n
(physics) A dual version of the Wilson loop in which the electromagnetic potential A is replaced by its electromagnetic dual Aᵐᵃᵍ, where the exterior derivative of A is equal to the Hodge dual of the exterior derivative of Aᵐᵃᵍ.
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(algebra) A group in which the group operation is commutative.
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(mathematics) An infinitesimal algebraic object associated with a groupoid
n
(mathematics, algebraic geometry, arithmetic geometry) A theory which describes the way in which the algebraic fundamental group G of an algebraic variety (or some related geometric object) V determines how V can be mapped into another geometric object W, under the assumption that G is very far from being abelian (commutative).
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(functional analysis) A normed vector space which is complete with respect to the norm, meaning that Cauchy sequences have well-defined limits that are points in the space.
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(linear algebra) In a vector space, a linearly independent set of vectors spanning the whole vector space.
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(mathematics) A generalized bundle in a Lie group
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(mathematics) The base space of a principal bundle with a homogeneous space as total space.
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(mathematics) Any of the elements of the Bloch space.
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(mathematics) The Chow groups of an algebraic variety over any field are algebro-geometric analogs of the homology of a topological space. The elements of the Chow group are formed out of subvarieties (so-called algebraic cycles) in a similar way to how simplicial or cellular homology groups are formed out of subcomplexes.
n
(algebra, mathematical physics) A unital associative algebra which generalizes the algebra of quaternions but which is not necessarily a division algebra; it is generated by a set of 𝛾ᵢ (with i ranging from, say, 1 to n) such that the square of each 𝛾ᵢ is fixed to be either +1 or −1, depending on each i, and such that any product 𝛾ᵢ𝛾ⱼ anticommutes when its factors are distinct (i.e., when i ne j).
n
(mathematics) The coequalizer of a mapping.
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(mathematics) The dual of a group.
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(mathematics) A system of quotient groups associated to a topological space.
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(mathematics) The vector space generated by all the column vectors of a matrix
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(algebra) A correspondence between a projective space and its dual.
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(linear algebra) In a quadratic form, a term which is not a squared variable; i.e., a term with more than one variable.
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(mathematics) A type of graph embedding used to study Riemann surfaces and to provide combinatorial invariants for the action of the absolute Galois group of the rational numbers.
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(mathematics) The vector space which comprises the set of linear functionals of a given vector space.
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(linear algebra) A term at any position in a matrix.
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(topology, of a topological space) The sum of even-dimensional Betti numbers minus the sum of odd-dimensional ones.
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(topology, literally, relatively rare) A space which is a factor (multiplicand) in a product space.
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(topology, Britain, Canada) Alternative spelling of fiber bundle [(American spelling, topology, category theory) A topological space E with a base space B and a fiber space F such that any point x ∈ B has a neighborhood N that is homeomorphic to the product space B × F (that is, the space is locally the product space B × F, although its global structure can be quite different).]
n
(linear algebra) A scalar field.
adj
(mathematics) (of a vector space) having a basis consisting of a finite number of elements.
n
(mathematics, linear algebra) A sequence of subspaces of a vector space, beginning with the null space and ending with the vector space itself, such that each member of the sequence (until the last) is a proper subspace of the next.
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(mathematics) A tool for studying symplectic geometry and low-dimensional topology. It is a novel invariant that arises as an infinite-dimensional analog of finite-dimensional Morse homology.
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(graph theory) A theorem stating that given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions so that no two adjacent regions have the same color (adjacent being defined as two regions sharing a boundary, not counting corners, in which three or more regions share a boundary).
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(mathematics) Any of seven symmetry groups of infinitely long ornamented bands.
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(mathematics) A space which parameterizes all linear subspaces of a vector space of a given dimension.
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(category theory) A structure on a category which makes the category's objects act like the open sets of a topological space.
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(approximation theory) A finite-dimensional subspace V of 𝒞(X, 𝕂), where X is a compact space and 𝕂 either the real numbers or the complex numbers, such that for any given f∈𝒞(X, 𝕂) there is exactly one element of V that approximates f "best", i.e. with minimum distance to f in supremum norm.
adj
(mathematics, graph theory, of a graph) Containing a Hamiltonian cycle.
n
(mathematics) The group of 3×3 upper triangular matrices of the form 1ac\0 1b\0 0 1\ under the operation of matrix multiplication, arising in the description of one-dimensional quantum-mechanical systems.
n
(linear algebra) A square matrix A with complex entries that is equal to its own conjugate transpose, i.e., such that A=A^†.
n
(mathematics) One aspect of the study of differential forms of a smooth manifold M. More specifically, it works out the consequences for the cohomology groups of M, with real coefficients, of the partial differential equation theory of generalised Laplacian operators associated to a Riemannian metric on M.
adj
(mathematics, of a topological space) That is not the union of two proper closed sets; such that every open set is dense.
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(mathematics) Any algebraic group equipped with a hyperoperation.
adj
(mathematics, of a group) Such that there exists a homomorphism from it to ℤ, the integers.
n
(mathematics) A vector space which is additionally equipped with an inner product.
n
(mathematics) Any block diagonal matrix whose blocks are Jordan blocks.
n
(dated, obsolete) that part of algebraic topology comprising what is now called topological K-theory.
n
(mathematics) A symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras.
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(loosely) The coset space G / H.
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A generalized version of the Laplace operator that can be used with Riemannian and pseudo-Riemannian manifolds.
