If two pure quantum states overlap by $(1/\sqrt{2}{)}^{1/n}$, their $n$-copy states have overlap $1/\sqrt{2}$. Each block spans an effective qubit, with orthonormal basis $e_0={\Psi }_1$ and $e_1=\sqrt{2}{\Psi }_2-{\Psi }_1$. Applying the two-block PBR exclusion measurement for zero and plus uses $2n$ independently prepared copies. A common single-copy ontic overlap of positive mass remains positive under this finite tensor power.

For the constant-entrainment axisymmetric plume equations $Q^{\prime }=2\alpha \sqrt{M}$, $M^{\prime }=FQ/M$ with $F>0$, pure balance means $M^{5/2}=5F{Q}^2/(8\alpha )$ at every height. This is a similarity relation between fluxes, not a requirement that the physical source have zero radius. A finite source on this curve is represented by a plume virtual origin. For fluxes normalized by pi and top-hat fields, the same relation is $g^{\prime }b/w^2=8\alpha /5$.

Let $f:{\mathbb{R}}^p\times {\mathbb{R}}^q\to \mathbb{R}$ be a differentiable convex function whose full gradient is $L$-Lipschitz continuous, with $L>0$. The alternating update first solves the implicit $y$ step and then the explicit $x$ step:The implicit step is the proximal operator of $\tau f(x,\cdot )$, so it exists uniquely. The gradient bound must control the full joint variable; smoothness alone supplies no global constant $L$.

For two updates, let $(p,q)$ be the difference of their mixed points $(x,y^{+})$ and $(a,b)$ the corresponding full gradient difference. The input and output differences are $(p,q+\tau b)$ and $(p-\tau a,q)$. The mixed-point norm identity for alternating updates and Baillon–Haddad theorem giveHence $T$ is a $(\tau L/2)$-averaged operator for $0<\tau L<2$. Its fixed points have both block gradients zero and are exactly the global minimizers of $f$. If a minimizer exists, Browder convergence theorem for averaged operators gives norm convergence of iteration in the stated finite-dimensional setting.

The proton–proton chain first produces deuterium by a weak reaction and then helium-3 by proton capture. pp I ends with ${}^3\mathrm{He}+{}^3\mathrm{He}\to {}^4\mathrm{He}+2p$. pp II passes through beryllium-7 electron capture to lithium-7 and then proton capture. pp III passes through boron-8 beta-plus decay and beryllium-8 breakup. Each branch produces one net helium-4 nucleus from four protons, but the neutrino spectrum and deposited heat differ. In pp II the net weak reactions involve one emitted positron and one captured electron.

For the leading periodic propulsive force of a periodic planar filament and power of a periodic planar filament functionals, the Euler-Lagrange equation for $F_x+\Gamma (\dot{W}-{\dot{W}}_0)$ gives $({\xi }_{\perp }-{\xi }_{\parallel })y_{xt}+\Gamma {\xi }_{\perp }{y}_{tt}=0$. For nonzero thrust, $y_t$ obeys an advection equation and the active travelling wave is $f(x-ct)$, with $c=({\xi }_{\perp }-{\xi }_{\parallel })/(\Gamma {\xi }_{\perp })$. A static periodic shape may be added without changing leading force or power. Stationarity is a necessary extremum condition, not by itself an existence theorem for a global maximum.

An additive constant leaves the target variance and accepted-sample mean variance unchanged, but changes the variance of unnormalized importance sampling because its random weights do not sum to one. With g uniform on (0,1), f(x)=2x and M=2, phi(x)=x gives importance variance 16/45 versus rejection proposal-budget variance 1/9. Replacing phi by x-1 gives importance variance 1/45 with the same rejection variance, reversing preference.

With values $v_i>0$ and effort costs $b_i^2$, the unique pure Nash equilibrium has the efforts displayed above. Interior first-order conditions give $v_i{b}_j/(b_1+b_2{)}^2=2b_i$, hence $b_1/b_2=\sqrt{v_1/v_2}$ and $(b_1+b_2{)}^2=\sqrt{v_1{v}_2}/2$. Against a positive rival effort, each payoff is a strictly concave function, so these conditions identify global best responses. On the axes, lowering a positive uncontested effort or adding a sufficiently small effort at the zero tie rules out equilibrium.

