Expected photon absorption by a receptor class. A simple relative model is $Q_i=\int L(\lambda )R(\lambda )S_i(\lambda )d\lambda$, with illumination $L$, object reflectance $R$ and effective spectral sensitivity $S_i$. Absolute catch also requires geometric and temporal factors.

Variability in a sensory receptor signal that limits discrimination. A difference in average receptor quantum catch does not alone establish a perceptually discriminable difference.

A rough contact may transmit both normal and tangential impulses. Conservation of linear and angular momentum fixes totals but does not specify their division between translation and rotation. Predicting outgoing states requires constitutive data such as normal restitution and a friction or tangential restitution law. Rough contact is not synonymous with adhesive contact.

Insert coherent-state completeness between short factors of $e^{-\beta (H-\mu N)}$. The adjacent bra and ket labels determine the normal-ordered symbol $H_N({\bar{\psi }}_{j+1},{\psi }_j)$, while overlaps generate the derivative term. Bosonic labels are complex and periodic in imaginary time; fermionic labels are independent Grassmann variables and antiperiodic. The continuum shorthand is defined by that temporal ordering and measure, not an arbitrary substitution of commuting fields for operators.^[ref]

Elastic deformation required to match differing natural lattice spacings across a coherent crystal interface. Its energy can affect precipitate shape and loss of coherency.

A liquid-water particle suspended in a cloud. Number, size distribution and liquid-water content are separate properties; increasing number at fixed total water shifts the characteristic radius downward.

Radiative cooling can promote condensation and turbulence; net heating can promote evaporation or sublimation. Cloud altitude alone does not determine lifetime, because optical thickness, environmental radiation, entrainment and moisture supply all matter.

For positive coordinate intercepts $A=(a,0,0)$, $B=(0,b,0)$ and $C=(0,0,c)$, their plane equation is $\mathbf{n}\cdot \mathbf{r}=1$. The closest point on a plane to the origin is $\mathbf{d}=\mathbf{n}/\Vert \mathbf{n}{\Vert }^2$, with distance $1/\Vert \mathbf{n}\Vert$. Its barycentric coordinate system coefficients are proportional to $a^{-2},b^{-2},c^{-2}$ and sum to one.

The projection is also the orthocenter of triangle $ABC$. Indeed $\mathbf{a}\cdot (\mathbf{b}-\mathbf{c})=0$, and $\mathbf{d}\cdot \mathbf{b}=\mathbf{d}\cdot \mathbf{c}$ because each equals $1/\Vert \mathbf{n}{\Vert }^2$. Hence $(\mathbf{a}-\mathbf{d})\cdot (\mathbf{b}-\mathbf{c})=0$. Cycling the vertices gives all three altitude directions. This connects an ambient-space orthogonal projection with intrinsic triangle geometry.

The ground-hyperfine magnetic-dipole transition between $F=1,m_F=0$ and $F=2,m_F=0$ has vanishing first-order Zeeman shift near zero field. It obeys $\Delta F=1,\Delta m_F=0$. Its residual quadratic shift follows from the Breit-Rabi formula, rather than from assigning both states permanently zero magnetic dipole moment.

For the two $m=0$ branches of the Breit-Rabi formula, their splitting is $\sqrt{{\Delta }_{\mathrm{hf}}^2+[(g_J-g_I){\mu }_{B}B{]}^2}$. Expanding at small field gives the quadratic frequency shift displayed above. Vanishing linear frequency sensitivity at zero field therefore does not eliminate field-gradient dephasing at a finite bias field.

The diagonal coherent state representation writes a single-mode density matrix in the form shown. Its weight may be a generalized distribution rather than an ordinary nonnegative probability density. A nonclassical state of light has no nonnegative measure of this form. This use of $P$ differs from the Wigner distribution; a nonnegative Wigner function does not by itself certify classicality.

A pro-crossover E3 ubiquitin ligase whose dosage can influence crossover number.

Its abundance and distribution help determine which recombination intermediates mature into class I crossovers. Consequently, a change in crossover number can reflect altered designation or maturation rather than an identical change in the number of initial DNA breaks. ^[ref]

A DNA sequence recognized by an activated hormone-regulated transcription factor. It connects ligand-dependent regulator activity to a particular target gene.

Let a planar Lagrangian have the form $L=a(r){\dot{r}}^2+b(r){\dot{\theta }}^2+c(r)$, with smooth coefficients and $b(r)>0$. Its cyclic coordinate $\theta$ has constant conjugate momentum $p_{\theta }=2b(r)\dot{\theta }$. For a circular orbit $r=R$, the radial Euler-Lagrange equation reduces to $b^{\prime }(R){\dot{\theta }}^2+c^{\prime }(R)=0$. If $b^{\prime }(R)\ne 0$, eliminating $\dot{\theta }$ gives the displayed condition. A real orbit requires $-c^{\prime }(R)/b^{\prime }(R)\ge 0$; the radial kinetic coefficient $a$ does not enter this constant-radius condition, although it affects radial perturbations. If $b^{\prime }(R)=0$, the original condition must be used rather than divided by zero.

For a spherical central body of radius $R$ with surface gravity $g_s=GM/R^2$, a circular test-particle orbit at radius $r$ has $v=\sqrt{g_s{R}^2/r}$. surface gravity is used to infer $GM$, not incorrectly kept constant at orbital height.

The intersection graph of the segments in a permutation diagram. Two segments cross exactly when their endpoint orders disagree. A set of mutually noncrossing segments corresponds to a monotone subsequence of the endpoint permutation.