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A pairing is called perfect if the above map is an isomorphism of R-modules and the other evaluation map ′: (,) is an isomorphism also. In nice cases, it suffices that just one of these be an isomorphism, e.g. when R is a field, M,N are finite dimensional vector spaces and L=R .
A pointed metric space is a pair (X,p) consisting of a metric space X and point p in X. A sequence ( X n , p n ) of pointed metric spaces converges to a pointed metric space ( Y , p ) if, for each R > 0, the sequence of closed R -balls around p n in X n converges to the closed R -ball around p in Y in the usual Gromov–Hausdorff sense.
The closest pair of points problem or closest pair problem is a problem of computational geometry: given points in metric space, find a pair of points with the smallest distance between them. The closest pair problem for points in the Euclidean plane [ 1 ] was among the first geometric problems that were treated at the origins of the systematic ...
A rectangular grid (top) and its image under a conformal map f (bottom). It is seen that f maps pairs of lines intersecting at 90° to pairs of curves still intersecting at 90°. A conformal map is a function which preserves angles locally. In the most common case the function has a domain and range in the complex plane. More formally, a map,
The Gelfand property is often satisfied by symmetric pairs. A pair (G, K) is called a symmetric pair if there exists an involutive automorphism θ of G such that K is a union of connected components of the group of θ-invariant elements: G θ. If G is a connected reductive group over R and K = G θ is a compact subgroup, then (G, K) is a ...
Informally, a particular map (esp. an isomorphism) between individual objects (not entire categories) is referred to as a "natural isomorphism", meaning implicitly that it is actually defined on the entire category, and defines a natural transformation of functors; formalizing this intuition was a motivating factor in the development of ...
Each pair (GX, ε X) is a terminal morphism from F to X in C; Each pair (FY, η Y) is an initial morphism from Y to G in D; In particular, the equations above allow one to define Φ, ε, and η in terms of any one of the three. However, the adjoint functors F and G alone are in general not sufficient to determine the adjunction. The equivalence ...
The point x = 0 in R p,q maps to n o in R p+1,q+1, so n o is identified as the (representation) vector of the point at the origin. A vector in R p+1,q+1 with a nonzero n ∞ coefficient, but a zero n o coefficient, must (considering the inverse map) be the image of an infinite vector in R p,q.