In the weighted inner product $⟨f,g⟩={\int }_0^{1}xf(x)g(x)dx$, the best approximation to $x^2$ in the span of $1,x$ is the displayed polynomial. The normal equations are $1/4-A/2-B/3=0$ and $1/5-A/3-B/4=0$. Their solution gives $A=-3/10$, $B=6/5$. The residual is orthogonal to that span, so the Pythagorean theorem proves minimality and uniqueness. Its squared norm is $1/600$. A weight vanishing at a single endpoint still gives a positive-definite inner product on continuous functions: a nonzero continuous function remains nonzero on an interval where the weight is positive.

For a bilinear form $B$ on a finite-dimensional vector space $V$, let $\Phi :V\to V^{\ast }$ send $u$ to $B(u,\cdot )$. Under the bidual evaluation map, $U^{\perp }$ is the annihilator of a vector subspace $\mathrm{im}(\Phi {|}_U)$, so $\dim U^{\perp }=\dim V-\mathrm{rank}(\Phi {|}_U)$. The rank-nullity theorem on $U$ now gives the displayed equality. In particular the sum of dimensions is at least $\dim V$. If $U\cap U^{\perp }=0$, the dimension formula for a sum of subspaces forces $V=U\oplus U^{\perp }$. The exact formula records the extra dimensions contributed by the left radical $\ker \Phi$, and does not assume an inner product.

Forsuppose $r_0>0$ satisfies $g(r_0)=a$. Then $r=r_0$ is a circular periodic orbit of period $2\pi$. Linearizing its radial equation givesso its transverse Floquet multiplier is $\exp [-2\pi r_0{g}^{\prime }(r_0)]$. It attracts radially if $g^{\prime }(r_0)>0$ and repels if $g^{\prime }(r_0)<0$; a vanishing derivative requires nonlinear terms.

For example, $g(r)=r^2$ and $a>0$ give $r_0=\sqrt{a}$, radial exponent $-2a$, and an attracting limit cycle. The angular direction is neutral because shifting the initial angle changes the phase of the same orbit. Attraction is therefore orbital stability with convergence to the orbit, rather than asymptotic convergence to one moving point. The argument uses only differentiability of $g$ and does not assume that it is a polynomial.

A periodic orbit has orbital asymptotic stability if nearby trajectories remain close and their distance to the orbit tends to zero. A persistent phase shift along the orbit is allowed.

For contingent claim payoff $(S_T-K{)}_{+}^2$ in the non-dividend Black-Scholes model, differentiating the discounted expected value gives $\Delta =2e^{-r\tau }{S}^{-1}\mathbb{E}[(S_T^2-K{S}_T)1_{\{S_T\ge K\}}]$. Completing the square in the Gaussian exponent computes each truncated lognormal moment and yields the displayed formula, where $d_1=(\log (S/K)+(r+{\sigma }^2/2)\tau )/(\sigma \sqrt{\tau })$. Its terminal limit is $2(S-K{)}_{+}$.

For a constraint written $g_i(x)\ge 0$, its violation is the nonnegative number $v_i(x)=\max \{-g_i(x),0\}$. It vanishes precisely when the constraint holds. An equality $h_j(x)=0$ has violation $|h_j(x)|$. A norm of all these violations measures total infeasibility; its numerical magnitude depends on the scaling of the constraint functions.

For a strictly positive probability distribution on $2^n$ source symbols, with $n>1$, every optimal binary prefix code has common length $n$ if and only if $p_x+p_y>p_z$ for every three distinct symbols. It suffices to compare the sum of the two smallest probabilities with the largest.

An optimal binary prefix tree has no node with only one child, because removing that detour would strictly shorten positive-probability words. It therefore satisfies equality in the Kraft inequality. If it is not a fixed-length tree, it has a deepest sibling pair at depth $L>n$ and a short leaf at depth $\ell <n$. Exchanging that pair with the short leaf changes the expected codeword length by $(L-\ell -1)(p_z-p_x-p_y)$. Strict positivity of every pair sum over a third probability makes this change negative, contradicting optimality.

Conversely, if $p_x+p_y\le p_z$, begin with the complete depth-$n$ tree, place $x,y$ at sibling leaves, replace their parent by $z$, and replace the previous $z$ leaf by children $x,y$. The new expected length is $n+p_x+p_y-p_z\le n$ and its lengths are unequal. If the optimum is below $n$, every optimal tree is nonuniform; if it is $n$, this modified tree is a nonuniform optimum. Thus equality in the probability condition already defeats the assertion that every optimum has fixed length.

The frontal nerve is a branch of the ophthalmic nerve within the orbit (anatomy). Its terminal branches include the supraorbital nerve and supratrochlear nerve, supplying sensation to forehead and adjacent scalp regions.

The lacrimal nerve is a branch of the ophthalmic nerve supplying sensation near the lacrimal gland and lateral upper eyelid. Secretory fibers can join its peripheral route from a different central origin. A nerve’s name does not imply one uniform fiber type.

The nasociliary nerve is a branch of the ophthalmic nerve carrying sensory pathways from orbital and nasal structures. Its branches include the long ciliary nerves and ethmoidal pathways. Anatomical branching helps connect a sensory territory to a central route.

The central retinal artery enters the retina through the optic-nerve route and supplies much of the inner retinal tissue. The outer retina depends strongly on the choroidal circulation. Separating these supplies explains why retinal layers need not respond identically to one vascular interruption.

The lacrimal artery is an orbital arterial branch associated with supply of the lacrimal gland and nearby tissues. Its glandular name does not restrict every branch to the gland itself. It participates in the wider network of orbital and eyelid supply.

The posterior ciliary arteries are ophthalmic branches supplying important parts of the ocular coats and associated structures. Short and long groups differ in course and distribution. Their supply is distinct from the inner-retinal territory of the central retinal artery.

The supraorbital artery is an ophthalmic branch that reaches the forehead region through the superior orbital margin. It supplies superficial tissues beyond the orbit. Its course illustrates how a vessel arising within an orbital supply system can serve an external facial territory.

The anterior ethmoidal artery is an ophthalmic-artery branch crossing toward anterior ethmoidal, nasal and neighboring tissues. Its route links orbital and nasal anatomy. It is distinct from the posterior ethmoidal branch despite their related names.

The posterior ethmoidal artery is an ophthalmic-artery branch supplying posterior ethmoidal and adjacent regions. Its course crosses the medial orbital boundary. The name identifies a regional branch rather than a primary vessel of the retina.

For a bounded skew-adjoint operator on a complex Hilbert space, the operator exponential converges in norm because its term norms are bounded by $\Vert T{\Vert }^n/n!$. Continuity of the adjoint operator gives $(e^T{)}^{\ast }=e^{T^{\ast }}=e^{-T}$. Absolute convergence permits the Cauchy product of $e^T$ and $e^{-T}$: its degree-$n$ coefficient is $T^n{\sum }_{j=0}^n(-1{)}^j/[j!(n-j)!]$, zero for $n>0$ and the identity for $n=0$. Both multiplication orders therefore give the identity, proving unitarity. The boundedness hypothesis makes this elementary power-series proof valid without domain issues for unbounded generators.

Extractum Nucis Vomicae is a concentrated nux-vomica extract. Its defined strength is needed to relate it to the source seed material.^[ref]

Tinctura Nucis Vomicae is a tincture of nux-vomica material. It contains a chemically mixed extract, not simply a stated mass of purified strychnine.^[ref]