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For a positive-definite weighted inner product on a compact interval, a least-squares polynomial is the orthogonal projection of a function onto the finite-dimensional space ${\Pi }_n$ of polynomials of degree at most $n$. In a basis ${\varphi }_0,\dots ,{\varphi }_n$, its coefficients solve the normal equations $Gc=d$, where $G_{ij}=({\varphi }_i,{\varphi }_j)$ and $d_i=(u,{\varphi }_i)$. The Gram matrix is positive definite: $z^{T}Gz=\Vert \sum z_i{\varphi }_i{\Vert }^2>0$ for $z\ne 0$. Thus the coefficients exist and are unique. The residual is orthogonal to every candidate polynomial, giving $\Vert u-q{\Vert }^2=\Vert u-p_n{\Vert }^2+\Vert p_n-q{\Vert }^2$ and proving optimality. An orthogonal basis makes these equations diagonal, as in a least-squares polynomial in an orthogonal-polynomial basis.

A corner of a polyhedron, stored with its spatial coordinates and incidences with boundary edges and faces. In a closed manifold boundary representation, the incident faces form a single cyclic fan. The vertices of a polyhedron and edges define a graph, but geometric embedding and valid face incidence require additional information.

A polyhedron that is also a convex set. A bounded full-dimensional convex hull of finitely many points is a convex polyhedron; each face lies in a supporting plane, with the whole solid in its inward half-space.

A polyhedron formed by joining two pyramids along a common polygonal base on opposite sides of its plane. The common base is internal; the external boundary consists of the two sets of triangular side faces. A triangular bipyramid has five vertices of a polyhedron, nine edges and six triangular faces.

Consider a homogeneous Poisson process of arrival rate $\lambda$, with each arrival staying for a fixed duration $d>0$. For any fixed arrival index $k\ge 2$, its visit overlaps another visit if the preceding interarrival time is below $d$ or the following interarrival time is below $d$. If the nearest preceding arrival has already left then all earlier ones have left; if the next arrival is too late then all later ones are too late. These two gaps are independent exponential random variables, so the probability of no overlap is $e^{-\lambda d}{e}^{-\lambda d}$. Taking its complement gives the formula. This is an unconditional statement for a fixed indexed arrival; conditioning on a fixed calendar closing time would change the problem.

Transferring the upper half-plane hyperbolic metric $|dw{|}^2/(\mathrm{Im}w{)}^2$ through the Cayley transform gives $d{s}^2=4|dz{|}^2/(1-|z{|}^2{)}^2$ on the unit disc. Radial integration gives distance $2\mathrm{artanh}r$ from zero to $r$. The alternative convention $d{\tilde{s}}^2=|dz{|}^2/(1-|z{|}^2{)}^2$ divides the metric tensor by four and every length by two, giving $\mathrm{artanh}r$. Both have the same isometries and unparametrized geodesics, but their Gaussian curvatures are respectively minus one and minus four. In general scaling a Riemannian metric by $c>0$ scales lengths by $\sqrt{c}$ and curvature by $1/c$.

A plastic flow rule prescribes the direction and rate of plastic straining once a yield condition is active. In a plastic-potential model its direction is the stress gradient of $g$, multiplied by a nonnegative plastic multiplier. An associated flow rule takes $g$ equal to the yield function. The stress-strain split, loading conditions and hardening evolution are needed along with this equation to close an elastoplastic model.

Medicinal bark obtained from specified species of Cinchona; a historical source of quinine and related cinchona alkaloids.

The medicinal seed of Physostigma venenosum; the 1867 article is Physostigmatis faba. Its apparently ordinary seed characters do not imply that it is edible.

The sliced and dried bulb specified as Scilla in the British Pharmacopoeia by the General Medical Council (1867), botanically attributed there to Urginea scilla, with Scilla maritima as another name. The medicinal organ is a bulb, not a root.

The concrete fat obtained from the ground seeds of Theobroma cacao by expression and heat; Oleum theobromæ in the 1867 standard.

The oil expressed from the seeds of Croton tiglium; a strongly irritant historical medicinal material.

A dried plant exudate used historically as an astringent; it must not be confused with the heartwood specified as red sandalwood.

Mustard seed materials used in historical preparations. Moistening crushed black-mustard seeds generates an irritant volatile product containing allyl isothiocyanate; the volatile product differs from the fixed seed oil.

Historical medicinal capsicum fruit material, distinct from a dose of an isolated modern drug.

Resinous material used in historical stimulating plasters.

Historical medicinal material from deadly nightshade. Leaf, root, extract and tincture are not interchangeable dose units.

Historical medicinal material from aconite plants, containing toxic alkaloids.

Historical seed material containing strychnine; the extract and tincture require their own specified strengths.

Mandrake is a plant of the genus Mandragora and the root material historically derived from it. Botanical identity and chemical composition distinguish it from unrelated plants sharing the common name.