A PROOF OF THE PROJECTIVE DESARGUES AXIOM IN THE DESARGUESIAN AFFINE PLANE

2004 ◽  
Vol 37 (4) ◽  
Author(s):  
Małgorzata Prażmowska
Keyword(s):  
Affine Plane ◽  
Desarguesian Affine Plane

scholarly journals On the Non-Existence of Certain Hyperovals in Dual André Planes of Order $2^{2h}$

10.37236/912 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Angela Aguglia ◽  
Luca Giuzzi
Keyword(s):  
Affine Plane ◽  
Dimension 2 ◽  
Desarguesian Affine Plane

No regular hyperoval of the Desarguesian affine plane $AG(2,2^{2h})$, with $h>1$, is inherited by a dual André plane of order $2^{2h}$ and dimension $2$ over its kernel.

scholarly journals BRUCK NETS AND PARTIAL SHERK PLANES

2017 ◽  
Vol 104 (1) ◽  
pp. 1-12
Author(s):  
JOHN BAMBERG ◽  
JOANNA B. FAWCETT ◽  
JESSE LANSDOWN
Keyword(s):  
Affine Plane ◽  
Finite Geometries ◽  
Affine Planes ◽  
Incidence Structures ◽  
Finite Affine ◽  
Odd Order ◽  
Finite Incidence ◽  
Desarguesian Affine Plane

In Bachmann [Aufbau der Geometrie aus dem Spiegelungsbegriff, Die Grundlehren der mathematischen Wissenschaften, Bd. XCVI (Springer, Berlin–Göttingen–Heidelberg, 1959)], it was shown that a finite metric plane is a Desarguesian affine plane of odd order equipped with a perpendicularity relation on lines and that the converse is also true. Sherk [‘Finite incidence structures with orthogonality’, Canad. J. Math.19 (1967), 1078–1083] generalised this result to characterise the finite affine planes of odd order by removing the ‘three reflections axioms’ from a metric plane. We show that one can obtain a larger class of natural finite geometries, the so-called Bruck nets of even degree, by weakening Sherk’s axioms to allow noncollinear points.

scholarly journals Ordered Desarguesian Affine Hjelmslev Planes

1978 ◽  
Vol 21 (2) ◽  
pp. 229-235 ◽  
Author(s):  
L. A. Thomas
Keyword(s):  
Local Ring ◽  
Division Ring ◽  
Affine Plane ◽  
Hjelmslev Plane ◽  
Zero Divisors ◽  
Affine Hjelmslev Plane ◽  
Desarguesian Affine Plane

A Desarguesian affine Hjelmslev plane (D.A.H. plane) may be coordinatized by an affine Hjelmslev ring (A.H. ring), which is a local ring whose radical is equal to the set of two-sided zero divisors and whose principal right ideals are totally ordered (cf. [3]). In his paper on ordered geometries [4], P. Scherk discussed the equivalence of an ordering of a Desarguesian affine plane with an ordering of its coordinatizing division ring. We shall define an ordered D.A.H. plane and follow Scherk's methods to extend his results to D.A.H. planes and their A.H. rings i.e., we shall show that a D.A.H. plane is ordered if and only if its A.H. ring is ordered. We shall also give an example of an ordered A.H. ring. Finally, we shall discuss some infinitesimal aspects of the radical of an ordered A.H. ring.

Cyclic Affine Planes

1952 ◽  
Vol 4 ◽  
pp. 295-301 ◽  
Author(s):  
A. J. Hoffman
Keyword(s):  
Cyclic Group ◽  
Infinite Number ◽  
Affine Plane ◽  
Principal Result ◽  
Affine Planes ◽  
Present Discussion ◽  
Desarguesian Affine Plane

Let Π be an affine plane which admits a collineation τ such that the cyclic group generated by τ leaves one point (say X) fixed, and is transitive on the set of all other points of Π. Such “cyclic affine planes” have been previously studied, especially in India, and the principal result relevant to the present discussion is the following theorem of Bose [2]: every finite Desarguesian affine plane is cyclic. The converse seems quite likely true, but no proof exists. In what follows, we shall prove several properties of cyclic affine planes which will imply that for an infinite number of values of n there is no such plane with n points on a line.

The Flag-Transitive Collineation Groups of the Finite Desarguesian Affine Planes

1964 ◽  
Vol 16 ◽  
pp. 443-472 ◽  
Author(s):  
David A. Foulser
Keyword(s):  
Positive Integer ◽  
Collineation Group ◽  
Affine Plane ◽  
Affine Planes ◽  
Collineation Groups ◽  
Point Line ◽  
Explicit Determination ◽  
Desarguesian Affine Plane

Let π be the Desarguesian affine plane of order n = pr, for p a prime and r a positive integer. A collineation group G of π is defined to be flag-transitive on π if G is transitive on the set of incident point-line pairs, or flags, of π. Further, G is doubly transitive on π if G is doubly transitive on the points of π. Clearly, G is flag transitive if G is doubly transitive on π.The purpose of the following study is the explicit determination of the flagtransitive and the doubly transitive collineation groups of π (I am indebted to D. G. Higman for suggesting this problem). The results can be summarized in Theorems 1′ and 2′ below (a complete description of the results is contained in Sections 12-15).

On a conjecture of Hughes

1967 ◽  
Vol 63 (3) ◽  
pp. 647-652 ◽  
Author(s):  
Judita Cofman
Keyword(s):  
Projective Plane ◽  
Permutation Group ◽  
Translation Plane ◽  
Collineation Group ◽  
Affine Plane ◽  
Finite Projective Plane ◽  
Transitive Permutation Group ◽  
Condition C

D. R. Hughes stated the following conjecture: If π is a finite projective plane satisfying the condition: (C)π contains a collineation group δ inducing a doubly transitive permutation group δ* on the points of a line g, fixed under δ, then the corresponding affine plane πg is a translation plane.

Bayesian Nash Equilibrium in Electricity Spot Markets: An Affine-Plane Approximation Approach

2021 ◽  
pp. 1-1
Author(s):  
Pranjal Pragya Verma ◽  
Mohammad Hesamzadeh ◽  
Ross Baldick ◽  
Darryl Biggar ◽  
K. Shanti Swarup ◽  
...  
Keyword(s):  
Nash Equilibrium ◽  
Affine Plane ◽  
Spot Markets ◽  
Approximation Approach ◽  
Bayesian Nash Equilibrium ◽  
Electricity Spot Markets
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