## Sasaki metric

Let M, g {\displaystyle M,g} be a Riemannian manifold, denote by τ: T M → M {\displaystyle \tau \colon \mathrm {T} M\to M} the tangent bundle over M {\displaystyle M}. The Sasaki metric g ^ {\displaystyle {\hat {g}}} on T M {\displaystyle \mathrm ...

## Vector bundle

In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X: to every point x of the space X we associate a vector space V in such a way that these vector ...

## Complex vector bundle

In mathematics, a complex vector bundle is a vector bundle whose fibers are complex vector spaces. Any complex vector bundle can be viewed as a real vector bundle through the restriction of scalars. Conversely, any real vector bundle E can be pro ...

## Dual bundle

In mathematics, the dual bundle of a vector bundle π: E → X is a vector bundle π ∗: E ∗ → X whose fibers are the dual spaces to the fibers of E. The dual bundle can be constructed using the associated bundle construction by taking the dual repres ...

## Flat vector bundle

Let π: E → X {\displaystyle \pi:E\to X} denote a flat vector bundle, and ∇: Γ X, E → Γ X, Ω X 1 ⊗ E {\displaystyle \nabla:\Gamma X,E\to \Gamma X,\Omega _{X}^{1}\otimes E} be the covariant derivative associated to the flat connection on E. Let Ω X ...

## Tautological bundle

In mathematics, the tautological bundle is a vector bundle occurring over a Grassmannian in a natural tautological way: the fiber of the bundle over a vector space V is V itself. In the case of projective space the tautological bundle is known as ...

## De Rham curve

In mathematics, a de Rham curve is a certain type of fractal curve named in honor of Georges de Rham. The Cantor function, Cesàro curve, Minkowskis question mark function, the Levy C curve, the blancmange curve and the Koch curve are all special ...

## Levy C curve

In mathematics, the Levy C curve is a self-similar fractal that was first described and whose differentiability properties were analysed by Ernesto Cesàro in 1906 and Georg Faber in 1910, but now bears the name of French mathematician Paul Levy, ...

## Multiplier algebra

In mathematics, the multiplier algebra, denoted by M, of a C*-algebra A is a unital C*-algebra which is the largest unital C*-algebra that contains A as an ideal in a "non-degenerate" way. It is the noncommutative generalization of Stone–Čech com ...

## Approximately finite-dimensional C*-algebra

In mathematics, an approximately finite-dimensional C*-algebra is a C*-algebra that is the inductive limit of a sequence of finite-dimensional C*-algebras. Approximate finite-dimensionality was first defined and described combinatorially by Ola B ...

## Bunce–Deddens algebra

In mathematics, a Bunce–Deddens algebra, named after John W. Bunce and James A. Deddens, is a certain type of direct limit of matrix algebras over the continuous functions on the circle. They are therefore examples of simple unital AT algebras. I ...

## Completely positive map

In mathematics a positive map is a map between C*-algebras that sends positive elements to positive elements. A completely positive map is one which satisfies a stronger, more robust condition.

## Cuntz algebra

In mathematics, the Cuntz algebra O n {\displaystyle {\mathcal {O}}_{n}}, named after Joachim Cuntz, is the universal C*-algebra generated by n isometries satisfying certain relations. The Cuntz algebras provided the first concrete examples of a ...

## Exact C*-algebra

A C*-algebra E is exact if, for any short exact sequence, 0 → A → f B → g C → 0 {\displaystyle 0\;{\xrightarrow {}}\;A\;{\xrightarrow {f}}\;B\;{\xrightarrow {g}}\;C\;{\xrightarrow {}}\;0} the sequence 0 → A ⊗ min E → f ⊗ id B ⊗ min E → g ⊗ id C ⊗ ...

## Graph C*-algebra

## Hereditary C*-subalgebra

In mathematics, a hereditary C*-subalgebra of a C*-algebra is a particular type of C*-subalgebra whose structure is closely related to that of the larger C*-algebra. A C*-subalgebra B of A is a hereditary C*-subalgebra if for all a ∈ A and b ∈ B ...

## K-graph C*-algebra

In mathematics, a k-graph is a countable category Λ {\displaystyle \Lambda } with domain and codomain maps r {\displaystyle r} and s {\displaystyle s}, together with a functor d: Λ → N k {\displaystyle d:\Lambda \to \mathbb {N} ^{k}} which satisf ...

In mathematics, the Kadison–Kastler metric is a metric on the space of C * -algebras on a fixed Hilbert space. It is the Hausdorff distance between the unit balls of the two C * -algebras, under the norm-induced metric on the space of all bounded ...

## Nuclear C*-algebra

In mathematics, a nuclear C*-algebra is a C*-algebra A such that the injective and projective C*-cross norms on A ⊗ B are the same for every C*-algebra B. This property was first studied by Takesaki under the name "Property T", which is not relat ...

## Toeplitz algebra

In operator algebras, the Toeplitz algebra is the C*-algebra generated by the unilateral shift on the Hilbert space l 2. Taking l 2 to be the Hardy space H 2, the Toeplitz algebra consists of elements of the form T f + K {\displaystyle T_{f}+K\;} ...

## Uniformly hyperfinite algebra

In mathematics, particularly in the theory of C*-algebras, a uniformly hyperfinite, or UHF, algebra is a C*-algebra that can be written as the closure, in the norm topology, of an increasing union of finite-dimensional full matrix algebras.

## Universal C*-algebra

In mathematics, a universal C*-algebra is a C*-algebra described in terms of generators and relations. In contrast to rings or algebras, where one can consider quotients by free rings to construct universal objects, C*-algebras must be realizable ...

