The complete simply connected Riemannian manifold of dimension $n$ with constant sectional curvature $-1$ is hyperbolic space. In the upper half-space model its metric is $d{s}^2=(d{x}_1^2+\cdots +d{x}_n^2)/x_n^2$ for $x_n>0$. Its ideal boundary in dimension three is the Riemann sphere.

In the Poincare disc model, a hyperbolic circle in the Poincare disc consists of points at one fixed hyperbolic distance from a center. At center zero it is the Euclidean circle $|z|=\tanh (R/2)$, since radial hyperbolic distance is $2\mathrm{artanh}|z|$. A Poincare disc automorphism sending a general center to zero transports this description. Conversely every Euclidean circle entirely inside the disc is such a circle: the product of inversion in that circle and unit-circle inversion has reciprocal fixed points, one inside; sending the inner fixed point to zero makes the circle concentric with the unit circle.

A tessellation is a covering by tiles with pairwise disjoint interiors; a locally finite tessellation meets any compact set in only finitely many tiles. In a tessellation by congruent compact hyperbolic polygons, the adjacency graph records side-crossing steps. The area growth $\mathrm{Area}B(r)=2\pi (\cosh r-1)$ and bounded tile diameter imply exponential growth of the number of tiles within $n$ steps: a unit geodesic segment meets a uniformly bounded number of tiles, by packing congruent interior balls, so a metric ball of radius proportional to $n$ lies in the step ball. Conversely the step ball lies in a metric ball of radius proportional to $n$. The regular right-angled pentagons with four around every vertex give the tiling $\{5,4\}$.

For a right-angled geodesic triangle in curvature minus one, with legs $a,b$ and hypotenuse $c$, the displayed identity replaces the Euclidean theorem. Normalize the right vertex to $i$ in the upper half-plane, one endpoint to $iy$ with $y>1$, and the other to $s+it$ with $s^2+t^2=1$ and $t>0$. The Hyperbolic distance in the Poincare half-plane gives $\cosh a=(y+y^{-1})/2$, $\cosh b=1/t$ and $\cosh c=(1+y^2)/(2yt)$, proving the identity. For small lengths its expansion gives $c^2=a^2+b^2$ to leading order.

For a continuously differentiable dynamical system $\dot{x}=f(x)$, a fixed point is hyperbolic if no eigenvalue of its Jacobian has zero real part. In a neighborhood of a hyperbolic fixed point, the nonlinear flow is locally topologically conjugate to its linearized flow. Thus the saddle, attracting or repelling character persists under the nonlinear terms. The theorem does not decide the nonlinear character of an equilibrium with purely imaginary or zero eigenvalues: a conserved Hamiltonian, Lyapunov function or higher-order analysis may be needed instead.

Gauge pressure is pressure measured relative to the surrounding atmospheric pressure; absolute pressure is instead measured relative to an ideal vacuum. At an open surface $p_{\mathrm{surface}}=p_{\mathrm{atmosphere}}$, so the constant-mass density hydrostatic pressure law becomes $p_g={\rho }_{f}gh$. Here $p_g=0$ at the surface denotes equality with atmospheric pressure. It does not denote zero absolute pressure. ^[ref]

A negative hydrogen ion is a hydrogen nucleus with two bound electrons. Its weakly bound extra electron permits absorption at visible and infrared wavelengths in cool stellar atmospheres.

Metallic hydrogen is a high-pressure conducting state of hydrogen. Its fluid form in the deep interiors of gas giants supplies a conducting medium for a planetary dynamo, unlike the iron-alloy conducting cores of terrestrial planets.

A spatially nonuniform system is globally unstable when a normal mode satisfying its global spatial boundary conditions has positive temporal growth rate. This differs from purely local temporal amplification or convective wave-packet instability. A locally absolutely unstable region can be necessary without being sufficient for a growing global mode of finite spatial extent.

A global hydrodynamic mode is a separated solution $a(x)e^{-i\omega t}$ satisfying the complete spatial boundary conditions of a linearized flow problem. Its eigenvalue depends on the spatial domain and variable coefficients, rather than solely on a local dispersion relation.

Ethane is the alkane ${\mathrm{C}}_2{\mathrm{H}}_6$, containing two tetrahedral carbon centers joined by a single bond. Rotation about that bond connects staggered and eclipsed conformations.

The staggered arrangement is lower in energy, so ethane provides a basic example of torsional preferences in a molecule whose connectivity remains unchanged. The rotational barrier is finite: a single-bond drawing does not mean every torsion angle has the same energy.

A hydraulic bore is a propagating abrupt depth change described as a moving hydraulic jump. In the shock frame, an admissible rightward through-flow goes from supercritical flow to subcritical flow, with the appropriate characteristic curves entering the jump.

