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Happy ending problem
The happy ending problem, also called the Erdős–Szekeres conjecture, asks how many points in general position in the plane guarantee n vertices of a convex polygon, named for the marriage of George Szekeres and Esther Klein.
Happy number
A happy number is a positive integer that reaches 1 when repeatedly replaced by the sum of the squares of its digits, for example 7, 13, and 19.
Harald Bohr
Harald Bohr (1887–1951) was a Danish mathematician, younger brother of physicist Niels Bohr, who founded the theory of almost periodic functions and worked on Dirichlet series.
Hard-core point process
A hard-core point process is a spatial stochastic model where no two points lie closer than a fixed minimum distance, modeling repulsive patterns like cells and trees.
Hardy inequality
The Hardy inequality is a result in analysis bounding averaged sequences or functions by a constant times the original, introduced by G. H. Hardy in 1920.
Hardy space
The Hardy space H^p, also called a Hardy class, is a space of analytic functions with bounded integral means, introduced by G. H. Hardy in 1915.
Hardy–Littlewood circle method
The Hardy–Littlewood circle method is an analytic number theory technique for counting integer representations as sums of powers or primes, arising from Hardy and Ramanujan's 1917 partition work.
Hardy–Littlewood conjecture
The Hardy–Littlewood conjecture, made by G. H. Hardy and John E. Littlewood in 1923, predicts how often patterns of primes occur and generalizes the twin prime conjecture.
Hardy–Littlewood maximal function
The Hardy–Littlewood maximal function is an operator in harmonic analysis that replaces a function with its supremal averages over balls, introduced by Hardy and Littlewood in 1930.
Harish-Chandra isomorphism
The Harish-Chandra isomorphism is an algebra isomorphism identifying the center of the universal enveloping algebra of a semisimple Lie algebra with Weyl-group-invariant polynomials on a Cartan subalgebra.
Harmonic analysis
Harmonic analysis is the branch of mathematics that decomposes functions and signals into sine and cosine waves, the idea behind Fourier series and the Fourier transform.
Harmonic balance
Harmonic balance, or the harmonic balance method, is a frequency-domain numerical method for the periodic steady-state response of nonlinear circuits, used for amplifiers, mixers, and oscillators.
Harmonic form
A harmonic form is a differential form on a Riemannian manifold lying in the kernel of the Hodge Laplacian, central to Hodge theory, which links analysis and topology.
Harmonic function
A harmonic function is a twice continuously differentiable function satisfying the Laplace equation, describing steady-state temperatures, electrostatic potentials, and ideal fluid flows; it obeys the mean value property.
Harmonic map
A harmonic map is a mapping between Riemannian manifolds that is a critical point of a natural energy integral, generalizing Dirichlet's integral; the theory was begun systematically by Eells and Sampson in 1964.
Harmonic mean
The harmonic mean is a type of average, one of the three Pythagorean means, calculated as the number of values divided by the sum of their reciprocals.
Harmonic measure
Harmonic measure is a probability measure on a domain's boundary giving where Brownian motion from an interior point first exits, equivalently how boundary values determine solutions of the Dirichlet problem.
Harmonic number
A harmonic number is the sum of the reciprocals of the first n positive integers, a partial sum of the divergent harmonic series that grows like the natural logarithm.
Harmonic series (mathematics)
The harmonic series is the infinite sum of 1 + 1/2 + 1/3 + 1/4 + …, which diverges even though its terms tend to zero.
Harris hawks optimization
Harris hawks optimization (HHO) is a population-based, gradient-free metaheuristic for continuous global optimization, introduced in 2019 and inspired by the cooperative hunting of Harris's hawks.
Harshad number
A Harshad number, also called a Niven number, is a positive integer divisible by the sum of its digits; 1729 qualifies because 1729 = 91 × 19, the sum of its digits.
Hartley transform
The Hartley transform is an integral transform introduced by Ralph Hartley in 1942 that, unlike the Fourier transform, gives a real-valued output for real signals.
Hasse diagram
A Hasse diagram is a drawing of a finite partially ordered set, showing an edge between two elements exactly when one covers the other, with greater elements higher.
Hasse norm theorem
The Hasse norm theorem, Hasse's theorem on cyclic extensions, is a 1931 result of Helmut Hasse: in a cyclic extension of number fields, being a norm locally everywhere is equivalent to being a global norm.
Hasse principle
The Hasse principle, also called the local-global principle, is a statement in number theory about when solutions to equations over the rational numbers follow from local solutions.
Hat monotile
The hat monotile is a 13-sided shape discovered by David Smith in 2023 that tiles the plane with one shape but never periodically, solving the long-standing einstein problem.
Hausdorff dimension
The Hausdorff dimension, also called the Hausdorff-Besicovitch dimension, is a possibly fractional measure of a set's size, introduced by Felix Hausdorff in 1918; the middle-third Cantor set has dimension log 2 / log 3.
Hausdorff distance
The Hausdorff distance, introduced by Felix Hausdorff in 1914, is a metric on compact subsets of a metric space, used in medical image segmentation and comparing point clouds.
Hausdorff maximal principle
The Hausdorff maximal principle, also called the Hausdorff maximality theorem, states that every chain in a partially ordered set is contained in a maximal chain, and is equivalent to Zorn's lemma.
Hausdorff measure
The Hausdorff measure is a family of measures introduced by Felix Hausdorff in 1918 that assigns sizes to sets of any dimension, including fractals, and extends the Lebesgue measures of length, area, and volume.