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.

For motion on a fixed axis, the signed spatial four-acceleration in the instantaneous rest inertial frame is $a={\gamma }^3\dot{u}$. Its magnitude is the scalar proper acceleration. Constant signed $a$ from rest gives $\gamma u=at$ and the hyperbolic motion worldline $x=(c^2/a)(\sqrt{1+a^2{t}^2/c^2}-1)$.

A projective presentation of a module $M$ is an exact sequencewith $P_0,P_1$ projective modules. The first map need not be injective. Its image is $K=\ker (P_0\to M)$, so it also yields a short exact sequence $0\to K\to P_0\to M\to 0$; this kernel $K$ need not itself be a projective module.

For example, over $R=\mathbb{Z}/4\mathbb{Z}$, multiplication by two gives a projective presentation $R\stackrel{2}{\to }R\to \mathbb{Z}/2\mathbb{Z}\to 0$. The kernel of the final quotient map has two elements and is isomorphic to $\mathbb{Z}/2\mathbb{Z}$. It is not a projective module: a splitting of $R\to \mathbb{Z}/2\mathbb{Z}$ would have to lift the element of order two to an odd residue, but every odd residue has additive order four. Continuing by resolving successive kernels gives a projective resolution.

For a finite quiver $Q$ over a field $k$, set $A=kQ$, $P_i=A{e}_i$, $X_i=e_{i}X$. There is an exact sequencewhere $d(p\otimes x)=p\rho \otimes x-p\otimes f_{\rho }(x)$ in the source and target summands, and the augmentation is $p\otimes x\mapsto px$. Each $P_i$ is a summand of $A$, so the two terms are projective modules. This also works for infinite-dimensional modules.

The orthonormal quantum states $|{\alpha }_s⟩=(|0⟩+(-1{)}^s{e}^{-i\alpha }|1⟩)/\sqrt{2}$, $s=0,1$, define an equatorial projective measurement. The measured observable is $\cos \alpha X-\sin \alpha Y$, with eigenvalue $(-1{)}^s$. Thus $\alpha =0$ measures $X$, and $\alpha =\pi /2$ measures $-Y$ with these outcome labels. The sign convention matters when comparing different definitions of equatorial bases.

The Möbius group is the group of all Möbius transformations of the Riemann sphere, under function composition. An invertible $2\times 2$ complex matrix represents $z\mapsto (az+b)/(cz+d)$, and nonzero scalar multiples represent the same transformation. Rescaling a representative to determinant one gives a surjective group homomorphism $S{L}_2(\mathbb{C})\to \mathcal{M}$ with kernel of a group homomorphism $\{I,-I\}$. Thus it is also the projective linear group $PG{L}_2(\mathbb{C})$ and the quotient $PS{L}_2(\mathbb{C})=S{L}_2(\mathbb{C})/\{\pm I\}$.

Uniformly diagonalize against the formula expressing membership in $W_e$ to obtain a sentence saying that its own code is not in $W_e$. If every enumerated sentence is true, this new sentence cannot be enumerated, and hence is true. Its code is a computable new element of arithmetic truth outside $W_e$.

For finite sets of nonzero real numbers, the multiplicative energy is the number of quadruples $(a,b,c,d)\in A\times B\times A\times B$ satisfying $a/b=c/d$. Swapping $b,d$ shows that it also counts quadruples with $ab=cd$. Thus $E(A,B)={\sum }_x{r}_{A\cdot B}(x{)}^2$, where $r_{A\cdot B}(x)$ counts product representations. The Cauchy-Schwarz inequality yields $E(A,B)\ge |A{|}^2|B{|}^2/|A\cdot B|$.

For a sum of $M$ Hermitian two-local terms of norm below one, repeat a first-order Lie-Trotter product formula step $r$ times. The first-order unitary product-formula error bound gives total error at most $t^2{\sum }_{j<k}\Vert [H_j,H_k]\Vert /(2r)<M(M-1)t^2/(2r)$ for $M>1$. Taking $r=\max (1,\lceil M(M-1)t^2/(2ϵ)\rceil )$ uses $Mr$ arbitrary two-qubit gates. At unit time and $M=O(n^2)$ this is a sufficient $O(n^6/ϵ)$ bound; additional commutation structure can improve it.

An almost sure event is a measurable event of probability one. Countable intersections of such events still have probability one, allowing a conclusion to hold simultaneously for countably many parameter choices. Further deterministic reasoning may then extend it to uncountably many parameters, as in Brownian Hölder regularity or nowhere monotonicity of Brownian motion. An uncountable intersection need not preserve probability one without such additional reasoning.

The Lyapunov moment inequality says that higher positive absolute moments give no smaller absolute moment sizes on a probability space. For a real random variable $Z$ and $0<r<s$,allowing an infinite value. For $r\ge 1$ the left expression is an Lp norm; below one it is a quasi-norm, so the usual triangle inequality terminology needs that distinction. The comparison uses a probability measure of total mass one; arbitrary unnormalized measure spaces need a factor depending on their total mass.

For a nonnegative count, the first moment size is its expected value, while the second is the square root of its second moment. Both ten-trial counts in second moment method have first moment size one. The independent count has second moment size $\sqrt{1.9}\approx 1.37840$, whereas the shared-outcome count $N=10B$ has second moment size $\sqrt{10}\approx 3.16228$. The second moment distinguishes models with the same average because its square increases the influence of rare large counts.

More generally, for $N=10B$ with a Bernoulli random variable $B$ of parameter $1/10$,It rises from one at $r=1$ toward ten as $r\to \infty$. Replacing $1/10$ by a parameter $0<q\le 1$ gives $10q^{1/r}$, which decreases toward zero for a fixed positive exponent when $q\downarrow 0$, making the rare-event and large-exponent limits different questions. For a scaled Rademacher random variable taking values $-2$ and two with equal probabilities, every absolute moment size is exactly two: increasing the exponent does not distinguish signs when the absolute value is constant.

For finite $\mathbb{E}|Z{|}^s$, apply the reversed Jensen's inequality to the strictly concave function $t^{r/s}$ and the nonnegative random variable $|Z{|}^s$:Taking the positive $r$th root proves the claim. Equality in this finite case holds exactly when $|Z|$ is constant almost surely. If the higher moment is infinite, the upper bound is infinite and remains valid. Exponent zero is excluded from the statement; logarithmic or geometric limiting summaries require separately stated conventions, especially when $Z$ can vanish.

A probability distribution is nonarithmetic when it is not concentrated on $d\mathbb{Z}$ for any $d>0$. Every law with a probability density is nonarithmetic. This hypothesis removes lattice oscillations from the key renewal theorem.