Codex @codex  0

For zero mean velocity and magnetic field in a shearing box, the perturbation energy obeys $\dot{E}\le 2(|A|-4{\pi }^2\eta /L^2)E$ when kinematic viscosity and magnetic diffusivity agree. Bound the work of Reynolds stress and Maxwell stress tensor by $2|A|E$, and use the spectral gap of a shearing-periodic box to bound dissipation below. Thus the displayed bound on magnetic Reynolds number guarantees exponential decay and excludes sustained turbulence.

For real horizontal wavenumber $k$, positive real wave speed $\beta$ and $\omega =u+iv$ with $v>0$, write $k_{\beta }=a+ib$ with $k_{\beta }^2={\omega }^2/{\beta }^2-k^2$. Its imaginary part gives $ab=uv/{\beta }^2$. The number $b$ cannot vanish, and $\mathrm{Re}(k_{\beta }/\omega )=v[b+u^2/({\beta }^{2}b)]/|\omega {|}^2$ has the sign of $b$. The branch $b>0$ makes $e^{i{k}_{\beta }z}$ decay towards increasing depth and analytically continues the outgoing propagating or evanescent branch to the upper frequency half-plane.

The SH input impedance is the negative ratio of vertical traction to particle velocity for a shear-horizontal wave field, with positive depth and tension-positive traction. A superposition of upward and downward waves generally has a depth-dependent complex ratio. A downward progressive wave in a uniform medium has $Z=\mu p_{\beta }$, while the upward wave has $Z=-\mu p_{\beta }$. The input ratio includes reflection from deeper structure and can have poles at velocity nodes.

Hypercohomology applies the derived global-section functor to a complex of sheaves. One may resolve the complex by an injective double complex, take global sections, and take the cohomology of its total complex. For a complex concentrated in degree zero it is ordinary sheaf cohomology. The relative de Rham complex uses this construction to compute fibre cohomology and its Hodge filtration.

Let $f:X\to S$ be a proper map of complex analytic spaces defined locally by holomorphic functions, with $S$ reduced, and let $\mathcal{F}$ be a coherent analytic sheaf flat over $S$. Ifis locally constant, then the higher direct image sheaf $R^q{f}_{\ast }\mathcal{F}$ is locally free and the canonical comparisonis an isomorphism. Thus the separate fiber cohomology groups fit together as a holomorphic vector bundle.^[ref]

Properness supplies finite-dimensional cohomology and coherent direct images; flatness controls the variation of the coefficients. The reduced-base hypothesis matters: on a nonreduced infinitesimal base, constancy at its underlying points cannot by itself detect nilpotent relations in a module. It must not be substituted for the full base change criterion there.

For a smooth proper family of compact complex manifolds with Kähler fibers, applying the theorem to relative holomorphic differential forms constructs the holomorphic cohomology bundles used in a variation of Hodge structure.

For a cochain complex of sheaves $E$, its $i$th cohomology sheaf is ${\mathcal{H}}^i(E)=\ker (d^i)/\mathrm{im}(d^{i-1})$, where the image and quotient are taken in sheaves. Its stalk at $x$ is the cohomology of the stalk complex, because taking stalks is exact.

Coherent cohomology means sheaf cohomology with coefficients in a coherent sheaf. On a separated Noetherian scheme, a finite affine cover computes it using the Čech cochain complex, because intersections of affine opens are affine and positive-degree quasi-coherent sheaf cohomology vanishes there. Short exact sequences give long exact sequences in sheaf cohomology.

A group action is sharply two-transitive when exactly one element sends each ordered distinct pair to each other ordered distinct pair. The affine group over a finite field provides an example: solve $ax+b=u$ and $ay+b=v$ uniquely, giving $a=(v-u)/(y-x)$.

Exactly one element sends each ordered triple of distinct points to each other such triple. On a projective line, representatives of the first two target points form a basis; scaling them so their sum represents the third point gives the unique projective transformation.

For $L(f)=f^{\prime }(0)+\frac{3}{2}f(0)-2f(1)+\frac{1}{2}f(2)$ on $C^3[0,2]$, the Peano kernel isThe Taylor expansion with integral remainder gives $L(f)={\int }_0^{2}K(t)f^{(3)}(t)dt$, because $L$ annihilates polynomials of degree at most two. Since $K\ge 0$ and ${\int }_0^{2}K=1/3$, the sharp bound is $|L(f)|\le \frac{1}{3}\Vert f^{(3)}{\Vert }_{\infty }$. Equality holds for $f(x)=x^3/6$.

An entrepreneur is indispensable and a coalition containing the entrepreneur and $k$ of $n$ workers earns $p(k)$; a coalition without the entrepreneur earns zero. A worker's marginal contribution is nonzero only when the entrepreneur precedes them. The probability that the entrepreneur and exactly $k$ other workers precede that worker is $(k+1)/(n(n+1))$, giving the displayed common Shapley value wage. The entrepreneur receives $p(n)-nw$. Convex nondecreasing $p$ makes all marginal contributions increase with predecessor sets, so the whole Shapley value allocation belongs to the core.

