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In mathematics, transversality is a notion that describes how spaces can intersect; transversality can be seen as the "opposite" of tangency, and plays a role in general position. It formalizes the idea of a generic intersection in differential topology. It is defined by considering the linearizations of the intersecting spaces at the points of ...
There are more powerful statements (collectively known as transversality theorems) that imply the parametric transversality theorem and are needed for more advanced applications. Informally, the "transversality theorem" states that the set of mappings that are transverse to a given submanifold is a dense open (or, in some cases, only a dense G ...
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Eisenbud, David; Harris, Joe (2016), 3264 and All That: A Second Course in Algebraic Geometry, Cambridge University Press, ISBN 978-1107602724; Kleiman, Steven L. (1974), "The transversality of a general translate", Compositio Mathematica, 28: 287– 297, MR 0360616
In dynamical systems theory, an area of pure mathematics, a Morse–Smale system is a smooth dynamical system whose non-wandering set consists of finitely many hyperbolic equilibrium points and hyperbolic periodic orbits and satisfying a transversality condition on the stable and unstable manifolds.
The proof depends on and is intimately related to the transversality properties of smooth manifolds—see Thom transversality theorem. By reversing this construction, John Milnor and Sergei Novikov (among many others) were able to answer questions about the existence and uniqueness of high-dimensional manifolds: this is now known as surgery theory.
In mathematics, particularly in combinatorics, given a family of sets, here called a collection C, a transversal (also called a cross-section [1] [2] [3]) is a set containing exactly one element from each member of the collection.
The latter is called a transversality condition for a fixed horizon problem. [7] It can be seen that the necessary conditions are identical to the ones stated above for the Hamiltonian. Thus the Hamiltonian can be understood as a device to generate the first-order necessary conditions. [8]