Mathematics. Dictionary
Showing university terms (400). Also available: all (649), school (249).
In algebraic topology and category theory, factorization homology is a variant of topological chiral homology, motivated by an application to topological quantum field theory and cobordism hypothesis in particular. It was introduced by David Ayala, John Francis, and Nick Rozenblyum.
In number theory, Fermat's little theorem states that if p is a prime number, then for any integer a, the number ap − a is an integer multiple of p. In the notation of modular arithmetic, this is expressed as a p ≡ a ( mod p ) . {\displaystyle a^{p}\equiv a{\pmod {p}}.} For example, if a = 2 and p = 7, then 27 = 128, and 128 − 2 = 126 = 7 × 18 is an integer multiple of 7. If a is not divisible by p; that is, if a is coprime to p, then Fermat's little theorem is equivalent to the statement that ap − 1 − 1 is an integer multiple of p, or in symbols: a p − 1 ≡ 1 ( mod p ) .…
In mathematics, and particularly topology, a fiber bundle (Commonwealth English: fibre bundle) is a space that is locally a product space, but globally may have a different topological structure. Specifically, the similarity between a space E {\displaystyle E} and a product space B × F {\displaystyle B\times F} is defined using a continuous surjective map, π : E → B , {\displaystyle \pi :E\to B,} that in small regions of E {\displaystyle E} behaves just like a projection from corresponding regions of B × F {\displaystyle B\times F} to B .…
A difference between the value of a function evaluated at a number, and the value of the same function evaluated at a different number, a fixed distance from the first.
A finite geometry is any geometric system that has only a finite number of points. The familiar Euclidean geometry is not finite, because a Euclidean line contains infinitely many points. A geometry based on the graphics displayed on a computer screen, where the pixels are considered to be the points, would be a finite geometry. While there are many systems that could be called finite geometries, attention is mostly paid to the finite projective and affine spaces because of their regularity and simplicity.…
In algebraic geometry, a finite morphism between two affine varieties X , Y {\displaystyle X,Y} is a dense regular map which induces isomorphic inclusion k [ Y ] ↪ k [ X ] {\displaystyle k\left[Y\right]\hookrightarrow k\left[X\right]} between their coordinate rings, such that k [ X ] {\displaystyle k\left[X\right]} is integral over k [ Y ] {\displaystyle k\left[Y\right]} . This definition can be extended to the quasi-projective varieties, such th…
In mathematics, especially in the area of algebra known as group theory, the Fitting subgroup F of a finite group G, named after Hans Fitting, is the unique largest normal nilpotent subgroup of G. Intuitively, it represents the smallest subgroup which "controls" the structure of G when G is solvable. When G is not solvable, a similar role is played by the generalized Fitting subgroup F*, which is generated by the Fitting subgroup and the components of G.…
In mathematics, especially homological algebra and other applications of abelian category theory, the five lemma is an important and widely used lemma about commutative diagrams. The five lemma is valid for all abelian categories, as well as the category of groups. The five lemma can be thought of as a combination of two other theorems, the four lemmas, which are dual to each other.
A value which is unchanged by a function or other mapping. Formally: a value x for which f(x) = x.
In mathematics, in particular in algebraic geometry, a flat morphism f from a scheme X to a scheme Y is a morphism such that the induced map on every stalk is a flat map of rings, i.e., f P : O Y , f ( P ) → O X , P {\displaystyle f_{P}\colon {\mathcal {O}}_{Y,f(P)}\to {\mathcal {O}}_{X,P}} is a flat map for all P in X. A map of rings A → B {\displaystyle A\to B} is called flat if it is a homomorphism that makes B a flat A-module. A morphism of schemes is called faithfully flat if it is both surjective and flat.…
A function that maps a real number to the largest integer that is not greater than it.
In mathematics, a p-dimensional foliation is a partition of a manifold into submanifolds, all of the same dimension p, locally modeled on the decomposition of Rn into the p-dimensional planes cut out by the equations x p + 1 = a p + 1 , … , x n = a n {\displaystyle x_{p+1}=a_{p+1},\ldots ,x_{n}=a_{n}} . The submanifolds are called the leaves of the foliation. The 3-sphere has a famous codimension-1 foliation called the Reeb foliation. The submanifolds are required to be connected and injectively immersed, but they are not required to be embedded.…
A mathematical set that has a non-integer and constant Hausdorff dimension, corresponding to a geometric figure or object that is self-similar at arbitrarily small scales and thus has infinite complexity.
