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A ket $|\psi ⟩$ denotes a vector representing a quantum state, and its bra $⟨\psi |$ is the linear functional taking $|\varphi ⟩$ to $⟨\psi |\varphi ⟩$. The map from a ket to its bra is conjugate-linear. In the physics convention, $⟨\psi |\varphi ⟩$ is conjugate-linear in $\psi$ and linear in $\varphi$. The adjoint operator is characterized by $⟨\psi |A\varphi ⟩=⟨A^{\ast }\psi |\varphi ⟩$. Products such as $|\psi ⟩⟨\varphi |$ are rank-one operators, while $⟨\varphi |\psi ⟩$ is a scalar inner product.

Let orthonormal energy eigenvectors be ${\Psi }_1,{\Psi }_2$, and let a measured observable have eigenvectors $v_{+}=\cos \alpha {\Psi }_1+\sin \alpha {\Psi }_2$ and $v_{-}=-\sin \alpha {\Psi }_1+\cos \alpha {\Psi }_2$. If the initial quantum state is $v_{-}$, unitary time evolution multiplies each energy component by $e^{-it{E}_j/\hslash }$. Its amplitude in $v_{+}$ is $\sin \alpha \cos \alpha (e^{-it{E}_2/\hslash }-e^{-it{E}_1/\hslash })$. The Born rule gives the displayed probability. The energy gap fixes the oscillation frequency, while the angle between the measurement and energy bases fixes its amplitude. Energy degeneracy or coinciding bases makes the transition probability vanish.

In the language naming addition, zero and every scalar multiplication over a fixed field, the theory of infinite underlying vector spaces eliminates quantifiers. In a conjunction of linear equations and inequations, an equation with a nonzero coefficient solves for the quantified variable. Otherwise the variable is excluded from only finitely many values, so infinitude supplies a witness. Disjunction and induction finish elimination. This works over finite fields with infinite dimension and over infinite fields with every nonzero dimension.

Morleyization adds a predicate symbol for each first-order formula and an axiom equating that predicate symbol with its defining logical formula. Every original first-order model has a unique expansion. Translating an expanded logical formula back to the original first-order language and then using its named predicate symbol gives quantifier elimination. Completeness of the original first-order theory implies completeness of the expanded first-order theory.

A projective quadric is the zero set of a nonzero homogeneous quadratic polynomial in projective space. Over a field of characteristic different from two, represent it by a symmetric matrix $B$, up to a nonzero scalar. The gradient at a representative $v$ is $2Bv$, so the singular locus is the projectivization of $\ker B$. Thus the quadric is nonsingular exactly when $B$ is invertible. In the complex projective plane, ranks three, two, and one give respectively a smooth projective conic, two distinct projective lines, and a double projective line.

For a symmetric positive-definite matrix $A$ and a nonzero vector $p$, minimizing $f(x+\alpha p)$ over $\alpha \in \mathbb{R}$, where $f(x)=x^{T}Ax/2-b^{T}x$, gives the displayed optimizer. Put $r=b-Ax$. Expansion gives $f(x+\alpha p)=f(x)-\alpha p^{T}r+{\alpha }^2{p}^{T}Ap/2$, a strictly convex quadratic function of $\alpha$. At its minimum,Strict decrease is therefore equivalent to $p^{T}r\ne 0$. This is minimization over a full line, so the optimizer can be negative. For an exact line search restricted to a nonnegative step size, the optimizer is instead $\max (0,{\alpha }_{\ast })$.

The quadratic form gradient gives $D{q}_x(h)=2x^{T}Ah$. Since $A$ is invertible, this linear map is nonzero exactly when $x\ne 0$. The regular level set theorem therefore makes every nonzero point of $C_{\alpha }$ a smooth point of dimension $n-1$. If $\alpha \ne 0$, the entire level set is smooth.

If $\alpha =0$ and the level set contains a nonzero point, the form is an indefinite quadratic form. Its only singular point is the origin. To prove this rather than merely observe a vanishing derivative, use Sylvester's law of inertia to write $q=|u{|}^2-|v{|}^2$ with both sign spaces nonzero. The null vectors $e_i\pm f_j$ span the whole ambient space. Their straight-line curves through the origin lie in the cone. If a relatively open neighborhood of the origin in the cone were an embedded submanifold, its tangent space would contain all these directions and hence have dimension $n$. It would then be open in the ambient space, impossible for the zero set of this nonzero quadratic form. Thus the smooth locus is $C_0\setminus \{0\}$.

Apply the product rule to the scalar quadratic form and transpose its last scalar term: $x^{T}Adx=(dx{)}^T{A}^{T}x$. Only the symmetric part of $A$ contributes to its value or its derivative with respect to $x$. A varying matrix also contributes the independent term $x^T(dA)x$. No assumption that $A$ is symmetric is needed. Repeating this rule and the chain rule computes mixed derivatives when vector and matrix arguments depend on separate parameters.

A QR shift is a scalar ${\mu }_k$ subtracted before the QR factorization in a QR algorithm step and restored after reversing the factors:The final identity uses the orthogonality of $Q_k$, proving that the eigenvalues are preserved. Choosing a shift near an eigenvalue can accelerate block deflation in the QR algorithm. A common choice is the current bottom-right entry, but it can stall, as in Exact-shift diagonalization of a two-state swap matrix.

For $A=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$, one shifted QR algorithm step with real QR shift $\mu$ gives, with a consistent choice of factor signs,To verify this, factor $A-\mu I=QR$ with $Q=(1+{\mu }^2{)}^{-1/2}\left(\begin{array}{cc}-\mu & -1\\ 1& -\mu \end{array}\right)$. Its columns are orthonormal, and $R=Q^T(A-\mu I)$ is upper triangular. Then $A_1=Q^{T}AQ$. Thus $A_1$ is diagonal exactly when $\mu =\pm 1$, the two eigenvalues. The bottom-right-entry shift is zero and leaves the matrix unchanged under the positive-diagonal factor convention. This illustrates why a poorly chosen QR shift can stall even a two-by-two symmetric eigenvalue computation.

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.