In terms of gauge theory on natural bundles, gauge fields are linear connections on a world manifold X, defined as Higgs field responsible for spontaneous symmetry breaking of general covariant transformations.
12.
Let us note that a true translation gauge field ( i . e ., an affine connection which yields a flat linear connection on TX ) is well defined only on a parallelizable manifold X.
13.
For instance, the Dirac operator in metric-affine gravitation theory in the presence of a general linear connection \ Gamma is well defined, and it depends just of the adapted connection \ Gamma ^ g.
14.
Given an affine connection \ Gamma on a vector bundle Y \ to X, let R and \ overline R be the curvatures of a connection \ Gamma and the associated linear connection \ overline \ Gamma, respectively.
15.
In comparison with General Relativity, dynamic variables of "'metric-affine gravitation theory "'are both a torsion where a linear connection obeys the condition that a covariant derivative of a metric equals zero.
16.
Given bundle coordinates ( 1 ) on TX, it takes the form ( 3 ) where the linear connection \ Gamma ( 4 ) and the basic soldering form \ sigma ( 6 ) are considered as independent variables.
17.
At the same time, any linear connection \ Gamma defines a principal adapted connection \ Gamma ^ g on a Lorentz reduced subbundle F ^ gX by its restriction to a Lorentz subalgebra of a Lie algebra of a general linear group GL ( 4, \ mathbb R ).
18.
Work by Hava Siegelmann and Eduardo D . Sontag has provided a proof that a specific recurrent architecture with rational valued weights ( as opposed to full precision real number-valued weights ) has the full power of a Universal Turing Machine using a finite number of neurons and standard linear connections.
19.
Conversely, every linear connection \ Gamma ( 4 ) on TX \ to X is extended to the affine one A \ Gamma on ATX which is given by the same expression ( 4 ) as \ Gamma with respect to the bundle coordinates ( 1 ) on ATX = TX, but it takes a form
20.
The impact of their work on the ley-hunting community was substantial, with one section moving in a more fully religious direction by declaring that leys could only be detected by intuition, and the other renouncing a ley line belief in favour of a more ethnographically-rooted analysis of linear connections in the landscape.
