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The Euclidean plane kinematics

El Said El Shinnawy (1979)

Aplikace matematiky

Restricting his considerations to the Euclidean plane, the author shows a method leading to the solution of the equivalence problem for all Lie groups of motions. Further, he presents all transitive one-parametric system of motions in the Euclidean plane.

The gap phenomenon in the dimension study of finite type systems

Boris Kruglikov (2012)

Open Mathematics

Several examples of gaps (lacunes) between dimensions of maximal and submaximal symmetric models are considered, which include investigation of number of independent linear and quadratic integrals of metrics and counting the symmetries of geometric structures and differential equations. A general result clarifying this effect in the case when the structure is associated to a vector distribution, is proposed.

The natural affinors on ( J r T * ) *

Włodzimierz M. Mikulski (2000)

Archivum Mathematicum

For natural numbers r and n 2 a complete classification of natural affinors on the natural bundle ( J r T * ) * dual to r -jet prolongation J r T * of the cotangent bundle over n -manifolds is given.

The natural operators lifting vector fields to generalized higher order tangent bundles

Włodzimierz M. Mikulski (2000)

Archivum Mathematicum

For natural numbers r and n and a real number a we construct a natural vector bundle T ( r ) , a over n -manifolds such that T ( r ) , 0 is the (classical) vector tangent bundle T ( r ) of order r . For integers r 1 and n 3 and a real number a < 0 we classify all natural operators T | M n T T ( r ) , a lifting vector fields from n -manifolds to T ( r ) , a .

The natural operators lifting vector fields to ( J r T * ) *

Włodzimierz M. Mikulski (2000)

Archivum Mathematicum

For integers r 2 and n 2 a complete classification of all natural operators A : T | M n T ( J r T * ) * lifting vector fields to vector fields on the natural bundle ( J r T * ) * dual to r -jet prolongation J r T * of the cotangent bundle over n -manifolds is given.

The natural operators T ( 0 , 0 ) T ( 1 , 1 ) T ( r )

Włodzimierz M. Mikulski (2003)

Colloquium Mathematicae

We study the problem of how a map f:M → ℝ on an n-manifold M induces canonically an affinor A ( f ) : T T ( r ) M T T ( r ) M on the vector r-tangent bundle T ( r ) M = ( J r ( M , ) ) * over M. This problem is reflected in the concept of natural operators A : T | f ( 0 , 0 ) T ( 1 , 1 ) T ( r ) . For integers r ≥ 1 and n ≥ 2 we prove that the space of all such operators is a free (r+1)²-dimensional module over ( T ( r ) ) and we construct explicitly a basis of this module.

The natural transformations T T ( r ) T T ( r )

Włodzimierz M. Mikulski (2000)

Archivum Mathematicum

For natural numbers r 2 and n a complete classification of natural transformations A : T T ( r ) T T ( r ) over n -manifolds is given, where T ( r ) is the linear r -tangent bundle functor.

The structure tensor and first order natural differential operators

Piotr Kobak (1992)

Archivum Mathematicum

The notion of a structure tensor of section of first order natural bundles with homogeneous standard fibre is introduced. Properties of the structure tensor operator are studied. The universal factorization property of the structure tensor operator is proved and used for classification of first order * -natural differential operators D ̲ : T × T ̲ T ̲ for n 3 .

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