In the unsquared convention, $F({\sum }_x{p}_x{\rho }_x,{\sum }_x{p}_x{\sigma }_x)\ge {\sum }_x{p}_{x}F({\rho }_x,{\sigma }_x)$. Use Uhlmann's theorem to choose component purifications with nonnegative maximizing overlaps, and then construct matching flagged purifications of a quantum ensemble. Their overlap is the weighted sum on the right; the fidelity of the mixtures maximizes over all such overlaps.

Replace the derivative in the Dirac action by the gauge covariant derivative $D_{\mu }={\partial }_{\mu }+iq{A}_{\mu }$. The interaction is then $-q{A}_{\mu }\bar{\Psi }{\gamma }^{\mu }\Psi$. With $\Psi \mapsto e^{-iq\alpha }\Psi$ and $A_{\mu }\mapsto A_{\mu }+{\partial }_{\mu }\alpha$, the covariant derivative transforms in the same way as the spinor, proving local gauge invariance.

In the large-color limit of QCD with $g_s^2{N}_c$ fixed, baryon masses scale as $N_c$ while meson masses remain of order one. A Skyrme model action with coefficients of order $N_c$ has soliton mass and rotational moment of inertia of order $N_c$, and rotor splittings of order $1/N_c$. This motivates a semiclassical mesonic description of baryons; it does not uniquely determine every term in a finite-color effective theory.

For $k$ equiprobable quantum code states decoded by a POVM $\{E_m\}$, testing whether the classical message label agrees with the decoded output gives a joint binary POVM with $F_0={\sum }_m|m⟩⟨m|\otimes E_m$. Its acceptance probability is the decoding success $1-ϵ$ on the joint classical-quantum state and $1/k$ on the product of its marginals. Applying quantum relative entropy monotonicity to this test gives the one-shot classical-quantum coding converse.

For an odd prime $p$, the least quadratic nonresidue $n(p)$ is the smallest positive integer whose nonzero residue class is a quadratic nonresidue modulo $p$.

Two square matrices $A,C$ are congruent when $C=P^{T}AP$ for an invertible matrix $P$. This is the transformation of the matrix of a bilinear form under change of basis, and also transforms a quadratic form under an invertible linear map on its coordinates. The matrices need not be symmetric. It differs from matrix similarity, which uses $P^{-1}AP$ for a linear operator. For real symmetric matrices, completing the square diagonalizes the corresponding quadratic form by matrix congruence.

For a field of characteristic different from two, adjoining a square root of a nonsquare gives every quadratic extension. If two such extensions are isomorphic over $K$, write $\sqrt{a}=x+y\sqrt{b}$; the vanishing of its cross coefficient forces $x=0$, giving $a/b=y^2$. This explains why one subtracts the identity class when counting extensions from the square-class group.

A resistive velocity term proportional to the speed squared removes energy at rate proportional to $|\omega {|}^3$. Its derivative at zero velocity vanishes, so it does not appear in the linearization of a dynamical system at rest. Nonlinear energy arguments can still prove asymptotic stability.

A verifier with clean ancillas and a designated output measurement induces an effect $0\le M\le I$ on the witness register. Its acceptance probability is $\mathrm{Tr}(M\rho )$, and the largest possible acceptance is $\Vert M\Vert$. Spectral bounds on $M$ make the soundness quantifier over all witnesses, including entangled repeated witnesses, explicit.

Independent verifier circuits on $r$ witness registers induce a threshold acceptance effect built from tensor products of $M$ and $I-M$, where $M$ is the quantum verifier acceptance operator. In a product eigenbasis of $M$, the effect's eigenvalues are threshold probabilities for independent Bernoulli random variables with parameters given by the selected eigenvalues of $M$. Bounding every parameter on a NO instance bounds the effect's operator norm, even if the actual submitted witness is entangled. Independence is used only for the spectral calculation, not assumed for the witness.

