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Hyperbolic functions
The hyperbolic functions, also called hyperbolic trigonometric functions, are elementary functions built from exponentials that play the role for the hyperbola that sine and cosine play for the circle, describing hanging cables and rapidity in relativity.
Hyperbolic geometry
Hyperbolic geometry, also called Lobachevsky geometry, is a non-Euclidean geometry with constant negative curvature in which triangles have angle sums below 180°, discovered independently by Lobachevsky and Bolyai in the 1820s.
Hyperbolic group
A hyperbolic group, also called a word-hyperbolic group, is a finitely generated group whose Cayley graph with the word metric is hyperbolic, meaning geodesic triangles are uniformly thin; introduced by Mikhail Gromov in 1987.
Hyperbolic lattices
Hyperbolic lattices are regular tessellations of the two-dimensional hyperbolic plane, a non-Euclidean space of uniform negative curvature, used as synthetic platforms for band theory, topology, and disorder physics.
Hyperbolic manifold
A hyperbolic manifold is a Riemannian manifold locally isometric to hyperbolic space of constant curvature -1, playing a central role in 3-dimensional topology and geometrization.
Hyperbolic paraboloid
A hyperbolic paraboloid is a saddle-shaped quadric surface, one of only two doubly ruled surfaces in three-dimensional space, with two families of straight lines lying entirely on it.
Hyperbolic plane
The hyperbolic plane is the unique complete, simply connected surface of constant Gauss curvature −1, studied through models like the Poincaré disk because it cannot sit in ordinary 3-space.
Hyperbolic surface
A hyperbolic surface is a two-dimensional surface with constant Gaussian curvature −1, locally like the hyperbolic plane, existing on closed orientable surfaces of genus at least 2.
Hypercomplex number
A hypercomplex number, also called a hypercomplex system, is an element of a finite-dimensional algebra with identity over the real numbers; examples include the quaternions, octonions, and Clifford algebras.
Hypercube
A hypercube, also called an n-cube or measure polytope, is the generalization of the ordinary three-dimensional cube to n dimensions; the sequence runs point, segment, square, cube, and tesseract, each step doubling the vertex count.
Hypercube graph
The hypercube graph, also called the n-cube or Boolean cube, is the graph formed by binary strings of length n, with edges between strings differing in one bit.
Hyperfinite type II factor
The hyperfinite type II₁ factor, usually written R, is a von Neumann algebra approximated by finite-dimensional matrix algebras, unique of its type, introduced by Murray and von Neumann in 1943.
Hyperfinite type II₁ factor
The hyperfinite type II₁ factor R is the unique infinite-dimensional von Neumann algebra that is a factor, carries a finite trace, and is a limit of matrix algebras.
Hypergeometric distribution
The hypergeometric distribution is a discrete probability distribution counting successes in a random sample drawn without replacement from a finite population, used in acceptance sampling and Fisher's exact test.
Hypergeometric function
The hypergeometric function, also known as the Gauss hypergeometric function or ₂F₁, is a special function defined by a power series that solves a broad class of differential equations.
Hypergeometric series
A hypergeometric series is a power series in which the ratio of consecutive coefficients is a rational function of the index, named by Euler in 1769 and made rigorous by Gauss in 1812.
Hypergraph
A hypergraph is a generalization of a graph in which an edge may join any number of vertices rather than exactly two; an ordinary graph is a 2-uniform case.
Hypergraph matching
Hypergraph matching is the problem of finding a largest collection of pairwise disjoint edges in a hypergraph; it is NP-hard when edges contain 3 or more vertices.
Hypergraph regularity method
The hypergraph regularity method is an extremal combinatorics technique decomposing any large hypergraph into pseudorandom blocks. It extends Szemerédi's regularity lemma and yields the hypergraph removal lemma.
Hypergraph removal lemma
The hypergraph removal lemma is a theorem of extremal combinatorics stating that a hypergraph with few copies of a fixed hypergraph F can be made F-free by deleting few edges.
Hyperplane
A hyperplane is a subspace of one dimension less than its space: a plane in 3D, a line in 2D, generalizing the plane to any dimension.
Hyperplane arrangement
A hyperplane arrangement is a finite collection of hyperplanes, codimension-one affine subspaces, in a space like R^d, studied for how they cut space into regions.
Hyperreal number
Hyperreal numbers, also called nonstandard real numbers or hyperreals, are an ordered field extending the real numbers with infinitesimals, used in Abraham Robinson's nonstandard analysis.
Hypersimplex
A hypersimplex Δd(k) is a convex polytope generalizing the simplex, the slice of the unit cube [0, 1]^d by a hyperplane summing to k, central in polytope and matroid theory.
Hypotenuse
The hypotenuse is the side of a right triangle opposite the right angle, and the longest side, given by the Pythagorean theorem and central to sine and cosine.
I-adic completion
The I-adic completion of a ring R at an ideal I is the inverse limit of the quotients R/Iⁿ, replacing polynomials by formal power series and yielding the p-adic integers.
I-SPY 2 trial
The I-SPY 2 trial is an adaptive phase 2 platform trial launched in 2010 testing new drugs added to standard chemotherapy in high-risk stage II or III breast cancer.
ICH E9: Statistical Principles for Clinical Trials
ICH E9 is the International Council for Harmonisation guideline setting statistical principles for clinical trial design, analysis, and interpretation; its 2019 addendum E9(R1) added the estimand framework.
Icosahedron
An icosahedron is a polyhedron with 20 faces; the regular icosahedron is the Platonic solid with 20 equilateral triangular faces, 30 edges, and 12 vertices, five meeting at each vertex.
Icosidodecahedron
The icosidodecahedron, also called the rectified dodecahedron or pentagonal gyrobirotunda, is an Archimedean solid with 20 triangular and 12 pentagonal faces, made by truncating a dodecahedron or icosahedron.