A core promoter sequence element recognized by TATA-binding protein. Its position and spacing relative to other elements can affect initiation geometry.

A core promoter element overlapping a transcription start site. In focused human gene promoters, its position has a defined relationship to initiation.^[ref]

The process by which a stalled outgoing shock in a collapsing stellar core regains sufficient energy to expel outer material. Neutrino heating and multidimensional flow can contribute, while dissociation and radiation drain energy. Collapse does not guarantee a successful ordinary supernova.

A type II supernova has conspicuous hydrogen features in its observed spectrum and is associated with collapse of a massive stellar core retaining a hydrogen-rich envelope. Its spectral label should not be confused with a unique light-curve shape or a unique remnant type.

A type Ib supernova is a stripped-envelope event with no strong hydrogen lines but identifiable helium features. The hydrogen-poor spectrum does not make it a thermonuclear Type Ia event.

A type Ic supernova lacks prominent hydrogen and helium features and is associated with a strongly stripped core-collapse progenitor. Line visibility depends on excitation and transfer as well as composition, so spectral classification is not an exact inventory of every element present.

Capture of small drifting solids by a planetary embryo, aided by dissipation through gas drag. Its efficiency depends on the Stokes number, particle-layer thickness, embryo mass and the supply of solids; it is not simply geometric sweeping of every passing particle.

A growth regime in which larger planetesimals have shorter fractional growth times. In a dispersion-dominated population with nearly mass-independent random speed and strong gravitational focusing, fixed material density gives $\dot{M}\propto M^{4/3}$. The displayed scaling is an illustrative limit, not a universal law once large bodies control the random velocities.

A stage in which several neighboring planetary embryos dominate stirring of surrounding planetesimals and grow in a self-regulated manner. Larger embryos heat their feeding populations more strongly, weakening gravitational focusing and preventing unlimited runaway planetesimal growth. A small number of embryos coexist with many much smaller bodies.

A model-dependent threshold beyond which a cooling gaseous envelope cannot remain on a slowly growing low-mass branch around a solid core. Opacity, solid accretion heating, ambient disk conditions and envelope composition affect the threshold. It is not a universal constant assigning one exact mass to every gas-giant progenitor.

Rapid envelope growth after cooling and envelope self-gravity allow gas capture to accelerate. It requires an available disk gas supply; eventually disk transport, gap formation or protoplanetary disk dispersal can limit growth. It differs from solid-body runaway planetesimal growth.

Gravitational binding is necessary but not sufficient for gas capture. A hydrostatic envelope contracts by losing energy on a Kelvin-Helmholtz scale $t_{\mathrm{KH}}\sim |E|/L$. Solid accretion heats it, while radiative transport depends on opacity and convection carries energy when the radiative gradient is unstable. The disk supply and outer tidal/thermal boundaries also matter.

Two added electrons move above a frozen filled Fermi sea and attract within a thin energy shell. The Pauli-blocked continuum has a logarithmic pair susceptibility, so an arbitrarily weak net attractive constant interaction produces a state below the two-particle threshold at zero temperature. This two-added-particle result is not by itself a full many-electron superconducting state.

A fragment formed by gravitational instability of an astrophysical disk must remain bound after it forms. Its own gravity competes with the surrounding tidal field, while heating, cooling and accretion change its size and internal support.

For a low-mass clump on a circular orbit around a dominant star, its Hill radius estimates the region over which its attraction can resist the local stellar tide. Cooling and contraction can make a fragment more compact, whereas inward migration can shrink that tidal region. A calculation demonstrating initial fragmentation therefore does not alone establish a long-lived giant planet: survival requires the subsequent thermal and orbital evolution as well.

For a finite point set in the plane, a convex polygon whose vertices of a polygon belong to the set and whose interior contains no point of the set. The convention must specify whether additional points on the boundary are allowed. A fan triangulation reduces emptiness to tests on constituent triangles, together with checks that internal diagonals contain no forbidden points.

A strictly convex polygon is a convex polygon whose interior angles are all strictly less than $180$ degrees.

An underfocus setting that balances objective-lens spherical aberration against defocus to provide a broad first phase-contrast band for a weak phase object. A common approximate convention is $\Delta f_S\simeq -1.2\sqrt{C_S\lambda }$ for positive $C_S$; numerical optimization conventions differ slightly.

Between consecutive continued fraction convergents, intermediate convergents have the form $(t{p}_{k-1}+p_{k-2})/(t{q}_{k-1}+q_{k-2})$ for $1\le t\le a_k$. The final value $t=a_k$ is the next convergent.

Choose an unassigned variable with the fewest currently legal values. No legal values detects failure immediately; exactly one legal value forces a choice. This ordering can reduce the search tree without changing the set of solutions.

An all-different constraint requires its variables to take pairwise distinct values. For domains $D(X)=D(Y)=\{1,2\}$ and $D(Z)=\{1,2,3\}$, the first two variables consume values one and two in every solution, forcing $Z=3$. Pairwise inequality propagation may miss this deduction, while a global propagator can use a matching in a bipartite graph between variables and values. Hall's marriage theorem supplies the feasibility condition: every subset of variables needs at least as many available values as variables. This constraint appears in schedules, assignments and numerical puzzles.