Compare the extinction-corrected apparent magnitudes of a cluster's main sequence with calibrated absolute magnitudes at matching colour or stellar parameters. The magnitude displacement estimates its distance modulus. Age, metallicity, helium abundance, reddening and unresolved binaries can move or broaden the sequence, so a colour-magnitude displacement is not uniquely a distance change.

The termination of the first-ascent red giant luminosity function near helium ignition provides a calibrated population-II distance indicator. In suitable passbands the tip luminosity varies relatively weakly over a useful old, metal-poor population range. Colour/composition corrections, foreground extinction, photometric completeness and contamination by other evolved stars must be controlled; an arbitrary brightest red star is not a reliable tip measurement.

An orbit model for a resolved maser disk combines angular positions, Doppler velocities, accelerations and sometimes proper motions to infer physical orbital dimensions and distance. For an ideal matched circular radius, $R=v^2/a_{\mathrm{orb}}$ and $d=R/\theta$; velocities and accelerations of unrelated maser spots cannot simply be inserted into that expression. Warps, inclination, eccentricity and feature identification enter real disk fits.^[ref]

For a subgroup $H$ of a group $G$ and $g\in G$, its right coset is $Hg=\{hg:h\in H\}$. The map $h\mapsto hg$ is a bijection from $H$ onto $Hg$, with inverse $x\mapsto x{g}^{-1}$; in particular all right cosets have the same cardinality as $H$. Multiplication by subgroup elements is on the left in this expression because $g$ is the factor on the right.

For a subgroup $H$ of a group $G$ and $g\in G$, its left coset is $gH=\{gh:h\in H\}$. Left multiplication $h\mapsto gh$ is a bijection from $H$ onto $gH$. Left and right cosets agree for every $g$ exactly when $H$ is a normal subgroup.

A projectile thrown across a rotating disk's diameter at speed $v$ relative to the thrower inherits the thrower's tangential velocity. During its approximate flight time $2a/v$, a receiver at the opposite rim moves tangentially in the opposite direction. The relative sideways displacement is consequently $4\omega a^2/v$ to leading order in $a\omega /v$. Integrating the Coriolis force in the disk's frame gives the same result.

The Sec23-Sec24 complex forms the inner COPII coat layer. Sec24 provides cargo-recognition sites, while Sec23 binds Sar1 GTPase and contributes to regulation of its GTPase cycle. The outer Sec13-Sec31 complex then organizes the coat scaffold. Distinct cargo-binding sites on Sec24 explain how one coat can capture different cargo classes rather than relying on a single universal recognition sequence. ^[ref]

The Sec13-Sec31 complex forms the outer layer of a COPII coat. Its assemblies provide a scaffold around a budding carrier, while the inner Sec23-Sec24 complex couples coat formation to Sar1 GTPase and cargo selection. Reconstitution using purified coat components and defined liposomes demonstrated that this machinery can generate coated carriers without the full complexity of a cell. ^[ref]

Coatomer is a multisubunit protein complex that forms the COPI coat on transport carriers. With an ARF GTPase, it helps select cargo and deform membranes in traffic associated with the Golgi apparatus, including retrieval toward the endoplasmic reticulum. Coat assembly helps generate a carrier; later targeting and membrane fusion require additional machinery. Retrieval prevents forward traffic from progressively draining resident components from earlier compartments. ^[ref]

Eight nearest neighbours at cube corners around a central site; fluorite structure cations have this arrangement.

Four nearest neighbours at tetrahedron vertices around a central site.

For $a,b>0$, the convolution of the normalized interval indicator $(2a{)}^{-1}{1}_{[-a,a]}$ with the two-sided exponential $(2b{)}^{-1}{e}^{-|x|/b}$ isSplit the defining integral at $t=x$ in the interior case; outside the interval the absolute value has a fixed sign. These elementary integrals prove the formula and its continuity at the joining points. With Fourier transform convention $\hat{w}(k)=\int w(x)e^{-ikx}dx$, the convolution theorem gives $\hat{w}(k)=\sin (ak)/[ak(1+b^2{k}^2)]$, using its continuous value at $k=0$. Parseval's identity can therefore turn the squared integral of this rational-trigonometric transform into a piecewise elementary real-space integral.

