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The conformal group of a metric manifold consists of diffeomorphisms $f$ with $f^{\ast }g={\Omega }^{2}g$ for a positive function $\Omega$. Its infinitesimal generators are Conformal Killing vector fields. The choice of manifold matters: in flat Euclidean dimension greater than two, special conformal transformations can be regular on the conformal compactification of Euclidean space while having finite poles in the uncompactified space.

For the lens $\mathrm{Im}z>0$ and $|z+i|<\sqrt{2}$, the Möbius transformation $w=(1+z)/(1-z)$ maps its real boundary segment to the positive real axis and its circular boundary arc to the ray with argument $\pi /4$. Its interior maps bijectively to $0<\arg w<\pi /4$. The fourth power therefore maps this lens conformally onto the upper half-plane. Composing with $(F-i)/(F+i)$ maps it onto the unit disc. The endpoints $\pm 1$ are boundary points, so the pole and zero introduce no interior singularity or branching.

A conformal map from the complex upper half-plane to a polygonal domain with finite prevertices $a_j\in \mathbb{R}$ and interior angles $\pi {\alpha }_j$ has the displayed derivative, with upper-half-plane branches of the powers. Integrating supplies the map, up to translation and nonzero scaling. For a slit boundary, repeated geometric vertices are treated as distinct boundary approaches: the two sides at the slit base have their respective angles, and a slit tip has angle $2\pi$, hence exponent one.

A storage unit holding part of a file's contents. The toy inode references two four-byte file data blocks for six bytes of contents: allocated capacity is $2\cdot 4=8$ bytes, while logical size is $4+2=6$ bytes. This counts only the two file data blocks, excluding storage for the inode and directory entries.

Information describing a file, such as type, owner, access permissions, size, timestamps and hard link count. In the toy layout, the inode carries these attributes; the directory entries carry the names. The distinction matters because renaming a file need not copy its contents.

A byte position measured from the start of a file, beginning at zero. A memory-mapped file combines a starting file offset with a displacement inside its mapping; this is distinct from an open file description's current read/write position.

A partial function is computable when a program returns its value on every input in its domain and runs forever on inputs outside its domain.

The S-m-n theorem says that fixed inputs can be compiled into program codes by a total computable transformation. For every $m,n\ge 1$, there is a total computable $s_n^m$ such thatwhenever either side is defined.

Fix an effective numbering ${\phi }_e$ of the partial computable functions with a universal partial computable function and the parameter compilation supplied by the S-m-n theorem. For every total computable function $f:\mathbb{N}\to \mathbb{N}$, there is a program index $e$ such thatHere $\simeq$ means equality as partial functions, including the inputs on which computation diverges. The conclusion concerns a program's behavior, not equality of program indices: $f(n)=n+1$ has no numerical fixed point, but still has this semantic fixed point.^[ref]

For a proof, the S-m-n theorem gives a total computable function $d$ compiling $a$ into a program that, on input $x$, first computes ${\phi }_a(a)$ and then runs the program with that resulting index on $x$. Thuswhere both sides diverge if the inner computation does. Choose an index $b$ for the total computable function $a\mapsto f(d(a))$. Setting $e=d(b)$ givesThe theorem makes rigorous a program's ability to use a description of itself. For example, let $f(a)$ compile a program that ignores its input and prints $a$. Its semantic fixed point is a program that prints its own index. This does not require a program to read its source file.

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 $-3/4$.