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At the surface of a perfect conductor, the tangential electric field and normal magnetic field vanish. An incident plane electromagnetic wave is therefore accompanied by a reflected wave whose tangential electric field cancels the incident field at the surface.

A semicubical parabola is a cusped cubic curve affinely equivalent to $y^2=x^3$.

For a smooth plane curve of degree $d\ge 4$, adjunction givesIts canonical map is induced by the $(d-3)$rd Veronese map and is therefore an embedding.

For a finite connected graph with at least one vertex, embedded as a plane graph, let $n$ count vertices, $m$ count edges, and $f$ count faces of a plane graph, including the outer face. ThenThe hypotheses refer to an actual embedding; regions in a crossing-filled graph drawing do not supply its $f$.^[ref]

In the plane graph $K_4$ pictured above, three bounded faces of a plane graph and the outer face give $4-6+4=2$. A square cycle graph gives $4-4+2=2$, while the graph path $A-B-C$ gives $3-2+1=2$. Changing a crossing into a new vertex changes the graph: the square-and-diagonals drawing then has five vertices, eight edges, and five faces of a plane graph, giving $5-8+5=2$.

A proof outline deletes edges on graph cycles. Each deletion preserves the connected graph and merges two faces of a plane graph, leaving $n-m+f$ unchanged. Eventually a tree remains, with $m=n-1$ and $f=1$. Adding the point at infinity identifies the plane picture with a sphere; the value two agrees with its Euler characteristic and the cell-count formula for Euler characteristic.

A planar dual graph has one vertex in each face of a fixed planar embedding and one dual edge crossing each primal edge.

This is the planar-embedding case of a dual graph.

For a place $v$ of a number field $K$, the local degree is $d_v=[K_v:{\mathbb{Q}}_v]$. Thus $d_v=1$ at a real place, $d_v=2$ at a complex place, and $d_v=e_v{f}_v$ at a finite place over $p$. For each rational place, the local degrees above it sum to $[K:\mathbb{Q}]$.

If a prime ideal $\mathfrak{p}$ of $K$ lies over the prime number $p$ with ramification index of a prime ideal $e_{\mathfrak{p}}$, its normalized absolute value isIt extends the usual p-adic absolute value on $\mathbb{Q}$. Its local degree is $[K_{\mathfrak{p}}:{\mathbb{Q}}_p]=e_{\mathfrak{p}}{f}_{\mathfrak{p}}$, where $f_{\mathfrak{p}}$ is the residue-field degree.

A real embedding $K\hookrightarrow \mathbb{R}$ gives a real Archimedean place of local degree $1$. A conjugate pair of complex embeddings $K\hookrightarrow \mathbb{C}$ gives a complex Archimedean place of local degree $2$.

Reflection symmetry $x\mapsto -x$ forces the reduced vector field to be odd in $x$, naturally producing paired nonzero equilibrium branches.

The normal form $\dot{x}=\mu x+x^3$ has a stable trivial equilibrium for $\mu <0$ and two unstable nonzero equilibria for $\mu <0$. They collide with the trivial branch at $\mu =0$, after which that branch is unstable.

The photon quadrupole is the leading anisotropy that sources linear polarization through Thomson scattering.

The photon dipole represents the bulk velocity of the radiation. In tight coupling to baryons, $v_b=-3{\Theta }_1$ under the convention used here.

The photon monopole is the direction-averaged fractional temperature perturbation. In Newtonian gauge its gravitationally observable effective temperature is the Sachs-Wolfe combination ${\Theta }_0+\Psi$.

The comoving sound horizon is the distance traveled by an acoustic wave,

The tight-coupling approximation expands the photon-baryon Boltzmann equations in powers of $k/\Gamma$ and $\mathcal{H}/\Gamma$, where $\Gamma$ is the Thomson scattering rate.

A symplectic form is a closed differential form of degree two whose value on every tangent space is a nondegenerate alternating bilinear form. In canonical position-momentum coordinates it is ${\sum }_{i}d{q}_i\wedge d{p}_i$.

This representation is a distinguished example in the representation theory of the symmetric group.

The permutation representation of $S_n$ on ${\mathbb{C}}^n$ decomposes asThe first summand is the trivial representation and the second is the $(n-1)$-dimensional standard representation.

Let $S_n$ act on the $r$-element subsets of $\{1,\dots ,n\}$, with permutation character ${\pi }_r$. If $0\le l\le k\le n/2$, thenbecause orbits of pairs are classified by intersection size $0,\dots ,l$.

Let $a,n$ be coprime integers with $n>1$, and let $d={\mathrm{ord}}_n(a)$ be their multiplicative order. For integers $x,y$,Negative exponents mean powers of the modular inverse of $a$. Thus periodic modular exponentiation has smallest positive period $d$, and $1,a,\dots ,a^{d-1}$ are distinct congruence classes.^[ref]

This lets a huge modular exponentiation reuse a short list: reduce its exponent modulo the multiplicative order, then look up the corresponding power. For example, $1000\equiv 1(\mathrm{mod}3)$ gives $2^{1000}\equiv 2(\mathrm{mod}7)$, while $1001\equiv 2(\mathrm{mod}3)$ gives $2^{1001}\equiv 4(\mathrm{mod}7)$. When the multiplicative order is unknown, Euler's theorem permits reduction modulo Euler's totient function, though that period may be longer. Repeated squaring remains useful when computing either the unreduced or the reduced power.

For a proof outline, multiply $a^x\equiv a^y$ by the modular inverse of $a^y$, leaving $a^{x-y}\equiv 1$. Apply Euclidean division to write $x-y=td+r$ with $0\le r<d$. Then $a^{x-y}\equiv a^r$; minimality of the multiplicative order forces $r=0$. The reverse implication follows from $a^d\equiv 1$. The coprime integers hypothesis enables cancellation: modulo eight, $2^3\equiv 2^4\equiv 0$, but this repetition supplies no multiplicative order for two.

For a Boolean function of independent coordinates and a randomized decision tree that determines it, the OSSS inequality bounds its variance by a sum of coordinate influences weighted by their revealment probabilities.