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A localized virial identity differentiates a weighted mass $\int \chi |u{|}^2$ and its associated momentum. Compactly supported or flattened weights retain the coercive interior contribution while replacing an infinite-variance assumption by controllable tail errors.

An $H^1$ solution of the nonlinear Schrödinger equation either exists globally forward in time or its $H^1$ norm, equivalently its gradient norm when mass is conserved, becomes unbounded at the finite endpoint of its maximal lifespan.

For the focusing power equation $i{u}_t+\Delta u+|u{|}^{p-1}u=0$, the conserved energy is

For sufficiently regular solutions of a gauge-invariant nonlinear Schrödinger equation,is independent of time.

In the convention used here, a Brzozowski NFA has one final state, every state can reach it, and each word labels a path to it from exactly one starting state.

The powerset construction turns an NFA with state set $Q$ into a DFA with state set $\mathcal{P}(Q)$, transitionand accepting subsets that meet the NFA final set.

Without epsilon transitions, defineA word is accepted when the set reached from the initial state meets the final set.

In $R=\mathbb{Z}[\sqrt{-5}]$, the idealhas additive index two. If $I=(a+b\sqrt{-5})$ were principal, multiplication by its generator would givewhich has no solution in integers. Thus $I$ is not principal.

The global phase symmetry of a Dirac field gives the conserved Dirac current $j^{\mu }=\bar{\psi }{\gamma }^{\mu }\psi$.

Spacetime translations give the canonical stress-energy tensor

For a point mass and $r>0$, writeThe vacuum scalar-curvature equation gives $r{f}^{\prime }+3f=0$, so $f=C/r^3$. The remaining vacuum equations fix $C=2GM$ in the corresponding Schwarzschild gauge.

For a planar velocity field in polar coordinates,

Shear stress is the tangential component of traction on a surface. For leading unidirectional flow $u(y)$ in a Newtonian fluid, $\tau =\mu du/dy$.

A negative-temperature system is hotter than every positive-temperature system. If heat $\delta Q$ passes from it to a positive-temperature body, the total entropy change is

Instantaneously reversing a Hamiltonian $H(B)$ satisfying $H(-B)=-H(B)$ leaves populations unchanged but rewrites them as a canonical distribution with ${\beta }^{\prime }=-\beta$. When the partition function is unchanged, the free energy changes sign and the heat capacity is unchanged.

Forwrite $k=n\pi /a+\kappa$. Near the $n$th Bragg crossing,At $\kappa =0$, the two branches differ by $2|V_n|$.

A damped-driven Navier-Stokes equation adds linear damping and external forcing. In two-dimensional steady vorticity form it reads

Rayleigh-Bénard convection is buoyancy-driven flow caused by heating a fluid from below. In an infinite-Prandtl-number reduction, velocity is determined instantaneously by a forced Stokes equation, while temperature satisfies an advection-diffusion equation.

For a periodic or boundary-tangent divergence-free vector field $u$, integration by parts givesConsequently $\int (u\cdot \nabla f)f=0$, the cancellation underlying many fluid energy estimates.

The Stokes operator is $A=-P_{\sigma }\Delta$ on divergence-free vector fields, where $P_{\sigma }$ is the Leray-Helmholtz projection. On a periodic domain it equals $-\Delta$ on its domain, and its inverse gains two Sobolev derivatives.