For $x=r\cos \theta$, $y=r\sin \theta$ and $r>0$, set $c=\cos \theta$, $s=\sin \theta$. The differential operators are ${\partial }_x=c{\partial }_r-(s/r){\partial }_{\theta }$ and ${\partial }_y=s{\partial }_r+(c/r){\partial }_{\theta }$. Applying them twice givesTheir sum is the Laplacian in polar coordinates. The first-derivative terms come from differentiating the coordinate coefficients, as required for second derivatives under a two-variable coordinate change. The origin is excluded from the coordinate formulas, even when the underlying Cartesian function is smooth there.

Condensation in a wetting confined geometry at a vapor pressure below planar-interface saturation. A concave liquid meniscus has lower equilibrium vapor pressure in the capillarity approximation. Gap size, contact angle and surface adsorption affect the actual onset.

Write $M(a,b,c)=\left(\begin{array}{ccc}1& a& b\\ 0& 1& c\\ 0& 0& 1\end{array}\right)$ with integer coordinates. Multiplication in this parametrization is $M(a,b,c)M(A,B,C)=M(a+A,b+B+aC,c+C)$. The subgroup $H=\{M(0,0,t):t\in \mathbb{Z}\}$ has right cosets obtained by multiplying on the left by its elements, so its right cosets are exactly the displayed sets with fixed $(a,b)$. In particular, right cosets are indexed by ${\mathbb{Z}}^2$. Left cosets differ: $M(a,b,c)H$ fixes $a$ and $b-ac$, since right multiplication changes $b$ by $at$ and $c$ by $t$. This also exhibits why $H$ is not a normal subgroup of the Integer Heisenberg group.

Ice embryos form without a foreign nucleating surface once thermal fluctuations can overcome the free-energy barrier on the relevant timescale. An apparent freezing threshold depends on droplet volume and measurement duration.

Ice nucleates on an inclusion already immersed in supercooled water. The interfacial nucleation barrier, inclusion properties, droplet volume and observation time affect freezing probability.

A suitable particle colliding with a supercooled droplet can trigger heterogeneous freezing. This is distinguished from nucleation around an inclusion already inside it.

Ice forms from ice-supersaturated vapour at a suitable foreign surface. The relevant temperature and surface properties differ from liquid-droplet immersion freezing.

For one positive carrier population, define $r_H$ by $R_H=r_H/(ep)$ with $e>0$. Different statistical averages of energy-dependent scattering affect hall conductivity and longitudinal electrical conductivity, so $r_H$ need not be one. The measured Hall mobility is ${\mu }_H=\sigma R_H=r_H{\mu }_{\mathrm{drift}}$. Multiple charge carriers require a more general model even before such a correction is made.

In a one-carrier Drude model, the Hall coefficient is $R_H=1/(nq)$. Combining its inferred electron density with atomic density ${\rho }_m{N}_A/M$ gives the displayed number of mobile electrons per atom. The molar mass and mass density must be supplied independently. Multiple bands, anisotropic carrier motion and Hall factors limit this simple interpretation; field reversal helps distinguish the transverse response from voltage offsets.

The distance over which second-order group velocity dispersion substantially changes a pulse of characteristic duration $T_0$. Width conventions and initial chirp affect numerical broadening factors; the displayed definition is the standard envelope-equation scale.

A pulse's frequency components accumulate a differential group delay $\Delta t=L|{\partial }_f(1/v_g)|\Delta f$. If that spread approaches the pulse repetition period $T_r$, neighbouring symbols overlap. The resulting length estimate depends on bandwidth, initial pulse width, chirp and the receiver's allowable intersymbol interference. For $n(f)=n_0+\tau (f-f_0)$, the group index is $n_0+\tau (2f-f_0)$ and ${\partial }_f(1/v_g)=2\tau /c$: differentiating the phase index alone loses a factor of two.

Overlap of a gas absorption spectrum with existing atmospheric absorption changes its incremental radiative effect. A strong band in an already opaque spectral interval can contribute less additional outgoing-flux reduction than a weaker band in a window. Pressure broadening, altitude, temperature and the source spectrum also matter.

A measure of the spatial scale of crystal grains. Its definition must be stated for elongated crystal grains; size, aspect ratio and orientation distribution have distinct physical effects.

A measure of crystal grain elongation. It changes directional boundary spacings and can influence anisotropic defect transport independently of average grain size.

A global warming potential compares equal-mass pulses. If ${ϵ}_i$ is the greenhouse gas radiative efficiency per ppb, an emitted mass produces a mixing-ratio increment inversely proportional to molar mass $M_i$. For two gases with exponential pulse removal, their GWP ratio at horizon $H$ is $({ϵ}_i/{ϵ}_j)(M_j/M_i){\tau }_i(1-e^{-H/{\tau }_i})/[{\tau }_j(1-e^{-H/{\tau }_j})]$. The common carbon-dioxide reference integral cancels. Comparing only the per-ppb coefficients would instead compare unequal emitted masses.

For a gas pulse with unchanged greenhouse gas radiative efficiency and decay $e^{-t/\tau }$, its integrated forcing through $H$ is proportional to $\tau (1-e^{-H/\tau })$. Comparing two such lifetimes at fixed carbon-dioxide reference gives the ratio of these two factors. A fractional lifetime increase is not generally the same fractional GWP increase at finite $H$.

When spin-flip scattering is weak, two spin populations have separately defined current responses and share a terminal electric field. Their electrical conductivities add: $\sigma ={\sigma }_{\uparrow }+{\sigma }_{\downarrow }$, so $\rho ={\rho }_{\uparrow }{\rho }_{\downarrow }/({\rho }_{\uparrow }+{\rho }_{\downarrow })$. Spin-dependent band structure and scattering set the channel parameters; being the majority population does not universally imply lower channel resistivity. The model is named for Nevill Francis Mott and is the independent-channel basis for many giant magnetoresistance models.

For an east-west baseline $b$ and phase-centre zenith angle $\theta$ in its vertical plane, path compensation subtracts $b\sin \theta$. An offset $\alpha$ leaves $\delta \ell =b[\sin (\theta +\alpha )-\sin \theta ]\simeq b\alpha \cos \theta$. The complex correlation therefore has phase $2\pi b\alpha \cos \theta /\lambda$, up to receiver-order convention. Equal cable lengths do not remove this sky-dependent path difference.

A. E. Ringwood contributed to using mineral structures as hosts for radioactive waste. His SYNROC work proposed incorporating waste elements within the crystal lattices of a synthetic assemblage of minerals, drawing on the long-term retention of related elements in natural materials.^[ref]

The physical idea is more specific than surrounding waste with an inert container: the waste constituents become part of the host phases. Evaluating the resulting material requires both radiation-damage behaviour and its response to chemical alteration.

G. R. Lumpkin studies the relationships between mineral chemistry, microstructure and damage from radioactive decay. Work on naturally occurring uranium-bearing material connects accumulated alpha decay damage with observations of composition and structure.^[ref]

Such natural samples provide long-duration evidence relevant to candidate host materials for radioactive elements. Their interpretation must account for composition, thermal history and alteration, rather than treating elapsed geological time alone as a controlled irradiation experiment.