For any tripartite density operator, expanding the three quantum mutual information terms in Von Neumann entropy gives $I(X:BD)=I(X:B)+I(XB:D)-I(B:D)$. This algebraic identity allows separate bounds on two information contributions, while nonnegativity of quantum relative entropy allows the last term to be dropped in an upper bound.

A local quantum channel on either subsystem cannot increase quantum mutual information. Indeed, apply the data-processing inequality for quantum relative entropy to $D({\rho }_{AB}\Vert {\rho }_A\otimes {\rho }_B)$; the channel maps the product of marginals to the product of the new marginals.

For a tripartite density operator, the quantum conditional mutual information is $I(A:C|B)=S(AB)+S(BC)-S(B)-S(ABC)$. It is nonnegative by Strong subadditivity of Von Neumann entropy. It measures the coherent information lost when a channel environment is discarded after an isometric dilation.

Two first-order structures are rank-n equivalent if they agree on all sentences of quantifier rank at most $n$. The tuple version compares formulas at fixed tuples. For finite relational vocabulary this is equivalent to a Duplicator winning strategy in the $n$-round Ehrenfeucht-Fraïssé game.

A quantifier-free sentence is a quantifier-free formula with no free variables. In a language whose only closed group terms denote zero, all quantifier-free sentences have the same truth value in every group, even when existential sentences distinguish them.

On $[0,1]$, the sum $Q_n$ of squared adjacent dyadic increments of standard Brownian motion has $\mathbb{E}{Q}_n=1$ and $\mathrm{var}(Q_n)=2^{1-n}$. Therefore $Q_n\to 1$ in $L^2$. On an interval of length $t$, scaling gives mean $t$ and variance $2t^2{2}^{-n}$. Independence is used within each grid, not between different grids.

A function with finite Hölder seminorm of exponent $\alpha >1/2$ has squared-increment sum on a uniform $N$-interval partition bounded by $C^2{N}^{1-2\alpha }$, hence tending to zero. This conflicts with a nonzero quadratic-variation limit. In particular dyadic quadratic variation of Brownian motion prevents a Brownian coordinate law supported on such a Hölder space. This argument alone gives no conclusion at the critical exponent $1/2$.

For observations of an Itô integral on a partition of an interval, the statistic ${\sum }_{i}g(t_{i-1})(X_{t_i}-X_{t_{i-1}}{)}^2$ estimates $\int g(t)\sigma (t{)}^{2}dt$. With a deterministic noise coefficient the squared increments are independent, and the fluctuation has variance $2{\sum }_{i}g(t_{i-1}{)}^2({\int }_{t_{i-1}}^{t_i}{\sigma }^2{)}^2$.

Completed squared increments form a nondecreasing step process. The continuous sum that includes the last partial increment differs by at most the square of the path's modulus of continuity over one mesh interval. Therefore both have the same limit in uniform convergence on compacts in probability. This proves monotonicity of the limiting quadratic variation without incorrectly asserting monotonicity of each continuous partial-increment sum.

Stop a continuous local martingale at increasing level-and-time stopping times to obtain bounded martingales. Their limits of discrete quadratic variation sums agree before the earlier stopping time, because the sums commute exactly with stopping. These continuous limits patch into a continuous adapted nondecreasing process. On each compact interval, the chance that the stopping time occurs early tends to zero, giving uniform convergence on compacts in probability for the original sums.

On a compact interval, squared increments of a continuous finite variation process are bounded by its modulus of continuity at the mesh size times its total variation, and therefore vanish. The Cauchy-Schwarz inequality bounds the mixed increment sum by the square root of the product of the two squared-increment sums. The local-martingale sum is bounded in probability, so the mixed term also vanishes.

The mean-vector prediction risk measures squared Euclidean norm error in estimating the deterministic response mean. For independent future noise with covariance matrix ${\sigma }^2{I}_n$, prediction of a new noisy response adds $n{\sigma }^2$ to this risk. A Gaussian orthogonal projection matrix $P$ of rank $k$ has risk $\Vert (I-P)\mu {\Vert }^2+k{\sigma }^2$.

A homeomorphic embedding of a metric space induces a homeomorphic embedding of its probability measures for their weak topologies. The inverse continuity follows from the open-set Portmanteau theorem, since every relatively open set is the trace of an ambient open set. This does not give surjectivity onto $P(Z)$ when points of $Z$ are outside the embedded image.

The dimensionless ratio $\Gamma =5F{Q}^2/(8\alpha M^{5/2})$ compares a plume to the constant-entrainment pure balance. Values one, below one and above one describe pure plume balance, a momentum-excess forced plume, and a momentum-deficient lazy plume, respectively. For fluxes normalized by pi and a top-hat profile, $\Gamma =5g^{\prime }b/(8\alpha w^2)$.

When $F$ and $\alpha$ are constant, eliminating height between $Q^{\prime }=2\alpha \sqrt{M}$ and $M^{\prime }=FQ/M$ gives $d(M^{5/2})/dQ=5FQ/(4\alpha )$. Thus $M^{5/2}-5F{Q}^2/(8\alpha )$ is constant. Its sign classifies forced and lazy departures from pure plume balance, and its vanishing makes a finite-source plume exactly self-similar after a virtual-origin shift.

For a smooth map between manifolds $f:M\to N$, the pullback vector bundle $f^{\ast }TN$ has fibre $T_{f(p)}N$ at $p\in M$. A smooth section is a vector field along a map. It allows one to differentiate maps and curves without requiring them to be invertible.

With $k>0$ active players of harmonic mean valuation ${\bar{v}}_k$, unit-cost Nash equilibrium effort satisfies $R+\delta =\frac{{\bar{v}}_k}{2k}[(k-1)+\sqrt{(k-1{)}^2+4k\delta /{\bar{v}}_k}]$. Sum the active-player first-order equations $b_i=(R+\delta )(1-(R+\delta )/v_i)$ to obtain the quadratic. If $\delta$ is at least the largest valuation, no one is active and $R=0$. With no outside effort, at least two players must be active.

For ordered positive valuations, rank $j$ is active exactly when $\delta <v_j(2-j+{\sum }_{i<j}{v}_j/v_i)$. These ranks form a prefix, and the number active is the largest qualifying rank, or zero if none qualifies. This follows by evaluating the decreasing equilibrium equation ${\sum }_i(1-S/v_i{)}_{+}-1+\delta /S=0$ at $S=v_j$. Strict inequality excludes zero-effort boundary players.

This smooth projective plane curve has no real rational points: a sum of three real squares is zero only when all three coordinates vanish, which is forbidden in projective space. It is therefore not isomorphic over $\mathbb{R}$ to the projective line, which has rational points. Over $\mathbb{C}$ it becomes a nonsingular conic isomorphic to a projective line. Thus the ground field changes whether a geometric genus-zero curve admits a projective-line parametrization.

A plane cubic is a projective plane curve defined by a nonzero homogeneous polynomial of degree three. Its arithmetic genus is one by the genus-degree formula, and its Hilbert polynomial is $3m$. A singular cubic can have normalization of smaller genus; the Hilbert polynomial measures the arithmetic genus.