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(linear algebra) A main diagonal.
n
(topology) A theorem which states that for any open covering of some given set in Euclidean n-space there is an open subcovering (of that same set) which is countable.
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(mathematics) Any function whose value on the sum of two elements is the sum of the values of the function on the two elements and whose value on the product of a scalar times an element is the scalar times the value of the function on the element: f(ax+by)=af(x)+bf(y)
n
(group theory) Any group that is isomorphic to a matrix group.
n
A mathematical model of a system based on the use of a linear operator.
adj
(of a real-valued function on a topological space) Such that, for each fixed number, the subspace of points whose images are at most that number is closed.
n
(mathematics) Function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces.
n
(more generally) a similarly multiply linear mapping Mʳ → R defined for a given module M over some commutative ring R.
n
(geometry, complex analysis) A transformation of the extended complex plane that is a rational function of the form f(z) = (az + b) / (cz + d), where a, b, c, d are complex numbers such that ad − bc ≠ 0; an automorphism of the complex projective line.
n
(mathematics) The kernel of a linear map between two vector spaces or two modules.
n
(linear algebra) A square matrix whose columns, considered as vectors, are orthonormal to each other. This implies that the transpose of such a matrix is also its inverse.
n
(mathematics, linear algebra) The tensor product of two vectors.
n
(mathematics) A topological semigroup that is algebraically a group.
n
(mathematics) A collection of non-negative-valued functions defined on a topological space, such that every point has only finitely many nonzero values, and these sum to one.
n
(mathematics) A vector whose components are Pauli matrices; e.g., ⃑𝜎=𝜎₁̂i+𝜎₂̂j+𝜎₃̂k.
adj
(mathematics) Of a field: being a complete topological field whose topology is induced by a nondiscrete valuation of rank 1.
n
(topology) An n-dimensional topological space with a distinguished element µ of its nth homology group such that taking the cap product with an element of the kth cohomology group yields an isomorphism to the (n − k)th cohomology group.
n
(mathematics, algebra) A finite algebra satisfying all the axioms for a skew field except multiplicative associativity and the existence of a multiplicative identity.
n
(linear algebra) A transformation matrix whose associated transformation is a projection.
n
The quotient of a Hilbert space by an equivalence relation which considers a pair of vectors equivalent iff they differ only by a factor which is a complex number.
n
(mathematics) A set containing the spectrum of an operator and the numbers that are "almost" eigenvalues.
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(linear algebra) A decomposition of a matrix into a product of a unitary matrix (orthogonal matrix) and an upper-triangular matrix.
n
(topology, complex analysis) The complex numbers extended with the number ∞; the complex plane (representation of the complex numbers as a Euclidean plane) extended with a single idealised point at infinity and consequently homeomorphic to a sphere in 3-dimensional Euclidean space.
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(linear algebra) The product of two vectors computed as the sum of the corresponding elements of the vectors, or, equivalently, as the product of the magnitudes of the vectors and the cosine of the angle between their directions.
adj
(linear algebra) Of a vector space, isomorphic to its dual space.
adj
(group theory) Of a subgroup A of a group G, having a subgroup B such that AB = G, and for any proper subgroup C of B, AC is a proper subgroup of G.
adj
(group theory, of a subgroup) That commutes with every subgroup whose order is relatively prime to its own.
adj
(topology, of a topological space) Whose regular open sets form a base.
n
(category theory) A category together with a choice of Grothendieck topology.
n
(mathematics) For two pointed spaces (i.e. topological spaces with distinguished basepoints) X and Y, the quotient of the product space X × Y under the identifications (x, y₀) ∼ (x₀, y) for all x ∈ X and y ∈ Y; usually denoted X ∧ Y.
n
(mathematics) A vector space of functions equipped with a norm that is a combination of Lp-norms of the function itself as well as its derivatives up to a given order.
n
(mathematics) A result concerning certain infinite-dimensional vector spaces, stating that any orthomodular form that has an infinite orthonormal sequence is a Hilbert space over the real numbers, complex numbers or quaternions.
n
(mathematics) The space of all linear combinations of something.
n
(mathematics, functional analysis) Of a bounded linear operator A, the set of scalar values λ such that the operator A—λI, where I denotes the identity operator, does not have a bounded inverse; intended as a generalisation of the linear algebra sense.
n
(computing theory) An algorithm for matrix multiplication.
adj
(mathematics, of the topology of a category) in which every representable contravariant functor is a sheaf
n
(mathematics, mathematical analysis) A surjection between diffeological spaces such that the target is identified as the pushforward of the source.
adj
(mathematics, of a Banach space X) Having every closed, infinite-codimensional subspace of X contained in an infinite-codimensional subspace which is complemented in X
adj
(mathematics) Such that each connected subspace is a singleton.
n
(functional analysis) A generalization of many standard function spaces such as Lᵖ spaces and Sobolev spaces.
n
(mathematics, knot theory) A link that is equivalent (under ambient isotopy) to finitely many disjoint circles in the plane.
n
(algebra, geometry, mathematics, topology) A set of elements called vectors, together with some field and operations called addition (mapping two vectors to a vector) and scalar multiplication (mapping a vector and an element in the field to a vector), satisfying a list of constraints.
n
(mathematics) A subgroup of the isometry group of a root system, generated by reflections through the hyperplanes orthogonal to the roots.
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