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Let $\alpha$ be an algebraic irrational number, let $S$ be a finite set of primes, and let $\epsilon >0$. There are only finitely many reduced fractions $p/q$ whose denominator is S-smooth number and which satisfyThis is the denominator-restricted consequence of the finite-place Schmidt subspace theorem.

A basic sequence in a Banach space is a sequence that is a Schauder basis for its closed linear span.

The nth coordinate functional of a Schauder basis is the bounded linear functional $e_n^{\ast }({\sum }_k{a}_k{e}_k)=a_n$.

The nth basis projection associated with a Schauder basis is

After analytic continuation in momentum, a pole at $k=i\kappa$ with $\kappa >0$ gives negative energy and spatial decay, and therefore represents a bound state.

In the Born approximation, scattering from identical atomic potentials translated to lattice sites factors into one atomic form factor times a finite sum of phases over the crystal sites.

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.