The radial vector field $B=e_r/r^2$ on punctured three-dimensional Euclidean space has flux $4\pi$ through every centered sphere. It has no vector potential smooth on an entire enclosing sphere: the Stokes theorem on a closed surface would force zero flux. Two overlapping patches suffice. In spherical coordinates take $A_{+}=(1-\cos \theta )e_{\varphi }/(r\sin \theta )$, regular away from the negative $z$ axis, and $A_{-}=-(1+\cos \theta )e_{\varphi }/(r\sin \theta )$, regular away from the positive $z$ axis. Their curls both equal $B$.

The Cartesian forms $A_{+}=(-y,x,0)/[r(r+z)]$ and $A_{-}=(y,-x,0)/[r(r-z)]$ show that the first is smooth at the north pole and the second at the south pole. On the overlap their difference is $A_{+}-A_{-}=2\nabla \varphi$, where the azimuth $\varphi$ is defined locally. Its multivaluedness around the axis explains the circulation difference $4\pi$. Splitting the sphere along a latitude, applying the Stokes theorem with $A_{+}$ on the north and $A_{-}$ on the south, and using their opposite boundary orientations recovers the total flux. A coordinate singularity in the tangential spherical basis does not by itself make a vanishing smooth vector field singular at a pole.

Diversification of a lineage into ecologically differentiated descendants. Colonization, ecological opportunity and reproductive barriers are distinct contributing processes.

A diversification rate describes change in the number of evolutionary lineages. Under a simple branching model with speciation rate $\lambda$ and extinction rate $\mu$, net diversification is $r=\lambda -\mu$. Equal net rates can therefore arise from slow turnover or from rapid speciation accompanied by rapid extinction.

An observed accumulation of living lineages does not uniquely identify both component rates. Fossils can supply additional evidence about extinct branches, with taphonomy and sampling affecting which histories are observed.

Ring formation favored by binding an anionic template that organizes the reactive precursors.

The epithelial lining of pulmonary gas-exchange units, containing differently specialized cell types.

Movement before air breathing that contributes to fluid-related mechanical stimulation of the developing lung. Physical loading and mature respiratory function are distinct.

locomotion by jumps. Skeletal geometry can constrain jumping capability but does not establish the exclusive behaviour of an extinct organism.

Fritz London and Heinz London formulated a phenomenological theory of the magnetic response of superconductors. Their equations make a static magnetic field decay over a finite London penetration depth, explaining field expulsion in the Meissner effect. This adds a physical condition beyond the acceleration of a perfectly conducting electron fluid. The London equations connect a measurable length scale with superconducting currents, providing a useful macroscopic description before a microscopic pairing theory was available.^[ref]

A physicist associated with the London dispersion force and London equations.

John Lennard-Jones used gas-transport measurements to constrain models of intermolecular forces. His 1924 work connects the temperature dependence of viscosity with an assumed force law. The familiar modern Lennard-Jones potential represents short-range repulsion and longer-range attraction as $V(r)=4ϵ[(\sigma /r{)}^{12}-(\sigma /r{)}^6]$. Its minimum at $r=2^{1/6}\sigma$ has depth $ϵ$, making the two parameters physically interpretable. This 12–6 form is a simplified interaction model, rather than an exact force law for every atom pair.^[ref]

A physicist associated with the Lennard-Jones potential.

An experimental physicist who coauthored a study of superconductivity in germanium-based cage compounds. Guest atoms occupy the cages, and their displacement can introduce substantial disorder. Applying hydrostatic pressure strongly changed the superconducting transition in one compound, providing a controlled way to study the relation between atomic arrangement, disorder and low-temperature electronic behavior.^[ref]

Homepage, portrait and biography: www.phy.cam.ac.uk/profile/prof-malte-grosche/

A physicist associated with low-temperature studies of CeAl3 and the discovery of heavy fermion behaviour.^[ref]

Institutional history: www.wmi.badw.de/the-institute/profile

A physicist who used neutron scattering to investigate lattice dynamics, including the phonon anomaly associated with the Peierls transition in TTF-TCNQ.^[ref]

Memorial biography: ieee-uffc.org/contact/gen-shirane

A physicist whose publications include the cold-atom Bloch oscillation experiment. His own publication list identifies the 1996 paper whose published byline uses Maxime Ben Dahan.

Publication list: www.phys.ens.fr/~dahan/Publications.html

An experimental physicist who, with collaborators, reported electron crystallization on liquid helium. A sheet of electrons above the liquid surface formed a strongly interacting, nearly two-dimensional system. Measuring the transition as density changed tested the competition between electrostatic interaction and thermal motion, providing a particularly clean setting for studying a crystal made from charges rather than atoms.^[ref]