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Hoeffding's inequality
Hoeffding's inequality, also called the Hoeffding bound, bounds the probability that a sum of bounded independent random variables deviates from its expectation using only the summands' ranges.
Hölder condition
A Hölder condition is a continuity requirement on a function, |f(x) − f(y)| ≤ A|x − y|^α with 0 < α ≤ 1; the case α = 1 is Lipschitz continuity.
Hölder's inequality
Hölder's inequality is a bound in analysis relating the integral of a product of functions to the product of their norms; it generalizes the Cauchy–Schwarz inequality.
Holomorphic function
A holomorphic function, also called a complex analytic function, is a complex-valued function differentiable at every point of an open set, the central object of complex analysis.
Holomorphic functional calculus
The holomorphic functional calculus, also called the Riesz–Dunford functional calculus, assigns to an element of a Banach algebra and a holomorphic function near its spectrum an element f(a) via a contour integral.
Holonomic function
A holonomic function, also called a D-finite function, satisfies a linear differential equation with polynomial coefficients; the class follows Zeilberger's holonomic systems approach and contains most special functions.
Holonomy
Holonomy, or the holonomy group, is a construction in differential geometry measuring how parallel transport around closed loops fails to return geometric data to its starting value.
Hom functor
The Hom functor is the category theory construction assigning to each pair of objects the set of morphisms between them, contravariant in the first argument.
Homeomorphism
A homeomorphism, also called a topological equivalence, is a continuous bijection between topological spaces whose inverse is continuous; homeomorphic spaces are identical from topology's standpoint.
Homoclinic orbit
A homoclinic orbit is a trajectory of a dynamical system that approaches the same equilibrium as time tends to both plus and minus infinity, first identified by Henri Poincaré.
Homogeneous coordinates
Homogeneous coordinates, also called projective coordinates, represent points of projective space by tuples defined only up to a nonzero scalar, letting points at infinity be written as ordinary tuples.
Homogeneous differential equation
A homogeneous differential equation is an ordinary differential equation that is either first-order with right-hand side depending only on y/x, or linear with zero forcing term.
Homogeneous function
A homogeneous function, also known as a homogeneous map, scales by a fixed power when all its arguments are scaled by the same factor, as in the Cobb–Douglas function.
Homogeneous space
A homogeneous space is a space on which a group acts transitively, so any point can be carried to any other by some group element.
Homological conjectures in commutative algebra
The homological conjectures are a web of related conjectures about homological properties of commutative rings, collected by Melvin Hochster in 1975; since 2016 perfectoid geometry has proved the direct summand conjecture.
Homology (mathematics)
Homology is a construction in algebraic topology that assigns to each topological space homology groups recording, dimension by dimension, cycles such as loops and voids that are not boundaries.
Homomorphism
A homomorphism is a map between two algebraic structures of the same kind, such as groups or rings, that preserves their operations, making it algebra's basic tool for comparing structures.
Homothety
A homothety, also called a central dilatation, is a geometric transformation with a center O that sends each point M to a point on line OM at a scaled distance.
Homotopy
Homotopy is a continuous deformation of one map between topological spaces into another, introduced by Henri Poincaré in 1895 as a central idea of algebraic topology.
Homotopy analysis method
The homotopy analysis method, developed by Shi-Jun Liao of Shanghai Jiaotong University in 1992, is a semi-analytical technique giving convergent series solutions of strongly nonlinear differential equations.
Homotopy category
The homotopy category Ho(C) of a category with weak equivalences is obtained by universally forcing them to become isomorphisms, turning homotopy theory into equivalence classes of morphisms.
Homotopy equivalence
A homotopy equivalence is a continuous map between topological spaces with a homotopy inverse; homotopy equivalent spaces share invariants like the fundamental group and homology.
Homotopy group
A homotopy group is an algebraic invariant assigning to a space a group of homotopy classes of maps from the n-sphere, introduced by Witold Hurewicz in 1935–36.
Homotopy method
A homotopy method, also called a continuation method, solves a system of equations by deforming an easy problem with known solutions into the target problem and tracking the solution paths.
Homotopy perturbation method
The homotopy perturbation method (HPM) is an analytical technique introduced by Ji-Huan He in 1999 that solves nonlinear differential equations without needing a small parameter.
Homotopy type theory
Homotopy type theory, also known as homotopical type theory or HoTT, combines Martin-Löf type theory with homotopy theory, with Voevodsky's univalence axiom and higher inductive types at its core.
Hook length formula
The hook length formula is a 1954 theorem of combinatorics giving the number of standard Young tableaux of a shape, and the dimensions of symmetric group representations.
Hopcroft–Karp algorithm
The Hopcroft–Karp algorithm, published by John E. Hopcroft and Richard M. Karp in 1973, finds a maximum matching in a bipartite graph in O(m√n) time.
Hopf algebra
A Hopf algebra is an algebraic structure that is both an algebra and a coalgebra with a compatible antipode map, generalizing groups and Lie algebras and underlying the theory of quantum groups.
Hopf bifurcation
The Hopf bifurcation, or Andronov-Hopf bifurcation, is a local bifurcation of dynamical systems in which an equilibrium changes stability and a small limit cycle emerges.