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An affine space, as with essentially any smooth Klein geometry, is a manifold equipped with a flat Cartan connection. More general affine manifolds or affine geometries are obtained easily by dropping the flatness condition expressed by the Maurer-Cartan equations. There are several ways to approach the definition and two will be given.
Let Y → X be an affine bundle modelled over a vector bundle Y → X. A connection Γ on Y → X is called the affine connection if it as a section Γ : Y → J 1 Y of the jet bundle J 1 Y → Y of Y is an affine bundle morphism over X. In particular, this is an affine connection on the tangent bundle TX of a smooth manifold X. (That is, the ...
These are examples of affine connections. There is also a concept of projective connection, of which the Schwarzian derivative in complex analysis is an instance. More generally, both affine and projective connections are types of Cartan connections. Using principal bundles, a connection can be realized as a Lie algebra-valued differential form.
In the mathematical field of differential geometry, the term linear connection can refer to either of the following overlapping concepts: . a connection on a vector bundle, often viewed as a differential operator (a Koszul connection or covariant derivative);
Guliyev and Ismailov [14] constructed a smooth sigmoidal activation function providing universal approximation property for two hidden layer feedforward neural networks with less units in hidden layers. [15] constructed single hidden layer networks with bounded width that are still universal approximators for univariate functions. However, this ...
Let X be an affine space over a field k, and V be its associated vector space. An affine transformation is a bijection f from X onto itself that is an affine map; this means that a linear map g from V to V is well defined by the equation () = (); here, as usual, the subtraction of two points denotes the free vector from the second point to the first one, and "well-defined" means that ...
In particular, any affine combination of the fixed points of a given affine transformation is also a fixed point of , so the set of fixed points of forms an affine space (in 3D: a line or a plane, and the trivial cases, a point or the whole space).
In differential geometry, an affine manifold is a differentiable manifold equipped with a flat, torsion-free connection. Equivalently, it is a manifold that is (if connected) covered by an open subset of R n {\displaystyle {\mathbb {R} }^{n}} , with monodromy acting by affine transformations .