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Two distinct hyperbolic lines are parallel, in the limiting-parallel convention, when they do not intersect in the hyperbolic plane and have exactly one common ideal endpoint. In the upper half-plane model, distinct vertical lines are parallel, sharing infinity. Disjoint lines with no common ideal endpoint are instead ultraparallel hyperbolic lines; only these have a common perpendicular of ultraparallel hyperbolic lines.

An ideal endpoint of a hyperbolic line is one of its two limiting points on the boundary at infinity. In the Poincare disc model these lie on the unit circle; in the upper half-plane model they lie on $\mathbb{R}\cup \{\infty \}$. These are boundary points, not points of the hyperbolic plane. Any two distinct ideal endpoints determine a unique hyperbolic line.

A hydrogen-burning shell converts hydrogen into helium in a layer surrounding a hydrogen-exhausted stellar core. It moves outward in enclosed mass as it supplies fresh helium to that core; its movement in radius need not have the same sign.

Here $\mathbf{F},\mathbf{G}$ are forces and torques exerted by the body on the fluid, or the external force and torque needed to maintain its motion. The Lorentz reciprocal theorem for Stokes flow makes the six-dimensional hydrodynamic resistance matrix symmetric. The power identity $\mathbf{F}\cdot \mathbf{U}+\mathbf{G}\cdot \mathbf{\Omega }=2\mu \int e:edV$ makes it a positive-definite matrix for a body moving in otherwise stationary unbounded fluid.

The HAWAII-2RG architecture has a $2048\times 2048$ grid with an 18-micrometre pitch. For infrared work its readout circuit is normally bonded to a mercury cadmium telluride absorber. Reference pixels track electronic offsets and an independently addressed guide window can be read alongside the science array.

A nondestructive read samples a pixel’s accumulated signal without resetting it. Repeated samples permit Fowler sampling and up-the-ramp sampling, and can help identify saturation and cosmic-ray charge steps.

A detector reference pixel has no illuminated photosensitive connection and samples readout offsets. Reference-pixel subtraction can suppress common electronic drifts but does not measure the sky background or all pixel-dependent errors.

For a nonreal regular Rayleigh equation for inviscid shear flow mode vanishing at the walls, put $F=\hat{w}/(U-c)$. The equation becomes $[(U-c{)}^2{F}^{\prime }{]}^{\prime }-k^2(U-c{)}^{2}F=0$. Multiplication by $\overline{F}$ and integration by parts gives the displayed identity. With $Q=|F^{\prime }{|}^2+k^2|F{|}^2$, its imaginary and real parts give $\int UQ=c_r\int Q$ and $\int U^{2}Q=|c{|}^2\int Q$. Since $(U-U_{\min })(U_{\max }-U)\ge 0$, these imply Howard's semicircle theorem.

The ideal centre of a horocycle is its point of tangency to the boundary at infinity; a horizontal horocycle in the upper half-plane model has centre infinity. Sending the centre to infinity by an isometry makes the horocycle horizontal. Orthogonal hyperbolic lines then become vertical, proving that a hyperbolic line meets a horocycle orthogonally exactly when it has the centre as an ideal endpoint.

Two horocycles have a unique common orthogonal hyperbolic line when their ideal centres of horocycles differ: it is the line with those two ideal endpoints. If the centres agree, an isometry puts both horocycles on horizontal lines; every vertical hyperbolic line meets both orthogonally, so there are infinitely many. This classification remains valid for intersecting or tangent horocycles.

A domain satisfies the interior sphere condition at a boundary point if an open ball contained in the domain is tangent to its boundary there. It gives the local barrier geometry for the Hopf boundary point lemma. The condition at a single point suffices for the corresponding local derivative conclusion.

Let $v\le 0$ be $C^1$ on a connected domain and $C^2$ away from its zero set, with $\Delta v+v=0$ wherever $v<0$. If negative and zero points coexist, an interior Euclidean ball in $\{v<0\}$ touches its zero set. On that Euclidean ball $\Delta v=-v>0$, so the Hopf boundary point lemma forces a nonzero derivative at an interior maximum, contrary to $Dv=0$. Thus $v$ is identically zero or strictly negative. Mere Lipschitz continuity does not suffice: $-\max \{\sin x_1,0\}$ on $(-1,1{)}^n$ has both negative and zero regions.

