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The operators $S_{\pm }=S_x\pm i{S}_y$ satisfy

The spin-$s$ irreducible representation has dimension $2s+1$, with basis $|s,m⟩$ for $m=-s,-s+1,\dots ,s$ and Casimir eigenvalue ${\hslash }^{2}s(s+1)$.

For $\mathbf{n}=(\sin \theta ,0,\cos \theta )$, the normalized eigenstates of $\mathbf{n}\cdot \mathbf{S}$ arewith eigenvalues $+\hslash /2$ and $-\hslash /2$.

For $N$ noninteracting units with energies $-ϵ,0,ϵ$, put $x=\beta ϵ$. Thenand

A vacuum cavity of radius $a(t)$ in an infinite inviscid incompressible fluid of density $\rho$ and far-field pressure $p_0$ obeysIf $a(0)=a_0$ and $\dot{a}(0)=0$, its collapsing branch satisfies

The convex hull of finitely many unit-circle points contains zero exactly when the points do not lie in any open semicircle, equivalently when their shortest containing arc has length at least $\pi$.

For $U_{\gamma }=\mathrm{diag}(1,e^{i\gamma })$ with $0\le \gamma <2\pi$,

A Laplacian term in the infected-population equation models unbiased random migration of infective individuals.

Mass-action transmission at unit nondimensional rate transfers population from susceptible to infected compartments at rate $SI$.

A directed spanning tree rooted towards a vertex $r$ is a spanning tree whose edges are oriented so that every other vertex has a unique directed path to $r$.

A uniform spanning tree is sampled uniformly from all spanning trees of a finite connected graph.

This is the sequence space of sequences with a finite limit.

The space $c$ consists of convergent scalar sequences with the supremum norm.

If $\epsilon :\mathbb{R}\to [0,\infty )$ satisfies $\epsilon (t)\le M|t|$, thenThus these square-root perturbations converge uniformly to $|x|$.

With the supremum norm, $C_0({\mathbb{R}}^d)$ is complete. A uniform Cauchy sequence has a uniform continuous limit, and one member controls the limit uniformly outside a large ball. Every member is also uniformly continuous: use uniform continuity on a large compact ball and smallness of the function beyond it.

A plane wave in a homogeneous linear medium satisfies

A group is nilpotent when it has a finite central series, equivalently when its lower central series reaches the identity.

A weak solution satisfies an integrated identity in which derivatives are transferred to test functions. It can therefore exist with fewer classical derivatives.

A classical solution has enough ordinary derivatives for the differential equation and its data to hold pointwise. For the heat equation, $u_t$ and $u_{xx}$ should be continuous in the interior, with the specified initial and boundary traces. A weak solution instead imposes an integrated identity, allowing less regular data; uniqueness must always be stated for the solution class used.

This is the two-dimensional form of Ladyzhenskaya's inequality. For $u\in C_c^{\infty }({\mathbb{R}}^2)$,Apply the fundamental theorem of calculus to $|u{|}^2$ in each coordinate, multiply the resulting one-dimensional bounds, integrate, and use Cauchy--Schwarz.

If $s>d/2$, then $H^s({\mathbb{R}}^d)$ embeds continuously into bounded continuous functions.