1,733 articles
Gelfond's constant
Gelfond's constant is the transcendental number e^π, approximately 23.1407, proven transcendental in 1934 by Aleksandr Gelfond and independently by Theodor Schneider in 1935, settling Hilbert's seventh problem.
General linear group
The general linear group GL_n(R) is the group of all invertible n by n matrices over a ring R, forming a group under matrix multiplication.
General number field sieve
The general number field sieve (GNFS) is the fastest known classical algorithm for factoring arbitrary large integers, generalized by Buhler, Lenstra, and Pomerance in 1993 and used for RSA-250.
General theory of Banach algebras
The general theory of Banach algebras, which began in 1941, examines complete normed algebras with submultiplicative norms. It details concepts like closed ideals, quotients, and the Gelfand transform.
Generalizations of Fibonacci numbers
Generalizations of Fibonacci numbers, including the k-bonacci or k-step Fibonacci numbers, are integer sequences that extend the Fibonacci recurrence by summing more preceding terms, giving tribonacci and tetranacci numbers.
Generalized Cartan matrix
A generalized Cartan matrix, plural Cartan matrices, is a square integer matrix with diagonal entries 2 and non-positive off-diagonal entries; each one determines a Kac-Moody Lie algebra.
Generalized continuum hypothesis
The generalized continuum hypothesis (GCH) is the set-theoretic statement that each infinite cardinal's power set has the next larger cardinality; Gödel and Cohen proved it independent of ZFC.
Generalized eigenvector
A generalized eigenvector of a matrix A is a nonzero vector v with (A − λI)^p v = 0 for some p; ordinary eigenvectors are the case p = 1.
Generalized Kac–Moody algebra
A generalized Kac–Moody algebra, also called a Borcherds or BKM algebra, is an infinite-dimensional Lie algebra introduced by Richard Borcherds in 1988, generalizing Kac–Moody algebras by allowing imaginary simple roots.
Generalized mean
The generalized mean, also called the power mean or Hölder mean, is a family of averaging functions parameterized by an exponent, including the arithmetic, geometric, and harmonic means.
Generalized minimal residual method
The generalized minimal residual method (GMRES) is an iterative Krylov subspace method for solving large sparse nonsymmetric linear systems, proposed by Saad and Schultz in 1986.
Generalized Riemann hypothesis
The generalized Riemann hypothesis (GRH) is an unproved conjecture stating that all nontrivial zeros of every Dirichlet L-function lie on the critical line where the real part is 1/2.
Generalized Verma module
The generalized Verma module, also called a parabolic Verma module, is a Lie algebra representation induced from a module over a parabolic subalgebra, generalizing the Verma module.
Genus of a curve
The genus of a curve is a nonnegative integer measuring an algebraic curve's complexity; over the complex numbers it equals the number of handles of the curve as a Riemann surface.
GeoGebra
GeoGebra is a free, open-source suite of interactive mathematics software combining geometry, algebra, and calculus, founded by Markus Hohenwarter in 2001 and used by over 100 million students.
Geometric algebra
Geometric algebra is the geometric interpretation of Clifford algebra, which Clifford combined with Grassmann's exterior algebra in 1878. It defines vectors and volumes as elements within a unified system.
Geometric complexity theory
Geometric complexity theory is a program in computational complexity theory, founded by Ketan Mulmuley and Milind Sohoni in 2001, that uses algebraic geometry and representation theory to prove lower bounds such as P ≠ NP.
Geometric progression
A geometric progression, also called a geometric sequence, is a sequence of numbers where each term is the previous term multiplied by a fixed common ratio.
Geometry of numbers
The geometry of numbers, or geometric number theory, is a branch of number theory founded by Hermann Minkowski, studying how lattice points interact with convex bodies.
Gerolamo Cardano
Gerolamo Cardano, also Girolamo Cardano, was an Italian physician, mathematician, and astrologer whose 1545 Ars Magna gave the first published solutions of cubic and quartic equations.
Gershgorin circle theorem
The Gershgorin circle theorem is a result in matrix analysis stating that every eigenvalue of a square matrix lies in disks centered at the diagonal entries, published in 1931.
Gilbert–Varshamov bound
The Gilbert–Varshamov bound is a lower bound in coding theory: proved by Gilbert in 1952 and Varshamov in 1957, it guarantees codes of a certain size exist.
Gimel function
The gimel function, named after the Hebrew letter gimel, maps each infinite cardinal κ to κ^cf(κ), where cf(κ) is κ's cofinality, and determines cardinal exponentiation.
Givens rotation
A Givens rotation is a plane rotation in numerical linear algebra, acting on two coordinates of a vector to turn one chosen entry into zero.
Glaisher's theorem
Glaisher's theorem is an 1883 result stating that partitions into parts not divisible by d equal partitions with no part repeated d or more times.
Global dimension
Global dimension, also called global homological dimension, is the supremum of the projective dimensions of modules over a ring, measuring how far the ring is from semisimple.
Global field
A global field is a field that is either a number field, a finite extension of the rationals, or a function field over a finite field.
GNU MPFR
GNU MPFR (Multiple Precision Floating-Point Reliable) is a portable C library for arbitrary-precision floating-point arithmetic, built on GNU MP and providing correctly rounded, reproducible results.
GNU Octave
GNU Octave is a free, high-level programming language and interactive environment for numerical computations, with syntax mostly compatible with Matlab. It runs on Linux, macOS, and Windows.
Golay code
The Golay codes are two perfect error-correcting block codes, one binary and one ternary, introduced by Marcel Golay in 1949 and used for error control in the Voyager mission.