An alpha-squared dynamo generates a mean magnetic field using an alpha effect without requiring an additional large-scale shear coupling. For $\mathbf{\alpha }={\alpha }_0\mathrm{diag}(1,1,0)$ and a transverse mean field varying as $e^{iKz}$, the growth rates are $-\eta K^2\pm {\alpha }_{0}K$. A nonzero ${\alpha }_0$ therefore allows growth for sufficiently long wavelengths, provided the domain and scale separation permit them.

For $q=e^{i\pi /6}$ define $I_0(C)={\int }_0^{\infty }\mathrm{Ai}(q(s-C))ds$ and $I_1(C)={\int }_0^{\infty }s\mathrm{Ai}(q(s-C))ds$. Both converge by Airy function decay. Integrating its differential equation gives $I_1-C{I}_0=iq\mathrm{Ai}\prime (-qC)$. These moments convert a decaying vorticity profile into its velocity and displacement for three-layer long-wave shear-flow matching.

For an affine scheme $X=\mathrm{Spec}A$, its ring of global sections is $A$. A morphism of schemes between affine schemes is determined contravariantly by its map on these rings. Consequently a morphism between affine schemes is an isomorphism of schemes exactly when that ring map is an isomorphism. The affine hypothesis is essential, as the punctured affine plane shows.

A Noetherian scheme is affine exactly when every coherent ideal sheaf has zero first sheaf cohomology. One direction is vanishing of quasi-coherent cohomology on an affine scheme. For the converse, ideal-sheaf vanishing produces affine principal neighbourhoods from ideal-sheaf vanishing. A finite such cover yields a unit-ideal certificate from a principal affine cover, and those affine charts glue to the spectrum of a commutative ring of global sections. The criterion, including the finite-type ideal version for quasi-compact quasi-separated schemes, is recorded in Stacks Project, Section 30.3.

Identify $T_{p}M$ with coordinate vectors using a basis and invert the affine exponential map near zero. Radial geodesics then have coordinates $sX$. Their equations imply ${\Gamma }^{\lambda }{}_{\mu \nu }(p)X^{\mu }{X}^{\nu }=0$ for every $X$, hence the displayed vanishing of the symmetric part. The antisymmetric part may survive when the torsion tensor is nonzero. For a general connection the Levi-Civita connection coefficients and first metric derivatives need not vanish in these coordinates.

The excess measures the minimum degree on which an admissible Steenrod monomial can be nonzero. Strict excess below $n$ picks out polynomial generators in the mod-two cohomology of $K(\mathbb{Z}/2,n)$; equality can correspond to a square, rather than a new generator.

Adiabatic invariance of an orbital action preserves action values during slow regular changes, but frequencies $\partial H/\partial J_i$ can vary. Closure requires rational frequency ratios, which need not persist. A slowly added harmonic component in a Kepler orbit potential produces generic apsidal precession.

Every entanglement-breaking channel $\mathcal{E}$ has additive Holevo capacity with every finite-dimensional quantum channel $\mathcal{N}$. A measure-and-prepare channel factorization exposes a classical measurement outcome. The quantum mutual information balance identity and data processing for quantum mutual information then bound the joint output Holevo quantity by the sum of the individual capacities, using the conditional input ensemble after a local measurement for the other channel. Independent product ensembles attain the reverse inequality.

Choose successive points and, after each choice, thin the unused reservoir to decide all subsets of the finite chosen prefix. There are only finitely many stems to handle at each stage. The final diagonal infinite set decides every one of its finite stems, because its relevant tail lies in the reservoir chosen when the stem's largest point was selected. The argument uses the hereditary decisions from acceptance and rejection of finite stems.

Suppose a reservoir rejects $s$ and decides every extension $s\cup \{a\}$. Only finitely many of those one-point extensions can have accepting tails. If infinitely many did, collect their new points into $C$; every infinite subset of $C$ begins with an accepting successor, so $[s,C]$ would lie in the family. That would make $C$ accept $s$, contradicting rejection. This finite-obstruction fact lets a second fusion preserve rejection of every finite stem.

The tensor product of Lie algebra representations decomposes as $R(2,0)\otimes R(2,0)=R(4,0)\oplus R(2,1)\oplus R(0,2)$, of dimensions $15,15,6$. The exterior square is $R(2,1)$ and the symmetric square is $R(4,0)\oplus R(0,2)$. Highest-weight vectors and the dimension formula determine the decomposition.

