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Bézout's identity
Bézout's identity, also called Bézout's lemma, is a theorem of elementary number theory stating that the gcd of two integers is an integer linear combination of them.
Bézout's theorem
Bézout's theorem is a result in algebraic geometry stating that two projective plane curves of degrees m and n intersect in exactly mn points, counted with multiplicity.
Bfloat16
Bfloat16 (Brain Floating Point, BF16) is a 16-bit floating-point format for machine learning, introduced by Google Brain on TPUs, with FP32's range but lower precision.
BGG category O
BGG category O is a subcategory of modules over a complex semisimple Lie algebra serving as the standard setting for highest weight representation theory, introduced by Bernstein and the Gelfands in the 1970s.
Bhāskara I
Bhāskara I, also known as Bhaskara the First, was a 7th-century Indian mathematician and astronomer whose 629 commentary on Āryabhaṭa's Āryabhaṭīya is the earliest detailed prose exposition of mathematical astronomy in India.
Bhāskara II (भास्कर)
Bhāskara II, also called Bhāskarāchārya or Bhaskara the Learned, was an Indian mathematician and astronomer (1114–1185) who led the Ujjain observatory and wrote the Lilavati.
Bhattacharyya distance
The Bhattacharyya distance is a divergence measure between two probability distributions, defined as the negative logarithm of the Bhattacharyya coefficient and introduced by A. Bhattacharyya in 1943.
Bialgebra
A bialgebra (also bigebra) is a vector space with both an algebra structure and a compatible coalgebra structure, the common ancestor of Hopf algebras and quantum groups.
Bianchi identities
The Bianchi identities are two differential identities satisfied by the curvature and torsion of a connection on a manifold, named for Luigi Bianchi, who proved them in 1902.
Bias (statistics)
Statistical bias is a systematic tendency for an estimate or measurement to fall above or below its true value, arising from systematic rather than random error.
Bias of an estimator
The bias of an estimator, also called estimator bias, is the difference between the estimator's expected value and the true parameter value; an estimator with zero bias is unbiased.
Bicategory
A bicategory is a weak 2-category in which composition of morphisms is associative and has identities only up to invertible 2-morphisms, introduced by Jean Bénabou in 1967.
Bicomplex number
A bicomplex number is a four-dimensional generalization of the complex number, written with two independent imaginary units; unlike quaternions it is commutative but has zero divisors.
Bidirectional search
Bidirectional search is a graph search algorithm that runs two simultaneous searches, from the start and the goal, meeting in the middle to find a shortest path.
Bidual
The bidual (second dual) of a vector space X is the dual of its dual, X = (X), containing X via a canonical embedding; reflexivity means this embedding is onto.
Bifactor model
The bifactor model is a factor analysis model where every item loads on one general factor and at most one orthogonal group factor, introduced by Holzinger and Swineford in 1937.
Bifurcation analysis
Bifurcation analysis is a mathematical and computational method for studying how a dynamical system's qualitative behavior changes as parameters vary, tracing branches of equilibria and periodic orbits through numerical continuation.
Bifurcation theory
Bifurcation theory is the branch of dynamical systems theory describing qualitative changes in a system's behavior as a control parameter varies, such as equilibria giving way to oscillations.
Big O notation
Big O notation, also called Landau notation, is a mathematical notation describing how a function grows by bounding it above by a constant multiple of another function.
Biharmonic equation
The biharmonic equation is the fourth-order elliptic partial differential equation Δ²u = f, whose solutions are biharmonic functions; it governs thin-plate bending and Stokes flow.
Bijection
A bijection, also called a one-to-one correspondence or bijective function, is a function between two sets that is both injective and surjective, pairing the sets element by element with nothing left over.
Bijection, injection and surjection
Injection, surjection, and bijection are three properties of a function defined by how its outputs cover its codomain, used to compare the sizes of sets, including infinite ones.
Bijective proof
A bijective proof is a combinatorial technique that proves two counting expressions equal by constructing an explicit one-to-one, onto correspondence, prized for its clarity and generalizations.
Bilevel optimization
Bilevel optimization is a mathematical framework in which one optimization problem is nested inside another as a constraint, so the leader must anticipate the follower's best response, with applications in transportation, security, and machine learning.
Bilevel programming
Bilevel programming, also called bilevel optimization or the Stackelberg game, is mathematical programming with one optimization problem nested in another's constraints, used in hyperparameter optimization and neural architecture search.
Bilinear form
A bilinear form is a function on a vector space that is linear in each argument separately, generalizing the dot product and underlying quadratic forms.
Bilinear interpolation
Bilinear interpolation, also known as bilinear filtering, is a resampling method that estimates a value inside a rectangular grid cell from its four corner values, using two linear interpolations.
Bilinear model
The bilinear model is a nonlinear time series model that extends ARMA by adding cross terms between past observations and past innovations, so shocks depend on the series' state.
Billion
Billion, also called one billion, is the number 1,000,000,000 (a thousand million) in American and British English, though older European usage gives it the value 10^12.
Bimodule
A bimodule is an abelian group carrying a left module structure over one ring and a right module structure over another, with the two actions commuting.