Representations of SL(2,4) on translation planes of even order

A representation for SL(2,4) In characteristic 2 is given which leads to a group theoretic construction of the Dempwolff plane of order 16. Using this construction,it is further shown that the Dempwolff plane may be obtained by the derivation of the semifield plane of order 16 and kern GF(2).     

Uniqueness of the inversive plane of order 7

A search has shown that any inversive plane of order 7 must be isomorphic with the Miquelian plane. The method used was explicitly set out in a companion paper, which was concerned with a plane of order 5.     

Pasch geometric spaces inducing finite projective planes

Summary A geometric space over a geometric sfield of dimension three induces a protective plane. A relation between the order of the projective plane and that of the geometric sfield is obtained. For a particular order of the sfield, the induced projective plane is shown to be desarguesian.     

Nonexistence of certain planes of order 10 and 12

It is shown here that no plane of order 12 exists which has a group of elations of order 12. It is also shown that no plane of order 10 exists which has a multiplicative loop that is a group.     

A connection between Fano configurations and Minkowski planes of odd order

It is shown in this paper that a pair of points contained in a Fano configuration in a projective plane of odd order cannot induce a Minkowski plane. From this result we derive that no pair of points in the Hughes plane of order 9 can induce a Minkowski plane.     

A characterization of Hering's plane of order 27

Hering's translation plane of order 27 has been characterized by its order and the fact that it admits SL(2, 13) in its translation complement (see [1]). We show that, aside from the Desarguian plane and a Generalized, André plane, it is the only plane of order 27 which admits a subgroup of SL(2, 13) of order 13×12.Partially supported by FONDECYT 0343 and ANDES FOUNDATION.     

A class of unitals of order q which can be embedded in two different planes of order q2

By deriving the desarguesian plane of order q² for every prime power q a unital of order q is constructed which can be embedded in both the Hall plane and the dual of the Hall plane of order q² which are non-isomorphic projective planes. The representation of translation planes in the fourdimensional projective space of J. André and F. Buekenhouts construction of unitals in these planes are used. It is shown that the full automorphism groups of these unitals are just the collineation groups inherited from the classical unitals.     

Translation planes admitting a pair of index three homology groups

It is shown that a translation plane of order which admits two homology groups of order must in fact admit symmetric homology groups of this order. It is further shown that a plane admitting such symmetric index 3 homology groups is, with a finite number of exceptions, a generalized André plane. A list of the possibly exceptional orders is determined. Received 20 March 2000.     

On the automorphism group of inversive planes of odd order

The structure of the fix bundle free automorphism groups of inversive planes of odd order is determined. As a special case of our main result, the automorphism groups with a transitive action on the points of an inversive plane of odd order are essentially determined, and the plane is shown to be miquelian when these have no non-trivial normal subgroups of odd order.     

On equations of the state of stress in a plate of variable thickness

Method of asymptotic integration of three-dimensional equations of the theory of elasticity is used to construct the internal state of stress in a plate of variable thickness /1/. It is shown that it can generally be described by a system of differential equations of eighth order in the components of the displacement vector of the points of a plane projected inside the plate, with the equations of flexure and of plane state of stress not separated. In a particular case when the face surfaces are symmetrical with respect to this plane, the flexure and the plane state of stress are described by separate equations. The accuracy of the equations obtained is of the order of square of relative thickness of the plate away from the edge and other distortion lines of the stress state.

The boundary layer is not considered and conditions at the plate edge are not formulated.     

Some basic properties of planar Singer groups

A planar Singer group is a collineation group of a finite (in this article) projective plane acting regularly on the points of the plane. Theorem 1 gives a characterization of abelian planar Singer groups. This leads to a necessary and sufficient condition for an inner automorphism to be a multiplier. The Sylow 2-structure of a multiplier group and some of its consequences are given in Theorem 3. One important result in studying multipliers of an abelian Singer group is the existence of a common fixed line. We extend this to an arbitrary planar Singer group in Theorem 4. Theorem 5 studies the order of an abelian group of multiplers. If this order equals to the order of the plane plus 1, then the number of points of the plane is a prime. If this order is odd, then it is at most the planar order plus 1.Partially supported by a NSA grant.     

论Bézier曲线的仿射不变量

本文的目的是按照[1]的理论找出n次平面Bezier曲线的内在仿射不变量,特别是,对于3次Bezier曲线的保凸性作出其充要条件的几何解释。对于一般的情况下的保凸性问题,至今还没有解决。著者仅在4次的场合详尽地讨论了曲线段上是否存在拐点的分析的(而不是几何的)充要条件,而最后举出几个实例,以说明特征多角形的凸性是充分条件,而不是必要条件。     

On Projective Planes of Order 15 Admitting a Collineation of Order 7

In this article we prove that there is no projective plane of order 15 admitting a collineation group of order 21. C. Y. Ho proved that there is no projective plane of order 15 admitting a collineation group of order 49. But his proof is incorrect. We also correct his error. The conclusion remains the same. We used a computer for our research.     

A partial plane of order 6 constructed from the icosahedron

We show that a maximal partial plane of order 6 with 31 lines and a maximal pure partial plane of order 6 with 25 lines can be constructed from the icosahedron and the Petersen graph. To Daniel R. Hughes, to commemorate his 80th birthday, August 7th, 2007.     

The nonexistence of a certain Steiner system S(3, 12, 112)

Although the automorphism group of a projective plane of order 10, if one exists, must be very small, such a plane could be the derived design of a Steiner system S(3, 12, 112) with a larger group. There are several reasons why the Frobenius group of order 56 is a promising candidate for the latter group. However, in this paper it is shown that there is no S(3, 12, 112) which is fixed by this Frobenius group.     

Generalized symmetric planes

In this paper we relate the theory of stable planes to the theory of generalized symmetric spaces in the sense of differential geometry where the symmetries may be of arbitrary order. This leads to the notion of a generalized symmetric plane. We assign to every generalized symmetric plane an associated infinitesimal model and show that the associated infinitesimal model essentially determines a generalized symmetric plane up to global isomorphism. In particular, every generalized symmetric plane with an abelian group of transvections is a topological translation plane.     

On extendable planes of order ten

It is well known that the only way of extending a projective plane of order n (conceivable orders are 2, 4 and 10) is adjoining a set of hyperovals to the given projective plane. A converse is proved in this note. It is shown as a corollary that the existence of an extendable plane of order 10 is equivalent to the existence of a quasi-symmetric 2-(111, 12, 10)-design.     

Collineations of projective planes of order 10, part I

No projective plane of order 10 has a collineation group of order 9 which fixes a 12-arc, a set of twelve points no three collinear, as a set. This fact, proved in Part I, is used to prove in Part II that the full collineation group of any projective plane of order 10 has order 1, 3, or 5.     

Duality in stable planes and related closure and kernel operations

We compare two constructions that dualize a stable plane in some sense, namely the dual plane and the opposite plane. Applying both constructions one after another we obtain a closure or kernel operation, depending on the order of execution.We examine the effect of these constructions on the automorphism group and apply our results in order to compute the automorphism groups of the complex cylinder plane, the complex united cylinder plane, and their duals. Beside the complex projective, affine, and punctured projective plane these planes are in fact the most homogeneous four-dimensional stable planes, as will be shown elsewhere [1].Supported by Studienstiftung des deutschen Volkes.