Let $K_s$ be a computable increasing finite-stage enumeration of the halting problem set $K$. Color $x<y<z$ by whether the approximations below $x$ differ at stages $y,z$. This is a recursive two-color partition of triples. An infinite homogeneous set cannot have color one: for fixed $x$, the finite approximation below $x$ eventually stabilizes. If its color is zero, choose $x$ from that set above an input $n$, then another member $y>x$; agreement with arbitrarily large later stages gives $K(n)=K_y(n)$. Thus every infinite homogeneous set computes $K$, and none is a recursive set. This makes the failure of an effective version of Ramsey's theorem explicit.

Let $K_s$ be increasing finite stage approximations to a noncomputable computably enumerable set $K$. For $x<y<z$, colour the triple zero exactly when $K_y\cap [0,x]=K_z\cap [0,x]$. An infinite homogeneous set has colour zero, since each bounded approximation eventually stabilizes. Its first two elements above any requested bound recover the final approximation at that bound, so the homogeneous set computes $K$. Thus no infinite homogeneous set is computable. This illustrates the difference between a Ramsey's theorem existence result and an effective construction.

At order $e^4$, the open electron line can carry an internal-electron self-energy, a correction at either photon vertex, or a virtual photon spanning both vertices. Include the crossed external-photon order for each class. External-electron self-energies and external-photon vacuum polarization insertions appear in unamputated functions and are treated by LSZ reduction and field residues in physical amplitudes. A closed loop with three photon vertices cancels between orientations by Furry's theorem. Renormalized calculations also include the mass, field and vertex counterterms of the same perturbative order.

The Kompaneets equation describes frequency diffusion of a mean photon occupation number $n$ through repeated nonrelativistic Compton scattering. For thermal electrons, with $x_e=h\nu /(k_B{T}_e)$ and $dy=(k_B{T}_e/m_e{c}^2)n_e{\sigma }_{T}cdt$, its full homogeneous form is ${\partial }_{y}n=x_e^{-2}{\partial }_{x_e}[x_e^4({\partial }_{x_e}n+n+n^2)]$. The drift and stimulated terms permit a thermal stationary spectrum. In the hot-electron limit $T_e\gg T_{\gamma }$, with low photon energies, they can be neglected relative to the diffusion term; any fixed linear frequency variable $x$ then gives ${\partial }_{y}n=x^{-2}{\partial }_x(x^4{\partial }_{x}n)$. The quantity $n$ is not the frequency-integrated photon number density.

Effective conductivity describes the averaged response of a microscopic conductivity field to a macroscopic temperature or electric-potential gradient. Here the positive conductivity flux is $a\nabla u$; physical heat flux has the opposite sign. For periodic media it is computed by cell correctors, and can be a tensor even for isotropic constituents.

The indicator correlation gives the probability of specified phases at two points. Statistical homogeneity makes it depend on separation; isotropy makes it depend only on its length. In particular $p_{11}(0)=p_1$, so the connected correlation at zero separation is $p_1-p_1^2=p_1{p}_2$.

The volume fraction of a constituent is its occupied volume divided by total sample volume. Fractions sum to one for a partition into phases. In a statistically homogeneous medium the phase fraction also equals the probability that a typical point lies in that phase, under the appropriate spatial or ensemble averaging assumptions.

The effective elastic energy is the minimum averaged strain energy density under prescribed affine boundary displacement. For differentiable stable elastic response without body force, varying the prescribed mean strain gives $D{W}^{\mathrm{eff}}(E)=⟨\sigma ⟩$. The result follows by integration by parts, because the interior variation with zero boundary data does no work against an equilibrated stress tensor.

An elastic laminate has planar perfectly bonded layers with a common normal. Its piecewise constant strains differ by $\mathrm{sym}(a\otimes n)$, while its normal traction is common to every layer. For positive elastic stiffness tensors, these conditions and the mean strain uniquely determine the layer strains: the difference of two solutions has zero mean and zero total elastic work, and positivity forces it to vanish.

Böttcher's equation asks for a holomorphic coordinate conjugating a map with a superattracting fixed point to a monomial. If $f(z)=a{z}^m+O(z^{m+1})$ with $m\ge 2$ and $a\ne 0$, a normalized local solution has $\varphi (z)=bz+O(z^2)$ with $b^{m-1}=a$. The Böttcher theorem asserts the existence and uniqueness of this local coordinate after choosing $b$.

