6,052 articles
Fractal analysis
Fractal analysis is a family of mathematical and statistical methods that quantify self-similar, scale-invariant structure in images, point sets, and time series, used in physics, biomedical imaging, and machine learning.
Fractal dimension
The fractal dimension measures how an object's size grows as measurement scale shrinks, taking non-integer values for irregular sets; Mandelbrot estimated Britain's coastline at 1.25 in 1967.
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.
Fractional Brownian motion
Fractional Brownian motion is a centered Gaussian stochastic process, defined by Mandelbrot and Van Ness in 1968, that generalizes Brownian motion with correlated increments governed by a Hurst parameter.
Fractional calculus
Fractional calculus is the branch of mathematical analysis that extends differentiation and integration to arbitrary, non-integer order, with more than fifteen competing definitions, including Riemann–Liouville and Caputo.
Fractional differential equation
A fractional differential equation is a differential equation in which the derivative is taken to a non-integer order, making it a tool for modeling systems with memory.
Fractional diffusion equation
A fractional diffusion equation is a partial differential equation that replaces an integer-order derivative with a fractional one to model anomalous diffusion in porous media, finance, biology, and hydrology.
Fractional factorial design
A fractional factorial design is a designed experiment running only a selected fraction of a full factorial's treatment combinations, trading estimable effects for large run savings in industrial screening.
Fractional Fourier transform
The fractional Fourier transform (FrFT) is a linear integral operator generalizing the Fourier transform, rotating a signal's time-frequency representation through an arbitrary angle, used in signal processing and optics.
Fractional Laplacian
The fractional Laplacian is a nonlocal pseudo-differential operator, written (-Δ)^(α/2) for 0 < α < 2, that generalizes the Laplacian to fractional powers and appears in models of anomalous diffusion and stable Lévy processes.
Fractional-order model
A fractional-order model is a mathematical model whose derivatives or integrals appear at non-integer order, describing systems whose response depends on their entire past history.
Frailty model
A frailty model is a survival analysis model extending proportional hazards models with a multiplicative random effect representing unobserved heterogeneity, such as clustering or shared risk factors.
Frame bundle
In differential geometry, the frame bundle of a vector bundle is the bundle whose fiber over each point is the set of all bases of that fiber.
Frank–Wolfe algorithm
The Frank–Wolfe algorithm, also called the conditional gradient method, is a projection-free first-order method for minimizing a smooth convex function over a compact convex set.
Fréchet derivative
The Fréchet derivative is the derivative of a map between Banach spaces, a bounded linear operator approximating it locally, named after Maurice Fréchet, who introduced metric spaces in 1906.
Fréchet regression
Fréchet regression is a statistical method for regressing random objects, such as probability distributions, covariance matrices, or shapes, on Euclidean predictors, introduced by Petersen and Müller in 2019.
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.
Fredholm determinant
The Fredholm determinant, introduced by Erik Ivar Fredholm in 1903, generalizes the ordinary determinant to operators of the form identity plus trace class on a Hilbert space.
Fredholm index
The Fredholm index of a Fredholm operator between Banach spaces is dim ker T minus dim coker T, an integer that stays stable under compact perturbations.
Fredholm integral equation
A Fredholm integral equation is an integral equation in which the unknown appears under an integral with fixed limits, in a first kind and a second kind.
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 boundary problem
A free boundary problem is a boundary value problem in which part of the domain's boundary is unknown and must be solved for along with the PDE solution.
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 logic
Free logic, named by Karel Lambert in 1960, is a family of first-order systems that drops the classical assumption that every singular term denotes an existing object.
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.