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1

Automorphisms of Witt rings of finite fields

Stępień M. R.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

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2011

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Vol. 16

67--70

EN

The problem of general description of the group of automorphisms of any Witt ring W seems to be very difficult to solve. However, there are many types of Witt rings, which automorphism are described precisely (e.g. [1], [2], [4], [5], [6],[7], [8]). In our paper we characterize automorphisms of abstract Witt rings (cf. [3]) isomorphic to powers of Witt rings of quadratic forms with coefficients in finite fields with characteristic different from 2.

2

Automatic search of automorphisms of Witt rings

Stępień L., Stępień M. R.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

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2011

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Vol. 16

141--146

EN

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.

3

Automorphisms of Witt rings and quaternionic structures

Stępień M. R.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

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2011

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Vol. 10, nr 1

231-237

EN

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.

4

Commutative loops of exponent 3 with x.(x.y)2=y2

Armanious M.

Demonstratio Mathematica

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2002

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Vol. 35, nr 3

469-475

EN

It is well known that the class of Hall triple systems [5], Steiner triple systems in which each triangle generates an affine plane over GF(3), corresponds to the class of commutative Moufang loops of exponent 3 [6]. In this paper, we extend the class of algebras to the class of all commutative loops of exponent 3 satisfying the identity x.(x.y)2=y2, corresponding to the class of all Steiner triple systems. Such a commutative loop of exponent 3 with x . (x o y)2 = y2 is polynomially equivalent to a squag.

5

A note on (alpha)-derivations on semiprime rings

Thaheem A., Samman M.

Demonstratio Mathematica

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2001

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Vol. 34, nr 4

783-788

EN

In this note we investigate some properties of a-derivations on prime and semiprime rings. We establish some identities for a commuting a-derivation d on a semiprime ring R and show that d maps R into its center and obtain some well-known results as a consequence. We also generalize Posner's theorem on the composition of derivations for a-derivations and as an application resolve a functional equation of automorphisms on certain prime rings.

6

Inversive closure of metric affine space and its automorphisms

Oryszczyn H.

Demonstratio Mathematica

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1999

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Vol. 32, nr 1

151-155

7

Barycentric transformations

Mazouzi S.

Demonstratio Mathematica

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1999

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Vol. 32, nr 2

287-296

EN

We present in this note a new family of automorphisms on spaces of holomorphic functions called Barycentric Transformations. Beside the theoretical aspect of these transformations we shall use them to solve explicitly barycentric differential equations of the form.

8

Central automorphisms of Laguerre planes

Makowiecka H., Matraś A.

Demonstratio Mathematica

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1998

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Vol. 31, nr 4

753-763

EN

Any automorphism of a Benz plane having at least one fixed point induces a collineation on the projective extension of the residual affine plane with reference to this point. When this collineation is a central automorphism , the initial automorphism is called the central automorphism (or central-axial automorphism, cf.([3]). In this paper we present an analytical description of central automorphisms of a miguelian Laguerre planes with the characteristic different from two. This description is applied to find transitive groups of homotheties and translations of types occuring in the classification theorems of R. Kleinewillinghofer ([2]). Some examples over an arbitrary commutative field are constructed, the other over the finite field Z3 and z5. It is interesting that two types of the Kleinewillinghofer classification ( [2] ) appear only as automorphism subgroups of finite plane of order three or five. This will give a clear characterization of these planes. Throughaut we assume that the characteristic of a plane is not equal to two.