The total space of a real vector bundle $E\to M$ has the exact sequence $0\to {\pi }^{\ast }E\to TE\to {\pi }^{\ast }TM\to 0$. Taking determinant line bundles gives the displayed identity. If $M$ is an orientable smooth manifold, the total space is orientable exactly when $\det E$ is trivial. Sufficiency follows by pullback; necessity follows by restricting the total-space determinant identity to the zero section. For the cotangent bundle, base and fibre coordinate determinants cancel, giving an orientation even when the base is not orientable.

The rank of a terminal node is zero; the rank of any node is the supremum of the successor ranks of its immediate extensions. The rank of the empty node gives one convention for tree height. Every countable well-founded tree has countable rank. Changing between empty-root rank and successive leaf-deletion order changes harmless endpoint conventions.

For an ordered basis $\mathcal{B}=(b_1,\dots ,b_n)$, the coordinate vector $[v{]}_{\mathcal{B}}=(a_1,\dots ,a_n{)}^T$ records the unique coefficients satisfying $v={\sum }_i{a}_i{b}_i$. Change of basis changes this representation while preserving the underlying vector.

For $b_1=(1,0{)}^T$, $b_2=(1,1{)}^T$, the vector $v=(3,1{)}^T$ has coordinate vector $(2,1{)}^T$, since $2b_1+b_2=v$. These are the same coefficients extracted by the dual basis in its worked example. The vector is independent of the chosen basis; its list of coordinates is not. In an orthonormal eigenbasis, the coordinates additionally separate the action of a self-adjoint operator into independent scalar multiplications.

Give ${\mathbb{R}}^2$ the lexicographic order and the order topology. Every point $(a,b)$ has an interval neighbourhood between $(a,b-\epsilon )$ and $(a,b+\epsilon )$, lying entirely in the vertical line $\{a\}\times \mathbb{R}$. These intervals give each such line its usual real topology, and every line is open. Thus this space is a disjoint union of uncountably many open copies of the real line and is not a second-countable space: any basis of a topology must supply a different member inside each of these disjoint nonempty open sets.

A strict order-preserving function preserves strict comparisons. Between strict partial orders it may identify incomparable elements. Between strict total orders it is necessarily injective, and in a finite chain it is exactly an increasing injection. It need not reflect comparisons between incomparable elements of general partial orders.

An inflationary map on a partially ordered set satisfies $x\le f(x)$ for every $x$. This condition is distinct from being order-preserving.

An angular coordinate around the celestial sphere measured along the Earth's orbital reference plane. An Earth-based ecliptic longitude specifies a viewing direction; it does not by itself give a body's distance or its Sun-based position.

An armillary sphere is an astronomical model made from rings representing reference circles on the celestial sphere. Rotating the rings makes relations among the sky's reference planes visible in three dimensions. A central globe can represent the Earth, while inclined rings represent the apparent solar path and other celestial circles.

Royal Museums Greenwich records a 1602 instrument with eight brass rings, a horizon ring, movable solar and lunar ring systems and a small terrestrial globe. Its nested parts make the reference geometry tangible rather than merely drawing intersecting circles on paper.^[ref]

An armillary sphere is useful for explaining why an angular coordinate depends on a chosen reference plane. It represents directions and geometric relations; its ring sizes should not be interpreted as a scale model of the distances to the Sun, Moon or stars.

For radial acceleration $-\mu /r^2+C/r^3$, the Binet equation gives $u^{\prime \prime }+(1+C/h^2)u=\mu /h^2$, where $h$ is specific angular momentum. A regular bound orbit has $h^2+C>0$, $a=-\mu /(2\mathcal{E})$ and $e^2=1+2\mathcal{E}(h^2+C)/{\mu }^2$. Its radius is $a(1-e^2)/[1+e\cos (\alpha \theta )]$, with $\alpha =\sqrt{1+C/h^2}$. The apsidal precession per radial period is $2\pi ({\alpha }^{-1}-1)$, which reduces to the displayed expression for a small perturbation. Negative $C$ produces prograde precession; positive $C$ produces retrograde precession. The dimensionless coefficient is $C/(\mu a)$, since $C$ has units of length to the fourth power divided by time squared.

In the gravitational potential $-\mu (1/r+a/r^2)$, a regular noncircular bound orbit requires $L^2>2\mu a$ and has $u=\mu (L^2-2\mu a{)}^{-1}[1+e\cos \nu (\psi -{\psi }_0)]$ with $0<e<1$. Each radial period advances the angle by $2\pi /\nu$. The full position and velocity return after finitely many radial periods precisely when $\nu$ is rational. Irrational $\nu$ gives a dense rosette on the two-frequency invariant torus. Rationality permits a fifth orbit label fixing the apsidal phase on that resonant angular momentum family, but does not give global superintegrability for all nearby angular momenta. When $a=0$, every regular bound Kepler orbit is closed and the Laplace-Runge-Lenz vector supplies the globally defined additional integral.

