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1

Some examples of denumerable pseudo-ordered fields and their applications to geometry

Jakóbowski J.

Demonstratio Mathematica

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2013

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Vol. 46, nr 2

247--256

EN

The paper deals with the following problems concerning pseudo-ordered denumerable fields: (i) a connection between Fermat’s two squares theorem and the unique pseudo-order in a finite field; (ii) properties of a proper pseudo-order determined by any prime number in the field of rational numbers; (iii) existence of a proper pseudo-order in every subfield of the sequence used to obtain the field of constructible numbers; (iv) some brief of applications of the latter pseudo-orders to construct new algebraic and geometric structures. In particular, we extend the known construction of finite nearfields or quasifields given by e.g. W. A. Pierce or P. Dembowski – to infinite cases.

2

Ellipses numbers and geometric measure representations

Richter W.-D.

Journal of Applied Analysis

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2011

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Vol. 17, nr 2

165-179

EN

Ellipses will be considered as subsets of suitably defined Minkowski planes in such a way that, additionally to the well-known area content property A(r) = Π (a,b) r 2, the number Π (a,b) = abΠ reflects a generalized circumference property U (a,b)(r) = 2Π (a,b) r of the ellipses E (a,b)(r) with main axes of lengths 2ra and 2rb, respectively. In this sense, the number Π (a,b) is an ellipse number w.r.t. the Minkowski functional r of the reference set E (a,b)(1). This approach is closely connected with a generalization of the method of indivisibles and avoids elliptical integrals. Further, several properties of both a generalized arc-length measure and the ellipses numbers will be discussed, e.g. disintegration of the Lebesgue measure and an elliptically contoured Gaussian measure indivisiblen representation, wherein the ellipses numbers occur in a natural way as norming constants.

3

A residual skewaffine plane of a Mobius or Minkowski plane

Matraś A.

Demonstratio Mathematica

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2007

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Vol. 40, nr 4

913-924

EN

For Mobius and Minkowski planes of characteristic different from 2 a residual skewaffine piane associated with any point p is constructed. Following the construction given by Andre (cf. [1]) we obtain the residual piane as the group space of some normally transitive group of automorphisms fixing p. This is a skewaffine piane without straight lines in the Mobius case and with two families of straight lines in the Minkowski case.

4

Some generalization of nearaffine planes

Jakóbowski J.

Bulletin of the Polish Academy of Sciences. Mathematics

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2005

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Vol. 53, no 1

87-97

EN

There are three kinds of Benz planes: Mobius planes, Laguerre planes and Minkowski planes. A Minkowski plane satisfying an additional axiom is connected with some other structure called a nearaffine plane. We construct an analogous structure for a Laguerre plane. Moreover, our description is common for both cases.

5

Nearaffine planes related to pseudo-ordered fields

Jakóbowski J.

Bulletin of the Polish Academy of Sciences. Mathematics

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2002

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Vol. 50, no 3

345-360

EN

The constructions of affine planes and Minkowski planes related to pseudo-ordered fields are given in [3] and [2], respectively. We here give some analogous construction for nearaffine planes. Like before, we shall use some functions f, g and determine some conditions on f, g, necessary and suffcient to get the required plane. The Veblen postulate has a particular meaning in nearaffine planes, so it is also considered in the work. Some special cases like the field of the reals and finite fields of odd order are investigated, too. We give some examples of such nearaffine planes and consider their particular automorphisms. Every Minkowski plane related to pseudo-ordered field F determines a nearaffine plane connected with F [2, Proposition 1, p. 187]. But only weaker version of the reciprocal statement is true, i.e. a nearaffine plane related to a pseudo-ordered field determines a hyperbola structure (i.e. Minkowski plane without touching axiom).

6

Multicentral automorphisms in geometries of circles

Jakóbowski J., Matraś A.

Bulletin of the Polish Academy of Sciences. Mathematics

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2001

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Vol. 49, no 4

417-432

EN

We consider three types of geometries of circles (Moebius plane, Laguerre plane and Minkowski plane, cf. [4) with respect to so-called multicentral automorphisms. An automorphism [phi] of any geometry of circles is central if it has a fix point P and [phi] becomes a central collineation in the derived projective plane M(P). For any central automorphism [phi] we try to establish the whole set of points R such that [phi] becomes a central collineation in M(R.). Than [phi] is called multicentral if this set contains at least two points. Moreover, [phi] is proper if existing of a point [R is not equal to P], is not caused by the fact that [phi] is central in M(P). There is no proper multicentral automorphism in a Moebius plane. The most interesting proper multicentral automorphisms are involutorial mappings: double homotheties in Minkowski planes, and (sigma, tau)homologies in Laguerre planes. We give some examples.