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A multiplicative subset $S\subseteq A$ with $1\in S$ and $0\notin S$ of a ring $A$ satisfies the right Ore condition if for every $a\in A$ and $s\in S$ there are $u\in S$ and $b\in A$ such that $au=sb$. For a noncommutative domain, taking $S=A\setminus \{0\}$ gives a ring of right fractions $a{s}^{-1}$; every nonzero fraction is invertible. The usual additional denominator reversibility condition is automatic when the elements of $S$ are nonzero in a noncommutative domain.

A real vector field $f$ is dissipative in an inner product iffor all $x,y$. Two solutions of the ordinary differential equation $y^{\prime }=f(y)$ satisfyThus their distance never increases. The linear case $f(y)=Ay$ recovers the dissipative operator condition. B-stability asks whether a Runge-Kutta method preserves this contraction for every positive step size.

For a limit ordinal $\kappa$, a subset $C\subseteq \kappa$ is club if it is unbounded and contains each limit below $\kappa$ of an increasing sequence from $C$. At an uncountable regular cardinal $\kappa$, intersections of fewer than $\kappa$ club sets are club: above a starting point cycle through the sets and take suprema, using regularity to stay below $\kappa$; closure puts the resulting limit in all of them. Club sets provide the reflecting ranks in worldly cardinals below an inaccessible cardinal.

For observations $X_1,\dots ,X_n$, the nearest neighbour distance of order $k$ at $x$ is the $k$th order statistic of the distances $|X_i-x|$. An atomless distance distribution function ensures ties occur with probability zero.

For independent and identically distributed random variables with distribution function $F$, the sample maximum $M_n={\max }_{1\le i\le n}{X}_i$ has distribution function $F(x{)}^n$.

With constant positive kinetic coefficient $\Gamma$ and mobility $M$, two independent relaxation channels giveThe deterministic mobility operator is $\Gamma -M{\nabla }^2$. Each channel separately satisfies microscopic reversibility when its noise obeys the corresponding Model A fluctuation-dissipation relation or fluctuation-dissipation relation for a conserved flux.

A vector order parameter has several components transforming together as a vector under spatial rotations. A polar order parameter is an example. Its functional derivative $\delta F/\delta \mathbf{p}$ has one component for each component of the order parameter.

The angular diffraction-limited resolution of a telescope scales as $\lambda /D$, with wavelength $\lambda$ and illuminated diameter $D$. Larger diameter resolves smaller angular structure, while longer wavelength makes atmospheric phase correction easier.

A reflecting telescope forms an image using curved mirrors. Its primary and secondary shapes, spacing and aperture stop determine its optical aberrations, access to the focus, and obstruction.

A Schmidt camera combines a spherical primary mirror with a thin aspheric corrector at the mirror’s centre of curvature. The corrector compensates spherical aberration; the symmetric stop location gives a wide useful field. The native focal surface is curved and lies inside the instrument.

Active optics uses controlled support forces and alignment to correct slow changes in telescope figure, particularly gravity and thermal deformation. It permits thin large primary mirrors. Adaptive optics instead corrects rapid atmospheric turbulence.

A lightweight mirror uses a thin optical face backed by ribs or a honeycomb structure. The deep backing provides rigidity with less mass than a solid disk and facilitates thermal equilibration.

A segmented mirror assembles smaller mirrors into a large aperture. Sensors and actuators maintain position and figure; coherent imaging additionally requires mirror segment phasing.

The Large Binocular Telescope has two 8.4-metre primary mirrors with lightweight honeycomb structures.

The two Keck telescopes each use a ten-metre primary made from 36 hexagonal segments.

The Thirty Meter Telescope design uses a thirty-metre primary with 492 hexagonal segments. It was among the planned extremely large telescopes in the 2018 examination context.

The Giant Magellan Telescope design combines seven 8.4-metre primary mirrors, including off-axis segments. It was a planned extremely large telescope in the 2018 examination context.