For a smooth simple compressible system, $C_V=T(\partial S/\partial T{)}_V$ and the Maxwell relation $(\partial S/\partial V{)}_T=(\partial p/\partial T{)}_V$ give $(\partial C_V/\partial V{)}_T=T({\partial }^{2}p/\partial T^2{)}_V$. Thus an equation of state $p=a(V)T+b(V)$ implies that $C_V$ depends on temperature alone within a smooth phase. It does not require $C_V$ to be temperature independent.

A transition between oscillator energy eigenstates that adds one quantum changes kinetic and potential expectation energies by $\hslash \omega /2$ each. A sudden unobserved momentum kick instead initially adds its energy only to kinetic motion and leaves the position distribution unchanged. Coherent superpositions distinguish these descriptions. Averaging over motional outcomes gives a recoil-scale total energy; time averaging or dephasing gives equal kinetic and potential shares. Broad-line transition rates need not be exactly equal to bare overlap probabilities if finite spectral weights matter.

For a harmonic function on the planar annulus $a<r<b$, the radial amplitude of a sine or cosine Fourier mode of positive integer index $n$ is a linear combination of $r^n$ and $r^{-n}$. If its values at $a,b$ are $A_n,B_n$, respectively, the displayed hyperbolic-sine formula gives that combination. It follows either by solving the two boundary equations or by putting $s=\log (r/a)$, which changes the radial equation to $R_{ss}-n^{2}R=0$. The zero angular mode is instead affine in $\log r$:Superposing the modes gives the harmonic function with suitable prescribed Dirichlet boundary conditions. For a fixed interior radius, high angular modes from either boundary are exponentially suppressed. Piecewise smooth boundary data are attained at continuity points, while jumps have the usual mean-value Fourier limit.

At wavelengths much longer than the lattice spacing, $u_{j+1}-u_j\simeq a{\partial }_{x}u$ gives $L=\int dx[\rho ({\partial }_{t}u{)}^2-K({\partial }_{x}u{)}^2]/2$. The field obeys a wave equation with sound speed $v=\sqrt{K/\rho }$. Its conjugate momentum density is $\Pi =\rho {\partial }_{t}u$ and canonical quantization gives $[u(x),\Pi (x^{\prime })]=i\hslash \delta (x-x^{\prime })$. The lattice remains the ultraviolet cutoff of this long-wave theory.

The uniform displacement $X=N^{-1}{\sum }_j{u}_j$ and total momentum $P={\sum }_j{p}_j$ satisfy $[X,P]=i\hslash$. The nonzero-frequency phonon expansion alone yields the site commutator $i\hslash ({\delta }_{jl}-1/N)$; including $X,P$ restores the missing uniform part. The thermal free-coordinate factor requires a specified volume, pinning or a fixed centre of mass. Substituting zero frequency into an oscillator partition function instead produces a spurious divergence without defining the physical zero mode.

A quantum harmonic chain has quadratic couplings between canonical coordinates and momenta. A normal mode transformation turns it into independent quantum harmonic oscillators. A continuum elastic action with time coefficient $\chi$ and spatial stiffness $\kappa$ has long-wave frequency ${\omega }^2=(\kappa /\chi )q^2$, the structure of a smooth condensate phase action.

Reordering an orthonormal basis changes its matrix by a permutation similarity transformation. The off-diagonal block can move to different row and column indices without changing any energy eigenvalue. Matrix entries must therefore be compared in the same declared basis order, not only by their visual positions.

For dimensionless depth $\nu =a\sqrt{2m{V}_0}/\hslash$, each negative branch of $z\cot z=-\sqrt{{\nu }^2-z^2}$ gives one bound state. The $n$th state exists exactly when $\nu >(n-1/2)\pi$. At equality its exterior decay constant vanishes and the state is not square integrable. In particular, exactly two bound states means $3\pi /2<\nu \le 5\pi /2$. This strict lower and inclusive upper endpoint convention counts negative-energy eigenstates, excluding a zero-energy threshold solution.

A Cretaceous fossil hagfish preserving evidence of specialized soft-tissue anatomy, including slime-producing structures.

A gymnosperm genus used in comparisons of reproductive regulatory genes.^[ref]

The standard free-energy change for moving a species between specified phases or solvents. For a consistently standardized equilibrium distribution coefficient, $\Delta G^{\circ }=-RT\ln K$. Pure-solid lattice dissolution is a separate thermodynamic step.

For fixed composition and pressure, $\partial (G/T)/\partial T=-H/T^2$. Applying the identity to standard reaction quantities connects equilibrium temperature dependence to reaction enthalpy.

At imposed temperature and pressure, with fixed species amounts and only pressure-volume work, increasing total system-plus-reservoir entropy is equivalent to decreasing $G=U-TS+pV$. Unconstrained particle exchange requires the reservoir chemical potentials to be included; minimizing $G$ over arbitrary particle number is a different problem.

For a homogeneous extensive simple-compressible phase, $U=TS-pV+\mu N$ implies $G=\mu N$. A mixture instead obeys $G={\sum }_i{\mu }_i{N}_i$. Surface and other nonextensive contributions require extra terms.

The Gibbs energy of reaction for specified standard states. It determines the dimensionless equilibrium constant through ${\Delta }_r{G}^{\circ }=-RT\ln K$.

Suppose a contest uses $N_0$ trials initially and ends there with probability $1-\alpha$. Conditional on an initial tie, independent further rounds each use $\ell$ trials and resolve the contest with a fixed probability $0<r\le 1$. The number $J$ of further rounds, conditional on that tie, has a geometric distribution: $\mathrm{Pr}(J=j\mid \text{tie})=(1-r{)}^{j-1}r$ for $j\ge 1$. Thus the total number $N$ of trials has an atom $1-\alpha$ at $N_0$ and the displayed probabilities thereafter. The geometric series sums to $\alpha$, so termination has probability one, and the expected value is $\mathbb{E}[N]=N_0+\ell \alpha /r$. The displayed distribution requires independence of the future rounds from the initial tie.

A chronostratigraphic framework organizing Earth history into named intervals. Numerical boundary ages and the uncertainty of a particular fossil date are separate matters.

Change in relative continental positions through geological time, as explained within plate tectonics. Ancient connections must be reconstructed rather than read from modern maps.

Material deposited by physical, biological or chemical processes that may later be altered into sedimentary rock.

Mineralogy studies the composition, structure and properties of minerals. A chemical analysis identifies which elements are present, while X-ray diffraction constrains their periodic atomic arrangement. These measurements answer different questions: substances with the same overall composition can have different crystal structures.

Microscopy and physical properties provide complementary evidence. Mineral assemblages and textures help reconstruct rock formation, deformation and fluid interaction. Identification must allow for solid solutions, in which composition varies within one structural family, and for alteration that changes only part of a grain.