Wyniki wyszukiwania - BazTech

1

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

Jakóbowski J.

Demonstratio Mathematica

|

2013

|

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

|

2011

|

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

Some generalization of nearaffine planes

Jakóbowski J.

Bulletin of the Polish Academy of Sciences. Mathematics

|

2005

|

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.

4

Theorems on products of central collineations with distinct centers or axes applied to the Benz planes

Jakóbowski J.

Demonstratio Mathematica

|

2004

|

Vol. 37, nr 3

639--653

EN

There are three kinds of the Benz planes: Mobius planes, Laguerre planes and Minkowski planes [2, 3, 7]. In any Benz plane an automorphism φ is central if φ has a fixed point P and becomes a central collineation in the projective derived plane induced by P. Such central automorphisms have been considered by many authors (cf. [8,13, 11, 12, 10]), in particular the automorphism groups were classified. Usually product of two central collineations without common center or common axis is not central. But in some special cases it is central [4]. In this paper we apply theorems concerning such special cases - to the Benz planes.