A cofibration is a map with the homotopy extension property. An inclusion of a CW complex into another as a subcomplex is a cofibration; collapsing that subcomplex gives the quotient used in the K-theory six-term exact sequence.

If $f$ is a modular form on a finite-index subgroup and $\gamma$ is rational with positive determinant, then $f{|}_k\gamma$ is bounded near infinity. Choose $\sigma \in S{L}_2(\mathbb{Z})$ with $\sigma \infty =\gamma \infty$. The matrix ${\sigma }^{-1}\gamma$ is upper triangular and sends $z$ to $Az+B$ with $A>0$. Thus the existing cusp expansion of $f{|}_k\sigma$ stays bounded under this substitution. Rational conjugation of finite-index modular subgroups gives a positive period for the translate, so boundedness gives a removable singularity in its cusp parameter. Apply this to $\gamma \rho$ for every $\rho \in S{L}_2(\mathbb{Z})$ to obtain holomorphy at every cusp of the new subgroup. Vanishing at the cusps is preserved too.

On an oriented Riemannian manifold, the codifferential convention $\delta =(-1{)}^{n(p+1)+1}\ast d\ast$ and ${\ast }^2=(-1{)}^{p(n-p)}$ give $d\ast \alpha =(-1{)}^p\ast \delta \alpha$ and $\delta \ast \alpha =(-1{)}^{p+1}\ast d\alpha$. Applying these twice proves the displayed identity. Thus the Hodge star operator preserves harmonic differential forms even without compactness.

For a real one-form $\alpha$, the Hodge Laplacian differs from the rough Laplacian by the action of Ricci curvature: $(\mathrm{Ric}\cdot \alpha )(X)=\alpha ({\mathrm{Ric}}^{♯}X)$, where $g({\mathrm{Ric}}^{♯}X,Y)=\mathrm{Ric}(X,Y)$. Equivalently, with the nonnegative scalar Laplacian convention,

A Hochschild cocycle is a cochain $f$ with $\delta f=0$ in the Hochschild cochain complex. For degree two, this is $a\mu (b,c)-\mu (ab,c)+\mu (a,bc)-\mu (a,b)c=0$. It is the condition for associativity of a square-zero extension of an algebra.

For a unital associative algebra, normalized positive-degree cochains vanish whenever an argument is $1$. They form a subcomplex of the Hochschild cochain complex computing the same Hochschild cohomology. The unit-insertion contracting homotopy on degenerate bar terms proves the normalization equivalence. A normalized degree-two Hochschild cocycle defines a unital square-zero extension of an algebra.

For $f\in C^p(A,A)$ and $g\in C^q(A,A)$, set $(f⌣g)(a_1,\dots ,a_{p+q})=f(a_1,\dots ,a_p)g(a_{p+1},\dots ,a_{p+q})$. This associative cochain multiplication descends to Hochschild cohomology, where it gives a graded commutative algebra. It is one operation in the Gerstenhaber algebra structure.

If the members of a possibly infinite set family $\mathcal{F}$ have size at most $r$, there is a finite subset ${\mathcal{F}}_0\subseteq \mathcal{F}$ of size at most $M(r,s)$ with exactly the same hitting sets of size at most $s$. Use mathematical induction on $s$. For nonempty $\mathcal{F}$, choose $A_0\in \mathcal{F}$. When $s=0$ this one member suffices. Otherwise, for each $x\in A_0$, retain an inductive kernel for ${\mathcal{F}}_x=\{A\in \mathcal{F}:x\notin A\}$ at parameter $s-1$, and take their set union together with $A_0$. Its size is at most $1+rM(r,s-1)=M(r,s)$. A hitting set $H$ for this kernel meets $A_0$ at some $x$; $H\setminus \{x\}$ meets the kernel for ${\mathcal{F}}_x$, hence all of ${\mathcal{F}}_x$, while $x$ meets the other members of $\mathcal{F}$. If $A_0$ is empty no hitting set exists, and if $\mathcal{F}$ is empty its kernel is empty. Only finite branching is used, so no finiteness assumption on $\mathcal{F}$ is needed.