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The control state $q$, unread input word $w$, and stack contents $\alpha$ of a pushdown automaton. A fixed convention must say which end of $\alpha$ is the top. With the top on the left, replacing top $X$ by $\gamma$ changes $X\beta$ into $\gamma \beta$.

A pushdown automaton accepts a word if some finite run consumes all its input and reaches a designated final control state, regardless of the remaining stack contents. Reaching a final control state while leaving unread input is insufficient.

The unique positive square root of a density operator is a Hilbert-Schmidt operator $W$ with $\Vert W{\Vert }_{\mathrm{HS}}^2=\mathrm{tr}\rho =1$. It reproduces expectations by $\mathrm{tr}(A\rho )=\mathrm{tr}(W^{\ast }AW)$. In chosen orthonormal bases, vectorizing its entries gives a normalized bipartite vector $|W⟩⟩={\sum }_{r,s}{W}_{rs}|r⟩|s⟩$ whose reduced density operator is $W{W}^{\ast }=\rho$. Thus the positive operator amplitude and the vector purification are related but distinct objects.

More generally a coefficient matrix $C$ purifies $\rho$ when $C{C}^{\ast }=\rho$. For square nonsingular $C$, the matrix $U=C^{-1}\sqrt{\rho }$ obeys $U{U}^{\ast }=I$ and is unitary, so $\sqrt{\rho }=CU$. In an eigenbasis of $\rho$, $C=\mathrm{diag}(\sqrt{p_r})U^{\ast }$; its conjugated unitary columns give the second-factor Schmidt decomposition vectors. This is the matrix form of unitary freedom of purification.

Dilatation of the pupil; it is a sign requiring interpretation in context, rather than an identification of a unique poison.

This formal language is not a context-free language. For pumping length $p$, use $a^p{b}^{p^2}{c}^p$. A pumpable window of length at most $p$ cannot meet both outer blocks. If pumping adds $\alpha ,\beta ,\gamma$ copies of $a,b,c$, then $\alpha \gamma =0$ and $1\le \alpha +\beta +\gamma \le p$. Preserving the product constraint would require $\beta =p(\alpha +\gamma )$, impossible both when $\alpha +\gamma =0$ and when it is positive. This contradicts the pumping lemma for context-free languages.

In propositional logic with the usual two truth values, negating a proposition twice preserves its truth value. Thus the complement of a set taken twice, in one fixed universal set, is the original set.

A propositional formula $\phi$ semantically entails $\psi$ when every Boolean valuation satisfying $\phi$ satisfies $\psi$. Equivalently, $\phi \wedge \neg \psi$ is an unsatisfiable Boolean formula. Deciding semantic entailment is co-NP-complete: a counterexample Boolean valuation certifies failure in polynomial time, while $\theta \mapsto ⟨\theta ,p\wedge \neg p⟩$ reduces unsatisfiability to semantic entailment.

An indivisible statement treated as either true or false. In a labelled transition system, an atomic proposition $p$ holds at state $s$ exactly when $p\in L(s)$.

From Boolean clauses containing complementary Boolean literals, derive the Boolean clause formed by removing that complementary pair and taking the remaining Boolean literals. For example, resolving $p\vee \neg q$ with $\neg p\vee \neg r$ gives $\neg q\vee \neg r$. Backward proof search with a positive-headed Horn clause and an all-negative query is a restricted use of this rule.

A propositional variable is an atomic symbol assigned a truth value by a Boolean valuation. A propositional formula is built from these symbols by logical connectives. The variables carry no quantifiers over individuals; their values may be chosen independently.

An inference rule deriving a clause from two clauses containing complementary literals. In propositional logic the displayed rule eliminates the atom $A$. For first-order clauses, complementary atoms can first be made identical by a suitable first-order substitution.

A Hilbert proof system specifies axiom schemes and inference rules, commonly including modus ponens. A formal proof is a finite sequence of axiom instances, permitted assumptions and conclusions obtained from preceding lines by those rules. Soundness and completeness compare this syntactic derivability with semantic validity or semantic entailment. These properties concern the particular chosen axioms and rules.

