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Let $c$ be a bounded real measurable coefficient on an open set. Nonnegativity of $c$ clearly makes its gradient form nonnegative. Conversely, if $c<0$ on a set of positive measure, choose an interior Lebesgue point $x_0$ where $c(x_0)<0$. For a nonconstant compactly supported smooth function $\psi$, set $v_{\epsilon }(x)=\epsilon \psi ((x-x_0)/\epsilon )$. Change variables and use the Lebesgue differentiation theorem to obtain ${\epsilon }^{-n}\int c|\nabla v_{\epsilon }{|}^2\to c(x_0)\int |\nabla \psi {|}^2<0$, a contradiction. Thus the linear operator represented by $a(u,v)=\int c\nabla u\cdot \nabla v$ is monotone exactly when $c\ge 0$ almost everywhere.

An edge record stores endpoint vertices of a polyhedron, its two incident faces, and predecessor and successor edges for each face. These four boundary-neighbour references are the wings. Face and vertex of a polyhedron records also carry entry references for traversal. Local edits must preserve all reciprocal incidence references and orientations.

A mesh representation with two oppositely directed records per ordinary edge. Each record stores its origin vertex of a polyhedron, incident face, next face-boundary record and opposite record; previous records are often cached. The opposite record's origin gives the destination. This usage differs from a dangling half-edge of a graph.

A boundary condition at spatial infinity specifies limiting behaviour as the spatial coordinate leaves every bounded set, such as ${\lim }_{x\to +\infty }u(x,t)=g(t)$ on a half-line. A pointwise limit for each time is different from uniform control over a time interval. Maximum-principle and uniqueness proofs on unbounded domains must use a suitable uniform boundedness, growth restriction or barrier hypothesis to justify passage from finite domains to infinity.

An input combination at which a Boolean function's specification imposes no required output value. Karnaugh map minimization may assign either value independently to simplify the implemented Boolean circuit. The resulting circuit still assigns a definite output to every input. An unused finite-state machine encoding is therefore an optimization opportunity, but its eventual transitions must be checked if recovery from that encoding matters.

In Boolean algebra, the product $yz$ is the consensus of $xy$ and $\bar{x}z$. If $yz=1$, either $x=1$ and $xy=1$, or $x=0$ and $\bar{x}z=1$, so adding the consensus does not change the Boolean function. It can change a circuit's transient behavior: while $x$ switches, the two original product terms may briefly both be zero because of unequal propagation delays, but $yz$ remains one. Keeping the logically redundant consensus product can therefore remove a static-one logic hazard.

A directed acyclic graph representing a Boolean function. A decision node tests a Boolean variable and has zero and one successors; evaluation follows the corresponding arrows until a terminal node returns a Boolean value. Sharing subgraphs can make a binary decision diagram much smaller than a binary decision tree.

The Boolean function obtained by assigning a fixed Boolean value $b$ to a chosen Boolean variable $x$ in $f$. The original Boolean function satisfies $f=(\neg x\wedge f{|}_{x=0})\vee (x\wedge f{|}_{x=1})$.

A set of Boolean functions is functionally complete if function composition using those operations can express every positive-arity Boolean function. The NAND gate alone suffices: tie its inputs to obtain negation, negate its output to obtain logical conjunction, and apply De Morgan's laws to obtain logical disjunction; then use disjunctive normal form.

For three events in one classical probability space, the inclusion $A\setminus C\subset (A\setminus B)\cup (B\setminus C)$ and the union bound prove the displayed inequality. It is therefore an unconditional theorem of classical probability. In a polarization Bell test, a model assigning simultaneous setting-independent pass/fail outcomes to three possible analyzer directions must obey it. Equal unbiased marginals and quantum pair correlations instead give $P(A\setminus B)=\frac{1}{2}{\sin }^2({\theta }_A-{\theta }_B)$. Directions $0,\gamma ,2\gamma$ violate the resulting constraint for $0<\gamma <\pi /4$, because $2{\sin }^2\gamma <{\sin }^{2}2\gamma$. The conclusion excludes that common classical joint model. Conditional probabilities from sequential measurements cannot be inserted without their marginal factors or treated as if all three contexts shared an already established joint distribution.

A blood count quantifies one or more cell populations in a blood sample. Counts of red blood cells, white blood cells and platelets describe different populations, usually reported per unit volume rather than as a total number throughout the body.

A complete blood count combines several measurements, including cell counts and indices of cell size or composition. A differential count further separates types of white blood cells. These distinctions matter because two reported numbers may describe different components of the same specimen.^[ref]

For steady incompressible flow and irrotational flow of constant mass density, with no body force, this formula gives the hydrodynamic torque about the origin per unit span on an impermeable body. Orient its boundary counterclockwise, and let positive torque be counterclockwise. The proof uses $M=\mathrm{Re}{\oint }_{C}pzd\bar{z}$, the Bernoulli equation $p=P_{\ast }-\rho |w^{\prime }{|}^2/2$, and the tangency identity $|w^{\prime }{|}^{2}d\bar{z}=(w^{\prime }{)}^{2}dz$. The real part of the constant-pressure term vanishes because $\mathrm{Re}(zd\bar{z})=d|z{|}^2/2$. The full complex integral of $zd\bar{z}$ need not vanish.

Place a held point vortex of circulation $\Gamma$ at real position $b>a$ outside a circular cylinder $|z|=a$, and prescribe circulation $-\Gamma$ about the cylinder. An opposite image at $c=a^2/b$ gives complex velocity $q(z)=-i\Gamma (1/(z-b)-1/(z-c))/(2\pi )$. On the positively oriented cylinder contour, the only pole of $q^2$ inside is at $c$, with residue $-{\Gamma }^2/(2{\pi }^2(b-c))$. The residue theorem and Blasius force theorem give $F_x-i{F}_y=\rho {\Gamma }^2/(2\pi (b-c))$. Thus $F_y=0$ and $F_x=\rho {\Gamma }^{2}b/(2\pi (b^2-a^2))>0$: the force on the cylinder points toward the vortex. The specified cylinder circulation is part of the hypothesis; a zero-circulation cylinder has an additional central image term.

For a two-dimensional steady ideal flow outside a body, let $q=w^{\prime }$ be the complex velocity. Suppose $q$ is holomorphic in the exterior and $q^2$ has a Laurent series near infinity, with coefficient $a_{-1}$ of $z^{-1}$. Orient the body contour counterclockwise and assume the usual impermeability and Bernoulli hypotheses of the Blasius force theorem. Deforming the contour to a large circle gives $\oint q^{2}dz=2\pi i{a}_{-1}$. Substitution in Blasius' formula proves $F_x-i{F}_y=-\pi \rho a_{-1}$.

In particular, $q(z)=q_{\infty }+O(z^{-2})$ makes the force zero: its square has no reciprocal term. This criterion handles branch-cut obstacles by applying the contour in their analytic exterior, rather than separately assigning forces to endpoint singularities.

The Milne-Thomson circle theorem places an image dipole inside a cylinder at the inverse point of an external dipole. In the residue of the squared complex velocity, only the cross term between the analytic external velocity and the image double pole contributes. The product of their conjugate dipole strengths is independent of orientation, so the cylinder force is radial toward the external dipole.

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.