## Category of elements

In category theory, if C is a category and F: C → S e t {\displaystyle F:C\to \mathbf {Set} } is a set-valued functor, the category of elements of F e l ⁡ {\displaystyle \mathop {\rm {el}} } is the category defined as follows: Objects are pairs A ...

## Density theorem (category theory)

In category theory, a branch of mathematics, the density theorem states that every presheaf of sets is a colimit of representable presheaves in a canonical way. For example, by definition, a simplicial set is a presheaf on the simplex category Δ ...

## Generalized function

In mathematics, generalized functions, or distributions, are objects extending the notion of functions. There is more than one recognized theory. Generalized functions are especially useful in making discontinuous functions more like smooth funct ...

## Boehmians

In mathematics, Boehmians are objects obtained by an abstract algebraic construction of "quotients of sequences." The original construction was motivated by regular operators introduced by T. K. Boehme. Regular operators are a subclass of Mikusin ...

## Multiscale Greens function

Multiscale Greens function is a generalized and extended version of the classical Greens function technique for solving mathematical equations. The main application of the MSGF technique is in modeling of nanomaterials. These materials are very s ...

## Singularity function

Singularity functions are a class of discontinuous functions that contain singularities, i.e. they are discontinuous at their singular points. Singularity functions have been heavily studied in the field of mathematics under the alternative names ...

## White noise analysis

In probability theory, a branch of mathematics, white noise analysis is a framework for infinite-dimensional and stochastic calculus, based on the Gaussian white noise probability space, to be compared with Malliavin calculus based on the Wiener ...

## Anti-function

## Local inverse

The local inverse is a kind of inverse function or matrix inverse used in image and signal processing, as well as other general areas of mathematics. The concept of local inverse came from interior reconstruction of the CT image. One of the inter ...

## Superrigidity

In mathematics, in the theory of discrete groups, superrigidity is a concept designed to show how a linear representation ρ of a discrete group Γ inside an algebraic group G can, under some circumstances, be as good as a representation of G itsel ...

## Weight-of-conflict conjecture

Weight-of-conflict conjecture was proposed by Glenn Shafer in his book on the Dempster–Shafer theory titled A Mathematical Theory of Evidence. It states that if Q 1 {\displaystyle Q_{1}} and Q 2 {\displaystyle Q_{2}} are commonality functions for ...

## Free probability theory

## Conditional dependence

In probability theory, conditional dependence is a relationship between two or more events that are dependent when a third event occurs. For example, if A and B are two events that individually increase the probability of a third event C, and do ...

## Conditional independence

## Subindependence

In probability theory and statistics, subindependence is a weak form of independence. Two random variables X and Y are said to be subindependent if the characteristic function of their sum is equal to the product of their marginal characteristic ...

## Polya urn model

In statistics, a Polya urn model, named after George Polya, is a type of statistical model used as an idealized mental exercise framework, unifying many treatments. In an urn model, objects of real interest are represented as colored balls in an ...

## Probabilistic relevance model (BM25)

## Dunford–Pettis property

In functional analysis, the Dunford–Pettis property, named after Nelson Dunford and B. J. Pettis, is a property of a Banach space stating that all weakly compact operators from this space into another Banach space are completely continuous. Many ...

## Infinite-dimensional holomorphy

In mathematics, infinite-dimensional holomorphy is a branch of functional analysis. It is concerned with generalizations of the concept of holomorphic function to functions defined and taking values in complex Banach spaces, typically of infinite ...

## List of Banach spaces

In the mathematical field of functional analysis, Banach spaces are among the most important objects of study. In other areas of mathematical analysis, most spaces which arise in practice turn out to be Banach spaces as well.

## Lorentz space

In mathematical analysis, Lorentz spaces, introduced by George G. Lorentz in the 1950s, are generalisations of the more familiar L p {\displaystyle L^{p}} spaces. The Lorentz spaces are denoted by L p, q {\displaystyle L^{p,q}}. Like the L p {\di ...

## Lp sum

In mathematics, and specifically in functional analysis, the L p sum of a family of Banach spaces is a way of turning a subset of the product set of the members of the family into a Banach space in its own right. The construction is motivated by ...

## Method of continuity

In the mathematics of Banach spaces, the method of continuity provides sufficient conditions for deducing the invertibility of one bounded linear operator from that of another, related operator.

## Opial property

In mathematics, the Opial property is an abstract property of Banach spaces that plays an important role in the study of weak convergence of iterates of mappings of Banach spaces, and of the asymptotic behaviour of nonlinear semigroups. The prope ...

## Polynomially reflexive space

In mathematics, a polynomially reflexive space is a Banach space X, on which the space of all polynomials in each degree is a reflexive space. Given a multilinear functional M n of degree n that is, M n is n -linear, we can define a polynomial p ...

## Schauder basis

In mathematics, a Schauder basis or countable basis is similar to the usual basis of a vector space; the difference is that Hamel bases use linear combinations that are finite sums, while for Schauder bases they may be infinite sums. This makes S ...

## Tsirelson space

In mathematics, especially in functional analysis, the Tsirelson space is the first example of a Banach space in which neither an ℓ p space nor a c 0 space can be embedded. The Tsirelson space is reflexive. It was introduced by B. S. Tsirelson in ...

## Abel equation of the first kind

In mathematics, an Abel equation of the first kind, named after Niels Henrik Abel, is any ordinary differential equation that is cubic in the unknown function. In other words, it is an equation of the form y ′ = f 3 x y 3 + f 2 x y 2 + f 1 x y + ...