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Fixed-point arithmetic
Fixed-point arithmetic is a method of computing with real numbers by representing each value as an integer scaled by a fixed factor, with the radix point implicit.
Flat dimension
Flat dimension, also called weak dimension, is a homological invariant of a module over a ring, the length of the shortest resolution of the module by flat modules.
Flat module
A flat module is one whose tensoring preserves exactness of sequences; Serre introduced flatness in his 1956 GAGA paper, and Grothendieck recognized its geometric significance.
Flat morphism
A flat morphism is a morphism of schemes whose local ring maps make every local ring of the source a flat module over the target.
Floor and ceiling functions
The floor and ceiling functions, also called the greatest integer function, round a real number down or up to an integer, with notation introduced by Kenneth Iverson in 1962.
FOIL method
The FOIL method, or foil rule, is a mnemonic in elementary algebra for multiplying two binomials, standing for First, Outer, Inner, Last; it is a special case of the distributive property first named in 1929.
FORM (symbolic manipulation system)
FORM is an open-source symbolic manipulation system created by Jos Vermaseren of NIKHEF, started in 1984 and released as open source in 2010, used for large quantum field theory calculations.
Formal scheme
A formal scheme is an algebraic geometry object built from the formal spectrum Spf(A) of a complete adic ring, retaining infinitesimal information to all orders.
Formula for primes
A formula for primes is an expression that generates prime numbers, such as Mills' 1947 constant formula, Willans' 1964 nth-prime formula, and a 26-variable polynomial.
Fourier–Mukai transform
The Fourier–Mukai transform is a functor between the derived categories of coherent sheaves on smooth projective varieties that is an exact equivalence, introduced by Shigeru Mukai in 1981.
Fraction
A fraction is a number written a over b, representing a parts of a whole divided into b equal parts, with a the numerator and b the denominator.
Fréchet space
A Fréchet space, named for Maurice Fréchet, is a complete metrizable locally convex topological vector space whose topology is given by a countable family of seminorms.
Free abelian group
A free abelian group is an abelian group that is a direct sum of copies of the integers, or equivalently an abelian group with a basis.
Free algebra
A free algebra is an algebraic structure with a set of generators such that every map of the generators into any algebra of the class extends to a homomorphism.
Free Boolean algebra
A free Boolean algebra is a Boolean algebra generated by a set X so that every function from X to any Boolean algebra extends uniquely to a homomorphism.
Free convolution
Free convolution is a binary operation on probability measures giving the distribution of sums or products of freely independent non-commuting random variables, introduced by Dan-Virgil Voiculescu in 1986.
Free group
A free group is a group whose generators satisfy no relations beyond the group axioms, written F_n for n generators, with rank-1 F_1 the only abelian one.
Free Lie algebra
A free Lie algebra is a Lie algebra on a free generating set, the Lie-algebra analog of a free group, with any map of the generators extending to a homomorphism.
Free module
A free module, also called a free vector space, is a module with a basis, so every element is uniquely a linear combination of basis elements.
Free monoid
The free monoid on a set is the monoid of all finite words over it, with concatenation as the operation and the empty word as identity.
Free object
A free object is an algebraic structure built from a set of generators with no relations beyond those the structure type forces, so maps of the generators extend uniquely.
Free probability
Free probability is a branch of probability theory in which random variables are noncommuting operators and independence is modeled on free products, introduced by Dan-Virgil Voiculescu in the 1980s.
Free product
The free product of groups is a group built from them by identifying only their identity elements, so elements of different factors do not commute; it is the coproduct in the category of groups.
Frobenius algebra
A Frobenius algebra is a finite-dimensional algebra over a field carrying a nondegenerate bilinear form compatible with multiplication, with examples including group algebras and matrix rings.
Frobenius coin problem
The Frobenius coin problem, also called the coin problem or money-changing problem, asks for the largest integer not expressible as a nonnegative combination of given coprime integers.
Frobenius group
A Frobenius group is a transitive permutation group in which no non-identity element fixes more than one point, studied by Georg Frobenius in 1900 and 1901 using his new theory of group characters.
Frobenius norm
The Frobenius norm, also called the Hilbert-Schmidt norm, is a matrix norm defined as the square root of the sum of squared absolute entries, named for Ferdinand Georg Frobenius.
Frobenius reciprocity
Frobenius reciprocity is a theorem of representation theory stating that induction and restriction between a group and a subgroup are adjoint, introduced by Frobenius in 1898.
Frobenius theorem (real division algebras)
The Frobenius theorem is a result in abstract algebra stating that the only finite-dimensional associative division algebras over the real numbers are R, C, and the quaternions, proved by Ferdinand Georg Frobenius in 1877.
Functional equation (L-function)
The functional equation of an L-function is a symmetry relating its values at s to those at 1-s, one of the defining analytic properties of L-functions.