If holding one unit of a commodity costs a constant cash amount $\delta dt$ for storage, its gain is $dS-\delta dt$. A delta hedge long a derivative and short $V_S$ commodity units has riskless drift $V_t+\frac{1}{2}{\sigma }^2{S}^2{V}_{SS}+\delta V_S$ and value $V-S{V}_S$. Equating the drift to the bank return gives the displayed Black-Scholes equation. This is an additive cash carry term, unlike a proportional dividend or storage yield.

The corresponding pricing diffusion is $d\hat{S}=(r\hat{S}+\delta )dt+\sigma \hat{S}dW$. A fixed-strike forward has value $S+\delta (1-e^{-r\tau })/r-K{e}^{-r\tau }$, and put-call parity gives that same value for a matching call minus put. Replication buys the commodity and a bank reserve equal to the present value of its future storage payments. For $r=0$, the reserve is $\delta \tau$.

Differentiating the displayed transformation shows that it preserves the Black-Scholes equation. The first derivative coefficient fixes the exponent $1-2r/{\sigma }^2$, which also makes the zero-order coefficient correct. Subtracting this reflected image of a European call option from the call itself enforces a zero boundary at $S=B$. For strike price $K\ge B$, the reflected maturity contingent claim payoff vanishes on $S>B$, giving the no-rebate down-and-out option formula. Prices using this reflection are conditional on the barrier not having been hit already.

The bisection method preserves some root in a sign-changing bracket but does not select a fixed root when several are present. For the displayed polynomial, endpoint and midpoint roots are returned immediately. If $2<b<3$, the first midpoint lies in $(1/2,1)$ with negative function value, so the retained bracket contains only zero. If $b=3$, the first midpoint is exactly one. If $3<b\le 4$, that midpoint lies in $(1,3/2]$ with positive function value, so the retained bracket contains only two. At $b=2$ the right endpoint itself is the root two.

Thus arbitrarily small changes of the initial endpoint across $b=3$ change the limiting root from zero to two, with one selected exactly at the transition. The guarantee concerns distance to a root inside the retained interval, rather than continuous dependence of that root's identity on the starting data.

Two expressions are equal when they denote the same mathematical object. An equality test in a program implements a specified comparison of represented values; for example, approximate floating-point arithmetic comparison is not identical to exact equality of real numbers.

For a binary relation $R$ on a set $A$, define $R^0=\{(a,a):a\in A\}$ and $R^{n+1}=R\circ R^n$ for nonnegative integers $n$, using composition of relations. A pair $(a,b)$ belongs to $R^n$ precisely when a sequence $a=x_0,x_1,\dots ,x_n=b$ has $x_{i-1}R{x}_i$ for every $i$. This follows by mathematical induction, splitting off one step in the recurrence. Thus $R^n$ records paths with exactly $n$ steps, rather than at most $n$ steps. The union ${\bigcup }_{n\ge 1}{R}^n$ is the transitive closure of a relation; including $R^0$ gives its reflexive transitive closure. The powers of a single relation agree under the two common composition-order conventions.

Reversing every ordered pair of a binary relation. Swapping the two components is its own inverse and therefore preserves distinctness of a representation by a list of distinct pairs.

A binary relation is right Euclidean when any two successors of the same element are related in the indicated order. A reflexive relation with this property is an equivalence relation: reflexivity gives symmetry, and symmetry followed by the Euclidean property gives transitivity. Without reflexivity, even the empty binary relation is Euclidean.

A weighted quadratic Bézier curve with positive middle weight $w$ retains endpoint interpolation and endpoint tangent directions while changing the bulge. For a circular arc of angle $\theta$ less than $180^{\circ }$, choose endpoints on the circle, their tangent intersection as the middle control point, and $w=\cos (\theta /2)$. Unlike an ordinary nonconstant polynomial Bézier curve, this rational representation can describe a circular arc exactly.

Use the convention $K_m={\sum }_{|j|\le m}(1-|j|/(m+1))e^{ijt}$. Multiplying $K_{n-1}$ by $2n\sin (nt)$ gives Fourier coefficients $-ik$ for $|k|\le n$. A trigonometric polynomial of degree at most $n$ has no other frequencies, so convolution proves the displayed identity. Positivity and integral one of the Fejér kernel then give $\Vert P^{\prime }{\Vert }_{\infty }\le 2n\Vert P{\Vert }_{\infty }$. This is a useful elementary estimate; the sharp Bernstein inequality for trigonometric polynomials improves $2n$ to $n$.

In the area Hilbert space of the unit disk, conjugates of Bergman space functions vanishing at zero are orthogonal to all holomorphic functions, since ${\int }_{D}hgdA=\pi h(0)g(0)$. This follows from the angular mean-value property and the Cauchy-Schwarz inequality, which makes the product integrable. But the radial function $|z{|}^2-1/2$ is also orthogonal to every holomorphic function: its radial integral against the constant angular mean is zero. It is not conjugate holomorphic, since it is real-valued and nonconstant. Thus the conjugate-holomorphic family is a proper subset of the orthogonal complement.

Extractum Belladonnae is a concentrated belladonna extract. It is distinct from an equal mass of leaf or root material.^[ref]

Tinctura Belladonnae is a belladonna tincture. Solvent extraction produces a mixture of constituents rather than a single purified alkaloid.^[ref]

Emplastrum Belladonnae is a belladonna-containing plaster. Its supporting mass and plant-derived constituents describe different parts of the preparation.^[ref]

Linimentum Belladonnae is a belladonna liniment. The external liquid form is part of its pharmaceutical identity.^[ref]

Unguentum Belladonnae is a belladonna ointment. Its semisolid formulation differs from a belladonna tincture or extract.^[ref]

A beetle of the family Meloidae, several members of which supply cantharidin.

A family of open subsets is a subbasis of a topology when its finite intersections form a basis of a topology, including the empty intersection equal to the entire space. For the product topology, inverse images of open sets under the product projections form the standard subbasis of a topology. A map into the space is continuous if the inverse image of every subbasic set is open.

For the normalized discrete Fourier transform basis over ${\mathbb{Z}}_N$, products of coefficient functions satisfy $f_m{f}_n=N^{-1/2}{f}_{m+n}$. Taking adjoints of the multiplication rule for the basis-copying isometry gives the displayed identity, with addition modulo $N$. Its $N$ output terms are orthonormal, so the prefactor is required by norm preservation. Each reduced quantum state is $I/N$, making the output a maximally entangled state for $N\ge 2$. This explains concretely why copying one basis is incompatible with copying its Fourier basis.

The unique scalar coefficients in the finite linear combination expressing a vector in a chosen basis. Spanning gives existence, while linear independence makes the difference of two proposed representations have all zero coefficients. Ordering a finite basis turns these coefficients into a coordinate vector.

Pressure changes the melting temperature of ice according to the Clapeyron equation. Ice pushed against an obstacle can melt on its high-pressure side and refreeze downstream where pressure is lower. Conduction of the released latent heat back towards the melting region permits motion around the obstacle. Small obstacles favor this heat-transfer mechanism; larger ones can be bypassed mainly by ice deformation.