The hydrogenic sequence of bound relative electron–hole energies $E_n=E_g-R^{\ast }/n^2$ below the free-pair threshold, where $R^{\ast }=({\mu }^{\ast }/m_e)R_{\mathrm{H}}/{\epsilon }_r^2$. The effective-mass, screened-Coulomb approximation and optical selection rules limit the model.

The characteristic relative-motion length of a hydrogenic Wannier-Mott exciton, with dielectric screening and exciton reduced mass ${\mu }^{\ast }$. Comparing a quantum dot's dimensions with this scale helps distinguish strong carrier confinement from confinement of a comparatively intact bound pair.

In the ideal screened-Coulomb model confined to a plane, the relative-motion energies are $-R^{\ast }/(\nu -1/2{)}^2$ for $\nu =1,2,\dots$, where $R^{\ast }=(\mu /m_e)13.6\mathrm{eV}/{\epsilon }_r^2$. The ground-state binding is $4R^{\ast }$ rather than the three-dimensional value $R^{\ast }$. Finite-width wells need not be exactly planar.

In the envelope approximation, a quantum well confines electron and hole motion perpendicular to the plane. Averaging their screened attraction over the confinement envelopes gives the displayed effective in-plane potential. Its narrow-well limit approaches the ideal two-dimensional exciton; finite width softens the short-distance attraction. Dielectric mismatch, nonparabolic bands and strong electron–hole confinement mixing require more than this single-subband model.

stress is a linear causal functional of deformation history: in simple shear, $\sigma (t)={\int }_{-\infty }^{t}G(t-t^{\prime })\dot{\gamma }(t^{\prime })d{t}^{\prime }$. Superposition holds and complex viscoelastic modulus is amplitude-independent within the small-deformation regime.

Vinyl chloride is chloroethene, $\mathrm{C}{\mathrm{H}}_2=\mathrm{CHCl}$. It is a small vinylic halide whose carbon-carbon double bond can participate in addition polymerization.

Polymerization converts many vinyl chloride molecules into polyvinyl chloride, with chlorine-bearing repeating units along the carbon backbone. The resulting polymer has physical properties very different from those of the molecular monomer.

A sigma-H adduct forms by nucleophilic addition at an aromatic carbon that initially bears hydrogen. Both the original hydrogen and the incoming nucleophile remain attached at that carbon before the rearomatization steps.

The harmonic-oscillator entropy contribution for one vibrational coordinate, with $x={\theta }_{\mathrm{vib}}/T$. Contributions from independent normal modes add, with one term for each component of a degenerate mode. A temperature-independent zero-point energy does not change this entropy.

For a rectangle of sides a and b with fixed edges, separation of variables gives $\psi =A\sin (p\pi x/a)\sin (q\pi y/b)\cos (\omega t+\varphi )$ with positive integers p and q. Its angular frequency is $\omega =c\pi \sqrt{p^2/a^2+q^2/b^2}$. Zero integers produce the trivial displacement, not a lower mode. The p−1 and q−1 interior nodal lines divide alternating-sign lobes.

A vertebra anterior to the sacrum. Counts depend on identifying the sacral articulation, not just numbering visible centra.

The first vertebra, associated with the articulation of the head and axial column.

The second vertebra; its specialization varies across vertebrate lineages.

A column vector is represented by an $n\times 1$ matrix; a row vector is represented by a $1\times n$ matrix. Transposition exchanges the two layouts. If $A$ is $m\times n$, multiplication $Ax$ takes an $n$-component column to an $m$-component column.

A row acting on a column gives a scalar, while a column times a row gives an outer product. In coordinate-free terms a row naturally represents a linear functional once a basis is chosen; it is not automatically the same object as a vector. For complex inner products, the corresponding row is often the conjugate transpose, not just the transpose.

For endpoints with position vectors $p_i,p_j$ in an affine Euclidean SPACE, the directed edge vector from $i$ to $j$ is $e_{ij}=p_j-p_i$. Reversing orientation changes its sign, while translating both endpoints leaves it unchanged.

For a triangle, $e_{12}+e_{23}+e_{31}=0$. Its side lengths are the norms of the corresponding edge vectors, and dot products determine angles. Choosing an orientation is essential: an undirected edge alone specifies the pair of opposite vectors, rather than one of them. Edge differences also describe how a scalar or vector field changes across a mesh.

For a finite population $x_1,\dots ,x_n$ with arithmetic mean $\bar{x}$, its variance is $\frac{1}{n}{\sum }_{i=1}^n(x_i-\bar{x}{)}^2$. It averages squared deviations over the whole population.

The molar enthalpy difference between vapor and liquid at specified conditions, evaluated on coexistence when it is a latent heat. Treating it as constant and treating the vapor as ideal gives $\ln (P_2/P_1)=(\Delta H_{\mathrm{vap}}/R)(1/T_1-1/T_2)$.

For a one-dimensional subband $E=E_0+\alpha k^2$ with $\alpha >0$, the density of states varies as $(E-E_0{)}^{-1/2}$ just above its edge. Optical peaks also depend on the joint density of states, selection rules and excitons.

A vacancy population above its thermal-equilibrium value at the specified temperature. It can drive net vacancy absorption by growing cavities when competing sinks and emission permit.