\partial P/\partial T{)}_V-P$ fixes the volume-dependent term $-a{N}^2/V$ but leaves an arbitrary function of temperature. Classical monatomic mean-field modelling adds $3N{k}_{B}T/2$, giving $U=3N{k}_{B}T/2-a{N}^2/V+C$.

A mean-field fluid model with excluded-volume and attractive-interaction parameters, described by the Van der Waals equation. Its pressure law alone does not fix every thermal property: the caloric closure of the van der Waals equation supplies the additional internal-energy assumption.

A first-forbidden unique beta decay has $|\Delta J|=2$ with a parity change. Its leading lepton shape factor contains two additional powers of momentum relative to an allowed beta decay. In the massless, no-Coulomb limit, ${\int }_0^Q{E}^2(Q-E{)}^2[E^2+(Q-E{)}^2]dE=2Q^7/105$. Thus its microscopic reduced nuclear matrix element multiplies a seventh-power phase-space factor, rather than the fifth power in the Sargent rule. A fifth-power comparison across different forbiddenness classes instead extracts convention-dependent effective strengths.

In a paired nuclear shell model with the post-28 orbital order $2p_{3/2},1f_{5/2},2p_{1/2}$, proton counts 32 and 38 fill $2p_{3/2}$ and $1f_{5/2}$, so the odd 39th proton occupies $2p_{1/2}$. With paired valence neutrons this gives a $1/2^{-}$ yttrium-91 ground-state prediction. The daughter has paired 40 protons and one neutron beyond the 50-neutron closure in $2d_{5/2}$, giving $5/2^{+}$. The inferred ground-state beta transition is therefore first-forbidden unique. This is a stated paired-shell model deduction; an endpoint diagram alone does not determine the spin assignment or an evaluated isotope mass.

When $s^2\gg 4mg\ell I_1\cos \theta$, the two nonvertical steady-precession roots are approximately $mg\ell /s$ and $s/(I_1\cos \theta )-mg\ell /s$. These asymptotic descriptions do not apply at a singular denominator without taking the original quadratic limit.

For a uniform circular stellar disc, the first zero of $J_1$ at $x=3.831706\dots$ gives $Y_0=1.21967\lambda /\alpha$ when $\alpha$ is full angular diameter. In contrast a one-dimensional top-hat angular profile of full width $\alpha$ has a sinc envelope with $Y_0=\lambda /\alpha$. Diameter estimates therefore require a brightness-profile convention. A binary additionally multiplies the envelope by its separation-dependent interference factor.

If resisting force per unit moving length is $\beta v$, the total resistance is $\beta l\dot{y}$ and $\ddot{y}+\Gamma \dot{y}+{\omega }_0^{2}y=0$, where $\Gamma =\beta /(\rho A)$. For $\Gamma <2{\omega }_0$, an initial velocity $v_0$ from zero displacement gives $y=(v_0/{\omega }_d)e^{-\Gamma t/2}\sin {\omega }_{d}t$, with ${\omega }_d^2={\omega }_0^2-{\Gamma }^2/4$. Its quality parameter is $Q={\omega }_0/\Gamma$, and the amplitude envelope decays at half the energy-decay exponent. The effective damping model is an assumption, not a derivation of detailed viscous flow near the wall.

A rotating gravitational perturbation drives an epicyclic disk response where $m(\Omega -{\Omega }_p)=\pm \kappa$. The launched density waves carry angular momentum. For a planet, inner and outer waves exert torques of opposite signs, and their imbalance contributes to migration.

The orbital torque associated with disk material near the planet's corotation radius. Vortensity and entropy gradients matter because gas exchanged across the orbit carries different conserved properties. Horseshoe turns can make the response nonlinear even while the planet remains in the low-mass migration regime.

A location where the net disk torque on a planet vanishes and, for a stable trap, restoring torque changes sign across the location. A torque zero alone does not guarantee stable trapping as the disk or planet evolves.

For a fixed-mass planet in a circular orbit of radius $a$, let $q=M_p/M_{\star }$ be the planet-to-star mass ratio, $\Sigma$ the local surface density, and $h=H/a$ the disk aspect ratio defined by disk scale height $H$. The local angular velocity is $\Omega =\sqrt{G{M}_{\star }/a^3}$. Suppose the torque on the planet iswhere the dimensionless coefficient $C$ can have either sign and depends on the disk model.

The planet's orbital angular momentum is $L=M_p{a}^2\Omega =M_p\sqrt{G{M}_{\star }a}$. Its derivative gives $\dot{L}=(M_{p}a\Omega /2)\dot{a}=\Gamma$, hencePositive $\Gamma$ in this convention produces outward planetary migration; negative $\Gamma$ produces inward planetary migration. The torque exerted on the disk has the opposite sign. The radial evolution timescale is half $L/|\Gamma |$, because $L\propto \sqrt{a}$. This expression assumes the low-mass Type I migration regime and the stated torque law, rather than a universal direction or rate.

The scale $M_{\mathrm{th}}=h^3{M}_{\star }$ makes the Bondi radius comparable with the disk scale height. It organizes when planet-induced disturbances become nonlinear; it is not identical to an exact universal gap-opening mass.

If the differential Zeeman energy $|w|=|{\mu }_{0}B(g_1-g_2)|$ greatly exceeds $|\Delta |$, the two mixed levels are $-\Delta /4\pm |w|/2+O({\Delta }^2/|w|)$ and approach opposite-spin product states. Four energies are generically distinct. Equal signed moments have $w=0$, so that block remains singlet/triplet rather than being unmixed by a stronger field.

If $g_2=-g_1$ in a shared magneton convention, the Zeeman energy of both parallel-spin states cancels. Their energy stays $\Delta /4$, with exact degeneracy two for every field. Opposite charges alone do not enforce this equality because the masses and magnetic factors may differ.