A procedure passed as an argument to another procedure. For a nested procedure under lexical scope, the argument must preserve both its code and its defining environment, for example as a programming closure.

Here $\alpha >0$ and $q,p,\gamma$ are real functions of time. Differentiation in position gives ${\psi }_x=[-\alpha (x-q)+ip/\hslash ]\psi$. The standard probability current is thereforeFor a coherent state moving in a quadratic Hamiltonian, the centre follows the classical equation $\dot{q}=p/m$, so the current is the normalized Gaussian probability density transported at its centre velocity. The factor $\sqrt{\alpha /\pi }$ is essential: omitting it changes the current by a dimensional constant and violates the continuity equation for the normalized density. A spatially constant phase does not affect the current.

Let two first-order models satisfy a complete theory and suppose every element of the first is the value of a closed term. Send each such value to the value of the same first-order term in the second first-order model. Completeness decides all logical equality first-order sentences, making this map well-defined and injective. It also decides every logical formula with closed terms substituted, making the map elementary. Thus the first first-order model is prime.

The prime ideals $P$ of $\mathbb{Z}[t]$ with $P\cap \mathbb{Z}=\{0\}$ are the zero ideal and the principal ideals $(f)$ generated by nonconstant primitive polynomials that are irreducible in $\mathbb{Q}[t]$.

To prove the description, invert the nonzero integers to obtain the localization $\mathbb{Q}[t]$. The extension of an ideal $P$ consists of the polynomials $f/n$ with $f\in P$ and $n>0$. Its contraction is $P$: if $nh\in P$ and $n\ne 0$, the defining condition puts $n\notin P$, so primeness gives $h\in P$. The prime ideal correspondence for localization makes this extension a proper prime ideal of $\mathbb{Q}[t]$.

If $P\ne 0$, its extension has a nonzero generator because $\mathbb{Q}[t]$ is a principal ideal domain. Dividing this generator by its rational polynomial content gives an integer-coefficient primitive polynomial $f$ generating the same extended ideal. Divisibility descent by Gauss lemma for polynomials then gives $P=(f)$ in $\mathbb{Z}[t]$. The proper prime ideal $(f)$ in $\mathbb{Q}[t]$ makes $f$ irreducible there. Conversely, any such polynomial generates a prime ideal in $\mathbb{Q}[t]$ whose contraction is $(f)\subset \mathbb{Z}[t]$ and whose intersection with $\mathbb{Z}$ is zero.

The intersection condition is essential: $(2,t)$ is a maximal ideal of $\mathbb{Z}[t]$, since its quotient ring is ${\mathbb{F}}_2$, but it is not principal. A common divisor of $2$ and $t$ must be a unit, which would generate the whole ring.

Along a boundary-layer streamline directed by increasing $s$, an adverse pressure gradient is a rise in pressure in the direction of flow. The associated pressure gradient force opposes the velocity. In a steady inviscid outer stream the Euler equation gives $UdU/ds=-{\rho }^{-1}dp/ds$, so a positive gradient decelerates a positive outer velocity. Within a viscous boundary layer the slower fluid near the wall has less streamwise momentum and can reverse while the outer stream still moves forward. The onset of flow separation is therefore associated with vanishing and then reversed wall shear stress. A pressure rise does not by itself locate separation: the full boundary-layer solution and upstream history also matter.

A pressure increment applied to a confined fluid in equilibrium is transmitted throughout it. The pressure increment is a scalar, not a force vector copied in one direction. Comparing different heights also requires the hydrostatic pressure difference. Finite-speed pressure wave propagation precedes the new equilibrium.

An instrument measuring atmospheric pressure. A mercury barometer compares the hydrostatic pressure of a measured mercury column with the pressure above it. Its equation is $p_{\mathrm{atm}}=p_t+\rho gh$, with $h$ the vertical separation of the liquid levels.