In linear algebra, a frame of an inner product space is a generalization of a basis of a vector space to sets that may be linearly dependent. In the terminology of signal processing, a frame provides a redundant, stable way of representing a signal. Frames are used in error detection and correction and the design and analysis of filter banks and more generally in applied mathematics, computer science, and engineering.
In mathematics, a frame bundle is a principal fiber bundle F ( E ) {\displaystyle F(E)} associated with any vector bundle E {\displaystyle E} . The fiber of F ( E ) {\displaystyle F(E)} over a point x {\displaystyle x} is the set of all ordered bases, or frames, for E x {\displaystyle E_{x}} . The general linear group acts naturally on F ( E ) {\displaystyle F(E)} via a change of basis, giving the frame bundle the structure of a principal G L ( k , R ) {\displaystyle \mathrm {GL} (k,\mathbb {R} )} -bundle (where k is the rank of E {\displaystyle E} ).…
In mathematics, particularly in group theory, the Frattini subgroup Φ ( G ) {\displaystyle \Phi (G)} of a group G is the intersection of all maximal subgroups of G. For the case that G has no maximal subgroups, for example the trivial group {e} or a Prüfer group, it is defined by Φ ( G ) = G {\displaystyle \Phi (G)=G} . It is analogous to the Jacobson radical in the theory of rings, and intuitively can be thought of as the subgroup of "small elements" (see the "non-generator" characterization below).…
In mathematics, Fredholm operators are certain operators that arise in the Fredholm theory of integral equations. They are named in honour of Erik Ivar Fredholm. By definition, a Fredholm operator is a bounded linear operator T : X → Y between two Banach spaces with finite-dimensional kernel ker T {\displaystyle \ker T} and finite-dimensional (algebraic) cokernel coker T = Y / ran T {\displaystyle \operatorname {coker} T=Y/\operatorname {ran} T} , and with closed range ran T {\displaystyle \operatorname {ran} T} .…
In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subset such that every element of the group can be uniquely expressed as an integer combination of finitely many basis elements. For instance, the two-dimensional integer lattice forms a free abelian group, with coordinatewise addition as its operation, and with the two points (1, 0) and (0, 1) as its basis.…
In mathematics, the free factor complex (sometimes also called the complex of free factors) is a free group counterpart of the notion of the curve complex of a finite type surface. The free factor complex was originally introduced in a 1998 paper of Allen Hatcher and Karen Vogtmann. Like the curve complex, the free factor complex is known to be Gromov-hyperbolic. The free factor complex plays a significant role in the study of large-scale geometr…
In mathematics, the free group F S {\displaystyle F_{S}} over a given set S {\displaystyle S} consists of all words that can be built from members of S {\displaystyle S} , considering two words to be different unless their equality follows from the group axioms (e.g. s t = s u u − 1 t {\displaystyle st=suu^{-1}t} but s ≠ t − 1 {\displaystyle s\neq t^{-1}} for s , t , u ∈ S {\displaystyle s,t,u\in S} ). The members of S {\displaystyle S} are called generators of F S {\displaystyle F_{S}} , and the number of generators is the rank of the free group.…
In mathematics, and specifically in the field of homotopy theory, the Freudenthal suspension theorem is the fundamental result leading to the concept of stabilization of homotopy groups and ultimately to stable homotopy theory. It explains the behavior of simultaneously taking suspensions and increasing the index of the homotopy groups of the space in question. It was proved in 1937 by Hans Freudenthal. The theorem is a corollary of the homotopy…
In the mathematical field of differential geometry, a Frobenius manifold, introduced by Dubrovin, is a flat Riemannian manifold with a certain compatible multiplicative structure on the tangent space. The concept generalizes the notion of Frobenius algebra to tangent bundles. Frobenius manifolds occur naturally in the subject of symplectic topology, more specifically quantum cohomology. The broadest definition is in the category of Riemannian supermanifolds.…
In mathematics, and in particular representation theory, Frobenius reciprocity is a theorem expressing a duality between the process of restricting and inducing. It can be used to leverage knowledge about representations of a subgroup to find and classify representations of "large" groups that contain them. It is named for Ferdinand Georg Frobenius, the inventor of the representation theory of finite groups.
The branch of mathematics dealing with infinite-dimensional vector spaces, whose elements are actually functions, as well as generalizations such as Banach spaces and Hilbert spaces.
A category homomorphism; a morphism from a source category to a target category which maps objects to objects and arrows to arrows (either covariantly or contravariantly), in such a way as to preserve morphism composition and identities.
For a specified topological space, the group whose elements are homotopy classes of loops (images of some arbitrary closed interval whose endpoints are both mapped to a designated point) and whose group operation is concatenation.
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