Group the nine physical qubits into three blocks and write $B_r$ for the product of three Pauli X gates in block $r$. The commuting stabilizer generators $B_1{B}_2$ and $B_2{B}_3$ have eigenvalue $+1$ on both logical codewords. A Pauli Z gate in blocks one, two or three gives syndromes $(-,+)$, $(-,-)$ or $(+,-)$ respectively. Apply a Z gate to any one qubit of the indicated block to recover the logical state. Z errors at distinct locations within a block act identically on the code, so this correction is degenerate.

The Rankine-Hugoniot condition expresses conservation laws across a moving discontinuity of a weak solution. Across a discontinuity $x=\sigma t$ in $u_t+f(u{)}_x=0$, with left and right states $u_{-}$ and $u_{+}$, conservation requiresWriting $[u]=u_{+}-u_{-}$ and $[f(u)]=f(u_{+})-f(u_{-})$ gives $\sigma [u]=[f(u)]$. The same relation holds componentwise for a system of conservation laws; in compressible flow, conservation of mass, momentum and energy supplies the jump conditions for a shock wave. Integrating the weak solution identity over a thin space-time region crossing the discontinuity yields this balance between the swept-up conserved quantity and the difference of fluxes.

The normal velocity is reduced by $q$ in the strong-shock limit while tangential velocity is unchanged. If the upstream speed is $U$ and the shock normal makes angle $\beta$ with its direction, then $u_X=U[1-(1-q){\cos }^2\beta ]$ and $u_Y=\pm U(1-q)\cos \beta \sin \beta$. Eliminating the angle gives the displayed circle, centered at $\gamma U/(\gamma +1)$ with radius $U/(\gamma +1)$. It is the limiting shock polar in downstream velocity space.

At a saddle-focus homoclinic orbit, the leading Shilnikov return map is $f(x)=A{x}^{\delta }\cos (q\log x+\Phi )$, $x>0$, with $q>0$. If $1/2<\delta <1$, its fixed points solve $\cos (q\log x+\Phi )=x^{1-\delta }/A$. Infinitely many solutions accumulate at zero near the zeros of the cosine, with successive size ratios tending to $e^{-\pi /q}$. The derivative is $A{x}^{\delta -1}[\delta \cos (q\log x+\Phi )-q\sin (q\log x+\Phi )]$, so their expanding multiplier is unbounded. Narrow monotone strips near successive oscillations map across a common interval; inverse branches contract, supplying arbitrarily long symbolic itineraries and infinitely many saddle periodic orbits. Varying the splitting parameter shifts the graph vertically, producing accumulating fold and period-doubling thresholds and stable windows. The condition $\delta >1/2$ makes the local flow volume-contracting, whereas $\delta <1$ makes its saddle value positive.

For independent observations $Y_i=\theta +E_i$ with $E_i$ having exponential distribution of known rate $\lambda >0$, the likelihood increases in $\theta$ until the smallest observation, so the maximum-likelihood estimator is $Y_{(1)}$. Its survival function is $\mathbb{P}(Y_{(1)}-\theta >t)=e^{-n\lambda t}$ for $t\ge 0$. Therefore $n\lambda (Y_{(1)}-\theta )$ has exactly the unit-rate exponential distribution for every sample size. This gives statistical consistency and a one-sided nonnormal limit with rate $n$, illustrating how a support endpoint changes usual regular likelihood asymptotics.

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.

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.