For $N$ equal narrow slits, adjacent phase increment $\delta$ gives amplitude $\sin (N\delta /2)/\sin (\delta /2)$ relative to one slit. Near a principal maximum, set $u=N\delta /2$; for large $N$ the normalized intensity approaches $(\sin u/u{)}^2$. Approximating the first subsidiary maximum by $u=3\pi /2$ gives principal-to-subsidiary intensity ratio $9{\pi }^2/4$. The exact large-array extremum satisfies $\tan u=u$ in $(\pi ,2\pi )$ and gives a slightly different ratio, about $21.2$. The slit envelope can further modify observed peak heights.

An electret is a dielectric retaining long-lived electric polarization or trapped electric charge after its preparation field is removed. The retained charge distribution can supply a persistent electric field without continuous external polarization power.

An electret condenser microphone uses this property to bias a variable capacitor. Sound moves a diaphragm, changing capacitance and producing an electrical signal in the readout circuit. The electret supplies the polarization condition; it does not provide unlimited electrical energy, and the readout can still require power. Temperature, charge leakage and material history affect long-term stability. ^[ref]

For an infinitely long circular dielectric cylinder in a uniform transverse field in vacuum, the inside electrostatic potential is linear in the transverse coordinate. Matching the tangential electric field and normal electric displacement field to the outside dipolar solution gives the displayed uniform internal field. The result assumes no free interface charge and neglects end effects; an exterior dielectric changes the permittivity ratio.

The density of available one-particle states multiplied by their mean occupation. For equilibrium electrons, $Z(E)=D(E)f(E)$ with the Fermi-Dirac distribution $f$. The integral of $Z$ counts electrons; the integral of $D$ counts available states.

A nonanalyticity of the density of states associated with a stationary point of a band dispersion relation. Its form depends on dimension and on whether the stationary point is an extremum or a saddle.

The number of lattice-vibration modes per unit angular frequency. For a three-dimensional isotropic monatomic crystal with three linear acoustic branches of a common speed $v_s$, wavevector-shell counting gives $g(\omega )=3V{\omega }^2/(2{\pi }^2{v}_s^3)$ below a suitable cutoff. Different branch speeds replace $3v_s^{-3}$ by $v_L^{-3}+2v_T^{-3}$.

Periodic boundaries on a cube give one spinless state per $(2\pi /L{)}^3$ in wave-vector space. Counting the full spherical shell gives $d\mathcal{N}/V=k^{2}dk/(2{\pi }^2)$. Hard-wall counting in the positive octant gives the same leading volume term.

For a spinless nonrelativistic particle in a central potential, $g(ϵ)=(2m{)}^{3/2}(4{\pi }^2{\hslash }^3{)}^{-1}{\int }_{U<ϵ}\sqrt{ϵ-U}{d}^{3}r$. Spherical symmetry reduces this to a radial integral over every allowed shell. It is a leading phase-space approximation.

For an area $A$ with periodic in-plane boundary conditions, one polarization contributes $A2\pi kdk/(2\pi {)}^2$ normal modes of the electromagnetic field. For linear dispersion relation $\omega =ck$ and effective physical volume $Ad$, this gives the density of states in the title. No extra factor two for polarization is allowed. Opposite in-plane directions are already counted in the full circle of wavevectors.

For the continuum dispersion ${\epsilon }_{\mathbf{k}}={\hslash }^2(k_x^2+k_y^2+k_z^2/\gamma )/(2m)$ with $\gamma >0$, substituting $q_z=k_z/\sqrt{\gamma }$ makes the energy isotropic and gives Jacobian $\sqrt{\gamma }$. Every momentum-state counting integral therefore acquires that factor. Per internal state and unit volume, $N=\sqrt{\gamma }(2m/{\hslash }^2{)}^{3/2}\sqrt{\epsilon }/(4{\pi }^2)$. The result assumes an unrestricted three-dimensional continuum; increasing this mass anisotropy is not the same operation as imposing a two-dimensional Brillouin zone and a finite interlayer band width.

The in-degree of a vertex $v$ in a directed graph is the number of edges directed into it, counting multiplicities when parallel edges are allowed. A directed loop contributes one to both the in-degree and the outdegree.

For edges $a\to b$, $c\to b$, and $b\to c$, the vertex $b$ has in-degree two and outdegree one. In every finite directed graph, the sum of the in-degrees equals the number of edges, since every edge has exactly one destination. It also equals the sum of the outdegrees.

A binary decision tree represents a procedure whose internal nodes ask two-outcome questions and whose leaves specify results. A path records the answers obtained so far; its length counts the questions on that execution. Distinguishing $N$ possibilities requires at least $N$ leaves, so a depth-$d$ binary tree can distinguish at most $2^d$ outcomes. This gives $d\ge \lceil {\log }_{2}N\rceil$ for an exact worst-case identification problem. The bound relies on the questions really having only two outcomes; comparisons with an equality branch need a stated convention.