For a lossless polytropic flow with $\gamma >1$, negligible gravity and a sonic throat, the Bernoulli function is $u^2/2+c_s^2/(\gamma -1)=(\gamma +1)c_{\ast }^2/[2(\gamma -1)]$. On an expanding supersonic branch for which $\rho \to 0$, the specific enthalpy vanishes and the limiting velocity is the displayed value. If area grows as $C{r}^2$, mass conservation further gives $\rho \sim \dot{M}/(C{u}_{\infty }{r}^2)$. A subsonic pressure-confined branch need not have this asymptotic behavior.

The one-dimensional trivial representation, in which every symmetry operation acts as $+1$. An integral over a symmetry-invariant domain can be nonzero only when its integrand contains such a component. Molecular character tables often label it $A_1$ or $A_g$.

A parallel beam incident at elevation $\alpha$ has horizontal irradiance $F_h=F_{\perp }\sin \alpha$. Its path optical depth in a plane-parallel atmosphere is ${\tau }_v/\sin \alpha$. Confusing these projections changes both attenuation and intercepted power.

The number of inversions of a word: pairs of positions $i<j$ with $w_i>w_j$. Unlike an inversion of a permutation, this definition permits repeated letters in the word.

The inversion number of a permutation $a_1\cdots a_n$ isFor 3142, the inverted pairs of values are $(3,1),(3,2),(4,2)$, giving three. The inversion number ranges from zero for increasing order to $\left(\genfrac{}{}{0ex}{}{n}{2}\right)$ for decreasing order; it is the minimum number of adjacent transpositions needed to sort the permutation. Its parity gives the sign of a permutation, $(-1{)}^{\mathrm{inv}(a)}$. The same pair-counting formula defines the word inversion number when repeated symbols are allowed.

For real $-1\le x\le 1$, the principal arccosine is the unique angle $\theta \in [0,\pi ]$ satisfying $\cos \theta =x$. Restricting to this interval makes the cosine one-to-one. Differentiating the identity on $-1<x<1$ gives $d(\arccos x)/dx=-1/\sqrt{1-x^2}$. Other solutions of $\cos \theta =x$ are $\theta =\pm \arccos x+2\pi n$, with integer $n$; an initial condition can select one of these branches.

At $B=-L^2/(2m)$ the centrifugal and force potentials cancel. With zero radial speed, any radius is a circular solution, but there is no radial restoring force; radial-velocity perturbations are not confined.

A genetic element that can spread through a transmission or replication advantage even when it imposes costs on the organism carrying it.

Transmission distortion favouring an element over the Mendelian expectation. In a strict usage this acts during meiosis; broader usage also includes biased survival of gametes.

Two canonically normalized complex scalar fields with potential $V=m^2{\Phi }^{†}\Phi +\lambda ({\Phi }^{†}\Phi {)}^2$ are four real fields with a radial potential. Consequently the isolated action is invariant under the orthogonal group $O(4)$, not only under complex-linear $U(2)$ transformations. Writing ${\varphi }_1=(q_1+i{q}_2)/\sqrt{2}$ and ${\varphi }_2=(q_3+i{q}_4)/\sqrt{2}$ makes the kinetic term ${\sum }_A(\partial q_A{)}^2/2$ and the potential a function of ${\sum }_A{q}_A^2$. The six real-plane rotations give currents $j_{AB}^{\mu }=q_B{\partial }^{\mu }{q}_A-q_A{\partial }^{\mu }{q}_B$. Gauge or Yukawa couplings can break this larger symmetry, so it should not be asserted for an arbitrary interacting doublet.

For a simple compressible system at fixed composition and particle number, the internal pressure is the partial derivative of internal energy with respect to volume at fixed temperature. From $dU=TdS-pdV$ and the Maxwell relation $(\partial S/\partial V{)}_T=(\partial p/\partial T{)}_V$, it has the measurable expression in the title. It is distinct from the mechanical pressure $p$.

For an ideal gas, $p=nRT/V$ gives ${\pi }_T=0$, so its internal energy depends only on temperature at fixed amount of gas. For the Dieterici equation $p=nRT\exp [-an/(VRT)]/(V-nb)$, direct differentiation givesThus an equation of state determines this volume sensitivity even without determining the full caloric function $U(T,V)$.

An existing charge distribution polarizes a neighbouring molecule, changing its electron density and interaction energy. The resulting interaction depends on the molecular polarizability and local electric field.

Short-range repulsion associated with the antisymmetry requirement for overlapping electronic states, related to the Pauli exclusion principle. It competes with attractive electrostatic and London dispersion forces at small intermolecular separations.

Two benzene molecules associated by dispersion and electrostatic interactions. T-shaped and parallel-displaced geometries are important low-energy arrangements; a single permanent dipole–dipole model is inadequate.

An electrostatic interaction involving an electric quadrupole moment, the second-order term in a multipole expansion. About a specified origin, use the conventionHere $q_a$ is an electric charge, ${\mathbf{r}}_a$ its position, and ${\delta }_{ij}$ the Kronecker delta. Far from the electric charges, the quadrupole contribution to electrostatic potential iswhere ${\epsilon }_0$ is vacuum permittivity. Thus a point electric charge has quadrupole interaction potential energy proportional to $r^{-3}$ at fixed orientation. A second electric quadrupole moment couples to second derivatives of this electrostatic potential, giving quadrupole–quadrupole potential energy proportional to $r^{-5}$ at fixed orientations. These are far-field terms, rather than the whole intermolecular force at close contact.

Benzene has no permanent electric dipole moment by symmetry, but its electric quadrupole moment need not vanish. Its electrostatic interaction therefore depends on the relative molecular orientations even without permanent electric dipole moments. Published electric quadrupole moment conventions can differ by a factor of two; the tensor convention above must be matched before inserting tabulated values.^[ref]

Electrons reflected by two magnetic interfaces can mediate an interaction across a nonmagnetic metal spacer. The phase accumulated on traversing the spacer varies with its thickness, producing oscillatory contributions to $J(t)$ from Fermi surface spanning vectors. In the sign convention in the title, positive $J$ favours parallel moments and negative $J$ antiparallel moments. Multiple periods, interface phase shifts, disorder and finite temperature prevent a universal sign rule based on thickness alone.

A Rayleigh interferometer compares optical paths through a sample and a reference. A path-length difference $\Delta \ell$ shifts the interference phase by $2\pi \Delta \ell /\lambda$. In analytical ultracentrifugation, sample and reference sectors produce concentration-dependent fringe shifts, allowing radial concentration scans without requiring the solute to absorb at the probe wavelength.^[ref]

Measurement using wave interference: relative path or phase changes are inferred from the resulting signal. The interacting fields must have sufficient optical coherence in the measured modes for the intended interference signal.

A move removes one counter from each nonempty pile and forms a new pile from the removed counters. Thus the move acts on an integer partition of a fixed total.

An integer partition in which every part is a power of two. Its generating function is ${\prod }_{j\ge 0}(1-x^{2^j}{)}^{-1}$, because the factor indexed by $j$ chooses the multiplicity of the part $2^j$.