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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.
Kirchhoff's theorem (graph theory)
Kirchhoff's theorem, also called the matrix-tree theorem, states that the number of spanning trees of a finite graph equals any cofactor of its Laplacian matrix.
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.
Kite (geometry)
In geometry, a kite, also called a geometric kite, is a quadrilateral with two distinct pairs of equal adjacent sides. It may be convex or, in the concave case, called a dart.
Kleene star
The Kleene star, also called Kleene closure, is an operation that maps a language L to all strings formed by concatenating zero or more strings from L.
Kleene's O
Kleene's O is a set of natural numbers, called ordinal notations, that names the constructive ordinals; it was introduced by Stephen Cole Kleene in 1938.
Kleene's recursion theorem
Kleene's recursion theorem, also called the fixed point theorem, is a result in computability theory published by Stephen Kleene in 1938, implying self-reproducing programs called quines.
Klein bottle
The Klein bottle is a closed, one-sided surface with no inside or outside, described by Felix Klein in 1882 and impossible to place in space without self-intersection.
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.
Kleinian group
A Kleinian group is a discrete group of Möbius transformations of the sphere, equivalently isometries of hyperbolic 3-space, created by Poincaré and Klein in the 1880s.
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.
Knapsack problem
The knapsack problem, also called the rucksack problem, is a combinatorial optimization problem in which the most valuable items are packed within a fixed capacity, solvable by dynamic programming.
Kneser graph
The Kneser graph K(n, k) has the k-subsets of {1, ..., n} as vertices, with edges between disjoint sets; introduced by Lovász in 1978, K(5, 2) is the Petersen graph.
Knight's tour
A knight's tour, or knight's tour problem, is a sequence of chess knight moves visiting every square exactly once; a closed tour returns a knight's move from the start.
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.
Knot theory
Knot theory, also called the theory of knots, is the branch of mathematics studying closed loops in 3-dimensional space, asking when two knots are equivalent without cutting.
Knowledge base
A knowledge base is a structured store of facts, rules, and relationships that computers query and reason over, spanning expert systems like MYCIN and modern knowledge graphs such as Google's.
Knowledge-based systems
A knowledge-based system (KBS) is a computer program that solves problems using a stored knowledge base and a separate inference engine, as in expert systems.
Knuth–Bendix completion
Knuth–Bendix completion is an algorithm that tries to convert a set of equations into a confluent, terminating rewrite system, deciding the word problem by normal forms.
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.
Koch snowflake
The Koch snowflake, also called the Von Koch snowflake, is a fractal curve attributed to Helge von Koch in 1904, with infinite perimeter but finite area.
Koch's postulates
Koch's postulates, also called the Henle–Koch postulates, are four criteria for establishing that a specific microorganism causes an infectious disease, formulated by German physician Robert Koch in 1890.
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.
Kolmogorov complexity
Kolmogorov complexity, also known as algorithmic entropy or program-size complexity, is the length of the shortest program producing a given string, a measure discovered independently by Solomonoff, Kolmogorov, and Chaitin.
Kolmogorov equations
The Kolmogorov equations are the forward and backward differential equations, introduced by Andrei Kolmogorov in 1931, that describe how the transition probabilities of a Markov process change over time.
Kolmogorov extension theorem
The Kolmogorov extension theorem, also called the Daniell–Kolmogorov theorem, is a result in probability theory: consistent finite-dimensional distributions determine a probability measure on the product space.
Kolmogorov–Arnold–Moser theorem
The Kolmogorov–Arnold–Moser theorem, or KAM theorem, states that most invariant tori of an integrable Hamiltonian system survive a small perturbation. Kolmogorov stated it in 1954; Arnold and Moser proved it in the early 1960s.