Miroslav Kureš

Local Approach to Higher-Order Contact Elements

The local description of regular velocities and contact elements is covered by the paper. This description including an algebraic geometry view is newly prolonged to Weil velocities and Weil contact elements. An application in physics is emphasized.

Fixed point subalgebras of Weil algebras: from geometric to algebraic questions

The paper is a survey of some results about Weil algebras applicable in differential geometry, especially in some classification questions on bundles of generalized velocities and contact elements. Mainly, a number of claims concerning the form of subalgebras of fixed points of various Weil algebras are demonstrated.

Higher Order Frames and Material Symmetries

The paper is focused on mathematical approach to materials with microstructure, the special case of them are Cosserat media. Frame bundles are described with respect to their role in continuum mechanics and the structure jet groups. General (r-th order) holonomic, nonholonomic and semiholonomic microstructure configuration are introduces and the algebraic approach using Weil algebras is presented, too.

Some Remarks to Higher Order Frames Occurring in Continuum Mechanics

Frame bundles are described with respect to their role in continuum mechanics, the structure jet groups are studied and some expressions in local coordinates are derived. It is introduced the general (r-th order) microstructure configuration and it is suggested what needs to be investigated in subsequent research.

Some properties of orders of quaternion algebras with regard to the discrete norm

Quaternion algebras ((-1,b)/Q) are investigated and isomorphisms between them are described. Furthermore, the orders of these algebras are presented and the uniqueness of the discrete norm for such orders is proved.

Two general examples of Weil algebras having the group of automorphisms connected

The paper is devoted to the algebraic background of Weil contact elements: in particular, we investigate an eventual absence of orientation reversing maps by the description of Weil algebras having their group of automorphisms with only one connected component.

Jet Theory in Microstructures

We present jet approach to microstructures, in particular to Cosserat continuum. We compare classical methods with the representation of Cosserat bodies by frame bundles. We demonstrate that nonholonomic, semiholonomic and holonomic bundles occur in such description.

On the S-operation in polynomial rings

Abstract S-polynomials used in the Buchbergers algorithm for Gröbner basis, are introduced in a more general situation. For polynomials viewed as mappings from a monoid to a ring, we define the S-operation and derive its properties. We also demonstrate its application on polynomials on a ring which is not a field.