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Kummer's equation isIts solution analytic at zero is the confluent hypergeometric function of the first kindWhen $b$ is not an integer, a second local solution is

If $q_1\le q_2$, then between consecutive zeros of a nontrivial solution ofthere is a zero of every nontrivial solution of $y^{\prime \prime }+q_2(x)y=0$, unless the two coefficients agree throughout that interval.

If $Z\sim N_p(0,I_p)$, then $Z^{T}Z={\sum }_{j=1}^p{Z}_j^2\sim {\chi }_p^2$.

A scheme is semi-separated when the intersection of any two affine open subsets is affine, equivalently when its diagonal is affine. This condition makes affine covers acyclic for quasi-coherent sheaves.

The affine plane with doubled origin is formed by gluing two copies of ${\mathbb{A}}_k^2$ by the identity away from the origin. The overlap is the punctured affine plane, so this scheme is nonseparated and the two-open affine cover is not acyclic for the structure sheaf.

The gluing resembles a non-Hausdorff manifold example, but this node concerns an algebraic scheme and its nonseparatedness, not a real manifold.

A morphism of schemes is a morphism of locally ringed spaces. On affine schemes it is contravariantly equivalent to a homomorphism of their coordinate rings.

A Noetherian scheme has a finite cover by affine schemes $\mathrm{Spec}{A}_i$ with each $A_i$ a Noetherian ring. Equivalently, it is quasi-compact and locally Noetherian.

The spectrum $\mathrm{Spec}A$ of a commutative ring $A$ is the set of its prime ideals, with closed sets $V(I)=\{\mathfrak{p}:I\subseteq \mathfrak{p}\}$ and a structure sheaf whose sections locally look like fractions.

A scheme is nonreduced when its structure sheaf contains a nonzero nilpotent element. For an affine scheme $\mathrm{Spec}(A/I)$, this is equivalent to the ideal $I$ not being radical.

The Simon absorption lemma turns a scale-local estimate containing a sufficiently small multiple of the same quantity on a larger ball into a uniform estimate. Its boundary form uses balls intersected with a half-space and is the covering step that absorbs local boundary Hölder seminorms.

For a uniformly elliptic equation on a half-ball with $C^{0,\alpha }$ coefficients and $C^{2,\alpha }$ Dirichlet data $\phi$ on its flat face,

For ${\Omega }^{\prime }⋐\Omega$ and a uniformly elliptic operator with $C^{0,\alpha }$ coefficients,The constant deteriorates as ${\Omega }^{\prime }$ approaches the boundary.

The elementary estimate ${\sum }_{j=0}^D\left(\genfrac{}{}{0ex}{}{n}{j}\right)\le (n+1{)}^D$ converts finite VC dimension into a polynomial bound on the number of sample labelings.

An SH-wave is polarized horizontally and perpendicular to the vertical propagation plane.

An SV-wave is polarized in the vertical plane containing its wavevector and perpendicular to that wavevector.

In a transverse plane wave, displacement is perpendicular to the wavevector and its divergence vanishes.

The one-arm probability $\pi (r)$ is the probability that a specified vertex has an open path to distance $r$.

The beta function of a coupling $g$ isIt describes how the running coupling changes when the renormalization scale changes.

A minimizing geodesic realizes the Riemannian distance between its endpoints. A geodesic segment is locally minimizing, but need not remain globally minimizing beyond a cut point.