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Congruence subgroup
A congruence subgroup is a subgroup of the modular group SL2(Z) containing all matrices congruent to the identity modulo some integer N. It underlies modular forms.
Conjugacy class
A conjugacy class is the set of all elements of a group obtainable from one another by conjugation, a = gbg⁻¹; the classes partition the group and underlie the class equation and character theory.
Conjugate gradient method
The conjugate gradient (CG) method is an iterative algorithm for solving symmetric positive definite linear systems Ax = b, introduced by Magnus Hestenes and Eduard Stiefel in 1952.
Conjugate transpose
The conjugate transpose, also known as the Hermitian transpose or Hermitian conjugate, is the matrix obtained by transposing a complex matrix and taking the complex conjugate of each entry, written A, A^H, or A†.
Connes classification of type III factors
The Connes classification of type III factors is the 1973 partition of type III von Neumann factors into III₀, IIIλ, and III₁, defined by Alain Connes via the S and T invariants from modular theory.
Connes embedding problem
The Connes embedding problem, also called Connes' embedding conjecture, asked whether every tracial von Neumann algebra embeds into an ultrapower of the hyperfinite II1 factor; it was refuted in 2020.
Constructible number
A constructible number, also called a Euclidean number, is a real number that can be built as a length using only an unmarked straightedge and compass.
Constructible universe
The constructible universe, or Gödel's constructible universe, is a class L of sets defined by Kurt Gödel in 1938 to prove the consistency of the axiom of choice.
Construction and structure of Kac–Moody algebras
Kac–Moody algebras are generally infinite-dimensional Lie algebras built from a generalized Cartan matrix by generators and relations, introduced independently by Victor Kac and Robert Moody in the late 1960s.
Construction of the complex numbers
The construction of the complex numbers builds C rigorously from R, as Hamilton's ordered pairs, 2×2 matrices, or the quotient R[x]/(x²+1), all isomorphic fields containing a square root of −1.
Construction of the real numbers
The construction of the real numbers, or construction of the reals, builds the real numbers from the rationals, chiefly through Cantor's Cauchy sequences and Dedekind's cuts, both published in 1872.
Content (polynomial)
In algebra, the content of a polynomial is the greatest common divisor of its coefficients; dividing by it leaves the primitive part of the polynomial.
Continued fraction
A continued fraction is an expression built from nested denominators representing any real number, finite exactly for rationals; its convergents give best rational approximations, underlying results like the 355/113 approximation of π.
Continuum hypothesis
The continuum hypothesis, listed by Hilbert in 1900 as his first problem, states that no cardinality lies strictly between that of the natural numbers and the real numbers.
Conversion between quaternions and Euler angles
Conversion between quaternions and Euler angles is the mathematical transformation between unit quaternions and Euler angles, two equivalent representations of a 3D rotation, with gimbal lock making one direction ambiguous.
Conway group
The Conway groups are three sporadic simple groups, Co1, Co2, and Co3, discovered by John Conway around 1968 as symmetry groups of the 24-dimensional Leech lattice.
Coprime integers
Coprime integers, also called relatively prime, mutually prime, or strangers, are integers whose greatest common divisor is 1, such as 8 and 15; about 61 percent of random pairs are coprime.
Coproduct
A coproduct is a category-theoretic construction that combines objects by mapping into the result, dual to the product and realized as disjoint unions, free products, and direct sums.
Core model
The core model, usually written K, is a canonical inner model of set theory, built by Dodd and Jensen to carry as much large-cardinal structure as a stated bound allows.
Coset
A coset, left coset or right coset, is a translate gH or Hg of a subgroup H inside a group G; equal-sized cosets partition G, yielding Lagrange's theorem.
Cosine similarity
Cosine similarity is a measure of similarity between two nonzero vectors, the cosine of the angle between them; it runs from 1 for the same direction to -1 for opposite.
Cotangent complex
The cotangent complex is a homological invariant of a ring map or scheme morphism that replaces Kähler differentials for nonsmooth maps and controls first-order deformation theory.
Covariance and contravariance of vectors
Covariance and contravariance, including contravariant vectors, describe the two ways vector components transform under a change of coordinates; the metric tensor converts between the two types.
Covering lemma
A covering lemma is a theorem of set theory stating that, absent large cardinals, every uncountable set of ordinals can be covered by a same-sized set in L.
Coxeter group
A Coxeter group, or Coxeter system, is a group generated by involutions satisfying relations (r_i r_j)^{n_ij} = 1, introduced by H. S. M. Coxeter in 1934 through reflection groups.
Cramér's conjecture
Cramér's conjecture, proposed by Swedish probabilist Harald Cramér in 1936, states that prime gaps near large numbers grow no faster than a constant times the square of the logarithm.
Cramer's rule
Cramer's rule, also known as Cramer's law or Cramer's method, solves a square system of linear equations by writing each unknown as a ratio of determinants.
Crank of a partition
The crank of a partition, also called Dyson's crank, is a partition statistic defined by Andrews and Garvan in 1988 that gives a combinatorial explanation of Ramanujan's three partition congruences.
Cross product
The cross product, also called the vector product, is an operation on two vectors in three-dimensional space that produces a third vector perpendicular to both, with direction fixed by the right-hand rule.
Crossed product of von Neumann algebras
The crossed product of von Neumann algebras, or Verschränktes Produkt, is a von Neumann algebra M ⋊ G built from M and a group action, analogous to the semidirect product.