cartilage with mineral deposited in its matrix. Mineralization does not convert its histological identity automatically into bone.

An open carborane cage with the nido skeletal-pair count. Mild alkyne–borane reactions can preserve adjacency of newly incorporated carbon atoms.

A closed deltahedral carborane. Thermal conversion of nido precursors may change hydrogen content or cage size as well as carbon positions.

The open-face dianion $[{\mathrm{C}}_2{\mathrm{B}}_9{\mathrm{H}}_{11}{]}^{2-}$ can bind through its five-membered open face as an ${\eta }^5$, six-electron donor. Capping this face with a metal fragment completes a 12-vertex metallacarborane; two faces can also support a bis(dicarbollide) metal donor.

A carborane-derived cage with one or more metal vertices. An external ligand such as cyclopentadienyl is not counted as five additional cage vertices.

A cerium(III)-assisted borohydride reduction often used to favour 1,2-reduction of an enone to an allylic alcohol. Actual selectivity remains substrate- and condition-dependent.

Conversion of an aldehyde or ketone carbonyl into a methylene through hydrazone formation and base-promoted nitrogen extrusion. The strongly basic, often heated conditions must be compatible with the other functional groups.

A carbon nanotube comprising one rolled graphene wall. The wrapping indices determine its allowed circumferential electronic wave vectors and therefore its metallic or semiconducting band structure.

A carbon nanotube comprising several nested graphitic walls. Each shell can have its own wrapping indices; their difference alone is not a structural defect.

A plot of optical transition energies against carbon nanotube diameter, usually organized by wrapping indices and electronic type. It relates optical and resonant Raman spectroscopy observations to nanotube structure.

A dense population of approximately aligned carbon nanotubes growing from a support. Collective crowding can favor alignment even when the support does not epitaxially select every tube orientation.

A strength estimate formed by dividing the maximum axial force supported by load-bearing carbon bonds by the tube mass per length. Equal bond loading and a defect-free, fully participating shell are strong assumptions; network fibres can be limited by intertube shear and sliding.

For a nearly flat interface, the extra surface energy in the small-slope approximation is $(\gamma /2)\int |\nabla h{|}^{2}dA$. Each nonzero Fourier mode therefore has thermal variance $⟨|h_{\mathbf{q}}{|}^2⟩=k_{B}T/(A\gamma q^2)$ in a convention with height reconstructed as the sum of its mode amplitudes. Summing over a two-dimensional annulus of wavevectors givesThe smallest wavevector is set by the lateral size or pinning, and the largest by the breakdown of the continuum description, commonly a molecular length. A fixed mean height removes the zero mode. Gravity supplies an additional restoring term and changes the long-wavelength cutoff when it is retained. Roughness here means a height variance; its square root is the root mean square amplitude, even when the mean height fluctuation is zero.

For a circular capillary of inner radius $r$, balance the vertical force $2\pi r\gamma \cos \theta$ against the hydrostatic column weight $\rho g\pi r^{2}h$. The displayed signed height follows, neglecting meniscus-volume corrections. A wetting liquid with $\theta <\pi /2$ rises, whereas a nonwetting liquid with $\theta >\pi /2$ is depressed. The contact angle is measured through the liquid. The expression uses radius rather than diameter and presumes equilibrium at the specified wall chemistry.

Two wide vertical plates separated by gap $d$ provide two contact lines per unit width. Upward surface-tension force $2\gamma \cos \theta$ balances column weight $\Delta \rho gdh$. Compared with a circular tube of radius $r$, the height ratio is $r/d$ for the same fluid and contact angle. Equal plate gap and tube diameter give half the tube rise.

For thin parallel circular plates of radius $a$ and separation $L$, the quasistatic axial field $E_z(t)$ induces $H_{\varphi }={ϵ}_{0}r{\dot{E}}_z/2$. Integrating cycle-averaged vacuum energy densities for sinusoidal drive gives ${\overline{U}}_B/{\overline{U}}_E=(\omega a/c{)}^2/8$. The same fields have radial Poynting flux $S_r=-{ϵ}_{0}r{E}_z{\dot{E}}_z/2$, whose leading net inward power equals the time derivative of stored electric energy. The small ratio diagnoses the quasistatic regime.

Replace two well-separated cylindrical conductors by opposite axial line charges. Their surface-to-surface potential difference is $\lambda [\ln (d/a_1)+\ln (d/a_2)]/(2\pi ϵ)$, giving the displayed capacitance per unit length. The approximation needs $d\gg a_1,a_2$ and a homogeneous surrounding permittivity.

For an isolated smooth conducting sphere of radius $a$, Gauss's law gives surface electric field $E_s=V/a$. Combining its capacitance $4\pi {ϵ}_{0}a$ with capacitor energy $C{V}^2/2$ gives the displayed bound when the surrounding medium has dielectric strength $E_{\mathrm{bd}}$ and approximately vacuum permittivity. Nearby conductors, surface roughness and discharge geometry can lower the practical limit.