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For the Hermite differential equation $f^{\prime \prime }-2\xi f^{\prime }+(ϵ-1)f=0$ with real $ϵ$, the wavefunction $e^{-{\xi }^2/2}f$ is square integrable on the real line only if $f$ is a polynomial. To prove this without assuming that even and odd tails cannot cancel, project a square-integrable solution onto its even and odd parts first. Each projection remains square integrable and satisfies the same equation.

In either nonterminating parity sequence, $b_j=a_{2j+p}$ obeysThe tail has constant complex phase. For any fixed $0<\delta <1/2$, comparison with $(1-\delta )/(j+1)$ bounds its coefficient magnitudes below by $K(1-\delta {)}^j/j!$ for sufficiently large $j$. Thus on the positive real axis the tail dominates a positive multiple of ${\xi }^p{e}^{(1-\delta ){\xi }^2}$ after subtraction of finitely many terms. The Gaussian function factor leaves exponential growth $e^{(1/2-\delta ){\xi }^2}$, contradicting the square-integrable condition. Hence each nonzero parity sequence terminates. Its top coefficient forces $ϵ=2n+1$, giving Hermite oscillator quantization and a Hermite polynomial of degree $n$.

At a stationary point of the power penalty, define the displayed multiplier estimates. Its stationarity equation becomes $\nabla f-D{c}^{T}y=0$. If $\mu \downarrow 0$ and these multipliers are bounded, then each $c_i\to 0$. Along any convergent primal subsequence, a further multiplier subsequence converges and yields equality-constrained first-order KKT conditions at the limit. This is a statement about finite cluster points: bounded multipliers alone do not prove convergence or boundedness of the primal iterates.

In uniform gravity a body's gravitational potential energy is its mass times gravitational acceleration times the Cartesian height of its centre of mass. If that height is represented by an axis distance $h$ and tilt angle $\theta$, it is $h\cos \theta$ and the energy becomes $Mgh\cos \theta$. Treating $h$ simultaneously as a Cartesian height would introduce an extra, unjustified cosine.

Historical pharmaceutical preparation; its composition and clinical use must be identified by edition. ^[ref]

Near an equilibrium, a positive-definite scalar function has value zero at that point and is strictly positive at every other nearby point. This scalar notion differs from positive definiteness of a kernel.

For a nonzero polynomial $g\in \mathbb{Q}[t]$, choose a positive integer $n$ clearing its denominators and set $c(g)=c(ng)/n$, where the polynomial content of an integer-coefficient polynomial is the positive greatest common divisor of its coefficients. This is independent of $n$: if $f/n=h/m$, then $mf=nh$, hence $mc(f)=nc(h)$.

The integer-coefficient product formula from Gauss lemma for polynomials givesMoreover $g/c(g)$ is an integer-coefficient primitive polynomial: if $g=f/n$, then $g/c(g)=f/c(f)$. For example,This normalization proves a useful divisibility descent. If $f\in \mathbb{Z}[t]$ is primitive, $g\in \mathbb{Z}[t]$ and $g=fq$ for $q\in \mathbb{Q}[t]$, then $c(q)=c(g)$ is an integer. Since $q/c(q)\in \mathbb{Z}[t]$, it follows that $q\in \mathbb{Z}[t]$.

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.