The parton distribution function is a number density. Integrating the quark distribution minus the antiquark distribution counts the net flavour charge of the hadron, since every sea quark-antiquark pair contributes zero. The corresponding flavour-vector conserved current counts particles minus antiparticles. For a proton, the net up/down numbers are two and one; for a neutron they are one and two. The rule fixes the difference of distributions, not their sum, and does not imply absence of a sea.

Replacing the derivative in $x^{\prime }=-Kx$ by a central finite difference gives $x_{n+1}+2Kh{x}_n-x_{n-1}=0$. Its principal root is ${\rho }_{+}=e^{-\mathrm{arsinh}(Kh)}$, while the parasitic amplification root is ${\rho }_{-}=-e^{\mathrm{arsinh}(Kh)}$. On a fixed time interval, $A_h{\rho }_{+}^n+B_h{\rho }_{-}^n$ converges uniformly to $C{e}^{-Kt}$ exactly when $A_h\to C$ and $B_h\to 0$. A fixed nonzero parasitic amplitude alternates between two incompatible limits, but a vanishing starting error can converge despite the root's modulus exceeding one. This distinguishes fixed-time numerical convergence from long-time absolute stability.

For $y^{\prime \prime }+({\omega }^2+ϵ\cos t)y=0$, use the method of multiple scales with $T=ϵt$ and $y_0=a(T)\cos (t/2)+b(T)\sin (t/2)$. Removing secular terms gives $a_T=(k-1/2)b$ and $b_T=-(k+1/2)a$. The slow eigenvalues satisfy ${\lambda }^2=1/4-k^2$, so an exponentially growing mode exists inside the displayed leading instability interval. At its leading endpoints the slow matrix is nilpotent; higher-order corrections are needed to determine the exact edges. Floquet theory distinguishes this exponential instability from algebraic growth at a repeated multiplier.

Define $x^{\bullet }=x$, $(\lambda x.M{)}^{\bullet }=\lambda x.M^{\bullet }$, and develop an application by contracting its root when that root is syntactically a beta-redex: $((\lambda x.M)N{)}^{\bullet }=M^{\bullet }[x:=N^{\bullet }]$. Otherwise $(MN{)}^{\bullet }=M^{\bullet }{N}^{\bullet }$. Newly created root redexes need not be contracted. Induction with the substitution lemma proves that if $M\Rightarrow N$, then $N\Rightarrow M^{\bullet }$. Thus any two parallel reducts share this reduct.

For a claim count distribution in the Panjer claim-count class and independent nonnegative integer severities with mass $f_j$, the probability generating function composition is $G_S=G_N\circ G_X$. Combining it with $(1-az)G_N^{\prime }=(a+b)G_N$ gives $(1-a{G}_X)G_S^{\prime }=(a+b)G_S{G}_X^{\prime }$. Equating coefficients yields the displayed recursion. Zero severities change both the initial mass and the denominator. For nondegenerate proper Panjer count laws $a<1$, so the denominator is positive. An identically zero count is handled directly if its redundant parameter choice makes the denominator vanish.

A tube of neighboring elastic rays follows the wavefront normal. If $J$ is its infinitesimal transverse area, its expansion is $d\log J/ds=\nabla \cdot \mathbf{n}$ along arclength. The P-wavefront discontinuity transport law therefore gives the displayed conserved flux. Geometrical spreading, mass density and wave speed determine the leading jump wave amplitude together; the rule is valid before the ray tube reaches a caustic.

The spherical source potential $f(t-r/\alpha )/r$ gives leading longitudinal displacement field $-f^{\prime }(t-r/\alpha )/(\alpha r)$. Apply the local solid-fluid P-wave transmission coefficient to its potential at a smooth interface. In hyperbolic interface collimation of P-waves, transmitted rays are parallel, so ray-tube conservation for a P-wave jump makes the leading coefficient constant along each vertical ray. This yields the displayed jump for the first nonzero derivative discontinuity of the waveform. Smooth transverse derivatives multiply $f$ rather than $f^{\prime }$ and are one order more regular. The construction presupposes the indicated one-sided derivative jump exists; a source smooth to every order may have no jump.

The gametophyte inside an angiosperm's ovule, containing an ovum and, in the common arrangement, a central biological cell with two polar cell nuclei.^[ref]

With local thermal noise matched to spatially varying drag, the overdamped probability density obeys the displayed diffusion equation. Its equivalent Itô diffusion is $dX=D^{\prime }(X)dt+\sqrt{2D(X)}dW$, since the two terms in the Fokker-Planck equation combine to ${\partial }_x(D{P}_x)$. Thus $d⟨X⟩/dt=⟨D^{\prime }(X)⟩$, even with no applied force. The drift points toward greater diffusivity, but a uniform density has zero flux: the drift is balanced by the spatial variation of the noise. Holding the force-noise amplitude fixed instead would not produce this isothermal Fickian equation.

For thermal-time magnetoconvection, a physical oscillatory neutral mode has ${\omega }^2=\sigma (1-\zeta )Q_{\kappa }{\pi }^2/(1+\sigma )-{\zeta }^2(k^2+{\pi }^2{)}^2>0$. The steady and oscillatory neutral curves meet when this frequency tends to zero. Thus $0<\zeta <1$ is necessary and the displayed relation locates the merger. A formal oscillatory-curve minimum with negative frequency squared is not a physical onset.

A vector bundle trivialization is orthogonal when its coordinate frame is orthonormal for the fiber metric, so each fiber is identified isometrically with Euclidean space. Overlap changes then take values in the orthogonal group. Local orthogonal trivializations exist even when no global frame does.

This is a frame of a real vector bundle with a positive-definite fiber metric for which the basis vectors are orthonormal. The Gram-Schmidt process applied to any smooth local frame produces such frames smoothly. They are distinguished from Lorentzian orthonormal frames, whose metric has mixed signature.