Deleting a vertex from a Hamiltonian cycle leaves a spanning path, so every travelling-salesman tour is a one-tree. Minimizing the same edge cost over one-trees therefore gives a lower bound for the travelling salesman problem. A connected one-tree with every degree two is itself a Hamiltonian cycle, and a minimizing such one-tree certifies the tour optimum.

The vitelline duct connects the embryonic intestine to the yolk sac. In normal human development this connection narrows and disappears as the gut and surrounding structures mature.

Persistence of all or part of the connection can produce different anatomical remnants. Following the original connection explains why a remnant can occur at the intestinal side, the umbilical side or along the intervening route.^[ref]

Water intended for human consumption.

Examination of water for specified microorganisms or microbial indicators using validated procedures. Chemical clarity alone cannot establish microbiological safety.

Investigation of a water supply and its surroundings to identify plausible contamination routes and failures of protective barriers.

The historical operational fraction of Ammonia released by alkaline permanganate after preliminary removal of free Ammonia. It is neither a specific albumen measurement nor necessarily all organic nitrogen.

The capacity of water to consume soap, chiefly because of dissolved calcium and magnesium ions. Quantitative results are commonly expressed as a calcium carbonate equivalent; this does not mean all the hardness originally existed as dissolved calcium carbonate.

Water purified by distillation, used as a defined solvent in many pharmaceutical preparations.

Water condensed and precipitated from the atmosphere. It normally has little dissolved calcium or magnesium before contamination or contact with mineral-bearing surfaces; softness does not imply purity.

Pregnancy-associated vomiting is vomiting occurring in relation to pregnancy. Its presence alone does not specify the severity or establish that every episode has the same cause. Nutritional consequences and fluid loss are separate dimensions from the count of episodes.

Retching consists of coordinated expulsive movements associated with vomiting without necessarily ejecting contents. It is an observable motor pattern, distinct from nausea as a subjective experience. Different stages can therefore be measured separately in studies of emesis.

River discharge is volume flux across a channel section. For approximately uniform cross-sectional velocity it equals flow area times mean velocity. Its time dependence at a fixed station is a hydrograph.

The volume occupied by a specified gas at stated temperature and pressure.

A historical unit of volume; in the British apothecaries' system, 480 minims make one fluid ounce. It is not generally interchangeable with an uncalibrated drop.

For steady upward bubbles in a stationary liquid, assume fixed bubble mass, quadratic drag force, negligible gas weight and inertia, and negligible wall force. Then $a\propto {\rho }_g^{-1/3}$. Hydrostatic mixture support and terminal drag force balance give $v^2\propto a(1-\alpha )$. The gas mass flux $J=\alpha {\rho }_{g}v$ is constant, hence $J^2\propto {\alpha }^2(1-\alpha ){\rho }_g^{5/3}$. Comparing at a reference point proves the displayed relation. The approximation requires a small gas/liquid mass density ratio and macroscopic variation scales; it need not hold near a gas-dominated endpoint or rapid acceleration region.

For a straight pipe with cross-section $0<x<a$, $|y|<x/\sqrt{3}$, steady unidirectional incompressible flow reduces the Navier-Stokes equations to $\mu {\nabla }_{\perp }^{2}w=p_z$, with constant axial pressure gradient. The displayed cubic polynomial vanishes on every wall, and its transverse Laplacian is $p_z/\mu$. The axial drag per unit length on each of the three equal walls is $-p_z{a}^2/(3\sqrt{3})$. This follows by integrating the viscous traction on one wall; summing the three forces equals the pressure gradient times the triangular area.

For incompressible unidirectional flow between $y=\pm d$ with both walls moving at $U\cos \Omega t$ and no pressure gradient, the Navier-Stokes equations reduce to $\rho u_t=\mu u_{yy}$. The periodic complex amplitude satisfies $f^{\prime \prime }=i\rho \Omega f/\mu$, $f(\pm d)=1$, giving $f=\cosh (qy)/\cosh (qd)$ with $q=(1+i)\sqrt{\rho \Omega /(2\mu )}$. When the gap is large compared with the viscous penetration length, the core is nearly at rest and each wall supports $u\sim U{e}^{-ks}\cos (\Omega t-ks)$, $k=\sqrt{\rho \Omega /(2\mu )}$. Without initial data this specifies the periodic response; decaying zero-wall transients may be superposed.

A steady two-dimensional purely radial incompressible Newtonian fluid flow has $u_r=g(\theta )/r$ by the continuity equation. The polar Navier-Stokes equation givesEquality of the mixed pressure derivatives yields the displayed third-order equation. On fixed walls $\theta =\pm \alpha$, no-slip boundary conditions require $g(\pm \alpha )=0$. A source with volume flux $Q$ per unit out-of-plane length requires ${\int }_{-\alpha }^{\alpha }gd\theta =Q$. Consequently $g=QG$ leaves the single inertial parameter $ϵ=Q/\nu$ in $G^{(3)}+4G^{\prime }+2ϵG{G}^{\prime }=0$. The origin is a source singularity; the radial ansatz describes the wedge away from that point, not the detailed source apparatus.