A conjugate acid-base pair consists of two chemical species related by transfer of one proton. The conjugate acid has one more proton and one more positive unit of electric charge than the conjugate base.

The conjugate acid of a chemical base is the chemical species formed when it accepts one proton. This is a relationship between partners, rather than a claim that the partner is a strong acid.

For $b>0$, the conformal map $\cosh (\pi z/b)$ takes $\{\mathrm{Re}z>0,0<\mathrm{Im}z<b\}$ bijectively onto the upper half-plane. The exponential maps this half-strip onto the upper half-plane exterior to the unit circle; then $\cosh (\pi z/b)=(w+w^{-1})/2$ is a scaled Joukowski map. All critical points of the entire map are $z=inb$, mapping alternately to $1$ and $-1$.

Conformal geometry studies a metric tensor only up to multiplication by a nowhere-zero positive scalar function on a real slice. A conformal class determines angles in Riemannian signature and null directions in Lorentzian signature. The complexified flat four-dimensional conformal geometry is encoded by the Klein quadric, whose incidence relation describes null separation.

The sine maps $\{-\pi /2<\mathrm{Re}z<\pi /2,\mathrm{Im}z>0\}$ bijectively onto the upper half-plane: its imaginary part is $\cos (\mathrm{Re}z)\sinh (\mathrm{Im}z)>0$, and the equality of two sine values forces either equal arguments modulo $2\pi$ or arguments summing to an odd multiple of $\pi$. Only equality is possible in this domain. For surjectivity, solve $t^2-2iwt-1=0$ for $t=e^{iz}$. When $\mathrm{Im}w>0$, exactly one root has modulus below one and it has positive real part. The inverse $z=-i\log t$ therefore belongs to the half-strip. Composing with $(w-i)/(w+i)$ gives a conformal map onto the unit disc.

For the crescent inside $|z-2i|<2$ and outside $|z-i|\le 1$, inversion $w=1/z$ gives the horizontal strip $-1/2<\mathrm{Im}w<-1/4$. The affine map $\zeta =4w+2i$ sends it to $0<\mathrm{Im}\zeta <1$, and $e^{\pi \zeta }$ maps it bijectively onto the upper half-plane. A Cayley transform then maps it onto the unit disc. Tangency, rather than a separate enclosed hole, is why this domain is simply connected.

On $\mathbb{C}\setminus [-1,1]$, take the principal square root with positive real part. The Möbius transformation $(z-1)/(z+1)$ has image $(\mathbb{C}\setminus (-\infty ,0])\setminus \{1\}$. The principal square root therefore maps to the right half-plane with $1$ omitted, and the final Möbius transformation gives exactly $0<|w|<1$. The inverse is $z=-(w+w^{-1})/2$. Adding $\infty$ to the source on the Riemann sphere fills the missing value $w=0$ and gives a conformal map onto the full disk. Without $\infty$, a loop surrounding the segment has nonzero winding number, so the source is not a simply connected space and cannot be conformally equivalent to a disk.

For the upper half disk $|z|<2$ slit along $z=iy$, $0<y\le 1$, squaring maps onto $|w|<4$ minus $[-1,4)$. After scaling the radius-four disc, the disk automorphism $v=4(w+1)/(w+16)$ sends this slit to $[0,1)$. Its square root with argument in $(0,\pi )$ maps onto the upper half unit disc. There $u+u^{-1}$ maps bijectively onto the lower half-plane: its imaginary part is $\mathrm{Im}u(1-|u{|}^{-2})<0$, and the two roots of its inverse equation are reciprocal, with exactly one inside the unit circle. The final Möbius map gives the displayed conformal disk bijection.

Two intersecting circles bound a lens. A Möbius transformation sending their intersection points to zero and infinity sends the two boundary arcs to rays, and the lens to a sector. A suitable power straightens the sector to a half-plane. The chosen sector and its branch must be identified by an interior test point; merely squaring an arbitrary boundary-straightening map need not be injective on the intended domain. A Cayley transform then gives a conformal bijection to a disc.

The local angle-preservation property of a conformal map. At a regular point its derivative is a positive scaling times an orthogonal linear map. Consequently composition multiplies gradient lengths by that scaling, as follows directly from the chain rule.

If $g$ is rescaled to $\tilde{g}={\Omega }^{2}g$, a null geodesic retains its unparametrized path. For its original affine tangent $k$, the changed connection gives ${\tilde{\nabla }}_{k}k=2k(\log \Omega )k$. Rescaling the tangent by ${\Omega }^{-2}$ cancels this term and proves the displayed affine-parameter relation, up to a constant multiplier. A conformal boundary may therefore be reached at finite unphysical affine parameter while physical affine parameter is infinite.

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.