6,052 articles
Focused proof
A focused proof is a cut-free sequent calculus proof organized into alternating phases around a single formula in focus, shown complete by Andreoli's 1992 focalization theorem.
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.
Fokker–Planck equation
The Fokker–Planck equation, or Kolmogorov forward equation, is a partial differential equation describing how the probability distribution of a stochastic process with drift and diffusion evolves over time.
Foliation
In differential geometry, a foliation partitions a smooth manifold into same-dimensional submanifolds called leaves, a notion introduced in the 1940s by Georges Reeb and Charles Ehresmann.
Forcing (mathematics)
Forcing, also called the forcing method, is a technique in mathematical logic, devised by Paul Cohen in 1963, for proving independence results in set theory.
Ford–Fulkerson algorithm
The Ford–Fulkerson algorithm, also called the Ford–Fulkerson method, computes maximum flow in a network by repeatedly augmenting along paths; it was proposed in 1956 by Ford and Fulkerson.
Forecasting
Forecasting, also known as statistical forecasting, is the process of estimating a future value or state, like tomorrow's weather or demand, using historical data and structured methods.
Forest plot
A forest plot, also known as a blobbogram, is a graphical display used in meta-analysis that shows each study's effect estimate and confidence interval alongside the pooled estimate.
Forking (model theory)
Forking is a dependence relation on types in model theory, introduced by Saharon Shelah, whose negation, non-forking, serves as an abstract independence relation generalizing linear and algebraic independence.
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 grammar
A formal grammar (grammar formalism) is a finite set of rewriting rules that generates the strings of a formal language, classifying syntax in the Chomsky hierarchy.
Formal language
A formal language is a possibly infinite set of finite strings over an alphabet, defined purely by form rather than meaning, usually specified by a formal grammar such as a context-free grammar.
Formal power series
A formal power series is an infinite sum whose coefficients lie in a fixed ring, treated as an algebraic object without asking whether it converges.
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.
Formal science
A formal science is a discipline, such as mathematics, logic, and theoretical computer science, that establishes its claims by proof within defined symbolic systems rather than by experiment.
Formal system
A formal system, or formal logical system, is a set of axioms and inference rules for generating theorems, with proofs checkable mechanically; it underlies Gödel's incompleteness theorems.
Formula
A formula is an expression in mathematical symbols stating a fact, rule, or relationship among quantities; in logic it is a well-formed expression denoting a truth value.
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.
Forward error
Forward error is the difference between the output a numerical computation produces and the exact output, bounded by the condition number times the backward error.
Forward–backward algorithm
The forward–backward algorithm is a dynamic-programming method for hidden Markov models that computes the posterior probability of each hidden state at each time step, and it forms the computational core of Baum–Welch training.
Foundations of mathematics
Foundations of mathematics, also known as foundations of math, studies the logical and set-theoretic frameworks underlying mathematics, from Russell's paradox and Gödel's incompleteness theorems to modern proof assistants.
Four color theorem
The four color theorem, also known as the four color problem, states that any planar map needs at most four colors so that bordering regions differ, proved by computer in 1976.
Four-dimensional space
Four-dimensional space, or tetraspace, is a geometric space in which four independent coordinates are needed to specify the position of a point, generalizing ordinary three-dimensional space.
Fourier analysis
Fourier analysis, also called Fourier theory, is the branch of mathematics that expresses functions as sums of sines and cosines, founded by Joseph Fourier's 1807 work on heat conduction.
Fourier multiplier
A Fourier multiplier is a linear operator acting on a function by multiplying its Fourier transform by a fixed symbol and transforming back, equivalently convolution with a kernel.
Fourier series
A Fourier series, also called a Fourier expansion, represents a periodic function as an infinite sum of sines and cosines at multiples of the fundamental frequency.
Fourier spectral method
A Fourier spectral method solves partial differential equations by expanding the solution in a Fourier series, differentiating via the coefficients, and achieving exponential accuracy for smooth periodic problems.
Fourier transform
The Fourier transform, also called Fourier transformation, is a mathematical operation that converts a function of time or space into a function of frequency, named for Joseph Fourier and central to signal processing and differential equations.
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.
Fractal
A fractal is a geometric object whose structure repeats across scales, so magnifying a part reveals detail resembling the whole; the term was coined by Benoît Mandelbrot in 1975.