For $u=u_c[1+e\cos \nu (\psi -{\psi }_0)]$ with $\nu >0$ and $e\ge 1$, the positive-radius scattering branch has $|\nu (\psi -{\psi }_0)|<\arccos (-1/e)\le \pi$. Thus the principal complex argument in the displayed expression is continuous throughout that branch, unlike at an apocentre of a bound orbit. Its derivative is $\nu$, so the angle modulo $2\pi$, or the associated unit apsidal vector in the orbital plane, is a stationary integral of motion fixing the periapsis direction. Together with energy and three components of specific angular momentum, it gives five regular scattering labels even for irrational $\nu$. Periodicity is therefore necessary for five orbit labels in recurrent bound motion, not in general for escaping trajectories.

If $\tau ⪯\sigma$, choose $m\in {\sigma }^{\vee }\cap M$ with $\tau =\sigma \cap m^{\perp }$. Then $U_{\tau }=D({\chi }^m)\subset U_{\sigma }$. Its complement is the union of orbit closures indexed by faces of $\sigma$ that are not faces of $\tau$.

// Target: geometry-and-topology.bigb

Work with closed points over an algebraically closed field $k$. For $p\in \mathrm{Spec}k[C\cap M]$, the exponents $m$ for which ${\chi }^m(p)\ne 0$ form a semigroup face: a product is nonzero exactly when both factors are nonzero. In a saturated polyhedral monoid these exponents are precisely the toric lattice points of one face of $C$.

The nonzero values define a homomorphism from the group generated by that face to $k^{\ast }$. Extending it to the full toric lattice shows that points with the same support face differ by the action of the dense algebraic torus. They therefore form a single orbit.

For ${\mathbb{A}}^2$, the possibilities are both coordinates nonzero, one specified coordinate nonzero and the other zero, or both zero. These are the dense algebraic torus, the two punctured axes and the origin. The closed-point/field convention avoids confusing these geometric orbits with arbitrary scheme points or rational-point descent.^[ref]

// Target: mathematical-optimization.bigb

An adapted process with integrable coordinates is a martingale exactly when the displayed equality holds for every bounded stopping time. For the forward implication, expand $M_T-M_0={\sum }_{k=1}^N{1}_{\{T\ge k\}}(M_k-M_{k-1})$ and condition each term on ${\mathcal{F}}_{k-1}$. Conversely, for $A\in {\mathcal{F}}_n$ choose $T=n+1$ on $A$ and $T=n$ elsewhere. Subtract the equality for the constant stopping time $n$ to obtain $\mathbb{E}[1_A(M_{n+1}-M_n)]=0$, which is the defining conditional expectation identity.

For an integrable supermartingale with the displayed increment bound, $|X_{T\wedge n}|\le |X_0|+KT$. Thus the stopped values are dominated by one integrable random variable, including $X_T$. Apply optional stopping at the bounded times $T\wedge n$ and then the dominated convergence theorem. For a martingale the same proof gives equality.

An approximate design is a probability measure on the design region, with weight $w_i$ specifying a proportion of observations at $x_i$. For $N$ independent observations and common error variance ${\sigma }^2$, a linear regression with vector $f(x)$ has normalized information matrix of an experimental design $M(\xi )=\int f(x)f(x{)}^{T}d\xi (x)$; its least-squares estimator has covariance ${\sigma }^{2}M(\xi {)}^{-1}/N$ when the allocations realize these proportions. Exact designs restrict $N{w}_i$ to integers.

A D-optimal design maximizes the determinant of the information matrix of an experimental design, equivalently minimizing the determinant of the covariance matrix of a fixed-size least-squares estimator. It minimizes the volume of a fixed-level confidence ellipsoid for the model coefficients under the usual Gaussian model.

A G-optimal design minimizes the maximum design sensitivity function over the full design region. Thus it minimizes the worst variance of a fitted regression mean, excluding the separate noise variance of a new observation. The sensitivity average is $p$, so its supremum is at least $p$.

For continuous regressors spanning a $p$-dimensional space on a compact design region, and approximate designs with nonsingular information matrix of an experimental design, the following are equivalent: D-optimal design, G-optimal design, and ${\sup }_{x}d(x,\xi )=p$. A D-optimal design has nonpositive directional derivatives $d(x,\xi )-p$. Conversely, if every such derivative is nonpositive, concavity of $\log \det$ bounds every competing information matrix by the tangent at $M(\xi )$, proving D-optimality. The sensitivity average is $p$, and a D-optimal design exists by compactness of the information-matrix set and the spanning assumption; hence the optimal G-value is $p$. Sensitivity equals $p$ at every support point of an optimal design. The equivalence concerns approximate allocations; integer restrictions can change exact-design optima.

A dioptre, also spelled diopter, is the unit of optical power equal to one reciprocal metre. In $\Phi =1/f$, convert focal length to metres before taking the reciprocal: $f=25\mathrm{cm}=0.25\mathrm{m}$ gives $\Phi =4\mathrm{D}$, not $0.04\mathrm{D}$. A dioptre measures focusing strength, distinct from the transverse magnification of an image.