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For the Zeldovich approximation $\mathbf{x}=\mathbf{q}-b{\nabla }_q\psi$, let ${\lambda }_i$ be the eigenvalues of the potential Hessian, with this sign convention. Before shell crossing, mass conservation gives the displayed density. The first positive eigenvalue reaches $b{\lambda }_i=1$ and collapses one direction, creating a sheetlike cosmological caustic. Extrapolating straight trajectories after crossing lets particles pass through without self-consistent gravitational trapping, so the predicted multistream sheet spreads excessively. Collisionless gravitational evolution keeps matter more tightly bound; the linear displacement extrapolation cannot determine its late thickness reliably.

For a solenoidal velocity amplitude $\mathbf{v}$ with $\mathbf{k}=(k_x,-\Omega t{k}_x,k_z)$ and $\dot{\mathbf{v}}+\Omega v_y\hat{\mathbf{x}}=-i\mathbf{k}q$, differentiating $\mathbf{k}\cdot \mathbf{v}=0$ gives $q=2i\Omega k_x{v}_y/K^2$, where $K^2=|\mathbf{k}{|}^2$. Thus ${\dot{v}}_y=2\Omega k_x{k}_y{v}_y/K^2$ and ${\dot{v}}_z=2\Omega k_x{k}_z{v}_y/K^2$. For the real helicity quantity $H=i{k}_x(v_y{v}_z^{\ast }-v_z{v}_y^{\ast })$, the mixed forcing terms cancel, leaving $\dot{H}=2\Omega k_x{k}_{y}H/K^2=-{\dot{K}}^{2}H/K^2$. ConsequentlyThe invariant is a consequence of inviscid linear shear dynamics, not a claim that $H$ itself is constant under the shear.

For background shear ${\mathbf{u}}_0=-2Ax{\mathbf{e}}_y$, take ${\mathbf{u}}^{\prime }=\mathrm{Re}[\mathbf{v}(t)e^{i\mathbf{k}(t)\cdot \mathbf{r}}]$ with real $\mathbf{k}$ and $\mathbf{k}\cdot \mathbf{v}=0$. Then ${\mathbf{u}}^{\prime }\cdot \nabla {\mathbf{u}}^{\prime }=0$ pointwise, including conjugate terms. Choosing ${\dot{k}}_x=2A{k}_y$, ${\dot{k}}_y={\dot{k}}_z=0$ cancels background phase advection. The resulting linear amplitude equations are therefore exact finite-amplitude solutions of the incompressible Navier-Stokes equation in this unbounded or compatible shearing-periodic setting.

// Target: gravitational-instability-of-an-astrophysical-disk.bigb

For background velocity $-2Ax{\mathbf{e}}_y$, opposite radial faces of a shearing box are identified after the displayed azimuthal shift, modulo $L_y$. Products and spatial derivatives inherit this condition. Their radial surface integrals cancel because a translation preserves an integral over a periodic azimuthal interval.

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.

For a proper holomorphic map and a coherent sheaf flat over the base, local constancy of the fibre-cohomology dimension makes its higher direct image locally free and gives the displayed base-change identification. Applied to relative holomorphic forms in a smooth proper Kähler family, this constructs the holomorphic cohomology bundles entering its 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.