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The pair (P, η) defines the structure of an affine geometry on M, making it into an affine manifold. The affine Lie algebra aff(n) splits as a semidirect product of R n and gl(n) and so η may be written as a pair (θ, ω) where θ takes values in R n and ω takes values in gl(n).
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 ...
It is induced, in a canonical manner, from the affine connection. It can also be regarded as the gauge field generated by local Lorentz transformations . In some canonical formulations of general relativity, a spin connection is defined on spatial slices and can also be regarded as the gauge field generated by local rotations .
In the mathematical field of differential geometry, a Cartan connection is a flexible generalization of the notion of an affine connection.It may also be regarded as a specialization of the general concept of a principal connection, in which the geometry of the principal bundle is tied to the geometry of the base manifold using a solder form.
Affine gauge theory is classical gauge theory where gauge fields are affine connections on the tangent bundle over a smooth manifold.For instance, these are gauge theory of dislocations in continuous media when =, the generalization of metric-affine gravitation theory when is a world manifold and, in particular, gauge theory of the fifth force.
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 ...
An affine bundle is a fiber bundle with a general affine structure group (,) of affine transformations of its typical fiber of dimension . This structure group always is reducible to a general linear group G L ( m , R ) {\displaystyle GL(m,\mathbb {R} )} , i.e., an affine bundle admits an atlas with linear transition functions.
The sinh-Gordon model is the affine Toda field theory with the generalized Cartan matrix ( 2 − 2 − 2 2 ) {\displaystyle {\begin{pmatrix}2&-2\\-2&2\end{pmatrix}}} and a positive value for β after we project out a component of φ which decouples.