An operational measure often estimated from normalized ribosome footprints relative to RNA abundance. It reflects occupancy per transcript and is not automatically a direct rate of completed protein production.

At fluid Toomre parameter $Q=1$, the local spiral-wave dispersion relation becomes $n^2({\Omega }_P-\Omega {)}^2=(a_0|k|-\kappa {)}^2$. The minus-sign, longer-wavelength root is nonnegative only between the two Lindblad resonance conditions. The plus-sign, shorter-wavelength root remains positive outside that interval. Thus this dispersion relation alone does not prove that every density wave is confined between Lindblad resonances; an additional physical restriction is needed.

For constant circular speed $v_0$, angular velocity is $v_0/r$ and radial epicyclic frequency is $\sqrt{2}{v}_0/r$. Constant effective random speed and surface density then give $Q=r_Q/r$. The local axisymmetric fluid model is stable inside $r_Q$ and unstable outside it. This statement is conditional on the thin-disk local approximation, rather than a proof of global or collisionless stability.

Changing the trap frequency changes the equilibrium spacing proportionally to ${\omega }^{-2/3}$ without shifting the centre. The old equilibrium displacement from the new one is parallel to the breathing eigenvector. Starting at rest therefore excites that mode alone at linear order; the outer charges oscillate equally and oppositely while the central charge stays fixed.

Two narrow incoherent lines of powers $J_1,J_2$ and wave-number difference $\Delta k$ give visibility $\sqrt{J_1^2+J_2^2+2J_1{J}_2\cos (2\Delta k\delta )}/(J_1+J_2)$ in a Michelson scan of one-way arm difference $\delta$. The minimum is $|J_1-J_2|/(J_1+J_2)$, and revivals repeat in $\pi /|\Delta k|$. Finite linewidth damps this envelope.

A microstructure in which carbon redistribution and carbide formation reduce the supersaturation of a martensitic matrix. Plate or lath morphology can remain while local strain and thermodynamic phase composition evolve. The term describes a material state, not a universal guarantee of a particular toughness.

At an equilibrium minimum, $d{F}_{\min }/dT={\partial }_{T}F$ because ${\partial }_{\varphi }F=0$. For $0<\gamma <1$, the entropy decrease at coexistence is $S_0-S_{+}=c(1-\gamma )T^{\ast }({\varphi }^{\ast }{)}^2$. Multiplying by $T^{\ast }$ gives the positive latent heat released on cooling. The units are total heat or heat density according to the normalization of $F$.

When $\gamma \ge 1$ in the stated model, the broken, nonzero minimum remains globally preferred at every finite temperature. A metastable zero state can still exist for $T>T_0$, but its lifetime and a nucleation temperature are kinetic questions. Its disappearance at the spinodal point is not an equilibrium coexistence temperature.

The 15 Cartesian displacements of tetrahedral methane have characters $(15,0,-1,-1,3)$ on classes $(E,8C_3,3C_2,6S_4,6{\sigma }_d)$. Removing translations $T_2$ and rotations $T_1$ gives $(9,0,1,-1,3)$ and the stated decomposition. There are nine vibrational coordinates in four frequency families.

The Taylor remainder in this form evaluates one higher derivative at an intermediate point $\xi$ between the expansion point $a$ and $x$. If $|f^{(n+1)}|\le M$ on that interval, then $|R_n(x)|\le M|x-a{|}^{n+1}/(n+1)!$. This bounds truncation error without knowing the particular intermediate point. The differentiability hypotheses are those of the Taylor theorem with Lagrange remainder.

For a nonzero real constant $a$, the tangent line to $y=e^{ax}$ at $x_{\ast }$ passes through the origin precisely when $e^{a{x}_{\ast }}-x_{\ast }a{e}^{a{x}_{\ast }}=0$. Thus $x_{\ast }=1/a$, $y_{\ast }=e$, and its slope is $k=ae$. This gives the unique such tangent line. If $a=0$, the graph is the horizontal line $y=1$ and no tangent line through the origin exists. The equality between the derivative and the slope from the origin, $f^{\prime }(x_{\ast })=f(x_{\ast })/x_{\ast }$, is useful for other differentiable functions too.

A line tangent to two curves or to a curve at two distinct points. For functions $g_1,g_2$, contact at $x_1,x_2$ requires equal derivatives and equal intercepts: $g_1^{\prime }(x_1)=g_2^{\prime }(x_2)=m$ and $g_1(x_1)-m{x}_1=g_2(x_2)-m{x}_2$. The common-tangent construction uses the lower supporting line for phase coexistence.

The identical two-atom motif adds a factor $1+\exp [i\pi (h+k+l)/2]$ to the FCC lattice structure factor. Among even-index FCC diffraction reflections, only those with $h+k+l$ divisible by4 survive. The allowed FCC reflection200 therefore vanishes for equal motif scattering amplitudes.

The displayed conjugacy classes have sizes one, one, two, two and two. The identity and half-turn are central elements. Reflection sends $r$ to $r^{-1}$, pairing the two quarter-turns. Conjugating a reflection by $r$ sends $r^{k}s$ to $r^{k+2}s$; conjugating by $s$ sends it to $r^{-k}s$. Thus the parity of $k$ is invariant under conjugation, separating the axis reflections from the diagonal reflections, and each displayed pair is one conjugacy class.

In the order of the conjugacy classes of the symmetry group of a square displayed there, its irreducible complex character table isThe four one-dimensional representations assign independent signs to $r$ and $s$. The remaining row is the character of the ordinary planar representation: a rotation has trace $2\cos \theta$ and a reflection has trace zero. Weighted by class sizes, these rows satisfy character orthogonality. Their squared dimensions sum to eight, so they exhaust the irreducible representations.

Enhancement of molecular excitation and Raman emission by the local optical response of a nanostructure. Pump and emitted frequencies both enter the enhancement factor.

A change in Raman response from adsorption-induced electronic coupling or charge-transfer resonance, distinct from enhancement of the external optical field alone.

The area of a planar oriented region with a sign determined by orientation. For a triangle with cyclically ordered points $A,B,C$, it is $\frac{1}{2}((B-A)\times (C-A))$, using the two-dimensional cross product; reversing the order reverses the sign.

A self-similar energy-conserving strong spherical explosion into a uniform ambient density has $R_s=\xi (E{t}^2/{\rho }_0{)}^{1/5}$ and $v_s=2R_s/(5t)$. The dimensionless coefficient depends on the equation of state. Radiative losses, nonuniform media or continuous energy injection alter the similarity solution.