Systemic feedback restricting additional nodulation after symbiotic signaling has begun. Root-to-shoot and shoot-to-root communication coordinates local initiation with whole-plant status.

A wavefunction with extent $a$ and many nearly equally spaced internal nodes of a wavefunction has local wavevector $k\sim \pi n/a$. Thus $⟨K⟩={\hslash }^2\int |{\psi }^{\prime }{|}^2/(2m\int |\psi {|}^2)$ scales as ${\hslash }^2{n}^2/(m{a}^2)$. Exact coefficients require the envelope and boundary conditions. An order-one node-count offset is needed for low states: a nodeless localized state still has positive kinetic energy by the Heisenberg uncertainty principle.

For a diatomic molecular axis $\mathbf{n}$ and unit electric-field polarization $\mathbf{e}$, the axial and summed two transverse dipole transition directions have the displayed geometrical factors. Radial matrix elements differ between resonances. If axes have polar tilt $\beta$ and random azimuth, with electric-field angle $\alpha$ to the surface normal, the mean axial factor is ${\cos }^2\beta {\cos }^2\alpha +\frac{1}{2}{\sin }^2\beta {\sin }^2\alpha$. Polarization-dependent near-edge absorption therefore constrains an orientation distribution, subject to substrate hybridization and instrumental normalization.

A uniform thin ring of radius $a$ and mass $M$ has axial gravitational field $g_z=-GMz/(a^2+z^2{)}^{3/2}$. Every ring element is the same distance from the axial point; transverse contributions cancel and axial projection supplies the factor $z/\sqrt{a^2+z^2}$. This also follows by differentiating the axial potential $\Phi =-GM/\sqrt{a^2+z^2}$. Near the centre $g_z\simeq -GMz/a^3$, whereas far away it approaches the point-mass field.

An infinite plane sheet of surface mass density $\Sigma$ has constant gravitational-field magnitude $2\pi G\Sigma$ on either side, directed toward the sheet. A Gaussian pillbox has equal inward flux on its two caps and enclosed mass $\Sigma A$, so $-2gA=-4\pi G\Sigma A$. A uniform infinite slab has the same exterior formula with $\Sigma =\rho h$. Translating an infinite sheet does not change its exterior field, whereas adding a slab changes its surface mass density.

The reduction of reactor reactivity caused by neutron absorption by xenon-135. In a simplified model its abundance is fed by radioactive iodine decay, while its removal is by radioactive decay and neutron capture. Changes in these pathways can delay its response to a change of neutron flux.

For real $\kappa \ne 0$, this Green function inverts $d^2/d{x}^2-{\kappa }^2$ on $[0,1]$ with homogeneous Neumann boundary conditions. It is continuous and its first derivative has jump one at $x=\xi$. At $\kappa =0$, a Green function with the same unit Dirac delta source and zero boundary derivatives does not exist: integrating its defining equation would give zero equal to one. The equation $y^{\prime \prime }=f$ with these boundary conditions instead requires ${\int }_0^{1}f(x)dx=0$ and determines $y$ only up to a constant.

Maintenance or recovery of specified network performance after a disturbance, including its transient response and recovery time.

Alternative routes or components can preserve function after failure, provided they do not share the same relevant failure mode.

For $-V_0\cos Qx$, the Fourier coupling between the two free states at $k=\pm Q/2$ is $-V_0/2$. The split energies are $E_Q/4\mp V_0/2$, with $E_Q={\hslash }^2{Q}^2/(2m)$. The direct face gap is $V_0$, not $2V_0$.

For a harmonic magnetic dipole observed with kr of order unity, its quasistatic, induction and radiation fields scale as $r^{-3}$, $k{r}^{-2}$ and $k^2{r}^{-1}$ and need not be cleanly separated experimentally. The radiated-power law remains an asymptotic flux statement, but $P/(4\pi r^2)$ alone does not specify the local detected field. Detectability further depends on directional pattern, plasma propagation, antenna coupling, bandwidth, integration and noise. A tiny nonzero vacuum radiation term does not by itself establish either a practical detection or an absolute impossibility.