For an additive covariant left-exact functor on module categories, choose an injective resolution $M\to I^{\bullet }$ and set $R^{i}F(M)=H^i(F(I^{\bullet }))$. For a contravariant functor, use a projective resolution instead, as in contravariant right derived functor. Resolution comparison and homotopies give independence and functoriality; short exact sequences give long exact derived-functor sequences. The zeroth functor is naturally $F$.

An exact augmented cochain complex with injective module terms. Module categories have enough injectives: coinduction $J={\mathrm{Hom}}_{\mathbb{Z}}(A,\mathbb{Q}/\mathbb{Z})$ is injective by its character adjunction, and the maps $m\mapsto (a\mapsto \varphi (am))$, over all additive characters $\varphi$ of $M$, embed $M$ into a product of copies of $J$. Products of injectives are injective by coordinatewise extension. Repeat on cokernels to obtain a resolution.

For an additive right-exact functor on module categories, apply it to a projective resolution and take homology. Lifts of module maps exist by projectivity; any two lifts are chain homotopic, giving independence and functoriality. The zeroth derived functor is naturally $F$, positive degrees vanish on projective modules, and the horseshoe lemma produces a long exact homology sequence from each short exact sequence. A Tor functor is the left derived functor of a tensor product of modules functor.

The derived category of sheaves is formed by localizing cochain complexes of sheaves at quasi-isomorphisms, equivalently localizing their homotopy category. The bounded, bounded below and bounded above variants are denoted $D^b(X)$, $D^{+}(X)$ and $D^{-}(X)$. Cohomology sheaves are invariant under quasi-isomorphism and therefore define functors on these categories. Constructible bounded complexes form a natural subcategory on which Verdier duality reverses arrows and, under the usual finiteness hypotheses, is an involution.

Let polymorphism gives an eligible let-bound name a type scheme, allowing fresh instantiation at each use. In the pure Hindley-Milner type system, generalize the inferred type's type variables that are free in that type and not free in the current typing context. The identity function example under type variable meets this condition. A lambda abstraction's parameter receives one monomorphic type shared by its uses. Languages with mutation, such as Standard ML with ML reference types, need additional restrictions on generalization.

Algorithm W infers data types for a pure lambda calculus with let polymorphism. Write $W(\Gamma ,e)=(S,\tau )$, where $S$ is a type substitution and $\tau$ the inferred data type. Composition $S_2{S}_1$ means applying $S_1$ first. The following presentation uses type schemes explicitly.^[ref]

For a program identifier, take a fresh instance of its type scheme in the typing context. For $\lambda x.e$, introduce fresh $\alpha$, infer $(S,\tau )=W(\Gamma [x:\alpha ],e)$, and return $(S,S\alpha \to \tau )$.

For an application $e_1{e}_2$, compute $(S_1,{\tau }_1)=W(\Gamma ,e_1)$ and $(S_2,{\tau }_2)=W(S_1\Gamma ,e_2)$. With fresh $\beta$, let $U$ be the most general unifier of $S_2{\tau }_1$ and ${\tau }_2\to \beta$. Return $(U{S}_2{S}_1,U\beta )$. An incompatible base data type or failed occurs check rejects the expression.

For let x = e1 in e2, first infer $(S_1,{\tau }_1)$. Generalize exactly those type variables of ${\tau }_1$ not free in $S_1\Gamma$, producing a type scheme $\sigma$. Infer $(S_2,{\tau }_2)$ for $e_2$ under $S_1\Gamma [x:\sigma ]$ and return $(S_2{S}_1,{\tau }_2)$.

For example, let id = \x -> x in let a = id 3 in id True has a Boolean data type result. The identity function receives type scheme $\forall \alpha .\alpha \to \alpha$, then separate uses instantiate it at integer and Boolean data types. A parameter to a lambda abstraction instead has one shared monomorphic type; treating that parameter like a let-bound type scheme would accept programs outside this type system.

The Euler-Lagrange equation on an open interval can be derived using compactly supported variations even when an endpoint is singular. A formal solution with singular endpoint data need not make the improper action finite. For example, $y=x+x^{-1}$ solves $(x^4{y}^{\prime \prime }{)}^{\prime \prime }=0$, but substituting into $x^4(y^{\prime \prime }{)}^2+4y^2{y}^{\prime }$ gives $4x^2+4-4x^{-4}$, whose integral down to zero diverges. Stationarity of a finite functional then requires an additional regularization or admissible-space specification.

Local search repeatedly improves a feasible solution within a specified neighborhood until no improving move remains. For an allocation, neighbors can transfer an item or exchange items between bidders while preserving the partition. A local optimum need not be a global optimum; restarting from several initial solutions can explore different basins.