For positive dense-fluid reduced gravity $G$ and upward coordinate $z$, an unstable gradient $G_z>0$ supplies buoyant work of order $d^2{G}_z$ over mixing length $d$. velocity $u\sim d\sqrt{G_z}$ and eddy diffusivity $K\sim ud$ follow. Order-one constants depend on the turbulence model. This closure is for an unstable gradient; its square root must not be applied to a stable negative gradient.

A syntheme partitions six points into three duads, equivalently a perfect matching of the complete graph. There are $6!/(2^{3}3!)=15$ synthemes. Every duad belongs to three. Two edge-disjoint synthemes extend to a unique total of synthemes: their union is a six-cycle, whose complement has a unique factorization into three matchings.

For a finite nonempty subset of an abelian group with doubling constant at most $K$, the Plünnecke-Ruzsa inequality gives $|3A-2A|\le K^5|A|$. Applying the Ruzsa covering lemma to $2(A-A)$ using $A$ gives $2T\subseteq D+T$ for $T=A-A$ and $|D|=q\le K^5$. Thus $\ell T\subseteq (\ell -1)D+T$, and counting multiplicities in the finite set $D$ proves the displayed polynomial bound. For fixed $K>1$, it is eventually at most $K^{ϵ\ell }|A|$ for every $ϵ>0$.

A central division algebra over $K$ is a finite-dimensional division algebra whose center is exactly $K$. It is a central simple algebra. Every Brauer group class has a unique central division algebra representative up to isomorphism.

A source volume $Q$ makes roof outflow exceed floor inflow by $Q$. At zero floor inflow, an entraining plume cannot support a steady cold lower layer; the warm layer reaches the floor. With reduced gravity $g^{\prime }=B/Q$, the entire hydrostatic head drives roof discharge $Q=A_e\sqrt{g^{\prime }H}$. This gives the stated threshold. Equal splitting of pressure head is inappropriate at blocking. The same result in physical-area notation is $Q_c=(2C_d^2{A}_{\mathrm{geom}}^{2}BH{)}^{1/3}$.

With a unit cash asset, independent Gaussian price increments of mean $\mu$ and positive variance ${\sigma }^2$, and exponential terminal utility $-e^{-\gamma x}$, the optimal predictable risky holding is the displayed constant number of shares. The one-step exponential-loss exponent is $-\gamma \mu \pi +{\gamma }^2{\sigma }^2{\pi }^2/2$; completing its square gives a minimal multiplier $e^{-{\mu }^2/(2{\sigma }^2)}$. Backward induction proves optimality among adapted strategies and gives value $-e^{-\gamma x-T{\mu }^2/(2{\sigma }^2)}$.

Orthogonality of integer-frequency Fourier modes gives this identity: the integral is one for $n=m$ and zero otherwise. It converts integer step-count constraints into products of generating functions, as in the dilute-hopping lattice propagator.

The least divisor of a character's modulus through whose unit-group reduction the Dirichlet character factors. The Chinese remainder theorem reduces its construction to choosing the least exponent at each prime. Surjectivity of reduction on units makes the descended primitive Dirichlet character unique.

The parity determines the gamma factor and theta transformation in the completed Dirichlet L-function. It is the sign of the character at minus one.

A closed directed path whose vertices are distinct except for the common start and endpoint is a directed cycle. In a residual network, augmenting around a directed cycle preserves flow balance. A negative total cost on such a cycle gives an improving displacement for a minimum-cost flow.

A denominator such as $k\cdot \omega$ which may become small for large integer vectors $k$. Inverting a directional derivative on a torus divides the nonzero Fourier coefficients by $ik\cdot \omega$. A Diophantine condition bounds this loss polynomially, so exponential Fourier decay still gives convergence on a smaller complex strip.

Let $W$ consist of real-linear maps $\alpha$ from smooth vector fields to smooth functions such that $X_p=0$ implies $\alpha (X)(p)=0$. These are exactly evaluations by smooth differential one-forms. Define ${\omega }_p(v)=\alpha (X)(p)$ using any global field with $X_p=v$; the vanishing condition makes it well-defined. Bump-extended local frame vectors make its coefficients smooth. Also $\alpha (fX)(p)=f(p)\alpha (X)(p)$, so real linearity and pointwise vanishing automatically give $C^{\infty }$-linearity. The evaluation correspondence is a natural module isomorphism.

For a spin-degenerate band, the total density of states includes both spin species and is twice the single-spin density of states. Per lattice site, $\nu (ϵ)=2N^{-1}{\sum }_k\delta (ϵ-{ϵ}_k)$. Using a separate explicit spin sum then requires a compensating factor one half when converting momentum sums. The normalization matters in the Stoner criterion.

If $\mu <\theta =\mathrm{cf}(\theta )$ are infinite cardinals and $|[\alpha {]}^{<\mu }|<\theta$ for every $\alpha <\theta$, every $\theta$-sized family of sets of size less than $\mu$ has a $\theta$-sized delta-system subfamily.

The number of edges of a directed graph directed away from a given vertex. Assigning directions to undirected edges is an orientation of a graph.

The two-layer quasi-geostrophic potential vorticity coupling defines $R_d^{-2}=F_1+F_2$. With $F_i=f_0^2/(g\prime H_i)$ this gives the stated radius. The baroclinic energy ratio and deformation scale depends on $(l/R_d{)}^2$, so the thinner layer matters when the depths are strongly unequal.

For a pure state $a|0⟩|E_0⟩+b|1⟩|E_1⟩$ with normalized environment states, the upper-right device reduced density matrix entry is $a{b}^{\ast }⟨E_1|E_0⟩$. Distinguishability of the conditional environment states suppresses device coherence. States differing only by a global phase have unit-modulus overlap and do not cause quantum decoherence.

An unpolarized decay of a massive spin-one particle averages the squared amplitude over three parent spin states. Its polarization sum is $-{\eta }_{\mu \nu }+p_{\mu }{p}_{\nu }/m^2$. For a conserved external current the longitudinal $p_{\mu }{p}_{\nu }$ term drops out, but this does not replace the spin average by $1/2$: the physical massive parent still has three states.

A slender planar plume in a saturated porous medium has buoyancy velocity $w=\beta (T-T_0)$, with $\beta ={\rho }_{0}g\alpha \Pi /\mu$. The leading heat equation retains both advection components and lateral thermal conduction: $u{T}_x+w{T}_z=\kappa T_{xx}$. Its convective heat input per span is conserved, $F_h=C_h\int w(T-T_0)dx$. Defining $Q=\beta F_h/C_h$ gives width proportional to $z^{2/3}$ and vertical speed/temperature excess proportional to $z^{-1/3}$.

A cyclic interval of length $r$ is the set of $r$ consecutive positions in a cyclic ordering. For $1\le r<n$, there are $n$ distinct intervals, one ending at each position. Under a uniformly random permutation, a fixed $r$-set appears as an interval with probability $n/\left(\genfrac{}{}{0ex}{}{n}{r}\right)$. This is the counting mechanism of the Katona circle method.