An all-pay contest with allocations $x_i=b_i/({\sum }_j{b}_j+\delta )$ includes fixed outside effort $\delta \ge 0$. A player with unit effort cost has payoff $v_i{x}_i-b_i$. The outside effort represents an ineligible fixed competitor and may leave some prize probability unallocated to eligible players. At equilibrium, a player is active exactly when its value exceeds total eligible effort plus outside effort.

Quaternionic projective space is the space of right quaternionic lines in ${\mathbb{H}}^{n+1}$. It is $S^{4n+3}$ modulo the right action of unit quaternions, with one cell in each dimension $0,4,\dots ,4n$. It is a compact smooth manifold of real dimension $4n$; ${\mathbb{HP}}^1\cong S^4$. The quaternionic tautological line bundle and its sphere bundle give the multiplication in its cohomology ring of quaternionic projective space.

For a local ring $A$, every map from $\mathrm{Spec}A$ into projective space lands in one standard chart: the inverse image of a chart containing the closed point is the whole spectrum. This gives homogeneous coordinates with at least one unit entry, unique up to a common unit. For a nonlocal ring, coordinates may instead merely generate the unit ideal, and general maps can involve a nontrivial invertible sheaf quotient.

An object is indecomposable projective if every epic map from a coproduct to it has a component which is split epic. In a presheaf category, representables have this property by evaluation of the identity. Conversely the canonical representable cover shows such an object is a retract of a representable; if idempotents split in the indexing category, it is itself representable.

A set-indexed coproduct in a category of projective objects in a category is projective. Restrict a map out of the coproduct to each summand, lift each restriction through the given epimorphism, and assemble the lifts by the coproduct universal property. Choosing an arbitrary family of lifts uses the usual axiom of choice. The empty coproduct, an initial object, is projective as well.

For a locally small category $\mathcal{C}$, every representable functor $\mathcal{C}(A,-)$ is a projective object in a category in $[\mathcal{C},\mathbf{Set}]$. The Yoneda lemma identifies maps from this functor with elements at $A$, and the pointwise epimorphism in a functor category criterion lets such an element be lifted through any epimorphism.

A map that is both a model fibration and a quasi-isomorphism. The homology isomorphism together with degree-one surjectivity forces surjectivity in degree zero. Equivalently it has the right lifting property against the generating cofibrations for nonnegative chain complexes.

The sphere chain complex inclusions into disk chain complexes, together with the separate degree-zero cell, generate the model cofibrations by cell attachment and retracts. Their right-injective class is exactly the acyclic fibration in the projective chain-complex model structure.

For a free massive Proca field in the mostly-plus Minkowski metric, the covariant inverse of the quadratic action and the canonical time-ordered product of vector potentials differ locally. With Fourier transform $e^{iq\cdot (x-y)}$, the covariant Proca propagator is $-i({\eta }_{ab}+q_a{q}_b/m^2)/(q^2+m^2-i0)$, while the literal canonical two-point correlation function subtracts $i{\delta }_a^0{\delta }_b^0/m^2$ in momentum space. The time component is nondynamical; its physical-mode numerator is $|\mathbf{q}{|}^2/m^2$, rather than $-1+(q^0{)}^2/m^2$. The difference is an instantaneous Dirac delta function, not another propagating state. Both expressions agree away from coincidence. This distinction matters when identifying a covariant Green function with a canonical correlator in a constrained field theory.

A zero-probability event has probability measure zero. It need not be empty: for a continuously distributed random variable, equality with a specified value is such an event. The complementary event occurs almost surely.

A probability measure is tight if compact sets carry arbitrarily nearly all its mass. Every Borel probability measure on a Polish space is tight. The stronger family condition is uniform tightness.

A winning probability is the probability that a specified participant receives the prize, conditional on the submitted efforts and the allocation rule's randomization.

An oracle computation queries only finitely many positions. If the oracle remains unchanged below its use, the same computation has the same output. In a finite-injury priority construction, a successful requirement imposes this bound on lower priorities; the induction that higher priorities stop acting then makes the disagreement permanent.