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Arithmetic group
An arithmetic group is a group of integer matrices arising as the integer points of a linear algebraic group over the rationals, with SL(n, Z) the prototype.
Arithmetic logic unit
An arithmetic logic unit (ALU) is a digital circuit within a CPU or GPU's execution unit that performs arithmetic and bitwise logical operations on binary data.
Arithmetic of abelian varieties
The arithmetic of abelian varieties is the branch of arithmetic geometry studying rational points, torsion, and L-functions of abelian varieties, centered on the Mordell–Weil finiteness theorem.
Arithmetic progression
An arithmetic progression, also called an arithmetic sequence, is a sequence of numbers in which consecutive terms differ by a constant, the common difference; its nth term is a + (n − 1)d.
Arithmetic topology
Arithmetic topology is a branch of mathematics studying the analogy that knots are like prime numbers and 3-manifolds are like number rings, named in the mid-1990s.
Arithmetic-geometric mean
The arithmetic-geometric mean is the common limit reached by repeatedly taking the arithmetic and geometric means of two numbers, converging quadratically and underlying the fastest known algorithms for computing π.
Arithmetical set
An arithmetical set is a set of natural numbers definable by a first-order formula over addition and multiplication, belonging to some level of the arithmetical hierarchy introduced by Kleene in 1943.
Ars Magna (Cardano)
Ars Magna, the Great Art, is a book on algebra by Girolamo Cardano published at Nuremberg in 1545, containing the first printed solutions of the cubic and quartic equations.
Artin reciprocity law
The Artin reciprocity law, Emil Artin's 1927 theorem, is the central result of class field theory, identifying Galois groups of abelian extensions with class groups.
Artin–Hasse exponential
The Artin–Hasse exponential is a formal power series associated to a prime p whose coefficients are all p-adic integers, unlike the ordinary exponential, and is a tool in p-adic analysis.
Artinian ring
An Artinian ring is a ring satisfying the descending chain condition on ideals, named after Emil Artin, who studied such rings around 1928. Examples include every finite ring.
Aryabhata (आर्यभट)
Aryabhata (आर्यभट, also Aryabhata I) was an Indian mathematician and astronomer, 476–550 CE, whose 499 CE Sanskrit treatise the Āryabhaṭīya became the founding text of classical Indian mathematics.
Associated graded ring
The associated graded ring, or associated graded algebra, is a graded ring built from a filtered ring's successive quotients, often turning a complicated ring into a polynomial ring.
Association scheme
An association scheme is a finite set whose ordered pairs are partitioned into regular associate classes, whose adjacency matrices span an algebra, central to algebraic combinatorics.
Associative property
The associative property, also called the associative law, says regrouping operands does not change a binary operation's result; addition and multiplication have it, while subtraction and division do not.
Atan2
Atan2, also called Arctan2, is the two-argument arctangent function returning the angle to a point (x, y) in [−π, +π], fixing the quadrant from both signs.
Atiyah–Singer index theorem
The Atiyah–Singer index theorem, proved by Michael Atiyah and Isadore Singer in 1963, equates the analytical index of an elliptic operator with a topological index.
Atomic domain
An atomic domain is an integral domain in which every nonzero nonunit can be written as a finite product of irreducible elements, though the factorization need not be unique.
Automorphic representation
An automorphic representation is an irreducible representation occurring in the space of automorphic forms, packaging modular forms and their L-functions into representation theory for the Langlands program.
Automorphism group
An automorphism group is the group formed by all automorphisms of a mathematical object, such as a group, field, or graph, with composition as the operation.
Baby-step giant-step
Baby-step giant-step is a deterministic algorithm introduced by Daniel Shanks in 1971 that computes discrete logarithms in a cyclic group in square-root time, at the cost of comparable storage.
Babylonian cuneiform numerals
Babylonian cuneiform numerals are the base 60 place-value number system of ancient Mesopotamia, written with just two wedge signs into clay and surviving today in minutes and seconds.
Bailey–Borwein–Plouffe formula
The Bailey–Borwein–Plouffe formula (BBP) is a 1995 series for π that computes hexadecimal or binary digits of π starting at any position without earlier digits.
Baker–Campbell–Hausdorff formula
The Baker–Campbell–Hausdorff formula, or BCH formula, expresses log(exp(X)exp(Y)) as a series of nested Lie brackets with rational coefficients, converging for X and Y close to zero and used across physics and control theory.
Baker's theorem
Baker's theorem is a result in transcendental number theory giving explicit lower bounds for nonzero linear forms in logarithms of algebraic numbers, proved by Alan Baker in 1966.
Balanced ternary
Balanced ternary is a base-3 numeral system whose digits are −1, 0, and +1, so negative numbers need no separate sign; it drove the Soviet Setun computer of 1958.
Banach algebra cohomology
Banach algebra cohomology, also called cohomology of Banach algebras, is the cohomology theory for Banach algebras using bounded cochains, introduced by Barry Edward Johnson in 1972.
Banach function algebra
A Banach function algebra is a complete normed subalgebra of continuous functions that separates points and contains constants, studied with Gelfand theory as a commutative Banach algebra.
Bar complex
The bar complex of an associative algebra A is the chain complex whose n-th term is the tensor power A⊗(n+1), with a differential multiplying adjacent factors with alternating signs.
Bareiss algorithm
The Bareiss algorithm, also known as Montante's method, is a fraction-free form of Gaussian elimination that keeps integer matrices exact, published by Erwin Bareiss in 1968.