For $f_c(z)=z^2+c$, the Mandelbrot set is $\mathcal{M}=\{c:\{f_c^n(0){\}}_{n\ge 0}\text{ is bounded}\}$. Thus it records the nonescaping finite critical orbit. The connected Julia set criterion for a polynomial says that the corresponding filled Julia set is connected exactly for these quadratic parameters. Parameters with an attracting fixed point form the main cardioid: its multiplier $\lambda$ gives the fixed point $\lambda /2$ and parameter $c=\lambda /2-{\lambda }^2/4$, with $|\lambda |<1$. Its boundary has a cusp at $c=1/4$. There the critical orbit increases along the real interval to the parabolic fixed point $1/2$, so the cusp belongs to the set. The real section extends from $-2$ to $1/4$; the familiar period-two bulb is attached at $-3/4$.

A point $p$ is periodic for an iterated map $R$ if $R^n(p)=p$ for some positive integer $n$. Its exact period is the least such $n$, and the distinct points $p,R(p),\dots ,R^{n-1}(p)$ form its periodic orbit. For a holomorphic map its periodic-orbit multiplier is $(R^n{)}^{\prime }(p)$, computed in a local coordinate; at infinity use a reciprocal coordinate. The chain rule makes this the cyclic product of derivatives along the orbit, so it is the same at every point of that orbit.

A subset $E$ is completely invariant for a rational map if $R(E)=E=R^{-1}(E)$. This asserts invariance in both directions, including every preimage, rather than just invariance of one chosen inverse branch. For a nonconstant sphere map, the two inclusions $R(E)\subseteq E$ and $R^{-1}(E)\subseteq E$ already imply the equalities, because the map is surjective. The Fatou set and Julia set are completely invariant. A Fatou component need not itself be completely invariant even though the union of all components is.

Holomorphic maps $R$ and $S$ are holomorphically conjugate if a biholomorphic coordinate change $h$ satisfies $h\circ R=S\circ h$. Iterating gives $h\circ R^n=S^n\circ h$, so periodic points and their exact periods correspond. Differentiating at a periodic point gives equality of return multipliers, since the nonzero derivative of $h$ cancels. Locally uniform convergence and normality transfer through the coordinate change, so it also transports Fatou sets and Julia sets. Between entire Riemann spheres the biholomorphic change is a Möbius transformation; an affine change is sufficient for many polynomial normalizations.

Rectify outputs of simple cells with similar orientations and varying phases or positions, then sum or pool them. Opposite-polarity subunits allow response to both light and dark features. Pooling gives approximate phase and position tolerance over the subunit coverage; its extent and contrast dependence depend on the pooling rule.

Two oriented linear filters in quadrature have responses $A\cos \phi$ and $A\sin \phi$ to a suitably matched narrowband grating. Their squared sum is $A^2$, independent of phase $\phi$, while orientation and spatial-frequency selectivity remain. A nonlinear output function or normalization can transform this energy into firing rate. Ideal quadrature gives an exact phase identity for the matched stimulus; finite approximate filter pairs need not have perfect invariance for arbitrary images.

Let $K\subset \mathbb{C}$ be compact, and let $f$ be holomorphic on a neighborhood of $K$. Choose a set of allowed poles in the Riemann sphere outside $K$, meeting every connected component of its complement. Then rational functions with poles only in that allowed set approximate $f$ uniformly on $K$. The point at infinity may serve as the pole in the unbounded component.

If $\mathbb{C}\setminus K$ is connected, one can allow only infinity, giving the polynomial Runge theorem. If $K$ is the unit circle, its complement has an inside and an outside component. The function $1/z$ is holomorphic near $K$ and is already a rational approximant with its pole in the hole; it cannot be uniformly approximated there by polynomials, since their contour integrals vanish while $\oint dz/z=2\pi i$.

The proof first approximates a Cauchy integral by a finite rational sum with nearby poles. The pole-moving polynomial approximation argument transports poles through complementary components. This explains why complement geometry, rather than just smoothness of $f$, governs the allowed approximation class.

An elliptic integral is an integral of a rational function of $x$ and $\sqrt{P(x)}$, where $P$ is a cubic or quartic polynomial with distinct roots. The standard first, second and third kinds describe the essential normal forms. Choosing the full endpoint yields complete versions such as the complete elliptic integral of the first kind.

If a function is holomorphic on a simply connected domain, its integral around every closed piecewise smooth contour in that domain vanishes. In particular, contours with common endpoints may be deformed through the domain without changing the integral.

The right half-plane consists of complex numbers with positive real part. Its boundary is the imaginary axis. For example, $1+i$ is inside, $i$ is on the boundary, and $-1+i$ is outside. Rotation $z\mapsto iz$ takes the right half-plane onto the upper half-plane; the Cayley transform between the half-plane and disk uses $R(z)=(z-1)/(z+1)$ for this convention.