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Diffraction is interference produced when waves scatter from spatial structure. A periodic structure concentrates the scattered intensity at reciprocal-lattice conditions.

Elastic scattering preserves the total kinetic energy. For one particle scattered by a fixed target, the incoming and outgoing wavevectors therefore have equal magnitudes.

The momentum transfer in a scattering process is the difference between the incoming and outgoing wavevectors, commonly $Q=k-k^{\prime }$ up to a sign convention and a factor of $\hslash$.

When the singular free energy in one correlation volume is of order one, $\alpha =2-d\nu$. Hyperscaling can fail above the upper critical dimension because of a dangerously irrelevant coupling.

If a nontrivial uniform estimateholds for all integrable $L^p$ functions on ${\mathbb{R}}^n$, dilation by every $\lambda >0$ forcesThus $q$ must be the conjugate exponent of $p$.

Scalar anisotropic stress is the trace-free scalar part of the spatial stress perturbation. Under a common convention it enters as $\delta T^i{}_j=\delta P{\delta }_j^i+{\partial }^i{\partial }_j{\pi }^S$.

The longitudinal velocity perturbation of a cosmological fluid can be represented by a scalar velocity potential through $\delta T^0{}_i=(\bar{\rho }+\bar{P}){\partial }_i\delta u$.

Synchronous gauge sets the lapse and shift perturbations to zero, $h_{0\mu }=0$. Its coordinates follow freely falling observers, but residual coordinate freedom produces gauge modes that must be distinguished from physical perturbations.

For $p+e^{-}\leftrightarrow H+\gamma$, the hydrogen binding energy is the rest-energy defect

Because equilibrium reactions balance chemical potentials and photons have zero chemical potential,The rest-mass difference in the Maxwell--Boltzmann densities then supplies the Boltzmann factor $e^{-I_A/(k_{B}T)}$.

If $K\subset \mathbb{C}$ is compact with connected complement and $f$ is holomorphic near $K$, then for each $\epsilon >0$ there is a polynomial $p$ such that ${\sup }_K|p-f|<\epsilon$.

The order is the highest power through which the one-step expansion matches the exact Taylor expansion.

An implicit Runge-Kutta method defines one or more stages through equations involving those same stages.

The local truncation error is the defect left when exact solution values are inserted into a numerical update. For a one-step method started from the exact solution,A local error $O(h^{p+1})$ generally leads to global error $O(h^p)$ under a uniform stability bound.

A one-sheet hyperboloid is a connected, doubly ruled surface, diffeomorphic to a cylinder.

An object has rotational symmetry when a nontrivial rotation matrix leaves it unchanged.

Potential vorticity combines vorticity with vortex stretching and density stratification. It also applies to nonrotating stratified flow through Ertel potential vorticity. In the inviscid shallow water equations,where $\zeta$ is relative vorticity and $h$ is layer thickness.

The Coriolis parameter is $f=2\Omega \sin \phi$ for planetary rotation rate $\Omega$ and latitude $\phi$. It controls the vertical component of the Coriolis acceleration acting on horizontal flow.

A baroclinic Rossby wave has nontrivial vertical structure. Constant stratification and rigid boundaries add a positive vertical-mode contribution to the denominator of its dispersion relation.

A barotropic Rossby wave has no vertical shear. For horizontal wavenumbers $k,l$, its quasi-geostrophic frequency is $\omega =-\beta k/(k^2+l^2)$.