The lower boundary of the convex hull of points in the plane: the portion visible from points sufficiently far in the negative vertical direction. Sorting its vertices by horizontal coordinate gives a chain with nondecreasing edge slopes; collinear interior vertices can be omitted.

The greatest convex function lying below a given function, when such an envelope is considered in the specified domain. A convex combination of phase compositions and their Gibbs free energies constructs candidate mixtures; the equilibrium free energy is their lower convex envelope. For two differentiable coexisting branches, the supporting line is a common tangent.

The outer parallel body of a convex body $K\subset {\mathbb{R}}^n$ at distance $r\ge 0$ is $K_r=K+rB$, the Minkowski sum with the radius-$r$ closed ball. Equivalently it is the set of points at distance at most $r$ from $K$. A planar rectangle of sides $a,b$ expands into a rounded rectangle with area $ab+2r(a+b)+\pi r^2$: side strips give the linear term and four quarter-disks the quadratic term.

For a velocity field $\mathbf{u}=U(r)e_{\varphi }$ in spherical coordinates, the convective acceleration is $(\mathbf{u}\cdot \nabla )\mathbf{u}=-U(r{)}^2(e_r+\cot \theta e_{\theta })/r$, away from the polar axis. Each streamline is a circle of cylindrical radius $r\sin \theta$, so this is centripetal acceleration of magnitude $U^2/(r\sin \theta )$ directed toward the polar axis. The terms containing $U^{\prime }(r)$ cancel because a particle on such a streamline stays at fixed $r$ and $\theta$.

Given consecutive continued fraction convergents $p_{k-1}/q_{k-1}$ and $p_k/q_k$, the intermediate fractions $(t{p}_k+p_{k-1})/(t{q}_k+q_{k-1})$ for $1\le t\le a_{k+1}$ are semiconvergents, where $a_{k+1}$ is the next partial quotient. The endpoint is the next convergent. These fractions matter in best rational approximation when a denominator bound cuts between consecutive convergents.

Fix an irrational number $\alpha =[0;a_1,a_2,\dots ]$ and let $q_k$ be its convergent denominators, with $q_{-1}=0,q_0=1$. Ostrowski numeration writes each nonnegative integer uniquely as a finite sum $N={\sum }_{k\ge 0}{b}_{k+1}{q}_k$, subject to $0\le b_1<a_1$, $0\le b_{k+1}\le a_{k+1}$ for $k\ge 1$, and $b_{k+1}=a_{k+1}\Rightarrow b_k=0$.

The digit restrictions remove the ambiguities caused by $q_{k+1}=a_{k+1}{q}_k+q_{k-1}$. For $\alpha =(\sqrt{5}-1)/2$, all partial quotients equal one, so this becomes the representation by nonconsecutive Fibonacci numbers, with the duplicate initial weight removed. The numeration is useful for linking integer expansions to irrational rotations and Sturmian words.

The partial quotients of a simple continued fraction $x=[a_0;a_1,a_2,\dots ]$ are its integer coefficients: $a_0=\lfloor x\rfloor$ and $a_i\ge 1$ for $i\ge 1$. Repeatedly subtract the integer part and invert the positive remainder to obtain the next coefficient.

For $17/7$, this gives $[2;2,3]$, since $17/7=2+1/(2+1/3)$. For $\sqrt{2}$, it gives $[1;2,2,\dots ]$. Truncation after $a_k$ gives a continued fraction convergent. A large next partial quotient makes the current convergent particularly accurate because the next reciprocal tail is small.

For the continued fraction convergent $p_k/q_k=[a_0;a_1,\dots ,a_k]$ in lowest terms, $q_k>0$ is its convergent denominator. With $q_{-2}=1$ and $q_{-1}=0$, it satisfies $q_k=a_k{q}_{k-1}+q_{k-2}$.

For $[1;2,2,\dots ]=\sqrt{2}$, the first denominators are $1,2,5,12,29$. Their growth measures the increasing cost of representing a better best rational approximation. The determinant identity $p_k{q}_{k-1}-p_{k-1}{q}_k=(-1{)}^{k-1}$ proves that neighboring continued fraction convergents are reduced and tightly linked; in particular, their difference has absolute value $1/(q_k{q}_{k-1})$.