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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 ...
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 ...
In category theory, a branch of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain.
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.
Let : be a smooth map between smooth manifolds and .Then there is an associated linear map from the space of 1-forms on (the linear space of sections of the cotangent bundle) to the space of 1-forms on .
−1/5 0 The first row of coefficients at the bottom of the table gives the fifth-order accurate method, and the second row gives the fourth-order accurate method.
The numerical solution to the linear test equation decays to zero if | r(z) | < 1 with z = hλ. The set of such z is called the domain of absolute stability. In particular, the method is said to be absolute stable if all z with Re(z) < 0 are in the domain of absolute stability. The stability function of an explicit Runge–Kutta method is a ...
The Runge–Kutta–Fehlberg method has two methods of orders 5 and 4; it is sometimes dubbed RKF45 . Its extended Butcher Tableau is: / / / / / / / / / / / / / / / / / / / / / / / / / / The first row of b coefficients gives the fifth-order accurate solution, and the second row has order four.