Starting from completeness $2/3$ and soundness $1/3$, run an odd number $r$ of parallel copies and take majority. The QMA parallel repetition with entangled witnesses argument proves soundness for arbitrary repeated witnesses. Exponential Markov's inequality gives majority error at most $(2\sqrt{2}/3{)}^r$. Hence $O(\log (1/ϵ))$ copies suffice for error $ϵ$, with polynomial overhead when the target error is inverse-polynomial or exponentially small in input length.

For quantum witness embedding $J$, verifier unitary $U$ and accepting projector $\Pi$, the witness acceptance operator is the positive contraction $F=J^{†}{U}^{†}\Pi UJ$. The maximum acceptance over normalized quantum witnesses is ${\lambda }_{\max }(F)$ by the Rayleigh quotient. Restricting to basis quantum witnesses only tests its diagonal entries and need not find that maximum.

For $\rho ={\sum }_x{p}_x{\rho }_x$, choose component purifications $|u_x{⟩}_{AR}$ and an orthonormal flag register. The vector $|U⟩={\sum }_x\sqrt{p_x}|u_x⟩|x⟩$ is normalized and reduces to $\rho$ after tracing out $R$ and the flag. Orthogonal labels remove the cross terms. This construction turns mixture identities into pure-state overlap arguments and proves joint concavity of quantum fidelity.

A strategy is strictly dominated when another strategy gives a strictly larger payoff against every opposing strategy. A strictly dominated strategy cannot receive positive probability in a Nash equilibrium, because moving that probability to its dominator increases expected payoff.

An algebraic root can depend on a parameter through fractional powers even when the polynomial coefficients have ordinary power series. For example, the roots of $t^3={ϵ}^2$ are on the scale ${ϵ}^{2/3}$. Dominant balance for algebraic roots identifies the exponent, and local expansion of the scaled root supplies successive coefficients.

A Ricci-flat manifold has vanishing Ricci tensor. This is weaker than vanishing Riemann curvature tensor: the latter describes complete local flatness. A Kerr black hole supplies a Lorentzian manifold that is Ricci flat in its vacuum region but has nonzero Riemann curvature tensor.

For a real Hilbert space, $E(x)=\alpha J(cx-y)+⟨x,z⟩$, $\alpha >0$ and $c\ne 0$, identify $z$ with its vector by the Riesz representation theorem. Completing the square in $\Vert x-q{\Vert }^2/2+⟨x,z⟩$ shifts the input to $q-z$. Substitute $v=cx-y$ and multiply by $c^2$ to obtain the displayed formula, valid also for negative $c$. If $c=0$ and $J(-y)<\infty$, the answer is $q-z$; if $J(-y)=\infty$, the objective is identically infinite and no proper proximal problem remains.

On a Banach space one can define the set of minimizers of $E(x)+\Vert x-q{\Vert }^2/2$, but existence and uniqueness need additional geometry or coercivity hypotheses. The optimality condition is $0\in \partial E(x)+\mathcal{J}(x-q)$, with the duality mapping $\mathcal{J}=\partial (\Vert \cdot {\Vert }^2/2)$. The Hilbert completion-of-square shift by a dual vector does not generalize unchanged. On $({\mathbb{R}}^2,\Vert \cdot {\Vert }_1)$ with $E(x)=x_1+x_2$ and $q=0$, every $x_1,x_2\le 0$ with $x_1+x_2=-1$ is a minimizer; the vector $(-1,-1)$ is not one.

The proximal map of $\tau \Vert \cdot {\Vert }_2$ sends $x$ to $(1-\tau /\Vert x{\Vert }_2{)}_{+}x$, with zero assigned at $x=0$. This removes vectors inside the radius-$\tau$ ball and shortens all larger vectors by a fixed radial distance. It is useful for groupwise sparsity.

An implication is derivable from a theory precisely when its conclusion is derivable after adjoining its antecedent. Together with classical contradiction rules, this identifies failure to prove a proposition with consistency after adjoining its negation.