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Automorphisms of Witt rings and quaternionic structures

100%

Stępień M. R.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

2011

tom Vol. 10, nr 1

231-237

M. Marshall introduced the notion of quaternionic structure and he showed that the categories of Witt rings and quaternionic structures are naturally equivalent. Quaternionic structures turn out to bea useful tool for the investigation of Witt rings, since it suffices to handle the structure of a group. In our paper we shall describe precisely the one-to-one correspondence between automorphisms of quaternionic structures and strong the automorphisms of Witt rings.

Automatic search of automorphisms of Witt rings

63%

Stępień L. , Stępień M. R.

tom Vol. 16

141--146

The investigation of strong automorphisms of Witt rings is a difficult task because of variety of their structures. Cordes Theorem, known in literature as Harrison-Cordes criterion (cf. [1, Proposition 2.2], [3, Harrison's Criterion]), makes the task of describing all the strong automorphisms of a given (abstract) Witt ring W = (G, R) easier. By this theorem, it suffices to find all such automorphisms ơ of the group G that map the distiguished element -1 of the group G into itself (i.e. ơ(-1) = -1) in which the value sets of 1-fold Pfister forms are preserved in the following sense: ơ(D(1, α)) = D(1, ơ(α)) for all α ∈ G. We use the above criterion and the well-known structure of the group G as a vector space over two-element field F2 for searching all automorphisms of this group. Then we check Harrison-Cordes criterion for found automorpisms and obtain all the automorpisms of a Witt ring W. The task is easy for small rings (with small groups G). For searching of all strong automorpisms of bigger Witt rings we use a computer which automatizes the procedure described above. We present the algorithm for finding strong automorphisms of a Witt rings with finite group G and show how this algorithm can be optimized.

On strong automorphisms of direct products of Witt rings. Part 1

63%

Stępień M. R. , Stępień L.

Journal of Applied Mathematics and Computational Mechanics

2013

tom Vol. 12, nr 3

123--133

The notion of Witt ring is fundamental in bilinear algebra. Automorphisms of Witt rings have been investigated until recent years. In this paper we consider Witt rings which are direct products of finite number of other Witt rings. We shall present a necessary condition in order to group of all strong automorphisms of direct product of Witt rings be a direct product of groups of strong automorphisms of Witt rings which are factors in the direct product. Subsequently, there are considered some examples of Witt rings, where described condition is fulfilled.

On teaching of geometric transformations in school

51%

Stępień L. , Stępień M. R. , Ziółkowski M.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

2014

tom Vol. 19

153--164

The current core curriculum in mathematics for lower secondary school (3-rd educational level in Poland) omits formal definitions of concepts related to geometric transformations in the plane and is based on their intuitive sense. Practice shows that the current approach makes teaching very difficult and the students solve the typical tasks, not understanding the meaning of geometrical concepts. The article contains basic concepts connected with geometric transformations and examples of geometric tasks that are solved in the third and also in the fourth educational level in an intuitive way, sometimes deviating or even incompatible with the mathematical definition. We show how they could be solved in easier way with introducing definitions of geometric transformations in a simple and understandable for students way sometimes using vector calculus. We take into account isometries: reflection and point symmetry, rotation and translation and similarities with particular consideration on homothetic transformation.