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Algebraic number theory
Algebraic number theory studies algebraic integers in finite-degree number fields over Q, using rings of integers, ideal class groups, Dirichlet's unit theorem, class field theory, and zeta and L-functions; it arose from Fermat's last theorem and differs from elementary number theory by restoring unique factorization through ideals.
Algebraic quantum field theory
Algebraic quantum field theory, also called local quantum physics, is a mathematical framework for relativistic quantum physics proposed by Rudolf Haag and Daniel Kastler in 1964.
Algebraic Riccati equation
The algebraic Riccati equation, also called the matrix Riccati equation, is a nonlinear matrix equation central to optimal control, including the linear quadratic regulator and Kalman filtering.
Algebraic stack
An algebraic stack, also called an Artin stack, is a stack in groupoids over schemes that locally looks like a scheme, used for moduli problems.
Algebraic structure
An algebraic structure, or algebraic system, is a set equipped with operations satisfying specified axioms; examples include groups, rings, fields, lattices, and vector spaces, studied collectively by universal algebra.
Algebraic variety
An algebraic variety, also called an algebraic set, is the set of solutions of polynomial equations over a field and the central object of algebraic geometry.
Algebraically closed field
An algebraically closed field is a field in which every non-constant polynomial with coefficients in the field has a root in it, like the complex numbers.
Almost integer
An almost integer is a real number very close to, but not equal to, an integer. The flagship example, e^(π√163), differs from 262537412640768744 by less than 10^-12.
Alternating group
The alternating group A_n is the group of even permutations of n objects, of order n!/2, and is simple, with no proper nontrivial normal subgroups, for every n at least 5.
Alternative algebra
Alternative algebra, also known as alternative ring, is an algebra in which every two elements generate an associative subalgebra; the octonions are the standard non-associative example.
AM–GM inequality
The AM–GM inequality (inequality of arithmetic and geometric means) states that the arithmetic mean of positive numbers is never below their geometric mean, with equality only when all numbers are equal.
Amenable Banach algebra
An amenable Banach algebra is a Banach algebra in which every bounded derivation into a dual bimodule is inner; the notion was introduced by B. E. Johnson in 1972.
Amenable group
An amenable group is a group admitting a left-invariant mean on bounded functions, introduced by John von Neumann in 1929 while studying the Banach–Tarski paradox.
Amicable numbers
Amicable numbers, or amicable pairs, are two numbers each equal to the sum of the other's proper divisors; the smallest pair, 220 and 284, was known to Thābit ibn Qurra in the ninth century.
Ample line bundle
An ample line bundle is a line bundle whose some positive tensor power is very ample, embedding the variety in projective space; it is the standard algebraic positivity notion.
Anabelian geometry
Anabelian geometry, also known as the anabelian conjecture, is a branch of arithmetic geometry, proposed by Alexander Grothendieck in 1983, that reconstructs geometric objects from their fundamental groups.
Analytic number theory
Analytic number theory is the branch of mathematics that applies real, complex, and harmonic analysis to problems about the integers, above all the distribution of prime numbers.
Anasazi (software)
Anasazi is an extensible, interoperable framework for large-scale eigenvalue algorithms developed as a package within the Trilinos project, providing iterative solvers including block Davidson, LOBPCG, and Block Krylov-Schur.
Apéry's constant
Apéry's constant, the value of the Riemann zeta function at 3, equals about 1.2020569 and is the sum of the reciprocals of the cubes of the positive integers.
Applications of p-adic numbers
Applications of p-adic numbers are uses of p-adic number systems outside core p-adic analysis, in physics, cryptography and coding theory, data analysis, and dynamical systems, dating to 1987.
Approximations of π
Approximations of π, also called numerical approximations of pi, are values used in place of π, from Archimedes' 22/7 around 250 BC to a 2025 record of 314 trillion digits.
Arakelov theory
Arakelov theory, also called Arakelov geometry, is an intersection theory on arithmetic varieties introduced by Suren Arakelov in 1974 and extended by Gillet and Soulé in 1990.
Arb (software)
Arb is a C library for arbitrary-precision real and complex arithmetic that tracks numerical errors using ball arithmetic, created by Fredrik Johansson and merged into FLINT in 2023.
Arbitrary-precision arithmetic
Arbitrary-precision arithmetic, also known as bignum arithmetic, is computation on numbers whose precision is limited only by machine memory, supporting integers, rationals, and floats in cryptography, computer algebra, and record calculations.
Archimedean property
The Archimedean property, also known as Eudoxus' axiom, is a property of ordered fields stating that no element exceeds every natural number; Archimedes stated it in On the Sphere and Cylinder.
Argument (complex analysis)
The argument of a complex number, also called the phase, is the angle it makes with the positive real axis, defined only up to multiples of 2π.
Arithmetic
Arithmetic is the branch of mathematics that studies numbers and their operations, chiefly addition, subtraction, multiplication, and division, with roots going back to ancient Egypt.
Arithmetic coding
Arithmetic coding is a lossless data compression method that represents an entire message as a subinterval between 0 and 1, achieving better compression than Huffman coding at higher computational cost.
Arithmetic combinatorics
Arithmetic combinatorics, also called additive combinatorics, is a branch of mathematics at the intersection of combinatorics and number theory that studies sets under addition and multiplication.
Arithmetic dynamics
Arithmetic dynamics is a branch of mathematics, established around 2000 by Joseph H. Silverman, that reformulates number theory problems as questions about iterated rational maps.