Formation of a tissue lumen through polarized membrane organization, transport and related cell behaviours. Hollowing and apoptotic cavitation are distinct possible mechanisms.

An idealized model balancing localized production, diffusion and degradation. In one dimension at steady state, away from the source, $D{c}^{\prime \prime }-kc=0$ gives $c(x)=c_0{e}^{-x/\lambda }$ with $\lambda =\sqrt{D/k}$. Tissue geometry and time-dependent source distributions can invalidate a simple point-source interpretation.

A monotone stack whose stored values are kept in increasing order from bottom to top. An implementation specifies whether equality is allowed; popping values greater than or equal to a new value leaves strictly increasing stored values.

Excitation of one quantum of a normal mode from the vibrational ground state. For an infrared-active mode in the linear-dipole harmonic approximation, this is the allowed absorption underlying the usual fundamental band. A vibrational overtone excites two or more quanta from the ground state.

A transition starting from a thermally populated vibrationally excited state. Its intensity depends on the lower-state population through the Boltzmann distribution. Anharmonicity and coupling between normal modes can shift it away from the corresponding vibrational fundamental.

Spectroscopy of simultaneous changes of vibrational and rotational state. The transition wavenumbers combine vibrational energy differences with rotational constant terms. The allowed rotational changes depend on the molecule and the transition type.

The angular momentum associated with motion in a degenerate bending normal mode of a linear molecule. A one-quantum bend has $l=\pm 1$, and the total rotational quantum number obeys $J\ge |l|$.

A joint electronic and vibrational quantum state of a molecule. In a product approximation its irreducible representation is the tensor product of group representations of the electronic and vibrational factors.

Coupling between electronic and nuclear vibrational motion. Dependence of an electronic electric dipole transition moment on a normal mode coordinate can give intensity to a transition whose purely electronic matrix element vanishes at the equilibrium geometry.

Coordinates belonging to inequivalent irreducible representations cannot have an invariant bilinear cross term. Two copies of the same real irreducible representation can: their componentwise dot product is invariant. Diagonalizing the multiplicity-space force-constant matrix produces the normal modes. Symmetry alone therefore does not require two $T_2$ coordinate sets to have zero coupling before that diagonalization.

In the quadratic vibrational term expression $E_v/(hc)={\tilde{\omega }}_e(v+1/2)-{\tilde{\omega }}_e{x}_e(v+1/2{)}^2$, the constant $x_e$ is dimensionless, while ${\tilde{\omega }}_e{x}_e$ has units of spectroscopic wavenumber. A positive value makes adjacent level spacings decrease. For an exact Morse oscillator, $D_e/(hc)={\tilde{\omega }}_e/(4x_e)$. Using only two low-lying transitions to extrapolate this identity to a real molecule assumes the higher-order anharmonic terms are negligible all the way to dissociation.

A mass-weighted displacement coordinate along an eigenvector of the molecular harmonic force-constant matrix after rigid translations and rotations are removed. An orthonormal normal-mode basis gives kinetic energy $\frac{1}{2}{\sum }_i{P}_i^2$ and harmonic potential $\frac{1}{2}{\sum }_i{\omega }_i^2{Q}_i^2$. This use of normal coordinate concerns molecular vibrations, rather than normal coordinates of a spacetime metric.

Loss of excess vibrational excitation to other molecular degrees of freedom or to the surroundings. It can occur within one electronic state without photon emission, reducing the energy available to subsequent fluorescence.

The dependence of rotational energy on the vibrational state. For a diatomic molecule a common leading approximation is $B_v=B_e-{\alpha }_e(v+1/2)$, so vibrational satellites can have different rotational line spacings.

A model of conjugated $\pi$ molecular orbitals with an orthonormal atomic basis, one-centre energies ${\alpha }_i$, and resonance matrix elements ${\beta }_{ij}$ between selected neighbouring sites. Solving $H\mathbf{c}=E\mathbf{c}$ gives orbital energies and coefficients. Its simplicity makes symmetry and electron filling explicit, but quantitative charge predictions can depend strongly on the chosen parameters and omitted orbital relaxation.

Mixing between symmetry-compatible s-derived and p-derived orbital combinations. It shifts energies and changes orbital shapes; in a centrosymmetric diatomic, gerade and ungerade combinations do not mix with one another.

A diagram linking reactant and product orbitals by a symmetry conserved along a stipulated reaction path. Orbitals of the same symmetry avoid crossing in a generic one-parameter problem; orbitals of different symmetry can cross. Correlation constrains connectivity, not quantitative barrier heights.

A correspondence between molecular fragments based on the number, approximate symmetry, shape, energy and occupation of frontier orbitals. Equal total electron count alone is insufficient; the comparison predicts transferable bonding possibilities rather than identical chemistry.

A microtubule-associated molecular motor family. Many transport cargos toward plus ends, although direction and function vary among family members.