${\mathrm{B}}_5{\mathrm{H}}_9$, a nido borane with a square-pyramidal boron skeleton. A bridging proton can be removed to form $[{\mathrm{B}}_5{\mathrm{H}}_8{]}^{-}$.

$[{\mathrm{B}}_5{\mathrm{H}}_8{]}^{-}$, the bridge-deprotonated conjugate base of pentaborane(9). Its five-vertex framework has seven skeletal pairs and is nido.

$[{\mathrm{B}}_6{\mathrm{H}}_{11}{]}^{-}$, an arachno borane anion with nine skeletal pairs. A localized BH3-adduct drawing does not describe its lowest-energy fluxional cage adequately.

${\mathrm{B}}_6{\mathrm{H}}_{12}$, a six-vertex arachno borane with nine skeletal pairs. Bridge deprotonation gives the undecahydrohexaborate anion.

A borane-derived cluster containing skeletal carbon vertices. In a simple neutral count, a CH vertex contributes three skeletal electrons while BH contributes two.

$[{\mathrm{B}}_{12}{\mathrm{H}}_{12}{]}^{2-}$ has 26 skeletal electrons and a closo icosahedral B12 framework.

A mammalian embryonic stage with an outer trophectoderm, a cavity and an inner cell mass.

A black hole formed in the early universe rather than as the endpoint of a star. It is not subject to the ordinary stellar-remnant baryon budget in the same way as a black hole formed from a baryonic progenitor. Formation scenarios and observational constraints depend strongly on its mass.

Conditional adjustment of care in response to a partner, offspring or the environment. A matching response can reflect information about brood demand rather than absence of conflict.

For an infinitely long thin strip of width $w$ carrying uniform longitudinal current $I$, the magnetic field a perpendicular distance $z>0$ above its centre has magnitude $B={\mu }_{0}I\arctan (w/(2z))/(\pi w)$. Its near-sheet limit is ${\mu }_{0}I/(2w)$ and its distant line-current limit is ${\mu }_{0}I/(2\pi z)$.

For planar charge rotating rigidly about its centre, $\mathbf{v}=\mathbf{\Omega }\times \mathbf{r}$. The magnetic contribution at the centre is $dB={\mu }_0\Omega dq/(4\pi r)$, along the rotation axis. Integrate over the actual charge distribution; charges on a rotating polygon do not follow the sides of a stationary wire circuit.

Division of one cell into two. Ciliates can increase cell number through this vegetative process independently of a conjugation event.

A Franel sum adds fixed powers of the binomial coefficients in one row. A common convention uses the cubic sumFor example, $F_2=1^3+2^3+1^3=10$. More generally, the exponent can be specified explicitly as $F_n^{(r)}={\sum }_{k=0}^n{\left(\genfrac{}{}{0ex}{}{n}{k}\right)}^r$. For a positive integer $r$, each term counts ordered $r$-tuples of subsets of an $n$-element set that all have the same cardinality $k$. Summing over $k$ counts all such tuples. For $r=2$, Vandermonde's identity reduces the sum to the central binomial coefficient $\left(\genfrac{}{}{0ex}{}{2n}{n}\right)$; the cubic case is a distinct sequence.

A Kurie plot removes known electron phase-space and Coulomb factors from a beta spectrum and takes a square root. For negligible neutrino mass/recoil and constant nuclear shape factor, its ordinate is proportional to $Q-T$, giving a straight line whose kinetic-energy intercept is the beta endpoint. The precise factors depend on whether the measured spectrum is per momentum or per energy. Forbidden transitions generally need an additional shape correction.

For ground-to-ground beta-minus decay of bare nuclei, the nuclear mass difference supplies the electron rest energy as well as the kinetic energy shared among products. Neglecting recoil and neutrino mass, $T_{\max }=Q_{{\beta }^{-}}$. For neutral atomic masses the additional final atomic electron cancels the emitted electron rest-mass subtraction, giving $Q\simeq [M_{\mathrm{atom}}(i)-M_{\mathrm{atom}}(f)]c^2$ up to electron-binding corrections. A daughter excitation reduces the endpoint by its excitation energy.

An isotope of beryllium with 4 protons and 6 neutrons. Its nuclear spin concerns the coupled nuclear quantum state, not the electronic configuration.

An eigenstate of the rotating-frame Rabi Hamiltonian returns to the same laboratory ray after a full drive period. Its quantum phase includes both the rotating-frame energy and the sign of a full spin rotation. For spin $s$ and magnetic label $\mu$, the factor is $(-1{)}^{2s}{e}^{-i\mu \Omega T}$. Expanding the rotating-frame splitting to first order in the drive frequency and subtracting the adiabatic dynamical phase leaves $-\mu$ times the oriented solid angle of the field loop. Dropping the full-rotation sign gives a spurious half-integer-spin offset.

For $n\ge 1$, the von Staudt-Clausen theorem statesConsequently the positive denominator of the Bernoulli number $B_{2n}$ in lowest terms is exactly the squarefree product of the primes satisfying $p-1\mid 2n$. This gives its denominator without computing its numerator.^[ref]

For $B_4$, the relevant primes are $2,3,5$. Their reciprocals sum to $31/30$, and $B_4=-1/30$, so the displayed expression is $1$. The theorem concerns even positive indices; $B_1=-1/2$ is handled separately. It explains the recurring small prime factors in denominators of formulas for sums of powers.

An sp3 carbon directly attached to an aromatic ring.