Physics, mathematics, and technology

# Physics, mathematics, and technology

### Mathematics and computer science

#### Hierarchies of smooth manifolds up to zeroth and first orders

Abstract

Hierarchies of smooth manifolds in the form of sequences are given. The sequence of zero order consists of the parallelized manifold, Lie group and Abelian group of Lie. Each of three sequences of the 1st order for the ho-lonomic, semi-holonomic and the non- holonomic smooth manifolds includes base of the parallelized bundle of linear coframes, in other words, base of space of expanded affine connection, base of space of affine connection torsion-free and affine space.

#### Giving the 1st order affine connection by means of the 2nd order vector-valued forms

Abstract

Affine connection is given by 2nd order vectors called horizontal. Vertical and horizontal forms of 2nd order are entered for 1st order affine connection. It is proved that symmetric affine connection in the bundle of tangent linear frames defines vertical linear operator (a vertical vertical-valued form of 2nd order for 1st order affine connection) from 2nd order tangent space into 1st order tangent space to a manifold. It is shown that affine connection in bundle of tangent lin-ear frames defines linear operator from 1st order cotangent space (space of forms of degree 1) in cotangent space of 2nd order. It is proved that affine connection in bundle of tangent linear frames defines horizontal linear operator (2nd order hor-izontal horizontal-valued form of 1st order affine connection) in 2nd order tan-gent bundle. It is shown that the second usual differential of a point of a mani-fold it is possible to present as sum of vertical and horizontal projectors.

### Theoretical and experimental physics

#### Inertial oscillations induced by the propagation of the Vis-tula Lagoon waters in the coastal zone of the Southeastern Baltic Sea

Abstract

A mathematical model of the mechanism of anti-symmetric vortex at the confluence of fresh water to sea water, observed, in particular, the output of the Baltic Canal connecting Visla Lagoon to the Baltic Sea is constructed. It is shown in particular that the main reason the vortex formation in this case is the Coriolis force. Also analytically calculate the exact dependence of circula-tion rates from time to time for the three simplest forms of "language" of the intrusion of fresh water.