Covariant loop gravity comes out of the canonical analysis of the Palatini action and the use of the Dirac brackets arising from dealing with the second class constraints (“simplicity” constraints). Within this framework, we underline a quantization ambiguity due to the existence of a family of possible Lorentz connections. We show the existence of a Lorentz connection generalizing the Ashtekar...

Covariant loop gravity comes out of the canonical analysis of the Palatini action and the use of the Dirac brackets arising from dealing with the second class constraints (“simplicity” constraints). Within this framework, we underline a quantization ambiguity due to the existence of a family of possible Lorentz connections. We show the existence of a Lorentz connection generalizing the Ashtekar...

A contact path geometry is a family of paths in a contact manifold each of which is everywhere tangent to the contact distribution and such that given a point and a one-dimensional subspace of the contact distribution at that point there is a unique path of the family passing through the given point and tangent to the given subspace. A contact projective structure is a contact path geometry the...

A contact path geometry is a family of paths in a contact manifold each of which is everywhere tangent to the contact distribution and such that given a point and a one-dimensional subspace of the contact distribution at that point there is a unique path of the family passing through the given point and tangent to the given subspace. A contact projective structure is a contact path geometry the...

There is a well–understood connection between polynomials and certain simple algebraic dynamical systems. In this connection, the Mahler measure corresponds to the topological entropy, Kronecker’s Theorem relates ergodicity to positivity of entropy, approximants to the Mahler measure are related to growth rates of periodic points, and Lehmer’s problem is related to the existence of algebraic mo...

In this paper, we consider canonical F-planar mappings of spaces with affine connection onto m-symmetric spaces. We obtained the fundamental equations these in form a closed system Chauchy-type covariant derivatives. Furthermore, established number essential parameters on which its general solution depends.

In [Ch1], Chern gives a generalization of projective geometry by considering foliations on the Grassman bundle of p-planes Gr(p, R) → R by p-dimensional submanifolds that are integrals of the canonical contact differential system. The equivalence method yields an sl(n + 1, R)valued Cartan connection whose curvature captures the geometry of such foliation. In the flat case, the space of leaves o...

A network structure for canonical coordinate decomposition is presented. The network consists of two single-layer linear subnetworks that together extract the canonical coordinates of two data channels. The connection weights of the networks are trained by a stochastic gradient descent learning algorithm. Each subnetwork features a hierarchical set of lateral connections among its outputs. The ...