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.

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

Automorphisms of Hilbert's non-desarguesian affine plane and its projective closure

2007 ◽  
Vol 7 (4) ◽  
Author(s):  
Thomas Schneider ◽  
Markus Stroppel
Keyword(s):  
Affine Plane ◽  
Projective Closure ◽  
Desarguesian Affine Plane

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.

One-factorisations of complete graphs arising from hyperbolae in the Desarguesian affine plane

2019 ◽  
Vol 110 (1) ◽  
Author(s):  
Nicola Pace ◽  
Angelo Sonnino
Keyword(s):  
Affine Plane ◽  
Complete Graphs ◽  
Desarguesian Affine Plane

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).

scholarly journals Nash resolution for binomial varieties as Euclidean division. A priori termination bound, polynomial complexity in essential dimension 2

2012 ◽  
Vol 231 (6) ◽  
pp. 3389-3428 ◽  
Author(s):  
Dima Grigoriev ◽  
Pierre D. Milman
Keyword(s):  
Polynomial Complexity ◽  
A Priori ◽  
Essential Dimension ◽  
Dimension 2

scholarly journals Counting and equidistribution in quaternionic Heisenberg groups

2021 ◽  
pp. 1-38
Author(s):  
JOUNI PARKKONEN ◽  
FRÉDÉRIC PAULIN
Keyword(s):  
Hyperbolic Geometry ◽  
Quaternion Algebra ◽  
Rational Points ◽  
Hyperbolic Spaces ◽  
Heisenberg Groups ◽  
Arithmetic Groups ◽  
Hermitian Forms ◽  
Dimension 2 ◽  
Counting Formula ◽  
The Relationship

Abstract We develop the relationship between quaternionic hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on quaternionic hyperbolic spaces, especially in dimension 2. We prove a Mertens counting formula for the rational points over a definite quaternion algebra A over ${\mathbb{Q}}$ in the light cone of quaternionic Hermitian forms, as well as a Neville equidistribution theorem of the set of rational points over A in quaternionic Heisenberg groups.

scholarly journals GRADED TWISTED CALABI–YAU ALGEBRAS ARE GENERALIZED ARTIN–SCHELTER REGULAR

2021 ◽  
pp. 1-54
Author(s):  
MANUEL L. REYES ◽  
DANIEL ROGALSKI
Keyword(s):  
Regularity Property ◽  
Graded Algebra ◽  
General Study ◽  
Locally Finite ◽  
Tensor Algebras ◽  
Finite Dimensional ◽  
Dimension 2 ◽  
Separable Algebras ◽  
Gk Dimension

Abstract This is a general study of twisted Calabi–Yau algebras that are $\mathbb {N}$ -graded and locally finite-dimensional, with the following major results. We prove that a locally finite graded algebra is twisted Calabi–Yau if and only if it is separable modulo its graded radical and satisfies one of several suitable generalizations of the Artin–Schelter regularity property, adapted from the work of Martinez-Villa as well as Minamoto and Mori. We characterize twisted Calabi–Yau algebras of dimension 0 as separable k-algebras, and we similarly characterize graded twisted Calabi–Yau algebras of dimension 1 as tensor algebras of certain invertible bimodules over separable algebras. Finally, we prove that a graded twisted Calabi–Yau algebra of dimension 2 is noetherian if and only if it has finite GK dimension.

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