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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.