6,052 articles
Topological entropy
Topological entropy is a nonnegative number attached to a topological dynamical system that measures the exponential growth rate of distinguishable orbits, introduced by Adler, Konheim, and McAndrew in the mid-1960s.
Topological graph theory
Topological graph theory is the branch of graph theory that studies how graphs can be embedded, or drawn without edge crossings, on surfaces such as the sphere and torus.
Topological Hochschild homology
Topological Hochschild homology (THH) is a homology theory for ring spectra introduced by Marcel Bökstedt in 1985 as a spectrum-level analog of Hochschild homology, and is the target of trace maps from algebraic K-theory.
Topological K-theory
Topological K-theory is a generalized cohomology theory assigning to a space the Grothendieck group of its vector bundles, introduced by Atiyah and Hirzebruch in 1961.
Topological space
A topological space, also called a topological structure, is a set with a collection of open sets capturing nearness and continuity without distance, the basic object of point-set topology.
Topological vector space
A topological vector space (TVS) is a vector space with a topology making vector addition and scalar multiplication continuous, the basic setting of functional analysis beyond normed spaces.
Topology
Topology is the branch of mathematics that studies properties of figures and spaces preserved under homeomorphisms, with roots in Euler's 1736 Königsberg bridge work and Poincaré's founding of algebraic topology in 1895.
Topos
A topos (plural: topoi or toposes) is a category behaving like a category of sheaves. Grothendieck toposes are sheaves on a site; elementary toposes, defined by Lawvere and Tierney, are axiomatized categories.
Torelli theorem
The Torelli theorem is a theorem of algebraic geometry stating that a smooth complex algebraic curve is determined up to isomorphism by its canonically polarized Jacobian, proved by Ruggiero Torelli in 1913.
Toric variety
A toric variety is a normal algebraic variety containing an algebraic torus (C)^r as a dense open subset; normal toric varieties are classified by combinatorial objects called fans.
Torsion (algebra)
Torsion in algebra is the property of a module element killed by a nonzero ring element; every finite abelian group is a torsion module over the integers.
Torsion subgroup
The torsion subgroup of an abelian group G is the subgroup of all elements of finite order; a finite abelian group equals its own torsion subgroup.
Torsion tensor
The torsion tensor, introduced by Élie Cartan in 1922, measures the failure of an affine connection to be symmetric; a connection with zero torsion is called torsion-free.
Torsion-free module
A torsion-free module is a module over an integral domain in which no nonzero element is killed by a nonzero ring element, generalizing torsion-free abelian groups.
Torsor (algebraic geometry)
A torsor in algebraic geometry, also called a principal homogeneous space, is a scheme with a simply transitive group-scheme action, locally trivial over a covering of the base.
Torus
A torus is a doughnut-shaped surface, or surface of revolution, made by rotating a circle about a coplanar axis; the standard ring torus has genus 1.
Torus knot
A torus knot is a knot that can be drawn on the surface of an unknotted torus without crossing itself, denoted T(p, q); the trefoil is the simplest example.
Total derivative
The total derivative is the derivative of a multivariable function taken as a whole, a linear map giving the best linear approximation to its change near a point.
Total least squares
Total least squares is an estimation method for fitting a model when every variable is subject to error, minimizing the smallest correction to both the data matrix and observations.
Total order
A total order, also called a linear order, is a relation on a set in which every pair of elements is comparable, as with the usual order on numbers.
Total variation
Total variation is a numerical measure of how much a function, measure, or image oscillates, introduced by Camille Jordan to study pointwise convergence of Fourier series.
Total variation distance
The total variation distance is a metric on probability measures, defined as the largest discrepancy between the probabilities two distributions assign to any one event.
Trace (linear algebra)
The trace of a square matrix, also called the matrix trace, is the sum of its main diagonal entries; it is basis-independent and equals the sum of the eigenvalues.
Trace monoid
A trace monoid, also called a free partially commutative monoid, is a monoid obtained from a free monoid by declaring certain pairs of generators to commute; its elements, called traces, model concurrent computation.
Trachtenberg system
The Trachtenberg system, or Trachtenberg method, is a set of rapid arithmetic rules devised by Jakow Trachtenberg, a Russian-born engineer who developed it in a Nazi concentration camp around 1944.
Tracy-Widom distribution
The Tracy-Widom distribution is a probability distribution describing fluctuations of the largest eigenvalue of a large random matrix, found by Craig Tracy and Harold Widom in 1994.
Transcendence degree
The transcendence degree of a field extension L/K, written trdeg(L/K), is the size of a maximal set of elements of L algebraically independent over K, measuring how many transcendental quantities the extension contains.
Transcendental number
A transcendental number is a complex number that is not a root of any polynomial with integer coefficients, such as e, proved transcendental by Hermite in 1873.
Transfer entropy
Transfer entropy is an information-theoretic measure of directed information flow between time series, introduced by Thomas Schreiber in 2000 to detect asymmetric, possibly nonlinear coupling beyond what the target's own past explains.
Transfer function model
The transfer function model is a time-series regression method that estimates how an input series dynamically influences an output, compressing a distributed lag into a few parameters.