A zero-inflated Poisson regression specifies a structural-zero probability $\pi$ and a susceptible-component Poisson distribution mean $\mu$. In this parametrization,The mean is $(1-\pi )\mu$. Logistic predictors for $\pi$ and logarithmic predictors for $\mu$ allow covariates to affect component membership and count intensity differently. A susceptible individual can still generate a zero count; susceptibility is not equivalent to an observed positive response.

An oscillating tableau is a sequence of Young diagrams, each obtained from its predecessor by adding or removing one box. These are paths in the Young branching graph with both directions allowed. A path of length $\ell$ from the empty diagram to $\lambda ⊢n$, with $\ell -n=2m\ge 0$, has count $\ell !f_{\lambda }/(2^{m}n!m!)$. This differs from a saturated upward path, whose length must equal $n$.

When the plus factor has an analytic continuation to the forcing pole, its value there can be subtracted to split the forcing between analytic half-planes. For instance,The second quotient has a removable pole and belongs to the plus expression; the first keeps the prescribed minus-side forcing pole. After Wiener-Hopf factorization, analytic continuation identifies an entire remainder, whose value is fixed by growth and edge conditions. The pole prescription must be stated together with the Fourier transform convention.

In a finite-dimensional short exact sequence $0\to N\to E\to \mathbb{C}\to 0$ of representations of a complex semisimple Lie algebra, the generalized zero eigenspace of the Casimir operator maps onto the trivial quotient. All its irreducible composition factors have zero Casimir eigenvalue, so they are trivial. The action is therefore strictly upper triangular and has solvable image; because a semisimple Lie algebra is a perfect Lie algebra, that image is zero. Every lift in this generalized zero eigenspace is invariant, giving a split short exact sequence.

If a submodule $U$ of a Lie algebra representation $V$ admits an intertwining map $P:V\to U$ with $P{|}_U=I_U$, then $P^2=P$ and its kernel is an invariant complement. For a complex semisimple Lie algebra, the Hom representation on maps $V\to U$ whose restriction is a scalar identity reduces finding $P$ to splitting of a trivial quotient for a semisimple Lie algebra.

For a kappa-tree, retain nodes whose extension heights are unbounded in $\kappa$. The retained nodes are predecessor-closed. Each level remains nonempty: otherwise regularity would bound the union of fewer than $\kappa$ bounded extension sets. The same argument above a retained node ensures retained extensions at every later level. Singular height invalidates this proof and the unrestricted conclusion.

The spot variance is the instantaneous squared noise coefficient $\sigma (t{)}^2$ of an Itô integral. It can be estimated by averaging the weighted realized variance over a shrinking time window. A Hölder continuity assumption bounds the smoothing bias.

Consider the five-point discretization of the uniformly elliptic operator $-\Delta +c$ with $c\ge 0$ on the unit square, with homogeneous Dirichlet boundary conditions. Let $h=1/(n+1)$ and $1\le k,l\le n$. In the discrete sine basis, the weighted Jacobi method has iteration matrix eigenvaluesEach one-dimensional second difference contributes $4{\sin }^2(\pi kh/2)/h^2$, and the matrix diagonal is $4/h^2+c$, giving the displayed ratio. Its value is strictly between zero and two for every finite grid. Thus $0<\omega \le 1$ puts every iteration eigenvalue strictly inside the unit circle and gives convergence on every grid. At $\omega =1$, the highest-frequency factors approach minus one as $h\to 0$, so convergence holds while high-frequency smoothing is poor.

For an existing convection roll of intensity $r/g_0$, the amplitude of a new orientation at relative angle $\vartheta$ grows at rate $r[1-g(\vartheta )/g_0]$. Rotation permits $g(\vartheta )\ne g(-\vartheta )$. If one oblique orientation has $g(\vartheta )<g_0$, the convection roll is unstable immediately above a supercritical stationary threshold. This is the Küppers–Lortz mechanism. Successive orientation replacements can produce time-dependent convection roll switching; a universal attractor does not follow from the two-mode calculation. The perturbing orientation requires three-dimensional disturbances even if the initial convection roll is two-dimensional.