2016 Issue №2

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.

For describe the hydrodynamic and thermal processes in the ladle in the time of blowing argon Steel developed a mathematical model of convective heat transfer in the Boussinesq approximation. We give a statement of the problem of this description, the translation model in dimensionless criterial form diver-gent representation in the form (Θ, ω, Ψ)-system.

Norden inner normalization fields of basic structural Λ-, L-, H-sub-bundle of hyperband H-distribution are constructed in 2nd order differential neighborhood of affine space An by means of symmetric fundamental tensors of 1st order.

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.

A program of the general course of mathematical modeling coordinated with modeling of transport flows is submitted.

SH-distribution and the theorem of the existence in frame of order zero are given. Invariant normalizations, matchings Bompiani-Pantazi and equipments in order of E. Cartan of major structural sub-bundles are built.