An explicitly specified measure of moderator relative to fuel, for example hydrogen atoms per heavy-metal atom in a water-cooled fuel lattice. Atom, mass and volume ratios are distinct quantities. The ratio changes the competition between slowing down, absorption and leakage.

The time integral of scalar neutron flux, with inverse-area units. Multiplication by a constant microscopic reaction collision cross-section gives a dimensionless exposure parameter. A single integrated exposure does not specify how the neutron energy distribution varied during irradiation.

For scalar nematic order parameter $S=⟨(3{\cos }^2\theta -1)/2⟩$, the displayed angle has ${\cos }^2{\theta }_{\mathrm{eff}}=⟨{\cos }^2\theta ⟩$. It summarizes a second moment, not the literal mean angle. For example an isotropic orientation distribution has $S=0$, giving ${\theta }_{\mathrm{eff}}=\arccos (1/\sqrt{3})$; the average acute angle to its unoriented axis is instead one radian, obtained by integrating $\theta \sin \theta$ over $[0,\pi /2]$.

For a narrow nematic liquid crystal orientation distribution about its unoriented nematic director, expand $S=⟨(3{\cos }^2\theta -1)/2⟩=1-3⟨{\theta }^2⟩/2+O(⟨{\theta }^4⟩)$. This proves the displayed root-mean-square width. It does not determine the full distribution or a full width at half maximum. If the two small transverse tilt components are independent equal-variance Gaussians with variance ${\sigma }^2$, then $⟨{\theta }^2⟩\simeq 2{\sigma }^2$, so $\sigma \simeq \sqrt{(1-S)/3}$.

An apolar axis model with pair energy $-({\mathbf{s}}_i\cdot {\mathbf{s}}_j{)}^2$ is invariant when each molecular representative is independently reversed. The vector average changes under these replacements, but the nematic order parameter made from second moments does not. In an unbiased statistical ensemble the vector average vanishes; an individual finite representative need not have zero average. For independently chosen signs of a perfectly aligned axis, $⟨|N^{-1}{\sum }_i{\mathbf{s}}_i{|}^2⟩=1/N$. This distinguishes exact head-tail invariance of an observable from vanishing only after an average.^[ref]

The same stopping rule connects a waiting count to a fixed-length count. Let $N=X+r$ be the trial number of the $r$th success, and let $B_n$ count successes in the first $n$ trials, so $B_n$ has a binomial distribution. For any nonnegative integer $n$,The target is still unmet after $n$ trials exactly when fewer than $r$ successes have occurred. With $r=2$, $p=1/2$ and $n=4$, this gives $\mathbb{P}(N>4)=(1+4)/16=5/16$; equivalently, $\mathbb{P}(X\le 2)=1-5/16=11/16$, matching $1/4+1/4+3/16$. At $p=1$, the event identity still holds, with $N=r$ and $B_n=n$ deterministically.

For twice continuously differentiable coordinate functions $u,v$ and an outer scalar function $f$, apply the multivariable chain rule twice along an original coordinate $s$. The first three terms differentiate the outer function, while the last two differentiate the coordinate coefficients. They cannot be omitted for a nonlinear coordinate map. In matrix notation this is the Hessian matrix formula $H_{f\circ g}=J_g^T{H}_f{J}_g+f_u{H}_u+f_v{H}_v$ for $g=(u,v)$. The extra terms vanish for an affine map.

For a pair of distinct bosonic particle and antiparticle modes, the normalized state obeying $(a+q{b}^{†})|\psi ⟩=(b+q{a}^{†})|\psi ⟩=0$ is $\sqrt{1-|q{|}^2}{\sum }_{n\ge 0}(-q{)}^n|n,n⟩$, for $|q|<1$. It contains equal occupations and is generated by a pair-creation exponential. Infinite products require a separate implementability or regulator check.

Pair creation in two bosonic modes gives the normalized state displayed. The two occupations coincide in each term, while either mode alone has a geometric number distribution. At small $r$, the leading nonvacuum term is a single pair. This optical construction does not require particle-antiparticle modes.

Linearizing a uniform Kerr-fibre background couples opposite frequency sidebands. With $i{A}_z-{\beta }_2{A}_{TT}/2+\gamma |A{|}^{2}A=0$, the displayed dispersion relation permits exponential growth precisely when ${\beta }_2\gamma <0$ and $0<{\Omega }^2<-4\gamma P/{\beta }_2$. For positive $\gamma$, this is anomalous group velocity dispersion. complex conjugation requires the paired complex-mode ansatz on the unstable branch. amplitude and power growth exponents differ by a factor of two.

If a cascade determines only absolute spin differences relative to a known reference spin, reflection of all inferred spins around that reference can preserve the same multipole pattern. Nonnegative-spin constraints, independently known spins, angular correlations or polarization measurements may break the ambiguity. A set of line energies and lowest-multipole labels need not do so.

For a circular low-mass companion and a semi-amplitude threshold $K_{\min }$, $a_{\max }=G{M}_p^2{\sin }^{2}i/(M_{\star }{K}_{\min }^2)$. The edge-on case is the largest geometrically accessible distance. Adequate baseline and cadence are separate requirements, and a full peak-to-peak detection threshold is twice the corresponding semi-amplitude.

At fixed observed period and semi-amplitude, $M_p\sin i\propto M_{\star }^{2/3}$ and $a\propto M_{\star }^{1/3}$. At fixed planet mass and amplitude threshold, the largest circular orbit instead scales as $M_{\star }^{-1}$. A revised stellar radius affects transit-derived planet radii but not these leading dynamical relations.

A naturally occurring solid substance with a characteristic composition range and internal structure. Synthetic analogues can be engineered as hosts for selected waste constituents.

A bicontinuous domain morphology containing two interpenetrating, nonintersecting networks of one component within the other. In diblock copolymers it can appear between lamellar and cylindrical morphologies in part of the phase diagram.

A domain structure resembling lamellae pierced by an ordered array of perforations. It is distinct from an unperforated lamellar phase and from a double gyroid.

The transition between a compositionally disordered state and an ordered microdomain state as effective segregation strength increases. The parameter $\chi N$ combines unlike-segment interaction preference and chain length; its transition value depends on composition and model assumptions. A bulk thermodynamic equilibrium boundary is not by itself an evaporation or vitrification kinetic model.

An ordered morphology in which a minority block forms approximately spherical domains in a continuous majority-block matrix. The lattice packing and phase stability depend on segregation strength, composition and chain statistics. The label S in a phase diagram denotes a morphology, not a spherical shape of the entire specimen.

A microdomain pattern of approximately parallel cylinders arranged on a two-dimensional hexagonal lattice in the other block's matrix. It is conventionally labelled H in many diblock phase diagrams. Cylinder orientation and defects are additional properties beyond the local morphology.

Alternating sheet-like domains of the components of a microphase-separated material. In diblock systems the two layer thicknesses can be unequal and reflect unequal block fractions. Confinement, surface wetting and kinetic trapping can produce layered films outside the thermodynamic equilibrium bulk lamellar region.