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Kaczmarz method
The Kaczmarz method, also called the Kaczmarz algorithm, is an iterative method for solving systems of linear equations by projecting onto one equation's hyperplane at a time, widely used in computed tomography as ART.
Kadison–Singer problem
A 1959 problem posed by Richard Kadison and Isadore Singer asking whether every pure state on the diagonal operators extends uniquely to all bounded operators, solved by Marcus, Spielman, and Srivastava in 2013.
Kahan summation algorithm
The Kahan summation algorithm, also called compensated summation, is a method for adding floating-point numbers that captures each addition's rounding error in a compensation variable, cutting accumulated error; William Kahan published it in 1965.
Kan extension
A Kan extension is a category theory construction that extends a functor along another functor in a universal way, introduced by Daniel Kan in 1958.
Kaprekar number
A Kaprekar number is a positive integer whose square can be split into two parts that add up to the original number, such as 45.
Karnaugh map
A Karnaugh map (K-map) is a grid-based method for simplifying Boolean algebra expressions, invented by Maurice Karnaugh at Bell Labs in 1953 for telephone switching circuits.
Kempner function
The Kempner function, also called the Smarandache function, gives the smallest positive integer m such that n divides m!; it was introduced by Aubrey J. Kempner in 1918.
Kernel (algebra)
The kernel (also called the kernel of a homomorphism) in algebra is the set of elements a homomorphism sends to zero, a submodule measuring injectivity.
Kernel (linear algebra)
The kernel of a linear map, also called its null space, is the set of all domain vectors the map sends to zero; it is a linear subspace of the domain.
Khatri-Rao product
The Khatri-Rao product, or column-wise Kronecker product, is a matrix product that applies the Kronecker product column by column, introduced by Khatri and Rao in 1968.
Killing form
The Killing form is a symmetric bilinear form on a Lie algebra, defined as a trace in the adjoint representation, used to test solvability and semisimplicity.
Kirillov orbit method
The Kirillov orbit method, also called the orbit method, parametrizes the irreducible unitary representations of a Lie group by its coadjoint orbits; A. A. Kirillov introduced it in 1962.
Klein four-group
The Klein four-group is the smallest non-cyclic group, a 4-element abelian group in which every element is its own inverse, isomorphic to C2 × C2.
Kleisli category
A Kleisli category is a category built from a monad, keeping the base category's objects and taking morphisms X to Y to be morphisms X to T(Y).
Kloosterman sum
A Kloosterman sum is an exponential sum over invertible residues modulo an integer, introduced by Hendrik Kloosterman in 1926 to represent integers by quaternary quadratic forms.
KMS state
A KMS state, also known as a KMS condition, is a state on a C-algebra with time evolution whose analytic boundary condition characterizes thermal equilibrium at inverse temperature beta.
Knizhnik–Zamolodchikov equations
The Knizhnik–Zamolodchikov equations, or KZ equations, are a holonomic system of linear partial differential equations written down by Vladimir Knizhnik and Alexander Zamolodchikov in 1984 for conformal blocks of the Wess–Zumino–Witten model.
Knuth's up-arrow notation
Knuth's up-arrow notation, introduced by Donald Knuth in 1976, writes extremely large integers by extending exponentiation to tetration and higher repeated operations, and defines Graham's number.
Kodaira dimension
Kodaira dimension, also called the Iitaka–Kodaira dimension and written κ, is a birational invariant of algebraic varieties that measures the growth of pluricanonical sections and is the main invariant in the Enriques classification of surfaces.
Kodaira vanishing theorem
The Kodaira vanishing theorem, proved by Kunihiko Kodaira in 1953, states that for a positive line bundle on a compact complex manifold, higher cohomology vanishes.
König's theorem (set theory)
König's theorem is a result in cardinal arithmetic stating that if κᵢ < λᵢ for all i, then the sum of the κᵢ is strictly less than the product of the λᵢ.
Kostant partition function
The Kostant partition function is a representation theory function counting the ways a weight can be written as a sum of positive roots, introduced by Bertram Kostant in 1958.
Koszul complex
The Koszul complex is a chain complex built from the exterior algebra of a free module over a commutative ring, introduced by Jean-Louis Koszul in 1950 to detect regular sequences.
Kronecker delta
The Kronecker delta, written δij, is a mathematical symbol that equals 1 when its two indices are equal and 0 otherwise, serving as the index-notation form of the identity matrix.
Kronecker product
The Kronecker product, also called the tensor product of matrices, replaces each entry of a matrix A with A's entry times matrix B, yielding a block matrix.
Kronecker–Weber theorem
The Kronecker–Weber theorem, sometimes called the Kronecker–Weber–Hilbert theorem, states that every finite abelian extension of the rationals lies inside a cyclotomic field, an early result of class field theory proved by Hilbert in 1896.
Krull dimension
The Krull dimension of a commutative ring is the supremum of the lengths of chains of prime ideals, defined by Wolfgang Krull in 1928 as the standard notion of dimension in commutative algebra.
Krylov subspace
A Krylov subspace is the vector space spanned by b, Ab, A²b, and so on for a matrix A, named for Aleksey Krylov in 1931.
Krylov subspace methods
Krylov subspace methods are iterative projection methods for large linear systems, eigenvalue problems, and matrix functions, building approximations from repeated matrix-vector products; examples include CG, GMRES, and BiCGSTAB.
Kummer theory
Kummer theory, created by Ernst Eduard Kummer, classifies cyclic extensions of a field obtained by adjoining nth roots of elements, assuming the field contains roots of unity.