The branch points are $\pm 1$; infinity has two unbranched meromorphic choices. On the complex plane slit along $[-1,1]$, normalization $f(z)\sim z$ at infinity selects a holomorphic square root with boundary values $f(x+i0)=i\sqrt{1-x^2}$ and $f(x-i0)=-i\sqrt{1-x^2}$ for $-1<x<1$. On the exterior real intervals, $f(x)=\mathrm{sgn}(x)\sqrt{x^2-1}$. A counterclockwise contour integral around the finite branch cut satisfies $\oint d\zeta /f(\zeta )=2\pi i$: the upper lip runs right to left, the lower lip left to right, and the endpoint arcs contribute $O(\sqrt{\epsilon })$. The opposite global branch reverses these signs.

H3K27 trimethylation adds three methyl groups to lysine 27 of histone H3, conventionally H3K27me3. It is associated with Polycomb-linked repression of developmental genes in many eukaryotes. Deposition and removal help regulate the persistence of a chromatin state. A broad marked domain describes a molecular state; demonstrating that the mark is required for repression at a particular gene needs a perturbation that separates it from associated factors. ^[ref]

H3K4 methylation modifies lysine 4 of histone H3 with one, two or three methyl groups. The states have different genomic distributions; trimethylation is often enriched near active transcription start regions. A mark can recruit binding factors or reflect ongoing regulatory activity, so enrichment does not by itself prove that the modification initiated transcription. State, genomic location and experimental perturbation are all needed for a mechanistic claim. ^[ref]

A plant histone methyltransferase participating in the reinforcing relationship between H3K9 methylation and methylated DNA.

A plant histone methyltransferase participating in the reinforcing relationship between H3K9 methylation and methylated DNA.

A plant histone methyltransferase participating in the reinforcing relationship between H3K9 methylation and methylated DNA.

A chromatin complex that catalyses H3K27 methylation, with locus-specific recruitment and developmental functions.

Its catalytic activity deposits H3K27 methylation, including the trimethylated state associated with many repressed developmental genes. Recruitment and recognition of modified chromatin can stabilize a regulatory state while retaining the possibility of later reversal. ^[ref]

A plant histone deacetylase participating in chromatin silencing, including nucleolar dominance.

Removal of acetyl groups can alter histone interactions and recruitment of chromatin-associated factors. Its association with silencing should be interpreted at the specified loci and developmental stage rather than as a universal consequence of every histone deacetylation event. ^[ref]

The hip is the joint region connecting the femur to the pelvis. The femoral head articulates with the acetabulum, while capsule, ligaments and muscles stabilize movement. Its deeper socket favors a different mobility-stability balance from the shallower shoulder joint.

The knee is the joint region linking thigh and leg, principally through femoral, tibial and patellar articulations. Its menisci, ligaments and muscles guide motion and distribute load. It permits substantial flexion and extension while requiring stability against translation and rotation.

A foot is the distal part of a hindlimb, including structures beyond the ankle in the familiar human arrangement. Its bones, joints and soft tissues distribute loads and provide leverage. The whole foot’s behavior cannot be reduced to the action of one joint.

The slope of $\log [\theta /(1-\theta )]$ against $\log [L]$ in a Hill equation description. A value above one supports positive cooperative binding under the model's assumptions. It does not by itself count occupied sites or identify a molecular mechanism.

A Hill activation function is an increasing saturating input function for phenomenological regulation of gene expression. For positive $K,n$, its value is one-half at $u=K$. The Hill coefficient controls steepness; it need not equal a literal molecular binding-site count.

The positive concentration giving half of the maximal response in a Hill equation. This effective threshold need not equal an individual binding site's microscopic dissociation constant.

In the multiplicative set of Hilbert numbers, a squarefree Hilbert number is one divisible by no square $h^2$ of a Hilbert number $h>1$. This is weaker than being a squarefree integer: $9$ is a squarefree Hilbert number because its square divisor $9=3^2$ has $3$ outside the Hilbert numbers.

The spore type developing into a male gametophyte. Its production is distinct from the later production of sperm.

A megaspore is the spore type that develops into a female gametophyte in a heterosporous life cycle. It is distinct from an ovum: the megaspore begins a developmental generation, whereas an ovum is a gamete that fuses during fertilization.

The corresponding male-producing spore is a microspore. Following both spore types helps connect heterospory to the strongly reduced gametophytes of seed plants.

Extension of a repressive chromatin domain beyond its initial nucleation site. Reader–writer positive feedback can propagate local marks, while histone turnover, opposing activities and chromatin boundaries constrain the domain.