For $p>0$, $Q=\sqrt{2p}\sinh q$ and $P=\sqrt{2p}\cosh q$ satisfy $\{Q,P\}=1$. They transform $p\cosh 2q$ into $(Q^2+P^2)/2$. Their inverse is $p=(P^2-Q^2)/2$ and $q=\mathrm{artanh}(Q/P)$, so the chart covers only $P>|Q|$, not the whole oscillator plane. Oscillator evolution beyond that wedge is not a global real continuation of the original variables.

For a free molecule of mass $m$ in a large three-dimensional volume $V$, replacing the translational state sum by an integral gives the displayed partition function. For identical classical molecules the many-molecule canonical partition function is $q_{\mathrm{trans}}^N/N!$, not simply $q_{\mathrm{trans}}^N$. Differentiating its Helmholtz free energy at fixed volume gives the ideal-gas translational entropy.

The rotational partition function sums Boltzmann factors over rotational energy levels, including their degeneracies and the appropriate indistinguishability convention. For a linear rigid rotor at high temperature, $q_{\mathrm{rot}}=8{\pi }^{2}IkT/(\sigma h^2)$, with moment of inertia $I$ and rotational symmetry number $\sigma$.

A vibrational partition function sums over molecular vibrational energy levels. A single quantum harmonic oscillator, with energies measured relative to its ground level and vibrational temperature ${\theta }_v=h\nu /k$, has $q_{\mathrm{vib}}=(1-e^{-{\theta }_v/T}{)}^{-1}$. Retaining its absolute zero-point energy multiplies this by $e^{-{\theta }_v/(2T)}$. That factor shifts the internal energy but contributes no entropy.

The electronic partition function includes electronic excitation energies and degeneracies $g_i$. If only an exactly degenerate ground manifold of size $g$ is populated, $q_{\mathrm{el}}=g$ and its molar entropy is $R\log g$. Resolved splittings require a state sum rather than replacing all components by one degeneracy.

The single-molecule state sum $q={\sum }_s{e}^{-\beta E_s}$ for specified energy references and external constraints. In a dilute classical ideal gas, indistinguishability gives $Q_N=q^N/N!$; translation supplies the factor proportional to volume.

Products of noncommuting linear operators depend on their order. For position and momentum, $xp-px=i\hslash$; substituting classical commuting expressions into a product can therefore lose an additive quantum term. For example, expansion of the oscillator product $a^{†}a$ includes a commutator contribution $-1/2$, leading to the correct zero-point energy in $H=\hslash \omega (a^{†}a+1/2)$.

An actin-rich cellular protrusion associated with localized extracellular matrix degradation during invasion. Formation, matrix digestion and whole-cell invasion are related but distinct experimental endpoints.

Invasion by connected or coordinated groups of cancer biological cells rather than only isolated migrants. Leader and follower biological cells can perform different tasks.^[ref]

Strong dependence of an established tumour on an oncogenic activity. Dependence can change as the tumour acquires cooperating or bypass alterations.

differential scanning calorimetry measures differences in heat flow between sample and reference during a controlled temperature scan. Transition peaks can reveal cooperative lipid-chain melting, with scan rate, composition and reversibility affecting interpretation.

The Bürgi-Dunitz trajectory describes the preferred approach of a nucleophile toward a carbonyl carbon, usually at an obtuse angle to the carbon-oxygen bond, often near $107^{\circ }$. It is a geometrical trend, not a fixed angle obeyed by every reacting system.

An oblique approach balances favorable interaction with the carbonyl antibonding orbital against repulsion from occupied orbitals. Crystallographic observations of close contacts motivated this picture, linking static molecular structures with a possible path of nucleophilic addition. Steric constraints and the other substituents can modify the accessible trajectory.

A flat, unsupported film's bending and tension energies give modal stiffness $\kappa q^4+\sigma q^2$. Negative tension represents compression, destabilizing sufficiently long waves. A finite sheet restricts allowed wave numbers, while a supporting foundation adds stiffness and can select a wrinkle wavelength. This is generic continuum mechanics, independent of biological membrane structure.

A moss is a bryophyte with a prominent gametophyte and a dependent spore-producing sporophyte. The familiar LEAFY shoot and the capsule-bearing stalk represent different generations of the same life cycle.

A liverwort is a bryophyte whose dominant free-living generation is the gametophyte. Some forms are flattened, while others bear small leaf-like structures. This diversity makes external appearance alone an unreliable definition of the group.

A hornwort is a bryophyte with a photosynthetic gametophyte and an elongated horn-like sporophyte. The two generations differ in structure and dependence; the visible horn is not the whole life cycle.