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.

These critical exponents describe the leading singular powers of percolation probability, percolation susceptibility, correlation length and critical cluster distributions near the percolation critical probability. Their existence and exact values require theorems or scaling hypotheses for the particular model; they are not consequences of the definition of the threshold. For $\theta (p)=P_p(0\leftrightarrow \infty )$ and $\chi (p)={\mathbb{E}}_p|C(0)|$, the definitions are $\theta (p)=(p-p_c{)}^{\beta +o(1)}$ above $p_c$ and $\chi (p)=(p_c-p{)}^{-\gamma +o(1)}$ below $p_c$. The correlation length has $\xi (p)=|p-p_c{|}^{-\nu +o(1)}$. At criticality, the root-cluster tail is $P(|C(0)|\ge s)=s^{-1/\delta +o(1)}$; two-point connectivity has leading power $|x{|}^{-(d-2+\eta )}$; and the expected number per site of size-$s$ clusters has leading power $s^{-\tau }$. The last distribution differs from the size-weighted distribution seen at a specified site. Relations such as $\gamma =(2-\eta )\nu$ and $2\beta +\gamma =d\nu$ have specific scaling regimes, and the latter hyperscaling relation fails in ordinary mean-field regimes above the upper critical dimension of percolation.

For the FP ghost vacuum annihilated by $b_{n\ge 0}$ and $c_{n>0}$, the FP ghost level-one and level-two descendant coefficients are $-2$ and $-17$. Matter contributes ${\alpha }_0^2$ and $2{\alpha }_0^2+D/2$. Virasoro closure from BRST nilpotence at level one fixes the total highest weight to ${\alpha }_0^2/2-1$; level two then requires $D=26$. Imposing ground-state BRST closure gives ${\alpha }_0^2=2$ and the open bosonic tachyon. Interpret the coefficients with the stated ghost zero-mode pairing.

In a convention where the CP operation squares to one on the relevant bosonic state space, a CP eigenstate has eigenvalue $+1$ or $-1$. If the Hamiltonian commutes with CP, its nondegenerate eigenstates can also be CP eigenstates. In the neutral-kaon convention $CP|K^0⟩=-|{\bar{K}}^0⟩$, the antisymmetric flavor combination is CP even and the symmetric one CP odd. Rephasing flavor states changes the written combinations, not the invariant content.

Let $X$ be path-connected, locally path-connected and semilocally simply connected, and let $\alpha :{\pi }_1(X,x_0)\twoheadrightarrow \mathbb{Z}$ be a surjective group homomorphism. The classification of connected covering spaces supplies a connected covering space corresponding to the normal subgroup $\ker \alpha$. Its deck transformation group isChoose the deck transformation $T$ corresponding to $1\in \mathbb{Z}$. On each integral homology group, define $t\cdot z=T_{\ast }z$. Since $T_{\ast }$ is an automorphism, negative powers act too, making the homology group a module over the Laurent polynomial ring $\mathbb{Z}[t,t^{-1}]$. This records how a homology class changes when moved one sheet along the covering space.

For $X=S^1$ and the identity homomorphism ${\pi }_1(S^1)\cong \mathbb{Z}\to \mathbb{Z}$, the covering space is $\mathbb{R}\to S^1$, $u\mapsto e^{2\pi iu}$, with $T(u)=u+1$. Its only nonzero integral homology group is $H_0\cong \mathbb{Z}$, on which $t$ acts as the identity; as a Laurent polynomial ring module, it is $\mathbb{Z}[t,t^{-1}]/(t-1)$.