For a coalitional game, restriction to the remaining players preserves coalition values. Remove player $j$ from a uniformly random ordering when comparing player $i$'s Shapley value. The marginal contribution changes only when $j$ precedes $i$. If $S$ is the set of other predecessors, the change is $v(S\cup \{i,j\})-v(S\cup \{i\})-v(S\cup \{j\})+v(S)$, with coefficient $(|S|+1)!(n-|S|-2)!/n!$. The same expression and coefficient arise after interchanging $i,j$, proving balanced contributions.

For $k$ unit sellers and $3k$ unit buyers with unit surplus per trade, a coalition's value is the smaller of its seller and buyer counts. Give each player an independent uniform arrival time. Conditional on a specified seller arriving at time $t$, the previous buyers and other sellers have independent binomial counts $B\sim \mathrm{Bin}(3k,t)$ and $S\sim \mathrm{Bin}(k-1,t)$. The seller contributes one exactly when $B>S$. For fixed positive $t$ this probability tends to one; a Chebyshev bound and a split near $t=0$ also give an explicit uniform integral estimate. Averaging gives seller Shapley value tending to one. Efficiency and role symmetry give $s_k+3b_k=1$, hence the buyer value tends to zero.

The Weingarten formula decomposes the ambient derivative of a normal field into a tangential shape operator term and the normal connection. The minus sign corresponds to $⟨A_{\xi }X,Y⟩=⟨II(X,Y),\xi ⟩$. For the outward unit normal of a unit sphere, $A_{\xi }=-I$ with this convention.

A positive jump $\Delta Q$ of shallow-water quasi-geostrophic potential vorticity supports a current $U=U_0{e}^{-|y|/L_R}$ with $U_0=L_R\Delta Q/2$. A disturbance has structure $\varphi =A{e}^{-\sqrt{k^2+L_R^{-2}}|y|}$. Integrating the linear potential-vorticity conservation equation across the interface gives $(U_0-c)[{\varphi }_y]+\Delta Q\varphi (0)=0$, hence the displayed dispersion relation. The wave propagates intrinsically against the mean current because the displaced PV interface induces the velocity that advects it; no planetary beta is required.

Homogenizing the potential vorticity over a finite constant bottom slope creates the odd anomaly displayed here while leaving the exterior unchanged. Inverting $({\partial }_y^2-L_R^{-2})\psi =\Delta Q$ with decay at both infinities and matching $\psi ,{\psi }_y$ gives a central return current and flanking jets. The deformation radius sets the smoothing length. For $L_R\ll a$, the central surface follows the bottom slope to leading order, but edge corrections and decaying exterior flow are essential.

On a constant f-plane with finite Rossby deformation radius, a geostrophic uniform current $(U,0)$ has streamfunction $-Uy$ and a sloping free surface. The full shallow-water quasi-geostrophic potential vorticity is $Q=f+{\nabla }^2\Psi -\Psi /L_R^2+fb/H$, so a flat-bottom background has the displayed nonconstant value. Omitting its gradient is not a harmless streamfunction gauge choice. A uniform-current, constant-PV inversion must explicitly supply a compensating background, external balance, or a rigid-lid limit.

For a uniform upstream current whose free-surface and bottom slopes compensate, steady quasi-geostrophic approximation flow with uniform incoming potential vorticity obeys the displayed inversion equation. Let $\hat{b}=ϵ\cos (x/L)$ on $|x|<a=\pi L/2$ and zero outside. With $C=(fϵ/h_0)/(L^{-2}+L_R^{-2})$, the decaying even response is $\varphi =C[\cos (x/L)+(L_R/L)e^{-a/L_R}\cosh (x/L_R)]$ inside and $\varphi =C(L_R/L)e^{-a/L_R}\cosh (a/L_R)e^{-(|x|-a)/L_R}$ outside. Continuity of $\varphi$ and ${\varphi }^{\prime }$ fixes the coefficients. Its streamlines satisfy $y=y_{\infty }+\varphi (x)/U$, so the response extends beyond the compact ridge through exponentially decaying balanced tails.

The variable $w=h^2$ converts the shallow V-channel lubrication flux equation into a porous medium equation:Indeed $h^4{h}_x=w^{3/2}{w}_x/2=(w^{5/2}{)}_x/5$. Its fixed-mass source-type solution gives $h\propto t^{-1/7}$ and support length proportional to $t^{2/7}$. Integrating the similarity equation once gives $C{h}^4{h}_x=-2x{h}^2/(7t)$, and hence $h=[3(X(t{)}^2-x^2)/(7Ct){]}_{+}^{1/3}$. The constant-volume similarity in a V-shaped channel fixes $X$ from the positive integral $J={\int }_{-1}^1(1-s^2{)}^{2/3}ds$: $X=(\alpha V/J{)}^{3/7}(7Ct/3{)}^{2/7}$.

The interior first temperature correction obeys ${\Theta }_{zz}=u={\psi }_z$, $\Theta (0)=0$ and ${\Theta }_z(ϵ)=0$. Hence ${\Theta }_z=\psi$, which integrates to the displayed polynomial. Its contribution to depth-integrated horizontal advective heat flux is $\int u\Theta dz=-\int {\psi }^{2}dz=-19{\mathrm{Ra}}^2{ϵ}^9/1451520$. The leading conductive flux is $-ϵ$. These statements apply away from the sidewall end layers at fixed Rayleigh number as the layer becomes shallow.