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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.
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.
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.
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.
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.
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.
Householder transformation
The Householder transformation, also called a Householder reflection, is a symmetric orthogonal matrix reflecting vectors across a hyperplane, widely used in QR factorization since Alston Householder introduced it in 1958.
HP Prime
The HP Prime is a graphing calculator with a computer algebra system, color multi-touch screen, and rechargeable battery, introduced by Hewlett-Packard in autumn 2013 and still produced in a revised G2 version.
Hume's principle
Hume's principle is an abstraction principle in the philosophy of arithmetic stating that numbers of things are equal exactly when the things can be paired one to one.
Hurwitz's theorem (composition algebras)
Hurwitz's theorem is Adolf Hurwitz's 1898 classification of real composition algebras: their dimensions are only 1, 2, 4, or 8, namely the reals, complexes, quaternions, and octonions.
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.
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.
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.
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.
Ideal (ring theory)
An ideal in ring theory is a subset of a ring closed under addition and multiplication by any ring element, introduced by Richard Dedekind in 1871.
Ideal class group
The ideal class group of a Dedekind domain measures the failure of unique factorization; it is trivial exactly when the domain is a principal ideal domain.
Idele class group
The idele class group of a global field K is the quotient of the idele group by the diagonal K×, recovering ideal class groups and receiving the reciprocity map.
Idempotence
Idempotence is the property of an element e of a ring satisfying e² = e, equivalently e(1 − e) = 0; multiplication by e acts like a projection.
Identity (mathematics)
In mathematics, an identity is an element that leaves every other element unchanged when combined with it, or an equation, such as a + 0 = a, true for every value of its variables.
Identity matrix
The identity matrix, also called the unit matrix, is the square matrix with ones on the diagonal and zeros elsewhere that leaves any matrix unchanged under multiplication.
IEEE 754
IEEE 754, the IEEE Standard for Binary Floating-Point Arithmetic, is a technical standard first published in 1985 that defines floating-point formats, rounding rules, and exception handling in computing.
Imaginary number
An imaginary number, also called a purely imaginary number, is a number whose square is negative, such as multiples of the imaginary unit i, where i squared equals −1.
Imaginary unit
The imaginary unit is the number i, also known as the square root of minus one, whose square is -1; it extends the real numbers to the complex numbers.
Inaccessible cardinal
An inaccessible cardinal is an uncountable cardinal that cannot be reached from smaller cardinals by successor, union, and power set operations, and whose existence ZFC cannot prove.
Incidence matrix
An incidence matrix is a matrix recording which elements of one set are incident with elements of another, with rows for vertices and columns for edges.
Incomplete gamma function
The incomplete gamma functions γ(a,z) and Γ(a,z) are the lower and upper parts of the gamma integral, used in statistics as the gamma distribution's cumulative probabilities.
Incomplete LU factorization
An incomplete LU factorization (ILU) approximates a sparse matrix as LU plus a discarded remainder R, keeping factors sparse and serving mainly as a preconditioner for Krylov methods like GMRES.