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In mathematics, the vertical line test is a visual way to determine if a curve is a graph of a function or not. A function can only have one output, y , for each unique input, x . If a vertical line intersects a curve on an xy -plane more than once then for one value of x the curve has more than one value of y , and so, the curve does not ...
The pullback bundle is an example that bridges the notion of a pullback as precomposition, and the notion of a pullback as a Cartesian square.In that example, the base space of a fiber bundle is pulled back, in the sense of precomposition, above.
This is also the case in the category of differentiable manifolds. A special case is the pullback of two fiber bundles E 1, E 2 → B. In this case E 1 × E 2 is a fiber bundle over B × B, and pulling back along the diagonal map B → B × B gives a space homeomorphic (diffeomorphic) to E 1 × B E 2, which is a fiber bundle over B.
As a special case, note that if F is a linear form (or (0,1)-tensor) on W, so that F is an element of W ∗, the dual space of W, then Φ ∗ F is an element of V ∗, and so pullback by Φ defines a linear map between dual spaces which acts in the opposite direction to the linear map Φ itself:
For example, a two-stage method has order 2 if b 1 + b 2 = 1, b 2 c 2 = 1/2, and b 2 a 21 = 1/2. [8] Note that a popular condition for determining coefficients is [ 8 ] ∑ j = 1 i − 1 a i j = c i for i = 2 , … , s . {\displaystyle \sum _{j=1}^{i-1}a_{ij}=c_{i}{\text{ for }}i=2,\ldots ,s.}
An introduction to categorical approaches to algebraic topology: the focus is on the algebra, and assumes a topological background. Ronald Brown "Topology and Groupoids" pdf available Gives an account of some categorical methods in topology, use the fundamental groupoid on a set of base points to give a generalisation of the Seifert-van Kampen ...
Let (,) be a topological space; let () denote the Borel σ-algebra on , i.e. the smallest sigma algebra on that contains all open sets . Let be a measure on (, ()) Then the support (or spectrum) of is defined as the set of all points in for which every open neighbourhood of has positive measure: ():= {: (() >)}.
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