1,733 articles
Hermite polynomials
Hermite polynomials, also called the probabilists' Hermite polynomials in one of two conventions, are classical orthogonal polynomials on the real line with a Gaussian weight, central to the quantum harmonic oscillator.
Hermitian matrix
A Hermitian matrix, also called a self-adjoint matrix, is a complex square matrix that equals its own conjugate transpose; its eigenvalues are always real, and it is unitarily diagonalizable.
Hessian matrix
The Hessian matrix, or Hesse matrix, is the square matrix of a function's second-order partial derivatives, used to classify critical points as minima, maxima, or saddle points.
Hexadecimal
Hexadecimal, also known as base 16 or hex, is a numeral system using sixteen digits, 0–9 and A–F, serving in computing as a compact shorthand for binary.
Heyting algebra
A Heyting algebra, also called a pseudo-Boolean algebra, is a bounded lattice with an implication operation, serving as the algebraic semantics of intuitionistic logic as Boolean algebras do for classical logic.
Higher-dimensional class field theory
Higher-dimensional class field theory extends abelian class field theory to higher local fields and arithmetic schemes, due to Parshin and Kato, replacing the multiplicative group with Milnor K-groups.
Higher-order singular value decomposition
The higher-order singular value decomposition (HOSVD), also known as higher-order SVD, is a multilinear generalization of the matrix SVD, introduced by De Lathauwer, De Moor, and Vandewalle in 2000.
Highest-weight theory
Highest-weight theory is the branch of representation theory classifying representations of semisimple Lie algebras and Lie groups by a single linear functional, the highest weight, attached to each irreducible representation.
Highly composite number
A highly composite number, also called an anti-prime or maximally divisible number, is a positive integer with more divisors than any smaller one, first analyzed by Ramanujan in 1915.
Hilbert C*-module
A Hilbert C-module is a right module over a C-algebra with an algebra-valued inner product, generalizing Hilbert spaces and central to KK-theory and noncommutative geometry.
Hilbert class field
The Hilbert class field of a number field K is its unique maximal unramified abelian extension, with Galois group isomorphic to K's ideal class group.
Hilbert matrix
The Hilbert matrix is the square matrix with entries 1/(i + j − 1), the standard example of a badly ill-conditioned matrix in numerical linear algebra.
Hilbert symbol
The Hilbert symbol, also called the norm residue symbol, is a pairing on a local field that tests whether an element is a norm, central to reciprocity laws.
Hilbert–Pólya conjecture
The Hilbert–Pólya conjecture in number theory proposes that the Riemann zeta function's non-trivial zeros are eigenvalues of a self-adjoint operator, which would prove the Riemann hypothesis.
Hilbert's basis theorem
Hilbert's basis theorem is a theorem of David Hilbert, proved in 1890, stating that if a ring R is Noetherian, then the polynomial ring R[x] is also Noetherian.
Hilbert's Nullstellensatz
Hilbert's Nullstellensatz, German for "zero locus theorem," is David Hilbert's 1893 theorem bridging commutative algebra and algebraic geometry: over an algebraically closed field, ideals of polynomials correspond precisely to the sets of their common zeros.
Hilbert's Theorem 90
Hilbert's Theorem 90 is an algebra theorem stating that for a finite cyclic field extension, every norm-one element is a quotient y/σ(y), proved as Satz 90 of Hilbert's 1897 Zahlbericht.
Hilbert's twelfth problem
Hilbert's twelfth problem, also known as Kronecker's Jugendtraum, was posed by David Hilbert in 1900 and asks for explicit constructions of all abelian extensions of a number field.
Hill cipher
The Hill cipher is a classical polygraphic cipher invented by Lester S. Hill in 1929 that encrypts blocks of letters using an invertible key matrix modulo 26.
Hippasus (ππασος)
Hippasus of Metapontum was an early Pythagorean philosopher active in the first half of the fifth century BCE, known for music theory and legends of a divulged secret.
History of algebra
The history of algebra traces techniques for finding unknown quantities, from Babylonian tablets around 1800 BC through al-Khwarizmi, who gave the subject its name, to the abstract structures of the twentieth century.
History of homological algebra
The history of homological algebra traces the study of homology and cohomology from Hilbert's 1890 syzygy theorem to Cartan and Eilenberg's founding 1956 treatise and Grothendieck's 1957 abelian categories.
History of Kac–Moody algebra theory
The history of Kac–Moody algebra theory traces the infinite-dimensional Lie algebras defined independently by Victor Kac and Robert Moody in 1967–68, through Macdonald's identities and physics applications.
History of mathematics
The history of mathematics is the record of how counting, measurement, calculation, and proof developed, from prehistoric notched bones through Mesopotamia, Egypt, Greece, India, China, the Islamic world, and Europe to the modern era.
History of non-associative algebra
The history of non-associative algebra traces algebraic systems where multiplication need not associate, beginning with Hamilton's 1843 quaternions and Graves's octonions and maturing through Hurwitz's 1898 theorem and Zel'manov's 1983 classification.
History of quaternions
The history of quaternions begins with William Rowan Hamilton's 1843 discovery of a four-dimensional number system whose multiplication does not commute, carved into a Dublin bridge.
History of real and complex numbers
The history of real and complex numbers traces how complex numbers were used centuries before their definition, while real numbers got rigorous foundations only in the 1800s.
History of the classification of finite simple groups
The history of the classification of finite simple groups traces the effort from Galois's concept to the 2004 completion of a proof spanning over 10,000 pages.
Hochschild homology and cohomology
Hochschild homology and cohomology are pairs of derived-functor invariants of associative algebras, introduced by Gerhard Hochschild in 1945 and now extended to categories, schemes, and spectra.
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.